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Question 5 multiple-choice

In quantum many-body physics, calculating transport coefficients such as shear viscosity in strongly coupled gauge theories often requires sophisticated nonperturbative techniques. Recent advances in computational methods have enabled studies of these properties in low-dimensional lattice models. In a (2+1)-dimensional SU(2) gauge theory studied on a small honeycomb lattice, which result for the shear viscosity to entropy density ratio (η/s) is consistent with predictions from gauge/gravity duality for strongly coupled quantum systems? 1) η/s = 0 2) η/s = 1 3) η/s = π 4) η/s = 1/4π 5) η/s = 2π 6) η/s = 1/2 7) η/s = 4/π

✓ Correct Answer:

The correct answer is 4) η/s = 1/4π.

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Question 6 multiple-choice

In quantum error correction, duality transformations and encoding schemes are often used to construct codes with desirable properties, such as locality preservation and covariance under certain Hamiltonians. The Ising spin chain serves as a key example where these techniques enable robust and analyzable codes relevant to quantum computing and physics. Which property is achieved by mapping the Ising spin chain Hamiltonian to a noninteracting Hamiltonian via a duality transformation and encoding, as described for quantum error-correcting code construction? 1) The code becomes immune to all possible single-qubit errors. 2) The code distance is doubled relative to the original model. 3) The Hamiltonian terms must commute for error correction to be analyzed. 4) Local operators in the original model map to similarly local operators in the transformed space. 5) The code can universally correct any number of erasures regardless of their location. 6) The worst-case error scaling becomes exponential in system size. 7) The encoding prevents any logical operator from acting nontrivially on the code space.

✓ Correct Answer:

The correct answer is 4) Local operators in the original model map to similarly local operators in the transformed space..

📚 Reference Text:

properties of the code. To construct the code formally, we observe that, under a well-known duality transformation [61,62] , the Ising spin chain Hamiltonian (51) is equivalent to the noninteracting Hamiltonian ˜H¼Xn i¼2σZ i; ð52Þ which, except for the first site i¼1, is the same as the magnetization operator considered in our discussion of thethermodynamic code in Sec. VI C. This correspondence is established directly on the basis states fj ⃗xig ¼ fj x 1i⊗ /C1/C1/C1⊗jxnigof the full Hilbert space, where ⃗xis a bit string and the states jxi¼0;1iare eigenstates of the σZ ioperator. We encode this bit string into another bit string ⃗sð ⃗xÞthat provides the value of the first bit and all pairwise consecutiveparities, namely, s 1¼x1andsi¼xi−1þxiðmod 2Þfor 2≤i≤n. Clearly, this mapping is one to one and onto acting on the set of all bit strings of length n.D e n o t eb y U the unitary on the n-site Hilbert space that implements this transformation, i.e., U¼P ⃗xj ⃗sð ⃗xÞih ⃗xj.W es e et h a t σZ ij ⃗sð ⃗xÞi ¼ ð−1Þsij ⃗sð ⃗xÞ i¼ð−1Þxi−1þxij ⃗sð ⃗xÞi and, thus, U†σZ iU¼σZ i−1σZ i. This result implies that U†˜HU ¼HIsing.CONTINUOUS SYMMETRIES AND APPROXIMATE QUANTUM … PHYS. REV . X 10,041018 (2020) 041018-15 The Hamiltonian ˜His exactly the same operator as the magnetization operator in the thermodynamic code ofSec. VI C on the n−1the sites labeled by i¼2;…;n, with a dummy site at i¼1. Using the code words j0i 1⊗ jhn−1mi2…nwhich force the dummy site to the constant state j0i, we still have a code with the same asymptotic proper- ties as before (the dummy site cannot impact negatively theerror-correction properties of the code). Similarly, anyoperator supported on d−1consecutive sites is mapped under the unitary Uto an operator that has support on at most dsites. Hence, the erasure of d−1consecutive sites in the mapped model corresponds to erasure of at most d consecutive sites in the original model, which is correct-able. Our construction, thus, defines a code that is covariantwith respect to the Ising Hamiltonian, with distance d−1 and with ϵ worstðN∘EÞ≤Oð1=nÞ. Applying our bound directly to the newly constructed code, where any d−1consecutive sites can be erased with probability 1=ðn−dþ2Þ≈1=n, yields as before ϵworstðN∘EÞ≥Ωð1=nÞ. Although the overlapping terms in Eq. (51)all commute pairwise, we emphasize that Theorem 2 does not requirethese terms to commute. In consequence, one can study theerror-correction accuracy of codes that are defined usingmore complicated many-body models [8], going beyond the rather simplistic model presented here. Being able to characterize the performance of codes in which subsystems interact is helpful for potential applica-tions of our results. For example, in quantum metrology, theprobe detecting a weak signal might contain many mutuallyinteracting particles. In the AdS/CFT correspondence (seeSec. IX), the charge is the energy of a conformal field theory; in a lattice regularization of this theory, thesubsystems are lattice sites with strong nearest-neighbor interactions. VII. CODES WITH A UNIVERSAL TRANSVERSAL GATE SET Our second main technical contribution is a robust version of the Eastin-Knill theorem

Question 7 multiple-choice

In quantum computing, the hidden subgroup problem (HSP) is a central challenge whose efficient solutions can unlock powerful algorithms for various group-theoretic problems. Techniques such as Gröbner bases, entangled measurements, and pretty good measurement (PGM) have played important roles in advancing quantum algorithms for HSP, especially over nonabelian groups. Which group structure allows the hidden subgroup problem to be solved efficiently in poly(log p) time on a quantum computer, given r is fixed, and is directly connected to efficient Gröbner basis computation and quantum sampling? 1) Dihedral groups D_n 2) Cyclic groups Z_p 3) Abelian groups of order p^n 4) Zn_p ⋊ Z2 groups 5) Symmetric groups S_n 6) Zr_p ⋊ Zp groups with fixed r 7) General non-semidirect product groups

✓ Correct Answer:

The correct answer is 6) Zr_p ⋊ Zp groups with fixed r.

📚 Reference Text:

our purposes, a Gr¨ obne r basis can be computed eﬃciently. Since we consider only the cases in which there are at most 2 so lutions, elimination is also eﬃcient, giving an overall poly(log p) time algorithm for computing a list of matrix sum solutions , and hence for quantum sampling from those solutions. Collecting these results, we ﬁnd Theorem 6. The hidden subgroup problem over any group of the form Zr p⋊Zpwithrﬁxed can be solved in time poly(logp)on a quantum computer. 7 Discussion In this paper, we have studied the pretty good measurement fo r semidirect product groups of the formA⋊ZpwithAabelian and pprime. We found that the pgmis closely connected to the matrix sum problem, and we exploited this connection to ﬁnd eﬃcient quantum algorithms for certain metacyclic groups (Section 5) as well as all groups of the for mZr p⋊Zpwithrﬁxed (Section 6). The latter algorithm demonstrates that entangled measurement s may be useful for eﬃciently solving the nonabelian hsp. Aside from the fact that these particular nonabelian groups admit eﬃcient quantum algorithms, our results suggest two general directions for further inve stigation. First, in the standard approach to the nonabelian hsp, it would be helpful to have a better understanding of when en tangled measurementsarenecessaryandwhentheycanbeimplemented togiveeﬃcientalgorithms. Second, for thehspor for other problems that can be viewed as quantum state dist inguishability problems, identifying an optimal measurement (or considering a parti cularly nice measurement such as the pgm, which in general may or may not be optimal) can be used as a pri nciple for discovering new quantum algorithms. While the reduction of Lemma 1 combined with the pgmapproach outlined in Section 4 appears to eﬃciently solve the hspin most of the semidirect product groups where eﬃcient algor ithms are known, thereisonenotableexception. Thegroups Zn p⋊Z2(forwhichaneﬃcient quantumalgorithm is given in[10]) give risetoasubsetsumproblemover Zn p, which appearstobeessentially as diﬃcult as the subset sum problem over ZNarising from the dihedral group. Thus, it would be interesti ng to understand what allows the algorithm in [10] to be eﬃcient even though the matrix sum problem is (apparently) hard. Of course, there are many nonabelian groups that are semidir ect products of nonabelian groups, orthatcannotbenontrivially decomposedintosemidirectp roductsatall. The pgmapproachiswell 14 deﬁned for any group, so it would be interesting to explore th e approach in such cases, regardless of whether the pgmis optimal. Acknowledgments We thank Carlos Mochon and Frank Verstrate for helpful discu ssions of Theorem 3. AMC received support from the National Science Fo undation under Grant No. EIA- 0086038. References [1] N. Alon and J. H. Spencer. The Probabilistic Method . Wiley Interscience, New York, 2nd edition, 2000. [2] D. Bacon, A. M. Childs, and W. van Dam. Optimal measuremen ts for the dihedral hidden subgroup problem. arXiv:quant-ph/0501044. [3] R. Beals. Quantum computation of Fourier transforms ove r symmetric groups. In Proc. 29th Annual ACM Symposium on Theory of Computing , pages 48–53, 1997. [4] D. Boneh and R. J. Lipton. Quantum cryptanalysis of hidde n linear functions. In Lecture Notes

Question 8 multiple-choice

In quantum information theory, the classification of multipartite entangled states relies on both representation theory and sophisticated geometric frameworks. Understanding how group actions partition state spaces is central to this classification. Which statement accurately describes the role of the local unitary group SU(d₁) ×.. × SU(dₙ) when acting on the complex projective space CP^{D-1} of quantum states for a system with n subsystems of dimensions (d₁,.., dₙ)? 1) It identifies states that differ only by global phase, resulting in a set of orthogonal basis states. 2) It produces a decomposition of the Hilbert space into irreducible subspaces corresponding to each subsystem. 3) It combines all possible tensor product states into a single orbit representing total entanglement. 4) It partitions the projective space into orbits, each corresponding to a distinct entanglement class invariant under local unitary transformations. 5) It restricts physical states to those containing the identity representation in their tensor products. 6) It transforms entangled states into separable states through normalization and phase adjustment. 7) It defines the dimension of the projective space by the product of subsystem dimensions.

✓ Correct Answer:

The correct answer is 4) It partitions the projective space into orbits, each corresponding to a distinct entanglement class invariant under local unitary transformations..

📚 Reference Text:

database. Forn≥4 it is straightforward to see that suﬃciency can’t hold. To see this, note that if there exist representations with dimensions {d1,d2,d3,d4}whose tensor product contains the identity, then the tensor product of any two of the represen tations must contain a rep- resentation whose dimension is the same as a representation in the t ensor product of the other two representations. But this cannot work for dimensions ( 2,2,2,7) (for which LME states exist), since 2 ×2 can contain no representation of dimension larger than 4, while fro m Corollary 3.6, we ﬁnd that 2 ×7 can contain no representation of dimension smaller than 6. 4 Background: The geometry of LME states In this review section, we describe how the set of LME states has tw o natural geometrical formulations that turn out to be equivalent to each other. The ﬁrs t is related to symplec- tic geometry and the other is related to algebraic geometry and geo metric invariant theory. Physically, these two perspectives relate to two seemingly diﬀerent classiﬁcation problems for quantum states. This material has been discussed previously in the literature; see for example [Br02], [BB02], [Kly02], [Kly07, §3], [GW10], [SOK12], [Wall08, §4], [Walt14] and [SOK14]. Additional reviews on the classiﬁcation of multipart entangle ment and the geom- etry of multipart Hilbert spaces can be found in [ES14] and [BZ17], res pectively. Readers already familiar with these geometrical preliminaries can skip directly t o section 5. 22 4.1 Geometry of the space of states We begin with the full space of (unnormalized) states. For dimension s (d1,...,dn), the Hilbert space H=H1⊗···⊗H nis the complex vector space CDwithD=d1d2···dnwhose coordinates can be taken as the coeﬃcients ψi1...indeﬁning the state. The inequivalent physical states can be represented as equivalence classes of vec tors with unit norm with two states identiﬁed if they are multiplicatively related by a phase. Eq uivalently, we can work with unnormalized states, omitting the zero vector and identif ying states related via multiplication by a complex scalar. This deﬁnes the complex projective spaceP(H) = CPD−1. 4.1.1 Entanglement structure and the action of K= SU(d1)×···×SU(dn) It is an interesting question to characterize the possible entanglem ent structures that such states can have. By “entanglement structure” we mean propert ies of a state that are un- aﬀected by local unitary transformations; that is, unitary trans formations that act inde- pendently on each tensor factor of the Hilbert space. These tran sformations correspond to changes of basis for the individual subsystems. Mathematically, th e change of basis oper- ations correspond to the group ˜K=U(d1)× ··· ×U(dn) acting on H. Without loss of generality, we may consider the smaller group K= SU(d1)×···×SU(dn) since the action of˜KandKon the projective space P(H) =CPD−1have the same orbits. Each state in P(H) will be on some orbit of K. The space of these orbits then represents the space of possible entanglement structures. To parameterize the space of these orbits, we can deﬁne a set of coordinates which are polynomials in ψandψ†invariant under the

Question 9 multiple-choice

Quantum algorithms often rely on amplitude amplification and the Quantum Fourier Transform (QFT) to solve problems in cryptography and computational complexity. Ensuring that these algorithms succeed with certainty, rather than high probability, is a major technical challenge. Which of the following techniques enables amplitude amplification to work exactly (with guaranteed success), even when the success probability of eigenvalue estimation depends on the input instance? 1) Increasing the number of qubits in the quantum register 2) Performing the Quantum Fourier Transform twice in succession 3) Uniformizing the success probability by randomly shifting the input state and correcting afterwards 4) Measuring the quantum state after every operation 5) Using classical post-processing to identify correct eigenvalues 6) Applying an unstructured Grover search to the problem 7) Replacing phase estimation with brute-force search algorithms

✓ Correct Answer:

The correct answer is 3) Uniformizing the success probability by randomly shifting the input state and correcting afterwards.

📚 Reference Text:

eiϕto the good subspace and leave its orthogonal complement unchan ged. So how can we check whether a number x′is the right eigenvalue of |Ψx/an}⌊ra⌋k⌉tri}ht, thus whether x′=x? This can be done because the eigenstate |Ψx/an}⌊ra⌋k⌉tri}htis still available exactly. Thus given a state of the form |Ψx/an}⌊ra⌋k⌉tri}ht/summationtext x′cx,x′|x′,gx,x′/an}⌊ra⌋k⌉tri}ht, we can check the second register against the ﬁrst one. To do this we apply the re verse of the steps in eq. 1 to these two registers, thus: |x′,Ψx/an}⌊ra⌋k⌉tri}ht → | x′,Ψx−x′/an}⌊ra⌋k⌉tri}ht → | x′,θx−x′/an}⌊ra⌋k⌉tri}ht where in the second step we only act on the second register. The st ate|Ψ0/an}⌊ra⌋k⌉tri}htis mapped back to |0/an}⌊ra⌋k⌉tri}ht, while for x′/n⌉}ationslash=xwe get some state |θx−x′/an}⌊ra⌋k⌉tri}htorthogonal to |0/an}⌊ra⌋k⌉tri}ht. We can now apply the phase eiϕto the|0/an}⌊ra⌋k⌉tri}htstate and undo the previous operations. 3 2.2 “Uniformising” the success probability One obstacle to using amplitude ampliﬁcation to make algorithms exact is that the success probability of the “heuristic” algorithm Amust be known. But this probability may depend on the (unknown) instance of the problem. I n our case the success probability of eigenvalue estimation on |Ψx/an}⌊ra⌋k⌉tri}htindeed does depend on x. Wecanﬁx this problembymodifying Asuchthatthe new successprobability will become instance independent and equal to the average over all instances for the original A. To do this uniformisation we pick an integer runiformly at random from {0,1,...p−1}and replace |Ψx/an}⌊ra⌋k⌉tri}htwith|Ψx+r/an}⌊ra⌋k⌉tri}ht, which is just a rephasing. We keep a record of rand subtract it again from the result of eigenvalue estimation. To do this with a unitary Awe will need an additional register for r, but this is no problem, as we have already included the possibility that eigenvalue estimation (eq. 2) also generates some unwanted g arbagegx,x′. So now exact amplitude ampliﬁcation will allow us to do |Ψx,0/an}⌊ra⌋k⌉tri}ht → | Ψx/an}⌊ra⌋k⌉tri}ht|x,gx,x/an}⌊ra⌋k⌉tri}ht. To get rid of the “garbage” we can do the usual trick of copying the wanted resultxinto an additional “save” register and then undoing the previous st eps. In total this will lead to six applications of Afor an exact QFFT. In summary, the construction of an exact QFFT relies on making eige nvalue estimation (on Fourier states |Ψx/an}⌊ra⌋k⌉tri}ht) exact. The essential observations are that eigenvalue estimation leaves the eigenstate |Ψx/an}⌊ra⌋k⌉tri}htexactly unchanged and so it can be used for the checking stage of amplitude ampliﬁcation. Furth ermore we used that the success probability of estimating xfrom|Ψx/an}⌊ra⌋k⌉tri}htcan rather easily be “uniformised” across all x= 0...p−1. 3 An exact discrete logarithm algorithm An exact algorithm for the QFFT leads in a straightforwardmanner t o an exact algorithm for the discrete logarithm algorithm of the same order. Th is was also observed for ﬁnite ﬁelds of prime order by Brassard and Høyer [2] (Theorem 12). For smooth orders (only small prime factors) the problem can easily be solved classically. Here we give a quick review for the case when the or der is a large prime (see also [12], section 2.2.3). In a discrete logarithm problem we are given an element αwhich generates a cyclic group of some ﬁnite order, here a prime. Thus

Question 10 multiple-choice

In liquid-state NMR quantum computing, precise modeling of multi-spin molecular systems is essential for accurate simulation of quantum algorithms and interpretation of experimental results. Understanding how different nuclear spins interact and relax within a molecule is a key aspect of this domain. When simulating the quantum dynamics of an alanine molecule for NMR quantum computing experiments, which approach correctly accounts for the influence of hydrogen spins on the carbon-13 qubits during a Quantum Fourier Transform experiment? 1) Hydrogen spins are actively flipped during the experiment and require time-dependent modeling in the Hamiltonian. 2) Hydrogen spins are omitted entirely from the simulation because their couplings are negligible. 3) Hydrogen spins are included as coherent superpositions in the initial density matrix for the carbon system. 4) Hydrogen spins influence carbon-13 qubits only through dipolar couplings that must be averaged over molecular motion. 5) Hydrogen spins are treated as constants of motion, resulting in an incoherent mixture of 16 independent 3-spin Hamiltonians for the carbons, each corresponding to a different hydrogen configuration. 6) Hydrogen spins are modeled by their effect on the nitrogen-14 quadrupolar relaxation only. 7) Hydrogen spins are assumed to have the same relaxation dynamics as carbon-13 qubits and contribute equally to decoherence.

✓ Correct Answer:

The correct answer is 5) Hydrogen spins are treated as constants of motion, resulting in an incoherent mixture of 16 independent 3-spin Hamiltonians for the carbons, each corresponding to a different hydrogen configuration..

📚 Reference Text:

Hamiltonian. •The Hilbert space of the larger system with which the given system int eracts coher- ently, and the Hamiltonian describing these interactions. •The relaxation superoperator of the system, and some knowledge of how the larger system relaxes. •Bounds on the precision of the classical control ﬁelds applied. 12 •The distribution of incoherent variations in these ﬁelds across the s ample. Both the ﬁelds and the positions of the spins can be treated classica lly. Since the larger system contains only a total of eight spins, we can simulate it exactly . Further information, for example the correlation times or full spectral densities of the n oise generators driving relaxation [32], could have been included in the model, but the above pr oved adequate to explain most of the experimental observations for the QFT. The ﬁrst attribute is the Hamiltonian for the three carbon-13 spins of alanine used as qubits for theQFT, which hasthe formof Hsgiven in Eqn. (19). The larger system includes, in addition to the carbons, the four (nonexchanging) hydrogens a nd the spin-1 nitrogen-14 nucleus in alanine. The larger system’s Hamiltonian has the form HS=Hs+π4/summationdisplay i=1νi Hσi H,z+πνNσN,z+π 24/summationdisplay j>i=1Ji,j H,H/vector σi H·/vector σj H+ (22) π 24/summationdisplay i=13/summationdisplay j=1Ji,j C,Hσi H,zσj C,z+π 23/summationdisplay i=1Ji N,CσN,zσi C,z+π 24/summationdisplay j=1Jj N,HσN,zσj H,z, where we have used the well-known fact that couplings between spin s with distinct gyro- magnetic ratios can be truncated to just the secular (i.e. σzσz) part. The spin-1 nitrogen has an electric quadrupole moment and hence a s hortT1, so its coupling to the other spins cannot be observed. As a result, the nit rogen can be omitted from the Hamiltonian, although it still plays a role in the relaxation of th e other spins. The spin-lattice ( T1) relaxation time of the hydrogen atoms, on the other hand, is longe r than the experiment, so here we must include the additional frequency s hifts that depend upon their spin states. In other words, we take the σH,zto be constants of the motion, and treat the carbons as an incoherent mixture of 24= 16 independent 3-spin systems, each with their resonance frequencies shifted by one of the 16 possible sums of±JC,H/2. Because we start with the spins in the high-temperature equilibrium state, each of these 16 independent evolutions contributes equally to the simulated density matrix. The computer search for the strongly modulating control sequen ces (pulses) is quite demanding, and it is important to keep the Hilbert space as small as po ssible during the associated simulations. For this reason the simulations described in t his paper ignored the four hydrogen atomsinthe alaninemolecule, which were left alignedwit h themain magnetic ﬁeld during the actual experiments but which nonetheless have cou plings to the carbons on 13 the order of 150Hz. A better way, which we subsequently implement ed and is described here for completeness, would be to average the results of 16 simulations over each of the 3-spin Hamiltonians Hδ s,c=π3/summationdisplay i=1/parenleftBig νi C+4/summationdisplay j=1(−1)δj

Question 11 multiple-choice

In computational molecular spectroscopy, various quantum chemical and simulation techniques are used to predict vibrational frequencies and interpret solvent effects on small molecule spectra. The interplay between theoretical methods and experimental measurements is critical for understanding hydrogen bonding and molecular interactions in aqueous environments. Which computational approach utilizes a composite of CCSD reference frequencies with a QM(AM1)/MM perturbation potential to achieve both high accuracy and computational efficiency in predicting vibrational spectra? 1) QVP1 method 2) Dipole Autocorrelation Function Fourier transform 3) Harmonic Normal Mode Analysis with HF basis 4) Spectral Expansion using MP2 reference 5) DAF using CCSD exclusively 6) QM/MM simulation with AM1 only 7) Two-dimensional vibrational Schrödinger equation on M06-2X surface

✓ Correct Answer:

The correct answer is 1) QVP1 method.

📚 Reference Text:

(black) and computed vibrational spectra of the carbonyl stretching region of acetone in water. The spectrum from the first-order quantum vibration perturbation (QVP1) calculation employing a composite of the CCSD(T) reference frequency and a QM(AM1)/MM perturbation potential is shown in red, and that from the Fourier transform of the molecular dipole autocorrelation function determined using AM1 directly in QM/MM simulations is given in blue.Xue et al. Page 24 J Chem Theory Comput . Author manuscript; available in PMC 2018 February 01. Author Manuscript Author Manuscript Author Manuscript Author Manuscript Author Manuscript Author Manuscript Author Manuscript Author ManuscriptXue et al. Page 25 Table 1 Computed HCl Stretching Vibrational Frequency ωCl−H and the Shift ΔωCl−H of the HCl(H 2O) Complex Using Harmonic Normal Mode Analysis (NMA), Fourier Transform of the Dipole Autocorrelation Function (DAF), and the Spectral Expansion (eq 6) of the Frequency-Frequency Time Correlation Function from the QVP1 Method, along with the Experimental Value (Vibrational Frequencies Given in cm−1) method basis set ωCl−H ΔωCl−Ha HFQb 3033.0 309.5 NMA15 MP2Tb 2791.0 67.5 CCSD(T)Db 2782.0 58.5 DAF M06-2X cc-pVTZ 2837.6 114.1 QVP1 M06-2X cc-pVTZ 2653.0 −70.5 M06-2X/CCSD(T)-F12bcc-pVTZ/cc-pVTZ-F12c2724.8 1.3 CCSD(T)-F12b cc-pVTZ-F12 2737.6 14.1 EXP15,16 2723.5 aΔωCl−H = ωCl−H − ωexp. baug-cc-pVXZ, X = D,T,Q. cM06-2X/cc-pVTZ+CCSD(T)-F12b/cc-pVTZ-F12. J Chem Theory Comput . Author manuscript; available in PMC 2018 February 01. Author Manuscript Author Manuscript Author Manuscript Author ManuscriptXue et al. Page 26 Table 2 Computed H−Cl Stretching Vibrational Frequencies (cm−1) for (HCl) 2(H2O) by Solving Two Independent One-Dimensional (E 1D) Schrodinger Equations and a Two-Dimensional (E 2D) Vibrational Schrodinger Equationa Mode E1D E2D Δν Q1 2617.0 2612.3 4.7 Q2 2692.6 2693.7 −1.1 aThe potential energy surfaces were determined using M06-2X/cc-pVTZ. Q 1 and Q 2 represent, respectively, the H−Cl stretch associated with the HCl molecule directly hydrogen-bonded to water and hydrogen-bonded to the probe HCl molecule (Figure 5). J Chem Theory Comput . Author manuscript; available in PMC 2018 February 01.

Question 12 multiple-choice

Photonic quantum processors utilizing qudits—quantum systems with more than two levels—are advancing the scalability and flexibility of quantum computing. These devices often integrate multiple quantum logic gates, entanglement sources, and precise optical components on a single chip. Which feature uniquely enables a photonic multiqudit quantum processing unit to achieve fully reconfigurable and programmable initialization, manipulation, and analysis of multidimensional qudit states? 1) Integration of only polarization-encoded photon sources 2) Use of superconducting nanowire single-photon detectors 3) Implementation of time-bin encoding exclusively 4) Utilization of only single-qubit gates on a photonic chip 5) Limitation to probabilistic two-level entanglement 6) Absence of on-chip phase-shifters 7) Integration of arbitrary state preparation, multiqudit controlled logic gates (MVCU), and measurement for single and two-qudit states

✓ Correct Answer:

The correct answer is 7) Integration of arbitrary state preparation, multiqudit controlled logic gates (MVCU), and measurement for single and two-qudit states.

📚 Reference Text:

e 2shows the core of a multiqudit processor, i.e, the multiqudit multi-value con- trolled logic gate, which is realis ed by the following three steps: generation of the multiphoton multidimensional Greenberger- Horne-Zeilinger entangled state GHZ ji nþ1;d45,46,w h i c he n a b l e st h e entangling operations between the multiqudit states; Hilbert space expansion of each qudit in y-register to form an entire space of d2n, that locally allows individual and ar bitrary single-qudit operations56; coherent compression of the entire state back to a dnspace57.T h e s e sequences of operations result in a mu ltiqudit multi-value controlled- unitary (MVCU) gate as1ﬃﬃ dp∑d/C01 j¼0kj/C12/C12/C12E /C10Qn i¼1Oi;jϕ/C12/C12/C11 i,w h e r e kj/C12/C12/C12E in the auxiliary x-register presents the logical state in the j-th mode (for simplicity it is denoted as j/C12/C12/C11 ), and Oi,jin the data y-register refers to an arbitrary local o p e r a t i o no nt h eq u d i ts t a t e ϕ/C12/C12/C11 ithat is initialised by the Piqudit generator. Such multiqudit MVCU gate works with a (1/ d) success probability regardless of n(see Supple- mentary Note 3 and Supplementary F ig. 1). The quantum circuits in Fig.2a, b provide a scheme of implementing multiqudit quantum Fourier algorithms in the scalable Kitaev ’s framework26–29. Figure 2c illustrates the integrated photonic quantum circuits for a two-ququart version of qudit-based quantum processing unit ( d- QPU). It was fabricated in silico n using the complementary metal- oxide-semiconductor (CMOS) process with the 248nm deep ultraviolet lithography (see a device image in Fig. 2d). The processor allows the generation of a path-enc oded two-ququart entangled state of GHZji2;4(i.e., the 4-dimensional generalised Bell state of Bell ji4), by a coherent excitation of four integrated spontaneous four-wave- mixing (SFWM) sources. It is followed by the sequences of processes of“space expansion –local operation –coherent compression" for the realisation of d-QPU, see Fig. 2b. The d-QPU chip monolithically integrates the core capabilities and fu nctionalities, including arbitrary single-ququart preparation ( P), arbitrary two-ququart MVCU operation (that presents a d-ary generalisation of two-qubit controlled-unitary operation), and arbitrary single-ququart measure- ment ( M). Though on-chip generatio n, manipulation and measure- ment of entangled qudit states have been reported36,t h i sw o r k demonstrate the key abilities to in itialize, manipulate, and analyze qudit states and gates in a fully recon ﬁgurable and reprogrammable manner, providing a major technological advance for qudit quantum computing. In Fig. 2d it shows one of the largest-scale programmable quantum photonic chip having 451 photonic components, including 116 recon ﬁgurable phase-shifters (see their characterisations in Fig. 2c i n s e t s ) .T h et w o - p h o t o n sd e t e c t i o nr a t ea tt h em a g n i t u d eo f1 03/s was measured in the two-ququart device, which is six orders higher than that in a four-qubit device (note

Question 13 multiple-choice

In quantum computing, the Hidden Subgroup Problem (HSP) is central to the development of efficient algorithms and involves distinguishing between quantum states associated with different subgroups. Measurement strategies, such as the Pretty Good Measurement (PGM), play a crucial role in solving HSPs, especially for groups with special symmetries. Which measurement strategy is proven to be optimal for identifying hidden subgroups when the subgroups are conjugate to a fixed subgroup and are sampled uniformly? 1) Maximum Likelihood Measurement 2) Projective Measurement onto Irreducible Representations 3) Pretty Good Measurement (PGM) 4) Von Neumann Measurement 5) Adaptive Bayesian Measurement 6) Swap Test Measurement 7) Holevo-Helstrom Measurement

✓ Correct Answer:

The correct answer is 3) Pretty Good Measurement (PGM).

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generalized measurement can be described by a set of positive operators that sum to identity. If EHis the measurement operator corresponding to our guessing that the hidden subgroup is H, then the optimal measurement problem in the single copy case maxim izes the probability of successfully identifying the subgroups, psucc=/summationdisplay H∈Sub(G)pHTr[ρHEH], (4) subject to EHbeing a valid generalized measurement: EH≥0 and/summationtext H∈Sub(G)EH=I. Here Sub( G) is the set of subgroups of G. This is the well-known problem of state distinction. The state distinc tion problem was considered in the seventies by Holevo [20], and Yuen, Kennedy, and Lax [21] whe re necessary and suﬃcient conditions for the optimal measurement were derived for the generic state distinctio n problem. Applied to the above formulation of the hidden subgroup problem, these necessary and suﬃcient condition s yield that a measurement EHis optimal if and only if /summationdisplay H∈Sub(G)pHEHρH=/summationdisplay H∈Sub(G)pHρHEH, /summationdisplay H∈Sub(G)pHEHρH≥pH′ρH′,for allH′∈Sub(G). (5) We may also consider the multi-copy optimal measurement, where we query the function rtimes. In that case we produce rcopies of the hidden subgroup state ρH, and the optimality condition is obtained by simply replacing ρH byρ⊗r Hin Eq. (5). In [15], Ip considered the question of identifying the optimal measur ement for the abelian HSP when each of the subgroups of Gis given with equal a priori probability. Here we brieﬂy review this resu lt. Deﬁne PH:=|G| |H|ρH. (6) The optimal single copy measurement for the abelian HSP is then given by the measurement operators (recursively deﬁned) EH:=PH−/summationdisplay J⊃HEJ. (7) 3 In the case of rcopies, the optimal measurement is given by the similar expression E(r) H:=P⊗r H−/summationdisplay J⊃HE(r) J. (8) Finally Ip also showedthat for the HSP overthe dihedral group, the aboveexpressionis not the optimal measurement. Following Ip’s work, Bacon, Childs, and van Dam considered the optima l measurement for the HSP over the dihedral [16] and the Heisenberg groups [14]. In both of these case s, the measurement turned out to be the pretty good measurement (PGM), so named because it does a good job of id entifying states [17]. If the state ρiis given with probability pi, then the PGM is a measurement consisting of measurement operat orsEideﬁned as Ei:=piM−1/2ρiM−1/2, M:=/summationdisplay ipiρi, (9) where the inverse square root M−1/2is taken over the support of M. Bacon, Childs, and van Dam showed that the PGM was optimal for the dihedral and Heisenberg HSPs for both the single and multi-copy case and when the subgroups were restricted to certain order 2 or order psubgroups (which is enough to be able to solve the HSP over all subgroups due to a generalization of a theorem of Ettinger and H øyer [14, 16, 22].) Finally, Moore and Russell showed that under certain further cond itions the PGM is guaranteed to be optimal [18]. In particular they showed that if the hidden subgroup is restricted to come from the set of subgroups conjugate to a ﬁxed subgroup, then for the single copy case, if these subgroups are sampled with uniform

Question 14 multiple-choice

In Group Field Theory, quantum gravity condensates are studied as analogues of Bose-Einstein condensates, aiming to understand how macroscopic spacetime geometry can emerge from microscopic quantum building blocks. The behavior of condensate population and the mathematical representation of their quantum states are crucial in analyzing different interaction regimes. What is a key indication that non-Fock representations are required to accurately describe quantum gravity condensates in Group Field Theory? 1) The presence of only combinatorially nonlocal interaction terms 2) The unbounded growth ("blow-up") of the condensate population number operator in the strong interaction regime 3) The dominance of the highest geometric configuration in operator spectra 4) The invariance of condensate population across all interaction strengths 5) The vanishing expectation values of geometric operators for all solutions 6) The absence of any interaction terms in the model 7) The exclusive validity of the Bogoliubov ansatz in strong nonlinear regimes

✓ Correct Answer:

The correct answer is 2) The unbounded growth ("blow-up") of the condensate population number operator in the strong interaction regime.

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Question 15 multiple-choice

In the study of finite 2-groups, the structure of derived subgroups and maximal subgroups is crucial for understanding group classification, especially when certain subgroups are abelian or metacyclic. The relationship between the Frattini subgroup, derived subgroup, and properties of involutions often determines the possible isomorphism types of these groups. Which of the following statements accurately describes the derived subgroup G' of a finite nonabelian 2-group G with three abelian maximal subgroups and abelian quotient types, given that G is not metacyclic? 1) G' is cyclic of order 8 and equals the center of G. 2) G' is elementary abelian of order 8 and disjoint from the Frattini subgroup Φ. 3) G' is trivial, implying G is abelian. 4) G' is contained in every maximal subgroup but not in the Frattini subgroup. 5) G' is isomorphic to C₄ and equal to the commutator subgroup of every maximal subgroup. 6) G' is isomorphic to C₂ × C₂ (elementary abelian of order 4) and is contained in the Frattini subgroup Φ. 7) G' is metacyclic and has order 2, coinciding with the center of G.

✓ Correct Answer:

The correct answer is 6) G' is isomorphic to C₂ × C₂ (elementary abelian of order 4) and is contained in the Frattini subgroup Φ..

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≤Aso that H1//an}bracketle{tz/an}bracketri}htandH2//an}bracketle{tz/an}bracketri}htare two distinct FINITE 2-GROUPS 67 abelian maximal subgroups in G//an}bracketle{tz/an}bracketri}ht. It follows that H//an}bracketle{tz/an}bracketri}htis also abelian and soH′=/an}bracketle{tz/an}bracketri}htsinceHis nonabelian. If d( H) = 2, then (by Proposition 1.1) H would be minimal nonabelian, a contradiction. Thus d( H)≥3. IfG//an}bracketle{tz/an}bracketri}htis abelian, then G′=/an}bracketle{tz/an}bracketri}htand so G=H1CG(H1) which gives d(G) = 3, a contradiction. Hence G//an}bracketle{tz/an}bracketri}htis nonabelian and so ( G//an}bracketle{tz/an}bracketri}ht)′∼=C2 sinceG//an}bracketle{tz/an}bracketri}hthas three distinct abelian maximal subgroups. Thus |G′|= 4 and G′≤A= Φ(G). Taking h1∈H1−Aandh2∈H2−A, we have /an}bracketle{th1, h2/an}bracketri}ht=G and so s= [h1, h2]∈G′−/an}bracketle{tz/an}bracketri}ht. IfG′∼=C4, then zis a square in H1andH2and so both H1andH2are metacyclic (Proposition 1.1). Since d( H)≥3,Ghas exactly one nonmetacyclic maximal subgroup. But then Theor em 87.12 in [2] forn= 2 implies that Ghas an abelian maximal subgroup, a contradiction. Thus G′=/an}bracketle{ts, z/an}bracketri}ht∼=E4, where sis an involution. If s∈Z(G), then G//an}bracketle{ts/an}bracketri}ht would be abelian (because /an}bracketle{th1, h2/an}bracketri}ht=Gands= [h1, h2]), a contradiction. Hence s/ne}ati≀nslash∈Z(G) and so s/ne}ati≀nslash∈Z(H1) ors/ne}ati≀nslash∈Z(H2). Without loss of generality we may assume that s/ne}ati≀nslash∈Z(H1). Suppose that zis a square in H1, i.e., there is v∈H1such that v2=z. Suppose at the moment that v∈H1−Ain which case /an}bracketle{tv, G′/an}bracketri}ht=/an}bracketle{tv, s/an}bracketri}ht∼=D8 sincesv=sz. It follows that /an}bracketle{tv, G′/an}bracketri}ht=H1. Since C G(H1)≤H1(otherwise, d(G) = 3 ), Gwould be of maximal class (Proposition 1.7), a contradictio n (noting that 2-groups of maximal class have a cyclic maximal subgroup). Thus v∈A= Φ(G). In that case both H1andH2are metacyclic (Proposition 1.1) which together with d( H)≥3 allows us to use §87, part 20in [2]. If Ghas a normal elementary abelian subgroup of order 8, then we get g roups in part (a) of our theorem. If Ghas no normal elementary abelian subgroup of order 8, then we get the group of order 25given in part (b) of our theorem. Now we assume that zis not a square in H1which implies that H1is nonmetacyclic (Proposition 1.1). If h1is an involution, then sh1=szshows that /an}bracketle{th1, s/an}bracketri}ht∼=D8and so /an}bracketle{th1, s/an}bracketri}ht=H1is metacyclic, a contradiction. Hence o(h1) = 2n,n≥2. Set u=h2n−1 1so that u∈Z(H1) and u/ne}ati≀nslash∈G′sincezis not a square in H1andsh1=sz. We have E= Ω1(H1) =/an}bracketle{tz, s, u /an}bracketri}ht∼=E8,E/triangleleftequalG andE≤Awhich implies that H2is nonmetacyclic and therefore zis also not a square in H2. Since H1=E/an}bracketle{th1/an}bracketri}ht=/an}bracketle{th1, s/an}bracketri}ht, we have |H1|= 2n+2,|G|= 2n+3 andA=/an}bracketle{th2 1, s, z/an}bracketri}htis abelian of order 2n+1and type (2n−1,2,2). Also, H1is a splitting extension of G′by/an}bracketle{th1/an}bracketri}ht∼=C2n,n≥2. Since d( G/G′) = 2, we get thatG/G′is abelian of type (2n,2). We get G=FH1, where F∩H1=G′ and|F:G′|= 2 so that F=/an}bracketle{tG′, x/an}bracketri}htwith o( x)≤4. In fact, x2∈ /an}bracketle{tz/an}bracketri}ht. Indeed, if Fis not elementary abelian, then ℧1(F)∼=C2and℧1(F)≤G′. ButF/triangleleftequalGand so ℧1(F)≤Z(G) which implies that ℧1(F) = /an}bracketle{tz/an}bracketri}htsince G′/ne}ati≀nslash≤Z(G). Since /an}bracketle{txh1/an}bracketri}htG′/G′is another cyclic subgroup of index 2 in G/G′ (distinct from H1/G′),M=/an}bracketle{tG′, xh1/an}bracketri}htis a maximal subgroup of Gdistinct fromH1andM′=/an}bracketle{tz/an}bracketri}ht(since H′ 2=H′=/an}bracketle{tz/an}bracketri}ht). IfG′≤Z(M), then Mwould be abelian, a contradiction. We get sxh1=szand so M=/an}bracketle{txh1, s/an}bracketri}htis minimal nonabelian which gives M=H2. We may set h2=xh1, where [ h1, h2] =s 68 Z. BO ˇZIKOV AND Z. JANKO andG′=/an}bracketle{ts, z/an}bracketri}ht. From sxh1=szfollows sx= (sxh1)h−1 1= (sz)h−1 1=sh−1 1z= (sz)z=s and so Fis abelian. From s=

Question 16 multiple-choice

In computational algebra and coding theory, reconstructing hidden structures often involves analyzing polynomial spaces over finite fields and leveraging properties of linear independence, sample spanning, and coset partitioning. Understanding bounds on subset sizes and the uniqueness of solutions is critical when solving equations and designing effective algorithms in these contexts. In the setting of polynomials of degree at most $ p-1 $ in $ n $ variables over $ \mathbb{Z}_p $, which of the following statements accurately describes the upper bound on the ratio $|R_k|/|V_k|$ for certain subspaces and associated sets, as established using coset partitioning arguments? 1) The ratio is always equal to $ 1 $ for all $ k $. 2) The ratio cannot exceed $ 1/2 $ for any $ k $. 3) The ratio is strictly less than $ 1/p $ for all $ k $. 4) The ratio is unbounded and can grow with $ n $. 5) The ratio is minimized when $ k = p-1 $. 6) The ratio is at most $ (p-1)/p $ for $ k = 1, \ldots, p-1 $. 7) The ratio equals $ (k-1)/k $ for all $ k $.

✓ Correct Answer:

The correct answer is 6) The ratio is at most $ (p-1)/p $ for $ k = 1, \ldots, p-1 $..

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Question 17 multiple-choice

Quantum computing with qudits, which are quantum systems with more than two levels, enables algorithms to utilize higher-dimensional quantum registers. This approach can significantly impact resource requirements and error scaling for key algorithms such as phase estimation. In the phase-estimation algorithm using qudit circuits, which advantage does increasing the qudit dimension d provide for quantum computing? 1) It allows the use of fewer controlled operations in the algorithm. 2) It eliminates the need for an inverse quantum Fourier transform. 3) It makes SWAP gates unnecessary for output reordering. 4) It enables phase estimation without requiring two registers. 5) It removes the requirement for eigenvector input states. 6) It exponentially reduces error rates and decreases the number of physical resources required for a given accuracy. 7) It prevents the need for quantum multiplexers during phase readout.

✓ Correct Answer:

The correct answer is 6) It exponentially reduces error rates and decreases the number of physical resources required for a given accuracy..

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transform and ensure the correct sequence of j1j2/jn,as e r i e so fS W A P gates are applied at the end, which are not explicitly drawninFigure 10 . The QFT developed in qudit system offers a crucial subroutine for many quantum algorithm using qudits. Qudit QFT offerssuperior approximations where the magnitude of the errordecreases exponentially with dand the smaller error bounds are smaller [ 165]; which outperforms the binary case [ 34]. 3.2.2 Phase-Estimation Algorithm With Qudits With the qudit quantum Fourier transform, we are able to generalize the PEA to qudit circuits [ 26]. Similar to the PEA using qubit, the PEA in the qudit system is composed by tworegisters of qudits. The ﬁrst register contains tqudits and t depends on the accuracy we want for the estimation. Weassume that we can perform a unitary operation Uto an arbitrary number of times using qudit gates and generate itseigenvector |u〉and store it using the second register ’s qudits [ 17]. We want to calculate the eigenvalue of |u〉where U|u〉/equalse 2πir|u〉 by estimating the phase factor r. The following derivations follow those in Ref. 26. For convenience, we rewrite the rational number ras r/equalsR/dt/equals⎥summationdisplayt k/equals0Rl/dl/equals0.R1R2/Rt. (100) As shown in Figure 11A , each qudit in the ﬁrst register passes through the generalized Hadamard gate H≡F(d,d). For the lth qudit of the ﬁrst register, we have F(d,d)|0l〉/equals1⎥radicaltpext⎥radicaltpext d√⎥summationdisplayd−1 kl/equals0|kl〉. (101) Then the lthqudit is used to control the operation Udt−lon the target qudits of the state |u〉in the second register, which gives CUdl−1|k〉⊗|u〉/equals|k〉⎥parenleftBigUdt−l⎥parenrightBigk|u〉/equalse2πikdt−lr|k〉⊗|u〉. (102) Note that the function of the controlled operation CUdt−lcan be considered as a “quantum multiplexer ”[24, 87, 139 ]. After executing all the controlled operations on the qudits, the quditsystem state turns out to be ⎛⎝⎥productdisplayt l/equals1⊗1⎥radicaltpext⎥radicaltpext d√⎥summationdisplayd−1 kl/equals0e2πikldt−lr|kl〉⎞⎠⊗|u〉. (103) Therefore, through a process called the “phase kick-back ”, the state of the ﬁrst register receives the phase factor and becomes |Register 1 〉/equals1 dt/2⎥summationdisplaydt−1 k/equals0e2πirk|k〉. (104) The eigenvalue rwhich is represented by the state |R〉can be derived by applying the inverse QFT to the qudits in the ﬁrst register: F−1⎥parenleftbigd,dt⎥parenrightbig|Register 1 〉/equals|R〉. (105) The whole process of PEA is shown in Figure 11B . To obtain the phase r/equalsR/dtexactly, we can measure the state of the ﬁrst register in the computational basis. The PEA in qudit system provides a signi ﬁcant improvement in the number of the required qudits and the error rate decreases Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504 14Wang et al. Qudits and High-Dimensional Quantum Computing exponentially as the qudit dimension increases [ 129]. A long list of PEA applications includes Shor ’s factorization algorithm [ 142]; simulation of quantum systems [ 1]; solving linear equations [ 69, 128]; and quantum counting [ 147]. To give some examples, a quantum simulator utilizing the PEA algorithm has been used to calculate the molecular ground-state energies [ 8] and to obtain the energy spectra of molecular systems [ 13, 41, 42, 84, 154 ]. Recently, a method to

Question 18 multiple-choice

In mathematical physics, topological field theories connect algebraic structures like categories and algebras with the topology of manifolds, often leading to deep insights in representation theory and geometry. Categorical approaches to 2D and 4D topological field theories have advanced understanding of quantum invariants and dualities. Which algebraic structure is essential for constructing category-valued 2D topological field theories via factorization homology, and also underlies the quantum symmetries present in braided tensor categories? 1) Commutative Hopf algebras 2) Symmetric monoidal categories 3) Modules for Drinfeld–Jimbo quantum groups 4) Derived categories of coherent sheaves 5) Real reductive Lie groups 6) Vertex operator algebras 7) Supercommutative algebras

✓ Correct Answer:

The correct answer is 3) Modules for Drinfeld–Jimbo quantum groups.

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Question 19 multiple-choice

Modular tensor categories (MTCs) play a fundamental role in mathematical physics, especially in topological quantum field theory and quantum computing. The properties of modularity and unitarity, as well as specific fusion rules and the structure of the S-matrix, are central to their classification and physical applicability. In the MTC of type Z(A1) at ℓ=5, which fusion rule characterizes the simple object X₁, and what key implication does this have for the category's use in topological quantum computing? 1) X₁ ⊗ X₁ = X₁ ⊕ 1₁ ⊕ X₁; the category is non-modular 2) X₁ ⊗ X₁ = 1₁; the category is abelian 3) X₁ ⊗ X₁ = 1₁ ⊕ X₁; the category supports non-abelian anyon models 4) X₁ ⊗ X₁ = X₁ ⊕ X₁; the category is not unitary 5) X₁ ⊗ X₁ = 0; the category is trivial 6) X₁ ⊗ X₁ = 1₁ ⊕ 1₁; the S-matrix is singular 7) X₁ ⊗ X₁ = 1₁ ⊕ X₁ ⊕ X₁; the category is non-unitary

✓ Correct Answer:

The correct answer is 3) X₁ ⊗ X₁ = 1₁ ⊕ X₁; the category supports non-abelian anyon models.

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sub- category. We only give enough information to discuss the mod ularity and unitarity of the category. 5.1. Type Z(A1)atℓ= 5.The following MTC is obtained from C(sl2,5, eπi/5) by taking the subcategory of modules with integer highest we ights. There are two simple objects 1 1,and X1satisfying fusion rules: X1⊗X1= 1 1⊕X1and 1 1 ⊗Xi=Xi. TheS-matrix is S=/parenleftigg 11+√ 5 2 1+√ 5 2−1/parenrightigg and the twists: θ0= 1,θ1=e4πi/5. It is clear that det( S)/ne}ationslash= 0, and it follows from [ W] that the category is unitary (notice that the categorical dimensions are both positive). 5.2. Type B2at9th Roots of Unity. Consider the pre-modular categories C(so5,9, ejπi/9) with gcd(18 , j) = 1. There are 12 inequivalent isomorphism classes of simple objects. The simple iso-classes of objects are lab elled by ( λ1, λ2)∈1 2(N2) withλ1≥λ2. The twist coeﬃcients for Xλisq/angbracketleftλ+2ρ,λ/angbracketrightwhere the form is twice the usual Euclidean form. The obstruction to modularity men tioned in the proof of Theorem 4.3 is labelled by γ:=1 2(5,5) The categorical dimension function is: dλ:=[2(λ1+λ2+ 2)][2( λ1−λ2+ 1)][2 λ1+ 3][2 λ2+ 1] [4][3][2][1]. One checks that the simple object Xγis indeed the cause of the singularity of the S-matrix, that is, Sγ,λ=dγdλfor all λ. Thus this category is not modular by Brugui` eres’ criterion, Theorem 4.1. Now let us consider the subcategory of C(so5,9, ejπi/9) with gcd(18 , j) = 1 generated by the simple objects labelled by integer weights : {(0,0),(1,0),(2,0),(1,1),(2,1),(2,2)}. The braiding and twists from the full category restrict, so t he entries of the S- matrix are computed from Formula (2.2). Taking the ordering of simple objects above, we denote the categorical dimensions by di0≤i≤5. The fusion matrix corresponding to (1 ,0) is: N1:= 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 . It is not hard to show that N1determines the other ﬁve fusion matrices by observing thatN1has six distinct eigenvalues and the fusion matrices commut e. There are a total of six categories corresponding to the six possible va lues of q. To describe the S-matrices we let αbe a primitive 18th root of unity, and set r1=−α−α2+α5, r2=α+α2−α4andr3=α4−α5. Then we get the following S-matrices (for the 14 ERIC C. ROWELL 6 choices of α): 1r2r31−1r1 r21 1 r1−r31 r31 1 r2−r11 1r1r21−1r3 −1−r3−r1−1 1 −r2 r11 1 r3−r21 . One checks that det( S)/ne}ationslash= 0 for any α, so these categories aremodular. A bit of Galois theory shows that there are only three distinct Sfor the six choices of α. Notice that it is already clear that the ﬁrst column of Sis never positive, since both 1 and −1 appear regardless of the choice of α. So none of these categories is unitary. References [A] H.H. Andersen, Tensor products of quantized tilting modules , Comm. Math. Phys. 149(1991), 149-159. [AP] H.H. Andersen and J. Paradowski, Fusion categories arising from

Question 20 multiple-choice

Group theory and computational complexity are deeply intertwined in modern mathematical physics, particularly in the study of fault-tolerant quantum computation using anyons and topological phases. Understanding the structure of groups and their representations is essential for analyzing both the hardness of computational problems and the robustness of quantum algorithms. Which of the following statements accurately describes a property of simple groups that has implications for the existence of non-trivial homomorphisms in the context of free group evaluation homomorphisms? 1) Simple groups always have non-trivial normal subgroups, enabling trivial homomorphisms from free groups. 2) Any homomorphism from a free group to a simple group is necessarily trivial due to the group's structure. 3) The kernel of every evaluation homomorphism from a free group to a simple group is equal to the entire free group. 4) Simple groups cannot be used to construct normal subgroups via evaluation homomorphisms from free groups. 5) For simple groups, non-trivial evaluation homomorphisms from free groups must exist, implying there is always a word mapping non-trivially under some homomorphism. 6) All evaluation homomorphisms from free groups to simple groups must be isomorphisms. 7) Simple groups have abelian structure, guaranteeing the normality of all their subgroups.

✓ Correct Answer:

The correct answer is 5) For simple groups, non-trivial evaluation homomorphisms from free groups must exist, implying there is always a word mapping non-trivially under some homomorphism..

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have |{φ∈H(K(Kℓ c);C,G)|ψ(xi) =ci}|≥ 2ℓ|{φ∈H(K(Kℓ c);C,G)→C|ψ(xi) =di}|. (Observe that, as presented, this is only eﬃcient if G=Anhas constant size.) Lemma 6. Consider the alternating group on msymbolsAmand consider any conjugacy class with at least 4ﬁxed points, say, m−3,m−2,m−1andm. Since the conjugacy class is non-trivial, it has a kcycle withk≥2. Pickαin this class to have (1 2... k)in thatkcycle. Now, consider the equation y=α(1k)(m−2m−3)y. This equation has at least two solutions. Proof. It can easily be checked that two solutions are given by eleme nts that diﬀer from αonly in the k cycle. The ﬁrst one contains (1 2 ... k−1m) and the second (1 2 ... k−1m−1) in thatkcycle. Lemma 7. LetGbe a ﬁnite group and Fkbe the free group on kgenerators{x1,...,x k}. For an element d= (d1,...,d k)∈Gk, letφdbe the evaluation homomorphism that carries xitodiand deﬁne Ad= kerφd={w∈Fk|φd(w) = 1}. For two elements c,d∈Gk, deﬁneCto be the subgroup generated by the ci,Dto be the subgroup generated by thedi, and Ad(c) ={φc(w)|w∈Ad}. Then 1.Ad(c)is normal in C. In particular, when the cigenerateG,Ad(c)is normal in G. 2. Ifcanddare not related by a homomorphism of G(which is to say that no homomorphism from Dto Ccarries each citodi) thenAd(c)/\e}ati≀\slash= 1. 21 Proof. For part 1, ker(φd) is certainly normal in Fksinceφdis a group homomorphism. Let Cbe the subgroup of Ggenerated by the ci, andg∈C; we wish to show that g−1Ad(c)g=Ad(c). Ifw∈Fkis a word in the free group for which φc(w) =g, we havew−1Adw=Adand hence that g−1Ad(c)g=φc(w−1Adw) =φc(Ad) =Ad(c), as desired. As for part 2, letCdenote the subgroup generated by {ci}andDthe subgroup generated by {di}. Observe that if Ad(c) is trivial we have ker φd⊂kerφc. In this case, the natural quotient map qc:D∼=F/kerφd→C∼=F/kerφc yields a homomorphism ψ:D→Cfor whichψ:di/ma√st≀→ci. To be precise, let qcdenote the quotient map qc:F/kerφd→F/kerφcand letiddenote the inverse of the isomorphism φdinduces from F/kerφdtoD; observe that id(di) =xi(kerφd). Then the map ψ=φc◦qc◦idis a homomorphism of DontoCthat carries eachdiontoci; see Figure 6. D F/kerφd F/kerφc C idφd qc φc Figure 6: The homomorphism ψ:di/ma√st≀→ci. The above lemma implies that if Gis simple then H1=G(H1cannot be trivial because canddare not related by an automorphism). This means that there exists a w ordwin variables xisuch thatw(c) =zand w(d) = 1. Proof of Theorem 4.The ﬁrst two follow from the deﬁnitions of BPP andSBP and the following lemmas. The # P-completeness follows from the fact that exact evaluation o f the success probability of a randomized computation is # P-complete. 6 Quantum computation with anyons Anyons are particles which exist in two dimensions and have e xotic statistics. Anyons are useful for quantum computation because quantum information can be stored on a s ystem of anyons in a non-local fashion. This means that local errors do not corrupt the quantum informati on and a computer based on anyons will be inherently fault tolerant to local errors. For a tutorial on quantum computation using anyons, see the notes by Preskill [ 30]. In this section, we give short introduction to anyons desc ribed by D(G) for a

Question 21 multiple-choice

Quaternions are four-dimensional mathematical entities that have played significant roles in physics, mathematics, and modern computational fields such as computer graphics and robotics. Their unique algebraic properties have influenced the development of group theory and the understanding of spatial rotations. Which of the following properties distinguishes the quaternion group Q8 from the cyclic group Z4 in the context of group theory and symmetry operations? 1) Q8 is abelian while Z4 is non-abelian 2) Q8 has non-commutative (non-abelian) multiplication, whereas Z4 is abelian 3) Q8 contains only elements of order 4, while Z4 contains elements of order 8 4) Z4 has elements corresponding to the imaginary units i, j, k, while Q8 does not 5) Q8 and Z4 both have subgroup structures identical to the symmetric group S4 6) Q8 is a cyclic group, while Z4 is not 7) Z4 represents rotations in three dimensions, while Q8 cannot be used to encode rotations

✓ Correct Answer:

The correct answer is 2) Q8 has non-commutative (non-abelian) multiplication, whereas Z4 is abelian.

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have three dimensions...The mathematical quaternion partakes of both these elements; in technical language it may be said to be “time plus space”, or “space plus time”: and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, for Space the Three, Might in the Chain of Symbols girdled be. Hamilton was the ﬁrst to use the terms “scalar” and “vector” in their modern sense when he treated quaternions as having scalar (real) and vector (imaginary) part; he referred to quaternions with zero scalar part as pure quaternions. Even though many physicists, including James Clerk Maxwell, used quaternions with en- thusiasm, they were eventually discarded in favor of the vector algebra promoted by William Gibbs and Olivier Heaviside. One of the reasons was that in most physical applications only the imagi- nary part of the quaternions was required, so they came to be viewed as unnecessarily complicated. In essence, the vector algebraists simply extracted the vector part of the quaternions, discarding the scalar part, and developed a more consistent notation. But the drawbacks may have outweighed the beneﬁts. It has been argued that if quaternions “had been used in the correct way from early on, they would have pre-empted many later physical developments that came about much more tortuously” [28]. 42 A comment made by Bertrand Russell on Plato’s Theory of Ideas [29] comes to mind: Any attempt to divide the world into portions, of which one is more ’real’ than the other, is doomed to failure. The heated controversy over the usefulness of quaternions continued for several generations after Hamilton’s death. 4.1.3 The Revival The impact of quaternions on the development of mathematical thought is undeniable: much of the work in the theory of algebras descended from Hamilton’s discovery. That quaternions arise naturally in standard quantum mechanics when time reversals are considered was ﬁrst noted by Dyson [31] in 1962. Since then, other physicists have formulated quaternionic generalizations of the postulates of quantum mechanics, quantum ﬁeld theory, and gravitational theory. Advantages of quaternionic forms have been exploited for attitude control in aeronautics and astronautics [34]. Most recently, quaternions have found extensive applications in the ﬁelds of 3D computer graphics, animations, and virtual reality; and it all began with a seminal 1985 paper by Shoemake [39]. The Euler-angle method for implementing 3D rotations is susceptible to Gimbal lock, where the order of rotations causes the failure of rotation to appear as expected. Quaternions provide a solution to the problem, allowing the programmer to rotate an object through an arbi- trary rotation axis and angle instead of a series of successive rotations. The errors accumulating due to matrix multiplication are avoided, as quaternions representing the axii of rotations are mul- tiplied instead. Moreover, quaternions can be interpolated, thus producing smooth and predictable rotation effects. 43 4.2 The Group Q4 The generalized quaternion group of order 2n,n3, is given by the presentation Q2n=hx;yjx2n1=1;x2n2=y2;y1xy=x1i (4.17) Forn=2, Q4=Z4=hx;yjx2=1;x=y2i=hxjx4=1i (4.18) The only subgroups of Z4are H1=f0g=h4i

Question 22 multiple-choice

Quantum algorithms for the hidden subgroup problem (HSP) play a crucial role in computational group theory and cryptography. Efficient solutions often depend on the structure of the groups involved, influencing both algorithm design and cryptographic security. Which of the following group-theoretical properties allows certain instances of the non-Abelian hidden subgroup problem to be solved efficiently with quantum algorithms? 1) Having a simple group structure 2) Containing an infinite cyclic subgroup 3) Possessing a large center 4) Being nilpotent of high class 5) Admitting a non-normal subgroup of small order 6) Exhibiting a non-trivial direct product decomposition 7) Having an elementary Abelian normal 2-subgroup of small index

✓ Correct Answer:

The correct answer is 7) Having an elementary Abelian normal 2-subgroup of small index.

📚 Reference Text:

arXiv:quant-ph/0102014v1 2 Feb 2001Eﬃcient quantum algorithms for some instances of the non-Ab elian hidden subgroup problem∗ G´ abor Ivanyos†Fr´ ed´ eric Magniez‡Miklos Santha§ October 15, 2018 Abstract Inthispaperweshowthatcertainspecialcasesofthehidden subg roupproblemcanbe solved in polynomial time by a quantum algorithm. These special cases involve ﬁnding hidden normal subgroups of solvable groups and permutation groups, ﬁnding hidd en subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group. 1 Introduction A growing trend in recent years in quantum computing is to cas t quantum algorithms in a group theoretical setting. Group theory provides a unifying fram ework for several quantum algorithms, clariﬁes their key ingredients, and therefore contributes to a better understanding why they can, in some context, be more eﬃcient than the best known classica l ones. The most important unifying problem of group theory for the p urpose of quantum algorithms turned out to be the hidden subgroup problem (HSP) which can be cast in the following broad terms. Let Gbe a ﬁnite group (given by generators), and let Hbe a subgroup of G. We are given (by an oracle) a function fmapping Ginto a ﬁnite set such that fis constant and distinct on diﬀerent left cosets of H, and our task is to determine the unknown subgroup H. While no classical algorithm is known to solve this problem i n time faster than polynomial in the order of the group, the biggest success of quantum comp uting until now is that it can be solved by a quantum algorithm eﬃciently , which means in time polynomial in the logarithm of the order of G,whenever the group is Abelian. The main tool for this solutio n is the (approximate) quantum Fourier transform which can be eﬃciently implement ed by a quantum algorithm [17]. Simon’s algorithm for ﬁnding an xor-mask [26], Shor’s semin al factorization and discrete logarithm ﬁnding algorithms [25], Boneh and Lipton’s algorithm for ﬁn ding hidden linear functions [6] are all special cases of this general solution, as well as the alg orithm of Kitaev [17] for the Abelian stabilizer problem, which was the ﬁrst problem set in a gener al group theoretical framework. That all these problems are special cases of the HSP, and that an eﬃ cient solution comes easily once an eﬃcient Fourier transform is at our disposal, was realize d and formalized by several people, ∗Research partially supported by the EU 5th framework progra ms QAIP IST-1999-11234, and RAND-APX, IST- 1999-14036, by OTKA Grant No. 30132, and by an NWO-OTKA grant . †Computer and Automation Institute, Hungarian Academy of Sc iences, L´ agym´ anyosi u. 11., H-1111 Budapest, Hungary, e-mail: Gabor.Ivanyos@sztaki.hu . ‡CNRS–LRI, UMR 8623 Universit´ e Paris–Sud, 91405 Orsay, Fra nce, e-mail: magniez@lri.fr . §CNRS–LRI, UMR 8623 Universit´ e Paris–Sud, 91405 Orsay, Fra nce, e-mail: santha@lri.fr . 1 including Brassard and Høyer [7], Mosca and Ekert [22] and Jo zsa [15]. An excellent description of the general solution can be

Question 23 multiple-choice

Quantum computing exploits unique principles of quantum mechanics, such as superposition and entanglement, to tackle computational problems that are intractable for classical computers. Its theoretical foundation includes important concepts in quantum information science and has profound implications for cryptography and algorithm design. Which theorem fundamentally asserts that it is impossible to produce an exact copy of an arbitrary unknown quantum state, shaping the security and limitations of quantum information transmission? 1) Bell’s theorem 2) No-cloning theorem 3) Superposition principle 4) Church-Turing thesis 5) Quantum entanglement theorem 6) Grover’s algorithm 7) Unitarity principle

✓ Correct Answer:

The correct answer is 2) No-cloning theorem.

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Question 24 multiple-choice

Categorification in knot theory uses advanced algebraic structures, such as foams and higher representation theory, to generalize classical invariants and explore deep links between topology and quantum groups. Modified foam categories are important for improving functorial properties and robustness of categorified knot invariants. Which modification in the foam category for sl(2) is crucial for achieving better functoriality in Khovanov homology, and what analogous approach is suggested for sl(3) categories? 1) Blanchet’s modified foams, with analogous modifications proposed for sl(3) categories 2) Drinfeld double construction, with dual versions for sl(3) 3) Witten’s topological twists, generalized for sl(3) 4) MOY calculus enhancements, extended to sl(3) settings 5) Colored Jones polynomial extensions, applied to sl(3) algebra 6) Use of matrix factorizations, adapted for sl(3) foams 7) Temperley-Lieb algebra modifications, transferred to sl(3) context

✓ Correct Answer:

The correct answer is 1) Blanchet’s modified foams, with analogous modifications proposed for sl(3) categories.

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Question 25 multiple-choice

Quantum cryptanalysis explores how quantum algorithms and memory models can accelerate attacks on lattice-based cryptosystems, including those built upon the Learning with Errors (LWE) problem. Techniques such as quantum random access classical memory (QRACM), quantum amplitude estimation, and the Quantum Fourier Transform (QFT) are central to these advancements. Which of the following improvements is enabled when quantum-augmented dual lattice attacks on LWE employ quantum random access classical memory (QRACM) with unit cost? 1) Reduction in the modulus size required for secure cryptosystems 2) Elimination of the need for discrete Gaussian sampling 3) Exponential speedup in classical memory usage 4) Replacement of FFT with Grover search for coefficient identification 5) Ability to solve all lattice problems in polynomial time 6) Quantum speedup in searching FFT coefficients above a threshold by leveraging sparsity and amplitude estimation 7) Guarantee of security for all post-quantum cryptographic schemes

✓ Correct Answer:

The correct answer is 6) Quantum speedup in searching FFT coefficients above a threshold by leveraging sparsity and amplitude estimation.

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Question 26 multiple-choice

In quantum topology and mathematical physics, projective and linear representations of the mapping class group play a crucial role in understanding topological quantum field theories (TQFTs) and their associated quantum symmetries. The interplay between group cohomology, roots of unity, and geometric constructions provides subtle distinctions between different representation types. Under what condition does the projective representation of the mapping class group arising from an abelian TQFT associated with a q-deformation of U(1) become linearizable? 1) When q is a primitive even root of unity 2) When the underlying surface is non-orientable 3) When the mapping class group is abelian 4) When the Heisenberg group is replaced by a finite cyclic group 5) When q is an odd root of unity 6) When Lagrangian correspondences are not used in the construction 7) When the Schrödinger representation is infinite-dimensional

✓ Correct Answer:

The correct answer is 5) When q is an odd root of unity.

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Question 27 multiple-choice

In the representation theory of symmetric groups, Young tableaux, tabloids, and Specht modules are fundamental concepts used to construct and understand irreducible representations. The interplay between group actions on tableaux and algebraic structures underlies important results in both mathematics and quantum physics. Which statement correctly characterizes the algebraic behavior of swap relations in irreducible representations associated with two-row Young tableaux, specifically those of the form λ = [n−k, k]? 1) The swap relations vanish under evaluation in these representations, reflecting a simpler symmetry structure. 2) The swap relations generate nontrivial constraints that prevent the irreducibility of Specht modules for two-row tableaux. 3) The swap relations always result in incompatible algebraic structures, regardless of the tableau shape. 4) The swap relations lead to the formation of additional rows in the corresponding Young diagrams. 5) The swap relations have coefficients that depend only on the parity of the number of columns. 6) The swap relations in two-row tableaux produce polytabloids with alternating signs in every entry. 7) The swap relations correspond to the action of the Quantum Monte Carlo Hamiltonian on multipartite irreducible representations.

✓ Correct Answer:

The correct answer is 1) The swap relations vanish under evaluation in these representations, reflecting a simpler symmetry structure..

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tableau of shapeλ⊢nis a filling of the Young di- agramλby integers 1 ,2,...,n such that each box is assigned a unique integer. The action of the groupSnon a Young tableau tfollows by letting π∈Snact on the entries of t. Permut- ing entries within each row of a tableau tgives us an equivalence class of tableaux – a tabloid {t1,t2,...,tk}:={t}. (3.5) Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 22 The group action extends to tabloids as π{t}={π(t1),π(t2),...,π (tk)}. The so-called Specht irreducible module VλofC[Sn] is spanned by polytabloids et=/summationdisplay π∈Ctsgn (π)π{t} (3.6) wheretranges over Young tableau of shape λ, andCtis the set of permutations that permute elements only within the columns of t. For a 3-row irrep given by λ= [λ1,λ2,λ3], letTbe the Young tableau T=1 2··· ··· λ1 λ1+1... ... λ 2 λ2+1... λ 3.(3.7) andeTthe corresponding polytabloid. Consider the transpositions (1 λ1+ 1), (1λ2+ 1), (λ1+ 1λ2+ 1) all of which permute elements within the first column of Tand are thus contained in CT. Note that (1λ1)eT=e(1λ1)T (3.8) and on expanding the right-hand side of Eq. (3.8) we see that Tappears with coefficient sgn ((1λ1+ 1)) =−1. By the exact same reasoning, the coefficients of Tin (1λ2)eTand (λ1λ2)eTalso equal−1. Now, each of (1 λ1+ 1)(λ1+ 1λ2+ 1) = (1λ2+ 1λl+ 1) and (λ1+ 1λ2+ 1)(1λ1+ 1) = (1λ1+ 1λ2+ 1) is an even permutation contained in CT. Repeating the above for (1 λ1+ 1)(λ1+ 1λ2+ 1)eTand (λ1+ 1λ2+ 1)(1λ1+ 1)eT, we see that the coefficient of Tin these must be +1. It follows that [(1λ1+ 1) + (1λ2+ 1) + (λ1+ 1λ2+ 1)−1]eT̸={(1λ1+ 1),(λ1+ 1λ2+ 1)}eT(3.9) in the irrep specified by λ, and more generally in any irrep with 3 or more rows. That is, sijsjk+sjksij−(sij+sjk+sik−1) does not vanish under the evaluation sij=ρλ(i j) for the above irrep λ. Thus we have proved that the swap relations are incompatible with a ≥3 row Young Tableaux. For the converse, consider distinct indices i,j,k and a tabloid Tof shape [n−k,k]. Letπ∈CT. We distinguish two cases: (a)i,j,k lie in the same row of π{T}. Then each of the terms sij,sjk,sikacts as the identity on π{T}, whence /parenleftbigsijsjk+sjksij−(sij+sjk+sik−1)/parenrightbig(π{T}) = 0. (3.10) (b)i,jlie in the same row of π{T}, butkdoes not. Then, sij(π{T}) =π{T}, sijsjk(π{T}) =sik(π{T}), sjksij(π{T}) =sjk(π{T}). As before, this implies Eq. (3.10). This shows that for each π∈CT, Eq. (3.10) holds, whence sijsjk+sjksij−(sij+sjk+ sik−1) vanishes under the evaluation sij=ρλ(i j) for any irrep λ= [n−k,k]. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 23 Theorem 3.6 ([Pro07 , Theorem, §6.1]).The algebras Aswap n andMswap n are isomorphic. More precisely, the natural map Aswap n→Mswap n, sij∝⇕⊣√∫⊔≀→Swapij is an isomorphism. Proof. An immediate corollary of Proposition 3.5. The isomorphism between the Symbolic Swap Algebra and Swap Matrix Algebra given above lets us identify the QMC Hamiltonian associated with a graph Gwith an element of the Symbolic Swap Algebra. We make this clear in the following definition. Definition 3.7. For any graph gdefine the symbolic QMC Hamiltonian

Question 28 multiple-choice

In the study of abelian p-groups, the structure of automorphism groups and characteristic subgroups are crucial for understanding group extensions and classifications. Properties such as divisibility, direct decompositions, and the behavior of homocyclic components impact the nature of possible group extensions. Which statement correctly describes the necessary and sufficient condition for the automorphism group Coo to equal Aut A in an abelian p-group A with divisible part D and reduced part R? 1) Coo = Aut A if and only if Cpoo is contained in Aut D or in Aut R. 2) Coo = Aut A only if Cpoo is a non-central subgroup of A. 3) Coo = Aut A if and only if every extension of A by C^oo is non-nilpotent. 4) Coo = Aut A provided that D is reduced and R is divisible. 5) Coo = Aut A if and only if Cpoo equals Aut(A/C) for all characteristic subgroups C. 6) Coo = Aut A if and only if every homomorphism from Coo to Aut A is surjective. 7) Coo = Aut A only if A has no homocyclic components.

✓ Correct Answer:

The correct answer is 1) Coo = Aut A if and only if Cpoo is contained in Aut D or in Aut R..

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if Coo = Aut A, the corresponding split extension would be a nilpotent /?-group containing a non-central Cpoo, which is impossible. Conversely, if (ii) holds then every homomorphism from Coo to Aut A is trivial so that every extension G of A by C^oo is a central extension and thus nilpotent. In fact it is abelian since G has locally cyclic central factor-group. The equivalence of (iii) and (iv) is immediate since D and every homocyclic factor of L are direct factors of A. That (ii) implies (iii) is a direct consequence of (2.8). The hard part of this theorem is the proof that (iii) implies (ii). We need some lemmas to enable us to do it. LEMMA 3.4.1. Let C be a characteristic subgroup of an abelian group A and suppose that AI C is a p-group. //Coo = Aut A, then either Cpoo ^ Aut C or Cpoo ^ Aut(v4/C). https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press 1030 J. BUCKLEY AND J. WIEGOLD Proof. Every automorphism of A induces one on C and one on A/C. Furthermore, the obvious map 0:Aut A -> Aut C X Aut(A/C) has kernel the stability group of the chain 1 â C S A, which is isomorphic to Hom(A/C, C). Since A/C is a /?-group it is easily shown that Hom(v4/C, C) has no elements of infinites-height and so ker 0 contains no Coo. Thus Cp0o â im 6 ^ Aut C X Aut(,4/C). But any Coo in the direct product Aut A X Aut(,4/C) cannot be in the kernels of both projections, so Cpoo g Aut A or C^oo ê A\xt(A/C). COROLLARY. Let A be an abelian p-group with divisible part Z), so that A = D X R where R is reduced. Then Cœ = Aut A if and only if Cpoo â Aut D or Cpoo g Aut R. LEMMA 3.4.2. Let A be an abelian p-group, L a basic subgroup of A, and let Ln be the homocyclic component ofL of exponent p n. Then for each «èl, Z>|Z>2 • • • LnÂP is characteristic in A. Proof. Recall that A = L\L 2 . . . L„A*, where A* = AP"L* and L* = (ULjm > n). We shall show that (L XL2 . . . L n)a is contained in LXL2 . . . LnAp for each n ^ 1 and each a e Aut A. For each JC G LXL2 . . . L tv x = x xyxay , where xx G LjL2 . . . L, 7, _yj G L* and a & A. Since x /;" = x px = 1 it follows that (y]apY = I- So ><' = a~p2n and jf G L*. But L* is a direct factor of the pure subgroup L and so it is itself pure in A. Thus yP\ = yPi> where _y2 G L*. https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press NILPOTENT EXTENSIONS 103 1 Since (y^2P"f = i, it follows that y\yïP is

Question 29 multiple-choice

In the study of moduli spaces arising from group actions on vector spaces, the non-emptiness and dimension of quotient spaces can be determined using recursive operations and stability functions. The structure and properties of these spaces often depend on integer parameters and explicit combinatorial loci. When analyzing the non-emptiness of quotient spaces parameterized by triples (A, b, c) with A ≥ 3, which of the following best describes the set SA of pairs (b, c) that must be checked for non-emptiness? 1) SA consists of all pairs (b, c) with b and c both greater than A 2) SA is infinite and contains all pairs (b, c) where bc > A 3) SA is the set of pairs (b, c) with b = c and both less than or equal to A 4) SA is a finite region near the curve bc = A, symmetric about b = c and contained in b < A, c < A 5) SA is the set of pairs (b, c) with b and c both equal to 1 6) SA contains all pairs (b, c) such that b divides c exactly 7) SA is the collection of all pairs (b, c) with b or c equal to zero

✓ Correct Answer:

The correct answer is 4) SA is a finite region near the curve bc = A, symmetric about b = c and contained in b < A, c < A.

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and that the quotient corresponding to (b,c,A)is nonempty. Proof.By part (3) of Proposition 5.3, ∆ <0 implies ∆ ≤ −2; this is also implied by d3≥d∗(d1,d2) from the deﬁnition of d∗. Thus, suppose that the quotient is non-empty for dimensions ( A,d2,d3) with ∆(A,d2,d3)≤ −2. Consider applying the operation γ: (A,B,C)→(A,AB−C,B) (5.14) repeatedly without reordering todeﬁneasequence (( A,B0,C0)≡(A,d2,d3),(A,B1,C1),...). Since we must have C0≥A0B0/2 for ∆<0, the initial step does not increase the sum of the elements. Further, all elements remain positive unless we end up on a triple for which Ci=ABi. Thus, repeating the operation γmust either bring us to a triple ( A,b,Ab) or to a triple (A,b,c) for which the operation γdoes not decrease the sum of the elements. In the latter case, we can show that bandcsatisfy b≤A 2c c≤A 2b bc≥A Abc −A2−b2−c2+4≤0. (5.15) The ﬁrst pair of inequalities for ccome from demanding that γapplied to ( A,b,c) does not decrease the sum and that ( A,b,c) came by acting with γon a triple with a sum that was 37 not smaller. The third inequality follows since we are assuming the quot ient is not empty. The fourth inequality is the statement that ∆ ≤ −2. If (A,B,C) descends via γto (A,b,c), the quotient corresponding to this triple must be non- empty, sothepair( b,c)isintheset SAdeﬁnedbytheproposition. Thus, anytriple( A,d2,d3) with ∆≤ −2 descends via the recursion step to ( A,k,Ak) or (A,b,c) with (b,c)∈SA. To determine the form of such triples ( A,d2,d3) explicitly, note that the inverse of the operation γis the operation Bi+1=CiCi+1=ACi−Bi (5.16) Combining these we ﬁnd that BiandCimust be successive terms in the sequence fi+1=Afi−fi−1, (5.17) where the allowed starting values are ( f0,f1) = (k,Ak) or (f0,f1)∈SA. Remark 5.5. The general terms in the sequence deﬁned by (5.12) can be given ex plicitly via a generating function as fi=f0+(f1−Af0)x 1−Ax+x2|xn. (5.18) ForA≥3, the region deﬁned by (5.13) covers a narrow band of the plane ne ar the curve bc=A, symmetric about the line b=cand contained in the region b < A,c < A. Thus, while the deﬁnition of SAstill involves the recursion from Theorem 5.1, we need only check a ﬁnite number of points (on the order of A) in the region (5.13) to determine the set, after which we can use Proposition 5.4 to give an explicit formula for all ∆ <0 triples (A,d2,d2) with a nonempty quotient. As examples, we have S3= (3,2),(2,2),(2,3) S4= (4,2),(3,2),(2,3),(2,4) S5= (5,2),(4,2),(2,4),(2,5) (5.19) We consider the special case A= 2 presently. Example 5.6. For dimensions (2 ,d2,d3) the quotient P(H)//Gis non-empty if and only if (d1,d2,d3) = (2,b,b) forb≥2 or (2,kb,(k+1)b) for positive integers k,bwithkb>1. The quotient is a single point except for ( d1,d2,d3) = (2,b,b) withb >3, in which case it has dimensionb−3. Proof.For this case, the range dn−1≤dn< d∗is empty, so the only non-empty quotients are those covered by Proposition 5.4. For A= 2, the deﬁnition (5.12) of the sequence in Proposition 5.4 may be written as fi+1−fi=fi−fi−1so the sequence is arithmetic.

Question 30 multiple-choice

In computational group theory, the complexity of decision problems often depends on the structural properties of the groups involved. Extensions of classical problems such as the Post correspondence problem to group settings reveal deep connections between group structure and algorithmic tractability. Which of the following statements accurately characterizes the computational complexity of the bounded Post correspondence problem (PCP) in non-elementary hyperbolic groups? 1) It is solvable in logarithmic space. 2) It is undecidable in all cases. 3) It is NP-complete. 4) It is polynomial-time solvable for arbitrary input size. 5) It is co-NP-hard but not NP-hard. 6) It is decidable only for abelian groups. 7) It is PSPACE-complete.

✓ Correct Answer:

The correct answer is 3) It is NP-complete..

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Question 31 multiple-choice

Hardware accelerators play a vital role in enabling efficient processing for both digital signal processing and cryptographic applications. Unified architectures that support multiple computational transforms can optimize resources for modern embedded systems. Which architectural feature is essential for a hardware accelerator to efficiently support both Fast Fourier Transform (FFT) for signal processing and Number Theoretic Transform (NTT) for lattice-based cryptography? 1) Implementation of high-speed floating-point multipliers 2) Addition of modular reduction circuitry and control logic modifications 3) Use of analog-to-digital converters for input transformation 4) Dedicated complex domain arithmetic units for NTT operations 5) Inclusion of error correction modules for cryptographic protocols 6) Separate data paths for FFT and NTT computation 7) Exclusive reliance on silicon area minimization techniques

✓ Correct Answer:

The correct answer is 2) Addition of modular reduction circuitry and control logic modifications.

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Question 32 multiple-choice

In the study of group theory and representation theory, the Modular Isomorphism Problem investigates the relationship between the structure of finite p-groups and their group algebras over fields of characteristic p. Understanding which properties of p-groups are determined by their group algebra is central to addressing this problem. Which group-theoretic property of a finite p-group is guaranteed to be determined by its group algebra over the prime field ${{\mathbb {F}}}_p$? 1) The order of every subgroup 2) The nilpotency class of the group 3) The number of elements of each order 4) The exponent of the group 5) The number of normal subgroups 6) The isomorphism type of the maximal abelian direct factor 7) The number of non-abelian simple factors

✓ Correct Answer:

The correct answer is 6) The isomorphism type of the maximal abelian direct factor.

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Question 33 multiple-choice

Quantum algorithms have revolutionized the way certain mathematical problems are solved, particularly in the domain of group-based cryptography. The Hidden Subgroup Problem (HSP) plays a pivotal role in determining the security of cryptosystems against quantum attacks. Which statement most accurately reflects the current understanding of efficient quantum algorithms for solving the Hidden Subgroup Problem (HSP) in various group structures? 1) Efficient quantum algorithms exist for HSP in all infinite groups commonly used in cryptography. 2) Efficient quantum algorithms for HSP are known for non-abelian groups but not for abelian groups. 3) Efficient quantum algorithms for HSP exist for finite abelian groups, but not for infinite or non-abelian groups. 4) Efficient quantum algorithms for HSP exist for braid groups and other infinite groups. 5) Efficient quantum algorithms for HSP are universally applicable to all group-based cryptosystems. 6) Efficient quantum algorithms for HSP are limited to finite non-abelian groups used in cryptography. 7) Efficient quantum algorithms for HSP do not exist for any group structure considered in cryptography.

✓ Correct Answer:

The correct answer is 3) Efficient quantum algorithms for HSP exist for finite abelian groups, but not for infinite or non-abelian groups..

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Question 34 multiple-choice

In number theory, the multiplicative group modulo n, denoted (Z/nZ)×, encodes the structure of invertible elements under multiplication modulo n. The classification of its Sylow p-subgroups and the properties of its subgroups have important implications in algebra, cryptography, and analytic number theory. Which of the following statements best describes a "maximally non-cyclic" multiplicative group (Z/nZ)×? 1) Every Sylow p-subgroup is cyclic for all p dividing n. 2) (Z/nZ)× is isomorphic to a cyclic group for all n. 3) All its subgroups are trivial. 4) All prime-power subgroups are elementary abelian groups, i.e., direct products of cyclic groups of order p. 5) Its order is always a prime number. 6) It contains no elements of order greater than 2. 7) It is isomorphic to the additive group of integers modulo n.

✓ Correct Answer:

The correct answer is 4) All prime-power subgroups are elementary abelian groups, i.e., direct products of cyclic groups of order p..

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Question 35 multiple-choice

Quantum algorithms for hidden shift problems often exploit group structures to achieve improved efficiency, particularly for groups relevant to cryptography and computational number theory. Efficient space and time complexity are crucial for practical implementation on quantum hardware. For the hidden shift problem over groups of the form $\mathbb{Z}_{2^t}^n$, which statement accurately describes the space complexity of a quantum algorithm that achieves polynomial running time in the dimension $n$? 1) Both classical and quantum space are exponential in $n$ 2) Classical space is quadratic in $n$, while quantum space is linear in $n \log(k)$ 3) Classical space is linear in $n$ and quantum space is quadratic in $n$ 4) Both classical and quantum space are linear in $n$ 5) Classical space is constant and quantum space is exponential in $n$ 6) Classical space is cubic in $n$ and quantum space is linear in $n$ 7) Classical space is linear in $n \log(k)$ and quantum space is quadratic in $n$

✓ Correct Answer:

The correct answer is 2) Classical space is quadratic in $n$, while quantum space is linear in $n \log(k)$.

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Question 36 multiple-choice

In computational quantum chemistry, efficient electronic structure calculations often rely on representing Kohn-Sham orbitals using localized basis sets on real-space grids. Methods for constructing such bases leverage mathematical properties of the density matrix and linear algebra techniques to enhance computational scalability. Which approach directly constructs localized orbitals by selecting linearly independent, localized columns from the density matrix using QR factorization with column pivoting, particularly optimizing for insulating systems? 1) Maximally Localized Wannier Functions (MLWFs) 2) Generalized Eigenvalue Decomposition (GED) 3) Density Functional Perturbation Theory (DFPT) 4) Projector-Augmented Wave (PAW) Method 5) Selected Columns of the Density Matrix (SCDM) 6) Fast Fourier Transform (FFT) Basis Construction 7) Linearized Augmented Plane Wave (LAPW) Method

✓ Correct Answer:

The correct answer is 5) Selected Columns of the Density Matrix (SCDM).

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^ iis often localized around an atom or a chemical bond. Hence working with ^ i's can reduce both the storage and the computational cost. Assume we have access to ^ j(r)'s evaluated at a set of discrete grid points frigN i=1. Letf!igN i=1be a set of positive integration weights associated with the grid points frigN i=1, then the discrete orthonormality condition is given by (2.3)NX i=1^ j(ri)^ j0(ri)!i=jj0: Let^ j= [^ j(r1);^ j(r2);:::; ^ j(rN)]Tbe a column vector, and ^ = [ ^ 1;:::; ^ ne] be a matrix of size Nne. We call ^ the real space representation of the Kohn-Sham orbitals and de ne diagonal matrix W= diag[!1;:::;! N]. Our method requires the Kohn-Sham orbitals to be be represented on a set of real space grid points. This is the case for a plane-wave basis set, as well as other representations such as nite di erences, nite elements and wavelets. For instance, if the Kohn-Sham orbitals are represented using the plane-wave basis functions, their real space representation can be obtained on a uniform grid eciently with the fast Fourier transform (FFT) technique and in this case !itakes the same constant value for alli. It is in this setting that our method is of particular interest. However, 4 this procedure is also applicable to other basis sets such as Gaussian type orbitals or numerical atomic orbitals when a real space representation of the basis functions is readily available. Therefore, our method is amenable to most electronic structure software packages. We de ne = W1 2^ such that the discrete orthonormality condition in Eq. (2.3) becomes =I, whereIis an identity matrix of size ne. We now seek a compressed basis for the span of , denoted by the set of vectors = [ 1;:::; ne] where each i is a sparse vector with spatially localized support after truncating entries with small magnitudes. In such case, iis called a localized vector. 2.2. Selected columns of the density matrix. As opposed to widely-used procedures such as MLWFs [29], the key di erence in the SCDM procedure is that the localized orbitals iare obtained directly from columns of the density matrix P= . The aforementioned nearsightedness principle states that, for insulating systems, each column of the matrix Pis localized. As a result, selecting any linearly independent subset of neof them will yield a localized basis for the span of :However, pickingnerandom columns of Pmay result in a poorly conditioned basis if, for example, there is too much overlap between the selected columns. Therefore, we would like a means for choosing a well conditioned set of columns, denoted C= fc1;c2;:::;c neg;to use as the localized basis. Intuitively we expect such a basis to select columns to minimize overlaps with each other when possible. This is algorithmically accomplished with a QRCP factorization (see, e.g., [17]). More speci cally, given a matrix Aa QRCP seeks to compute a permutation matrix such that the leading sub-matrices ( A)1;:::;k; :for any applicable kare as

Question 37 multiple-choice

In multipartite quantum information theory, the explicit construction and classification of locally maximally entangled (LME) states play a crucial role in understanding the structure of quantum state spaces. These constructions often rely on symmetry, arithmetic progressions, and tools like Schmidt decomposition and local unitary transformations. For triples of the form (2, b, b) with b ≥ 2 in the classification of LME states, what is the dimension of the quotient space of local unitary orbits corresponding to these states? 1) 0 2) b 3) b+1 4) b-3 5) 2b-1 6) b-2 7) 1

✓ Correct Answer:

The correct answer is 4) b-3.

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The conditions (5.13) imply b=c, so we ﬁnd S2={(b,b)|b≥2}. The proposition then gives that the non-empty triples are (2 ,fi,fi+1) with (f0,f1) = (b,2b) or (f0,f1) = (b,b). Since the sequence is arithmetic, we have explicitly that (2 ,b,b) withb≥2 and (2,kb,(k+ 1)b) withbk>2. Proposition 5.3, implies that the quotient is a point except in the cas e (2,b,b), where it has dimension b−3. This reproduces the results that we obtained by our explicit constr uction in section 2. 38 Acknowledgements We would like to thank Jason Bell, Patrick Hayden, Alex May, Robert Ra ussendorf, David Stephen, and Michael Walter for helpful comments and discussions . This work is supported in part by the Natural Sciences and Engineering Research Council o f Canada and by the Simons Foundation. A Explicitconstructionofall LME statesfor (d1,d2,d3) = (2,B,C) In this appendix, we provide details of the explicit construction of all LME states with (d1,d2,d3) = (2,B,C) with 2≤B≤C. In this case, the necessary conditions (2.6) require that B≤C≤2B . (A.1) Using the Schmidt decomposition, and performing a U(2) rotation on the ﬁrst factor, any state|Ψ/an}b∇acket∇i}htwith our desired properties can be written as |Ψ/an}b∇acket∇i}ht=1√ 2|1/an}b∇acket∇i}ht⊗ψ1 bc|b/an}b∇acket∇i}ht⊗|c/an}b∇acket∇i}ht+1√ 2|2/an}b∇acket∇i}ht⊗ψ2 bc|b/an}b∇acket∇i}ht⊗|c/an}b∇acket∇i}ht (A.2) whereψ1 bc|b/an}b∇acket∇i}ht⊗|c/an}b∇acket∇i}htandψ2 bc|b/an}b∇acket∇i}ht⊗|c/an}b∇acket∇i}htdeﬁne orthonormal states of HB⊗HCand summation over bandcis implied. Making use of the Schmidt decomposition on ψ1 bc|b/an}b∇acket∇i}ht ⊗ |c/an}b∇acket∇i}htand performing U(B) andU(C) rotations on the second and third factors, we can write ψ1 bc=/bracketleftbig D{√pi}0B×(C−B)/bracketrightbig ψ2 bc=/bracketleftbig IB×BJB×(C−B)/bracketrightbig (A.3) whereD{√pi}is the diagonal matrix with elements√pi. Orthogonality of ψ1andψ2requires that tr(ID{√pi}) = 0. (A.4) The condition that ρBis maximally mixed gives 1 2D{pi}+1 2II†+1 2JJ†=1 B1 1B×B (A.5) while the condition that ρCis maximally mixed gives 1 2D{pi}+1 2I†I=1 C1 1B×B (A.6) I†J= 0 (A.7) 1 2J†J=1 C1 1(C−B)×(C−B). (A.8) 39 CaseC=B We begin with the special case C=B. Here, the conditions collapse to II†=I†I=D{2 B−pi} (A.9) together with the normalization condition (A.4). Deﬁning Hermitian ma trices H+=1 2(I+I†)H−=−i 2(I −I†) (A.10) the ﬁrst equality in (A.9) gives [ H+,H−] = 0, so the matrices are simultaneously diagonal- izable. We can write H1=UD1U†H2=UD2U†(A.11) so that I=UD{zi}U†(A.12) whereD{zi}is some general complex diagonal matrix. The latter equality in (A.9) g ives that UD{|zi|2}U†=D{2 B−pi}. (A.13) Without loss of generality, we can assume that the eigenvalues piare ordered from largest to smallest and the eigenvalues in D{|zi|2}are ordered from smallest to largest. Then we must have zi=eiφi/radicalbigg 2 B−pi (A.14) andUmust commute with D{pi}. It istherefore block-diagonal, with blocks corresponding to blocksofequaleigenvaluesin D{pi}. However, recallingthatthelocalunitarytransformations on the two subsystems of size Bact as D{pi}→W†D{pi}V I →W†IV (A.15) we see that whenever such blocks exist, we can take local unitary t ransformations with W=V=U, to eliminate U, leaving I=D{eiφi√ 2 B−pi}(A.16) Wheneverpi= 0, we have residual local unitary transformations that can be us ed to set φi= 0 also. We can also set φi= 0 whenpi=2 B. Finally, the condition (A.4) gives that /summationdisplay ieiφi/radicalbigg pi(2 B−pi) = 0 (A.17) so we must choose the phases so that the complex numbers in the su m add to

Question 38 multiple-choice

Quantum computing algorithms leverage group theory to solve certain problems exponentially faster than classical algorithms, with the Hidden Subgroup Problem (HSP) occupying a central role in algorithmic breakthroughs and ongoing research. The distinction between abelian and nonabelian group structures is crucial for understanding the complexity and applicability of quantum solutions. Which problem, formulated as a nonabelian Hidden Subgroup Problem, has significant implications for complexity theory and cryptography due to the current lack of efficient quantum algorithms? 1) Integer factorization 2) Discrete logarithm computation 3) Pell’s equation solution 4) Class group computation 5) Graph isomorphism 6) Hidden translation in abelian groups 7) Stabilizer problem

✓ Correct Answer:

The correct answer is 5) Graph isomorphism.

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Sorbonne Paris-Cit´ e, Paris, France 75205 (frederic. magniez@univ-paris-diderot.fr). ¶C N R S ,L I A F A ,U n i v e r s i t ´ e Paris Diderot, Sorbonne Paris-Cit´ e, Paris, France 75205, and Cen- tre for Quantum Technologies, National University of Singapore, Singapore (miklos.santha@liafa.univ-paris-diderot.fr). /bardblSchool of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India 400005 (pgdsen@tcs.tifr.res.in). 1 2 FRIEDL, IVANYOS, MAGNIEZ, SANTHA, AND SEN While no classical algorithm can solve this problem with polynomial query com- plexityevenif Gisabelian, thebiggestsuccessofqua ntumcomputinguntilnowisthat it can be solved by a quantum algorithmeﬃciently for any abelian G. We will refer to this quantum algorithm as the standard algorithmfor Hidden Subgroup .T h em a i n tool for this solutionis Fourier sampling based on the (approximate)quantum Fourier transform for abelian groups which can be eﬃciently implemented quantumly [29]. Simon’s XOR-mask ﬁnding [42], Shor’s fac torization and discrete logarithm ﬁnding algorithms [41], and Kitaev’s algorithm [29] for the abelian stabilizer problem are all special cases of this general solution. Quantum algorithms of Hallgren [20, 21] andSchmidt and Vollmer [40] computing class groups and unit groups of number ﬁelds, including the solution of Pell’s equation, also follow along these lines. Finding an eﬃcient algorithm for Hidden Subgroup for nonabelian groups G is considered to be one of the most important challenges at present in quantum com- puting. Besides its intrinsic mathematical interest, the importance of this problem is enhanced by the fact that it contains as a special case the graph isomorphism problem. Unfortunately, although its query complexity is shown to be polynomial by Ettinger, Høyer, and Knill [14], nonabelian Hidden Subgroup seems to be much more diﬃcult than the abelian case. Although considerable eﬀort was spent on it in the last few years, only a small number of successes can be reported. They can be di- vided into twocategories. The standardabelianFourier samplingbasedalgorithmhasbeen extended to some nonabelian groups in [39, 22, 19, 16, 33, 12] using the quantum Fouriertransformoverthese (nonabelian) groups. Although eﬃcient quantum Fourier transform implementations are known for several nonabelian groups [8, 23, 37, 32], the power of the technique appears to be very limited. In a diﬀerent approach, Hid- den Subgroup was eﬃciently solved in the context of speciﬁc nonabelian black-box groups [5, 45] by [26] without using the Fourier transform on the group, and instead using Fourier transforms over abelian groups only. Similarly, only abelian Fourier transforms were used by [24, 6, 10, 27, 28] to solve the hidden subgroup problem insome speciﬁc kinds of nonabelian groups. See [11] for a more detailed review of hidden subgroup algorithms and related problems. In light of the apparent hardness of Hidden Subgroup in nonabelian groups, a natural line of research is to address subproblems of Hidden Subgroup which, in some groups, capture the main diﬃculty of the original problem. In a pioneeringpaper, Ettinger and Høyer [13], in the case of dihedral groups, implicitly considered another paradigmatic group problem, Hidden Translation .H e r ew ea r eg i

Question 39 multiple-choice

Quantum algorithms for lattice problems often rely on Fourier analysis over lattices and their quotient spaces, revealing structural information in the frequency domain. The interpretation of quantum measurement outcomes depends critically on probability distributions derived from normalized quantum states. In the quantum Fourier transform of a function defined over a lattice, what ensures that the weights assigned to dual lattice points correspond to valid probability values in quantum measurement? 1) The function is normalized to unit norm so that the squared magnitudes of Fourier coefficients are nonnegative and sum to one. 2) The dual lattice is chosen to be orthogonal to the original lattice basis vectors. 3) The Fourier transform is computed only over even-dimensional lattices. 4) The quantum state is post-selected on the origin of the dual lattice. 5) The normalization is performed after measurement to ensure proper probabilities. 6) The original function is required to be real-valued for valid probabilistic interpretation. 7) The computation uses only integer-valued lattice points.

✓ Correct Answer:

The correct answer is 1) The function is normalized to unit norm so that the squared magnitudes of Fourier coefficients are nonnegative and sum to one..

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Fourier transform of ψδ andψ, respectively: pδ(y) =˙bψδ(y)˛˛bψδ(y)¸ withbψδ=FδZmψ; p(y) =˙bψ(y)˛˛bψ(y)¸ withbψ=FRnψ. (6.8) It can be shown that pδis close top. Let us now focus on the distribution p(y) =˙bψ(y)˛˛bψ(y)¸ . We have ψ=wf,bψ=bw∗(FRmf), wherebw=FRmw. Sincefis a hidden subgroup oracle, we may regard it as a function on Rm/Land deﬁnebf=FRm/Lf. That is, bfu=Z Rm/Le2πi/angbracketleftx,u/angbracketrightf(x)dx foru∈L∗, f(x) =1 d(L)X u∈L∗e−2πi/angbracketleftx,u/angbracketrightbfu. (6.9) 301 The Fourier transform over Rmis (FRmf)(y) =Z Rme2πi/angbracketleftx,y/angbracketright 1 d(L)X u∈L∗e−2πi/angbracketleftx,u/angbracketrightbfu! dx =1 d(L)X u∈L∗bfuδ(y−u). It follows that bψ(y) =` bw∗(FRmf)´ (y) =1 d(L)X u∈L∗bfubw(y−u),(6.10) p(y) =1 d(L)2X u,u/prime∈L∗˙bfu/prime˛˛bfu¸ bw(y−u)bw(y−u/prime).(6.11) The last equation is complicated, but to have a good ap- proximation it is enough to keep the terms with u=u/prime. Let us consider the quantum state FRm|ψ/angbracketrightwhose wavefunction is given by Eq. (6.10). It consists of identically shaped peaks at the points u∈L∗. Each peak has a weight qu=˙bfu˛˛bfu¸ d(L)2=1 d(L)2Z e2πi/angbracketleftx−x/prime,u/angbracketright˙ f(x/prime)˛˛f(x)¸ dxdx/prime, where the integral is over (Rm/L)2. The numbers qucan be interpreted as probabilities because they are nonnegative and add up to 1. Indeed, let us normalize fto make a function of unit norm, g(x) =d(L)−1/2f(x). Then X u∈L∗qu=1 d(L)X u∈L∗˙ bgu˛˛bgu¸ =˙ bg˛˛bg¸ =/angbracketleftg|g/angbracketright= 1. 7. REFERENCES [Ban93] W. Banaszczyk. New bounds in some transference theorems in the geometry of numbers. Mathematische Annalen, 296(1):625–635, 1993. [BK93] Johannes Buchmann and Volker Kessler. Computing a reduced lattice basis from a generating system, 1993. Preprint, August 4, 1993. [BP87] Johannes Buchmann and Michael Pohst. Computing a lattice basis from a system of generating vectors. In Eurocal’87 , volume 378 ofLNCS, pages 54–63. Springer-Verlag, June 1987. [BV11] Zvika Brakerski and Vinod Vaikuntanathan. Fully homomorphic encryption from ring-LWE and security for key dependent messages. In Advances in cryptology—CRYPTO 2011 , volume 6841 of LNCS, pages 505–524. Springer, 2011. [Coh93] Henri Cohen. A course in computational algebraic number theory. Springer-Verlag New York, Inc., New York, NY, USA, 1993. [FIM+03] Katalin Friedl, Gabor Ivanyos, Frederic Magniez, Miklos Santha, and Pranab Sen. Hidden translation and orbit coset in quantum computing. In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, San Diego, CA, 9–11June 2003. [GH11] C. Gentry and S. Halevi. Implementing gentry’s fully-homomorphic encryption scheme. Eurocrypt 2011 , pages 132–150, 2011.[GKP01] Daniel Gottesman, Alexei Kitaev, and John Preskill. Encoding a qubit in an oscillator. Phys. Rev. A , 64:012310, Jun 2001. [GVL96] Gene H. Golub and Charles F. Van Loan. Matrix Computations . Johns Hopkins University Press, Baltimore, MD, 3rd edition, 1996. [Hal05] Sean Hallgren. Fast quantum algorithms for computing the unit group and class group of a number ﬁeld. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pages 468–474, 2005. [Hal07] Sean Hallgren. Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem. Journal of the ACM, 54(1):1–19, 2007. [HMR+10] Sean Hallgren, Cristopher Moore, Martin R¨otteler, Alexander Russell, and Pranab Sen. Limitations of quantum coset states for graph isomorphism. J. ACM, 57:34:1–34:33, November 2010. [Kit95] Alexei Kitaev. Quantum measurements and the abelian stabilizer problem, 1995. quant-ph/9511026. [KW08] Alexei Kitaev and William A. Webb.

Question 40 multiple-choice

In the representation theory of complex reductive algebraic groups such as GL(n), highest weight theory provides a systematic approach to classifying irreducible representations. The moment map and related techniques play a foundational role in geometric invariant theory and have important computational and theoretical applications. Which of the following statements best characterizes the role of highest weight vectors in the decomposition of representations of GL(n) and their connection to irreducibility? 1) Highest weight vectors are invariant under all diagonal matrices but not under upper-triangular matrices. 2) Highest weight vectors correspond to the lowest eigenvalues of the torus action in any representation. 3) Highest weight vectors are orthogonal to all other weight vectors in the weight space decomposition. 4) Highest weight vectors exist only for representations of abelian groups, not reductive groups. 5) Highest weight vectors determine the trace of representation matrices but not their irreducibility. 6) Highest weight vectors are not preserved under dualization of representations. 7) Highest weight vectors are eigenvectors under the action of the Borel subgroup, and their weights uniquely characterize irreducible representations.

✓ Correct Answer:

The correct answer is 7) Highest weight vectors are eigenvectors under the action of the Borel subgroup, and their weights uniquely characterize irreducible representations..

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supported by NSF grant CCF-1412958. 16 2 Geometric invariant theory In this section, we present some results from geometric invariant theory that will feature centrally in the analysis of our algorithm in Section 3. While stated for tensors, all results in this section can easilybeextendedtoarbitraryrationalrepresentationsofconnectedcomplexreductivealgebraic groups. Most of the results are well known and only some are new. All previously known results will be citedwith references and we will make suretohighlight the new components. Section 2.1 discussesbasicsofthehighestweighttheory. Section2.2givesaformaldeﬁnitionofthemoment mapandalsodiscussesthesocalled“shiftingtrick"thatreducestheproblemofmembershipin moment polytopes to a null cone problem. Section 2.3 considers degree bounds for highest weight vectorswhichareusedtoboundtheinitialrandomnessusedinAlgorithm1. Section2.4recallsa classical construction of highest weight vectors and uses this construction to prove bounds on their evaluations (crucial in the analysis of Algorithm 1). Section 2.5 develops a necessary and suﬃcient condition for Borel scalability (i.e., scaling using tuples of upper-triangular matrices). Asbefore,let GGLpn1q GLpndq,KUpn1qUpndq,VTenpn0;n1;:::;ndq Cn0bCn1b:::bCnd, and XPpVqaG-stable irreducible projective subvariety (e.g., an orbit closure). 2.1 Highest weight theory We ﬁrst recall the representation theory of GLpnq(see, e.g., [ FH13] for an introduction). Let W be a ﬁnite-dimensional GLpnq-representation, equipped with a Upnq-invariant inner product. Let TpnqGLpnqdenotethesubgroupconsistingofinvertiblediagonalmatrices,calledthe maximal torusofGLpnq. SinceTpnqis commutative, its action can be jointly diagonalized. Thus, any ﬁnite-dimensional GLpnq-representation Wcanbewrittenasadirectsumofso-called weightspaces , WÀ !Wp!q, whereTpnqacts on any vector wPWp!qasTw !pTqwfor allTPTpnq. Here,!is an integer vector and !pTq ±n j1T!j j;j. We write pWqfor the set of all weights that occur in W. Now letBpnqGLpnqdenote the Borel subgroup of invertible upper-triangular matrices, which contains Tpnq. Ahighest weight vector is a vectorwPWthat is an eigenvector of the Bpnq-action. Letdenote its weight, which is now called highest weight . Necessarily, 1¥¥n, i.e.,is ordered non-increasingly, and we have that Rw pRqwfor allRPBpnq, where pRq ±n j1Rj j;j. We denote by HWVpWqthe space of highest weight vectors in Wwith highest weight . Theirreducible representations of GLpnqcontain a unique (up to scalar multiple) highest weightvector andare characterized byits highestweight. Wewrite Vfor theirreducible representation (which we always equip with a K-invariant inner product, denoted x;y) andv forahighestweightvector(whichwechoosetobeofunitnorm). Thus, HWVpVqCvif, andzerootherwise. Itisknownthat Bt0xv;exppAtqvytrrAdiagpqsforallnn-matricesA. It can also be veriﬁed that GLpnqrvsUpnqrvs(in particular, this G-orbit is closed). The dual ofanirreduciblerepresentationisalsoirreduciblewithhighestweight Ò,sothatV V. We now consider the group GGLpn1q GLpndq. All the preceding notions generalize immediately by considering tuples or tensor products of the relevant objects, and we shall use similar notation. Thus, the maximal torus isTTpn1qTpndq, theBorel subgroup is 17 BBpn1qBpndq.Highest weight vectors satisfy Rw pRqw;where pRqd¹ i1ni¹ j1pRpiq j;jqpiq j (7) for all tuples R pRp1q;:::;Rpdqq PB, andweight vectors satisfy the same relation restricted toTB.Weightsandhighest weight are now tuples pp1q;:::;pdqqof integer vectors as before. Thesums°ni j1piq jarenecessarilyequalfor i1;:::;d,andwewilldenotethemby jj. Thus,{jjPPpn1;:::;ndq. We denote by HWVpWqthe space of highest weight vectors in aG-representation W. The irreducible representations of Gare again labeled by their highest weight and denoted by V. Indeed, they are simply given by tensor products of the corresponding GLpniq-representations,i.e., VVp1qb:::bVpdq;thesameholdsfortheirhighestweightvectors. For every tuple of matrices ApAp1q;:::;Apdqq(Apiqisnini), we have that Bt0xv;exppAtqvyn¸ i1trrApiqdiagppiqqs; (8) where exppAtq:exppAp1qtqb:::bexppApdqtq. As before, wewrite ppp1qq;:::;pp1qqq, so thatV V. 2.2 Moment map and shifting trick LetWbe aG-representation. The associated

Question 41 multiple-choice

Molecular simulation force fields are essential tools in computational chemistry for predicting physical properties such as bulk density and vaporization enthalpy. The accuracy of these models depends significantly on which types of atomic interactions they include. Which of the following best explains why omitting three-body interaction terms in a force field leads to systematic overestimation of bulk density and underestimation of vaporization enthalpy in organic molecules? 1) It causes the modeled molecules to adopt higher-energy conformations, reducing intermolecular attractions. 2) It leads to excessive hydrogen bonding, inflating both density and enthalpy values. 3) It neglects long-range electrostatic interactions, making the system less cohesive. 4) It fails to account for collective atomic interactions that moderate total system energy, resulting in denser packing and reduced energy required for vaporization. 5) It incorrectly assigns partial charges, creating artificial dipole moments. 6) It omits van der Waals forces entirely, causing molecules to repel each other. 7) It introduces quantum tunneling effects that disrupt normal phase behavior.

✓ Correct Answer:

The correct answer is 4) It fails to account for collective atomic interactions that moderate total system energy, resulting in denser packing and reduced energy required for vaporization..

📚 Reference Text:

the a priori origin of the former and the fact the the experimental bulk de nsity is one of the target properties on which transferable FFs are tu ned. Molecule ρ(kg/m3) GAFF17OPLS17CGenFF59QMD-FF Exp. C1 1375 1373 - 1420 147980- 148083 C2 1258 1200 1241 1328 131680- 132779 C3 1054 1176 1217 1261 121379 C4(273 K) 898 915 912 969 92480 N1 -1410 - 1439 144079 N2 -1265 - 1378 136979 B1 1326 1740 - 1813 166283- 167579 B2 1946 2474 2402 2570 246979- 248480 B3 1807 2335 2170 2311 216983- 217979 B4 1229 1408 1398 1446 134580- 135479 I1 -2042 - 2107 193679 I2 1599 1835 - 1818 169583- 173780 St. Dev. (kg/m3)254 94 47 98 - Table 7: Bulk density ( ρ, kg/m3): comparison between experiment and data computed from MD simulations performed with popular FFs and the QMD-FF. Unle ss otherwise stated, all values refer to T = 298 K. The other property usually employed in empirical FF tuning i s the vaporization en- thalpy. In Table 8 the ∆ Hvapcomputed and experimental values are compared for each compound. By looking at the standard deviations, reported a s usual in the last row, it 27 Molecule ∆Hvap(kJ/mol) GAFF17OPLS17CGenFF59QMD-FF Exp. C1 28.4 29.2 - 30.4 31.384 C2 26.5 23.4 25.3 31.028.879-29.985 C3 24.9 24.3 29.9 27.7 26.579 C4(273 K) 22.9 26.9 24.0 28.9 27.886 N1 -35.8 - 42.3 41.770 N2 -35.5 - 34.5 33.070 B1 18.5 25.2 - 25.5 22.879 B2 28.6 34.2 43.1 40.3 37.484 B3 34.7 48.8 40.1 51.3 41.784 B4 31.4 34.8 35.3 39.4 32.279 I1 -29.0 - 39.8 32.079 I2 33.7 33.8 - 40.6 34.179 St. Dev. (kJ/mol) 4.6 3.7 3.7 4.8 - Table 8: Vaporization enthalpy (∆ Hvap, kJ/mol): comparison between experiment and data computed from MD simulations performed with popular FFs and the QMD-F F. Unless otherwise stated, all values refer to T = 298 K. appears as the QMD-FF error is slightly larger than the one co mputed with GAFF and ∼1 kJ/mol worse than OPLS and CGenFF. Consistently with ρ, where most values were overestimated with respect of the experiment, there is a gen eral overestimation of the total energy of the system given by the QMD-FF. In the previou s parameterizations per- formed on pyridine32and benzene40molecules, this issue was traced back to the absence of three body eﬀects, given the pure two-body nature of the QM D-FF. As a matter of fact, the eﬀect of the inclusion of three-body interactions ,intheFFshasbeenrecently reportedtoin the FF, which has been recently44quantiﬁed to account for 15-20% of the total interaction energy of water bulk phase (hence causing not negligible inaccuracies), was found to contribute only for a 2-5%, when organic molecul es, more similar to the ones here considered, are taken into account.36In turn, consistently with the present results, tha absence of three-body eﬀects was reported36to cause a ∼5% overestimation of the bulk density and to lower the ∆ Hvapby 6 to 12%. Here, itthese inacurracies could be also connected to a only partial

Question 42 multiple-choice

Quantum computing architectures must efficiently implement universal gate sets while addressing error correction and hardware compatibility. Parity encoding offers unique advantages for logical operations and physical qubit connectivity. Which characteristic of parity encoding directly enables virtual all-to-all connectivity among logical qubits in quantum computing architectures? 1) Encoding logical qubit information in the parity of multiple physical qubits 2) Restricting interactions to nearest-neighbor qubits only 3) Utilizing only non-diagonal multi-qubit operators 4) Requiring enforcement of parity constraints at every computational step 5) Implementing logical gates exclusively with multi-qubit entangling operations 6) Limiting error correction resources to single-qubit bit-flip detection 7) Designing the architecture for quantum annealing without adaptation for universal computation

✓ Correct Answer:

The correct answer is 1) Encoding logical qubit information in the parity of multiple physical qubits.

📚 Reference Text:

Universal Parity Quantum Computing Michael Fellner ,1,2,*Anette Messinger ,2Kilian Ender ,1,2and Wolfgang Lechner1,2,† 1Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria 2Parity Quantum Computing GmbH, A-6020 Innsbruck, Austria (Received 19 May 2022; accepted 16 September 2022; published 27 October 2022) We propose a universal gate set for quantum computing with all-to-all connectivity and intrinsic robustness to bit-flip errors based on parity encoding. We show that logical controlled phase gate and Rz rotations can be implemented in parity encoding with single-qubit operations. Together with logical Rx rotations, implemented via nearest-neighbor controlled-NOT gates and an Rxrotation, these form a universal gate set. As the controlled phase gate requires only single-qubit rotations, the proposed schemehas advantages for several cornerstone quantum algorithms, e.g., the quantum Fourier transform. We present a method to switch between different encoding variants via partial on-the-fly encoding and decoding. DOI: 10.1103/PhysRevLett.129.180503 Designing quantum computers [1–17] and quantum algorithms [18–26]is a current grand challenge in science and engineering, motivated by the prospect of solving certain problems exponentially faster than any known classical algorithms [21]. However, the fundamental rules of quantum mechanics that make this new paradigm possible also impose fundamental restrictions. In contrast to classical information, quantum information cannot becopied, which is known as the no-cloning theorem, but only propagated [27]. Thus, quantum computers will not be able to follow the von Neumann architecture [28]with separated memory and computational unit. As the quantum CPUserves as memory and computational unit at the same time, connectivity between any quantum bits on the chip is required. In current standard approaches to gate-basedquantum computers, either these long-range interactions are implemented as physical interactions, which limits scalability, or quantum information is moved on the chipvia SWAP sequences, which requires a large overhead in gates. Although there are recent approaches toward qubit routing that address this issue [29,30] , exchanging infor- mation between qubits remains a challenging problem. In this Letter, we propose a novel universal quantum computing approach based on the Lechner-Hauke-Zoller (LHZ) architecture [31], which was originally designed for quantum annealing. In this parity-based paradigm, eachphysical qubit represents the parity of multiple logicalqubits. We extend the LHZ architecture, up to now onlyused for solving combinatorial optimization problems, to a universal quantum computing approach by providing auniversal gate set on parity-encoded states and with that, open up new possibilities for universal quantum compu- tation. These extensions include an additional row of data qubits added to the original LHZ layout to enable control of single logical qubits. We introduce logical operations, inparticular R xrotations, to establish a universal gate set in the logical space. As the parity constraints no longer needto be enforced throughout the computation, they can beutilized for error correction. As it only requires nearest-neighbor interactions between qubits on a square lattice chip, our proposal can beimplemented on state-of-the-art quantum devices, indepen-dent of the qubit platform. Suitable platforms are forexample superconducting qubits [15,32 –36], neutral atoms [37–39], or trapped ions [40–43]. We show that the parity transformation renders diagonal multiqubit operatorsbetween arbitrary logical

Question 43 multiple-choice

Shifted quantum affine algebras and their representations are central objects in modern mathematical physics, with deep connections to quantum integrable systems, cluster algebras, and supersymmetric gauge theories. These structures often involve studying categories of modules and their interplay with algebraic geometry and dualities. Which mathematical structure is conjectured to parametrize simple modules for non simply-laced truncations of shifted quantum affine algebras, supported by evidence from Baxter polynomiality in quantum integrable models? 1) The Weyl group of the original Lie algebra 2) The Langlands dual Lie algebra 3) The quantum torus algebra 4) The Hecke algebra of type A 5) The root lattice of the original Lie algebra 6) The classical universal enveloping algebra 7) The category of perverse sheaves on the flag variety

✓ Correct Answer:

The correct answer is 2) The Langlands dual Lie algebra.

📚 Reference Text:

Title: Representations of shifted quantum affine algebras Year: 2020 Paper ID: cb1fb8aa6c65366a57400881d34c9299c6007a11 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/cb1fb8aa6c65366a57400881d34c9299c6007a11 Abstract: We develop the representation theory of shifted quantum affine algebras $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and of their truncations which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gauge theories. Our direct approach is based on relations that we establish with the category $\mathcal{O}$ of representations of the quantum affine Borel algebra $\mathcal{U}_q(\hat{\mathfrak{b}})$ and on associated quantum integrable models we have previously studied. We introduce the category $\mathcal{O}^\mu$ of representations of $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and we classify its simple objects. For $\mathfrak{g} = sl_2$ we prove the existence of evaluation morphisms to $q$-oscillator algebras. We establish the existence of a fusion product and we get a ring structure on the sum of the Grothendieck groups $K_0(\mathcal{O}^\mu)$. We introduce induction and restriction functors to the category $\mathcal{O}$ of $\mathcal{U}_q(\mathfrak{b})$. As a by product we classify simple finite-dimensional representations of $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and we obtain a cluster algebra structure on the Grothendieck ring of finite-dimensional representations. We establish a necessary condition for a simple representation to descend to a truncation, which is also sufficient for $\mathfrak{g} = sl_2$. We introduce a related partial ordering on simple modules and we prove a truncation has only a finite number of simple representations. We state a conjecture on the parametrization of simple modules of a non simply-laced truncation in terms of the Langlands dual Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations proved by using the Baxter polynomiality of quantum integrable models.

Question 44 multiple-choice

In representation theory and quantum computation, the symmetric group Sn captures the symmetries of n indistinguishable objects, with its representations acting on spaces like (C^2)⊗n for qubits. Irreducible representations of Sn are classified by integer partitions and have important combinatorial and physical implications. Which of the following statements correctly describes the classification of irreducible representations of the symmetric group Sn? 1) Each irreducible representation corresponds to a unique element of Sn written in cycle notation. 2) Irreducible representations are classified by the set of all possible two-row Young diagrams for n. 3) Every irreducible representation is associated with a distinct Pauli operator on n qubits. 4) The number of irreducible representations equals 2^n, the dimension of the n-qubit Hilbert space. 5) Irreducible representations are determined solely by the number of cycles in a permutation. 6) Irreducible representations of Sn are classified by the integer partitions of n, often depicted using Young diagrams. 7) Each irreducible representation corresponds directly to a swap operator permuting two qubits.

✓ Correct Answer:

The correct answer is 6) Irreducible representations of Sn are classified by the integer partitions of n, often depicted using Young diagrams..

📚 Reference Text:

Representations of the Symmetric Group In this section we review standard facts about representations of the symmetric group which will be necessary to prove our results. 2.2.1 Preliminary definitions For any finite-dimensional vector space V, we use GL(V)to denote the group of invertible linear transformations from Vto itself. We use Snto denote the symmetric group, the group of permutations of nobjects. We will specify elements group using cycle notation, withedenoting the identity element. Recall that a representation of Snis a group homo- morphismρ:Sn→GL(V). The vector space Vis also referred to as an Sn-module or simply a module. The group algebra C[Sn]can be defined as an Sn-module as follows. Promote the elements{π1,π2,...,πn!}∈Snto basis vectors π1,π2,...,π n!with the multiplication rule πiπj=πkifπiπj=πk. Then C[Sn]is given by C[Sn] ={c1π1+c2π2+···+cn!πn!|cj∈C} (2.6) whereci∈Cfor alli, andSnacts on C[Sn]by left-multiplication. A representation (ρ,V)ofSnalso gives rise to a representation of the algebra C[Sn]via the homomorphism ˜ρ:C[Sn]→C[GL(V)]defined by its action ˜ρ/parenleftiggn!/summationdisplay i=1ciπi/parenrightigg =n!/summationdisplay i=1ciρ(πi). (2.7) It is common to use ρto refer to the representation of both the group and its group algebra. ASn-module in general decomposes into a number of Sn-submodules . A module V isirreducible if the only submodules of Vare itself and the trivial module {0}. For a Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 10 representation (ρ,V)of any finite group, it follows from Maschke’s Theorem that there exists a decomposition of Vas V=/circleplusdisplay λVλ (2.8) where each Vλis an irreducible module. We will be interested in matrix representations ρofSnwhereVis(C2)⊗n, which is to say thatρ(π)is a2n×2nmatrix for each π∈Sn. Maschke’s Theorem then implies that there is an invertible matrix Usuch that ρ(π) =U/bracketleftigg/circleplusdisplay λρλ(π)/bracketrightigg U−1, π∈Sn, (2.9) whereρλare irreducible representations or irreps of Sn. The decomposition of Eq. (2.8) into irreducible modules can thus be seen as a block-diagonalization of matrices in ρ(C[Sn])⊆ M2n(C). 2.2.2 Irreducible representations of Sn Irreducible representations of Snare in one-to-one correspondence with integer partitions ofn, λ= [λ1,λ2,...,λk], λ 1≥λ2≥···≥λk>0,k/summationdisplay i=1λi=n (2.10) which we will denote as λ⊢n. It is convenient to associate the partition λwith the Young diagram of shapeλ, which consists of krows indexed top to bottom such that the i-th row containsλiboxes. As an example, the partition [3,2]corresponds to the shape (2.11) Of crucial importance to us later in the paper will be those irreps ρ[n−k,k]that correspond to two row Young diagrams and the respective modules V[n−k,k], defined for k= 0,...,⌊n 2⌋. 2.3 Swap Matrices and Permutations In order to analyze the QMC Hamiltonian, we introduce the swap matrices Swapijas permutations on n-qubit states. Definition 2.2. The swap matrices Swapijare defined by their action on tensor products of|ψ1⟩,|ψ2⟩,...,|ψn⟩∈C2as follows: Swapij/parenleftig |ψ1⟩⊗···⊗|ψi⟩⊗···⊗|ψj⟩⊗···⊗|ψn⟩/parenrightig =|ψ1⟩⊗···⊗|ψj⟩⊗···⊗|ψi⟩⊗···⊗|ψn⟩. (2.12) The connection to Quantum Max Cut is made by noticing that the Swapijis the following linear combination of the Pauli matrices Swapij=1 2/parenleftig I+σi Xσj X+σi Yσj Y+σi Zσj Z/parenrightig . (2.13) Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 11 Proposition 2.3. The QMC Hamiltonian HGof Eq. (2.5) in terms of the Swapij’s defined in Eq. (2.12) is given

Question 45 multiple-choice

Quantum processors can be designed using multi-level quantum systems called qudits, which offer increased computational capacity compared to traditional qubits. Silicon-photonic integrated circuits enable scalable and robust implementation of such quantum devices. Which of the following features most directly enables a programmable quantum processor based on ququarts to achieve enhanced computational parallelism and improved resource efficiency compared to qubit-based systems? 1) Employing error-correcting codes optimized for binary logic gates 2) Utilizing superconducting transmon qubits for state manipulation 3) Integrating nitrogen-vacancy centers in diamond for coherence 4) Implementing single-photon detectors for readout 5) Encoding quantum information in four-level qudits, increasing information density per particle 6) Incorporating trapped ions with microwave control pulses 7) Using classical post-processing to simulate quantum algorithms

✓ Correct Answer:

The correct answer is 5) Encoding quantum information in four-level qudits, increasing information density per particle.

📚 Reference Text:

Title: A programmable qudit-based quantum processor Year: 2022 Paper ID: a4390a7f1c46ff8a7fdae863295d0b610023d977 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a4390a7f1c46ff8a7fdae863295d0b610023d977 Abstract: Controlling and programming quantum devices to process quantum information by the unit of quantum dit, i.e., qudit, provides the possibilities for noise-resilient quantum communications, delicate quantum molecular simulations, and efficient quantum computations, showing great potential to enhance the capabilities of qubit-based quantum technologies. Here, we report a programmable qudit-based quantum processor in silicon-photonic integrated circuits and demonstrate its enhancement of quantum computational parallelism. The processor monolithically integrates all the key functionalities and capabilities of initialisation, manipulation, and measurement of the two quantum quart (ququart) states and multi-value quantum-controlled logic gates with high-level fidelities. By reprogramming the configuration of the processor, we implemented the most basic quantum Fourier transform algorithms, all in quaternary, to benchmark the enhancement of quantum parallelism using qudits, which include generalised Deutsch-Jozsa and Bernstein-Vazirani algorithms, quaternary phase estimation and fast factorization algorithms. The monolithic integration and high programmability have allowed the implementations of more than one million high-fidelity preparations, operations and projections of qudit states in the processor. Our work shows an integrated photonic quantum technology for qudit-based quantum computing with enhanced capacity, accuracy, and efficiency, which could lead to the acceleration of building a large-scale quantum computer. Qudit-based quantum devices can outperform qubit-based ones, but a programmable qudit-based quantum computing device is still missing. Here, the authors fill this gap using a programmable silicon photonic chip employing ququart-based encoding, showing the scaling advantages compared to the qubit counterpart.

Question 46 multiple-choice

In quantum information theory, the concept of unitary t-designs is crucial for simulating random quantum operations and understanding the universality of gate sets. Representation theory of unitary groups underpins the mathematical structure of these designs and their connection to universal quantum computation. Which condition guarantees that an infinite closed subgroup G of U(d) is universal for quantum computation via unitary gates? 1) G contains all diagonal unitary matrices. 2) G is abelian and acts transitively on V. 3) G is a unitary 1-design and includes the identity. 4) G is a unitary 2-design and contains SU(d). 5) G consists only of anti-symmetric tensors. 6) G preserves the symmetric decomposition of V⊗t for all t. 7) G commutes with all elements in U(d).

✓ Correct Answer:

The correct answer is 4) G is a unitary 2-design and contains SU(d)..

📚 Reference Text:

in Proposition 3. LetVbe ad-dimensional vector space, and let U(V)be the corresponding unitary group, the group of all unitary trans- formations of V. By picking a basis, we may identify VwithCdandU(V)withU(d). As discussed above, we have the natural representation (τ,V)ofU(V)onV, where τis simply the identity map. This representation τis in fact irreducible. We then take the tensor product of tcopies of the natural representation, determining the tensor product representation (or diagonal representation )(τt,V⊗t), given by τt(U) =U⊗t. This has a decomposition into irreps for U(V), V⊗t∼=M iV⊕ni i, (B1) where, as above, each Viis irreducible and Vi∼=Vjiffi=j. Proposition 3 is concerned with the sum of the multiplicities,P ini. For t= 1,Vis already irreducible, as mentioned above, soP in1= 1. For t= 2,V⊗2decomposes into a direct sum of two inequivalent irreps, namely the symmetric and anti-symmetric tensors, soP ini= 2. Now let Gbe a closed subgroup G⊆U(V). The representation V⊗trestricts to a representation of G, which then has its own decomposition into irreps for G. In fact, it suffices to consider the restriction of each Vito a representation of G. The 23 content of Proposition 3 is that Gis at-design if and only if, for each Viappearing with nonzero multiplicity in (B1), Viis also irreducible as a representation of G. 3. Proof of Theorem 7 We now restate and prove Theorem 7. Theorem 10. LetV∼=Cd, with d≥2. LetGbe an infinite closed subgroup of U(d) =U(V). IfGis a unitary 2-design, thenSU(d)⊆G. In particular, any set of unitaries generating Gis a universal set. Proof. Since Gis an infinite closed subgroup of U(d), the closed subgroup theorem implies that it is an infinite compact Lie group, with a nonzero complexified Lie algebra g. By compactness, the Peter-Weyl theorem implies that G’s finite- dimensional representations are completely reducible. Since we assume that Gis a2-design, it is also a 1-design; which implies Vis an irrep for G(recall Proposition 3). Since dimV=d >1, this implies that Gis not abelian: equivalently, theG-subrepresentation [g,g]is nonzero. Further, since commutators are traceless, we have 0̸= [g,g]⊆sl(d). These are the main facts we will need below. By Proposition 3, since Gis a2-design, V ⊗ V decomposes into a direct sum of exactly two irreps for G. Note that by Schur’s lemma, this is equivalent to the following condition on the commutant: dimC(EndG(V ⊗ V )) = 2 .We now apply a chain of standard vector space isomorphisms: EndG(V ⊗ V )∼=((V ⊗ V )⊗(V ⊗ V )∗)G∼=((V ⊗ V∗)⊗(V ⊗ V∗)∗)G∼=EndG(V ⊗ V∗). (B2) In particular, this implies dimCEndG(V ⊗ V∗) = 2 .Now we follow Ref. 7 to show that this condition gives universality. Applying Schur’s lemma in the same way as above, if V ⊗ V∗∼=L iV⊕mi i is a decomposition of V ⊗ V∗into distinct irreps for G, then 2 = dim CEndG(V ⊗ V∗) =X im2 i. Since the miare positive integers, we conclude that V ⊗V∗decomposes into exactly two irreducible representations for G, each of multiplicity 1. Further, we can identify them exactly. Note that V ⊗

Question 47 multiple-choice

In quantum information processing, error characterization and correction often involve advanced mathematical techniques to ensure the physical validity of quantum operations. One important approach is fitting experimental quantum processes to models that satisfy complete positivity and trace-preservation constraints. Which of the following statements best explains the effect of imposing CPTP (completely positive, trace-preserving) constraints on an experimentally determined quantum supermatrix? 1) It eliminates all eigenvalues, making the supermatrix singular. 2) It causes the largest Kraus operator to vanish, altering the dominant process behavior. 3) It increases the number of negative eigenvalues, indicating reduced physical validity. 4) It ensures the operation is physically realizable and typically improves correlation with theoretical and simulated models. 5) It randomizes the action of the quantum superoperator, reducing benchmarking accuracy. 6) It removes all rotation-based errors without affecting correlation. 7) It transforms the supermatrix into a classical probability matrix without quantum properties.

✓ Correct Answer:

The correct answer is 4) It ensures the operation is physically realizable and typically improves correlation with theoretical and simulated models..

📚 Reference Text:

U1,expandU1,simincreased from 0.90 to 0.96 on left-multiplying U1by the Hermitian conjugate of this product of single-spin rotation operators, these rotations are the cumulative result of many small rotation errors and are not simply traced back to any single short-coming in the expe riments. For com- pleteness, the axes and angles of the single-spin rotations that be st ﬁt the error operator U∆=U† 1,simU1,expare also shown in Table III, together with the eigenvalues of the cor re- sponding corrected superoperators in Fig. 16. The product of th ese rotations similarly has a 0.97 correlation with U∆, but in this case one must right-multiply U1,expby the Hermitian conjugate of the product to correct it. The close co-incidence be tween the angles of rotation about the x-axis on spin 1 in the ﬁrst case and about the z-axis on spin 3 in the latter case is expected, since UQFTσ1 xU† QFT=σ3 z(recallσ1 xσ3 zis a ﬁxed point of the QFT). Finally, it is of interest to demonstrate that despite a substantial n umber of negative eigenvalues in the Choi matrix of the experimental supermatrix, it is not necessary to change TABLE III: The two triples of single-spin rotations that optimize the c orrelation coeﬃcient between the best unitary approximations to the largest Krau s operators of the experimental and simulated supermatrices, when the experi mental supermatrix is left or right multiplied by the unitary matrix corresponding to e ach one of these triples. Side Spin x,y&z-Directional Cosines of Rotation Axis Rotation Angle left 1 −0.992 0 .007 0 .123 36.7◦ left 2 −0.386 −0.243 0 .890 9.0◦ left 3 0.703 0 .701 −0.123 16.9◦ right 1 0.059 −0.031 0 .998 14.2◦ right 2 0.227 −0.512 0 .829 10.2◦ right 3 −0.092 −0.292 −0.952 38.3◦ 24 it much in order to obtain a supermatrix which represents a complete ly positive and trace- preserving superoperator. For this reason the supermatrix MCPTPwhich best-ﬁt the Choi matrix of the experimental supermatrix subject to the constrain t that it was both positive semideﬁnite and satisﬁed the trace-preservation conditions was c omputed as described in Section III. Although this procedure made essentially no change in t he largest Kraus op- erator (as expected), it did have a signiﬁcant eﬀect on the experim ental supermatrix as a whole. The correlations between this CPTP-ﬁt and the other super matrices that we have dealt with up to now are given in Table IV, along with those to the origina l experimental supermatrix for comparison. This shows that even though imposing the complete positivity constraint on the experimental observations did not change the s upermatrix very much, the change was distinctly in the right direction since it improved the corre lation with both the simulated and theoretical supermatrices. This is further conﬁrme d by Fig. 17, which shows the eigenvalues of MCPTP, along with those of the superoperator ¯U1,CPTP⊗U1,CPTPobtained from the best unitary approximation to its largest Kraus operator (scaled down so as to have the same trace as MCPTP), and those of

Question 48 multiple-choice

In quantum mechanics and linear algebra, the anti-symmetric subspace of a tensor product space is crucial for describing the behavior of fermions and constructing multi-particle wavefunctions. The anti-symmetric projector operator is central to extracting states that obey the Pauli exclusion principle. Which of the following statements about the anti-symmetric subspace ASymₖ(ℂᵈ) and its orthogonal projector P(d, k)_asym is correct? 1) The dimension of ASymₖ(ℂᵈ) is always dᵏ, regardless of k and d. 2) The anti-symmetric projector operator is neither idempotent nor self-adjoint. 3) Basis vectors of the anti-symmetric subspace can include repeated indices. 4) For d < k, the dimension of ASymₖ(ℂᵈ) equals k! (k factorial). 5) The anti-symmetric subspace is used primarily to describe bosons. 6) P(d, k)_asym projects onto the symmetric subspace. 7) The dimension of ASymₖ(ℂᵈ) is zero if d < k, and equals the binomial coefficient "d choose k" otherwise.

✓ Correct Answer:

The correct answer is 7) The dimension of ASymₖ(ℂᵈ) is zero if d < k, and equals the binomial coefficient "d choose k" otherwise..

📚 Reference Text:

introduce the anti-symmetric subspace. Definition 18 (Anti-symmetric subspace) .The anti-symmetric subspace is the set: ASymk(Cd):=/braceleftig |ψ⟩∈(Cd)⊗k:Vd(π)|ψ⟩= sgn (π)|ψ⟩ ∀π∈Sk/bracerightig , (74) where sgn (σ)denotes the sign of a permutation σ∈Sk. Similarly as before, we can define the operator: P(d,k) asym :=1 k!/summationdisplay π∈Sksgn (π)Vd(π). (75) and prove the following theorem: Theorem 19. The operator P(d,k) asym is the orthogonal projector on the anti-symmetric subspace ASymk(Cd). Proof. We have: Vd(σ)P(d,k) asym =1 k!/summationdisplay π∈Sksgn (π)Vd(σ)Vd(π) =1 k!/summationdisplay π∈Sksgn (π)Vd(σπ) (76) =1 k!/summationdisplay π∈Sksgn/parenleftbig σ−1π/parenrightbig Vd(π) = sgn/parenleftbig σ−1/parenrightbig P(d,k) asym (77) = sgn (σ)P(d,k) asym. (78) Similarly,P(d,k) asymVd(σ) = sgn (σ)P(d,k) asym. Using this, we can show that P(d,k)2 asym =P(d,k) asym: P2 asym =1 k!/summationdisplay π∈Sksgn (π)Vd(π)P(d,k) asym =1 k!/summationdisplay π∈Sk(sgn (π))2P(d,k) asym =P(d,k) asym. (79) We also have P(d,k)† asym =P(d,k) asym: P(d,k)† asym =1 k!/summationdisplay π∈Sksgn (π)V† d(π) =1 k!/summationdisplay π∈Sksgn (π)Vd(π−1) =1 k!/summationdisplay π∈Sksgn/parenleftbig π−1/parenrightbig Vd(π−1) (80) =1 k!/summationdisplay π∈Sksgn (π)Vd(π) =P(d,k) asym. (81) We can show that Im/parenleftig P(d,k) asym/parenrightig ⊆ASymk(Cd) as follows: for all |ψ⟩∈(Cd)⊗k, we haveP(d,k) asym|ψ⟩∈ ASymk(Cd), sinceVd(π)P(d,k) asym|ψ⟩= sgn (π)P(d,k) asym|ψ⟩for allπ∈Sk, where we used again that Vd(π)P(d,k) asym = sgn (π)P(d,k) asym. Moreover, we can also show that ASymk(Cd)⊆Im/parenleftig P(d,k) asym/parenrightig . In fact, if|ψ⟩∈ASymk(Cd), then we have: P(d,k) asym|ψ⟩=1 k!/summationdisplay π∈Sksgn (π)Vd(π)|ψ⟩=1 k!/summationdisplay π∈Sk(sgn (π))2|ψ⟩=|ψ⟩. (82) Therefore, we have shown that Im/parenleftig P(d,k) asym/parenrightig = ASymk(Cd), which implies that P(d,k) asym is the orthogonal projector on the anti-symmetric subspace ASymk(Cd). Next, we compute the dimension of the anti-symmetric subspace. Proposition 20 (Dimension of the anti-symmetric subspace) .Ifd≥k, we have: Tr/parenleftig P(d,k) asym/parenrightig = dim/parenleftbig ASymk(Cd)/parenrightbig =/parenleftbiggd k/parenrightbigg , (83) otherwise Tr/parenleftig P(d,k) asym/parenrightig = 0. 11 Proof. First, we have: ASymk(Cd) = Im/parenleftig P(d,k) asym/parenrightig = span/parenleftig P(d,k) asym|i1⟩⊗···⊗|ik⟩:∀i1,...,ik∈[d]/parenrightig . (84) To count the number of linearly independent vectors P(d,k) asym|i1⟩⊗···⊗|ik⟩, we observe that if there are at least two tensor factors of |i1⟩⊗···⊗|ik⟩with matching entry i.e. there exist l̸=m∈[k] such that il=im, then: P(d,k) asym|i1⟩⊗···⊗|ik⟩= 0. (85) This because: P(d,k) asym|i1⟩⊗···⊗|ik⟩=P(d,k) asymVd(τl,m)|i1⟩⊗···⊗|ik⟩ (86) = sgn (τl,m)P(d,k) asym|i1⟩⊗···⊗|ik⟩ (87) =−P(d,k) asym|i1⟩⊗···⊗|ik⟩, (88) where in the second equality we used P(d,k) asymVd(τl,m) = sgn (τl,m)P(d,k) asym (as shown in the proof of Theorem 19). Therefore, i1,...,ikmust all be distinct for P(d,k) asym|i1⟩⊗···⊗|ik⟩to be nonzero. This also implies that if d < k , then Tr/parenleftig P(d,k) asym/parenrightig = 0. Therefore, we now focus on the case d≥k. If for all m∈[d], there exist two vectors |i1⟩⊗···⊗|ik⟩and|j1⟩⊗···⊗|jk⟩with i1,j1...,ik,jk∈[d] such that both sets, {i1,...,ik}and{j1,...,jk}, containnm∈{0,1}elements equal tom, then there exists a permutation Vd(π) such that Vd(π)|i1⟩⊗···⊗|ik⟩=|j1⟩⊗···⊗|jk⟩. SinceP(d,k) asymVd(π) = sgn (π)P(d,k) asym, we haveP(d,k) asym|i1⟩⊗···⊗|ik⟩= sgn (π)P(d,k) asym|j1⟩⊗···⊗|jk⟩, and hence, P(d,k) asym|i1⟩⊗···⊗|ik⟩andP(d,k) asym|j1⟩⊗···⊗|jk⟩are linearly dependent. Thus, taking n1,...,nd∈{0,1}withn1+···+nd=k, we define|n1,...,nd⟩asP(d,k) asym|i1⟩⊗ ···⊗|ik⟩, where|i1⟩⊗···⊗|ik⟩is any computational basis vector such that, for each m∈[d], there existnmelementsj∈[k] such that ij=m. It is easy to see that such vectors are orthogonal and hence independent. Therefore, the dimension is given by the number of such independent vectors, which is equal to the number of ways to choose an

Question 49 multiple-choice

Quantum speed limits define the minimum time required to perform specific quantum gates, which is crucial for optimizing operations in neutral atom quantum computing. Various control configurations, such as sequential, parallel, and phase, influence both the achievable gate speed and error rates when implementing gates like the controlled-Z (CZ) gate. Which control configuration for the CZ gate in neutral atom systems both achieves the quantum speed limit of 350 ns and requires the fewest physical resources while automatically mapping the |↓↓⟩ state onto itself? 1) Sequential configuration with site-dependent amplitude controls 2) Parallel configuration with simultaneous amplitude variation 3) Sequential configuration with phase and amplitude modulation 4) Phase configuration with intrinsic state mapping 5) Parallel configuration with independent phase controls 6) Sequential configuration with non-site-specific fields 7) Parallel configuration with optimized initial guess fields

✓ Correct Answer:

The correct answer is 4) Phase configuration with intrinsic state mapping.

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ZZZ( γ) and ZZZZ( γ), cf. Eq. ( A6), have been obtained for the maximally entangling gate at γ=π/4. The results for the CNOT gate on neutral atoms have been obtained using site-dependent control ﬁelds for each atom. Eq. ( A6) and Ref. [ 65]. Its QSL TCZ QSL=700 ns is twice as long compared to the other two conﬁgurations. In order to compare the results from the three conﬁgura- tions in terms of how successful the optimization has been inﬁnding solutions, the histogram on the right side of Fig. 3(a) provides the probability density for obtaining ﬁnal errors ε T within certain ranges. For obtaining a solution with errors εT/lessorequalslantεmax, we ﬁnd the lowest probability for the “sequen- tial” conﬁguration (diamonds)—coinciding with the highestQSL—and the highest and almost identical probability forthe other two conﬁgurations. Among those two, the “phase”conﬁguration (crosses) needs to be emphasized in particular.From a physical perspective, setting /Omega1 ↓(t) andϕ↓(t) to zero automatically ensures that |↓↓/angbracketright is mapped onto itself—as required by the CZ gate. This is not automatically guaranteedby the other two conﬁgurations and the optimization needsto explicitly ensure it and therefore needs to solve a slightlymore complex optimization problem. However, the advan-tage of having one basis state automatically mapped correctlydoes not translate into an advantage regarding the QSL ofT CZ QSL=350 ns or the reachable error in general. Interestingly, both the “parallel” and “phase” conﬁgurations yield the sameachievable lowest errors for each gate time T. This is visually highlighted by the lines connecting the lowest errors per T in Fig. 3(a). Moreover, it should be stressed that these errors εTare reached for almost every set of initial guess ﬁelds, i.e., independent of the initial starting point within the problem’scontrol landscape, and obtained independently for both con-ﬁgurations. This supports the conjecture that T CZ QSL=350 ns is the actual QSL for a CZ gate and generally validates ourmethod in determining the QSL. From an optimal controlperspective, it is interesting to see that the ﬂexibility origi-nating from the extended set of available control ﬁelds in the“parallel” conﬁguration can not be turned into an advantagein error or time compared to the “phase” conﬁguration. Froma practical perspective, the latter is advantageous for experi-mental realizations as it requires fewer physical resources. While the three conﬁgurations discussed so far should only be viewed as examples, we did not ﬁnd any conﬁguration giv-ing rise to faster CZ gates. Hence, we assume T CZ QSL=350 ns to be the fundamental QSL across all conﬁgurations. A naturalcomparison for T CZ QSLwith a value from the literature would be the gate time from the analytical protocol introduced inRef. [ 20], especially because it uses the same control ﬁelds as 023026-9 BASILEWITSCH, DLASKA, AND LECHNER PHYSICAL REVIEW RESEARCH 6, 023026 (2024) the “phase” conﬁguration to implement the gate. For our pa- rameters, we ﬁnd TCZ lit≈340 ns as analytical gate time, which we assume identical with our QSL TCZ QSL=350 ns given the rather coarse sampling of gate times Tin Fig. 3(a). However, it should

Question 50 multiple-choice

Quantum sensing utilizes advanced protocols to enhance measurement sensitivity and spectral resolution, particularly when detecting complex signals composed of multiple frequencies. The choice of phase estimation algorithm significantly impacts both dynamic range and experimental efficiency. Which approach enables unambiguous phase measurement with high sensitivity in a single readout, leveraging quantum entanglement to demultiplex multiple spin signals onto distinct qubit outputs? 1) Classical Fourier transform analysis on a single qubit system 2) Standard Ramsey interferometry with repeated measurements 3) Machine learning-based iterative phase estimation 4) Bayesian adaptive phase estimation with multiple interrogations 5) Direct amplitude sampling using analog signal processing 6) Inverse quantum Fourier transform protocol implemented on a quantum register 7) Frequency filtering via classical lock-in amplification

✓ Correct Answer:

The correct answer is 6) Inverse quantum Fourier transform protocol implemented on a quantum register.

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measured. To achieve the best possible sensitivity in each of thetwo phase acquisition steps interrogation times τclose to the decoherence limit T 2of the sensor are used, which results inacquiring a maximum phase ϕ. On the other hand, such choice of τcould result in ϕmore than πand thereby in an ambiguity in determining the actual strength of the signal. This becomes ultimately important when dealing with a high dynamic range correlation signal with multiple frequency components of variousstrengths. In the case of a nanoscale NMR signal, which iscomposed of multiple frequencies f iand amplitudes aithis results in a beating of the free precession signal requiring a high dynamic range sensing method (see Fig. 1b). Thus, the problem at hand is to design a phase correlation measurement protocol that at the same time yields a large dynamic range and allows for high spectral resolution (i.e., long correlation times Tc). This known dynamic range - sensitivity1trade-off could be tackled by a family of phase estimation algorithms, including Bayessian, adaptive and machine learning sensing protocols1–8and the ones based on inverse quantum Fourier transform (QFT)9–11. As correla- tion measurements typically have a large time overhead ( Tc= 10–2 0m s )i nc o m p a r i s o nt ot h es e n s i n gt i m e s( τ=10–100μs)12(also see SI for times used in present manuscript), repetitive measure-ments within each correlation step required in the adaptive techniques may not be well suitable. Further, including the in ﬂuence of the quantum back-action in such repetitive measurements leadsto erroneous results on the measured dynamics 13,14.O nt h eo t h e r hand, phase estimation using the inverse QFT could resolve ambiguities in each run (measurement) of the protocol and provide a sensitivity that is only limited by the coherence time of the sensor.In this work, we advance the correlation measurement protocol with phase estimation based on the inverse QFT algorithm using the NV center and a three nuclear spin register. We utilize the QFT †and QFT as a convenient transformation to convert the acquired phase to the population basis of the register. To demonstrate and benchmark the scope of QFT-based phase estimation we perform here threedifferent experiments using both classical and quantum ﬁelds. We ﬁrst start by demonstrating the phase to population mapping by QFT and show how it differs from the standard Ramsey measure- ments which do the same. After this we benchmark the QFT † algorithm and perform high dynamic range sensing of an arti ﬁcial classical AC-signal generated by the RF source. Later we extend it to 13. Physikalisches Institut, IQST and Centre for Applied Quantum Technologies, University of Stuttgart, Stuttgart, Germany.2Max-Planck Institute for Solid State Research, Stuttgart, Germany.✉email: v.vorobyov@pi3.uni-stuttgart.de; j.wrachtrup@pi3.uni-stuttgart.dewww.nature.com/npjqi Published in partnership with The University of New South Wales1234567890():,; the correlation spectroscopy protocol of the nuclear spin-bath and demonstrate the demultiplexing of target spin-signals onto separate register qubits outputs. As the sensor readout performed only at the end of the protocol, an entanglement

Question 51 multiple-choice

Quantum algorithms have dramatically impacted the security landscape of cryptography, especially for schemes relying on algebraic problems such as the discrete logarithm. The distinction between groups and semigroups, and the computational complexity of related problems, is crucial for understanding potential quantum vulnerabilities. In the context of quantum computing and algebraic structures, which statement best explains why some cryptosystems based on discrete logarithms in semigroups are not secure against quantum attacks, while certain generalizations of the discrete logarithm problem remain hard in semigroups? 1) Quantum algorithms cannot efficiently solve any discrete logarithm problems in semigroups. 2) Semigroups always have the same computational properties as groups regarding discrete logarithm problems. 3) The lack of inverses in semigroups makes all discrete logarithm problems easy for quantum computers. 4) Quantum algorithms like Shor's can efficiently solve standard discrete logarithm problems in semigroups, undermining cryptosystems, but certain generalized or shifted versions may still be hard due to the structure of semigroups. 5) The security of semigroup-based cryptosystems is guaranteed because quantum algorithms only accelerate group-based problems. 6) Discrete logarithm problems in semigroups are always harder than those in groups for quantum computers. 7) Semigroups do not admit any cryptographically hard problems in the presence of quantum algorithms.

✓ Correct Answer:

The correct answer is 4) Quantum algorithms like Shor's can efficiently solve standard discrete logarithm problems in semigroups, undermining cryptosystems, but certain generalized or shifted versions may still be hard due to the structure of semigroups..

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Question 52 multiple-choice

Computational spectroscopy often relies on advanced quantum chemical methods to predict vibrational frequencies and simulate molecular spectra, balancing accuracy with computational cost. Composite approaches can leverage the strengths of different methods for efficient and precise results, especially in the study of hydrogen-bonded systems and solvatochromic effects. Which computational strategy most effectively reduces computational expense while maintaining accurate peak position predictions for vibrational spectra by combining a density functional theory approach for spectral shifts with a high-level coupled cluster method for reference frequencies? 1) Using only the M06-2X/cc-pVTZ density functional theory method for all calculations 2) Relying exclusively on the CCSD-F12b coupled cluster method for both spectral shifts and reference frequencies 3) Applying the QVP1 frozen-bath approximation without further refinement 4) Calculating spectral shifts and reference frequencies with the same low-level DFT method 5) Employing only ab initio molecular dynamics with CCSD throughout the simulation 6) Combining spectral shifts from CCSD-F12b and reference frequencies from M06-2X 7) Utilizing the composite CCSD/M06 method by applying DFT for spectral shifts and CCSD for reference frequencies

✓ Correct Answer:

The correct answer is 7) Utilizing the composite CCSD/M06 method by applying DFT for spectral shifts and CCSD for reference frequencies.

📚 Reference Text:

based on the QVP1 transition frequencies in green. The spectrum determined from the DAF shows a series of side peaks with an interval of about 50 cm−1, attributed to the coupling with rotations of the water molecule about the H–Cl axis. This effect is not included in the instantaneous frozen-bath approximation in the QVP1 model. Xue et al. Page 7 J Chem Theory Comput . Author manuscript; available in PMC 2018 February 01. Author Manuscript Author Manuscript Author Manuscript Author Manuscript The maximum peak position from the QVP1 method is located at 2653 cm−1, redshifted by 185 cm−1 from the strongest peak position (2838 cm−1) displayed on the spectrum based on the classical DAF (Table 1). Since M06-2X/cc-pVTZ was used in both approaches, consistent with the dynamic trajectory, the difference between the two methods is primarily due to nuclear quantum effects of the oscillator, which clearly have significant contributions to the computed vibrational frequencies in Figure 3a. Nevertheless, the full widths at half- maximum (fwhm) of the spectrum peak are similar from both methods (21 and 23 cm−1, respectively). Figure 3b compares the results obtained using three different electronic structure methods: M06-2X (green line), CCSD(T)-F12b (red line), and a composite of M06-2X spectral shift and the CCSD(T)-F12b frequency. In the latter approach, the CCSD(T)-F12b frequency for the reference state is used, but the spectral shift due to dynamic fluctuations of the solvent is determined at the M06-2X level using QVP1. We designate this method as CCSD(T)/M06 in Figure 3b, which significantly reduces the computational costs compared with that when the perturbation theory is also carried out at the CCSD(T)-F12b level. Figure 3b shows that the maximum peak position from QVP1 using CCSD(T)-F12b is only 14 cm−1 greater than the experimental value (2723.5 cm−1).16,17 The M06-2X frequency in the HCl(H 2O) complex is blue-shifted by 84.6 cm−1 from the coupled-cluster value, and this is mainly due to the difference in the reference state between the two methods: 80.6 cm−1 from 2650.5 cm−1 (CCSD(T)-F12b) to 2569.9 cm−1 (M06-2X). Thus, the solvatochromic shifts are not very sensitive to the quantum chemical models used, here, differing only by 4 cm−1 (84.6 vs 80.6 cm−1). Clearly, it can be a major time-saving approach to use density functional theory to estimate the spectral shift, but the reference frequency value is evaluated using coupled cluster theory (eq 1) since the latter only needs to be computed once. As the computational cost of the QVP method is equivalent to that required to determine the perturbation energy at the discrete points, allowing the instantaneous vibrational frequency of the oscillator to be determined on-the-fly of dynamics simulations. Indeed, the composite approach, employing the CCSD(T) zeroth order frequency (reference state) and QVP1 frequency shift from M06-2X (blue curve in Figure 3b), yields a maximum peak position at 2725 cm−1, only 13 cm−1 different from the CCSD(T)-F12b value (2738 cm−1). Interestingly, the composite result is in better accord with experimental results than CCSD(T)-F12b, perhaps due to fortuitous error cancellations. Furthermore, the computed

Question 53 multiple-choice

In the classification of finite p-groups of nilpotency class 3, characteristic matrices over finite fields are used to distinguish group types and compute invariants related to subgroup structure. Arithmetic properties like quadratic residues can influence the existence of specific group extensions. Which condition ensures that two finite p-groups of class 3 with characteristic matrices w and w(Ḡ) are isomorphic, based on matrix equivalence over the field Fp? 1) There exists a permutation matrix over Fp such that w(Ḡ) = P w P⁻¹ 2) There exists an invertible diagonal matrix X over Fp such that w(Ḡ) = X w diag(x₁₁⁻¹ x₂₂⁻¹, x₁₁⁻² x₂₂⁻¹) 3) The trace of w and w(Ḡ) are equal modulo p 4) The determinant of w is a quadratic residue modulo p 5) A scalar multiple relates w(Ḡ) and w over Fp 6) The entries of w and w(Ḡ) are pairwise congruent modulo p 7) Both groups have identical commutator subgroup orders

✓ Correct Answer:

The correct answer is 2) There exists an invertible diagonal matrix X over Fp such that w(Ḡ) = X w diag(x₁₁⁻¹ x₂₂⁻¹, x₁₁⁻² x₂₂⁻¹).

📚 Reference Text:

types give non-isomorphic groups, (i i)G∼=¯Gif and only if there exists X= diag(x11,x22), an invertible matrix over Fp, such that w(¯G) = Xw(G)diag(x−1 11x−1 22,x−2 11x−1 22). By Table 5.2, we get the groups of Type (M1)–(M9). /square w(G) Case X w (¯G) Group Remark (a) w22/negationslash= 0diag(z,w11) where w 22=νz2/parenleftbigg 1w12w−1 11z−1 0 ν/parenrightbigg (M1) s=−w12w−1 11z−1 (a) w22= 0,w12/negationslash= 0 diag( w−1 11w12,w11)/parenleftbigg 1 1 0 0/parenrightbigg (M2) (a) w22=w12= 0 diag(1 ,w11)/parenleftbigg 1 0 0 0/parenrightbigg (M3) (b) w21/negationslash= 0 diag( w21,w−1 21w12)/parenleftbigg0 1 1 0/parenrightbigg (M4) (b) w21= 0 diag(1 ,w12)/parenleftbigg 0 1 0 0/parenrightbigg (M5) (c) w21/negationslash= 0 diag( w21,1)/parenleftbigg0 0 1w22w−2 21/parenrightbigg (M6) if w22/negationslash= 0 (M7) if w22= 0t=w−1 22w2 21 (c) w21= 0,w22/negationslash= 0diag(1,z) where w 22=νz2/parenleftbigg 0 0 0ν/parenrightbigg (M8) (c) w21=w22= 0/parenleftbigg 0 0 0 0/parenrightbigg (M9) Table 5.2: The types in Theorem 5.8 6 The case Φ(G′)≤G3∼=C2 p Suppose that Gis a ﬁnite p-group with Φ( G′)≤G3∼=C2 p,G3≤Z(G) andG/G3∼= Mp(n,m,1), where n >1 forp= 2 and n≥m. Let G/G3=/an}bracketle{t¯a,¯b,¯c|¯apn=¯bpm= ¯cp= 1,[¯c,¯a] = [¯b,¯c] = 1/an}bracketri}ht. 24 Then, without loss of generality, we may assume that G=/an}bracketle{ta,b,c/an}bracketri}htwhere [a,b] =c. Let x= [b,c],y= [c,a]. Then G3=/an}bracketle{tx,y/an}bracketri}ht. Sinceapn∈G3, we may assume that apn= xw11yw12. By similar reasons, we may assume that bpm=xw21yw22andcp=xw31yw32. Letw(G) =/parenleftbiggw11w12 w21w22/parenrightbigg andv(G) =/parenleftbiggw31 w32/parenrightbigg . Then we get two matrices over Fp. w(G) is called a characteristic matrix of G, andv(G) is called a characteristic vector ofG. Notice that w(G) andv(G) will be changed if we change the generators a,b. We also call a,ba set of characteristic generators of w(G) andv(G). Theorem 6.1. Suppose that Gis a ﬁnite p-group such that Φ(G′)≤G3∼=C2 p,G3≤ Z(G)andG/G3∼=Mp(n,m,1). Letw(G) = (wij)be a characteristic matrix of G. Then the following conclusions hold: (1)IfImin= 1, thenm= 1; (2)IfImax= 2, thenn≤2; (3)Ifp= 2andm= 1, thenImin= 1; (4)Ifp >3andn=m= 1, thenImin/ne}ationslash= 1if and only if w11=w22=w12+w21= 0; (5)Ifp >2andn > m= 1, thenImin/ne}ationslash= 1if and only if w11=w12= 0; (6)Ifp >3andn=m= 1, thenImax/ne}ationslash= 2if and only if (w12+w21)2−4w11w22is a quadratic non-residue modular p; (7)Ifp >2,m= 1andn= 2, thenImax= 2if and only if w22/ne}ationslash= 0; (8)Ifp >2,n=m= 2andΦ(G′) = 1, thenImax= 2if and only if (w12+w21)2− 4w11w22is a quadratic non-residue modular p; (9)Ifp >2,n=m= 2and(w31,w32) = (0,1), thenImax= 2if and only if w11/ne}ationslash= 0and one of the following holds: (i) (w12+w21)2−4w11(w22+1)is a quadratic non-residue modular p;(ii)w21=w11w22+w11and(w12+w21)2= 4w21. ProofLetN=/an}bracketle{tb,ap,c,x,y/an}bracketri}htandMi=/an}bracketle{tabi,bp,c,x,y/an}bracketri}htwhere 0≤i≤p−1. TheN andMiare all maximal subgroups of G. (1) Assume that His anA1-subgroup of index p. Then d(H/H′) = 2. Since G/G3∈ A1,H/G3is abelian and hence H′≤G3. It follows that d(H/G3) = 2. Since Φ(G)≤H,d(Φ(G)/G3)≤2. Since Φ( G)/G3=/an}bracketle{t¯ap,¯bp,¯c/an}bracketri}htis of type ( pn−1,pm−1,p), we havem= 1. (2) LetK=/an}bracketle{tb,c/an}bracketri}ht. ThenK∈ A1and|G:K| ≥ |G:KG3|=pn. SinceImax= 2, n≤2. (3) Ifm= 1 and p= 2, then 1 = [ a,b2] =c2x. Hence c2=x. In this case, M1=/an}bracketle{tab,c/an}bracketri}ht ∈ A1. ThusImin= 1. (4) Ifn=m= 1 and p >3, then|G|=p5and 1 = [ a,bp] =cp. IfImin/ne}ationslash= 1, thenN∈ A2andMi∈ A2. Sincey/ne}ationslash∈ /an}bracketle{tb,c/an}bracketri}ht,w22= 0. By calculation, [ abi,c] =y−1xi and (abi)p=apbip=xw11+iw21yw12. Hence, ∀i,w11+iw21=−iw12. It follows that w11=w12+w21=

Question 54 multiple-choice

Multi-value (qudit-based) quantum computing utilizes quantum systems with more than two levels, enabling higher-dimensional encoding and potentially improved resource efficiency for quantum algorithms. Key algorithms like Deutsch-Jozsa and Bernstein-Vazirani can be generalized for these systems, offering new capabilities for function analysis and phase estimation. In a quaternary (d=4) quantum system implementing generalized phase estimation, which advantage is achieved over binary (qubit) systems when measuring the eigenphase of a randomised gate? 1) The ability to achieve higher gate fidelity without error correction 2) Reduced decoherence due to shorter gate times 3) Elimination of the need for photon coincidence counting 4) Increased computational speed by removing the oracle query 5) Measurement in a single algorithmic interaction regardless of system dimension 6) Automatic correction of Poissonian statistical uncertainties 7) Achieving the same computational accuracy with fewer algorithmic interactions due to higher-dimensional encoding

✓ Correct Answer:

The correct answer is 7) Achieving the same computational accuracy with fewer algorithmic interactions due to higher-dimensional encoding.

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The algorithm iteratively computes all mdits of the eigenphase backwardly, in which, notably, each dit is once estimated with d-ary accuracy. Figure 5b–d report measured eigenphases of 4-dimensional unitary matrices by quaternary phase estimation. We estimated the four eigenphases for three logic gates, i.e., a phase gate Z4, a Fourier gate F4Fig. 4 Implementations of generalised Deutsch-Jozsa and Bernstein- Vazirani algorithms in quaternary. a Quantum logical circuit for implementing the d-ary Deutsch-Jozsa and Bernstein-Vazirani algorithms. This circuit can be implemented by the scheme in Fig. 1a, b with an exchange of the xand yregisters. The task of the d-ary Deutsch-Jozsa algorithm is to determine an unknown multi-value function f: {0, 1,..., d−1} n→{0, 1,..., d−1} is either constant or balanced, while that of the d-ary Bernstein-Vazirani algorithm is to compute the close expression of a multi- value af ﬁne function f:A0⊕A1x1...⊕Anxn, using only a single call of quantum oracle. When dequals to 2, the two algorithms return to the original Deutsch ’s algorithms. The key part is the implementation of f(x)⊕dyby the MVCU gate. The outcome of the algorithms is measured in the computation basis of the x-register states. b–iMeasured probability distributions (normalised coincidence counts) of the x-register in the computational basis. Results in b–hdemonstrate that the d-ary Deutsch- Jozsa algorithm allows the determination of whether f(x) is constant ( b)o r balanced ( c–h). Results in b,c,i,hshow the d-ary Bernstein-Vazirani algorithm can determine the expression of af ﬁne functions f:b,f(x)i s constant and A1=0;c,f(x)i sa f ﬁne and A1=1;i,f(x)i sa f ﬁne and A1=2; h,f(x)i sa f ﬁne and A1=3; Dotted boxes in (b--i) refer to theoretical probability distributions. Experimental probability distributions (coloured bars) are obtained from photon coincidences, which are accumulated by 20s per measurement. The classical ﬁdelity Fcpresents the success probability of each measurement. In order to make the small error bars visible in the plots, they are plot by ± 3 σ. The values in parentheses refer to ± 1 σuncertainty. All error bars are estimated from photon Poissonian statistics.ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-28767-x 6 NATURE COMMUNICATIONS | (2022) 13:1166 | https://doi.org/10.1038/s41467-022-28767-x | www.nature.com/naturecommunications and a randomised gate Urandom (see their explicit forms in Supplementary). Each pie chart presents one dit measurement outcomes, and the area of each coloured sector denotes measured probability distributions in the computational basis of {0ji,1ji,2ji,3ji}, respectively. In Fig. 5b, c, the eigenphases of Z4 andF4gates can be described by a single dit. Figure 5ds h o w st h e computed eigenphases of the Urandom gate with an accuracy of 12 dits, by running the algorithm with a number of 12 interactions on thed- Q P U .I n s t e a d ,i nt h eq u b i t - b a s e dd e v i c e ,a c h i e v i n gt h es a m ecomputational accuracy of ± 4−12requires a number of 24 computational interactions. And t he achieved computational accura- cies of 12 quarts are suf ﬁcient for

Question 55 multiple-choice

Quantum Markov semigroups (QMS) are fundamental tools in the study of open quantum systems and their dynamics, often analyzed using advanced functional analytic and operator algebra techniques. The interplay between group symmetries and non-commutative functional spaces is central to transferring classical analytical results to the quantum domain. Which operator space is specifically required when estimating properties of non-primitive quantum Markov semigroups with non-trivial fixed-point algebras, rather than relying solely on Schatten Lp spaces? 1) Hilbert-Schmidt spaces 2) Trace-class operator spaces 3) Unconditioned Lp spaces 4) Conditioned or amalgamated Lp spaces 5) Banach lattice spaces 6) Self-adjoint operator spaces 7) Classical L∞ spaces

✓ Correct Answer:

The correct answer is 4) Conditioned or amalgamated Lp spaces.

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II.8.Unfortunately, it is not the decoherence time or any other interesting quantity for Litself which transfers to all the Ln, but the underlying group which gives the corresponding estimates. Thus, the choice of the group and the classical Markov semigroup on it are particularly important. III. N ONCOMMUTATIVE LpSPACES AND NORM TRANSFERENCE In this section we introduce the main conceptual ideas of this article that we called group transference techniques . These ideas and subsequent mathematical results are mostly contained in [6]. All the applications we study in this article are concerned with the properties of certain (non-commutative) functional Lp spaces. When studying primitive QMS, only the usual (normalized Schatten) Lpspaces are required. However, in our case we are interested in non-primitive QMS withnon-trivial ﬁxed-point algebra. As ﬁrst illustrated in [16], the relevant Lpspaces in this case are the conditioned or amalgamated Lpspaces. Furthermore, the transference techniques require to look at the ampliﬁcation of the classical semigroup (St)t≥0to the algebra L∞(G)⊗ B(H)(see Lemma II.1). This in turn makes it necessary to consider completely bounded version of the Lp spaces. All these notions are introduced in Section III-A. Section III-B is dedicated to the presentation of the transference techniques. We present them in a general framework, as we believe they can also be useful in other settings (see [6] for an other example of application in quantum information theory). Finally we specialize to QMS in Section III-C, where these transference techniques are applied to transfer estimates on the classical Markov kernel to the QMS. A.Lpnorms and entropies We are now going to introduce several Lpnorms and entropies related to von Neumann algebras. Although this may not be clear at ﬁrst sight, it turns out that many of them are just the sandwiched Rényi entropies [17], [18] in disguise, as we will clarify. In the following M is a ﬁnite von Neumann algebra and ∶M→Ca normalized normal, faithful, tracial state (i.e. (IH)=1). Let us recall the deﬁnition of the noncommutative Lp spaces via (15) /divides.alt0x/parallel.alt1Lp( )∶=[ (/divides.alt0x/divides.alt0p)]1/slash.leftp: (16) ThenLp(M; )≡Lp(M)is the completion of Mwith respect to this norm. Indeed, for 1≤p≤∞the space is a Banach space such that Lp(M; )∗=L^p(M; )holds for1 p+1 ^p=1and1≤p<∞. In this article we will focus on three types of von Neumann algebras: ●Our main example is M=Mm, the space of m×m matrices over the ﬁeld of complex numbers, and (x)≡ m(x)≡1 mTr(x). To keep the notations at a more abstract level, we shall most of the time refer to a ﬁnite dimensional Hilbert space Hand to the algebra of (bounded) linear operators B(H) and we denote by Lp(B(H))the corresponding non-commutative Lpspace. ●If(E;F;)is a probability space, where Fis a-algebra on the set Eanda probability distribution, then the set of bounded complex-valued functionM=L∞()is a von-Neumann algebra, ∶f/uni21A6E(f)is a normal, faithful and tracial state and the corresponding Lpspaces are the usual Lp(). ●The last key example in the transference principle is the algebra of bounded Mm-valued function on a probability space (E;F;),M=L∞(E;Mm), with trace given by ∶f/uni21A6/integral.dispE m(f(x))d(x): 7 For

Question 56 multiple-choice

Quantum attacks on code-based cryptosystems often leverage the Hidden Subgroup Problem (HSP) and strong Fourier sampling techniques. Understanding subgroup structure in permutation and linear groups is crucial for analyzing the quantum security of systems like McEliece. Which property of the subgroup relevant to McEliece-type cryptosystems ensures its indistinguishability by strong quantum Fourier sampling and thus underpins quantum resistance against HSP-based attacks? 1) Having both large size and large minimal degree relative to n and log n 2) Being abelian and cyclic 3) Consisting only of the identity permutation 4) Having a small automorphism group but a large scrambler group 5) Containing all possible permutations of code positions 6) Being generated by elements of order two 7) Having a determinant equal to one for all group members

✓ Correct Answer:

The correct answer is 1) Having both large size and large minimal degree relative to n and log n.

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LD|D2 |Sn|≤p(n)n2dn n!≤eO(√ n)n2dn nne−n≤eO(n)n(2d−1)n≤n−γn/2 for sufﬁciently large n, so long as γ<1−2d. By Theorem 1, DH≤4|H|2(e−αm+ n−γn). Theorem 3 generalizes Moore, Russell, and Schulman’s result [15] on strong Fourier sampling over Sn, which only applied in the case |H|=2. To relate our result to the results of Kempe et al. [9], observe that since log |Sn|=Θ(nlogn), the subgroup His indistinguishable by strong Fourier sampling if |H|2e−αm≤(nlogn)−ω(1)or, equiva- lently, if m≥(2/α)log|H|+ω(logn). 4.2 Applications to McEliece-type Cryptosystems Our main application of Theorem 1 is to show the limitations of strong Fourier sam- pling in attacking the McEliece-type cr yptosystems. Throughout this section, we ﬁx parameters n,k,qof a McEliece-type cryptosys tem, and ﬁx the underlying k×nmatrix Mof the system. Here, Mcan be a generator matrix or a parity check matrix of the q- ary linear code used in the cryptosystem. Note that the entries of Mare in a ﬁnite ﬁeld Fq/lscript⊃Fq(when Mis a generator matrix of a q-ary linear code, we must have /lscript=1). Recall that the canonical quantum attack ag ainst this McEliece cryptosystem in- volves the HSP over the wreath product group (GLk(Fq)×Sn)≀Z2; the hidden subgroup in this case is K=( (H0,s−1H0s),0)∪((H0s,s−1H0),1) (5) for some hidden element s∈GLk(Fq)×Sn. Here, H0is a subgroup of GLk(Fq)×Sn given by H0=/braceleftbig (A,P)∈GLk(Fq)×Sn|A−1MP=M/bracerightbig . (6) To understand the structure of the subgroup H0,w ed e ﬁ n et h e automorphism group of Mas Aut (M)def=/braceleftbig P∈Sn|SMP =Mfor some S∈GLk(Fq)/bracerightbig .Note that Aut (M)is a subgroup of the symmetric group Snand each element (A,P)∈H0must have P∈ Aut(M). This allows us to control the maximal normalized characters on Kthrough the minimal degree of Aut (M). Then applying Theorem 1, we show that Theorem 4. Assume qk2≤nanfor some constant 00is a constant. The proof of Theorem 4 follows the technical ideas discussed in the Introduction. The details can be found in [3]. Asqk2≤nan,w eh a v el o g/vextendsingle/vextendsingle(GLk(Fq)×Sn)≀Z2/vextendsingle/vextendsingle=O(logn!+logqk2)=O(nlogn). Hence, the subgroup Kis indistinguishable if |K|2e−δm≤(nlogn)−ω(1). The size of the subgroup Kis given by |K|=2|H0|2,a n d|H0|=|Aut(M)|×|Fix(M)|,w h e r e McEliece and Niederreiter Cryptosystems 773 Fix(M)def=/braceleftbig S∈GLk(Fq)|SM=M/bracerightbig is the set of scramblers ﬁxing M. To bound the size of Fix (M), we record an easy fact which can be obtained by the orbit-stabilizer formula: Fact. Letrbe the column rank of M.T h e n|Fix(M)|≤q/lscriptk(k−r). Proof. WLOG, assume the ﬁrst rcolumns of MareFq/lscript-linearly independent, and each remaining column is an Fq/lscript-linear combination of the ﬁrst rcolumns. Let Nbe the k×r matrix consisting of the ﬁrst rcolumns of M. Then we can decompose MasM=(N| NA),w h e r e Ais an r×(n−r)matrix with entries in Fq/lscript. Clearly, Fix (M)=Fix(N). Consider the action of GLk(Fq/lscript)on the set of k×rmatrices over Fq/lscript. Under this action, the stabilizer of Ncontains Fix (N), and the orbit of the matrix N, denoted Orb (N), consists of all k×rmatrices over Fq/lscriptwhose columns are Fq/lscript-linearly independent. Thus,|Orb(N)|=(q/lscriptk−1)(q/lscriptk−q/lscript)...(q/lscriptk−q/lscript(r−1)).

Question 57 multiple-choice

Topological quantum computing utilizes the braiding of anyons to perform fault-tolerant quantum operations, with algebraic structures like braid groups and the Temperley-Lieb algebra playing a central role in gate construction and entanglement generation. The ability to directly create multipartite entangled states enhances the efficiency of quantum circuit design. Which of the following statements best describes the significance of constructing braiding operators based on the Temperley-Lieb algebra for multi-qubit quantum systems? 1) They restrict quantum gate implementation to only two-qubit operations. 2) They eliminate the use of single-qubit gates in entanglement generation. 3) They generalize braiding operations to enable direct creation of multi-qubit entangled states from separable bases. 4) They decrease the fault-tolerance of topological quantum computers. 5) They replace the braid group theory with a purely statistical mechanical approach. 6) They are limited to generating only Bell states, not GHZ or cluster states. 7) They require classical information storage for maintaining topological protection.

✓ Correct Answer:

The correct answer is 3) They generalize braiding operations to enable direct creation of multi-qubit entangled states from separable bases..

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Question 58 multiple-choice

Quantum computing leverages principles from quantum mechanics to process information in ways fundamentally different from classical computing. Understanding the roles of quantum gates and physical implementation challenges is key to advancing the field. Which experimental setup is commonly used to demonstrate quantum interference and implement quantum logic gates using photon polarization? 1) Josephson junction 2) Mach-Zehnder interferometer 3) Bell test apparatus 4) Quantum dot array 5) Ion trap 6) Fabry-Pérot cavity 7) SQUID (Superconducting Quantum Interference Device)

✓ Correct Answer:

The correct answer is 2) Mach-Zehnder interferometer.

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Question 59 multiple-choice

The quantum Fourier transform (QFT) is an essential operation in quantum computing, especially for algorithms that exploit the structure of cyclic groups. Understanding performance bounds and resource requirements for QFT is crucial for quantum algorithm design and practical implementation. Which statement most accurately describes a key advantage of having explicit bounds on the number of qubits required for a given tolerance in quantum Fourier transform algorithms over cyclic groups? 1) It allows the QFT to be replaced with classical algorithms in all cases. 2) It ensures that the QFT can be performed without any noise or errors. 3) It guarantees exponential speedup for all problems using QFT. 4) It removes the necessity for simulation before hardware deployment. 5) It enables heuristic estimation of resource requirements rather than exact quantification. 6) It allows researchers and practitioners to precisely optimize quantum hardware usage for desired algorithmic precision. 7) It restricts the applicability of QFT to only large quantum computers.

✓ Correct Answer:

The correct answer is 6) It allows researchers and practitioners to precisely optimize quantum hardware usage for desired algorithmic precision..

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Question 60 multiple-choice

In graph theory and quantum computing, decomposing complex graphs into simpler structures like cliques allows for efficient analysis of Hamiltonians that govern quantum interactions. Specialized algorithms use hierarchical tree structures to systematically break down these graphs. Which statement best describes the recursive rule for computing the Hamiltonian at a vertex in a tree-clique decomposition of a graph? 1) The Hamiltonian is equal to the sum of Hamiltonians of its children plus the edge Hamiltonians incident to the vertex. 2) The Hamiltonian is obtained by multiplying the complete subgraph Hamiltonian by the number of tree leaves below the vertex. 3) The Hamiltonian is given by the union of Hamiltonians from all ancestors of the vertex in the decomposition tree. 4) The Hamiltonian excludes all interactions involving vertices outside the immediate subgraph of the node. 5) The Hamiltonian at each vertex is computed by taking the intersection of clique Hamiltonians from its children. 6) The Hamiltonian equals the complete graph Hamiltonian on the vertex's associated subgraph minus the sum of Hamiltonians of its children. 7) The Hamiltonian for each node is constant regardless of its position in the tree.

✓ Correct Answer:

The correct answer is 6) The Hamiltonian equals the complete graph Hamiltonian on the vertex's associated subgraph minus the sum of Hamiltonians of its children..

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of the algorithm first note that an n-vertex graph can be decomposed into its connected components in time O(n2) (using depth first or breadth first search). This is the dominant step in the inner loop of the algorithm, so the algorithm takes time at most O(|V(G(w))|2) to complete the inner loop that removes vertex wfrom activeTleaves . Next, note that every vertex in T Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 52 appears in activeTleaves at most once. Then, letting ddenote the depth of T, the total runtime of the algorithm is bounded by: O /summationdisplay w∈V(T)|V(G(w))|2 =O d/summationdisplay i=1/summationdisplay w:depth(w)=i|V(G(w))|2  (6.16) ≤O/parenleftiggd/summationdisplay i=1n2/parenrightigg ≤O(n3), (6.17) where the first inequality follows from Item (2) of Theorem 6.3 and the second inequality follows from Item (1). 6.2.2 Clique decompositions and Hamiltonians Finally we show how the tree graph decomposition can be used to write the QMC Hamil- tonian associated with a graph Gas a signed sum of clique Hamiltonians and smaller “residual” graph Hamiltonians. Theorem 6.5. LetGbe a graph and T(G) ={T,{G(v1),...,G (vn)}}be the tree-clique decomposition of G. Also, for any graph G, letK(G)denote the complete graph on the vertices ofG. Then the following claims hold: (1)For any vertex v∈Twith children c1,...,ckwe have HG(v)=HK(G(v))−/summationdisplay j∈{1,...,k}HG(cj) (6.18) (2)LetLdenote the set of leaf vertices in T, andRbe all non-leaf vertices. Also, for any vertex v∈T, letd(v)denote the depth of vertex vin the the tree, with the root vertex having depth d(v1) = 0 . Then we have HG=/summationdisplay r∈R(−1)d(r)HK(G(r))+/summationdisplay l∈L(−1)d(l)HG(l) (6.19) Proof. Part (1) of the theorem follows from Item (2) in Definition 6.2 which gives G(v)c=/uniondisplay j∈{1,...,k}G(cj) (6.20) and hence HG(v)=HK(G(v))−HG(v)c=HK(G(v))−/summationdisplay j∈{1,...,k}HG(cj). (6.21) Part (2) then follows from repeated application of (1), beginning with the root vertex ofT. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 53 6.2.3 Example of clique decomposition, Algorithm 6 Before giving an explicit example of Algorithm 6 we note any complete k-partite graph has a particularly simple tree clique decomposition which Algorithm 6 finds in one step. This is because a complete k-partite graph is by definition the complement of the disjoint union ofkdifferent complete graphs; these are the output of the algorithm. Now we give an explicit illustration of Algorithm 6 on a graph which is not a complete k-partite graph. 78 9103 2 1 6 54 Figure 1: This is the input graph to the algorithm. The outputtree Tis in Section 6.2.3 and the graphs G(vj) are in Figure 3. v1 v2 v5 v3 v4 Figure 2: This is the output Tfrom Algorithm 6 run on the graph in Figure 1. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 54 78 9103 2 1 6 54 G(v1)78 93 2 1 6 54 G(v2)3 2 1 6 54 G(v3)77 78 9 G(v4)77 10 G(v5) Figure 3: This is the list of graphs which are output. The leaf graph, the graphs on which the algorithm terminates, correspond to v3,v4, andv5. The first of these is a

Question 61 multiple-choice

Matrix product states (MPS) are powerful computational tools for simulating quantum many-body systems and have been increasingly applied to lattice gauge theories. In these simulations, truncating the Hilbert space to a finite set of gauge group representations is a crucial practical step. In the context of simulating the Schwinger model with MPS methods under a uniform electric background field, why is it valid to truncate the Hilbert space of gauge fields to a finite number of irreducible representations? 1) The weight of each representation in the ground state decays exponentially with its quadratic Casimir invariant, making higher representations negligible. 2) Only the lowest representation contributes to the physical properties of the model. 3) All gauge group representations have equal contribution in the ground state, making truncation arbitrary. 4) The gauge fields in the Schwinger model are inherently finite-dimensional. 5) Truncation is always valid for any gauge theory regardless of the background field. 6) The electric background field forces all higher representations to vanish. 7) The single-particle spectrum remains unchanged even without truncation.

✓ Correct Answer:

The correct answer is 1) The weight of each representation in the ground state decays exponentially with its quadratic Casimir invariant, making higher representations negligible..

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Question 62 multiple-choice

Quantum groups generalize classical Lie groups by introducing a deformation parameter q, leading to non-commutative algebraic structures and novel representation theory. The quantum universal enveloping algebra Uq(SL(n)) and related constructs play a central role in mathematical physics and non-commutative geometry. Which of the following statements about the quantum determinant and its role in the structure of SLq(2) is correct? 1) The quantum determinant in SLq(2) is defined identically to the classical determinant, without q-dependent coefficients. 2) In SLq(2), the matrix constructed from τF and its derivatives always belongs to the quantum group GLq(2). 3) The quantum determinant det_q always commutes with all elements of SLq(2). 4) The classical group identities (g-identities) are directly applicable without adjustment in the quantum group setting. 5) The quantum determinant for SLq(2) does not involve operator ordering or q-dependent terms. 6) The quantum determinant det_q in SLq(2) is formulated using q-dependent coefficients and specific operator ordering to maintain invariance in the non-commutative setting. 7) Matrices constructed from τF and its derivatives in SLq(2) always satisfy commutative multiplication properties.

✓ Correct Answer:

The correct answer is 6) The quantum determinant det_q in SLq(2) is formulated using q-dependent coefficients and specific operator ordering to maintain invariance in the non-commutative setting..

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restrict ourselves to the case of G=SLq(2) and introduce: H1 1=τF=a+b¯t+ct+dt¯t. (98) If H1 2=D¯tH1 1=b+dt, H2 1=DtH1 1=c+d¯t, H2 2=D¯tDtH1 1=d,(99) we see that Ha bis actually not lying in GLq(2) (for example, H1 2H2 1/ne}ationslash=H2 1H1 2), i.e. a matrix consisting of the τFand its derivatives, despite these are all elements of A(G), does not longer belong toGq. Thus, it is not reasonable to consider det qH(or the deﬁnition of Hshould be somehow modiﬁed). Instead the appropriate formula for the case ofSLq(2) looks like τF(2)= detqg=H1 1H2 2−qH1 2M− ¯tH2 1=τFDtD¯tτF−qD¯tτFM− tDtτF.(100) We shall not go into more discussion of transition from g-identity to H-identities, because it involves some art in the work with appropriate time-variables, and is not yet brought to a reasonably simple form. Instead we present a few more formulas, w hich can be illuminating for some readers. 5.3 Comments on the quantum case The ﬁrst thing we wish to give some more details on is the statement (9 3). The Lie algebra SL(n) is generated by operators T±αand Cartan operators Hβ, such that [Hβ,T±α] =±1 2(αβ)T±α. All elements of all representations are eigenfunctions of Hβ, 26 Hβ|λ/an}bracketri}ht=1 2(βλ)|λ/an}bracketri}ht. The highest weight of representation F(k)isµk. Vectors µk’s are “dual” to thesimplerootsαi,i= 1,...,r: (µiαj) =δij, andρ=1 2/summationtext α>0α=/summationtext iµi. Representation F(1)consists of the states ψi=T−(i−1)...T−2T−1ψ1, i= 1,...,n. (101) Moreover T−iψj=δijψi+1 (102) (thus, forT−=/summationtextr i=1T−i Ti −ψj=ψj+iand (60) follows), and λ(ψi) =µ1−α1−...−αi1. (103) HereT±i≡T±αiare generators, associated with the simple roots. Let us denote t he corre- sponding basis in Cartan algebra Hi=Hαi, andHi|λ/an}bracketri}ht=1 2(αiλ)|λ/an}bracketri}ht=λi|λ/an}bracketri}ht. Then λ(j) i≡λi(ψj) =1 2(δij−δi,j−1). (104) This formula, together with (101) and (105) implies that ||ψi||2= 1, and, since comultiplica- tion formula in the classical case is just ∆( T) =T⊗I+I⊗T, it is obvious that ψ[1...ψk]are all highest weight vectors (i.e are annihilated by all ∆ k(T+i) and, thus by all the ∆ k(T+α)). Quantum universal enveloping algebra Uq(SL(n)) is generated by T±iandq±Hiwith basic commutation relations ( aijis the Cartan matrix of SL(n)) qHiT±jq−Hi=q±aijT±j, [T+i,T−j] =δijq2Hi−q−2Hi q−q−1(105) and comultiplication law ∆(T±i) =qHi⊗T±i+T±i⊗q−Hi, ∆(q±Hi) =q±Hi⊗q±Hi.(106) Comultiplication formulas for T±αin the case of non-simple roots are corollaries of these and look more sophisticated. For example, for the “height 2” α, such that T±α=±[T±αi,T±αi+1], 27 we have ∆(T−α) =−[∆(Tαi),∆(Tαi+1)] =qHα⊗Tα+Tα⊗q−Hα+ +(q1/2−q−1/2)/bracketleftig (T−αi⊗T−αi+1)(qHi+1⊗q−Hi)−(T−αi+1⊗T−αi)(qHi⊗q−Hi+1)/bracketrightig .(107) Given multiplication formulas, one can easily check that indeed (93) is t rue. For example, forF(2): ∆(T+i)(ψ1ψ2−qψ2ψ1) =δi,1(qλ(1) 1ψ1ψ1−q1−λ(1) 1ψ1ψ1) = 0, (108) becauseλ(1) 1=1 2. Thus Ψ(2) 12≡ψ[1ψ2]qis indeed the highest weight vector. Similarly ( i

Question 63 multiple-choice

The Yang-Baxter equation plays a central role in mathematical physics and group theory, with set-theoretic solutions linked to specific algebraic structures. Classifying and constructing these solutions involves interpreting connections between groups and permutation symmetries. Which property precisely characterizes all involutive Yang-Baxter groups (IYB groups) identified through the correspondence with involutive non-degenerate set-theoretic solutions to the Yang-Baxter equation? 1) They are always simple non-abelian groups. 2) They are necessarily nilpotent groups of class one. 3) They contain a unique subgroup isomorphic to the free abelian group Faₙ. 4) They are precisely cyclic groups of prime order. 5) They must be infinite and non-solvable. 6) They are defined only for abelian groups. 7) They are all solvable groups.

✓ Correct Answer:

The correct answer is 7) They are all solvable groups..

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Question 64 multiple-choice

In multipartite quantum systems, the classification and construction of entangled states under local operations is fundamental for quantum information processing. Locally maximally entangled (LME) states, which generalize Bell states, play a crucial role in this domain. Which defining property distinguishes a locally maximally entangled (LME) state in a composite quantum system? 1) Each subsystem's reduced density matrix is proportional to the identity operator. 2) The global state is invariant under all local unitary transformations. 3) The state can be transformed into any other state by local operations and classical communication. 4) Each subsystem is in a pure quantum state. 5) The state maximizes the global entropy of the composite system. 6) The state is an eigenstate of the total Hamiltonian. 7) Each subsystem's reduced density matrix is diagonal with non-equal eigenvalues.

✓ Correct Answer:

The correct answer is 1) Each subsystem's reduced density matrix is proportional to the identity operator..

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. . . . . 23 4.1.1 Entanglement structure and the action of K= SU(d1)×···×SU(dn) 23 4.1.2 SLOCC orbits and the action of G= SL(d1,C)×···×SL(dn,C) . . . 23 4.2SLME/Kas a symplectic manifold . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3SLME/Kas a complex manifold . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.4 Gradient ﬂow to LME states . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7 4.5 Algebraic characterization of P(H)//G. . . . . . . . . . . . . . . . . . . . . 28 5 Understanding SLMEusing geometric invariant theory 31 5.1 Proof of theorems governing existence and dimension of SLME. . . . . . . . 33 5.2 Explicit results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1 A Explicit construction of all LME states for (d1,d2,d3) = (2,B,C) 39 A.1 Sudoku States for n= 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 B Representation theory construction of LME states for (d1,d2,d3) = (2,p,p) with prime p 44 B.1 Irreducible representations of UT(3,p) . . . . . . . . . . . . . . . . . . . . . 44 B.2 The group H=UT(3,p)⋊φZ2. . . . . . . . . . . . . . . . . . . . . . . . . 45 B.3 Irreducible representations of UT(3,p)⋊Z2. . . . . . . . . . . . . . . . . . 47 C Multipart systems for which the generic state has trivial s tabilizer 49 1 Introduction Consider a multipart quantum system whose pure states are vecto rs in a tensor product Hilbert space H=H1⊗···⊗H n with subsystems Hiof dimension di. We will say that a state in His locally maximally entangled (LME) if the reduced density matrix corresponding to ea ch elementary subsystem is a scalar multiple of the identity operator on that subsystem.1Fixing the dimension vector (d1,...,dn), we deﬁne SLME⊂ Hto be the subset of states which are locally maximally entangled: SLME=/braceleftbigg |Ψ/an}b∇acket∇i}ht ∈ H/vextendsingle/vextendsingle/vextendsingleρi≡tr¯i|Ψ/an}b∇acket∇i}ht/an}b∇acketle{tΨ|=1 di1 1/bracerightbigg . (1.1) LMEstates play animportant roleinmany applicationsof quantum mec hanics andquantum information theory. Their importance has been pointed out by many authors in the past, for example [Kly02, VDD03, Sc04]. Well-known examples of LME states include Bell states |Ψ/an}b∇acket∇i}ht=1√ d/summationdisplay i|i/an}b∇acket∇i}ht⊗|i/an}b∇acket∇i}ht ∈ Hd⊗Hd, (1.2) 1Such states are sometimes

Question 65 multiple-choice

Infinite abelian group theory explores properties of groups such as slenderness, torsion, and divisibility, with important connections to algebraic topology and set theory. The Baer–Specker group and p-adic numbers play central roles in classifying torsion-free abelian groups of rank one. Which criterion precisely characterizes a torsion-free abelian group of cardinality less than the continuum as slender? 1) It has only finite-rank free subgroups. 2) It has no non-trivial divisible subgroups. 3) It is isomorphic to a subgroup of the p-adic integers for some prime p. 4) Every homomorphism from the Baer–Specker group into it is injective. 5) It contains a copy of the Baer–Specker group as a subgroup. 6) It is generated by finitely many elements. 7) Its endomorphism ring is a discrete valuation ring.

✓ Correct Answer:

The correct answer is 2) It has no non-trivial divisible subgroups..

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Question 66 multiple-choice

Quantum computing devices often impose restrictions on which qubits can directly interact, necessitating circuit optimizations for efficient implementation of algorithms. The Quantum Fourier Transform (QFT) is a core subroutine in many quantum algorithms, and its gate count is a crucial factor on noisy intermediate-scale quantum (NISQ) hardware. When designing QFT circuits for quantum computers with linear nearest-neighbor connectivity, which outcome is most directly achieved by optimizing the circuit for the hardware's connectivity constraints? 1) Increased robustness against decoherence by using more Toffoli gates 2) Enhanced algorithmic speed by parallelizing all one-qubit gates 3) Higher accuracy in phase estimation due to longer circuit depth 4) Substantial reduction in the total number of required CNOT gates 5) Improved entanglement of non-adjacent qubits without additional gates 6) Elimination of noise in single-qubit measurement operations 7) Automatic error correction of all two-qubit gates

✓ Correct Answer:

The correct answer is 4) Substantial reduction in the total number of required CNOT gates.

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a transpiler23 to transform an input circuit into a circuit that satisfies the specific NN con- dition, which is required in each IBM quantum computer. In this section, we put our QFT circuits and the conventional QFT circuits (such as the circuit in Fig. 1) in the Qiskit transpiler for implementation on IBM quantum computers: (1) IBM_Nairobi, a 7-qubit quantum computer using the Falcon r5.11H processor, (2) IBMQ_Guadalupe, a 16-qubit quantum computer using the Falcon r4P processor, (3) IBM_Cairo, a 27-qubit quantum computer using the Falcon r5.11 processor, and (4) IBM_Washington, a 127-qubit quantum computer using the Eagle r1 processor10. We transpiled 3- to 7-qubit QFT on the IBM_Nairobi, 3- to 16-qubit QFT on the IBMQ_Guadalupe, 3- to 27-qubit QFT on the IBMQ_Cairo, and 3- to 127-qubit QFT on the IBM_Washington. Each QFT circuit is transpiled 100 times. Next, we chose the minimal number of CNOT gates required to syn - thesize the QFT and compared them. As a result, we confirmed that using our QFT circuit as input requires fewer CNOT gates than using the conventional QFT circuit for all cases. The results can be found in Fig. 8. Implementation of QFT on actual quantum hardware. We implemented QPE using a 3-qubit QFT on the IBM_Nairobi10 and the Rigetti-Aspen-1111, a 40-qubit superconducting quantum computer, to compare their performance. The connectivity between qubits used for the implementation of QPE can be found in Fig. 7. QPE is an algorithm for finding an eigenvalue of a unitary operator using a corresponding eigenstate and QFT. A brief explanation of QPE can be found in the “ Background ” section. In this study, we chose the unitary operator U and the corresponding eigenvector |u� as follows: We chose θ as 1/8, 2/8, 3/8, …, and 7/8. The QPE circuits are synthesized using our method. If we use a quantum computer without noise when implementing QPE, we can get the right results with one execution for each θ . However, the quantum computers we used are noisy. Therefore, we implemented QPE 1000 times for each θ on each quantum computer.(11) U=/parenleftbigg10 0e2πiθ/parenrightbigg ,|u�=/parenleftbigg 0 1/parenrightbiggTable 1. The number of CNOT gates in QFT circuits for LNN architecture. The first column represents the number of qubits in the QFT circuit, the second column represents our results, the third to the fifth columns represent the results of previous studies14,17,18, and the sixth column represents the improvement rate of our circuit compared to the best-known result.Ours Ref.14Ref.17Ref.18Improvement (%) n n2 + n − 4 (5/2)(n2 − 1) – – ~ 60 5 26 50 31 – ~ 16.13 6 38 75 48 – ~ 20.83 7 52 105 105 75 ~ 30.67 8 68 140 124 121 ~ 43.80 9 86 180 192 165 ~ 47.88 10 106 225 240 225 ~ 52.89 Figure 7. Qubit connectivity of quantum devices (a ) Circuit diagram of IBM_Nairobi10, showing the connectivity of qubits. Qubits labeled 1, 3, 5, and 4 are used to implement QPE. (b ) Partial circuit

Question 67 multiple-choice

In quantum field theory, the interaction between Dirac fermions and non-Abelian monopoles gives rise to novel symmetry structures and operator properties. The mathematical framework often involves complex groups and specialized operators that influence the classification of quantum states. Which statement best characterizes the role of the matrix F when the parameter A is real in a system with Dirac fermion doublets and non-Abelian monopole potential? 1) F generates time-reversal symmetry and always leaves the Hamiltonian invariant. 2) F belongs to SU(2) and acts as a symmetry generator for the Hamiltonian by transforming basis wave functions. 3) F is a projection operator onto orthogonal quantum states for all values of A. 4) F represents charge conjugation and maps fermions to antifermions exclusively. 5) F becomes non-unitary and fails to preserve inner products when A is real. 6) F implements parity inversion and does not affect the Hilbert space structure. 7) F diagonalizes the N operator for both real and complex values of A.

✓ Correct Answer:

The correct answer is 2) F belongs to SU(2) and acts as a symmetry generator for the Hamiltonian by transforming basis wave functions..

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Question 68 multiple-choice

Quantum annealing and Ising Hamiltonian models are central to applying quantum computing for hard optimization problems, including integer factorization and cryptanalysis. These techniques rely on encoding mathematical problems into spin systems compatible with quantum hardware. Which process allows the mapping of a classical integer factorization problem into an Ising Hamiltonian suitable for quantum annealing, enabling solutions via quantum hardware? 1) Representing integers as binary strings and using classical brute-force search 2) Applying modular exponentiation to transform the factorization constraints 3) Renaming problem variables and mapping them to the spin domain {-1, 1}, rewriting the cost function in terms of local fields and couplings 4) Utilizing Grover's algorithm to sample possible factors 5) Encoding the problem as a quantum circuit using universal gates 6) Reducing factorization to a discrete logarithm problem in a cyclic group 7) Implementing lattice-based cryptographic schemes for enhanced security

✓ Correct Answer:

The correct answer is 3) Renaming problem variables and mapping them to the spin domain {-1, 1}, rewriting the cost function in terms of local fields and couplings.

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respectively. Further, rename the variables p1,p2,q1,q2,c1,..., c4,t1,..., t4asv1,...,v 12, and replace vi=1−si 2(i=1,..., 12) to make the variables lie in the domain {−1,1}. Now, the above cost function fcan be rewritten as f(p1,p2,q1,q2,c1,c2,c3,c4)=2f/prime(s1,..., s12) where f/primeis given in Fig. 3. •Step 5 Now, we can review the above cost function as an Ising Hamiltonian with local ﬁelds, and the values of hiand Jijcan be derived accordingly (See Fig. 3for details). •Step 6 Solve the Ising Hamiltonian system by calling qbsolv , the Python library provided by D-Wave systems, and map the results back to the prime factorization ofN. To summary, the qubits needed for different implementations of quantum factor- ization based on QAC are shown in Table 3. Table 3 Different implementations of quantum factorization based on QAC Authors Year The largest integer can be factorized Qubits Li et al. [ 40] 2018 291,311 – Jang et al. [ 41] 2018 376,298 94 Wang et al. [ 33] 2019 1,005,973 89 Comparison of the largest integer can be factorized 123 Quantum algorithms for typical hard problems… Page 11 of 26 178 f(s1,s2,··· ,s12)=261 2s1+215 2s2+261 2s3+215 2s4−41s5−82s6+3 s7+6 s8−137 s9−81s10−107 s11−81s12 +2s1s2+7 9 s1s3+95 2s1s4−2s1s5−4s1s6−8s1s7−16s1s8−148 s1s9−84s1s10 +95 2s2s3+7 1 s2s4−8s2s5+−16s2s6+s2s7+2 s2s8+6 s2s9+6 s2s10−124 s2s11−84s2s12 +2s3s4−2s3s5−4s3s6−8s3s7−16s3s8−148 s3s9−84s3s12 −8s4s5−16s4s6+s4s7+2 s4s8+6 s4s9−84s4s10−124 s4s11+6 s4s12 +34 s5s6−4s5s7−8s5s8−8s5s9+s5s10+2 s5s11+s5s12 −8s6s7−16s6s8−16s6s9+2 s6s10+4 s6s11+2 s6s12 +34 s7s8−4s7s10−8s7s11−4s7s12 −8s8s1016s8s118s8s12 +s9s11 +794 hT=( σ(1) z,··· ,σ(12) z) = (130 .5,107 .5,130 .5,107 .5,−41,−82,3,6,−137 ,−81,−107 ,−81) J=⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝σ (1) z σ(2) z σ(3) z σ(4) z σ(5) z σ(6) z σ(7) z σ(8) z σ(9) z σ(10) z σ(11) z σ(12) z σ(1) z 02 7 9 4 7 .5 −2 −4 −8 −16 −148 −84 0 0 σ(2) z 04 7 .57 1 −8 −16 1 2 6 6 −124 −84 σ(3) z 02 −2 −4 −8 −16 −148 0 0 −84 σ(4) z 0 −8 −16 1 2 6 −84 −124 6 σ(5) z 03 4 −4 −8 −81 2 1 σ(6) z 0 −8 −16 −16 2 4 2 σ(7) z 03 4 0 −4 −8 −4 σ(8) z 00 −8 −16 −8 σ(9) z 0010 σ(10) z 000 σ(11) z 00 σ(12) z 0⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ Fig. 3 Ising Hamiltonian system for factoring 143 5 Quantum algorithms for discrete logarithmic problems LetG=/angbracketleftg/angbracketrightbe a cyclic group of order pand gbe a generator of G. The discrete logarithmic problem (DLP) over Gis to ﬁnd an integer rsuch that gr=yfor given y∈G. Difﬁe–Hellman key exchange protocol, ElGamal encryption and most elliptic curve cryptosystems are based on the difﬁculty of computing discrete logarithms. Atpresent, the best known classical algorithm for solving DLP is the so-called index- calculate method (ICM) that requires sub-exponential classical operations. In 1994, Shor [ 3] put forward a polynomial time quantum algorithm to solve the discrete logarithmic problem in group. Proos et al. [ 42] further extended Shor’s quan- tum DLP algorithm to elliptic curves [ 42–44]. In 2012, Myasnikov et al. proposed a quantum algorithm for the DLP over matrices of ﬁnite group rings [ 45].

Question 69 multiple-choice

Quantum error correction uses mathematical structures to protect quantum information against noise, often leveraging symmetries from group theory. Advanced code families can surpass traditional limitations by adapting subsystem dimensions and exploiting entanglement properties. Which statement correctly characterizes how infinite-dimensional quantum codes covariant with respect to Lie groups can evade the Eastin-Knill theorem? 1) They utilize classical error correction protocols to bypass symmetry restrictions. 2) They rely on randomized code constructions to approximate transversal gates for any unitary. 3) They use finite-dimensional qudit systems to achieve perfect error correction for continuous groups. 4) They encode logical information exclusively in product states to avoid group covariance constraints. 5) They exploit subsystem swapping to circumvent the need for group representations. 6) They allow continuous symmetry groups and perfect erasure correction by employing infinite-dimensional systems, making exact transversal gates possible. 7) They restrict code families only to discrete group symmetries without using infinite-dimensional subsystems.

✓ Correct Answer:

The correct answer is 6) They allow continuous symmetry groups and perfect erasure correction by employing infinite-dimensional systems, making exact transversal gates possible..

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subsystem dimension (Sec. VII). Leveraging tools from representation theory, we show that the lower bound on ϵworst becomes ϵworst≥1 2nmax ilndiþO/C181 ndL/C19 ; ð3Þ where diis the dimension of the ith physical subsystem. We also find lower bounds for the local subsystem dimension that depend on the number of logical qubits and the code ’s infidelity. This result also applies to the case when eachgate can be approximated with a discrete sequence oftransversal operations to arbitrary accuracy, as in the context of the Solovay-Kitaev theorem. Furthermore, using randomized constructions, we prove the existence of Uðd LÞ-covariant code families which achieve arbitrarily small infidelity in the limit of large subsystem dimension. In addition, we exhibit a simpleUðd LÞ-covariant code family, whose code words are generalized Wstates, such that ϵworst approaches zero as the number of subsystems napproaches infinity. D. Covariant codes from multiqubit codes We also construct codes that are covariant with respect to any group Gadmitting a Haar measure, where both the logical system and the physical subsystems transform as the regular representation of G(see Sec. VIII). In this con- struction, the dimension of each subsystem is the order jGj of the group when Gis finite and infinite when Gis a Lie group. Using this formalism, we construct natural generaliza- tions of well-known families of qubit codes, such as thebit-flip and phase-flip codes, with the qubits replaced by jGj-dimensional systems. These codes admit transversal logical gates representing each element of G. When Gis a Lie group, the qubits are replaced by infinite-dimensional systems such as rotors or oscillators. These infinite-dimensional codes circumvent the Eastin- Knill theorem —they are covariant with respect to a continuous symmetry group, yet erasure of a subsystemis perfectly correctable. III. SETUP AND NOTATION A. Approximate quantum error correction We define a code as a completely positive, trace- preserving map Ewhich assigns to each logical state on some abstract logical system La corresponding state on a physical system Aconsisting of nsubsystems A¼A 1⊗A2⊗/C1/C1/C1⊗An(Fig. 3). Unless explicitly stated otherwise, we consider in this paper codes that are isometric, i.e., thatmap any pure logical state jxi Lto a pure physical state fjψxiAg. The latter is called a code word . The span of all the code words fjψxiAgforms the code space . While typically encountered codes are isometric, it is useful in some instances to consider more general encoding maps such as encodings that symmetrize a state with a referenceframe [12]. The noise channel is the process to which the physical system is exposed, which might cause the informationencoded in it to get degraded. It is a quantum channel N A→Bmapping the physical system to physical system B. (The system Bmight be the same as A, but it might be different; for instance, Bmight include a register which remembers which type of error occurred or which sub- system was lost.) To study the approximate error-correction properties of a code, we need to quantify the approximation quality using distance measures between states and channels. Proximitybetween quantum states can be quantified

Question 70 multiple-choice

Matrix Lie groups are mathematical structures that combine group theory and differential geometry, appearing frequently in physics and engineering to model symmetries and continuous transformations. Compactness is a key property in the representation theory of Lie groups, ensuring mathematical tractability and physical relevance. Which property distinguishes compact matrix Lie groups from general matrix Lie groups, making their representation theory particularly "well behaved"? 1) Associativity of multiplication 2) Existence of an identity element 3) Being defined as a subgroup of GL(d, C) or GL(d, R) 4) Smooth manifold structure 5) Closure under group operations 6) Being both closed and bounded in the space of d×d matrices 7) Allowing infinite-dimensional representations

✓ Correct Answer:

The correct answer is 6) Being both closed and bounded in the space of d×d matrices.

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subgroupHof a groupGis a subset of Gwhich is also a group. Deﬁnition 3. Amatrix Lie group Gis a closed subgroup ofGL(d;C)(orGL(d;R)), where by closed we mean that ifAm2Gis a sequence of matrices with limm!1Am= A2GL(d;C), thenA2G. Matrix Lie groups have the key feature that they also formsmooth manifolds , or hypersurfaces. In practice, this means that we can parameterize matrix Lie groups using coordinates, and we can take derivatives along paths in the group just as one can compute tangent vectors along paths embedded in surfaces. The smooth manifold structure turns the potentially unwieldy problem of understanding groups with uncountably many elements into a tractable one: we get to use powerful tools from not just algebra, but analysis, geometry, and topology as well. It is worth mentioningthatingreatergenerality, aLiegroupisagroup Gthat is also a manifold, where the group multiplication and inversion are continuous operations. Indeed, there are Lie groups that are not matrix Lie groups, but most of the physically relevant examples like the unitary groups U(n), orthogonal groups O(n), or special linear groups SL(n) form matrix Lie groups, so we restrict our attention to this more concrete class.3We refer the reader to Box 14 at the end of this work where we present some important exam- ples of matrix Lie groups and their key properties. Now, ﬁnite discrete groups are “well behaved” in com- parison to their inﬁnite discrete cousins, in the sense that we can say many things about their representation the- ory: for instance, we know their representations are com- pletely reducible (Theorem 4), can always be converted into unitary representations (Theorem 5), and we know how many irreducible representations they have (Deﬁni- tion 13). While we do not deﬁne these terms now, they will be presented below, so do not worry if these do not make sense now. The analogous condition to yield “well behaved” matrix Lie groups is a sort of topological proxy for ﬁniteness: compactness . Compact matrix Lie groups will have the nicest representation theory, and luckily for quantum researchers, unitary groups are compact. Below we present a special version of the deﬁnition of compact- ness given by the Heine-Borel theorem. 3In fact, via a corollary of the Peter-Weyl theorem [59], every com- pact Lie group is a matrix Lie group. 9 Deﬁnition 4. A matrix Lie group Gis called compact if it is a closed and bounded subset of the vector space of the ddmatricesMd(C)(orMd(R)). It is useful to note that any putative Lie groups deﬁned byequationsandcontinuousoperations(likemultiplication or taking adjoints) can be readily shown to be closed: for instance, if we consider a sequence Umof unitary matri- ces converging to a matrix U, then since taking adjoints U7!Uyand multiplication U7!UyUare continuous op- erations, we can take the limit of the equation Uy mUm=1 asm!1to see that UyU=1, meaning the unitary group is closed. Boundedness can be readily checked from standard linear algebra knowledge: for instance, since the operator normkUk= 1, the unitary group is bounded. Thus, the unitary group U(d)(and consequently the spe- cial

Question 71 multiple-choice

In quantum computing, implementing gates such as the Quantum Fourier Transform (QFT) often involves combining controlled rotations and free evolution in pulse sequences. Optimizing these sequences is critical for achieving high-fidelity gate operations within minimal time. When synthesizing a QFT gate using radio frequency (rf) pulse sequences, which parameter choice ensures minimum total gate duration while maintaining positive evolution times for different global phase options? 1) Selecting the smallest possible rotation angles regardless of evolution time 2) Maximizing the amplitude of the rf pulses without optimizing time parameters 3) Using only single rf pulses and avoiding composite pulse techniques 4) Choosing global phases that result in negative evolution times 5) Ignoring direction cosines and optimizing only pulse amplitude 6) Setting all evolution times equal to zero for fast operation 7) Selecting solutions with positive evolution times t₁ and t₂ that minimize the sum Tₘ = t₁ + t₂ for each global phase

✓ Correct Answer:

The correct answer is 7) Selecting solutions with positive evolution times t₁ and t₂ that minimize the sum Tₘ = t₁ + t₂ for each global phase.

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y y x x r                    . (20) where { } exp( ) iI is the operator of non selective rotations by angle  around the axis  and exp( )t qit H   is the free evolution for time t. In the center of sequence (20), there is the rotation by the angle 22  around the axis with the direction cosines / and / along the axes x and z, respectively. The nonselective rotation s can be obtained using a simple or composite pulse of the rf field with a large amplitude [25]. Thus, to implemen t the QFT gate, it remains for us to determine the parameters of pulse sequence (20) by equating the sum of matrices (18) to (9). As a result , we obtain the system of equations 7               2216 12 1 12 2 2212 12 22112 1 2 1 2 2 22113 1 2 1 2 2 1 12 23cos 1 3cos 1 sin cos sin cos sin sin sin cos sin cos sin sin 3cos 1 3cos 1 sin cos sin cos sin cos sin cmmtt K E it t tt it t t t t t it t it t                                                   2216 12 os 3cos 1 3cos 1 tt              (21) The joint solution of these equations yields the desire d values of the parameter s (Table I). For each value of the global phase { / 6, 5 / 6, 9 / 6}    , we select the solution s with positive evolution times 1t and 2t that yield the minimum sum m 1 2T t t and one solution with time mT next in magnitude but with the same global phase . [If we neglect the rf pulse length in (20), the n the total sequence duration is m 1 2T t t ; this value is the minimum tim e for implementation of the QFT gate with the use of the method under consideration . Here, w e use the notation mT instead of cT , because this value is only a rough estimate of critical time cT determined in Section III]. TABLE I. Parameter for implementing eff mH (9) with the use of pulse sequenc e (20) m 1 2 3,,m m m     1t 2t mT /6 1, 0, 0 1, 1, -1 /2 /2 -0.905 -0.984 0.790 4.764 0.105 3.822 3.441 0.503 4.267 7.548 7.71 8.05 5

Question 72 multiple-choice

Quantum Fourier transforms (QFTs) play a central role in quantum algorithms for group-theoretic problems, with efficient circuit implementations crucial for practical applications. For certain non-abelian groups, such as the Heisenberg group Hp of order p³, specialized techniques are required to achieve efficient QFTs. Which of the following statements correctly describes a feature of the efficient quantum Fourier transform circuit construction for the Heisenberg group Hp? 1) The circuit leverages a subgroup tower approach, starting with the abelian normal subgroup N∞ ≅ Zp × Zp, and extends to the full group using a transversal and representation decomposition. 2) The circuit directly performs the QFT on the entire group without decomposing into subgroups or using representation theory. 3) The efficient circuit requires exponentially many quantum gates in p to implement the QFT for Hp. 4) The subgroup used in the construction is non-abelian and does not have a direct product structure. 5) The construction does not utilize binary string encoding for group elements in the quantum registers. 6) The QFT for Hp is performed by measurement-based quantum computation rather than unitary gates. 7) The representation decomposition is not needed for efficient implementation in non-abelian groups.

✓ Correct Answer:

The correct answer is 1) The circuit leverages a subgroup tower approach, starting with the abelian normal subgroup N∞ ≅ Zp × Zp, and extends to the full group using a transversal and representation decomposition..

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now weare doing ab elian Fourier sampling over anon-abelian group, wehavetoconsider theeffectofapplying Ftoacosetof tAi,j,wheret= (0,0,τ)andτ∈Fp. Note thattAi,j={(µ,µi,/parenleftbigµ 2/parenrightbig i+µj+τ) :µ∈Fp}. Weobtain F|tAi,j/an}bracketri}ht=1 p2/summationdisplay a,b,c∈Fpωcτ /summationdisplay µ∈Fpω(a+bi+cj−ci 2)µ+ciµ2 |a,b,c/an}bracketri}ht. Hence, the probability of observing a particular triple (a,b,c)isp−4|/summationtext µ∈Fpω(a+bi+cj−(ci)/2)µ+ciµ2|2. If c/ne}ationslash= 0, this is a quadratic Weil sum and we can use Fact 1 to conclude t hat the probability of observing (a,b,c)is given by p−3, independent of i,j. The probability of observing (a,b,c),c/ne}ationslash= 0is1−1 p. If c= 0, only terms of the form (−bi,b,0)show up. These terms do give information about i; however, the probability of observing such a term is1 p. Thus, the forgetful abelian method gives exponentially sm all information about i. 2.5 Efﬁcient quantum circuits forthe FourierTransform on Hp The fact that any QFTfor any ﬁnite group is a unitary matrix (when properly normal ized) makes this class of transformations an important source of transforma tions a quantum computer can carry out. The problem of ﬁnding efﬁcient implementations of QFTs interms of quantum circuits wasstudied previously, see [Høy97, Bea97, PRB99, HRTS03, MRR04]. From [MRR04, Theo rem 2] it follows that for any prime ptheQFTfor the Heisenberg group Hpcan by computed in polylog(p)operations. In the following we give an explicit description of an efﬁcient quantum circuit which computes QFTHp. First, note that we are interestedinarealizationonaquantumcomputer whichwork sonqubits. Thismeansthatwehavetoembed thestates and transformations intoaregister of size 2nfor somepositive integer n. Inthefollowing wewill assumethatnisthesmallestintegersuchthat p<2nandwewillidentifythegroupelements (x,y,z)∈ Hp withasubset ofthebinarystrings oflength 3n: ineachofthethreecomponents wechoosethebasisvectors |0/an}bracketri}ht,...,|p−1/an}bracketri}htto represent the respective component of the element (x,y,z). The following proposition shows that a QFTforHpcan be implemented efﬁciently in termsof elementary quantu m gates. Proposition 1 Letpbeprime,let HpbetheHeisenberggroupoforder p3andletIrr(Hp) ={χa,b: (a,b)∈ F2 p} ∪ {ρk:k= 1,...,p−1}denote the irreducible representations of Hp. Then the QFTforHpwith respect to Irr(Hp)can be computed using O(log3p)elementary quantum gates. Proof: First we consider the normal subgroup N∞⊳Hpand compute a Fourier transform for this abelian group. Thisgroupisisomorphictoadirectproduct oftwocyc licgroups, i.e., N∞∼=Zp×Zp. Theelements ofN∞are given by N∞={(0,y,z) :y,z∈Zp}, i.e., we can identify the elements of N∞with those binary strings of length 3nwhich have trivial support on the ﬁrst npositions. Note that the irreducible representations of N∞are given by ψa,bfora,b∈Zp, where ψa,b(0,y,z) := exp(2πi/p(ay+bz)) =ωay+bz p. SinceN∞isnormalthegroup Hpoperates ontheirreducible representations [CR62]. Weden otethisaction by “∗”, i.e., we have a map ∗:Hp×Irr(Hp)→Irr(Hp)which is explicitly given by (x,y,z)∗ψa,b= ψa,b−ax. Next, we choose as a transversal for N∞⊳Hpthe ordered list T= [(x,0,0) :x∈Zp]. We have to be able to efﬁciently implement the images of all irreduci ble representations of Hpevaluated at the 8 elements of T. This is required for the so-called ‘twiddle factors’ in the decomposition of QFTHpalong the subgroup tower {1}⊳N∞⊳Hp. Indeed, we construct a QFTadapted to this subgroup tower, see also [PRB99, MRR04]. We now use the following formula for imp lementing a QFTGwhich holds in the situation where wehave anabelian normal subgroup Nand an abelian factor group G/N: QFTHp=/parenleftbig 11|G/N|⊗QFTN/parenrightbig/parenleftBigg/circleplusdisplay t∈TΦ(t)/parenrightBigg/parenleftBig QFTG/N⊗11|N|/parenrightBig . HereΦdenotes an extension of the decomposition of the regular rep resentation of Ninto irreducibles. Denoting this direct sum by Λ, i.e.,Λ :=/circleplustext

Question 73 multiple-choice

The Hidden Subgroup Problem (HSP) plays a central role in quantum computing, as it underpins several problems where quantum algorithms can outperform classical ones. Recent research has investigated the extension of HSP beyond groups to more general algebraic structures. Which statement accurately reflects a significant finding regarding the generalization of the Hidden Subgroup Problem to universal algebras and the tractability of their 2-element powers? 1) There exist 2-element algebras for which quantum algorithms achieve super-polynomial speedup over classical algorithms. 2) All 2-element algebras are classically intractable, regardless of quantum tractability. 3) Quantum algorithms are known to efficiently solve the HSP for all non-abelian groups. 4) The generalization to universal algebras eliminates the distinction between classical and quantum tractability. 5) No classification of tractability has been established for 2-element algebras. 6) Super-polynomial speedup by quantum algorithms is only possible for algebras with more than two elements. 7) The HSP for abelian groups remains unsolved by both classical and quantum algorithms.

✓ Correct Answer:

The correct answer is 1) There exist 2-element algebras for which quantum algorithms achieve super-polynomial speedup over classical algorithms..

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arXiv:2001.11298v2 [cs.LO] 1 May 2020TheHiddenSubgroupProblemforUniversalAlgebras MatthewMoore TheUniversity ofKansas ma/t_thew.moore@ku.eduTaylor Walenczyk TheUniversity ofKansas tawalenczyk@ku.edu Abstract The Hidden Subgroup Problem (HSP) is a computational problem which includes as special cases integer factoriza- tion, the discrete logarithm problem, graph isomorphism, andtheshortestvectorproblem.Thecelebratedpolynomial- time quantum algorithms for factorization and the discrete logarithm are restricted versions of a generic polynomial- time quantum solution to the HSP for abeliangroups, but despite focused research no polynomial-time solution for general groups has yet been found. We propose a general- ization of the HSP to include arbitrary algebraic structures and analyze this new problem on powers of 2-element al- gebras. We prove a complete classiﬁcation of every such power as quantum tractable (i.e. polynomial-time), classi- callytractable,quantumintractable,orclassicallyintract able. In particular, we identify a class of algebras for which the generalized HSP exhibits super-polynomial speedup on a quantumcomputer comparedto a classical one. CCS Concepts: •Theory of computation →Quantum querycomplexity ;FiniteModelTheory ;Quantum com- plexity theory ;Problems, reductionsand completeness . Keywords: quantum computation, hidden subgroup prob- lem,hiddenkernelproblem,universalalgebra,clonetheory ACM ReferenceFormat: Matthew Moore and Taylor Walenczyk. 2020. The Hidden Sub- group Problem for Universal Algebras. In Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science (LIC S ’20), July 8–11, 2020, Saarbrücken, Germany. ACM, New York, NY, USA,12pages.h/t_tps://doi.org/10.1145/3373718.3394764 1 Introduction Quantum algorithms enable new methods of computation through the exploitation of the natural phenomena associ- atedwithquantumsystems.Sometimes,thismakespossible Permission to make digital or hard copies of all or part of this work f or personal or classroom use is granted without fee provided that cop ies are notmadeordistributedforproﬁtorcommercialadvantageandthatc opies bearthisnotice andthefullcitation on theﬁrst page. Copyrights fo r com- ponentsofthisworkownedbyothersthantheauthor(s)mustbehono red. Abstracting with credit is permitted. To copy otherwise, or republis h, to postonserversortoredistributetolists,requirespriorspeciﬁc permission and/or a fee.Request permissions frompermissions@acm.org. LICS ’20, July8–11, 2020, Saarbrücken,Germany © 2020Copyright heldby theowner/author(s).Publication rights licensed toACM. ACMISBN978-1-4503-7104-9/20/07...$15.00 h/t_tps://doi.org/10.1145/3373718.3394764eﬃcient (i.e. polynomial-time) solutions to computational problems for which no eﬃcient classical algorithm exists. Whilethereisnoproofthatquantumcomputationissuper- polynomiallyfasterthanclassical methods,thereis agrow- ingbodyofevidence in support of sucha claim. Perhaps the most famous quantum algorithm exhibiting super-polynomialspeedupoverclassicalonesisShor’salgo- rithmforintegerfactorization[ 18].Otherexamplesinclude Shor’s solution to the discrete logarithm problem [ 18], Si- mon’salgorithm[ 19],theunitandclassgroupalgorithms[ 6], and Pell’s equation and the principal ideal algorithm [ 6] (see [15] for more details). Surprisingly, these superﬁcially diverse problemsareallspecialcasesofaunifyingproblem calledthe HiddenSubgroupProblem . Given a groupG, a setX,and a function f:G→X,the functionfhidesa subgroupD≤Giff(/afii10069.ital)=f(h)if and only if/afii10069.italD=hD(i.e.fis constant exactly on the cosets of DinG). We follow the convention in Universal Algebra of distinguishing betweenanalgebraicstructure (e.g.agroup) anditsunderlyingset,using Gfortheformerand Gforthe latter. HiddenSubgroupProblem(HSP) Input: groupG, functionf:G→Xhiding somesubgroup Task:determine the subgroup D≤Gthatfhides ThegroupGmaybespeciﬁedeitherbyprovidingamultipli- cationtableorbyﬁxingafamily (Gn)n∈Nofgroups(e.g.the symmetric groups Sn) and providing an index n. Formally, these two kinds of speciﬁcation will determine diﬀerent in- put sizes to the problem. In any case, however, we always consider the input size to be lg |G|. The function fis given asanoracle(i.e.ablack-boxfunction).Typically, Dis‘deter- mined’bygenerators,butanyotherschemewhichuniquely identiﬁesDamongst allother subgroupsis permitted. WhenGisabelian, HSP(G)includes all of the problems inthepreviousparagraph,andageneralsolutioninthisset- ting is due to Kitaev [ 11], though it includes many of the aforementionedpreviously knownalgorithms.Thecasefor non-abelianGhasproven to bemore challenging. TheHSPfor non-abelianGhasbeensolvedinsomequite speciﬁc cases [ 1,3,5,7–9,14,17], but it remains open in general.

Question 74 multiple-choice

Quantum computing has revolutionized the study of computational complexity, especially for problems involving hidden subgroup structures in Abelian groups. Understanding the quantum query complexity for these problems is essential for evaluating the fundamental limits of quantum algorithms. Which statement accurately describes a key theoretical advancement in the quantum query complexity of Simon’s Problem and its generalization to Abelian groups? 1) It establishes that classical algorithms can match quantum algorithms for all hidden subgroup problems. 2) It proves that no lower bound exists for quantum query complexity in hidden subgroup problems. 3) It shows that the quantum query complexity for non-Abelian groups is constant. 4) It demonstrates that the quantum query complexity is always exponential for any group structure. 5) It provides the first nontrivial lower bound on quantum query complexity for Simon’s Problem and an optimal lower bound (up to a constant factor) for any Abelian group. 6) It concludes that hidden subgroup problems cannot be solved by quantum algorithms faster than classical ones. 7) It determines that quantum query complexity is irrelevant for problems involving group structures.

✓ Correct Answer:

The correct answer is 5) It provides the first nontrivial lower bound on quantum query complexity for Simon’s Problem and an optimal lower bound (up to a constant factor) for any Abelian group..

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Question 75 multiple-choice

Complexity theory examines the relationships between various computational problems and the classes that describe their difficulty. Understanding the placement of problems like Graph Isomorphism and group-theoretic challenges within these classes is fundamental to assessing their computational boundaries. Which of the following statements is true regarding the class SPP and its relationship with other complexity classes and lowness properties? 1) SPP contains all languages that are low for NP and is strictly larger than LWPP. 2) SPP is defined using gap-definable functions where the gap must be 0 for non-members and depends on an arbitrary function for members. 3) Every language in SPP is low for GapP, meaning adding it as an oracle does not increase the power of GapP. 4) SPP is equivalent to UP, and all problems in SPP have at most one accepting computation path. 5) Relativizing SPP to oracles destroys its containment relationships with UP and LWPP. 6) SPP is the same as PP, as both are characterized by gap-definable functions with positive gaps. 7) SPP only contains languages that are recognized by deterministic polynomial-time algorithms.

✓ Correct Answer:

The correct answer is 3) Every language in SPP is low for GapP, meaning adding it as an oracle does not increase the power of GapP..

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question in [13] (also see [7]). As a consequence it follows that GI is in and low for ⊕P (in fact, GI ∈Mod kP for eachk≥2), C =P etc. Previously, only a special case of Graph Isomorphism, namely Tournament Isomorphism, was known to be in ⊕P.1 What we prove is a more general result: we show that a generic problem FIND-GROUP is in FPSPPas a consequence of which GI and some other algo- rithmic problems on permutation groups that are not known to have polynomial- time algorithms also turn out to be in SPP. In particular, as another corollary, we show that the hidden subgroup problem ( HSP) over permutation groups is in FPSPP. The hidden subgroup problem is of interest in the area of quantum algorithms. 2 Preliminaries and Notation Let Σ = {0,1}be the ﬁnite alphabet. Let log denote logarithm to base 2. Let FP denote the class of polynomial-time computable functions and NP denotes all languages accepted by polynomial-time nondeterministic Turing machines. LetZdenotes the set of integers. A function f: Σ∗→Zis said to be gap- deﬁnable if there is an NP machine Msuch that, for each x∈Σ∗,f(x) is the diﬀerence between the number of accepting paths and the number of rejecting paths ofMon inputx. Let GapP denote the class of gap-deﬁnable functions [7]. For each NP machine MletgapMdenote the GapP function deﬁned by it. The language class PP is deﬁned as follows: Lis in PP if there is an f∈GapP such thatx∈Lif and only if f(x)>0. The language classes UP, SPP and LWPP are deﬁned using GapP func- tions [7].Lis in UP if there is an NP machine MacceptingLsuch thatM has at most one accepting path on any input. Lis in SPP if there is an NP machineMsuch thatx∈Limplies that gapM(x) = 1, andx/negationslash∈Limplies that 1Tournament Isomorphism in ⊕P follows because any tournament has an odd number of automorphisms. There are special cases of Graph Isomorphism, e.g. Graph Isomorphism for bounded-degree graphs or bounded genus graphs, that have polynomial-time algorithms. 2 gapM(x) = 0.Lis in LWPP if there are an NP machine Mandh∈FP such that x∈Limplies that gapM(x) =h(0|x|), andx/negationslash∈Limplies that gapM(x) = 0. For L∈SPP (or in LWPP) we say that the language Lisaccepted by the machine M. The containments UP ⊆SPP⊆LWPP is shown in [7]. We say that fis in GapPA, for oracle A⊆Σ∗, if there is an NPAmachine MAsuch that, for each x∈Σ∗,f(x) is the diﬀerence between the number of accepting paths and the number of rejecting paths of MAon inputx. For any oracle A, we can deﬁne the standard relativized classes UPA, SPPA, and LWPPAand we can easily see the containments UPA⊆SPPA⊆LWPPAfor any oracleA. We say that A⊆Σ∗islowfor PP if PPA= PP. In [7] it is shown that every language in LWPP is low for PP. Similarly, we say that A⊆Σ∗islowfor GapP if GapPA= GapP. Again, it is shown in [7] that Ais low for GapP if and only if A∈SPP. LetMbe an oracle NP machine, let A∈NP be accepted by an NP

Question 76 multiple-choice

Lattice gauge theories are important tools for simulating non-perturbative phenomena in high-energy and condensed matter physics. Quantum computing offers new avenues for implementing these models using discrete quantum systems like qubits. Which of the following statements best describes a key advantage of recasting gauge field theories as quantum link models in Minkowski space for quantum simulation? 1) It allows the use of bosonic operators exclusively for simulation. 2) It eliminates the need for gauge invariance in digital simulations. 3) It restricts simulations to classical computers rather than quantum devices. 4) It enables only static, equilibrium calculations without real-time dynamics. 5) It requires continuous variables for representing the gauge fields. 6) It makes real-time evolution feasible and compatible with quantum qubit algorithms. 7) It prevents the implementation of Suzuki-Trotter expansions in qubit circuits.

✓ Correct Answer:

The correct answer is 6) It makes real-time evolution feasible and compatible with quantum qubit algorithms..

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Question 77 multiple-choice

Quantum computing relies on maintaining quantum coherence, but real-world systems are often subject to decoherence from environmental interactions. Decoherence-free subspaces (DFS) and robust circuit designs are essential strategies for reliable quantum computation. Which approach enables the implementation of the Quantum Fourier Transform (QFT) in quantum networks where multiple qubits experience collective decoherence, thus improving the reliability of quantum algorithms? 1) Utilizing classical error correction codes alongside quantum circuits 2) Employing physical isolation of individual qubits in separate environments 3) Applying active error correction with frequent syndrome measurements 4) Encoding quantum information within decoherence-free subspaces (DFS) tailored for collective decoherence 5) Running quantum algorithms at extremely low temperatures to minimize noise 6) Increasing the number of ancillary qubits to detect and correct all errors 7) Using quantum teleportation to bypass environmental interactions

✓ Correct Answer:

The correct answer is 4) Encoding quantum information within decoherence-free subspaces (DFS) tailored for collective decoherence.

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Question 78 multiple-choice

Group symmetries play a fundamental role in quantum physics and quantum machine learning, where the invariance of labels under group actions ensures robust classification and meaningful physical properties. Both continuous and discrete groups underpin the classification of quantum states and operations on Hilbert spaces. Which statement accurately describes the role of local unitaries in the classification of multipartite entanglement in quantum systems? 1) Labels for multipartite entanglement are invariant under local unitaries acting independently on each qubit, corresponding to the group U(2) ×.. × U(2). 2) Local unitaries always change the entanglement classification of a quantum state. 3) Multipartite entanglement labels are only invariant under global orthogonal transformations, not under local unitaries. 4) The invariance of labels under local unitaries is specific to classical systems, not quantum systems. 5) Local unitaries are irrelevant in determining the separability of quantum states. 6) Entanglement measures are only invariant under the symmetric group Sn. 7) Only ground states in the Heisenberg XXX model exhibit invariance under local unitaries, not multipartite entangled states.

✓ Correct Answer:

The correct answer is 1) Labels for multipartite entanglement are invariant under local unitaries acting independently on each qubit, corresponding to the group U(2) ×.. × U(2)..

📚 Reference Text:

the unitary group on H. •Representation: U:G7!U(d), wheregi=UgiUy g. Orthogonal transformations: Consider a problem of classifying orthogonal (real-valued) states from Haar- random states. The dataset here is composed of states with label yi= 0(yi6= 0) ifiis a real-valued state (a Haar random state). Here, the labels yi= 0are invariant under the action of any orthogonal unitary, as conjugated a real-valued state by a real-valued unitary yields a real-valued state. Note that here f(UiUy) =f(i), but in general UiUy6=i. •States:Statesi2D(H). •Labels:Orthogonal yi= 0and mixed yi6= 0. •Group:G=O(d), the orthogonal group on H •Representation: U:G7!O(d), wheregi=UgiUy g Local unitary transformations and the XXX model: Consider the problem of classifying ground states of the Heisenberg XXXmodelH=JPn j=1(XjXj+1+YjYj+1+ZjZj+1). Here,yi= 0(yi= 1) ifiis a ferromagnetic (antiferromagnetic) ground state of HwithJ < 0(J > 0). SinceHcommutes with the total magnetization operators Sx=Pn j=1Xj,Sy=Pn j=1Yj,Sz=Pn j=1Zj, then the labels are invariant under the action of the same local unitary acting on all qubits. That is, f((Nn iU)i(Nn iUy)) =f(i)for any local unitaryU. •States:Ground states of the XXXchaini2D(H) •Labels:Ferromagnetic yi= 0and antiferromagnetic yi= 1 •Group:G=U(2) •Representation: U:G7!U(d), wheregi= (Ug Ug)i(Ug Ug)y Local unitary transformations and multipartite entanglement: Consider the problem of classifying pure quantum states according to the amount of multipartite entanglement they posses. Here, yi= 1if the states posses a large amount of multipartite entanglement (according to some measure), while yi= 0if the states are separable. Since local unitaries do not change the multipartite entanglement in a quantum state, then we have that f((Nn jUj)i(Nn jUy j)) =f(i)for any local unitary Ujacting on the j-th qubit. •States:Pure states i2D(H) •Labels:yi2[0;1], where 0 means separable and 1 means “highly entangled” •Group:G=U(2)U(2), (ntimes) •Representation: U:G7!U(d), where (g1;:::;gn)i= (Ug1 Ugn)i(Ug1 Ugn)y 42 Example 13: The usual suspects: commonly appearing discrete groups While a veritable zoo of group symmetries can (and do!) appear in the wild, any QML practicioner should be familiar with some especially common species. Consider this a beginner’s ﬁeld guide to some frequently appearing groups and a few classic applied locations they have been observed. Note that discrete group representations often appear in disguise as subgroups of continuous groups (which is inevitable, because as we described earlier, we care about unitary representations on Hilbert spaces). For example, a =2pulse instantiates a rotation by =2on the Bloch sphere for a single qubit—the possible actions of strings of =2pulses generate the group of rotations Z4. Discrete groups •Zn, the cyclic group of integers modulo n. –Type, size, # irreps: Abelian,jZnj=n, n irreps (given by roots of unity) –Where you might ﬁnd them: translations on periodic lattices, rotations of 2D regular polygons, roots of unity, parity transformations –Useful fact: Every ﬁnite abelian group is a direct product of cyclic groups, so ﬁnite abelian sym- metries can often be studied by restricting to cyclic groups. •Sn, the symmetric group on nletters (aka the group of all permutations on a set of size n). –Type, size, # irreps: nonabelian for n>3,jSnj=n!, integer partitions ofn ∗A tuple of positive integers

Question 79 multiple-choice

Finite W-algebras are algebraic structures that arise from Lie algebras through reductions and play a significant role in representation theory and quantum field theory. Modern approaches to their study involve cohomological and geometric methods, including BRST quantization and Poisson geometry. Which of the following statements best describes the role of the BRST differential in the quantization of finite W-algebras? 1) It splits into two commuting differentials, forming a double complex whose cohomology can be computed using spectral sequences. 2) It acts solely as a grading operator on the universal enveloping algebra without contributing to cohomology. 3) It defines a filtration that trivializes the algebraic structure, making spectral sequences unnecessary. 4) It generates finite dimensional irreducible representations directly, bypassing cohomological methods. 5) It identifies the Kirillov Poisson structure with the group manifold itself, removing the need for reduction. 6) It replaces the need for Hamiltonian reduction through explicit Fock space constructions. 7) It only applies to infinite W-algebras and cannot be used for finite cases.

✓ Correct Answer:

The correct answer is 1) It splits into two commuting differentials, forming a double complex whose cohomology can be computed using spectral sequences..

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systems are r eductions of a system describing a free particle moving on a group manifold. This a llows us to give general formulas for the solution space of such systems. In the second part of the paper we BRST quantize the ﬁnite Walgebras. The nontrivial part of this is of course calculating the BRST coh omology and its algebraic structure. Since the BRST diﬀerential is a sum of two other di ﬀerentials one can associate a double complex to the BRST complex. In order to ca lculate the BRST cohomology one can then use the theory of spectral sequences . There is a choice to be made between one out of two spectral sequences that one c an associate to a double complex. These spectral sequences must give the same ﬁnal answer for the BRST cohomology, as is well known from the theory of spectral sequences, but for the calculation it is crucial which one one takes. The choice we m ake is diﬀerent from the one made by Feigin and Frenkel and allows us to quantize any ﬁn iteWalgebra and reconstruct its algebraic structure. In order to make our co nstruction more explicit we calculate all ﬁnite quantum Walgebras that can be obtained from sl2,sl3andsl4. In the third and last part of the paper we discuss the represen tation theory of ﬁnite Walgebras. Crucial for this is a quantum version of the genera lized Miura transfor- mation which embeds any ﬁnite Walgebra into the universal enveloping algebra of some (semi)simple Lie algebra. An arbitrary representatio n of this Lie algebra there- fore immediately yields a representation of the ﬁnite Walgebra. This also allows us 4 to derive Fock realizations for arbitrary ﬁnite Walgebras since Fock realizations for simple Lie algebras are well known. This replaces the cumber some construction of W algebras as commutants of screening operators. As an illust rative example we realize the ﬁnite dimensional representations of the ﬁnite Walgebra ¯W(2) 3as a subrepresen- tation of certain Fock realizations. In principle this prov ides the ﬁrst term of a Fock space resolution of these representations [14]. We have tried to keep the paper as self-contained as possible and give full proofs of the main assertions. 2. Basics of Kirillov Poisson structures and Poisson reduct ion In order to make the paper reasonably selfcontained we brieﬂ y discuss in this sec- tion the Kirillov Poisson structure on a Lie algebra and Pois son reduction of Poisson manifolds. These two concepts will then be put together in th e remainder of the paper. 2.1. Kirillov Poisson structures Let (M,{.,.}) be a Poisson manifold, that is {.,.}is a Poisson bracket on the space C∞(M) ofC∞functions on M, andGa Lie group. Also let Φ : G×M→Mbe a smooth and proper action of GonMwhich preserves the Poisson structure, i.e. Φ∗ g{φ,ψ}={Φ∗ g(φ),Φ∗ g(ψ)} (2.1) where Φ∗ gis the pullback of Φ g:M→M. Physically this implies that if γ(t) is a solution of the equations of motion w.r.t. some G-invariant Hamiltonian H

Question 80 multiple-choice

Quantum topology investigates properties of 3-manifolds using quantum-inspired mathematical frameworks, often leveraging fusion categories to define computable invariants. Computational challenges in this field arise from both the quantum algebraic input and the underlying topological complexity of manifolds. Which statement accurately describes the computational complexity and parameterization of algorithms for computing state sum invariants from Tambara-Yamagami categories on triangulated 3-manifolds? 1) The computation is #P-hard in general, but becomes fixed parameter tractable when parameterized by the first Betti number with Z/2Z coefficients. 2) The computation is always polynomial-time regardless of the manifold's topological properties. 3) The computational hardness originates solely from the quantum algebraic structure of the fusion categories. 4) There are currently no known algorithms that achieve fixed parameter tractability for this problem. 5) The complexity depends exponentially on the width of the triangulation rather than any topological parameter. 6) The first Betti number cannot be computed efficiently from combinatorial data of the manifold. 7) State sum invariants from Tambara-Yamagami categories are only defined for manifolds with trivial first homology.

✓ Correct Answer:

The correct answer is 1) The computation is #P-hard in general, but becomes fixed parameter tractable when parameterized by the first Betti number with Z/2Z coefficients..

📚 Reference Text:

Title: An algorithm for Tambara-Yamagami quantum invariants of 3-manifolds, parameterized by the first Betti number Year: 2023 Paper ID: 937b79eace34a89a7b58a9274ca096b6d8316f24 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/937b79eace34a89a7b58a9274ca096b6d8316f24 Abstract: Quantum topology provides various frameworks for defining and computing invariants of manifolds inspired by quantum theory. One such framework of substantial interest in both mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum construction, which uses the data of a spherical fusion category to define topological invariants of triangulated 3-manifolds via tensor network contractions. In this work we analyze the computational complexity of state sum invariants of 3-manifolds derived from Tambara-Yamagami categories. While these categories are the simplest source of state sum invariants beyond finite abelian groups (whose invariants can be computed in polynomial time) their computational complexities are yet to be fully understood. We first establish that the invariants arising from even the smallest Tambara-Yamagami categories are #P-hard to compute, so that one expects the same to be true of the whole family. Our main result is then the existence of a fixed parameter tractable algorithm to compute these 3-manifold invariants, where the parameter is the first Betti number of the 3-manifold with Z/2Z coefficients. Contrary to other domains of computational topology, such as graphs on surfaces, very few hard problems in 3-manifold topology are known to admit FPT algorithms with a topological parameter. However, such algorithms are of particular interest as their complexity depends only polynomially on the combinatorial representation of the input, regardless of size or combinatorial width. Additionally, in the case of Betti numbers, the parameter itself is computable in polynomial time. Thus while one generally expects quantum invariants to be hard to compute classically, our results suggest that the hardness of computing state sum invariants from Tambara-Yamagami categories arises from classical 3-manifold topology rather than the quantum nature of the algebraic input.

Question 81 multiple-choice

In quantum computing, problems like the Dihedral Coset Problem (DCP) involve extracting hidden parameters from quantum states, and the design of quantum algorithms is constrained by foundational principles such as unitarity and the no-cloning theorem. Understanding these limitations is crucial for developing effective quantum algorithms. Which statement most accurately describes a fundamental limitation encountered when attempting to produce additional DCP samples or extract the hidden parameter a from existing quantum samples? 1) Using classical copying techniques, one can reliably duplicate DCP samples for any unknown parameter a. 2) The no-cloning theorem applies only to entangled quantum states and does not affect DCP samples. 3) Unitary quantum operations can always extract the parameter a into a register from any list of DCP samples. 4) If the parity of a is known, then any DCP sample can be cloned without restriction. 5) The linearity of quantum mechanics allows unlimited copying of quantum information encoded in coset states. 6) Quantum measurements are sufficient to deterministically compute a from a single DCP sample. 7) It is impossible to use unitary quantum operations to clone unknown DCP samples or extract a into a register without violating quantum mechanical principles such as linearity and the no-cloning theorem.

✓ Correct Answer:

The correct answer is 7) It is impossible to use unitary quantum operations to clone unknown DCP samples or extract a into a register without violating quantum mechanical principles such as linearity and the no-cloning theorem..

📚 Reference Text:

a α α(6.7) The sum of the observed exponents of the two registers gives a. Ifais known, then DCP samples for a, (∣ ⟩ ∣ ⟩)=+−ψx y x1 2,aααa α ;The dihedral hidden subgroup problem  15 are chosen from a set of mutually orthogonal states depending on the parameter a. By Proposition 6.1 , for each sample of the form (6.5), we can copy it to produce a sample of the form (6.6). □ Remark 6.8. Copying a DCP sample up to parity would allow us to determine the parity of a, and vice versa . Theorem 6.9. If a is unknown, there is no unitary operation, which from a list of DCP samples for a, copies an additional DCP sample for the same a, while leaving the list of DCP samples alone . Proof. Suppose that there is a unitary operator Uthat transforms ∣⟩ ∣⟩∣ ⟩ ∣⟩ ∣ ⟩ ∣() ⟩ ∣⟩∣ ⟩ ∣⟩ ∣⟩⋯= ⋯U A ψψ ψ ψ ψψ ψ ψ 0Σ ,aam a aaaam aa11(6.10) where (∣ ⟩ ∣ ⟩) == +−ψψ x y xaa ααa α ;1 2is a DCP sample for aﬁxed, andαrandomly chosen for each such state. We are supposing that Uperforms the aforementioned operation for any(unknown) a. Thus, we also have that ∣ ⟩ ∣⟩∣ ⟩ ∣⟩ ∣ ⟩∣() ⟩ ∣⟩∣ ⟩ ∣⟩ ∣⟩⋯= ⋯U A ψψ ψ ψ ψψ ψ ψ 0Σ ,bbm b bbbbm bb11(6.11) for any other b. Taking the inner product of both sides of (6.10 ) and (6.11), we deduce ⟨∣⟩⟨∣⟩ ⟨∣⟩⟨∣⟩⟨∣⟩ ⟨∣⟩ ⟨() ∣() ⟩⋯= ⋯ψ ψ ψψ ψ ψ ψ ψ ψψ ψ ψ ψ ψ ΣΣ .ab am bm ab ab am bm ab aabb11 11 2(6.12) However, there are choices of ψψ,ai bifor=im 1,…, , andψψ,ab, which do not satisfy (6.12 ) from (5.5). We may thus suppose without loss of generality that ⟨∣⟩≠ψψ 0,1ai bifor all=iN 1,…, , and hence, (6.12 ) becomes ⟨∣⟩⟨∣⟩ ⟨() ∣() ⟩=ψψ ψψ ψ ψ ΣΣ .ab ab aabb2 We obtain a contradiction again by choosing ψaandψbso that⟨∣⟩≠ψψ 0,1ab as then ∣⟨ ∣ ⟩∣=ψψ1 2,ab(6.13) ∣⟨ ∣ ⟩ ⟨ ( )∣ ( )⟩∣ ≤ ψψ ψ ψ ΣΣ1 4.ab aabb2 (6.14) □ The following is another proof of Theorem 5.1 using the connection with quantum cloning. Theorem 6.15. There is no unitary operation to compute the value of a into a register from a list of DCP samples for a . Proof. Suppose that there is a unitary operator U, which has the e ﬀect ∣⟩ ∣ ⟩∣ ⟩ ∣⟩ ∣ () ⟩ ∣⟩⋯=UA ψ ψ ψ a 0Σ ,aam aa1(6.16) i.e.,Utakes a list of DCP samples for ﬁxed but unknown a, a blank initialization state ∣⟩0, and an ancilla state ∣⟩A, and then computes ainto the blank register. Using an additional blank register and copying ∣⟩a, there is a unitary operator ′Uwith the e ﬀect: ∣⟩ ∣ ⟩ ∣ ⟩ ∣⟩ ∣⟩ ∣ ( ) ∣⟩ ∣⟩′⋯ =UAψ ψ ψ aa 00 Σ .aam aa1(6.17)

Question 82 multiple-choice

Quantum complexity theory investigates the boundaries between quantum and classical computational models, particularly through problems that highlight computational separations under certain promises. The k-fold Forrelation problem and Quantum Circulant Optimization are significant in illustrating these separations and their implications for complexity classes. Which statement accurately describes the consequence if a classical polynomial-time algorithm can solve Quantum Circulant Optimization efficiently under standard assumptions? 1) It proves that QNC is strictly harder than PromiseBQP. 2) It implies that PromiseBQP equals PromiseBPP, collapsing the quantum-classical separation for promise problems. 3) It shows that Forrelation is NP-complete. 4) It establishes that all quantum algorithms can be simulated by constant-depth classical circuits. 5) It demonstrates that quantum circuits are unnecessary for solving linear systems efficiently. 6) It means that PromiseQNC is disjoint from PromiseBPP. 7) It ensures that variational quantum algorithms outperform classical algorithms for all optimization problems.

✓ Correct Answer:

The correct answer is 2) It implies that PromiseBQP equals PromiseBPP, collapsing the quantum-classical separation for promise problems..

📚 Reference Text:

By reverse triangular inequality, we can further obtain |ˆα1| − |Φf1,f2,···,fk| 2≤ |ˆα1−Φf1,f2,···,fk|2≤ζ≤1 16⇒ |ˆα1| − |Φf1,f2,···,fk| ≤p ζ≤1 4. (C8) Hence, we see that the value of |ˆα1|is 1/4-close to |Φf1,f2,···,fk|. Given the promise that |Φf1,f2,···,fk|is either ≥3 5or ≤1 100, if|ˆα| ≥7 20, then |Φf1,f2,···,fk| ≥3 5; If|ˆα| ≤13 50, then |Φf1,f2,···,fk| ≤1 100. We see that k-foldForrelation can be reduced to QuantumCirculantOptim . If there exists a relativizing classical algorithm that can solve QuantumCirculantOptim in poly( κC, n) time, the same algorithm can solve k-foldForrelation equally efficiently, implying that PromiseBQP =PromiseBPP . Lastly, we note that the above proof works under the assumption that oracles and poly ndepth circuits are allowed in the preparation of UbinQuantumCirculantOptim . If only poly log ndepth is allowed for the preparation for Ub, only under the case where k∈poly log nisk-foldForrelation reducible to QuantumCirculantOptim . In this case, and disregarding the implementation depth of oracles, k-foldForrelation isQNC-complete, and we obtain the result that QuantumCirculantOptim isPromiseQNC -hard. Similarly, if there exists a classical algorithm that can solve QuantumCirculantOptim efficiently, the same algorithm can solve Forrelation equally efficiently, which implies that PromiseQNC ⊆PromiseBPP . [1] A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear systems of equations, Phys. Rev. Lett. 103, 150502 (2009). [2] M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nature Reviews Physics 3, 625 (2021). [3] X. Xu, J. Sun, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, Variational algorithms for linear algebra, Science Bulletin 66, 2181 (2021). [4] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, and P. J. Coles, Variational quantum linear solver (2020), arXiv:1909.05820 [quant-ph]. [5] J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018). [6] J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature Communications 9(2018). [7] H.-Y. Huang, K. Bharti, and P. Rebentrost, Near-term quantum algorithms for linear systems of equations with regression loss functions, New Journal of Physics 23, 113021 (2021). [8] E. Tang, A quantum-inspired classical algorithm for recommendation systems, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing , STOC 2019 (Association for Computing Machinery, New York, NY, USA, 2019) p. 217–228. [9] N.-H. Chia, H.-H. Lin, and C. Wang, Quantum-inspired sublinear classical algorithms for solving low-rank linear systems (2018), arXiv:1811.04852 [cs.DS]. [10] A. Gily´ en, S. Lloyd, and E. Tang, Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension (2018), arXiv:1811.04909 [cs.DS]. [11] A. Gily´ en, Z. Song, and E. Tang, An improved quantum-inspired algorithm for linear regression, Quantum 6, 754 (2022). [12] C. Shao and H. Xiang, Quantum circulant preconditioner for a linear system of equations, Phys. Rev. A 98, 062321 (2018). [13] Q. Zuo and T. Li, Fast and practical quantum-inspired classical algorithms for solving linear systems (2023),

Question 83 multiple-choice

In group theory and quantum mechanics, unitary and special unitary groups play a fundamental role in understanding symmetry transformations of complex vector spaces. The mathematical properties of these groups have significant implications for the types of representations and physical phenomena they can describe. Which statement is true regarding the group SU(d) for d ≥ 2? 1) SU(d) is compact, connected, and simply connected. 2) SU(d) is non-compact, disconnected, and abelian. 3) SU(d) is compact, not connected, and not simply connected. 4) SU(d) is abelian and consists only of diagonal matrices. 5) SU(d) contains all orthogonal transformations of real vector spaces. 6) SU(d) is isomorphic to the cyclic group Z_d. 7) SU(d) is non-compact but simply connected.

✓ Correct Answer:

The correct answer is 1) SU(d) is compact, connected, and simply connected..

📚 Reference Text:

= (1;:::;k)is called a partition if1++k=nand 1>:::k>0. Partitions, and thus irreps of Sn, are labeled by Young diagrams. –Where you might ﬁnd them: Symmetric and antisymmetric vectors, determinants, more general permutations on tensor indices, combinatorial identities –Useful fact: Schur-Weyl duality tells us that the tensor representations U nofU(d)on(Cd) ncan be decomposed into representations of U(d)andSn, whereU(d)acts on one CdandSnpermutes tensor indices. •Dn, the dihedral group of symmetries of the regular n-gon. –Type, size, # irreps: nonabelian for n>3,jDnj= 2n,(n+ 3)=2ifnodd, (n+ 6)=2ifneven –Where you might ﬁnd them: Molecular symmetry, ﬁnite subgroups of O(2) –Useful fact: Dnis generated by rotations of angle 2=nand reﬂections, and so on the complex plane is commonly thought of as the group generated by multiplication by e2i=nand complex conjugation. •Zn=Z Z, the additive group of integer vectors (k1;:::;kn) –Type, size: abelian,jZnj=1 –Where you might ﬁnd them: Fourier series, translations on inﬁnite lattices –Useful fact: The cyclic group Z`can be thought of a quotient group of the group of integers Z, or more geometrically, Z`is the group of translations on a ring with `sites (“periodic boundary conditions”). Likewise, if we quotient every Zin the translation group Zn, we get tori Z`1Z`2 Z`n. 43 Example 14: The usual suspects: commonly appearing continuous groups The ﬁeld guide continues with some commonly appearing continuous groups. These species hold privileged positions in physics as symmetries enjoyed by a variety of diﬀerential equations, and they are often detected at the level of their Lie algebra as “inﬁnitesimal symmetries”. While certainly the unitary groups are most important for QML, we would not put it past these other groups to sneakily appear in a variety of tasks. Perhaps you will tell us where you have caught them! Continuous groups •GL(d;C)andSL(d;C), the (complex) general and special linear groups –Topological info: Not compact, connected, simply connected –Lie algebra: gl(d) =Md(C), theddcomplex matrices, and sl(d) =fX2gl(d) : Tr[X] = 0g, the tracelessddcomplex matrices –Where you might ﬁnd them: Change of bases, so basically everywhere. –Useful fact: The complexiﬁed Lie algebras u(d) C=gl(d) C=gl(d)and su(d) C=sl(d) C= sl(d), which means their (complex) representation theory is the same, even though gl(d)6=u(d)and sl(d)6=su(d)(u(d);su(d)are real vector spaces but not complex vector spaces). •U(d)andSU(d), the unitary and special unitary groups –Topological info (for SU(d)):Compact, connected, simply connected ∗U(d)=SU(d)=U(1), the circle group. The isomorphism follows immediately from det :U(d)! U(1).U(d)is not simply connected. –Lie algebra: u(d) =fX2Md(C) :X=Xyg, theddskew-hermitian matrices, and su(d) = fX2u(d) : Tr[X] = 0g, the traceless ddskew-hermitian matrices. Note that u(d);su(d)are real vector spaces: e.g. if X2su(d), theniX62su(d). –Where you might ﬁnd them: Literally everywhere in quantum. –Useful fact: Wigner’s theorem assures us that every symmetry transformation on physical states which preserves the Hermitian inner product is either a unitary or antiunitary transformation. This in large part motivates the focus on unitary representations from a physical standpoint. •O(d)andSO(d), the orthogonal and special orthogonal groups –Topological info (for SO(d)):Compact, connected, not simply connected ∗Note that the orthogonal group O(d)consists of two disconnected copies of SO(d): matrices fR:R2SO(d)g, and matricesfSR:R2SO(d)and det(S)

Question 84 multiple-choice

Quantum algorithms such as Shor’s threaten classical cryptographic systems by making integer factorization much more efficient. The practical implementation of these algorithms depends heavily on the complexity of quantum hardware and the types of quantum gates required. Which modification to Shor’s quantum factorization algorithm could simplify hardware requirements by allowing the use of only simple phase rotation gates with states 0 and π? 1) Replacing modular exponentiation with classical arithmetic operations 2) Using Grover’s search algorithm in place of the period-finding subroutine 3) Substituting Toffoli gates for all controlled-NOT gates 4) Applying the SWAP gate for state preparation 5) Implementing the classical discrete Fourier transform 6) Using phase estimation with random unitary gates 7) Replacing the quantum Fourier transform with the quantum Hadamard transform in certain cases

✓ Correct Answer:

The correct answer is 7) Replacing the quantum Fourier transform with the quantum Hadamard transform in certain cases.

📚 Reference Text:

Question 85 multiple-choice

In the study of topological quantum order in lattice models, qudits are often associated with the edges of a lattice, and the ground state properties are defined via commuting projector Hamiltonians. These systems exhibit robust quantum error correcting capabilities due to their global, non-local nature. Which statement accurately describes the condition known as TQO-1 in topologically ordered, frustration-free Hamiltonians? 1) The ground state energy is minimized only globally, not locally. 2) The ground state manifold is unique for all choices of sublattice. 3) Projectors Pv and Pf do not commute on overlapping regions. 4) Local operators can distinguish all ground states on any sublattice. 5) All ground states have indistinguishable reduced density matrices on any small sublattice, so no local operator can differentiate between them. 6) The ground state projector acts only on the vertices of the lattice. 7) Topological degeneracy arises from boundary conditions rather than global constraints.

✓ Correct Answer:

The correct answer is 5) All ground states have indistinguishable reduced density matrices on any small sublattice, so no local operator can differentiate between them..

📚 Reference Text:

qudits are conventionally deﬁned to live on the edges of Λinstead of the vertices; for simplicity we use the same convention here.2We therefore associate to each edge e∈E(Λ)a qudit He=Cd,and take the total Hilbert space to be H=/circlemultiplytext e∈E(Λ)He. We consider Hamiltonians of the form H=/summationdisplay v∈V(Λ)(1−Pv) +/summationdisplay f∈F(Λ)(1−Pf), (5) wherePvis a projector that acts non-trivially only on edges which meet the vertex v, andPf is a projector that acts non-trivially only on the boundary edges of the plaquette f. We further demand that the Pv’s and thePf’s mutually commute and that the Hamiltonian be frustration free, i.e. that the ground states of Hare stabilized by each Pvand eachPf: Vg.s.={|ψ/angbracketright∈H:Pv|ψ/angbracketright=|ψ/angbracketrightandPf|ψ/angbracketright=|ψ/angbracketright,∀v∈V(Λ),f∈F(Λ)}. (6) Denote the projection onto Vg.s.byP, which can be written as P=/productdisplay v∈V(Λ)Pv/productdisplay f∈F(Λ)Pf. (7) LetAbe a sublattice of Λof size/lscript×/lscript, denote by V(A)◦the subset of V(A)that are in the interior ofA(which is of size (/lscript−2)×(/lscript−2)), and deﬁne PA=/productdisplay v∈V(A)◦Pv/productdisplay f∈F(A)Pf. (8) We can now state the deﬁnition of TQO that we will use. Deﬁnition 2.2 (Topological Quantum Order [5]) .A Hamiltonian which is frustration-free is said to have topological quantum order (TQO) if there is a constant α > 0such that for any /lscript×/lscript sublatticeAwith/lscript≤Lα, the following hold. 2The choice of whether the qudits live on the edges or vertices of the lattice is arbitrary and makes no diﬀerence to the deﬁnition. Accepted in Quantum2020-09-17, click title to verify. Published under CC-BY 4.0. 4 •TQO-1: For any operator Oacting onA, POP =cOP, (9) wherecOis some complex number. •TQO-2: IfBis the smallest square lattice whose interior properly contains A,3then Tr¯A(P) andTr¯A(PB)have the same kernel, where ¯Ais the complement of AinΛ. TQO-1 heuristically corresponds to the statement that a suﬃciently local operator cannot be used to distinguish between two orthogonal ground states because they diﬀer only in their global, “topological” properties. Furthermore, ground state denegeracy is “topologically protected” in systems satisfying TQO-1 in the sense that perturbations by local operators can induce energy levelsplittingonlynon-perturbatively, oratsomelargeorderinperturbationtheorywhichincreases with the size of the lattice. It is straightforward to show that TQO-1 is equivalent to the condition that all normalized ground states |ψ/angbracketright∈Vg.s.have the same reduced density matrix on A. TQO-2 is the statement that the local ground state spaces and the global one should agree. We emphasize that TQO-2 can be violated at regions with non-trivial topologies, which is why one restricts to square lattices. Remark 2.3. For our purposes, TQO-1 and QECC are morally interchangeable. Indeed, if H is any Hamiltonian4withPthe projection onto the ground space Vg.s., then the following are equivalent. 1. The Hamiltonian Hhas TQO-1. 2. The Hamiltonian Hprovides a QECC with code subspace Vg.s.. There exists an α>0such that the code can correct any error ρ/mapsto→/summationtext iEiρE† ifor which every combination E† iEjis supported on an /lscript×/lscriptsublatticeAwith/lscript≤Lα. In §3.2, we will prove a theorem for Kitaev’s ﬁnite group models which simultaneously implies TQO-1 and TQO-2, and so by the above remark also implies that the model furnishes a QECC. 2.3 Kitaev’s ﬁnite group lattice model We now turn to Kitaev’s

Question 86 multiple-choice

In quantum photonic computing, photon indistinguishability is central to achieving high-fidelity quantum operations, and errors in this property introduce specific challenges for modeling and simulation of quantum circuits. Specialized frameworks are used to map such errors to standard quantum error types for efficient analysis. Which approach enables efficient simulation of large-scale quantum circuits with low-order photon distinguishability errors by combining stabilizer formalism with minimal internal state tracking per photon? 1) Using only dual-rail projection to eliminate all internal state information 2) Applying amplitude damping channels to model photon loss exclusively 3) Employing a full density matrix representation for every photon in the circuit 4) Probabilistically applying Pauli errors and a distinguishability operator, tracking stabilizers and minimal internal state per photon 5) Disregarding all distinguishability errors after initial state preparation 6) Using only phase-flip errors to represent all error types 7) Modeling errors as purely classical random bit flips on output qubits

✓ Correct Answer:

The correct answer is 4) Probabilistically applying Pauli errors and a distinguishability operator, tracking stabilizers and minimal internal state per photon.

📚 Reference Text:

the cases in which the photon in mode 2i(respectively 2i+ 1) is distinguishable. We will also consider a second-order pair error , in which the input photons in both modes 2iand2i+ 1are distinguishable with the same internal state η1=|ψ1⟩⟨ψ1|withTr(η0η1) = 0 . We note that in Ref. 20, the output state of the Bell state generator was very nicely calculated by entirely tracing out the internal states of all photons. Here we choose not to trace out the internal degrees of freedom entirely, instead giv- ing a framework in which distinguishability errors on the input photons may be mapped to a combination of Pauli and distinguishability errors on the output state. We briefly introduce some notation for this purpose. For an operator A, letPA(γ) =1 2(γ+AγA†). Also let Dibe an operator that acts on modes 2i,2i+ 1 bya† 2i(η0)|⃗0⟩ 7→ a† 2i(η1)|⃗0⟩, a† 2i+1(η0)|⃗0⟩ 7→ a† 2i+1(η1)|⃗0⟩. This operator is simply for convenience of notation, and allows us to write partially- distinguishable mixed states by starting with the ideal state |B+⟩⟨B+|and applying appropriate operators. Theorem 4. Suppose the n-GHZ state generation protocol heralds success with measurement pattern ⃗ m= (m0, . . . , m 2n−1). Assume that all input photons except for those in input modes 2i,2i+ 1are ideal, with internal state η0=|ψ0⟩⟨ψ0|. 1. Suppose that input photons 2i,2i+ 1are in state (16) above. After dual-rail projection, the output state is 1 2PXi(PZi(|B+⟩⟨B+|) +Di(|B+⟩⟨B+|)). (17) 2. Suppose we have a pair error on pair i, so that input photons 2i,2i+ 1 are in the same internal state η1, fully distinguishable from the ideal internal state η0. Then the output state is PZi(Di(|B+⟩⟨B+|)), with no dual-rail projection required. Remark 5. We note that these results suggest a paradigm for computing the effect of low-order distinguishability errors on much larger circuits. Suppose we have a large circuit whose subroutines are n-GHZ state generation, single-qubit Clifford operations, Type I fusions, and k-GHZ state analyzers (for varying n, k). Also suppose that the distinguishability error rates are sufficiently low that low-order error approximations are reasonable. Theorem 4 shows that, under these circumstances, erroneous n-GHZ state generation may be simulated by taking the ideal state and probabilistically applying Pauli errors and an operator Dithat converts ideal photons to fully distinguishable ones. This sampled state may then be fed into the later unitary operations, fusion gates, etc.; it is straightforward to see how Pauli errors propagate through the circuit, as with standard Pauli frame calculations, and the results of Sec. II D 1 show how distinguishability errors interact with each operation. This implies the possibility of a generalized stabilizer-type simulation, in which one tracks both the stabilizers of the simulated state and, for each photon, a minimal amount of information about its internal state. This, of course, will be simplest if one uses an appropriately simple distinguishability error model, such as those discussed in Sec. II D 3 below. Further, the assumption that the errors are low-order is essential: otherwise, multiple interacting errors could “cancel,

Question 87 multiple-choice

Quantum neural networks leverage mathematical structures to efficiently represent and manipulate complex quantum data. Clifford algebras are particularly valued in this domain for their ability to encode geometric relationships and facilitate quantum computations. Which of the following statements accurately describes a key advantage of using Clifford algebras, represented via Pauli matrices, in the design of quantum neural networks? 1) They guarantee the universal approximation of all classical functions without the need for entanglement. 2) They exclusively facilitate linear transformations, limiting their applicability to simple datasets. 3) They enable the implementation of irreversible activation functions required for quantum learning. 4) They provide a natural framework for representing multidimensional data and capturing geometric properties crucial for quantum computing. 5) They replace the need for unitary operations in quantum algorithms by offering non-unitary alternatives. 6) They ensure that quantum neural networks operate entirely without data entanglement. 7) They directly generalize Boolean logic gates for classical neural network architectures.

✓ Correct Answer:

The correct answer is 4) They provide a natural framework for representing multidimensional data and capturing geometric properties crucial for quantum computing..

📚 Reference Text:

Question 88 multiple-choice

Quantum algorithms frequently employ advanced measurement strategies and circuit constructions to solve computationally hard problems. The relationship between efficient quantum circuit implementation and classical computational complexity is a central consideration in quantum algorithm design. Which statement accurately describes the implication of efficient quantum circuit implementation for a rank-one POVM associated with the subset sum problem? 1) Efficient implementation guarantees the subset sum problem becomes polynomial-time classically solvable. 2) Efficient implementation allows arbitrary sampling from all NP-complete problems. 3) Efficient implementation implies the rank of the Hilbert space must be exponentially large. 4) Efficient implementation requires all blocks of the unitary transformation to be uniquely determined. 5) Efficient implementation is always possible for any instance regardless of structure. 6) Efficient implementation of the optimal POVM enables efficient quantum sampling from subset sum solutions, effectively solving an NP-complete problem. 7) Efficient implementation makes Neumark’s theorem unnecessary for realizing POVMs.

✓ Correct Answer:

The correct answer is 6) Efficient implementation of the optimal POVM enables efficient quantum sampling from subset sum solutions, effectively solving an NP-complete problem..

📚 Reference Text:

perform this measurement in a particular way, ﬁrst measuring the labelxand then performing the povm {Ex j}j∈ZNconditioned on that label. Note that each Ex jis rank one, so that the povm {Ex j}j∈ZNfor ﬁxed x∈Zk Nis reﬁned into one-dimensional subspaces, removing much of the freedom in the implementation of the original povm {Ej}. For any given x∈Zk N, we consider the implementation of the povm {Ex j}j∈ZNby anx-dependent quantum circuit followed by a measurement in the computational basis to give the outcome j. In general, this circuit and measurement will act on a larger Hilbert space than is required to hold the original input. The quantum circuit will then correspond to some unitary operation Uxon the larger space. Without loss of generality, we can assume that the ﬁnal measurement is in a basis {|j/angbracketright}such that the values j∈ {0,1,...,N −1}indicate the measurement outcome Ej. According to Neumark’s theorem [43, Chap. 9.6], the unitary operator Uxhas the block form Ux=/parenleftbigg VxAx BxCx/parenrightbigg (65) where Vx:=1√ N/summationdisplay j,q∈ZNω−jq|j/angbracketright/angbracketleftSx q| (66) is a ﬁxed ( N×2k)-dimensional matrix whose columns are the (subnor- malized) vectors corresponding to the rank one povm elements {Ex j}, and Ax,Bx,Cxare arbitrary up to the requirement that Uxis unitary. It is convenient to perform a Fourier transform on the left, i.e., on the index j 18 DAVE BACON, ANDREW M. CHILDS, AND WIM VAN DAM (overZN, for the relevant values j∈ {0,1,...,N −1}), giving a unitary operator ˜Ux=/parenleftbigg˜VxAx ˜BxCx/parenrightbigg (67) with ˜Vx:=1 N/summationdisplay j,p,q∈ZNωj(p−q)|p/angbracketright/angbracketleftSx q| (68) =/summationdisplay p∈ZN|p/angbracketright/angbracketleftSx p|. (69) Clearly,Uxcan be implemented eﬃciently if and only if ˜Uxcan be imple- mented eﬃciently. Therefore, if we have an eﬃcient quantum circuit for the transformation (70) |p,x/angbracketright /mapsto→/braceleftBigg |Sx p,x/angbracketrightηx p>0 |ψx p/angbracketrightηx p= 0 where |ψx p/angbracketrightis any state allowed by the unitarity of ˜Ux(i.e., if we can eﬃ- ciently quantum sample from subset sum solutions for legal inputs), then by running this circuit in reverse, we can eﬃciently implement ˜Ux, and hence the measurement. Conversely, given the ability to implement the optimal povm by the mea- surement of xfollowed by an eﬃcient implementation of Ux, we can solve the subset sum problem. By running the quantum circuit for ˜Uxin the reverse direction, we can eﬃciently implement the transformation (70). Suppose we are trying to solve the subset sum problem for a legal instance ( x,t). Using (70), we can produce the state |Sx t/angbracketright, which upon measurement gives a uni- formly random subset of xsumming to t. On the other hand, if the instance is not legal, then we can easily check that the output does not correspond to a subset of xsumming to t. If we could eﬃciently implement the unitary operation ˜Uxfor anyk= poly(logN), then we could solve the subset sum problem eﬃciently even in the worst case. Since the subset sum decision problem is NP-complete, such an implementation seems unlikely. However, for the purpose of solving thedhsp , it is suﬃcient for each Nto consider a speciﬁc k=νlogN(with ν >1, according to Theorem 2) and

Question 89 multiple-choice

Quantum reductions are a foundational tool for understanding the relative hardness of cryptographic problems, especially in the development of secure post-quantum cryptosystems. Techniques such as the quantum Fourier transform and custom quantum circuits are often employed to construct reductions between problems like EDCP and LWE. Which operation is essential in both quantum reductions between the Extended Dihedral Coset Problem (EDCP) and Learning with Errors (LWE) for extracting structural information from quantum superpositions? 1) Quantum Fourier Transform (QFT) 2) Grover's search algorithm 3) Quantum phase estimation 4) Quantum error correction 5) Classical post-processing only 6) Quantum teleportation 7) Quantum amplitude amplification

✓ Correct Answer:

The correct answer is 1) Quantum Fourier Transform (QFT).

📚 Reference Text:

building quantum reductions between them. The EDCP problem over D Nis speciﬁed as follows: Given /lscriptmany registers in a normalized state corresponding to /summationdisplay j∈Ze−π|j|2 r2|j,(xi+j·s)mod N/angbracketright where xi∈Zn N(i=1,...,/lscript ), and s∈Zn Nis ﬁxed, the objective of EDCP is to ﬁnd the secret value s. (1) Quantum reduction from LWE to EDCP. An instance of LWE problem over the lattice L(A),A∈Zm×n q , can be reduced to an instance of EDCP problem over the dihedral group DN,N=2n, according to the following quantum steps (Fig. 6): •Step 1 Initialize the four registers with required qubits |ϕ1/angbracketright=| 0/angbracketright|0/angbracketright|0/angbracketright|0/angbracketright •Step 2 Perform QFT on the second register (normalization omitted) |ϕ2/angbracketright=/summationdisplay s∈Znq|0/angbracketright|s/angbracketright|0/angbracketright|0/angbracketright 123 Quantum algorithms for typical hard problems… Page 21 of 26 178 •Step 3 Apply GR02 algorithm [ 81] in the ﬁrst register, which is a quantum process to create a superposition state according to given probability distribution |ϕ3/angbracketright=/summationdisplay s∈Znq(/summationdisplay j∈Zρr(j)|j/angbracketright)|s/angbracketright|0/angbracketright|0/angbracketright •Step 4 Suppose that the quantum circuit Uf Uf|j/angbracketright|s/angbracketright|0/angbracketright→| j/angbracketright|s/angbracketright|As−jbmod q/angbracketright is at hand. Apply U fon the ﬁrst three registers |ϕ4/angbracketright=/summationdisplay s∈Znq,j∈Zρr(j)|j/angbracketright|s/angbracketright|As−j·As0−je0/angbracketright|0/angbracketright =/summationdisplay s∈Znq,j∈Zρr(j)|j/angbracketright|s+js0/angbracketright|As−je0/angbracketright|0/angbracketright •Step 5 Further, suppose that the quantum circuit Ug Ug|x/angbracketright|0/angbracketright→| x/angbracketright|x/z−wmod¯q/angbracketright is at hand, where ¯q=q/z=c. Apply U gon the last two registers, and we get |ϕ5/angbracketright=/summationdisplay s∈Znq,j∈Zρr(j)|j/angbracketright|s+j·s0/angbracketright|As−je0/angbracketright|g(As−je0)/angbracketright •Step 6 Measure the fourth register and discard it |ϕ6/angbracketright=/summationdisplay j∈Zρr(j)|j/angbracketright|s+j·s0/angbracketright|As−je0/angbracketright •Step 7 Apply Ufto the ﬁrst three registers, the third register gives 0 and discard it, and the state is of the form |ϕ7/angbracketright=/summationdisplay j∈Zρr(j)|j/angbracketright|s+j·s0/angbracketright •Step 8 Repeat the above procedure /lscripttimes, and we obtain /lscriptmany EDCP states with probability (1−1 k)m/lscript |ϕEDCP/angbracketright={/summationdisplay j∈Zρr(j)|j/angbracketright|s+j·s0/angbracketright}k≤/lscript where xk∈Znq. 123 178 Page 22 of 26 J. Suo et al. Fig. 7 From EDCP to LWE [80].FZnqis Fourier transform over Znq,a n d UED CP is the output stateρt(j)|j FZnq ULWE |x+j·s0FZnq ˆa (2) Quantum reduction from EDCP to LWE. The reverse quantum reduction from EDCP to LWE is given below (Fig. 7). •Step 1 Prepare the input state |ϕ1/angbracketright=/summationdisplay j∈Zρr(j)|j/angbracketright|x+j·s0mod q/angbracketright •Step 2 Apply QFT on the second register |ϕ2/angbracketright=/summationdisplay a∈Znq/summationdisplay j∈Zω/angbracketleft(x+j·s0),a/angbracketright q ·ρr(j)|j/angbracketright|a/angbracketright where ωq=e2πi q •Step 3 Measure the second register and obtained ak, omitting global phase ω/angbracketleftx,/hatwidea/angbracketright q |ϕ3/angbracketright=/summationdisplay j∈Zωj·/angbracketleft/hatwidea,s0/angbracketright q ·ρr(j)|j,/hatwidea/angbracketright •Step 4 Apply QFT on the ﬁrst register |ϕ4/angbracketright=/summationdisplay b∈Zq/summationdisplay j∈Zqωj·(/angbracketleft/hatwidea,s0/angbracketright+b) q ·ρr(j)|b/angbracketright Using Poisson summation formula to reorganize |ϕ4/angbracketright, then |ϕ4/angbracketright=/summationdisplay e∈Zρ1 2/parenleftbigge q/parenrightbigg |−/hatwidea,s0/angbracketright+emod q/angbracketright •Step 5 Measure the ﬁrst register, and we can obtain an LWE sample |ϕLWE/angbracketright=(−/hatwidea,/angbracketleft−/hatwidea,s0/angbracketright+ek) 9 Conclusion With the rapid development of quantum computing, it broke through the defense line of the classic cryptosystems, which makes the post-quantum cryptography become the frontier of research. In order to search the novel cryptography which is resistant to 123 Quantum algorithms for typical hard problems… Page 23 of 26 178 quantum attack, it is of great necessity to conduct a systematical analysis of the quan- tum algorithms that could solve the typical hard problems. In this paper, we start from the typical hard problems: integer factorization problem, discrete logarithmic problemand dihedral hidden subgroup problems in the public-key cryptosystem (respectively, RSA, ElGamal,

Question 90 multiple-choice

Hopf algebras constructed from quivers and categories are fundamental structures in modern algebra, with rich interactions between categorical properties, module categories, and specialized algebraic relations. The interplay between morphism composition, tensor products, and specialization at roots of unity yields diverse algebraic frameworks with applications in representation theory and quantum groups. Which property ensures that non-composable morphisms in the associative algebra Kc constructed from a category are represented by orthogonal elements, and what is the algebraic consequence of this? 1) Non-trivial kernel in the comultiplication, causing all morphisms to act identically 2) Existence of a two-sided ideal generated by identity morphisms, making all compositions non-zero 3) Direct product decomposition of Kc into simple modules, ensuring full reducibility 4) Orthogonality of non-composable morphisms, meaning their product is zero and respects the underlying categorical structure 5) Coassociativity of the tensor product, making all morphism pairs composable 6) Specialization at roots of unity, causing certain paths to collapse to the unit 7) Action of the free group by translations, associating every morphism to a unique matrix entry

✓ Correct Answer:

The correct answer is 4) Orthogonality of non-composable morphisms, meaning their product is zero and respects the underlying categorical structure.

📚 Reference Text:

In this situation Kc = 2tf is an associative iC-algebra: if/and g are in Kc their product^ is well defined byfg(γ) = Σ0α=y/(β)0(oc); the unit element is the sum of all the objects and two non-composable morphisms are orthogonal as elements of Kc. Notice that Jtf* = Kc is actually isomorphic to the algebra of infinite lower triangular matrices when C is deduced from A™. The proof of Theorem 2.1 suggests that Jf is a Hopf algebra provided that we consider the adequate tensor product as a receptacle for the comultiplication since Afa) = Σj+k = ί Sj ® ak + Xj+fe=i^J dj ® sk is not an element of Kc (x) Kc. Never- theless C x C is again obtained through a cofinite category and the expression above lies in the algebra KCxC of possibly infinite sums of couples (/?, α) = β (x) α. 466 C. Cibils Let us consider A: Kc -> KCxC given by (\ U/q j+k=i w as suggested by Lemma 3.1. This gives a well defined map on the entire algebra Kc since for/eKc the expression AfM, yj) = ( ) qukf(yuj+k) does not involve \ u A infinite sums. Notice that A is an algebra map and that the coassociativity has to be understood in the following sense: there is a map that we still denote A (x) 1 from κc«c to χcχcχc given by which is well defined. There is also an analogous map denoted 1 (x) zl and it is easy to check that (A (g) 1)A = (1 (x) A) A. Finally there is an antipode S and a counit,., m(m+l) β given by [S(/)](yΠ = ( - IΓ/ί^i-J <Γm(' m) ~^~ and ε(/) =/(yg) =/(s0) . An important fact that we shall use is that the category of ^-free finite rank left ^-modules still forms a monoidal category even if the receptacle of A is KCxC instead of Kc ® Kc. Indeed if M is a J^-free finite rank ^c-module almost all elements of C has zero action on it since M = ©;ezS;M. It follows that if M and N are two such modules the transformations of M (x)x N associated to elements of CxC are almost all zero and this enables us to consider M®KN as a KCxC- module. Using A: Kc -> KCxC we obtain the Xc-module structure on M ®KN. The free group on one generator acts on A% by translations of amplitude n for each n ^ 2; the orbits give the path algebra k(q)Zn. Actually specializing q to an nih root of unity allows to get a Hopf algebra structure on kZn which is of course the one we have considered before. Remark 3.12. Let 3? be the fc(g)-Hopf algebra attached to the quiver A™ and let Id be the two-sided ideal of 3? generated by the paths of length a. Specializing q to a primitive dth root of unity enables us to consider the /c-Hopf algebra 3Ί?(q)/Id, as in Proposition 3.3; its

Question 91 multiple-choice

In group theory, central extensions and Burnside groups are central topics in the study of infinite groups and group varieties. Understanding the relationships between centers, quotients, and cohomological invariants reveals deep structural properties of groups. Which construction allows any countable abelian group to be realized as the center of a group whose quotient is isomorphic to a free Burnside group of given rank and odd period? 1) A central extension using a countable abelian group as the center and a free Burnside group of odd period as the quotient 2) A direct product of a free Burnside group and a cyclic group of finite order 3) An amalgamated free product of abelian groups and finite cyclic groups 4) A semidirect product of a Burnside group with an infinite symmetric group 5) A wreath product of a countable abelian group with a finite cyclic group 6) A quotient of an infinite cyclic group by a finite abelian subgroup 7) An HNN extension with a free Burnside group as the base group

✓ Correct Answer:

The correct answer is 1) A central extension using a countable abelian group as the center and a free Burnside group of odd period as the quotient.

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Question 92 multiple-choice

In representation theory and symplectic geometry, moment polytopes encode geometric and algebraic information about group actions on vector spaces. The structure of these polytopes is closely related to the facets defined by certain elements associated with the representation. Which condition must be satisfied by a Ressayre element (H, z) to ensure that the associated matrix constructed from root operators and weight vectors is meaningful for describing a non-trivial facet of the moment polytope? 1) The matrix must be symmetric and positive definite. 2) The determinant of the matrix must be zero. 3) The matrix must have strictly positive eigenvalues. 4) The trace of the matrix must be equal to the dimension of the representation. 5) The determinant of the matrix must be non-zero. 6) The matrix must be diagonalizable over the complex numbers. 7) The rank of the matrix must be less than that of the weight space.

✓ Correct Answer:

The correct answer is 5) The determinant of the matrix must be non-zero..

📚 Reference Text:

Question 93 multiple-choice

In mathematical physics and representation theory, understanding the structure of Lie group representations and their reducibility is essential for analyzing symmetries in quantum systems. Compact Lie groups, operators, and commutation relations play foundational roles in this analysis. Which property of compact Lie groups is essential for the validity of Haar averaging, used to ensure that every finite-dimensional representation is equivalent to a unitary representation? 1) The group has a non-Abelian structure 2) The group has an infinite-dimensional center 3) The group contains raising and lowering operators 4) The group is simple but not semisimple 5) The group admits only reducible representations 6) The group has a discrete topology 7) The group has finite volume under its natural topology (compactness)

✓ Correct Answer:

The correct answer is 7) The group has finite volume under its natural topology (compactness).

📚 Reference Text:

be true forSU(2)): there are counterexamples for all SU(d) withd>3, whereinthesamedimensionalvectorspacemay have several inequivalent representations. Secondly, we see the powerful strategy for detecting equivalent representa- tions (partially aforementioned after Example 9), at play: diagonalize the representatives of Zin both representations (here, the maps Z 1+1 Zon Sym2(C2)andadZ()on su(2)),then use other operators and commutation relations to move between these eigenbases . Let us see how this strategy played out. We began the tensor example by diagonalizing the Zrepresentative in the tensor rep by choosing the basis j11i;j10i+j01i;j00i. In this basis, the Zrepresentative has respective eigenval- ues 2, 0, and -2. Then, the raising and lowering operators map between these eigenspaces, which looks like “rais- ing” or “lowering” the eigenvalue. It becomes clear that we can map any of these eigenspaces to each other by raising and/or lowering operators: thus, we have an irreducible subspace. When we got to the adjoint representation and saw thatadZ()had the same eigenvalues, this suggested a similar strategy should work. There is nothing inherently special about the Zoperator here–it just so happens to be fairly easy to diagonalize in both cases. This generalizes to more complicated Lie algebras by simultaneously diagonal- izing a commuting collection of operators (called a Cartan subalgebra ) and cleverly tracking their eigen-information using so-called “weights”. In practice, diagonalization in a given representation may be highly nontrivial. Checking the reducibility of any acquired invariant subspace may also be a diﬃcult task, and often requires either problem speciﬁc insight or any variety of representation theoretic tools. It cannot however be understated how crucial this strat- egy is. It undergirds the classiﬁcation of semisimple Lie al- gebras. Italsoelegantlyframestheanalysisofthequantum harmonic oscillator: there, we diagonalize the self-adjoint number operator ayato obtain eigenstates, and then use 25 the canonical commutation relations [x;p] =i~combined withpositivityof ayatomovebetweeneigenstatesvia“rais- ing and lowering” ladder operators. This is well trodden and exposited ground, and every reference we list contains thiscomputation. InHall[51], thiscomputationappearsin the section “Representations of sl2(C)” (recall that as com- plexiﬁed Lie algebras, su(2)=sl2(C), and so they have the same representation theory). Theorem 5 (Weyl’s unitary trick for complete reducibil- ity).LetGbe a compact Lie group. Then every ﬁnite di- mensional representation of Gis equivalent to a unitary representation, and by Theorem 4, completely reducible. While we omit the (brief) proof here, there are two im- portant messages from it to be mentioned: •The equivalence is established by performing a Haar averaging procedure15to the dot product h;ion V, much akin to that of the twirl operator (we will visit these in Sections VIIA, VIIB). These sorts of Haar averaging methods are of great importance across quantum computation and representation the- ory alike, since the Haar measure is the most “natu- ral” probability distribution on Lie groups. •Compactness is critical here: without it, the Haar measure of our Lie group jGjmay not be ﬁnite and the averaging procedure is nonsensical. The ﬁnal result we mention is perhaps the most critical weapon any researcher needs in their arsenal before using representation theory to fully tackle QML

Question 94 multiple-choice

In quantum information theory, symmetries are deeply connected to the structure and implementation of quantum channels. Programmable quantum processors are devices used to flexibly realize families of quantum operations, often leveraging properties such as group representations and associated mathematical tools. Under what condition does a programmable quantum processor implementing covariant quantum channels admit a measure-and-prepare form, simplifying its structure? 1) When the symmetry group is non-abelian and acts reducibly on the input space 2) When the tensor representation has only one irreducible component 3) When the commutant of the tensor representation is abelian 4) When the dimension of the program space equals the dimension of the input Hilbert space 5) When the symmetry group contains only unitary elements 6) When the Choi-Jamiołkowski state is maximally mixed 7) When every representation of the group is reducible

✓ Correct Answer:

The correct answer is 3) When the commutant of the tensor representation is abelian.

📚 Reference Text:

also hold for ﬁnite groups. Symmetries are of fundamental importance in physics, since they give rise to conserved quantities via Noether’s theorem [33]. In open systems, these symmetries arise as covariant quantum channels and are studied using tools from quantum information theory [11, 29]. From a practical point of view, symmetries often simplify problems and break the curse of dimen- sionality, thus making them amenable to rigorous analysis. For these reasons, covariant quantum channels appear in many diﬀerent settings, such as channel discrimination, ca- pacities and communication tasks (see Ref. [30] and the references therein). Since this is a special set, the question arises whether those channels can be implemented exactly by a programmable quantum processor. In this article, we show that an exact implementation is possible if the group acts irreducibly on the channel input. While we prove upper bounds on the program dimension for general representations U, we focus on the case in which U is irreducible. As any representation can be decomposed into irreducible ones, it is natural to start studying this scenario before investigating more general representations in future work. In Section 2, we present some preliminaries and our notation. We consider exact pro- grammability of group-covariant channels in Section 3, where we ﬁrst look at a method based on extreme points in Subsection 3.1. We show that processors have a particularly simple measure-and-prepare form if and only if the commutant of the tensor representation is abelian. This yields a program dimension equal to the number of irreducible representa- tions occurring in the direct sum decomposition of the tensor representation in Corollary 19. Subsection 3.2 discusses the structure of the commutant of the tensor representa- tion in more detail. We give a diﬀerent construction of covariant programmable quantum processors based on teleportation in Subsection 3.3. This construction is subsequently concatenated with a compression map which allows us to utilize the special structure of the Choi-Jamiołkowski states corresponding to the covariant channels. This leads to The- orem 25, where we show that we obtain a program dimension of at most the sum of the dimensions of the blocks occurring in the structure of the Choi-Jamiołkowski states. After the analysis of exact programmability, we consider an approximate version thereof (see Section 4). First, we provide approximate upper bounds in the case of arbitrary repre- sentationsU,Vin Proposition 29. They are in general worse than the exact bounds in Theorem 25, but they apply more generally. This result is the only one in which we con- sider arbitrary representations Uinstead of irreducible ones. In Theorem 31, we provide lower bounds on the program dimension of approximate covariant quantum processors. In particular, this shows that the construction in Theorem 25 is optimal for the exact case. 2 Preliminaries We use the following notation: Let d1,d2∈N. We denote the set of bounded linear operatorsH1→H 2withd1- andd2-dimensional Hilbert spaces H1andH2byB(H1)and B(H2), respectively. The set of all d1-dimensional density operators is D(H1) ={ρ∈B(H1)|ρ≥0,tr(ρ) = 1}, Accepted in Quantum2021-06-18, click title to verify. Published under

Question 95 multiple-choice

Quantum Markov semigroups (QMS) play a fundamental role in modeling the dynamics of open quantum systems, particularly when symmetries are present via group representations. The relationship between classical and quantum processes is often explored through the structure of fixed points and their connection to operator algebras. In the construction of a quantum Markov semigroup on operators over a Hilbert space H, arising from a projective unitary representation u(g) of a group G, which set precisely characterizes the fixed points of the semigroup? 1) All diagonal operators in B 2) The center of B 3) Operators invariant under all *-automorphisms of B 4) The set of all positive operators in B 5) The commutant of the projective representation, i.e., all X in B satisfying X u(g) = u(g) X for all g in G 6) The set of trace-class operators in B 7) Operators with support orthogonal to u(g) for all g in G

✓ Correct Answer:

The correct answer is 5) The commutant of the projective representation, i.e., all X in B satisfying X u(g) = u(g) X for all g in G.

📚 Reference Text:

of completness. We have, for any x∈B(H), ○Tt(x)(g) =u(g)∗/integral.dispGkt(h−1)u(h−1)xu(h)dG(h)u(g) =/integral.dispGkt(gg−1h−1)u((hg)−1)xu(hg)dG(h) =/integral.dispGkt(gh−1)u(h−1)xu(h)dG(h) =(St⊗idB(H))((x))(g): From the invariance of G, one can also easily verify that any QMS (Tt)t≥0transferred from (St)t≥0is doubly stochastic :T† t(d−1 HIH)=d−1 HIHfor anyt≥0. On the other hand, the reversibility of (St)t≥0is transferred to the QMS (Tt)t≥0: Lemma II.2. Assume that the Markov semigroup (St)t≥0 is reversible, or equivalently that kt(g)=kt(g−1)for any g∈G. Then any QMS (Tt)t≥0transferred from (St)t≥0 is self-adjoint with respect to d−1 HIH. Proof. The result follows from the simple calculation: /uni27E8x;Tt(y)/uni27E9HS =/integral.dispGkt(g−1)Tr(x∗u(g)∗yu(g))dG(g) =/integral.dispGkt(g−1)Tr((u(g)xu(g)∗)∗y)dG(g) =/integral.dispGkt(g−1)Tr((u(g)∗xu(g))∗y)dG(g) =/uni27E8Tt(x);y/uni27E9HS; where the third line follows from the identity kt(g)= kt(g−1)for allg∈G. Sinced−1 HIHis an invariant state of (Tt)t≥0, the setNfixof ﬁxed points is an algebra (see [9], [12], Theorem 6.12 of [10]), and is characterized as the commutant of the projective representation (see Theorem 6.13 of [10]): Nfix={∈B(H)/divides.alt0∀g∈G∶u(g)=u(g)}≡u(G)′: (5) By deﬁnition, it is also the algebra of ﬁxed points of the∗-automorphisms x/uni21A6u(g)∗xu(g),g∈G. This implies that the following commuting diagram holds: 3 B(H) Nfix L∞(G;B(H)) B(H):Efix EG HereB(H)in the lower right corner has to be understood as the subalgebra of constant value functions on Gwith value in B(H). In practice, we will only consider situations where the classical Markov semigroup (St)t≥0isprimitive , that is,Gis the unique invariant distribution and furthermore Stf/leftrightline→ t→+∞EG[f]=/integral.dispGf(g)dG(g): This does not imply that (Tt)t≥0is also primitive, however, it will always be ergodic as deﬁned in Equation (1). We now turn our attention to two special cases of the above construction. In both cases, we explicitly construct the Lindblad generator of the QMS. B. Diffusion Given a Riemannian manifold M, aHörmander system onM is a set of vector ﬁelds V= {V1;:::;Vm}such that, at each point p∈M, there exists an integer Ksuch that the iterated commutators /bracketleft.alt1Vi1;[Vi2;/uni22EF[Vik;⋅]]/bracketright.alt,k=1;:::;K , generate the tangent space TpM. Specializing to the case of a Lie groupG, a Hörmander system V={V1;:::;Vm}can more simply be deﬁned as a set of vectors in the Lie algebra, i.e. the tangent space at the neutral element e, such that for some K∈Nthe iterated commutators of order at most Kspan the whole tangent space. For ﬁxed j∈{1;:::;K}, we ﬁnd a geodesic gj(t)withgj(0)=e such that for any f∈C1(G) Vj(f)(h)=d dtf(gj(t)h)/divides.alt3 t=0: This leads to the corresponding left invariant classical generator LV∶=−/summation.disp jV2 j: (6) The generator LVgenerates a Markov semigroup Pt=e−tLVonL∞(G). Whenever Vis a basis for the Lie algebra, then LV≡−is the negative of the Laplacian and (Pt)t≥0is called the heat semigroup . Since the semigroup commutes with the right action of the group it is implemented by a right-invariant convolution kernel as in Equation (2) and it is reversible with respect to the Haar measure. Next, considering a projective representation g/uni21A6 u(g)ofGon some ﬁnite dimensional Hilbert space H, we want to ﬁnd the Lindblad generator of the QMS deﬁned by Equation (3). We ﬁrst observe that, for ﬁxed j∈{1;:::;K}and given the geodesic gjassociated tothe vector ﬁeld Vj,u(gj(t))is a one-parameter family of unitaries and hence d dtu(gj(t))/divides.alt3 t=0=iaj (7) whereaj∈B(H)is self-adjoint. This implies that, for anyx∈B(H), (Vj⊗idB(H)○(x)(g) =d dt(x)(gj(t)g)/divides.alt3

Question 96 multiple-choice

Transformation monoids are algebraic structures that generalize permutation groups by considering actions of monoids on sets, with important applications in combinatorics, representation theory, and theoretical computer science. Primitivity and orbital digraphs are key concepts for understanding the symmetry and structure of these actions. In the context of finite transformation monoids, which statement correctly describes the connection between primitivity and orbital digraphs as established by the monoid version of Higman's theorem? 1) A transformation monoid is primitive if and only if all its non-trivial orbital digraphs are connected. 2) A transformation monoid is primitive if and only if all its orbital digraphs are acyclic. 3) A transformation monoid is primitive if and only if every element acts as a permutation. 4) A transformation monoid is primitive if and only if its orbital digraphs have minimal rank. 5) A transformation monoid is primitive if and only if its transformations preserve every partition of the set. 6) A transformation monoid is primitive if and only if its orbital digraphs are bipartite. 7) A transformation monoid is primitive if and only if there exists a transformation mapping every element to a fixed point.

✓ Correct Answer:

The correct answer is 1) A transformation monoid is primitive if and only if all its non-trivial orbital digraphs are connected..

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Question 97 multiple-choice

The hidden subgroup problem (HSP) is fundamental in quantum computing, underpinning many algorithms that achieve exponential speedups. Solving HSP in nonabelian groups, such as semidirect products of cyclic groups, presents unique challenges and opportunities for algorithmic advancement. Which of the following statements correctly identifies a key technical strategy used by a polynomial-time quantum algorithm for the HSP over the group $\mathbb{Z}_{p^r} \rtimes \mathbb{Z}_{q^s}$, where $p$ and $q$ are odd primes? 1) Employing the nonabelian quantum Fourier transform to directly identify all subgroup structures 2) Utilizing quantum amplitude amplification to distinguish between normal and non-normal subgroups 3) Applying Grover’s search to enumerate hidden subgroups exhaustively 4) Reducing the problem to solving instances of the graph isomorphism problem 5) Using the abelian quantum Fourier transform and reducing the problem to finding cyclic subgroups 6) Implementing quantum walks to traverse the group’s Cayley graph 7) Leveraging classical algorithms for abelian subgroup decomposition

✓ Correct Answer:

The correct answer is 5) Using the abelian quantum Fourier transform and reducing the problem to finding cyclic subgroups.

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Question 98 multiple-choice

Quantum dot-cavity systems are central to advancing photonic quantum computing, enabling high-fidelity quantum logic gates and scalable implementations of algorithms such as the discrete quantum Fourier transform. Key physical parameters determine the efficiency and reliability of photon-based quantum operations. Which parameter regime is essential for ensuring strong coupling and high-fidelity interaction between photons and quantum dot-cavity systems in the implementation of deterministic quantum logic gates? 1) g ≫ (κ, γ) and κs ≪ κ 2) g ≪ (κ, γ) and κs ≫ κ 3) κ ≫ g and γ ≫ κs 4) γ ≫ g and κ ≪ κs 5) κs ≫ κ and g ≈ γ 6) g ≈ κ and γ ≈ κs 7) κ ≈ γ and g ≪ κs

✓ Correct Answer:

The correct answer is 1) g ≫ (κ, γ) and κs ≪ κ.

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trapped atoms used as qubits. Compared with the previous schemes15,17,21,22,25,27,31–33, our DQFT scheme consists of the QD-cavity systems (nonlinearly optical devices) to implement the CRk operations as the basic modules for the deterministic perfor - mance and simple expansion into multi-qubit DQFT. Moreover, in our scheme, the flying photons play the roles of qubits transferring quantum information, and QDs are the ancillary systems to efficiently perform the CRk operations for the applications of various quantum computations and algorithms. Also, In view of the experimental realization, for the high fidelity (condition of g ≫ (κ, γ) and κ s ≪ κ) of the interaction between a photon and the QD-cavity system, researchers have proposed a variety of experimental techniques, including the following: Rosenblum et al . 68 obtained the coupling strength as g /(κ + κs) ≈ 0.5 in a micropillar cavity (d = 1.5 μm) with a quality factor Q = 8800, and for Q = 40000, the coupling strength could be increased to g /(κ + κs) ≈ 2.474. Furthermore, Arnold et al .76 enhanced the quality factor, Q = 215000 (κ ≈ 6.2 μeV), in order to acquire a small side-leakage rate. Reitzensteina et al .77 demonstrated two methods (the etching process and improving the sample growth) to reduce the side-leakage rate, κs/κ, in an In0.6Ga0.4As optical cavity having g/(κ + κs) ≈ 2.4 and Q = 40000. Thus, the QD-cavity systems in our CRk gate can be reliably operated with high fidelity, according to the results of our analysis presented in Sec. 4. Moreover, in order to ensure the reliable interaction between a flying photon and stationary qubit (electron spin in QD), the initialization (elec-tron spin-superposition state) and manipulation of the spin state can be prepared by optical pumping or optical cooling 78, and can be acquired using pulsed magnetic resonance techniques, nanosecond microwave pulses, or picosecond/femtosecond optical pulses79–82. Therefore, we obtain the interaction of the CRk gate, using the QD-cavity system, with high fidelity for the DQFT scheme. Consequently, according to our analysis results, in practice (under vacuum noise and sideband Figure 6. Fidelities, F1 (QD1 gate: ω − ωc = κ/2) and F2 (QD2 gate: ω − ωc = 0) of the output states for the side- leakage rate κs/κ and coupling strength g/κ with γ/κ = 0.1 and ω ω=−c X under vacuum noise, N(ω), for the operation of the QD-dipole and leaky modes, S(ω) (sideband leakage and absorption). In the tables, the values of fidelities F 1 and F2 are listed for the differences in κs/κ with g/κ = 2.5, and also the differences in g/κ with κs/κ = 1.0. 10 Scientific RepoRtS | (2019) 9:12440 | https://doi.org/10.1038/s41598-019-48695-z www.nature.com/scientificreports www.nature.com/scientificreports/leakage), we can obtain high efficiency and reliable performance of the interaction between a photon and QD within a single-sided optical cavity (CRk gate). Our results also indicate that our DQFT scheme using CRk gates can be experimentally feasible, as well as scalable for multi-qubit DQFT. References 1. Kitaev, A. Quantum measurements and the

Question 99 multiple-choice

Quantum simulations of lattice systems often require adaptation of mathematical techniques, especially for non-standard system sizes. Discrete symmetries can play a crucial role in the evolution and observable dynamics of such systems. In a quantum simulation of a lattice system with L=33 sites, which statement best describes the role of discrete Lorentz symmetry in the observed quantum walk dynamics? 1) It guarantees energy conservation throughout the simulation. 2) It enables efficient encoding of quantum information using only edge sites. 3) It enforces thermal equilibrium by periodically redistributing probabilities. 4) It partitions lattice sites into equivalence classes with identical wavefunction values, leading to collapse and revival patterns. 5) It eliminates the need for numerical decomposition of the Fourier matrix for local implementation. 6) It suppresses all effects of Gaussian noise added to the Hamiltonian. 7) It ensures the tunneling matrix contains only nearest-neighbor terms.

✓ Correct Answer:

The correct answer is 4) It partitions lattice sites into equivalence classes with identical wavefunction values, leading to collapse and revival patterns..

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displayed in Table. I. Its corresponding tunneling ma- trix necessarily involves long-range terms. We apply the QQFT scheme to quantum simulation of this model. Since L=33 is not an integer power of 2, for experimental realization, we perform numerical decomposi- tion of the Fourier matrix38for local implementation of the QQFT. In order to benchmark the performance of our ap- proach, we investigate single-particle quantum walks, which can be measured using quantum microscopes in cold atom ex- periments55,56. We consider quantum walk of a single-particle initialized at site n=0. The time ( t) evolution of its wave- function is denoted as n(t). The spacetime crystallization is described by n(t) at discrete times t=sT, with san inte- ger and Ta time period determined by the Lorentz invariant energy dispersion49. The symmetric properties are captured by the matrixGns n(sT). The discrete Lorentz symmetry implies49, Gns=Gn0s0; (29)with s0 n0! =L s n! (mod L): (30) Here, 2 takes integer values, and the coordinates nand sare deﬁned by modulo L. On the spacetime lattice f(s;n)g with 0s;nL1, the transformation connects one site to another. The LLsites are then partitioned into a few equiva- lence classes in each of which the wavefunction Gnsmust have the same value. As shown in Fig. 3(a), Gdisplays a periodic collapse and revival pattern, which respects the Lorentz sym- metry (Eq. (29)). It has been established by one of the authors that the nontrivial quantum revival dynamics is a consequence of the Lorentz symmetry49. Considering the Li atom experiment setup described above, the total evolution time to simulate the Poincar ´e dynamics in Fig. 3 with system size L=33 is estimated to be 2 :7 seconds. This in principle can be improved by analytically decompos- ing the QQFT according to 33 =311. B. Robustness of simulations against noise To quantify the robustness against imperfections potentially existent in experiments, we add Gaussian noise to the Hamil- tonian sequence (Eq. (5)), replacing H[s] pby (1 +s)H[s] p, with sa random variable drawn from Gaussian distribution char- acterized by the standard deviation . The consequent e ects on the quantum spatiotemporal dynamics are shown in Fig. 3. With increasing the noise strength, we ﬁnd that the discrete Poincar ´e symmetry is gradually broken. The symmetric re- vival dynamics is evident even at a noise level of =5103. We provide a quantitative analysis of the noise-induced Poincar ´e symmetry breaking. If a particle is located at the siten1at the initial time t=0, we then use n1+n(t) to denote its wave function at the time t=sTand the site n1+n. Then, Pn1(s;n)= n1+n(t) 2denotes the probability of a particle hop- ping from the initial site n1to the site n1+nat the time sT. For 7 the conservation of probability,P nPn1(s;n)1 must hold for arbitrary n1ands. The translation symmetry guarantees that Pn1(s;n) is independent of n1. More important, the Lorentz symmetry guarantees P(s;n)=P(s0;n0) for ( s;n) and ( s0;n0) satisfying Eq. (30). The equivalence relation (30) partitions all the lattice sites f(s;n)ginto several equivalence classes, de- noted as C

Question 100 multiple-choice

Representation theory of algebraic groups studies modules and their bases using combinatorial and geometric methods. Demazure modules and moment polytopes are closely connected to the structure of Lie algebras and the computational complexity of related problems. Which of the following statements most accurately describes a key outcome of Lakshmibai's inductive construction for Demazure modules in terms of computational complexity and representation theory? 1) The construction provides a basis for Demazure modules, but requires exponential time due to the complexity of the Bruhat order. 2) The basis for Demazure modules cannot be extended to reducible modules or tensor products. 3) Moment polytope membership is undecidable because monomial bases cannot be computed efficiently. 4) The Bruhat order and reduced decomposition computations must be approximated, as exact computation is infeasible. 5) Lakshmibai monomial bases are only practical for small Weyl groups due to computational constraints. 6) The weight computations for Lakshmibai bases require non-polynomial time algorithms for general Lie algebras. 7) Lakshmibai's method produces explicit monomial bases for Demazure modules, and all steps—including Bruhat order, reduced decompositions, and tensor product decompositions—are computable in polynomial time, enabling efficient algorithms for moment polytope membership.

✓ Correct Answer:

The correct answer is 7) Lakshmibai's method produces explicit monomial bases for Demazure modules, and all steps—including Bruhat order, reduced decompositions, and tensor product decompositions—are computable in polynomial time, enabling efficient algorithms for moment polytope membership..

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Note that M;1=Cv, whileM;w`=M. We will now describe Lakshmibai’s inductive construction of sets M;wrof monomials of the form x=fns sfn1 1, with i2PS(G)and eachni>0, such that each B;wr:=(M;wr)vis a basis of the Demazure module M;wr. It will be useful to deﬁne w(x) :=s ss 1and (x) :=Ps i=1ni ifor any such monomial. For r= 0, we deﬁneM;w 0:=f1g. ThusB;wr=fvgin this case. For r>0, we consider (8) N;wr1:=fx2M;wr1:wr16s rw(x)g; wheredenotes the Bruhat order, noting that M;wr1has already been deﬁned by the induction hypothesis. For each x2N;wr1, lettxdenote the weight of (x)v with respect to sl r 2, which is a positive integer [ 29]. That is, tx= (x)(h r), since (x)is the gC-weight of(x)v. We ﬁnally deﬁne (9)M;wr:=M;wr1[ffi rx:x2N;wr1;i= 1;:::;txg; which is a disjoint union. Lakshmibai has shown that, for each r= 1;:::;`,B;wr= (M;wr)vis a basis of the Demazure module M;wr(see [29, Theorem 4.1] and its proof). In particular, B:=B;w`is a basis of the irreducible representation M=M, indexed by the monomials in M:=M;w`. We callBtheLakshmibai monomial basisofM. From the perspective of computational complexity, the crucial observation is thatBcan be constructed in polynomial time: Lemma10.GivenGandM=Mas speciﬁed in section 3.1, the set of monomials Mcan be constructed in polynomial time. Proof.Since the length `=`(w)of the longest Weyl group element is poly(hGi), it suﬃces to show that, for each r= 1;:::;`,M;wrcan be computed from M;wr1in time poly(hGi;hMi). We ﬁrst argue that N;wr1as deﬁned in (8) can be constructed in polynomial time. For this, we note that both wr1ands rw(x)for anyx2M;r1 are given by their reduced decompositions [ 29]. But for any two Weyl group elements w0;w002W, given by their reduced decompositions, it can be decided in poly(hGi) time whether w0w00(ifgis of classical type, this result can be deduced from [ 37, Theorems 5A, 5BC, 5D]; the ﬁve exceptional Lie algebras can be treated separately in constant time). Since #M;r1=dimM;wr1dimMhMi, it is now easy to see thatN;wr1can be constructed in time poly(hGi;hMi). After this,M;rcan be constructed via (9) likewise in poly(hGi;hMi). We now establish Lemmas 8 and 9: Proof of Lemma 8.Ifgis the compact real form of simple Lie algebra and Mis irreducible then this follows directly from Lemma 10: First compute Mand then add MEMBERSHIP IN MOMENT POLYTOPES IS IN NP AND CONP 13 the weight (x)of(x)vfor allx2Minto the multiset. If gis one-dimensional abelian and Mirreducible then there is only a single weight, which we already know from the speciﬁcation of M. Ifg=g1 gnis a direct sum of such Lie algebras and Mirreducible, then any irreducible representation is a tensor product of irreducible gi-representations for i= 1;:::;n, and the multiset of weights can be identiﬁed with the Cartesian product of the multiset of weights of its constituents, which can be computed in polynomial time. Finally, if Mis reducible we apply the above procedure to each irreducible summand in its speciﬁcation. Proof of Lemma 9.We may likewise assume that gCis simple and Mis irreducible, i.e.,M=Mfor some highest weight . It moreover suﬃces to show that the representation matrices of e ,f =e , andh for

Question 101 multiple-choice

Quantum phase estimation is a fundamental algorithm in quantum computing, used to estimate the eigenvalues of unitary operators with high precision. Managing errors, coherence, and auxiliary states are key considerations in the practical implementation and efficiency of such algorithms. Which statement most accurately describes the optimal runtime scaling for quantum phase estimation algorithms in terms of the promise gap α and error tolerance δ? 1) The runtime scales as O(α log(δ⁻¹)), improving as α increases. 2) The runtime is independent of both α and δ for sufficiently large systems. 3) The runtime scales quadratically with respect to α⁻¹ and logarithmically with δ⁻¹. 4) The runtime is determined only by the number of qubits, not by α or δ. 5) The runtime has an exponential dependence on δ⁻¹ and no dependence on α. 6) The runtime scales as O(α⁻¹ + log(δ⁻¹)), combining linear and logarithmic terms. 7) The runtime scales as O(2ⁿα⁻¹log(δ⁻¹)), and the α⁻¹ dependence is shown to be optimal based on approximate counting lower bounds.

✓ Correct Answer:

The correct answer is 7) The runtime scales as O(2ⁿα⁻¹log(δ⁻¹)), and the α⁻¹ dependence is shown to be optimal based on approximate counting lower bounds..

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to make the ampli- tudes always close to 0 or 1. We claim that all of these attempts fail, and furthermore that achieving coherent phase estimation without some kind of promise is impossible in principle. This is due to the polynomial method argument above: any coherent quantum algorithm’s output state’s amplitudes must be a continuous function of the phase, and continuous functions that are sometimes ≈0and sometimes≈1must somewhere have an intermediate value. Recall that uncomputation only works when the amplitudes are close to 0 or 1. The error depends on if the phase is close to this transition point or not, so the error must depend on the phase. This issue was not taken into account by [Ambainis10], [KP17], and [KLLP18], since all of these either implicitly or explicitly state that there is an algorithm that approximately performs |ψi/angbracketright|0n/angbracketright→|ψi/angbracketright|λi/angbracketrightin superposition while λi Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 6 is a deterministic computational basis state1. A more promising approach is detailed in [TaShma13], which, crudely speaking, shifts the transition points by a classically chosen random amount. Now whether or not a phase is close to a transition point is independent of the phase itself, making ampliﬁcation possible. [Ambainis10] describes a similar idea, calling it ‘unique-answer’ eigenvalue estimation. However, we claim that this only works for a single phase. If we consider, for example, a unitary whose phases are uniformly dis- tributed in [0,1)at a suﬃciently high density, then there will be a phase near a transition point for any choice of random shift. The only way to avoid the rounding promise is to sacriﬁce coherence and measure the output state. All of the algorithms in this paper, including textbook phase estimation, achieve an asymptotic runtime of O(2nα−1log(δ−1)), whereδis the error in diamond norm. Before we move on to the formal deﬁnition of the estimation task, we informally argue that the α−1dependence is optimal, via a reduction to approximate counting. We are given N items,Kof which are marked. Following the standard method for approximate counting [BHMT00], we construct a Grover unitary whose phases λjencode arcsin(/radicalbig K/N ). Given a(1,α)-rounding promise, computing ﬂoor (21λj)amounts to deciding if λj≤1−α/2 2or λj≥1+α/2 2given that one of these is the case. By shifting the λjaround appropriately we can thus decide if K≥(1/2 +Cα)NorK≤(1/2−Cα)N, for some constant Cobtained by linearising arcsin(/radicalbig K/N ). We have achieved approximate counting with a promise gap ∼α. Thus the Ω(α−1)lower bound on approximate counting [NW98] implies our runtime must be Ω(α−1). Equipped with the notion of a rounding promise, we can deﬁne our estimation tasks. Many algorithms in this paper produce some kind of garbage, which can be dealt with the uncompute trick [BBBV97]. Rather than repeat the analysis of uncomputation in every single proof, we present a modular framework where we can deal with uncomputation separately. Furthermore, some applications may require computing some function of the ﬁnal estimate, resulting in more garbage which also needs to be uncomputed. Rather than baking the uncomputation into

Question 102 multiple-choice

Isogeny-based cryptography leverages the mathematical structure of elliptic curves and their isogenies to construct protocols aimed at resisting attacks from both classical and quantum adversaries. Efficient and secure representations of isogenies are critical for the practicality and safety of such protocols. Which of the following statements best explains why suborder representations in isogeny-based cryptography fail to provide quantum resistance, even for isogenies of prime degree? 1) Quantum algorithms solving the hidden subgroup problem can recover the endomorphism ring from suborder representations, undermining security. 2) Suborder representations inherently disclose the full isogeny kernel, allowing direct computation of secret keys. 3) The use of suborder representations increases the degree of the isogeny, making brute-force search feasible for quantum computers. 4) Suborder representations rely on polynomials whose coefficients can be efficiently factored with quantum algorithms. 5) Even when using suborder representations, knowledge of only the domain curve is sufficient to reconstruct the shared secret. 6) Suborder representations are equivalent to ideal representations, which are not vulnerable to quantum attacks. 7) The suborder representation only protects against classical attacks and does not consider the algebraic structure exposed to quantum adversaries.

✓ Correct Answer:

The correct answer is 1) Quantum algorithms solving the hidden subgroup problem can recover the endomorphism ring from suborder representations, undermining security..

📚 Reference Text:

setting of pSIDH [ Ler22a]). Indeed, in that case, the standard ways to represent isogenies(with polynomials, or kernel points) are not compact or eﬃcient enough to match our deﬁnition. The Deuring correspondence gave us the tools to obtain eﬃcient re presenta- tions with a natural isogeny representation obtained by taking sφas the ideal Iφ corresponding to φ. This ideal representation matches Deﬁnition 2.2, however it 8 M. Chen, M. Imran, G.Ivanyos, P.Kutas, A.Leroux, C. Petit also reveals the endomorphism ring of E′. One of the motivations of Leroux in [Ler22a] to introduce another isogeny representation (called the subord er rep- resentation) is to have an isogeny representation that does not d irectly reveal the endomorphism ring of the codomain. This suborder representa tion matches our notion of weak isogeny representation as deﬁned in Deﬁnition 2.2 . The main contribution of this paper implies that the suborder representatio n does not hide the endomorphism ring of the codomain to a quantum computer, eve n when the degree is prime. Since then, Robert [ Rob22] suggested to use the techniques introduced to attack SIDH in order to obtain another isogeny representation (t his one not even requiring to reveal the endomorphism ring of the domain). Our analysis holds for any suborder representation, hence it also applies to Rob ert’s one. 2.5 The pSIDH key exchange As an application of the hardness of computing the endomorphism rin g from the suborder representation, Leroux introduced a key exchang e called pSIDH. The principle can be summarized as follows: use the evaluation algorith m for the suborder representation to perform an SIDH-like key exchange, but for isogenies of big prime degree. The SIDH and pSIDH key exchange both use the following commutative isogeny diagram: EBψA/d47/d47EAB E0φB/d79/d79 φA/d47/d47EAψB/d79/d79 In pSIDH, Alice and Bob’s secret keys are ideal representations fo r the iso- geniesφAandφB(or equivalently the endomorphism ring of the two curves EA andEB), and their associated public keys are the suborder representat ions for φAandφB. Leroux showed that the knowledge of End( EA) (resp. End( EB)) and the suborder representation of φB(resp.φA) was enough to compute the end curve EABfrom which the common secret can be derived eﬃciently. The mechan ism behind this computation is quite complicated and is not relevant for us since we target the key recovery problem. We refer to [ Ler22a] for more details. 2.6 The hidden subgroup problem The hidden subgroup problem (HSP for short) in a group Gis deﬁned as the problemofﬁnding a subgroup H≤Ggivena function fonGsatisfyingthat fis constant on the left cosets of Hand takes diﬀerent values on diﬀerent cosets, i.e., f(x) =f(y) if and only if x−1y∈H. There is also a right version of the hidden subgroupproblemwherethelevelsetsofthehidingfunction faretherightcosets ofH. As taking inverses in Gmaps left cosets to right cosets and vice versa, Title Suppressed Due to Excessive Length 9 the two versions of HSP are equivalent. (One just needs to replace the hiding function with its composition with taking inverses.) Although the equiv alence is straightforward, it is

Question 103 multiple-choice

Integrable systems often utilize matrix identities and τ-functions to describe their solutions, with connections extending to quantum groups and representation theory. The transition from classical to quantum cases involves introducing deformation parameters and new algebraic structures. Which construction is necessary when extending τ-functions from the classical group SL(n) to its quantum counterpart SLq(n) for q ≠ 1? 1) Replacing Schur polynomials with Legendre polynomials 2) Introducing time variables as complex conjugates 3) Utilizing ordinary determinants for all calculations 4) Imposing the trace condition instead of the determinant condition 5) Switching from difference to differential equations 6) Enforcing commutativity of all group elements 7) Introducing q-antisymmetrization and q-determinants

✓ Correct Answer:

The correct answer is 7) Introducing q-antisymmetrization and q-determinants.

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[β1...βk/parenrightigg H/parenleftiggα′ kα′ 1...α′ k−1 βk+1]β′ 1...β′ k−1/parenrightigg =H/parenleftiggα1...αk[α′ k β1...βkβk+1/parenrightigg H/parenleftiggα′ 1...αk−1]′ β′ 1...β′ k−1/parenrightigg .(77) Just like original (67) these are just matrix identities, valid for any Hα β. However, after the switch from gtoHwe, ﬁrst, essentially represented the equations in n-independent form and, second, opened the possibility to rewrite them in terms of time-deriv atives. 21 For example, in the simplest case of αi=i, i= 1,...,k′; βi=i, i= 1,...,k+1; α′ i=i, i= 1,...,k−1, α′ k=k+1; β′ i=i, i= 1,...,k−1(78) we get: H/parenleftigg1...k 1...k/parenrightigg H/parenleftiggk+1,1...k−1 k+1,1...k−1/parenrightigg − −H/parenleftigg1...k−1,k 1...k−1,k+1/parenrightigg H/parenleftiggk+1,1...k−1 k,1...,k−1/parenrightigg = =H/parenleftigg1...k+1 1...k+1/parenrightigg H/parenleftigg1...k−1 1...k−1/parenrightigg(79) (all other terms arising in the process of symmetrization vanish). T his in turn can be repre- sented through τ-functions: ∂1¯∂1τ(k)·τ(k)−¯∂1τ(k)∂τ(k)=τ(k+1)τ(k−1). (80) This is the usual lowest Toda-lattice equation. For ﬁnite nthe set of solutions is labeled by g∈SL(n) as a result of the additional constraints (75). We can now use the chance to illustrate the ambiguity of deﬁnition of τ-function, or, to put it diﬀerently, that in the choice of time-variables. Eq.(80) is actu ally a corollary of two statements: the basic identity (67) and the particular deﬁnition (1 ), which in this case implies (74) withP’s being ordinary Schur polynomials (71). At least, in this simple situatio n (of fundamental representations of SL(n)) one could deﬁne τ-function not by eq.(1), but just by eq.(73), with Hα β(s,¯s)−→ Hα β(s,¯s) =/summationdisplay i,jPi−α(s)gi jPj−β(¯s) (81) withanyset of independent functions (not even polynomials) Pα. Such τ(k) P= det 1≤α,β≤kHα β (82) 22 still remains a generating function for all matrix elements of G=SL(n) in representation F(k). This freedom should be kept in mind when dealing with “generalized τ-functions”. As a simple example, one can take Pα(s) to beq-Schur polynomials, /productdisplay ieq(sizi) =/summationdisplay jP(q) j(s)zj,or /productdisplay ieqi(sizi) =/summationdisplay jˆP(q) j(s)zj,(83) which satisfy (hereafter we denote D≡D(0)) DsiP(q) j(s) = (Ds1)iP(q) j(s) =P(q) j−i(s). (84) Then instead of (76) we would have: DsiHα β=Hα+i β, D¯siHα β=Hα β+i(85) and τ(k) P(q)(s,¯s|g) = det 1≤α,β≤kDα−1 s1Dβ−1 ¯s1H1 1(s,¯s). (86) So deﬁnedτ-function satisﬁes diﬀerence rather than diﬀerential equations [14, 15]: τ(k)·Ds1D¯s1τ(k)−Ds1τ(k)·D¯s1τ(k)=τ(k−1)·M+ s1M+ ¯s1τ(k+1), ... .(87) We emphasize, however, that, in a sense, this is just a redeﬁnition o f theSL(n), notSLq(n) τ-function, as generating function of matrix elements. In particula r, thisτ-function is a c- rather than q-number function. Still it would have something to do with SLq(n) group, but as a function of times, i.e. rather in spirit of the connection of q-hypergeometric functions to quantum groups (see, for example, [16]). 5.2 Approach to SLq(n) In order to extend this reasoning to the case of SLq(n) withq/ne}ationslash= 1 we need to go into some more details about the group structure. 23 The main thing we shall need is the notion of q-antisymmetrization, to be deﬁned as a sum over all perturbations, ([1,...,k]q) =/summationdisplay P(−q)degP(P(1),...,P(k)), (88) where degP= # of inversions in P. (89) The ﬁrst place to use this notion is the deﬁnition of q-determinant: detqA∼A[1 [1...An]q n]q=/summationdisplay P,P′(−q)degP+degP′/productdisplay aAP(a) P′(a). (90) Note that this is

Question 104 multiple-choice

In distributed quantum computing, efficient circuit compilation is essential for large-scale computations, especially when using hardware architectures that limit qubit interactions to nearest neighbors. Techniques that exploit circuit topology and strategic qubit placement can significantly improve performance and scalability. Which approach most effectively reduces the number of SWAP gates required in quantum circuits with Linear Nearest Neighbor (LNN) architecture when implementing algorithms characterized by high local connectivity and sparse full connectivity? 1) Increasing the number of non-local two-qubit gates 2) Utilizing solely fully connected qubit layouts 3) Leveraging dangling qubits to facilitate interactions 4) Applying only single-qubit gates throughout the circuit 5) Minimizing the use of ancillary qubits 6) Employing random qubit assignments 7) Restricting the quantum algorithm to classical subroutines

✓ Correct Answer:

The correct answer is 3) Leveraging dangling qubits to facilitate interactions.

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Question 105 multiple-choice

In supersymmetric theories inspired by string models, the properties of moduli fields and their interactions with the visible sector have significant implications for cosmology and particle physics, particularly regarding dark matter and collider searches. The mass spectrum of superpartners is closely tied to the mechanism by which supersymmetry breaking is mediated. Which supersymmetry breaking mediation scenario typically allows gaugino superpartners to be light enough for detection at the LHC, while requiring suppression of gravity-mediated scalar masses to realize this spectrum? 1) Gravity mediation with unsuppressed scalar masses 2) Gauge mediation with high messenger scales 3) Pure gravity mediation with Planck-scale couplings 4) Mini-split supersymmetry with heavy scalars and gauginos 5) Anomaly mediation (AMSB) or AMSB-like scenarios with suppressed gravity-mediated contributions 6) Gauge mediation with extremely light gravitinos 7) Pure moduli mediation without anomaly or gauge contributions

✓ Correct Answer:

The correct answer is 5) Anomaly mediation (AMSB) or AMSB-like scenarios with suppressed gravity-mediated contributions.

📚 Reference Text:

moduli are not completely understood, certain features do seem to be fairly universal. For example, moduli masses of m'm3=2are expected when the potential arises mainly from supersymmetry breaking [18]. Moduli may also have supersymmetric potentials [19], and m' m3=2is found in some cases [20, 21]. However, m'm3=2is still frequently obtained from supersymmetric potentials once the constraint of a very small vacuum energy is imposed [21]. Thus, a plausible generic expectation from string theory is that there exists at least one modulus eld with m'm3=2and MPl[6].2Other heavier moduli may be present, but since the lightest and most weakly-coupled modulus will decay the latest, it is expected to have the greatest impact on the present-day cosmology. Putting these two pieces together, acceptable reheating from string moduli suggests m' m3=2&100 TeV. This has important implications for the masses of the SM superpartner elds. Surveying the most popular mechanisms of supersymmetry breaking mediation [24], the typical size of the superpartner masses is msoft8 >< >:m3=2 gravity mediation LMPl M m3=2gauge mediation Lm 3=2 anomaly mediation(2) 1Such an early matter-dominated phase might also leave an observable signal in gravitational waves at multiple frequencies [11] or modify cosmological observables [12{15]. 2The LARGE Volume Scenario of Refs. [22, 23] is a notable exception to this. 3 whereLg2=(4)2is a typical loop factor and M MPl=Lis the mass of the gauge messengers. Of these mechanisms, only anomaly mediation (AMSB) allows for superpartners that are light enough to be directly observable at the Large Hadron Collider (LHC) [25, 26]. Contributions to the soft terms of similar size can also be generated by the moduli themselves [27, 28], or other sources [29{32]. However, for these AMSB and AMSB-like contributions to be dominant, the gravity-mediated contributions must be suppressed [25], which is non-trivial for the scalar soft masses [33{37]. An interesting intermediate scenario is mini-split supersymmetry where the dominant scalar soft masses come from direct gravity mediation with msoftm3=2, while the gaugino soft masses are AMSB-like [38{43]. Moduli reheating can also modify dark matter production [16, 44{46, 48]. A standard weakly-interacting massive particle (WIMP) will undergo thermal freeze-out at temper- atures near Tfom =20. If this is larger than the reheating temperature, the WIMP density will be strongly diluted by the entropy generated from moduli decays. On the other hand, DM can be created non-thermally as moduli decay products. A compelling picture of non-thermal dark matter arises very naturally for string-like moduli and an AMSB-like superpartner mass spectrum [16]. The lightest (viable) superpartner (LSP) in this case tends to be a wino-like neutralino. These annihilate too eciently to give the observed relic density through thermal freeze-out [49{51]. However, with moduli domination and reheating, the wino LSP can be created non-thermally in moduli decays, and the correct DM density is obtained for M2200 GeV and m'3000 TeV. This scenario works precisely because the wino annihilation cross section is larger than what is needed for thermal freeze-out. Unfortunately, such enhanced annihilation rates are strongly constrained by gamma-ray observations of the galactic centre by Fermi

Question 106 multiple-choice

Quantum algorithms leverage the principles of quantum mechanics to solve certain computational problems more efficiently than classical algorithms. Various paradigms exist, including hidden subgroup approaches, quantum walks, and topological methods, each offering distinct advantages for specific problem classes. Which quantum algorithm framework provides exponential speedup for integer factoring and discrete logarithm problems by exploiting Abelian group structures? 1) Quantum walk algorithms 2) Adiabatic algorithms 3) Abelian Hidden Subgroup algorithms 4) Topological quantum algorithms 5) Quantum simulation algorithms 6) Grover's amplitude amplification 7) Non-Abelian Hidden Subgroup algorithms

✓ Correct Answer:

The correct answer is 3) Abelian Hidden Subgroup algorithms.

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Question 107 multiple-choice

In the theory of quantum groups and representation theory, the universal fusion matrix plays a key role in describing how representations can be fused while respecting underlying algebraic structures. The ABRR equation uniquely determines this fusion matrix through a functional relation involving certain operators. Which property guarantees the existence of a unique solution to the ABRR equation for the universal fusion matrix J(λ) in both quantum group and Lie algebra settings? 1) The commutativity of the Cartan subalgebra elements 2) The invertibility of the R-matrix 3) The semisimplicity of the underlying algebra 4) The centrality of the Casimir operator 5) The surjectivity of the groupoid morphism 6) The recursive construction based on the structure of the root system 7) The symmetry of the Poisson bracket

✓ Correct Answer:

The correct answer is 6) The recursive construction based on the structure of the root system.

📚 Reference Text:

which equips X with the structure of a Poisson groupoid with base U. Moreover, it is easy to check that the Poisson groupoid Xis naturally isomorphic to Xr. 8 Theuniversal fusion matrix and theArnaudon- Buﬀenoir-Ragoucy-Roche equation 8.1. The ABRR equation. In [ABRR], Arnaudon, Buﬀenoir, Ragoucy and Roche give a general method for constructing the universal fusio n matrix J(λ), which lives in some completion of Uq(g)⊗2, i.e the unique element satisfying JV W(λ) =J(λ)|V⊗Wfor allV,W. A similar approach is suggested in [JKOS], based on the method of [Fr] LetU′(b±) be the kernel of the projection U(b±)→U(h). We use the same notations with the index qfor the quantum analogs of these objects. We setθ(λ) =λ+ρ−1 2/summationtext ix2 i∈Uhwhere as usual ρ=1 2/summationtext α∈∆+hαand (xi) is an orthonormal basis of h. SetR0=Rq−/summationtextxi⊗xi. It is known that R0∈1+U′ q(b+)⊗U′ q(b−). Theorem 8.1 ([ABRR]). The universal fusion matrix J(λ)ofUq(g)is the unique solution of the form 1+U′ q(b−)⊗U′ q(b+)of the equation J(λ)(1⊗q2θ(λ)) =R21 0(1⊗q2θ(λ))J(λ). (8.1) The universal fusion matrix J(λ)ofU(g)is the unique solution of the form 1+U′(b−)⊗U′(b+)of the equation [J(λ),1⊗θ(λ)] = (/summationdisplay α∈∆+e−α⊗eα)J(λ) (8.2) We will call these equations the ABRR equations forUq(g) and for g, re- spectively. Proof.Let us ﬁrst show the statement about uniqueness. Let T(λ)∈1 + U′ q(b−)⊗U′ q(b+) be any solution of (8.1). Then (R21 0)−1T(λ) = Ad (1 ⊗q2θ(λ))T(λ) ⇔((R21 0)−1−1)T(λ) = (Ad (1 ⊗q2θ(λ))−1)T(λ) ⇔T(λ) = 1+(Ad (1 ⊗q2θ(λ))−1)−1((R21 0)−1−1)T(λ) Now notice that (( R21 0)−1−1)∈U′ q(b−)⊗U′ q(b+). This implies that T(λ) can be recusively constructed as follows. Set T0(λ) = 1 and put Tn+1(λ) = 1+(Ad (1 ⊗q2θ(λ))−1)−1((R21 0)−1−1)Tn(λ). 23 Then lim n→∞Tn(λ) =T(λ) (the limit is in the sense of stabilization). In par- ticular there exists a unique solution to (8.1) of the given form. The proof in the rational case (i.e in the case of a simple Lie algebra g) is similar. In that case, the recursive construction is given by T0(λ) = 1 and Tn+1(λ) = 1−ad(1⊗θ(λ)−1)(/summationdisplay α∈∆+e−α⊗eα)Tn(λ). We now give a proof that the fusion matrix J(λ) actually satisﬁes the ABRR relation in the case of simple Lie algebras. The proof in the case of qua ntum groups is analogous but technically more challenging, and is given in App endix B. LetCbe the quadratic Casimir operator in the center of the universal en - veloping algebra Ug: C=/summationdisplay ix2 i+2ρ+2/summationdisplay α∈∆+e−αeα. ThenCacts on any highest weight representation of gof highest weight λby the scalar ( λ,λ+ 2ρ). Now let V,Wbe two ﬁnite-dimensional g-modules and letv∈V, w∈Wbe two homogeneous elements of weight wt( v) and wt( w). We compute the quantity F(λ) =/a\}bracketle{tv∗ λ−wt(v)−wt(w),Φw λ−wt(v)(C⊗1)Φv λvλ/a\}bracketri}ht in two diﬀerent ways. On one hand we have F= (λ−wt(v),λ−wt(v)+2ρ)J(λ)(w⊗v). (8.3) On the other hand, F=/a\}bracketle{tv∗ λ−wt(v)−wt(w),/braceleftbig 2((e−αeα)1+(e−αeα)2+(eα⊗e−α)12+(e−α⊗eα)12+ρ1+ρ2) +/summationdisplay i(x2 i)1+(x2 i)2+2(xi⊗xi)12/bracerightbig Φw λ−wt(v)Φv λvλ/a\}bracketri}ht. Sincev∗ λ−wt(v)−wt(w)is a highest weight vector, it is clear that /a\}bracketle{tv∗ λ−wt(v)−wt(w),(e−αeα)1Φw λ−wt(v)Φv λvλ/a\}bracketri}ht=/a\}bracketle{tv∗ λ−wt(v)−wt(w),(e−α⊗eα)12Φw λ−wt(v)Φv λvλ/a\}bracketri}ht= 0. Moreover, by the intertwining property again, we have (eα⊗e−α)12Φw λ−wt(v)Φv λvλ=−(e−αeα)2−(e−α⊗eα)23Φw λ−wt(v)Φv λvλ, (ρ1+ρ2)Φw λ−wt(v)Φv λvλ=−ρ3Φw λ−wt(v)Φv λvλ+Φw λ−wt(v)Φv λρvλ, ((x2 i)1+(x2 i)2+2(xi⊗xi)12)Φw λ−wt(v)Φv λvλ=−(2(xi⊗xi)13+2(xi⊗xi)23+(x2

Question 108 multiple-choice

Quantum computing leverages probabilistic phenomena to solve certain computational problems more efficiently than classical approaches. One notable application is integer factorization, which is fundamental to modern cryptography. Which of the following best describes how the Chernoff bound is utilized in quantum algorithms to reduce error probability when using majority voting across multiple runs? 1) It guarantees that the algorithm will always produce the correct result after a finite number of repetitions. 2) It provides a linear decrease in failure probability as the number of runs increases. 3) It determines the minimum number of runs required to achieve a 100% success rate. 4) It ensures that hardware errors are eliminated during repeated executions. 5) It allows the algorithm to deterministically amplify quantum interference effects during computation. 6) It bounds the probability of algorithmic failure by showing that, after k independent runs, the chance of majority error decreases exponentially with k. 7) It replaces the need for majority voting by enabling a single-run quantum algorithm to achieve arbitrary precision.

✓ Correct Answer:

The correct answer is 6) It bounds the probability of algorithmic failure by showing that, after k independent runs, the chance of majority error decreases exponentially with k..

📚 Reference Text:

probabilistic devices and can be viewed as a generalization of a probabilistic Turing machine. It is hardly surprising that quantum algorithms exhibit their probabilistic nature by producing a solution with some probabil- ity. Suppose we have an algorithm which succeeds with probability1 2+dfor some ﬁxed constant d. Then the probability of failure for this algorithm is1 2d. 21 The algorithm is run ktimes and we take the majority result. Then the probability that the algorithm fails is given by Pr=bk 2c å l=0k l1 2+dl1 2dkl (2.62) where lis the number of times the algorithm returns the correct answer. Forl=k 2 we have 1 2+dbk 2c1 2dkbk 2c =1 2+dbk 2c1 2ddk 2e =1+2d 2bk 2c12d 2dk 2e (2.63) 14d2 4k 2 = 14d2k 2 2k: Since Eq (2.62) contains at most 2kterms l=k 2 , the failure probability is bounded as follows: Pr2k2 4 14d2k 2 2k3 5= 14d2k 2: (2.64) Application of 1xex(2.65) produces 14d2k 2 e4d2k 2=e2d2k(2.66) and we obtain a Chernoff bound Pre2d2k: (2.67) Therefore, the algorithm succeeds with probability exponentially close to 1 in the number of trials. 22 To illustrate the point convincingly, suppose the algorithms fails with probability1 4, i.e., d=1 4. The approximate Chernoff bounds corresponding to an increasing number of algorithm repetitions are given in Table 2.2. Table 2.2: The Chernoff bounds for d=1 4 k e2d2k 10 0:286 40 0:007 100 3:73106 400 1:931022 1000 5:171055 When the probability drops below 1020, a failure of the computer itself is more likely than that of an algorithm. For 1000 runs, you are more likely to be hit by an asteroid while reading this sentence than to get a wrong answer from the algorithm! 23 CHAPTER 3 SHOR’S FACTORIZATION ALGORITHM The existence of a unique decomposition of any integer N>1 into a product of primes was formalized by Euclid around 300 BC in Propositions 30 and 32 of his Elements [18]. The ﬁrst complete proof was provided in 1801 by Carl Friedrich Gauss in his Disquisitiones Arithmeticae , where he remarked [19]: The problem of distinguishing prime numbers from composite numbers, and of re- solving the latter into their prime factors is known to be one of the most important and useful in arithmetic. The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated. Almost two centuries later Peter Shor discovered a quantum algorithm that factors an integer N with high probability in O n2log2nlog2log2n (3.1) steps where nis the number of bits needed to represent N, an exponential speedup over the best- known classical algorithm. 3.1 Reduction of Factorization to Order Finding We may assume that an integer Nwe wish to factor is odd, as factors of 2 can be easily removed. Moreover, since there exist efﬁcient classical tests for prime powers [30], we assume thatNis an odd non-prime power integer. Miller [42] showed that factorization reduces to order ﬁnding. 24 Choose a random

Question 109 multiple-choice

In quantum information theory, algebraic structures such as the Temperley-Lieb Algebra (TLA) and braid group representations play a central role in understanding and engineering entanglement in multi-qubit systems. The action of specific unitary operators constructed from tensor products of Hermitian matrices determines how quantum states are transformed and can lead to various degrees of entanglement. Which choice describes a situation in which the action of a unitary braiding operator on an n-qubit separable state results in the state remaining fully separable rather than entangled? 1) All Hermitian operators $ s_j $ are chosen as the Pauli $ \sigma_1 $ matrix in the tensor product construction 2) All Hermitian operators $ s_j $ are the identity matrix 3) All Hermitian operators $ s_j $ are the Hadamard matrix 4) Some $ s_j $ are Pauli matrices and others are identity 5) The parameter $ \phi $ is set to $ \pi $ in the construction 6) The normalization condition is not satisfied 7) The operators are chosen from a Fibonacci fusion category

✓ Correct Answer:

The correct answer is 2) All Hermitian operators $ s_j $ are the identity matrix.

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to generalize the above representa- tion of TLA to higher dimensions is as follows. Let e1=/parenleftbigg 1 0 0 0/parenrightbigg , e2=/parenleftbigg a20 0b2/parenrightbigg , e3=/parenleftbigg 0e−iφ eiφ0/parenrightbigg .(11) Deﬁne E(n,k) 1≡ ⊗k−1 j=1I⊗e1⊗n j=k+1I, (12) E(n,k) 2≡ ⊗k−1 j=1I⊗e2⊗n j=k+1I +ab⊗k−1 j=1sj⊗e3⊗n j=k+1sj,(13) where⊗m j=1sj=s1⊗s2⊗···⊗sm. Heresjis any Her- mitian operator satisfying s2 j= 1. For example, sjcan beI, any one of the Pauli matrices σm(m= 1,2,3), or the Hadamard matrix H. The integer nis the number of 2×2 matrices in the tensor products, and kindicates the position of e1, e2ande3. TheE(n,k) i’s are 2n×2nma- trices, and they reduce to (9) in the case n=k= 1. One can easily check that E(n,k) i’s satisfy (10). Hence, the op- eratorsh(n,k) i=dE(n,k) iform a 2n×2nmatrix realization ofTL3(d)4. A unitary braid group representation is then obtained from the hi’s by the Jones representation. Our new unitary braid representation generalizes the 2×2 matrices of (9) to 2n×2nmatrices of (13) within the TLATL3(d). Other routes of generalization are pos- sible. For instance, in [14] the 2 ×2 representation of TL3(d) were generalized to higher dimensional matrices forTLm(d) withm >3, where the dimension of repre- sentation varies with the number of strands maccording to the Fibonacci numbers, or with the number of inde- pendent bit-strings of certain path model proposed in [15]. Generalized GHZ states. – From now on we will be mainly concerned with the unitary braiding transfor- mation representing the action of the braid b1b2. This braiding operator can be evaluated to be b(n,k) 1b(n,k) 2 =⊗k−1 j=1I⊗/parenleftbigg da20 0db2+A−2/parenrightbigg ⊗n j=k+1I(14) 4See [9] for an n2×n2matrix realization of the TLA. The braiding operator (called the Yang-Baxter matrix in these works) was obtained there through aYang-Baxterization process. This latter process was also employed in [8], but not related to TLA, to generalize the Bell matrix to (2 n)2×(2n)2braid matrices.+⊗k−1 j=1sj⊗/parenleftbigg 0−e−iφA4dab eiφdab 0/parenrightbigg ⊗n j=k+1sj. Its action on the separable n-qubit states |a1a2...ak−10ak+1···an/an}bracketri}htand|a1a2...ak−11ak+1···an/an}bracketri}ht (aj= 0,1, j= 1,2,...,k−1,k+1,...n) is given by b(n,k) 1b(n,k) 2|a1a2...ak−10ak+1···an/an}bracketri}ht = (da2)|a1a2...ak−10ak+1···an/an}bracketri}ht + (eiφdab)|˜a1˜a2...˜ak−11˜ak+1···˜an/an}bracketri}ht,(15) and b(n,k) 1b(n,k) 2|a1a2...ak−11ak+1···an/an}bracketri}ht = (db2+A−2)|a1a2...ak−11ak+1···an/an}bracketri}ht + (−e−iφA4dab)|˜a1˜a2...˜ak−10˜ak+1···˜an/an}bracketri}ht,(16) where|˜aj/an}bracketri}ht ≡sj|aj/an}bracketri}ht(j= 1,...,k−1,k+1,...,n). Thus under the action of b(n,k) 1b(n,k) 2, the separable n-qubit state|a1a2...ak···an/an}bracketri}htis superimposed on the state |˜a1˜a2...˜ak···˜an/an}bracketri}htineithertheform(15)or(16), depend- ingonwhetherthe k-thqubit |ak/an}bracketri}htis|0/an}bracketri}htor|1/an}bracketri}ht. The states in (15) and (16) arenormalized, as ( da2)2+|eiφdab|2= 1, and|db2+A−2|2+|−e−iφA4dab|2= 1,which can be eas- ily checked. Depending on the choice of the set of sj’s, the resulting state (15) or (16) will have varying degrees of entanglement. In particular, if all sj=I, then the resulting state is separable, and b(n,k) 1b(n,k) 2is simply a local unitary transformation. We now consider a subclass of the representation ob- tained by setting φ= 0 in (13) (i.e., e3=σ1),sj=Ifor j < k, andsj=σ1forj > k. In this case, |˜aj/an}bracketri}ht=|aj/an}bracketri}ht forj k. Hence, un- der the action of B(n,k)≡b(n,k) 1b(n,k) 2(with the above- mentioned choice of the sj’s inb(n,k) iunderstood), the separablen-qubit state |a1a2...ak−1akak+1···an/an}bracketri}htis su- perimposed on the state |a1a2...ak−1¯ak¯ak+1···¯an/an}bracketri}htin either the form (15) or (16) (with the appropriate change in the ˜aj), depending on whether the

Question 110 multiple-choice

Computational invariant theory deals with the complexity of algorithms that analyze group actions on vector spaces, often distinguishing between commutative and non-commutative cases. Representation theory provides analytical tools for these problems, particularly when handling non-commutative group actions. Which feature is essential for guaranteeing polynomial-time performance in algorithms for non-commutative group actions, particularly when using highest weight vectors as potential functions? 1) Allowing arbitrary bit-length group elements in each iteration 2) Ignoring the difference between uniform and non-uniform cases 3) Relying solely on black-box applications of previous results without explicit constructions 4) Avoiding the use of representation theory in bounding algorithm complexity 5) Using matrix scaling techniques without modification for non-commutative groups 6) Utilizing degenerate spectra to simplify computations 7) Truncating group elements to polynomial bit-length so that runtime remains polynomial

✓ Correct Answer:

The correct answer is 7) Truncating group elements to polynomial bit-length so that runtime remains polynomial.

📚 Reference Text:

at most exponential in the input parameters. This is achieved by relying on Derksen’s degree bounds [Der01] (see Proposition 2.5). 1.7 Additional discussion We would like to point out two important distinctions between the analysis for matrix scaling in Section 1.5 and our analysis here. First is that, as we have seen, there is a major diﬀerence betweentheuniformandthenon-uniformversionsofourproblem-whilethiswasnotthecase for matrix scaling. This phenomenon is general and is a distinction between commutative and non-commutative group actions. It has to do with the fact that all irreducible representations ofcommutativegroupsareone-dimensional,whereasfornon-commutativegroupstheyarenot. Secondly, inthematrix scalinganalysis,the upperboundwas easytoobtain aswell. Whereas for us,theupperboundstepisthehardestandrequirestheuseofdeepresultsinrepresentationtheory. The upper bound steps were the cause of main diﬃculty in the papers [ GGOW16 ,BGO17,Fra18] aswell /two.superior/three.superiorandthisisonekeypointofdistinctionbetweencommutativeandnon-commutativegroup actions. We believe that our framework of using the highest weight vectors as potential functions for the analysis of analytic algorithms is the right way to approach moment polytope problems - even beyond the cases that we consider in this paper. Theapproachtakenin[ Fra18](forthecaseof d2)isoneofreducingthenon-uniformversion oftheproblemtotheuniformversion,whichwassolvedin[ GGOW16 ]forthecaseof d2(the reduction in [ GGOW17 ] is a simple special case of the reduction in [ Fra18]). The reduction is complicated and a bit ad hoc. We generalize this reduction to our setting ( d¡2) in Section 4, and providingasomewhatmoreprincipledviewofthereductionalongtheway. However,itstillseems rather specialized and mysterious compared to the general reduction in geometric invariant theory from the “non-uniform” to the “uniform” case (also known as the shifting trick , see Section 2.2). Wealso notethat applyingthe resultsof [ BGO17]to thereduction inSection 4in ablack-box manner does not yield our main theorem (Theorem 1.7) - the number of iterations would be exponential in the bit-complexity ofp, and we would even require an exponential number of bits for /two.superior/three.superiorIn some of the papers, lower bound is the hard step, due to the use of a dual kind of potential function. 14 the randomization step! To remedy these issues with the reduction in Section 4 one must delve intotherelationshipbetweenthereductionandtheinvariantpolynomials. Wewillsee,byfairly involved calculation, that invariant polynomials evaluated on the reduction will result in the same construction of highest weight vectors anyway. This teaches us two lessons: 1.Highest weight vectors are the only suitable potential functions in sight. Though it may have other conceptual beneﬁts, the reduction in Section 4 is no better than the shifting trick for the purpose of obtaining potential functions! 2.WehadtolookattheconstructionofhighestweightvectorsinSection2.4 beforecalculating them from the reduction - the calculation might not have been so easy a priori! Again, the classical construction of highest weight vectors saves the day. ItisinterestingtodiscusssomeofthesalientfeaturesandpossiblevariationsofAlgorithm1 (we expand on these points in the main text): •Iterations and randomness. The algorithm terminates after at most Tpoly maxd i0ni;d;b; 1{" iterations and uses log2pMq polypmaxd i0ni;d;bqbits of randomness. Forﬁxed oreven inversepolynomial "¡0, this ispolynomial inthe input size. In fact, this is better than the number of iterations in [ Fra18]: there, the number of iterations also depended on ppiq ni 1 . •Bit complexity : Togetanalgorithmwithtrulypolynomialruntime,oneneedstotruncate thegroupelements gpiq’suptopolynomialnumberofbitsafterthedecimalpoint. Weprovide an explanation on why this doesn’t aﬀect the performance of the algorithm in Section 3.3. •Degenerate spectra. Ifpiqis degenerate, i.e. piq jpiq

Question 111 multiple-choice

Post-quantum digital signature algorithms are designed to remain secure even when adversaries have access to quantum computers. Advanced approaches increasingly rely on algebraic structures and computational hardness assumptions that differ from classical cryptography. Which feature directly contributes to both the security and efficiency of a novel post-quantum signature scheme based on non-commutative algebras? 1) Use of integer factorization as the underlying hard problem 2) Reliance on traditional discrete logarithm techniques 3) Implementation on three-dimensional commutative algebras 4) Employment of four-dimensional sparse non-commutative algebras with quadratic equations over finite fields 5) Public key obfuscation solely through hash functions 6) Signature verification using linear equations over real numbers 7) Dependence on classical polynomial-time algorithms for signature generation

✓ Correct Answer:

The correct answer is 4) Employment of four-dimensional sparse non-commutative algebras with quadratic equations over finite fields.

📚 Reference Text:

Title: A NEW CONCEPT FOR DESIGNING POST-QUANTUM DIGITAL SIGNATURE ALGORITHMS ON NON-COMMUTATIVE ALGEBRAS Year: 2022 Paper ID: 955f69b24abc7bf27b89d292b2577fb612068af2 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/955f69b24abc7bf27b89d292b2577fb612068af2 Abstract: Purpose of work is the development of a new approach to designing post-quantum digital signature algorithms that are free from the shortcomings of known analogs – large sizes of the signature and public key. Research method is the use of power vector equations with multiple occurrences of the signature S as a signature verification equation. The computational difficulty of solving equations of the said type relatively the unknown value of S ensures the resistance of the signature scheme to attacks using S as a fitting parameter. The possibility of calculating the value of S by the secret key is provided by using the public key in the form of a set of secret elements of the hidden group, masked by performing left and right multiplications by matched invertible vectors. Results of the study include a new proposed concept for the development of post-quantum digital signature algorithms on non-commutative algebras, which use a hidden commutative group. One of its main differences is the use of a secret key in the form of a set of vectors, the knowledge of which makes it possible to calculate the correct signature value for the random powers present in the verification equation. The form of the latter defines a system of quadratic vector equations connecting the public key with the secret, which is reduced to a system of many quadratic equations with many unknowns, given over a finite field. The computational difficulty of finding a solution to the latter system determines the security of the algorithms developed within the framework of the proposed concept. A quantum computer is ineffective for solving this problem, therefore, the said algorithms are post-quantum. As analogs in construction, digital signature algorithms based on the computational difficulty of the hidden discrete logarithm problem are considered, however, the use of a hidden group and exponentiation operations represent only a general technique for ensuring the correctness of the signature schemes developed within the framework of the concept, and not for specifying a basic computationally difficult problem. To improve the performance of the signature generation and verifications procedures, the four-dimensional algebras defined by sparse basis vector multiplication tables are used as an algebraic support. The proposed concept is confirmed by the development of a specific post-quantum algorithm that provides a significant reduction in the size of the public key and signature in comparison with the finalists of the NIST global competition in the nomination of post-quantum digital signature algorithms. Practical relevance: The developed new concept for constructing post-quantum digital signature algorithms expands the areas of their application in conditions of limited computing resources

Question 112 multiple-choice

The hidden subgroup problem (HSP) is a central challenge in quantum computing, underpinning efficient solutions to key computational problems. Quantum measurement strategies and group structure play crucial roles in designing algorithms for both abelian and nonabelian groups. Which of the following statements accurately describes a significant advance in solving the hidden subgroup problem for nonabelian groups using quantum algorithms? 1) Efficient quantum algorithms for the dihedral group HSP have been discovered, enabling polynomial-time solutions to lattice problems. 2) The pretty good measurement (PGM) is always optimal for all nonabelian groups when distinguishing hidden subgroups. 3) Entangled measurements are unnecessary for efficiently solving nonabelian hidden subgroup problems with quantum algorithms. 4) Efficient quantum algorithms have been developed for nonabelian groups such as metacyclic groups and groups of the form Z_r^p ⋊ Z_p, leveraging optimal measurements and entanglement. 5) The HSP for symmetric groups has been solved in polynomial time, leading to efficient algorithms for graph isomorphism. 6) Shor’s algorithm provides efficient solutions to both abelian and nonabelian hidden subgroup problems. 7) Subexponential time algorithms for the dihedral group HSP rely exclusively on classical computation.

✓ Correct Answer:

The correct answer is 4) Efficient quantum algorithms have been developed for nonabelian groups such as metacyclic groups and groups of the form Z_r^p ⋊ Z_p, leveraging optimal measurements and entanglement..

📚 Reference Text:

arXiv:quant-ph/0504083v2 26 Apr 2005From optimal measurement to eﬃcient quantum algorithms for the hidden subgroup problem over semidirect product groups Dave Bacon∗ dabacon@santafe.eduAndrew M. Childs† amchilds@caltech.eduWim van Dam‡ vandam@cs.ucsb.edu Abstract We approach the hidden subgroup problem by performing the so-ca lled pretty good mea- surement on hidden subgroup states. For various groups that ca n be expressed as the semidirect product of an abelian group and a cyclic group, we show that the pre tty good measurement is optimal and that its probability of success and unitary implementatio n are closely related to an average-case algebraic problem. By solving this problem, we ﬁnd eﬃc ient quantum algorithms for a number of nonabelian hidden subgroup problems, including some for which no eﬃcient algorithm was previously known: certain metacyclic groups as well as all groups of the form Zr p⋊ Zpfor ﬁxedr(including the Heisenberg group, r= 2). In particular, our results show that entangled measurements across multiple copies of hidden subg roup states can be useful for eﬃciently solving the nonabelian hsp. 1 Introduction The hidden subgroup problem ( hsp) stands as one of the major challenges for quantum compu- tation. Shor’s discovery of an eﬃcient quantum algorithm fo r factoring and calculating discrete logarithms [30], which essentially solves the abelian hsp[4, 19], focused attention on the question of what computational problems might be solved asymptoticall y faster by quantum computers than by classical ones. In particular, we would like to understan d when the nonabelian hidden subgroup problem admits an eﬃcient quantum algorithm. Considerable progress on this question has been made since S hor’s discovery. Eﬃcient quantum algorithms have been found for the case where the hidden subg roup is promised to be normal and there is an eﬃcient quantum Fourier transform over the group [13], or where the group is “almost abelian” [12] or, more generally, “near-Hamiltonian” [11] . In addition, eﬃcient algorithms have been found for several groups that can be written as semidire ct products of abelian groups: the wreath product Zn 2≀Z2[28], certain groups of the form Zn pk⋊Z2for a ﬁxed prime power pk[10], q-hedral groups with qsuﬃciently large [21], and particular groups of the form Zpk⋊Zpwithpan odd prime [17]. Unfortunately, eﬃcient algorithms have been elusive for tw o cases with particularly signiﬁcant applications: the dihedral group and the symmetric group. A n eﬃcient algorithm for the hspover the symmetric group would lead to an eﬃcient algorithm for gr aph isomorphism [4, 7, 3, 16], and an eﬃcient algorithm for the dihedral hsp(based on the standard approach described in Section 2.1) would lead to an eﬃcient algorithm for certain lattice probl ems [26]. While no polynomial-time quantum algorithms for these problems are known, Kuperberg recently gave a subexponential (but superpolynomial) time and space algorithm for the dihedral hsp[20], and Regev improved the space requirement to be only polynomial [27]. ∗Santa Fe Institute, Santa Fe, NM 87501, USA †Institute for Quantum Information, California Institute o f Technology, Pasadena, CA 91125, USA ‡Department of Computer Science, University of California, Santa

Question 113 multiple-choice

Quantum phases on graphs expand the landscape of topological order by allowing for unconventional connectivity and topology, particularly when modeled with abelian gauge theories. Ground state degeneracy and entanglement entropy play key roles in distinguishing these phases and their potential applications in quantum information science. Which property best distinguishes a topologically ordered phase on a graph from one lacking topological order when using abelian gauge theory models? 1) Absence of long-range entanglement in all ground states 2) Ground state degeneracy that varies only with system size 3) Local order parameters sufficient to classify all phases 4) Presence of gapless excitations throughout the spectrum 5) A constant contribution to entanglement entropy reflecting non-local quantum correlations 6) Spontaneous breaking of global symmetries 7) Ground state uniqueness irrespective of topology

✓ Correct Answer:

The correct answer is 5) A constant contribution to entanglement entropy reflecting non-local quantum correlations.

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Question 114 multiple-choice

Quantum Fourier Transform (QFT) is a fundamental operation in quantum computing, enabling analysis and extraction of frequency components from quantum-encoded signals such as audio data. Simulating QFT circuits on classical hardware allows researchers to visualize frequency information and test quantum algorithms for signal processing applications. When a quantum circuit implementing the Quantum Fourier Transform is applied to qubits encoding sampled audio amplitudes and followed by measurement in the computational basis, what does the amplitude of each basis state after measurement most directly represent? 1) The phase shift introduced by each quantum gate in the circuit 2) The probability of observing a specific time-domain sample 3) The total energy of the quantum system 4) The strength of each frequency present in the original signal 5) The number of qubits used in the encoding 6) The fidelity of the quantum simulation 7) The error rate from gate imperfections

✓ Correct Answer:

The correct answer is 4) The strength of each frequency present in the original signal.

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QFT to quantum circuit, create measurement register, and draw visualization of circuit 2qft(audio qc, n qubits) 3audio qc.measure all() 4 5#Simulation 6sim = Aer.get backend(”aer simulator”) 7qobj = assemble(audio qc) 8counts = sim.run(qobj).result().get counts() 9qftcounts = get qftcounts(counts, n qubits)[: len(samples)//2] 10plothistogram(counts) Figure 22 Applying the QFT Since each qubit is measured in the Pauli-Z basis, all qubits will collapse into one of the two eigenstates of the Pauli-Z operator probabilities dictated by the superposition that we put the qubits in. The QFT on each qubit in a superposition has the e ect of causing the qubits collapse in a way that gives us the Fourier transform of the input amplitudes. 23 SciPost Physics Codebases Submission Figure 23 Histogram of the QFT Output of A440 5.1.5 Decoding the Result If the nal output of the state is given by : |ψ/angbracketright=/summationdisplay iai|i/angbracketright (52) whereiare the basis states, then the information we need to determine the frequency of the musical note is stored in the amplitude in ai. Here we will use Python to evaluate the top indices of our output. To get the value of frequency we have to multiply these peaks by framerate 2nqubitsHz. Python: 1#Calculate Frequencies and print top 2 frequencies 2topindices = np.argsort( −np.array(qft counts)) 3freqs = top indices ∗frame rate/2 ∗∗nqubits 4print (freqs[:2]) Figure 24 Obtaining the Frequency The top two indices computed for frequency were computed to be: [430.6640625,473.73046875] (53) 24 SciPost Physics Codebases Submission These were the top two frequencies output by our QFT circuit. These are very close to the actual frequency which we know to be 440Hz since we sampled the musical note A440. 5.2 Using the QFT on a Musical Chord Now that we have demonstrated using the QFT to nd a single musical note, the next step is to determine if we can use this method to nd multiple musical notes played together. Multiple notes played at the same time is known as a musical chord. Below is a F-major chord comprised of three notes: C3, F3, and A4 all played simultaneously. The resulting audio signal will be three frequencies in superposition. C3 corresponds to 130.81Hz, F3 to 174.61Hz, and A4 to 440.0Hz. Figure 25 F-Major Chord Following the same steps on the single note example, we can sample the audio of this chord played and obtain an array of amplitudes for each sample and plot the results. Figure 26 Output Audio Waveform of F-Major Chord One might notice that it would be dicult to discern the three distinct frequencies of the musical chord by looking at the graph alone to calculate the period or frequency. The QFT is useful here because it will transform this data from the time domain to the frequency domain where we will see amplitudes of the three individual frequencies clearly. Next, we encode the samples again onto our quantum system and apply the QFT. After simulating the circuit and outcome we are left with the following results. 25 SciPost Physics Codebases Submission Figure 27

Question 115 multiple-choice

Quantum computing enables the development of new algorithms for efficient data processing, including image and signal manipulation, by leveraging quantum parallelism and specialized transformations. Understanding the characteristics and limitations of these quantum algorithms is crucial for advancing practical applications. Which statement best describes a key advantage of quantum image interpolation algorithms over classical counterparts? 1) Quantum image interpolation algorithms rely exclusively on entanglement for efficiency. 2) They require exponentially more resources as image size increases. 3) Their computational complexity remains constant regardless of image size due to processing image subspaces in quantum superposition. 4) They cannot utilize techniques inspired by classical image processing such as compression. 5) They are limited to processing only classical data encoded in quantum circuits. 6) Quantum parallelism does not contribute to any speedup in image interpolation tasks. 7) These algorithms can only operate on fully decohered quantum states.

✓ Correct Answer:

The correct answer is 3) Their computational complexity remains constant regardless of image size due to processing image subspaces in quantum superposition..

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be eﬃciently resam- pledviainterpolationalgorithms. Itcanalsobeextended to any uploading technique that deals with non-classical band-limited quantum signals. Additionally, this inter- polation technique can be used on genuinely quantum states, but the interpolation will be extended along com- putational basis states which might fail to capture corre- lations that go beyond that. We have also showcased the power of processing quan- tum data in superposition. Encoding diﬀerent sets ofdata using label ancillas allows for the implementation of quantum transformations in parallel. Furthermore, by extending native ideas of natural image processing, used in the JPEG procedure for image compression, we can employ this parallel processing power of a quantum circuit to gain a substantial advantage. By processing subspaces of an image simultaneously using quantum su- perposition, wecanperforminterpolationinacomplexity that is constant with the original size of the image. We would like to further highlight the implementation of eﬃcient quantum transformations in order to provide quantum advantages to algorithms that might not ini- tially rely on them. After all, at the core of the eﬃciency of Shor’s factorization algorithm [34] is the use of the QFT. Looking into areas where these types of transfor- mations are extensively used may result in further av- enues for quantum advantage. Natural image and signal processingareﬁeldsthathaveevolvedaroundsuchtrans- formations, furtherworkmightborrowfromsuchwellun- derstood ﬁelds in order to enhance quantum algorithms in diﬀerent ways or explore interpolation techniques with even more advanced transformations. Compression tech- niques that rely on these transformations might also pro- vide quantum advantages. The code used to simulate the quantum circuits pre- sented is available online [35]. ACKNOWLEDGEMENTS The author would like to acknowledge J. I. Latorre and I. Roth for fruitful discussions. 7 [1] J. J. García-Ripoll, Quantum-inspired algorithms for multivariate analysis: from interpolation to partial dif- ferential equations, Quantum 5, 431 (2021). [2] P. García-Molina, J. Rodríguez-Mediavilla, and J. J. García-Ripoll, Quantum fourier analysis for multivariate functions and applications to a class of schrödinger-type partial diﬀerential equations, Physical Review A 105, 012433 (2022). [3] N. Ahmed, T. Natarajan, and K. R. Rao, Discrete co- sine transform, IEEE transactions on Computers 100, 90 (1974). [4] G. K. Wallace, The jpeg still picture compression stan- dard,IEEEtransactionsonconsumerelectronics 38,xviii (1992). [5] S. Efthymiou, S. Ramos-Calderer, C. Bravo-Prieto, A. Pérez-Salinas, D. García-Martín, A. Garcia-Saez, J. I. Latorre, and S. Carrazza, Qibo: a framework for quan- tum simulation with hardware acceleration, Quantum Science and Technology 7, 015018 (2021). [6] S. Efthymiou, S. Carrazza, C. Bravo-Prieto, Adrian- PerezSalinas, S. Ramos, D. García-Martín, M. Lazzarin, N. Zattarin, A. Pasquale, Paul, and J. Serrano, qi- boteam/qibo: Qibo 0.1.7 (2022). [7] L. Grover and T. Rudolph, Creating superpositions that correspond to eﬃciently integrable probability distribu- tions, arXiv preprint quant-ph/0208112 (2002). [8] V. V. Shende, S. S. Bullock, and I. L. Markov, Syn- thesis of quantum-logic circuits, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Sys- tems25, 1000 (2006). [9] A. Kitaev and W. A. Webb, Wavefunction prepara- tion and resampling using a quantum computer, arXiv preprint arXiv:0801.0342 (2008). [10] M. Plesch and Č. Brukner, Quantum-state preparation with

Question 116 multiple-choice

The group SU(2) plays a crucial role in quantum mechanics, especially for describing the symmetries of spin systems and the transformations of qubit states. Its representation theory connects mathematical structure to physical phenomena such as spin and quantum state rotations. Which property of SU(2) is directly responsible for the fact that a spin-½ particle requires a 720° rotation to return to its original quantum state, rather than 360°? 1) SU(2) is composed only of diagonal matrices 2) The Pauli matrices commute under multiplication 3) SU(2) is isomorphic to SO(3) 4) The trivial representation of SU(2) acts on all group elements 5) SU(2) is a double cover of SO(3), leading to the spinor sign flip after a 360° rotation 6) SU(2) matrices have purely imaginary eigenvalues 7) The fundamental representation of SU(2) acts on three-dimensional vectors

✓ Correct Answer:

The correct answer is 5) SU(2) is a double cover of SO(3), leading to the spinor sign flip after a 360° rotation.

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unitary group SU(2), and the QML task of classifying single-qubit states according to their purity (see Fig. 2(b)). Our choice of considering SU(2)is further motivated both for its privileged posi- tion in quantum physics as the symmetry group underlying spins and for its remarkably well behaved representation theory. Again, just like with discrete groups, we always have the trivial representation of Gon any vector space V, given byRg=12GL(V)for allg2G. All matrix Lie groups also have a so-called fundamen-Example 3: SU(2)and its fundamental rep (spin-1/2) Recall that the special unitary group is deﬁned as SU(2) =fU2GL(C2) :UyU=1and det(U) = 1g: Take 1 qubit V=C2. The fundamental represen- tationU:G!GL(V)is given by Ug=g, i.e., the matrix in Gis the same as its representative. While we are here, it is good to recall that any ma- trixUg2SU(2)can be expressed in terms of Pauli matricesX;Y;Zby Ug=c01+i(c1X+c2Y+c3Z); where (c0;c1;c2;c3)is a real vector with Euclidean normj(c0;c1;c2;c3)j= 1. Note that since this de- ﬁnes an invertible and continuous map between the 3-sphereS3R4andSU(2), the two spaces are homeomorphic. As special cases of this decomposi- tion, we have the rotations around Cartesian axes of angleof the Bloch sphere: RX() =eiX= 2= cos(=2)1isin(=2)X RY() =eiY=2= cos(=2)1isin(=2)Y RZ() =eiZ=2= cos(=2)1isin(=2)Z tal representation ordeﬁning representation6(Example 3), which means that the matrices in the group and the ma- trices in the representation coincide (up to isomorphism, which we will learn morally means “up to change of basis” in Deﬁnition 16). Try to note this important distinction: even though the matrices between the group Gand its rep- resentativesfRg:g2GgGL(V)are identical, we think of the abstract group and its representatives as conceptu- ally distinct. For instance, spin representations of angular momentum are, at their core, (irreducible) representations of the Lie group SU(2). In the case of spin- 1=2, this is ex- 6There yet another unfortunate clash of terminology here–many physicists commonly use these terms interchangeably, but mathe- maticians would say that the deﬁnition here is strictly the deﬁning representation, which for classical matrix Lie groups is one of the fundamental representations. Matrix Lie groups commonly have more fundamental representations than the deﬁning rep (e.g. the deﬁning rep 3and its dual 3forSU(3)), but in the case of SU(2), the deﬁning representation is the only fundamental representation. 12 actly the fundamental representation of SU(2)onV=C2 whereboththematricesandtheirrepresentativesareofthe formei ~where= (x;y;z)is a vector of real angles and= (X;Y;Z )is a vector whose entries are the Pauli matrices. But in higher spin, e.g., spin- 1, the group SU(2) is ﬁxed but the representation V=C3is not, and so the 33representative “spin matrices” take on the new form SX;SY;SZ. More light will be shed on the spin representa- tions and angular momentum once we get to Lie algebras: a pervading theme of Lie theory is that it is often easier to work on the Lie algebra (e.g., the spin matrices X;Y;Z with commutation relations) than on the Lie group (here, the rotation matrices ei). Herewenotethatthefactthattheparameter isdivided by a factor of two in Example 3 is closely related to an im- portant topological fact: SU(2)is the

Question 117 multiple-choice

Quantum algorithms leveraging group symmetries have become central to solving complex problems in quantum many-body physics, especially when classical computational resources are insufficient. The J1-J2 Heisenberg model is a key example where the interplay of frustration and symmetry determines both computational complexity and the nature of quantum phases. Which statement accurately describes why quantum speedup is expected to persist for solving highly frustrated regimes of the J1-J2 Heisenberg model associated with symmetric group operations? 1) Because classical FFT algorithms for Sn scale polynomially with the number of spins, making simulation straightforward. 2) Because quantum circuits for Sn always scale exponentially regardless of model complexity. 3) Because classical dequantization techniques can efficiently simulate all quantum time evolutions in group-theoretic models. 4) Because all ground states of frustrated J1-J2 models can be analytically constructed via Marshall-Lieb-Mattis theorems. 5) Because the regular representation of Sn admits low-dimensional classical embeddings for highly frustrated quantum phases. 6) Because classical algorithms exploiting Schur-Weyl duality achieve superexponential speedup over quantum algorithms in the frustrated regime. 7) Because classical simulation of quantum time evolution involving many Sn group elements in the highly frustrated regime remains intractable, ensuring quantum speedup persists.

✓ Correct Answer:

The correct answer is 7) Because classical simulation of quantum time evolution involving many Sn group elements in the highly frustrated regime remains intractable, ensuring quantum speedup persists..

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classically diﬃcult question and the best classical algorithms Sn-FFT requires a factorial complexity [ 35,36]a sSnhasn! group elements and so is the dimension of its regular represen- tation (see the Wedderburn theorem in the Appendix for more details). Therefore, comparing with the complexity ofSn-FFT, there is a superexponential quantum speed per query. However as a caveat, the entire Hilbert space of nqudits only scales exponentially with nand Snirreps decomposed from the system by Schur-Weyl duality also scales exponentially. Therefore, it would be more reason- able and cautious to refer to a superpolynomial quantum speedup for Sn-CQA. Except comparing with Sn-FFT, recent work from Refs. [ 10,81] proposes the notion of dequantization to compare the eﬃciencies of classical and quantum algo- rithms. Roughly speaking, with well-prepared quantum initial states, quantum algorithms can always be exponen- tially faster than the best counterpart classical algorithms. Assume classical algorithms also have eﬃcient access to input. If the output can now be evaluated with at most polynomially larger query complexity then the quantum analogy, it is said to be dequantized with no genuine quantum speedup. In our case, let us assume our initial states—Schur-basis elements |αT,μS/angbracketrightor their linear com- binations can be eﬃciently accessed with classical meth- ods. Even though, dequantization still unlikely happens. Except conducting Sn-FFT, matrix representations of σ∈ Sncan also be eﬃciently sampled [ 48,75], but the method works exclusively for a single group element. To sample U(p) CQA(θ)|αT,μS/angbracketrightprocessed after Sn-CQA from Eq. (3), the time evolution of CQA Hamiltonians is expanded and approximated by at least superpolynomially many Sngroup elements (see the Appendix for more details) and hence is still thought to be classically intractable. V.C[Sn] SYMMETRIES OF THE J1-J2 HEISENBERG HAMILTONIAN The spin-1 /2J1-J2Heisenberg model is deﬁned by the Hamiltonian: ˆHp=J1/summationdisplay /angbracketleftij/angbracketrightˆSi·ˆSj+J2/summationdisplay /angbracketleft/angbracketleftij/angbracketright/angbracketrightˆSi·ˆSj,( 8 ) where ˆSi=(ˆSx i,ˆSy i,ˆSz i)represents the spin operators at site iof the concerned lattice. The symbols /angbracketleft ···/angbracketright and /angbracketleft/angbracketleft··· /angbracketright/angbracketright indicate pairs of nearest- and next-nearest-neighbor sites, respectively. The J1-J2model has been the subject of intense research over its speculated novel spin-liquidphases at frustrated region [ 82]. The unfrustrated regime (J 2=0o r J1=0)for the antiferromagnetic Heisenberg model is characterized by the bipartite lattices, for which the sign structures of the respective ground states areanalytically given by the Marshall-Lieb-Mattis theorem [54–56]. As an important result, ground states of unfrus- trated bipartite models are proven to live in the Snirrep corresponding to the Young diagram λ=(n/2,n/2).B y Schur-Weyl duality, this subspace is often referred to as the direct sum of SU (2)-invariant subspaces with total spin J=0 in the context of physics [cf. Figs. 2(a) and2(b)]. With this fact, algorithms like Ref. [ 38] has been designed to enforce SU (2)symmetry at J=0 and solve Heisenberg models without frustration. The system is known to be highly frustrated when J1andJ2are comparable J2/J1≈0.5 [ 83] and near the region of two phase transitions from Neel ordering to the quantum paramagnetic phase and from quantum para- magnetic to colinear phase, where no exact solution is known. Moreover, little

Question 118 multiple-choice

In quantum information theory, locally maximally entangled (LME) states are pure quantum states in multipartite systems where every subsystem appears maximally mixed when others are traced out. The existence and structure of such states depend critically on the dimensions and number of subsystems involved. In which scenario does the dimension of the space of locally maximally entangled states modulo local unitary transformations and permutations exceed 3, indicating a rich variety of entanglement structures? 1) A system of five qubits 2) A system of three qubits 3) A system of two qutrits 4) A system of three subsystems each of dimension 2 5) A bipartite system of subsystems with unequal dimensions 6) A system of three subsystems each of dimension 3 7) A system of four qubits

✓ Correct Answer:

The correct answer is 1) A system of five qubits.

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dimension of the space SLME/K∼P(H)//G is larger than 3. 9 For qubit systems, this holds whenever n>4, so we reproduce the results in [GKW17]. For systems ofnqutrits, this holds for n>3. The condition also holds for any system of three or more subsystems of the same dimension d>3. These cases reproduce and generalize the results in [SWGK17]. Our general condition covers all remaining cases , where the subsystem dimensions are not equal. 2 Setup and simple cases We will consider pure states in a multipart Hilbert space H=H1⊗···⊗H n with subsystems Hiof dimension di. We can write a general state explicitly as |Ψ/an}b∇acket∇i}ht=/summationdisplay iψi1···in|i1/an}b∇acket∇i}ht⊗···⊗|in/an}b∇acket∇i}ht, (2.1) where states |ik/an}b∇acket∇i}htwith 1≤ik≤dkform an orthonormal basis of Hk. The density matrix for thekth subsystem is given by ρk= tr¯k|Ψ/an}b∇acket∇i}ht/an}b∇acketle{tΨ|or explicitly as ρjklk=/summationdisplay iψi1···jk···in(ψ∗)i1···lk···in(2.2) A locally maximally entangled state |Ψ/an}b∇acket∇i}htis deﬁned as a state such that for every kwe have ρjklk=1 dkδjklk. (2.3) 2.1 The Schmidt decomposition, bipartite systems, general nec- essary conditions For a bipartite system with H=HA⊗ H¯A, we can write a general (pure) state using the Schmidt decomposition as |Ψ/an}b∇acket∇i}ht=/summationdisplay i√pi|ψA i/an}b∇acket∇i}ht⊗|ψ¯A i/an}b∇acket∇i}ht, (2.4) where|ψA i/an}b∇acket∇i}htare orthonormal states in HA,|ψ¯A i/an}b∇acket∇i}htare orthonormal states in H¯A, andpiare positive real numbers with/summationtext ipi= 1. The density operator for the ﬁrst subsystem is ρ1=/summationdisplay ipi|ψA i/an}b∇acket∇i}ht/an}b∇acketle{tψA i|. (2.5) This is a multiple of the identity operator if and only if {|ψA i/an}b∇acket∇i}ht}form an orthonormal basis of HAandpi=1 dAfor alli. This is possible only if the dimension of HAis less than or equal 10 to the dimension of H¯A. Otherwise, we can’t have the required dAorthogonal states {|ψA i/an}b∇acket∇i}ht} since the number of these is equal to the number of orthonormal s tates|ψ¯A i/an}b∇acket∇i}htwhich is limited by the dimension of H¯A. We can consider a general system with dimensions ( d1,...,dn) as a bipartite system with HA=HkandH¯A=⊗i/ne}ationslash=kHi. Then a state |Ψ/an}b∇acket∇i}ht ∈ H=⊗Hican be LME only if ρk≡ρAis proportional to the identity; we have just seen that this requires dA≤d¯A, so we arrive at general necessary conditions dk≤/productdisplay i/ne}ationslash=kdi (2.6) for the existence of locally maximally mixed states. It has been sugge sted that these condi- tions are also suﬃcient, but we will see that this is not the case. 2.2 Explicit construction: n= 2 For a two-part system, the necessary conditions (2.6) give immedia tely thatd1=d2≡din order for there to exist LME states. The discussion in the previous subsection also implies that such a state can be written via the Schmidt decomposition as |Ψ/an}b∇acket∇i}ht=1√ d/summationdisplay i|ψ1 i/an}b∇acket∇i}ht⊗|ψ2 i/an}b∇acket∇i}ht (2.7) where|ψ1 i/an}b∇acket∇i}htand|ψ2 i/an}b∇acket∇i}htare orthonormal bases of H1andH2. Any state of this form is in SLME. The group of local unitary transformations in K= SU(d)×SU(d) allow us to independently rotate the bases for the two subsystems, so any LME state (2.7) can always be written as U1⊗U2|Ψ/an}b∇acket∇i}htBellwhere|Ψ/an}b∇acket∇i}htBellis the Bell state (1.2). We thus have the well-known result that forn= 2, LME states exist if and only if d1=d2, and in this case, the Bell state is the unique LME state up to local unitary transformations. 2.3 Explicit

Question 119 multiple-choice

In finite group theory, the study of character tables is crucial for understanding the representations and structure of complex groups, including non-split extensions. The group 26.S8, arising as a maximal subgroup of certain orthogonal groups over finite fields, is an example of such an intricate construction. Which subgroup serves as one of the inertia factors for the irreducible characters of the non-split group extension 26.S8? 1) S7 × S1 2) S5 × S3 3) S4 × S2 4) (S4 × S4):2 5) S3 × S6 6) S2 × S8 7) S6 × S4

✓ Correct Answer:

The correct answer is 4) (S4 × S4):2.

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Question 120 multiple-choice

Quantum annealing is a computational approach used to solve optimization problems by leveraging quantum mechanical effects, often compared to classical methods like simulated annealing. The implementation of time evolution in quantum simulations requires specialized techniques depending on hardware architecture. Which method allows quantum annealers to simulate real-time dynamics by encoding time evolution as an optimization problem over an enlarged Hilbert space? 1) Trotterization of the time evolution operator 2) Euclidean lattice discretization 3) Classical Monte Carlo sampling 4) Imaginary-time propagation 5) Feynman clock states with ancillary encoding 6) Grover's algorithm for state preparation 7) Adiabatic quantum gate decomposition

✓ Correct Answer:

The correct answer is 5) Feynman clock states with ancillary encoding.

📚 Reference Text:

theseintoourvariationalcalculation,therhsofEq.( 15)canrewrittenasacostfunction F=/angbracketleftψ|ˆH|ψ/angbracketright–η/angbracketleftψ|ψ/angbracketright=Nconf/summationdisplay α,βK/summationdisplay i,jQαβ,ijqα,iqβ,j, (17) Frommetal. EPJQuantumTechnology (2023) 10:31 Page8of19 where Qαβ,ij=2i+j–2K–2z(–1)δiK+δjKhαβ+δαβδij˜Qα,i, ˜Qα,i=2i–K–z+1(–1)δiKNconf/summationdisplay γa(z) γhγα, (18) hαβ=Hαβ–ηδαβ. Here, ηrepresents the tunable parameter multiplying the penalty term. In addition to givingthevariationalparametersaﬂoating-pointrepresentation,intheabovedeﬁnitionswe have already introduced the parameter z, which is used in the adaptive variational search method [ 10]. This procedure iteratively improves the estimates for the a (z+1) αby distributingthe KsamplingpointsaroundtheprecedingsolutiontotheQUBOproblem, a(z) α, a(z+1) α=a(z) α–qα,K 2z+K–1/summationdisplay i=1qα,i 2K–i+z, (19) startingat a(0) α=0.Theestimatefortheeigenstateisreﬁnedateachstep,hencethename “zooming” for this procedure. The number of zoom steps plays a signiﬁcant role in the overall computational cost of our calculation, as each reﬁnement requires calls to the quantumannealer. The quantum computations are done on the quantum annealing hardware Advan- tage_system5.1 fromD-Wave[ 11],whichisaccessibleviaitsAPID-WaveOcean[ 21]. Asoursystemsizesarestillsmall(c.f.Table 1),theresultscanbecomparedwiththeexact solutionusingtheHamiltonianEq.( 3)aswellaswithsimulatedannealingviatheOcean packageneal. As alternative, which we did not employ, D-Wave oﬀers hybrid solvers s.a.theKerberosSampler [21]thatattempttobreakdowntheoriginalQUBOmatrix into smaller pieces to be subsequently solved using classical or quantum hardware. This appears to be particularly useful for system sizes that cannot be embedded on currentlyavailableannealerarchitectures. It should be noted that for computations employing both simulated and quantum an- nealing,resultsstillhavea ηdependence,seeEq.( 18).Asalreadynoticedin[ 9,10],conver- gencetothetrueground-stateisachievedfor ηlyingintheactualvicinityoftheground- state energy, E 0(approaching from above). For practical reasons, one can determine the “suitable” ηfor a given Qby solving Eq. ( 18)i t e r a t i v e l yi n η, terminating the calculation when a certain convergence criterion, s.a. relative improvement in the solution, is ful- ﬁlled.Thisstrategyworkswellforlocalcomputationswithsimulatedannealingandcouldinprinciplebeemployedalsoforquantumannealing.Here,runtimeonthequantuman-nealeristhemajorconstraint. We ﬁnally comment on our setup when accessing the annealer via the provided soft- ware package. We use the quantum annealer in its forward annealing mode with defaultannealing schedule and annealing time t f=20μs. At least one more parameter needs to be provided by the user during the quantum annealing computations. This is what is re- ferred to as the chain strength. For our calculations, we ﬁnd automatic chain strengthtuning (default option) to be suﬃcient. Figure 3shows results from both simulated and Frommetal. EPJQuantumTechnology (2023) 10:31 Page9of19 Figure3 (Left):Ground-stateexpectationvaluesfor D3forthefullKogut-SusskindHamiltonian(red),electric term(black),andmagneticterm(blue)asafunctionoftheinverseHamiltoniancouplingsquared.Theopen symbolsrepresenttheminimumresultfromthequantumannealerwhiletheﬁlledsymbolswereobtained fromclassicalsimulatedannealing.Thecoloredband,althoughbarelyvisible,representsthemeanand samplestandarddeviationofthemeasurementsfromthequantumannealer.Simulationparametersforthe latterwere K=3withzmax=5zoomstepsand nreads=1000.(Right):Thesamequantitiesfor G=D4where thesamplestandarddeviationfromQAismuchlarger.Thisisduetothefactthatmorecomputingresources areneededtoaccuratelydeterminetheminimum.SimulationparametersforQAwere K=2withzmax=7 zoomstepsand nreads=2000 quantumannealing,fortheground-stateenergy /angbracketleftH/angbracketright(red)aswellastheexpectationvalue for the magnet part /angbracketleftHB/angbracketright(blue)and kinetic part /angbracketleftHE/angbracketright(black)for G=D3(left).When go- ing toG=D4(right), an increase in computational resources is needed due to the more complicatedenergylandscapeforthelargergroup,whichwehoweveronlypartiallymeetduetoruntimerestrictions. 3.2 Timeevolution OneofthemainmotivationsforworkingintheHamiltonianformulationoflatticegaugetheoriesistheabilitytoaccessreal-timedynamics.Thisstandsinstarkcontrasttomain-stream lattice calculations which work in Euclidean space and must perform an analyticcontinuationofnumericaldatainordertoaccessreal-timephysics.Inthegate-basedap-proach, the so-called Trotter approximation is employed to the time evolution operator, ˆU(T)=exp{–iTˆH},whichevolvesaninitialstatebyaﬁnitetime T[15].Thisapproxima- tionconsistsofreplacingthefull ˆUwithproductsofoperatorswhichevolvethesystemon a smaller time interval, δt. Corrections to this approximation typically scale with powers ofδt.Thisapproachtotime-evolutionofquantumstatesallowsforaneﬃcientsimulation ofthetheoryusinguniversalquantumcomputers. In order to solve this problem on the quantum annealer, however, one must reformu- late time evolution as an optimization problem. This can be done by the introduction ofFeynman clock states [ 22], a mechanism ﬁrst applied to quantum chemistry calculations in order to generate parallel-in-time quantum dynamics [ 23]. We thus have to introduce anancillaryquantumsystemwithstates |t/angbracketright,t=1,2,..., N twhereNtisthenumberoftime- slices in the time evolution. Tensoring this orthonormal state with our as-yet-unknownstatevector |ψ t/angbracketrightateachtimeslice,theproblemoftimeevolutionisequivalenttothemin- imizationofthefollowingfunctional L=Nt/summationdisplay t,t/prime=1/angbracketleftt/prime|/angbracketleftψt/prime|ˆC|ψt/angbracketright|t/angbracketright–η/parenleftBiggNt/summationdisplay t,t/prime=1/angbracketleftt/prime|/angbracketleftψt/prime|ψt/angbracketright|t/angbracketright–1/parenrightBigg , (20) Frommetal. EPJQuantumTechnology (2023) 10:31 Page10of19 where ˆC≡ˆC0+1 2/parenleftbig I⊗|t/angbracketright/angbracketleftt|+I⊗|t+1/angbracketright/angbracketleftt+1| –ˆUδt⊗|t+1/angbracketright/angbracketleftt|–ˆU† δt⊗|t/angbracketright/angbracketleftt+1|/parenrightbig . (21) Hereδt≡T/Ntisthestepsizeintime, ηisaLagrangemultiplieranalogoustoourprevious

Question 121 multiple-choice

Quantum algorithms for estimating Hamiltonian eigenvalues rely on efficient resource management, including the elimination of unwanted ancillary qubits (“garbage”) and complex phase factors. Advanced techniques use mathematical lemmas to refine both the output quality and the query complexity of these estimators. In constructing a quantum energy estimator that outputs clean states without phases or garbage, which sequence of techniques is employed to minimize the number of garbage registers while also eliminating phases, and what is the resulting query complexity bound assuming α is bounded away from 1? 1) First invoke Lemma 8 to eliminate garbage, then apply Lemma 7 to minimize garbage registers, resulting in query complexity O(α⁻¹ log(δ⁻¹) (2n + log(α⁻¹))) 2) First use Lemma 7 to minimize garbage registers, then Lemma 8 to eliminate phases, yielding query complexity O(nα log(δ) + log(α)) 3) Apply only Lemma 8 to remove all phases, with query complexity O(α log(n) + δ) 4) Use Theorem 15 directly without any lemmas, resulting in query complexity O(2ⁿα log(δ⁻¹)) 5) Alternate invoking Lemma 7 and Lemma 8 multiple times, leading to query complexity O(α⁻¹ n² log(δ)) 6) Apply Lemma 7 to eliminate garbage and Lemma 8 for phase removal, obtaining query complexity O(n log(α) log(δ)) 7) Discard all garbage after measurement without applying lemmas, giving query complexity O(α⁻¹ n log(log(δ)))

✓ Correct Answer:

The correct answer is 1) First invoke Lemma 8 to eliminate garbage, then apply Lemma 7 to minimize garbage registers, resulting in query complexity O(α⁻¹ log(δ⁻¹) (2n + log(α⁻¹))).

📚 Reference Text:

phases and garbage from Theorem 15 can be combined with Lemma 8 and Lemma 7 to make an energy estimator without phases and without garbage with query complexity bounded by: O/parenleftBig α−1log(δ−1)/parenleftBig 2n+ log/parenleftBig α−1/parenrightBig/parenrightBig/parenrightBig (173) Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 33 assuming that αis bounded away from 1 by a constant. Furthermore, even when there is no rounding promise, there exists an algorithm that, given an eigenstate |ψj/angbracketrightof the Hamiltonian λj, for anyδ >0performs a transformation δ-close in diamond norm to the map: |ψj/angbracketright/angbracketleftψj|→/parenleftBig p|ﬂoor(2nλj)/angbracketright/angbracketleftﬂoor(2nλj)|+ (1−p)|λ/prime j/angbracketright/angbracketleftλ/prime j|/parenrightBig |ψj/angbracketright/angbracketleftψj|(174) wherepis some probability and λ/prime j=ﬂoor(2nλj)−1mod 2nis an erroneous estimate. Just as in Corollary 13, the performance is the same except that 0< α < 1can be any constant. Proof.As with Corollary 13, we write ηkto make the kdependence explicit and demand accuracyδamp,k= (1−10−mamp)(δ2−k−1)2/8for thek’th bit. From Lemma 14 we obtain an asymptotic upper bound r(t,ε)∈O/parenleftbigt+ log(ε−1)/parenrightbig. Again, recall from Lemma 11 that Mηk→δamp∈O/parenleftBig η−1 klog(δ−1 amp)/parenrightBig . The asymptotic query complexity of the iterative energy estimator from Theorem 15 is then bounded by: O/parenleftBig (ηk−δcos)−1log(δ−1 amp,k)·(2n−k+ log(δcos,k))/parenrightBig (175) ≤O/parenleftBig (1−10−mcos)−1η−1 klog((1−10−mamp)−122(k+1)δ−28)·(2n−k+ log(10mcosη−1 k))/parenrightBig (176) ≤O/parenleftBig η−1 klog(2k+1δ−1)·(2n−k+ log(η−1 k))/parenrightBig (177) Next we invoke Lemma 8 to remove the phases and the garbage, doubling the query complexity. We do this before invoking Lemma 7, because Lemma 7 involves blowing up the number of garbage registers by a factor of n. While we could also invoke Lemma 7 and then invoke Lemma 3 to obtain a map without garbage, this would involve many garbage registers sitting around waiting to be uncomputed for a long time. If we invoke Lemma 8 ﬁrst we get rid of the garbage immediately. Finally, we invoke Lemma 7 to turn our iterative energy estimator without garbage and phases into a regular energy estimator without garbage and phases. As in Corollary 13, we observe that η0=α/2and fork >0we haveηkbounded from below by a constant. Then, the total query complexity is: O/parenleftBigg η−1 0log(20+1δ−1)/parenleftBig 2n−0+ log/parenleftBig η−1 0/parenrightBig/parenrightBig +n−1/summationdisplay k=1η−1 klog(2k+1δ−1)/parenleftBig 2n−k+ log/parenleftBig η−1 k/parenrightBig/parenrightBig/parenrightBigg (178) ≤O/parenleftBigg α−1log(δ−1)/parenleftBig 2n+ log/parenleftBig α−1/parenrightBig/parenrightBig +n−1/summationdisplay k=1(k+ log(δ−1))2n−k/parenrightBigg (179) ≤O/parenleftBig α−1log(δ−1)/parenleftBig 2n+ log/parenleftBig α−1/parenrightBig/parenrightBig + 2(2n−n−1) + (2n−2) log(δ−1)/parenrightBig (180) ≤O/parenleftBig α−1log(δ−1)/parenleftBig 2n+ log/parenleftBig α−1/parenrightBig/parenrightBig/parenrightBig (181) Next, we show that it is possible to implement a map that, given an eigenstate |φj/angbracketright, measures an estimate that is either ﬂoor (2nλj)or ﬂoor (2nλj)−1mod 2nwith some probability. Note that this is not the same algorithm as above. Just as with phase estimation, it is only the ﬁrst bit that actually needs the rounding promise, and all other bits are guaranteed to be deterministic. Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 34 The ﬁrst bit performs a map of the form: |0n/angbracketright|0...0/angbracketright|ψj/angbracketright→/parenleftBig√p|0/angbracketright|gar0,j/angbracketright+/radicalbig 1−p|1/angbracketright|gar1,j/angbracketright/parenrightBig |ψj/angbracketright (182) We immediately see that Lemma 8 cannot be used to perform uncomputation here, be- cause uncomputation only works when p= 1orp= 0. Instead, we simply measure the output register containing |0/angbracketrightor|1/angbracketrightand discard the garbage. This would damage any superposition over

Question 122 multiple-choice

In quantum information theory, the study of symmetries in multi-qubit systems often utilizes the representation theory of the symmetric group and matrix algebras constructed from its actions. Schur-Weyl duality provides a powerful framework for decomposing tensor product spaces and understanding their associated algebras. Which statement accurately describes the decomposition of the Swap Matrix Algebra $ M_\text{swap}^n $ based on representation theory? 1) It decomposes into direct sums indexed by all possible partitions of $ n $. 2) It is equivalent to the full group algebra $ \mathbb{C}[S_n] $ without quotients. 3) Its decomposition is indexed by single-row Young diagrams corresponding to trivial representations. 4) The swap algebra is irreducible and cannot be decomposed further for any $ n $. 5) It decomposes as a direct sum over the irreducible representations of $ S_n $ corresponding to two-row Young diagrams. 6) Its structure is determined solely by the action of $ \mathrm{GL}_2(\mathbb{C}) $ on $ (\mathbb{C}^2)^{\otimes n} $. 7) The decomposition involves only the alternating representation of the symmetric group.

✓ Correct Answer:

The correct answer is 5) It decomposes as a direct sum over the irreducible representations of $ S_n $ corresponding to two-row Young diagrams..

📚 Reference Text:

by HG=/summationdisplay (i,j)∈E(G)2wij/parenleftig I−Swapij/parenrightig (2.14) Now, observe that we can write the right-hand side of Eq. (2.12) as /vextendsingle/vextendsingle/vextendsingleψ(i j)−1(1)/angbracketrightig ⊗···⊗/vextendsingle/vextendsingle/vextendsingleψ(i j)−1(i)/angbracketrightig ⊗/vextendsingle/vextendsingle/vextendsingleψ(i j)−1(j)/angbracketrightig ⊗···⊗/vextendsingle/vextendsingle/vextendsingleψ(i j)−1(n)/angbracketrightig , (2.15) where (ij)∈Snis the transposition of objects iandj. This is in fact a representation of Snon(C2)⊗ndefined by ρn(π) (|ψ1⟩⊗···⊗|ψn⟩) =/vextendsingle/vextendsingle/vextendsingleψπ−1(1)/angbracketrightig ⊗···⊗/vextendsingle/vextendsingle/vextendsingleψπ−1(n)/angbracketrightig (2.16) Definition 2.4. The algebra ρn(C[Sn]), for the representation/parenleftbigρn,(C2)⊗n/parenrightbigin Eq. (2.16) , is called the Swap Matrix Algebra Mswap n. Example 2.5. For smalln, the swap algebra Mswap n can be explicitly determined. Firstly, since the swap matrices Swapijare defined in terms of a representation of Sn, there is a ∗-homomorphism C[Sn]→Mswap n,(i j)∝⇕⊣√∫⊔≀→Swapij, (2.17) whenceMswap n is isomorphic to a semisimple quotient of the group algebra C[Sn] of the symmetric group Sn. Thus representation theory of Sncan be used to study Mswap n. For instance,S3has two one-dimensional representations (the trivial one and the signature), and one two-dimensional one, hence C[S3]∼=C⊕C⊕M2(C). Since dimMswap 3 = 5, this immediately yields Mswap 3∼=C⊕M2(C). Similarly, classifying irreducible representations for S4gives C[S4]∼=C⊕C⊕M2(C)⊕M3(C)⊕M3(C), which together with dim Mswap 4 = 14 implies that Mswap 4∼=C⊕M2(C)⊕M3(C). The general characterization of the Swap Matrix Algebra follows from Schur-Weyl duality [EGH+11] ofSnandGL2(C), the group of invertible 2×2complex matrices. The natural representation of GL2(C)on(C2)⊗nis defined by the diagonal action of the group elements on tensor products of |ψ1⟩,|ψ2⟩,...|ψn⟩∈C2, ζn(g) (|ψ1⟩⊗|ψ2⟩⊗···⊗|ψn⟩) =g|ψ1⟩⊗g|ψ2⟩⊗···⊗g|ψn⟩, g∈GL2(C).(2.18) The irreducible modules of GL2(C)are indexed by two row Young diagrams with an unbounded number of boxes. We denote the former by L[n−k,k]forn∈Nandk= 0,...,⌊n 2⌋. In fact,L[n−k,k]is the space of all linear maps from V[n−k,k]to(C2)⊗nthat commute with the action of Sn, that is L[n−k,k]= HomSn(V[n−k,k],(C2)⊗n). (2.19) The following lemma is essentially a restatement of Schur-Weyl duality for SnandGL2(C). Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 12 Lemma 2.6. The algebras Mswap n andζn(C[GL 2(C)])are centralizers of each other inside End((C2)⊗n) =M2n(C). Moreover, the space (C2)⊗ndecomposes under the action of the direct product GL2(C)×Snas (C2)⊗n∼=⌊n 2⌋/circleplusdisplay k=0L[n−k,k]⊗V[n−k,k]. (2.20) Proof. See, e.g., [EGH+11, Sec. 5.19] or [Pro07, §6.1]. Since the action of SnonL[n−k,k]is trivial, as an Sn-module, the space (C2)⊗ndecom- poses by Lemma 2.6 into irreducible modules V[n−k,k]with multiplicities as follows: (C2)⊗n=⌊n 2⌋/circleplusdisplay k=0(V[n−k,k])dim(L[n−k,k]). (2.21) The Weyl character formula [EGH+11, Theorem 5.22.1] gives an explicit formula for dim(L[n−k,k]). Further, Lemma 2.6 immediately leads to the following characterization of Mswap n in terms of the irreducible representations of C[Sn]; cf. [Pro21, (1.12)]. Theorem 2.7. The Swap Matrix Algebra decomposes into the direct sum of simple algebras generated by the two row irreps of the symmetric group. That is, we have Mswap n∼=⌊n 2⌋/circleplusdisplay k=0ρ[n−k,k](C[Sn]). (2.22) Proof. This is immediate from Lemma 2.6 and Eq. (2.21). 2.4 Quantum Max Cut and Irreps Now we discuss how the decomposition of the Swap Matrix Algebra into irreps described in the previous section can be applied to calculations related to the eigenvalues of Quantum Max Cut Hamiltonians. Among other benefits the calculations in this section are essential to Section 6. Definition 2.8. Letλ⊢nbe any partition labeling an irrep of Sn, and letGbe

Question 123 multiple-choice

In descriptive data mining, subgroup discovery aims to identify meaningful subpopulations within complex networks, often using graph-based analysis techniques. Advanced methods seek to overcome the limitations of traditional approaches by incorporating behavioral interactions between nodes. Which methodological innovation most effectively addresses the challenge of uncovering indirect behavioral influences between nodes in non-uniform, user-generated datasets? 1) Relying exclusively on explicit graph connections between nodes 2) Applying uniform weighting to all node interactions regardless of behavior 3) Ignoring implicit relationships during subgroup classification 4) Using unweighted digraphs for subgroup analysis 5) Performing subgroup division based only on node degree centrality 6) Forming a weighted complete digraph that incorporates implicit behavioral interactions for subgroup discovery 7) Selecting subgroups based solely on demographic attributes

✓ Correct Answer:

The correct answer is 6) Forming a weighted complete digraph that incorporates implicit behavioral interactions for subgroup discovery.

📚 Reference Text:

Question 124 multiple-choice

In quantum computing, iterative estimators are crucial for algorithms that extract binary representations of eigenvalues, such as phase and energy estimation. Managing computational byproducts, often called garbage, and controlling error accumulation are essential for practical and scalable quantum algorithm design. Which statement best captures how the total error of a non-iterative estimator constructed by stitching together multiple coherent iterative estimators is controlled in such modular quantum algorithms? 1) The total error is the product of individual estimation errors at each bit. 2) The total error is equal to the largest error among all bits estimated. 3) The total error accumulates linearly with the number of bits extracted. 4) The total error is bounded by δ, leveraging the triangle inequality of the diamond norm across all bits. 5) The total error is always zero when garbage is uncomputed at the end of the process. 6) The total error is independent of the rounding promise used in the protocol. 7) The total error increases exponentially with the number of least significant bits estimated.

✓ Correct Answer:

The correct answer is 4) The total error is bounded by δ, leveraging the triangle inequality of the diamond norm across all bits..

📚 Reference Text:

idea into the modular framework of Deﬁnition 2 and Lemma 3. A ‘coherent iterative estimator’ obtains a single bit of the estimate, given access to all the previousbits. Severalinvocations of acoherent iterative estimator yielda regularestimator as in Deﬁnition 2. Furthermore, we can choose to uncompute the garbage at the very end using Lemma 3, or, as we will show, we can remove the garbage early, which prevents it from piling up. Deﬁnition 6. Acoherent iterative phase estimator is a protocol that, given some n, α, anyk∈{0,...,n−1}, and any error target δ>0, produces a quantum circuit involving calls to controlled- UandU†. If the unitary Usatisﬁes an (n,α)-rounding promise, then this circuit implements a quantum channel that is δ-close to some map that performs: |0/angbracketright|∆k/angbracketright|ψj/angbracketright→|bitk(λj)/angbracketright|∆k/angbracketright|ψj/angbracketright (74) Here bitk(λj)is the (k+ 1)’th least signiﬁcant bit of an n-bit binary expansion, and |∆k/angbracketright is ak-qubit register encoding the kleast signiﬁcant bits. (For example, if n= 4and λj= 0.1011...then bit 2(λj) = 0and∆2= 11.) Note that the target map is only constrained on a subspace of the input Hilbert space, and can be anything else on the rest. Ancoherent iterative energy estimator is the same thing, just with controlled- UH andU† Hqueries to some block-encoding UHof a Hamiltonian Hthat satisﬁes an (n,α)- rounding promise. Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 16 A coherent iterative estimator can also be with garbage and/or with phases, just like in Deﬁnition 2. Lemma 7. Stitching together coherent iterative estimators. Given a coherent iterative phase/energy estimator with query complexity Q(n,k/prime,α,δ/prime), we can construct a non-iterative phase/energy estimator with query complexity: n−1/summationdisplay k=0Q/parenleftBig n,k,α,δ·2−k−1/parenrightBig (75) The non-iterative estimator has phases if and only if the coherent iterative estimator has phases, and if the coherent iterative estimator has mqubits of garbage then the iterative estimator has nmqubits of garbage. Proof.We will combine nmany coherent iterative phase/energy estimators, for k= 0,1,2,..,n−1. The diamond norm satisﬁes a triangle inequality so if we let the k’th iterative estimator have an error δk:=δ2−k−1then the overall error will be: n−1/summationdisplay k=0δk≤δ 2n−1/summationdisplay k=02−k=δ 2(2−21−n)≤δ (76) So now all that is left is to observe that the exact iterative estimators chain together correctly. This should be clear by observing that for all k>0: |∆k/angbracketright=|bitk−1(λj)/angbracketright⊗...⊗|bit0(λj)/angbracketright (77) and|∆0/angbracketright= 1∈Csince when k= 0there are no less signiﬁcant bits. So the k’th iterative estimator takes the kleast signiﬁcant bits as input and computes one more bit, until ﬁnally atk=n−1we have: |bitn−1(λj)/angbracketright|∆n−1/angbracketright=|bitn−1(λj)/angbracketright⊗...⊗|bit0(λj)/angbracketright=|ﬂoor(2nλj)/angbracketright(78) The total query complexity is just the sum of the ninvocations of the iterative estimators fromk= 0,...,n−1with errorδk. If the iterative estimator has garbage, then the garbage from each of the ninvocations just piles up. Similarly, if the estimator is with phases, and has an eiϕj,kfor thek’th invocation, then the composition of the maps will have a phase/producttextn−1 k=0eiϕj,k. Lemma 8. Removing garbage and phases from iterative estimators. Given a coherent iterative phase/energy estimator with phases and/or garbage and with query com- plexityQ(n,k,α,δ/prime)that has a unitary implementation, we can construct a coherent it- erative phase/energy estimator

Question 125 multiple-choice

In the study of algebraic structures and their associated matrix theory, diagonalizability over localizations of the integers and the bijectivity of natural maps play critical roles in arithmetic geometry and the theory of moduli spaces. Special attention is often given to properties of symmetric matrices and the impact of localization at specific primes. Which statement best describes why symmetric matrices over the localization Zₗ may fail to be Zₗ-diagonalizable specifically at the prime l = 2, and how this issue can be resolved? 1) Because the determinant of any symmetric matrix over Z₂ is always zero, diagonalization is impossible. 2) Since invertible matrices do not exist over Z₂, symmetric matrices cannot be diagonalized. 3) The failure arises because all entries become units at l = 2, making diagonalization trivial. 4) Diagonalizability fails at l = 2 due to the presence of nontrivial zero divisors that obstruct invertibility. 5) At l = 2, only skew-symmetric matrices are diagonalizable, not symmetric ones. 6) Symmetric matrices over Z₂ may not be diagonalizable, but adding diagonal blocks with appropriate powers of 2 allows diagonalization. 7) The obstruction at l = 2 is due to the lack of a well-defined trace, which prevents diagonalization.

✓ Correct Answer:

The correct answer is 6) Symmetric matrices over Z₂ may not be diagonalizable, but adding diagonal blocks with appropriate powers of 2 allows diagonalization..

📚 Reference Text:

Question 126 multiple-choice

Quantum computing can utilize qudits, which are quantum systems with more than two levels, to enhance computational efficiency and scalability. Universal gate sets for qudits are essential for implementing arbitrary quantum algorithms in higher-dimensional systems. Which combination of gates forms a universal gate set for qutrits (three-level quantum systems), enabling the dense generation of SU(3) and the qutrit Clifford group? 1) CNOT, H, T 2) SUM₃, H₃, and gates from {P[0]₃, P[1]₃, P[2]₃} 3) Toffoli, Pauli-X, and SWAP 4) Fredkin, Hadamard, and π/8 5) Controlled-Z, Phase, and SUM₂ 6) SUM₄, H₄, and P[3]₄ 7) Pauli-Y, Hadamard, and Controlled-SUM

✓ Correct Answer:

The correct answer is 2) SUM₃, H₃, and gates from {P[0]₃, P[1]₃, P[2]₃}.

📚 Reference Text:

2 2N2). So the qudit method has a (log2d)2 scaling advantage over the qubit case. Furthermore, in this FIGURE 1 | The schematic circuit of Cm[Rd]with C2[Rd]and C2[Pd(p,q)]. The horizontal lines represent qudits. The auxiliary qudits initialized to |0〉is denoted by the red lines and the black lines denoting m controlling qudits. The two-qudit controlled gates is shown as theverticle lines. P d(p,q)is the permutation of |p〉and |q〉state, and Rdis either X(l) dorZd. Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504 3Wang et al. Qudits and High-Dimensional Quantum Computing reviewed method, for an arbitrary unitary U∈SU(N), from Eqs 7 and 8 Neigenoperators is needed and each can be decomposed with three rotations shown in Eq. 9 . Deriving from the appendix of Ref. 108;Uj,Ncan be decomposed with less than 3 dn−1multiple controlled operations. Finally, as Figure 1 has shown, Cm[Rd] needs mnumber of C2[Rd]andC2[Pd(p,q)].Udcan be composed with d−1 numbers of X(l) das in Eq. 3 . Therefore the total number of primitive operations Lin this decomposition method is L≤2N×3dn−1×n×(d−1)+N×n≤6nd2n+ndn. (12) It is clear that there is an extra factor of nreduction in the gate requirement as the number scale of this method is O(nN2). The other advantage is these primitive qudit gates can be easily implemented with fewer free parameters [ 108]. For qudit quantum computing, depending on the implementation platform, other universal quantum gate setscan be considered. For example, in a recent proposal fortopological quantum computing with metaplectic anyons, Cuiand Wang prove a universal gates set for qutrit and qupit systems, for a qupit being a qudit with pdimensions and pis an prime number larger than 3 [ 38]. The proposed universal set is a qudit analogy of the qubit universal set and it consists severalgeneralized qudit gates from the universal qubit set. The generalized Hadamard gate for qudits H dis Hd⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉/equals1⎥radicaltpext⎥radicaltpext d√⎥summationdisplayd−1 i/equals0ωij|i〉,j∈{0,1,2,... ,d−1}, (13) where ω:/equalse2πi/d .(14) The SUM dgate serves as a natural generalization of the CNOT gate SUM d|i,j〉/equals|i,i+j(modd)〉,i,j∈{0,1,2,... ,d−1}.(15) The Pauli σz, with the π/8 gate as its 4th root, can be generalized to Q[i]gates for qudits, Q[i]d⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉/equalsωδij⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉, (16) withωdeﬁned by Eq. 14 and the related P[i]gates are P[i]d⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉/equals⎥parenleftbig−ω2⎥parenrightbigδij⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉,i,j∈{0,1,2,... ,d−1}. (17) In general Q[i]pis always a power of P[i]pifpis an odd prime. The proposed gate set for the qutrit system is the sum gate SUM 3, the Hadamard gate H3and any gate from the set{P[0]3,P[1]3,P[2]3}.A sa na n a l o g u eo ft h es t a n d a r d universal set for qubit {CNOT ,H,T/equalsπ/8−gate}, the qutrit set generate the qutrit Cliffo rd group whereas the qubit set generate the qubit Clifford group (the de ﬁnition of the Clifford group can be found in Section 2.2.1 ). Whereas the rigorous proof can be found in Ref. 38; the proving process follows the idea introduced in Ref. 23that the gate SUM 3is imprimitive, and the Hadamard H3and any gate from {P[0]3,P[1]3,P[2]3} generates a dense subgroup of SU(3). Similarly, the proposed gate set for the qupit

Question 127 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) over real vector spaces rely on lattice theory, efficient state construction, and sampling techniques. Understanding how these components interact is essential for grasping the algorithmic framework in continuous group settings. Which of the following is a critical step in quantum algorithms for solving the Hidden Subgroup Problem over R^m that ensures efficient extraction of the hidden subgroup information? 1) Sampling an approximation to a random point in the reciprocal lattice L* using a Lipschitz function with compact support 2) Encoding the hidden subgroup directly as a classical bitstring prior to quantum state preparation 3) Restricting the group structure exclusively to finite Abelian groups without embedding discrete groups into continuous spaces 4) Avoiding the use of bounded condition numbers when analyzing reduced lattice bases 5) Constructing quantum states solely from discrete periodic functions without generalizing to continuous functions 6) Applying measurement exclusively in the standard basis rather than the Fourier basis 7) Prohibiting any parameter scaling to preserve original lattice geometry during embedding

✓ Correct Answer:

The correct answer is 1) Sampling an approximation to a random point in the reciprocal lattice L* using a Lipschitz function with compact support.

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improved analysis of [BK93] in Section 4.3. It can be shown that the condition number of a reduced basis is bounded so that the dual lattice, which is the hidden subgroup, can be computed. These provide the main ingredients in the full proof. Theorem 6.1. There is a polynomial time quantum algo- rithm for solving the HSP over Rm. In Section 6.2 we outline the steps of the algorithm and indicate how to derive the probability expression. First we show that the known cases of the Abelian HSP can be re- duced to the new continuous case in Deﬁnition 1.1 over the HSP instance Rm. 6.1 Application: Reduction to G=Rm Our HSP algorithm is applicable to Abelian groups of the formRk×Zl×(R/Z)s×H, whereHis ﬁnite. We call such groups “elementary”. The reduction to G=Rk×Zlis straightforward. In the case of interest, the hidden subgroup Lis a full-rank lattice in G⊆Rk×Rlsuch thatλ1(L∩Rk)/greaterorequalslant λandd(L)/lessorequalslantdfor some ﬁxed numbers λandd. We now describe the further reduction to the group ˜G=Rk+l. The main idea can be illustrated in the one-dimensional case, where the parameter λhas no meaning. We embed G=Zinto ˜G=Rin the standard way, set ν= 2−qfor someq/greaterorequalslant2, and deﬁne the R-oraclegin terms of the Z- oraclefas follows: ˛˛g(x)¸ =c0˛˛strν(t)¸ ⊗˛˛f(s)¸ + c1˛˛strν(t−1)¸ ⊗˛˛f(s+ 1)¸ , (6.2) wheres=⌊x⌋, t =x−s, c 0= cos`π2t´ , c1= sin`π 2t´ . It is clear that gis a continuous function. If fis a periodic function, then gis also periodic with the same period. To construct the state |g(x)/angbracketright using the original oracle f, we compute sandt, use them as parameters in the following sequence of operations, and “uncompute” sandt: |0/angbracketright /mapsto→X zcz|z/angbracketright /mapsto→X zcz|z/angbracketright ⊗˛˛f(s+z)¸ /mapsto→X zcz˛˛strν(t−z)¸ ⊗˛˛f(s+z)¸ , wherez∈ {0, 1}. The last step, |z/angbracketright /mapsto→ | strν(t−z)/angbracketright, re- quires that we discriminate between the states |strν(t)/angbracketrightand |strν(t−1)/angbracketright. This is easy because the supports of those states on the ν-grid do not overlap. Let us now consider the general case, G=Rk×Zl. The groupGis embedded in ˜G=Rk+lby scaling the Zfactors byλ. This is to guarantee that λ1(˜L)/greaterorequalslantλ, where ˜Lis the image ofLunder the embedding. The other condition on the new hidden subgroup reads: d(˜L)/lessorequalslant˜d, where ˜d=dλl. The generalization of Eq. (6.2) is straightforward: ˛˛g(x,x 1,...,x l)¸ =X z1,...,z l∈{0,1} lO j=1|ψ(xj,zj)/angbracketright! ⊗˛˛f(x, s(x 1,z1),...,s(x l,zl)¸ ,(6.3) 300 wheres(x,z ) =⌊x/λ⌋ +z,|ψ(x,z)/angbracketright= cos`π 2t´˛˛strν(t)¸ , witht=x/λ−s(x,z ). Note that the terms in the above sum are mutually or- thogonal vectors. It can be shown that ghas parameters ˜a2=a2+l`π 2νλ(1 +ν)´2, ˜r2=r2+l(2νλ)2and/epsilon1. 6.2 An HSP algorithm for the group Rm Letfbe an (a,r,ε) oracle function for some full-rank lat- ticeL⊆Rmsuch thatλ1(L)/greaterorequalslantλandd(L)/lessorequalslantd(see Def- inition 1.1). The core part of our algorithm is a sampling subroutine that generates an approximation to a random point of the reciprocal lattice L∗. It works under certain assumptions about the oracle parameters. Letω:R→Cbe some Lipschitz function with unit L2- norm supported on the interval [0, 1]. For example, ω(x) =(√2 sin(πx) for x∈[0,1], 0 otherwise.(6.4) Let us also choose a suﬃciently large number ∆ = 2q1and a suﬃciently small number

Question 128 multiple-choice

Representation theory of symmetric groups and invariant theory play a central role in quantum information, combinatorics, and mathematical physics, especially regarding algebras constructed from group actions and their decompositions. Understanding the structure and labeling of projectors within such algebras requires detailed knowledge of Young diagrams and combinatorial coefficients. In the algebra A(m,n), which is a subalgebra of the group algebra of the symmetric group Sm+n invariant under the adjoint action of Sm×Sn, the projectors in its center are explicitly labeled by which mathematical objects, and what key combinatorial property must these labels possess? 1) Pairs of partitions of m and n with coprime sizes 2) Single symmetric functions with nonzero Kronecker coefficients 3) Unordered sets of conjugacy classes in Sm+n with maximal cycle length 4) Pairs of Young diagrams with equal number of boxes and zero Littlewood-Richardson coefficient 5) Triples of irreducible representations with matching dimension 6) Young diagram triples (R, R1, R2) with nonzero Littlewood-Richardson coefficients 7) Necklaces corresponding to the minimal number of orbits under Sm×Sn action

✓ Correct Answer:

The correct answer is 6) Young diagram triples (R, R1, R2) with nonzero Littlewood-Richardson coefficients.

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generalisations of this task. For example Bob could send Alice a pair of Q-states, both with the same (R1,R2,R3)and with Kronecker coeﬃcient 1or greater than 1, and Alice could be tasked with determining the triple of Young diagrams and determining whether the two states sent by Bob are linearly independent. If they are, then this would show that the Kronecker coeﬃcient for the triple is greater than 1. This would require using QPE to determine the eigenvalues of elements in K(n)which are not necessarily central. The use of non-central elements of a permutation centraliser algebra to construct distinct multiplicity states (τ1,τ2) has been used [ 37] in the context of multi-matrix invariants (related to the algebra A(m,n) discussed in section 7). Holography. Following the similarity between the projector detection in this section and the detection in Z(C(Sn))from section 3 it is natural to ask if we can formulate a holographically dual gravitational detection problem for K(n). This algebra has been studied [ 30,31] in connection with invariant observables for tensor models. These in turn have been related [ 70] to the SYK models [ 71,72]. A holographic correspondence between SYK and near- AdS 2quantum gravity has been discussed (see the original papers [ 73–75] and a recent review which discusses the correspondence and associated subtleties [ 76]). It is an interesting question whether near-AdS2 gravity allows a classical gravity dual of the projector detection in K(n)analogous to the classical gravity dual of projector detection in Z(C(Sn))based on LLM geometries [ 11] which we described in section 4. 7 Detection of the Littlewood-Richardson projector Given two non negative integers mandn, there is a PCA A(m,n)which is relevant for the counting of polynomial functions of two matrices X,Yinvariant under adjoint action by unitary matrices U: (X,Y )→(UXU†,UYU†). We present a review of the background from [30]. A(m,n)is the subalgebra of C(Sm+n)made of the elements that are invariant under the adjoint action of Sm×Sn. There is a basis of A(m,n)labelled by multi-necklaces with mbeads of one colour and nbeads of another colour. These are in 1-1 correspondence with orbits in Sm+ngenerated by the conjugation action of γ∈Sm×Sn⊂Sm+n. Withr running over these necklaces, Er=/summationdisplay γ∈Sm×Snγσ(r)γ−1, (7.1) whereσ(r)∈Sm+nis a representative of the r’th orbit. One shows that A(m,n)is a semi-simple associative algebra and therefore admits a Wedderburn-Artin decomposition. It is proved that Dim(A(m,n)) =/summationdisplay R/turnstileleftm+n,R1/turnstileleftm,R 2/turnstileleftng(R1,R2,R)2(7.2) whereg(R,R 1,R2)is the so-called Littlewood-Richardson (LR) coeﬃcient. There is a Wedderburn-Artin basis of A(m,n)in the form QR R1,R2;µνwithR,R 1,R2 being Young diagrams with m+n,m,nboxes respectively. The Littlewood-Richardson – 28 – JHEP05(2023)191coeﬃcient determines the range of the indices µ,ν:1≤µ,ν≤g(R1,R2,R). Explicit formulae for QR R1,R2;µνare known in terms of matrix elements of permutations in the irrepRofSm+nalong with branching coeﬃcients for the reduction of the irrep Rinto representations of the subgroup Sm×Sn. The centreZ(A(m,n))is spanned by projectors labelled by triples (R1,R2,R3)with non-vanishing g(R1,R2,R). They can be written in terms of characters: PR R1,R2=dRdR1dR2 (m+n)!m!n!/summationdisplay σ∈Sm+n/summationdisplay σ1∈Sm/summationdisplay σ2∈SnχR(σ)χR1(σ1)χR2(σ2)σ(σ1◦σ2)(7.3) TASK: Identify a triple (R,R 1,R2)for a LR-projector PR R1,R2.We proceed again by making explicit

Question 129 multiple-choice

Isogeny-based cryptography leverages complex mathematical structures, such as quaternion algebras and maximal orders, to achieve security against classical and quantum attacks. Understanding how algorithmic techniques interact with parameters like powersmoothness is crucial for evaluating protocol security. Which algorithmic procedure is specifically designed to find an element in the maximal order O₀ of a quaternion algebra with a prescribed norm M, supporting the construction of isogenies with desired properties? 1) Cornacchia's algorithm 2) StrongApproximationF(N, µ₀) 3) KLTPM 4) Quantum Decomposition Algorithm 5) RepresentInteger′ R+Rj(N, A, B) 6) Lattice Reduction 7) RepresentIntegerO₀

✓ Correct Answer:

The correct answer is 7) RepresentIntegerO₀.

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As a direct consequence, th is breaks pSIDH quantumly since Nis a large prime in pSIDH. As mentioned previously, the IsERP is easy when Nis powersmooth. We brieﬂy discuss some approaches to solve the general case in Appendix D. Title Suppressed Due to Excessive Length 21 In Section 5.1, we give a summary of useful known algorithms and pro vide variantsfor RepresentInteger ,StrongApproximation andKLPTtobetteraccommo- date our speciﬁc application. Following this, we introduce our primary strategy for resolving the PQLP in Section 5.2. In Section 5.3, we deal with a crit ical technical point which we introduce as the Quaternion Decomposition problem. The crux of this problem, and indeed our main conceptual contribut ion, is the decomposition of σ0into elements that are easy to lift, and elements already possessing a powersmooth norm. Finally in Section 5.4, we provide a qu antum algorithm that solves the PQLP. 5.1 Algorithmic building blocks Our algorithm for the PQLP is founded on algorithmic building blocks initia lly introduced in [ KLPT14 ] and later extended in other work,such as [ FKL+20]. We provide a brief recap of these algorithms here, along with several n ew variants tailored to suit our requirements. We ﬁx logcpto be our powersmooth bound for some constant c, and this bound is inherently implied whenever we reference the term ‘powersmooth’. As in [KLPT14 ], for eachp, we ﬁx a special p-extremal maximal order O0. O0=/braceleftBigg Z/an}bracke⊔le{⊔i,1+j 2/an}bracke⊔ri}h⊔wherei2=−1,j2=−p, forp≡3 mod 4, Z/an}bracke⊔le{⊔1+i 2,j,ci+k q/an}bracke⊔ri}h⊔wherei2=−q,j2=−p,forp≡1 mod 4,(3) wherecis any root of x2+pmodq. In the second case, qis required to satisfy thatq≡3 mod 4 is a prime and/parenleftbig−p q/parenrightbig = 1. We add one extra condition that (q,N) = 1. Under the generalized Riemann hypothesis (GRH), the smallest qis of sizeO(log2p). Foreach O0, weidentify asuborderofthe form R+RjforR=Z[i] (note that wearemakingaslightlydiﬀerent choiceherethan the Rin[KLPT14 ] wherethey always take Rto the maximal order in Q(i)). For an element α=a+bi∈R, we useReR(α) to denote aandImR(α) to denote b. LetDdenote the index [O0:R+Rj], then D=/braceleftBigg 4,forp≡3 mod 4, 4q,forp≡1 mod 4.(4) We will now detail the algorithmic building blocks, sourced from [ KLPT14 ] or [FKL+20]. –Cornacchia (M): on an input M∈Z, outputs either ⊥ifMcannot be repre- sented by a ﬁxed quadratic form f(x,y), or a solution x,yto the equation M=f(x,y). –RepresentIntegerO0(M): on an input M > p, outputsγ∈ O0such that n(γ) =M. –StrongApproximationF(N,µ0): on inputs an integer F > pN4, a primeN andµ0∈Rj, outputsλ /∈NZandµ∈ O0of norm dividing Fsuch that µ=λµ0modNO0. 22 M. Chen, M. Imran, G.Ivanyos, P.Kutas, A.Leroux, C. Petit –KLTPM(I): on inputs an integer M >p3and an ideal I, outputs an equiva- lent idealJsuch thatn(J) =M. Let us denote the output γofRepresentIntegerO0(M) asC+DjwithC,D∈ R. To ﬁt our algorithm’s speciﬁc use case, we require not only that n(γ) is powersmooth, but also that C,Dsatisfy additional conditions relative to some inputsA,B∈R, which are determined by σ0from the PQLP. As a result, we introduce the following variant named RepresentInteger′ R+Rj(N,A,B). This variantnecessitatesmorerandomizedsteps to ﬁnd the desiredou tputsC,D∈R. Algorithm 1: RepresentInteger′ R+Rj(N,A,B) Input:An integer NandA,B∈R Output: C,D∈Rsuch that: i)

Question 130 multiple-choice

In knot theory, various methods are used to represent and classify links, including the use of braids and their closures. Certain theorems guarantee that all links can be represented using specific types of braid closures, which are foundational in the study of link invariants. Which theorem asserts that every link can be represented as the closure of a braid using trace closure? 1) Birman's theorem 2) Alexander's theorem 3) Markov's theorem 4) Hilden's theorem 5) Jones's theorem 6) Artin's theorem 7) Reidemeister's theorem

✓ Correct Answer:

The correct answer is 2) Alexander's theorem.

📚 Reference Text:

we say that they are ambient isotopic . Alink invariant is association of links to algebraic objects (numbers, grou ps, modules, etc.) that respects the equivalence relation of ambient isotopy: equivalent li nks must be associated with the same object. One well-studied framework for studying links is to represent t hem as certain canonical “closures” of braids and explore the algebraic properties of braids that preserve li nk equivalence in this representation. We consider two such closures here: trace closure and plat closure. 4.1.1 Trace closure Trace closure of a braid is obtained by joining the ends (top t o bottom) as shown in Figure 2a; Alexander’s theorem [ 4] asserts that any link can be obtained in this fashion. Of cou rse, a given link can be represented as the closure of many diﬀerent braids. It has been shown by Markov [ 25] that braids that yield equivalent knots under the trace clo sure are related by the Markov moves . These moves are θη←→ηθ, θ,η ∈Bn,and θ←→θσ±1 n−1, θ ∈Bn−1⊂Bn. AMarkov trace is a mapφ:Bn→Cthat is well-behaved with respect to these moves: φ(θη) =φ(ηθ), φ(θσn−1) =zφ(θ), φ(θσ−1 n−1) = ¯zφ(θ), 12 braid (a) The trace closure.braid (b) The plat closure. Figure 2: Braid closures. wherez=φ(σn−1) and ¯z=φ(σ−1 n−1). With such a trace map, one can construct a link invariant by deﬁning, for any braid θthat realizes the link, the quantity L(θ) = (z¯z)−(n−1)/2/parenleftbigg¯z z/parenrightbigge(θ)/2 φ(θ), (4.1) where the linking number, e(θ), is the sum of the exponents of the generators σiappearing in θ. It was shown by Tsohantjis and Gould [ 34] that the quantum doubles of ﬁnite groups yield link invaria nts by this approach. Speciﬁcally, let ( ρ,h) be an irrep of D(G) acting on the Hilbert space Vand letτdenote the natural representation of the braid group Bn, described in Section 2.2above, on the space V⊗n. Then the quantity Lρ,h(θ) = (dρ,h)−1/a\}bracketle{th/a\}bracketri}ht−e(θ) ρ Tr(τ(θ)) (4.2) is a link invariant, where /a\}bracketle{th/a\}bracketri}htρ=χρ(h)/dρandχρ,dρare the character and dimension of the irrep ρofZ(h). 4.1.2 Plat closure A related method to represent links by braids is the plat closure . This is deﬁned on braids with an even number of strands where one takes pairs of adjacent strands o n the top (and bottom) and joins them as in Figure 2b. Similar to trace closure, it can be shown that any link can be represented as the plat closure of some braid. Birman [ 7] proved an analogue of Markov’s theorem for the plat closure . Theorem 1 (Birman [ 7]).Given two braids in B2nwith the same plat closure, there exists a sequence of moves from one to the other such that each move is one of the fol lowing: •(Double coset move) θ←→h1θh2, whereh1,h2∈H2n(deﬁned below) and, •(Stabilization move) θ←→σ2nθ. Here,H2ndenotes the Hilden group , the subgroup of the braid group B2ngenerated by σ1, σ 2σ2 1σ2,andσ2iσ2i−1σ2i+1σ2i,∀i∈{1,...,n−1}. The quantum double of a ﬁnite group Glikewise yields link invariants by this approach. Consider the 2n-fold tensor power of an irrep Λ for

Question 131 multiple-choice

In quantum photonics, linear optical unitaries are implemented using beamsplitters and phase shifters, but achieving universality requires extending the gate set. Universality enables the approximation of any unitary operation on the system's Hilbert space. Which of the following modifications to a linear optics gate set is sufficient to achieve universality in quantum photonic circuits for m ≥ 3 modes? 1) Adding a second beamsplitter with a variable angle 2) Adding a single SNAP gate at a fixed mode, angle, and photon number 3) Introducing photon number-resolving detectors only 4) Including only additional phase shifters 5) Restricting to passive linear optics without ancilla photons 6) Using only displacement operations on each mode 7) Applying a controlled-Z gate between two modes

✓ Correct Answer:

The correct answer is 2) Adding a single SNAP gate at a fixed mode, angle, and photon number.

📚 Reference Text:

and universality Since Un,mis not universal in U(H), we instead ask whether it is a t-design for sufficiently large t. We first observe that the linear optical unitaries form a 1-design. This result is quite straightforward and is surely known; however, as we could not find an explicit statement in the literature, we provide the proof in Appendix B 4. Proposition 4. Letn, m≥2. The group Un,mof linear optical unitaries is a 1-design in U(H). However, Un,mis not a t-design for t≥2. This may be proven directly using representation theory, but it is also a trivial consequence of Theorem 7: Corollary 1. Letn, m≥2. Let Gbe a closed subgroup of U(H)extending the linear optical unitaries, Un,m⊆G⊆ U(H). IfGis a2-design in U(H), then Gis universal. In particular, since Un,mis not universal, it is not a t-design for t≥2. As a result of Corollary 1, if we want a 2-design extending Un,m, we are forced to work with a universal set. The natural question is whether we can at least obtain a nice generating set for U(H). Oszmaniec and Zimborás [8] showed that linear optics augmented by nearly any additional gate is sufficient to densely generate U(H). We further discuss this result in Appendix B 5. Theorem 8 ([8]).Consider the group Un,mof linear optical unitaries for n≥2photons in m≥2modes. Let V∈U(H) be any unitary gate with V̸∈Un,m. 1. For m > 2, the group generated by Un,mandVis universal. 2. For m= 2, define |ζ⟩=Pn a=0(−1)a|a, n−a⟩ ⊗ |n−a, a⟩.If [V⊗V,|ζ⟩⟨ζ|]̸= 0, (28) then the group generated by Un,mandVis universal. In Appendix B 6, we prove that the extension of linear optics by any single SNAP gate at a fixed (nontrivial) angle, on a fixed mode s, and acting upon a fixed photon number k, is sufficient to give universality: Theorem 9. Letn, m≥2,0≤k≤n,0≤s≤m−1. Let θ∈Rwithθ̸≡0 mod 2 π. Then the group generated by Un,mandSk,s(θ)is universal. In Corollary 1 of Ref. 11, Bouland proved that if m≥3, one may densely generate Un,m using copies of a single beamsplitter. Combined with Theorem 9, this implies that U(H)may be generated using a single SNAP gate (with fixed mode, angle, and photon number) and a single beamsplitter (acting on different pairs of modes). V . DISCUSSION In Sec. II, we discussed paradigms for the simulation of loss and distinguishability errors in photonics, and we applied these paradigms to investigate the impact of distinguishability errors on Type I fusion, generalized Type II fusion (the n-GHZ state analyzer), and n-GHZ state generation. For low-order errors, one may do the calculations by hand, as in Secs. II D 1 and II D 2. To incorporate higher-order errors in Bell state generation in Sec. II D 3, we turned to numerical simulation. In the longer term, it will be essential to simulate circuits larger than the BSG circuit, an obvious example being n-GHZ state generation. More generally, as in fusion-based quantum computation, one often wants to generate complex linear optical states by starting from small entangled states, such as GHZ

Question 132 multiple-choice

Quantum simulation platforms using cold atoms in optical lattices have enabled experimental realization of complex quantum models and advanced algorithmic operations. Programmable control technologies like digital-micromirror devices (DMDs) are expanding the possibilities for engineering quantum Hamiltonians and implementing novel unitary transformations. Which of the following statements accurately describes a unique advantage of the quadratic quantum Fourier transform (QQFT) protocol on cold atom optical lattice platforms? 1) It allows direct measurement of quantum entanglement without noise. 2) It enables simulation of superconductivity at room temperature. 3) It provides universal quantum computation in a single step. 4) It optimizes laser cooling techniques for higher atom densities. 5) It automatically implements error correction during quantum simulation. 6) It permits programmable engineering of Hamiltonians with complex symmetries and long-range interactions not easily accessible by conventional methods. 7) It eliminates decoherence effects in many-body quantum systems.

✓ Correct Answer:

The correct answer is 6) It permits programmable engineering of Hamiltonians with complex symmetries and long-range interactions not easily accessible by conventional methods..

📚 Reference Text:

Programmable Hamiltonian engineering with quadratic quantum Fourier transform Pei Wang,1,Zhijuan Huang,1,Xingze Qiu,2, 3,and Xiaopeng Li2, 3, 4,y 1Department of Physics, Zhejiang Normal University, Jinhua 321004, China 2State Key Laboratory of Surface Physics, Institute of Nanoelectronics and Quantum Computing, and Department of Physics, Fudan University, Shanghai 200438, China 3Shanghai Qi Zhi Institute, Shanghai 200030, China 4Shanghai Research Center for Quantum Sciences, Shanghai 201315, China (Dated: November 2, 2022) Quantum Fourier transform (QFT) is a widely used building block for quantum algorithms, whose scalable implementation is challenging in experiments. Here, we propose a protocol of quadratic quantum Fourier trans- form (QQFT), considering cold atoms conﬁned in an optical lattice. This QQFT is equivalent to QFT in the single-particle subspace, and produces a di erent unitary operation in the entire Hilbert space. We show this QQFT protocol can be implemented using programmable laser potential with the digital-micromirror-device techniques recently developed in experiments. The QQFT protocol enables programmable Hamiltonian en- gineering, and allows quantum simulations of Hamiltonian models, which are di cult to realize with con- ventional approaches. The ﬂexibility of our approach is demonstrated by performing quantum simulations of one-dimensional Poincar ´e crystal physics and two-dimensional topological ﬂat bands, where the QQFT protocol e ectively generates the required long-range tunnelings despite the locality of the cold atom system. We ﬁnd the discrete Poincar ´e symmetry and topological properties in the two examples respectively have robustness against a certain degree of noise that is potentially existent in the experimental realization. We expect this approach would open up wide opportunities for optical lattice based programmable quantum simulations. I. INTRODUCTION Quantum Fourier transform1has been widely used in con- structing e cient quantum algorithms, that have exponential quantum speedup over the classical computing2,3. The fa- mous example is Shor’s algorithm, where QFT is integrated in a quantum circuit to perform prime factorization of vi- tal importance to cryptography4. It has been combined with control unitary circuits constituting quantum phase estima- tion5, which can compute many-body Hamiltonian spectra2. It has also been applied to implement quantum phase kick- back in digital quantum simulations of Hamiltonian time evo- lution6. However, its experimental realization meets much challenge7–12, with present noisy-intermediate-scale-quantum devices13. Large scale implementation so-far has not been achieved. Cold atoms conﬁned in optical lattices provide a versa- tile platform for large-scale quantum simulations of quantum many-body physics. There has been tremendous progress in simulating strongly correlated equilibrium physics and exotic quantum dynamics in cold atom experiments. The Fermi- Hubbard model of fundamental importance to modeling cor- related electrons has been implemented with upto hundreds of atoms14–18. Mott-superﬂuid transition and its quantum criti- cality have been studied in lattices of di erent geometry and of di erent dimensionality19–21. Topological bands and geo- metrical Berry phases in momentum space22–28have been ob- served in artiﬁcial gauge ﬁeld lattices with the e ective mag- netic ﬁeld strength reaching the order of unit ﬂux quantum per unit cell. The dynamical phase transition from quantum ther- malization to many-body-localization have been investigated with both interaction

Question 133 multiple-choice

Quantum algorithms are often employed to solve group-theoretic problems that are computationally challenging for classical computers. Techniques such as oracle simulation, swap tests, and reductions between problems play pivotal roles in designing efficient quantum solutions. When solving the Hidden Subgroup Problem in the group Zn_p ⋉ Z2 using quantum algorithms, which approach enables a quantum polynomial-time solution by leveraging a reduction to a related problem, and what is the key aspect of this approach? 1) Directly applying the Quantum Fourier Transform to Zn_p ⋉ Z2 without any problem reduction 2) Utilizing Grover's search to find hidden elements in Zn_p ⋉ Z2 3) Reducing the Hidden Subgroup Problem in Zn_p ⋉ Z2 to the Hidden Translation Problem in Zn_p, allowing use of efficient quantum algorithms for translations 4) Mapping Zn_p ⋉ Z2 to a classical group and using classical subgroup enumeration 5) Employing amplitude amplification techniques to increase probability of finding subgroup generators 6) Treating Zn_p ⋉ Z2 as an abelian group and applying abelian HSP algorithms 7) Using quantum walks to identify subgroup structure through state evolution

✓ Correct Answer:

The correct answer is 3) Reducing the Hidden Subgroup Problem in Zn_p ⋉ Z2 to the Hidden Translation Problem in Zn_p, allowing use of efficient quantum algorithms for translations.

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apply Theorem 3.5 in this new setting. Let us now show how to simulate this new oracle access. From |x/angbracketright|b/angbracketright|0/angbracketrightS|0/angbracketrightSwe compute |(−1)bx/angbracketright|b/angbracketright|0/angbracketrightS|0/angbracketrightS, and then we call f0and get|(−1)bx/angbracketright|b/angbracketright|f0((−1)bx)/angbracketrightS|0/angbracketrightS. We multiply the ﬁrst register by ( −1) and call f1,w h i c hg i v e s |(−1)b+1x/angbracketright|b/angbracketright|f0((−1)bx)/angbracketrightS|f1((−1)b+1x)/angbracketrightS. Finally, we multiply the ﬁrst register by (−1)b+1a n ds w a pt h el a s tt w or e g i s t e r sw h e n b=1 . As there is a quantum reduction from Hidden Subgroup inZn p/multicloserightZ2toHidden Translation inZn pby the method of [13], we obtain the following corollary. Corollary 3.7. Letpbe a ﬁxed prime. Then Hidden Subgroup (Zn p/multicloserightZ2)can be solved in quantum time poly(n). The algorithm TranslationFinding c a na l s obee x t e n d e dt os o l v e Translating CosetinZn p. Corollary 3.8. Letpbe a prime. Let αbe a group action of Zn p.W h e n t=Ω (p(n+p)p−1log(1/ε)),Translating Coset (Zn p,α,t)c a nb es o l v e di nq u a n t u m time(n+p)O(p)log(1/ε)with error ε. Proof. Let the input of the Translating Coset (Zn p,α,t)b e(|φ0/angbracketright⊗t,|φ1/angbracketright⊗t). We can suppose w.l.o.g. that the stabilizers of |φ0/angbracketrightand|φ1/angbracketrightare trivial. Indeed the stabilizers can be computed by Proposition 2.1. If they are diﬀerent, then the algo- rithm obviously has to reject; otherwise we work in the factor group Zn p/G|φ0/angbracketright∼=Zn/prime p for some n/prime≤n. To be more speciﬁc, we can compute a ( Zp-basis for) a subgroup G1ofZn pwhich is a direct complement of G|φ0/angbracketrightby augmenting a basis for G|φ0/angbracketrightto a basis ofZn p, and we can actually work with G1in place of G. Forb=0,1, letfbbe the injective quantum function on Gdeﬁned by |fb(x)/angbracketright= |x·φb/angbracketrightfor every x∈G. If the translating coset of ( |φ0/angbracketright,|φ1/angbracketright)i se m p t y ,t h e n f0and HIDDEN TRANSLATION AND TRANSLATING COSET 13 f1have distinct ranges. Otherwise the translating coset of ( |φ0/angbracketright,|φ1/angbracketright) is a singleton {u},a n d(f0,f1) have the translation u. Thealgorithmfor Translating Coset oninput ( |φ0/angbracketright⊗t,|φ1/angbracketright⊗t)isthealgorithm TranslationFinding on input ( f0,f1) with a few modiﬁcations described below. The oracle access to ( f0,f1) is modiﬁed in the same way as in Corollary 3.6. We simulate theith query |x/angbracketright|b/angbracketright|0/angbracketrightS|0/angbracketrightSusing the ith copy of |φ0/angbracketright|φ1/angbracketright.T h et w or e g i s t e r s |0/angbracketrightS|0/angbracketrightS are swapped with |φb/angbracketright|φ1−b/angbracketright, and then we let act xon|φb/angbracketrightand (−x)o n|φ1−b/angbracketright. The equality tests in steps 0 and 9 are rep laced by the swap test [7, 18] iterated O(log(1/ε)) times. Finally, Nis multiplied by O(log(1 /ε)), and the algorithm re- jects whenever the algorithm TranslationFinding aborts or there is no solution in step 9. 4. Translating coset in solvable groups. 4.1. Preparation. 4.1.1. Quantization of the problems. LetGbe a black-box group with unique encoding, and let αbe a group action on Γ. We now describe quantum analogues of problems with classical outcomes, as unitary transformations whose outputs are basically uniform superpositions on the possible classical outcomes. We

Question 134 multiple-choice

Hybrid tensor networks are an emerging approach in quantum simulation, combining quantum resources with classical computational techniques to efficiently model complex quantum systems. These methods are particularly relevant for simulating large many-body wave functions with limited quantum hardware. Which key advantage does a hybrid tensor network offer for simulating large quantum systems on current intermediate-scale quantum computers? 1) Requires only classical resources for all calculations 2) Eliminates the need for tensor contractions entirely 3) Allows simulation of only small quantum systems due to device limitations 4) Restricts applications to quantum field theory exclusively 5) Guarantees fault-tolerance on noisy quantum devices 6) Uses only quantum hardware with no classical computation 7) Enables simulation of systems larger than the available quantum hardware by leveraging classical tensors alongside quantum states

✓ Correct Answer:

The correct answer is 7) Enables simulation of systems larger than the available quantum hardware by leveraging classical tensors alongside quantum states.

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Question 135 multiple-choice

Quantum algorithms offer new approaches to signal and image processing by leveraging the principles of superposition and parallelism, potentially enabling efficient manipulation of large-scale visual data. In classical methods, techniques like JPEG compression exploit localized correlations within images by processing blocks separately. Which feature distinguishes the JPEG-inspired quantum interpolation algorithm from classical bicubic interpolation, particularly in terms of computational complexity and parallelism? 1) It requires exponentially increasing circuit depth as image size grows. 2) It applies transformations sequentially to each pixel of the image. 3) Its performance metrics are consistently superior to classical bicubic interpolation for all images. 4) It does not exploit any blockwise correlations within images. 5) It relies on quantum measurement to maintain parallel processing advantage. 6) It achieves constant circuit depth regardless of image size by applying parallel transformations to all blocks encoded in the least significant qubits. 7) It compresses images using wavelet transforms instead of discrete cosine transforms.

✓ Correct Answer:

The correct answer is 6) It achieves constant circuit depth regardless of image size by applying parallel transformations to all blocks encoded in the least significant qubits..

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QCT can be applied to all members of the superposition insingularcost. However, weneedtokeepinmindthatif the data in superposition needs to then be extracted via measurements, theadvantagethattheparallelprocessing power introduces will be lost. IV. SUBSPACE-QCT RESAMPLING OF NATURAL IMAGES The power of parallel QFT computation can be ex- tended beyond actions on diﬀerent sets of data encoded in superposition. Subsets of a single data entry can also be processed in parallel. Correlations present in natural data tend to be local- ized within short distances. Indeed, natural images are usually built from large structures (in number of pix- els) that do not necessarily correlate with the rest of the picture. The JPEG protocol for image compression [4] exploits this fact by performing DCT onto 88blocks of the image. This way, the small DCT only capture short range correlations, and the following steps of the protocol can be described using 88matrices. A similar procedure can be applied in a quantum cir- cuit. All the 88pixel matrices of the image encoded in the proposed way are stored in the superposition of the 3 least signiﬁcant qubits. The rest of the system can be un- derstood in the same way we considered the label register for multi-layer images. Applying a transformation to the 3 least signiﬁcant qubits, a QCT in this case, achieves in constant depth the transformation of all subspaces of the image. To be precise, this JPEG-inspired interpolation algo- rithm using the DCT proceeds as follows. Given an im- age encoded into a quantum state in registers jxijyijli, perform the QCT on the last 3 qubits of registers jxi, and simultaneously in the last 3 qubits of register jyi. Introduce the ancilla qubits needed for the interpolationm=1Bicubic QFT n-QCT 3-QCT PSNR30.095 27.395 29.930 29.988 SSIM0.880 0.829 0.871 0.878 Table I. Comparison of the Peak Signal-to-Noise Ratio (PSNR) and Structural SIMilarity (SSIM) of the gray-scale cameraimage, after diﬀerent methods of interpolation. The image is downsized to half its original size ( m= 1) via a classical algorithm using the pixel area relation of the image implemented in OpenCV. The image is interpolated to the original size and compared to the base image. The classical bicubic inteprolation scheme yield the best PSNR and SSIM values, closely followed by the s=3-QCT. From the quantum algorithms, the best choice is the JPEG-inspired s=3-QCT method, it provides the best results and is the most eﬃcient option. after the third least signiﬁcant qubit on both registers. Undo the QCT transformation on the extended space. This achieves image interpolation at a depth constant with the original system size. Given a ﬁxed subspace 2s, s= 3in the JPEG-inspired procedure, this algorithm resamples an 2nsignal into a 2n+mspace in complexity O (s+m)2 , the complexity of the algorithm no longer depends on the size of the original image. A circuit de- picting this algorithm is sketched in Fig. 4b. This algorithm can be generalized to any subspace s, and can introduce improvements depending on the un- derlying structure of the signal. In particular, an

Question 136 multiple-choice

In quantum computing and group theory, the hidden subgroup problem (HSP) and its variants play a pivotal role in algorithm design and complexity theory. The structure and representations of finite groups are central to understanding these algorithms. Which mathematical property ensures that the regular representation of a finite group decomposes into a direct sum of all its irreducible representations, each appearing with multiplicity equal to its dimension? 1) The regular representation is reducible and decomposes according to the group's representation theory. 2) The group is necessarily Abelian, so all representations are one-dimensional. 3) The regular representation is always irreducible for non-Abelian groups. 4) Only groups with normal subgroups have decomposable regular representations. 5) The decomposition requires the group to be cyclic. 6) The decomposition occurs only for quantum groups, not classical finite groups. 7) The regular representation decomposes only if the group is infinite.

✓ Correct Answer:

The correct answer is 1) The regular representation is reducible and decomposes according to the group's representation theory..

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the hidden subgroup conjugacy problem, but less is known about the reverse relation. F irst it is clear that when the group is Abelian or when the subgroups are normal, the HSP is equ ivalent to the HSCP (since subgroups of Abelian groups and normal subgroups are con jugate only to themselves.) Recently Fenner and Zhang[45] have examined the diﬀerence betwe en the search and decision version of the HSP (in the search problem one is required to return t he hidden subgroup and in the decision subgroup one is required to distinguish whether th e hidden subgroup is trivial or not.) Their results imply that the HSCP and the HSP over per mutation groups are polynomial time equivalent. Similarly they show that for the dihedr al group, when the order of the group is the product of many small primes, then the HS CP and the HSP are polynomial time equivalent. In Section 6 we will show that the HSCP and HSP are quantum polynomial time equivalent for the Heisenberg group. Finally we note that the HSP can be decomposed into the HSCP along wit h what we call the hidden conjugate subgroup problem (HCSP). In the hidden con jugate subgroup, one is given a function which hides one of a set of subgroups all of which are conjugate to each other and one desires to identify the conjugate subgroup. For the single copy HCSP, the pretty good measurement was shown to be optimal by Moore and Russell[22]. 4 Symmetry Considerations and the HSP The hidden subgroup state of Eq. (2) possess a set of symmetries which allow us to, with- out loss of generality, perform a change of basis which exploits thes e symmetries. These symmetries are related to the regular representations of the gro up. 4.1 Regular Representations There are two regular representations of the group Gwhich we will be interested in, the left regular representation and the right regular representation. Bo th of these representations act on a Hilbert space with a basis labeled by the elements of the group G. Deﬁne the left regular representation via its action on basis states of this Hilbert space, RL(g)|g′/an}bracketri}ht=|gg′/an}bracketri}ht, (5) wheregg′is the element of Gobtained by multiplying gandg′. Similarly, deﬁne the right regular representation via RR(g)|g′/an}bracketri}ht=|g′g−1/an}bracketri}ht. (6) The regularrepresentationsare, in general, reducible represent ationsof G. In fact these repre- sentations are particularlyimportant in the representationtheor y ofﬁnite groups. The reason for this is that the regular representations are reducible into a dire ct sum of all irreducible representations (irreps) of the group G. Thus, it is possible to ﬁnd a basis in which RLacts as RL(g) =/circleplusdisplay µIdµ⊗Dµ(g), (7) D. Bacon 9 where the direct sum is over all irreps of the group G,Dµ(g) is theµth irrep evaluated at group element ganddµis the dimension of the irrep µ. A similar decomposition occurs for the right regular representation, RR(g) =/circleplusdisplay µDµ(g)⊗Idµ. (8) In fact, an elementary result of ﬁnite group representation theo ry tells us that the basis in

Question 137 multiple-choice

In the representation theory of unitary and symmetric groups, combinatorial objects such as Standard and Semi-Standard Young Tableaux (SYT and SSYT) play a crucial role in labeling basis vectors and understanding group actions. These tableaux have specific rules governing their entries and structure, which are fundamental to their application in mathematical physics and combinatorics. Which statement correctly describes a necessary property of Semi-Standard Young Tableaux (SSYT) with entries from {1, 2,.., d} as used to label basis vectors of irreducible representations of the unitary group U_d? 1) Entries must increase strictly across rows and columns. 2) No two boxes with different labels can appear in the same row. 3) Each label must appear exactly once in the tableau. 4) Columns are weakly increasing while rows are strictly increasing. 5) Rows are weakly increasing and columns are strictly increasing. 6) All boxes labeled with the same integer must be in the same row. 7) The number of boxes in each row must equal the number of boxes in each column.

✓ Correct Answer:

The correct answer is 5) Rows are weakly increasing and columns are strictly increasing..

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1interchanged, which can be seen to be a SYT. If kandk+ 1are in the same row, they must be next to each other and the action is given as (k,k+ 1)|T/angbracketright=|T/angbracketright, (2.7) and if they are in the same column (and necessarily in adjacent rows) the action is (k,k+ 1)|T/angbracketright=−|T/angbracketright. (2.8) For the unitary group, a subgroup tower that leads to a canonical basis is 1 =U1⊂U2⊂...Ud, whereUi is the unitary group acting on the i×iminor of the full d×dmatrix. This tower is determined once we x a basis for each qudit register. This tower, like the one for the symmetric group, gives rise to multiplicity-free branching and hence to a canonical basis. This canonical basis is called the Gelfand-Tsetlin basis. Given any irrepλofUdin the form of a Young diagram, one can obtain the diagrams in the restriction to Ud−1 by removing a box from end of each column in all possible ways. If two columns have the same length in λ and the choice is to remove a box from the left column, then one must also remove the box from the right column to ensure that a valid Young diagram is obtained. An example is given below. λ=→  ,,,  . (2.9) 5 Suppose we pick one and proceed with a choice down the tower, we would have the following possibility. →→→. (2.10) This sequence gives us a basis vector of the irrep λ. This can be encapsulated by putting numbers into the original irrep, which represent the stage before which the boxes are removed. For the above sequence the following numbering would hold. 112 233 3. (2.11) Notice that the rows are weakly increasing i.e., the numbers either increase or stay the same as we move right and the columns are strictly increasing. Such a Young diagram is called a semi-standard Young diagram (SSYT). SSYTs with numbers taken from the set [d]label basis vectors in the irrep of Ud. We will see below that SSYTs also play a role in certain induced representations of the symmetric group called permutation modules. Ecient encodings of these bases are constructed in [5] i.e., using poly( logd, n,log 1//epsilon1) bits. For a SSYT, the tuple that contains the number of boxes labeled by a given integer is called the content of the SSYT. For instance, in the example above, the content is (2,2,3)corresponding to 2boxes numbered one, 2boxes numbered two and 3boxes numbered three. SSYTs have an interesting structure that is useful in our algorithms. If we consider all the boxes con- taining a speci c number, we nd that no two of them appear in the same column. If we isolate these boxes, such a skew diagram is called a horizontal strip . An SSYT can be thought of as being composed of horizontal strips. It turns out that this composition can be made more precise as we describe in the next subsection. An example of an SSYT and the associated horizontal strips are given below. 112 233 3,11,2

Question 138 multiple-choice

In the representation theory of the symmetric group Sn, central elements in the group algebra and their action on irreducible representations play a crucial role in analyzing operators such as permutation-invariant Hamiltonians. The behavior of these operators can be described using character theory and combinatorial formulas. For the irreducible representation of Sn labeled by the partition λ = [n−k, k], what is the explicit formula for the scalar ηλ by which the sum over all transpositions acts on this irrep? 1) ηλ = k(n−k) 2) ηλ = n(n−1)/2 3) ηλ = k(n+1) − k^2 4) ηλ = 2k(n+1) − 2k^2 5) ηλ = n(n+1) − 2k^2 6) ηλ = 2k(n−1) − 2k^2 7) ηλ = k^2(n+1) − k

✓ Correct Answer:

The correct answer is 4) ηλ = 2k(n+1) − 2k^2.

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(2.32) whereηλis some scalar depending only on the irrep λandIis an identity matrix of the appropriate dimension. Proof. We begin by showing something stronger: let qKn=/summationdisplay (i,j)∈E(Kn)(i j)∈C[Sn] (2.33) denote the sum over all transpositions in Sn. Then, for any permutation π∈Sn, we have πqKnπ−1=π/summationdisplay i,j=1,...,n(i j)π−1=/summationdisplay i,j=1,...,n(π(i)π(j)) =qKn (2.34) from which it follows that the element qKnis central in the group algebra C[Sn]. In particular, for any irrep λ= [n−k,k], we see that ρλ(qKn) is central in the irreducible algebra of matrices ρλ(C[Sn]) acting on the vector space Vλas defined in Eq. (2.8). Similarly for any such ρλwe seeρλ(2/parenleftbign 2/parenrightbige−2qKn) =Hλ Knis anSn-linear map from the irreducible module Vλto itself and hence, by Schur’s lemma, is a scalar multiple of the identity matrix. This completes the proof. With a little bit of extra work, we can also compute the scalar multiple ηλfrom Eq. (2.32) associated with Hamiltonian of the clique in each irrep. We do this next. Lemma 2.12. Letηλbe as in Lemma 2.11. Then, for any integer nand irrepρ[n−k,k]we have η[n−k,k]= 2k(n+ 1)−2k2(2.35) Proof. We can find the value of ηλby computing two quantities: the dimension of the λirrep (i.e., the trace of ρλ(e)) and the trace of Hλ Kn. It is possible to compute both of these quantities with the Frobenius trace formula, but we shall save some effort by instead Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 16 using the hook length formula and Murnaghan-Nakayama rule. To deal with the many partial factorials appearing in this calculation, recall the falling factorial notation n[k]from Eq. (2.29). We shall also use the dimension of the [ n−k,k] irrep given in Lemma 2.10. For ease of future use we also write this formula as χ[s,t](e) =s−t+ 1 s+ 1/parenleftigg s+t t/parenrightigg =(s−t+ 1)(s+t)[t−1] t!. (2.36) Next we compute the trace of Hλ Kn. We can do this by computing the character χλ((i j)) of a single transposition ( i j) in theλirrep. Since all transpositions belong to the same conjugacy class, they will also have the same character. We now compute this character. The Murnaghan-Nakayama rule relates the character of a permutation in some irrep to the character of smaller permutations in irreps of the symmetric group on fewer elements. In the special case considered here, it gives χλ((i j)) =/summationdisplay ξ(−1)h(ξ)χλ/ξ(e) (2.37) where the sum runs over all ways of removing two adjacent boxes from the irrep λwhile leaving a valid Young diagram; h(ξ) is one if the boxes removed are stacked vertically, zero otherwise; and λ/ξis the resulting partition when the boxes are removed from λ. More explicitly, we write λ= [n−k,k] and then, for any n>3: χ[n−k,k]((i j)) =  χ[n−k−2,k](e) if k= 1 χ[n−k−2,k](e) +χ[n−k,k−2](e) if 2≤k≤n/2−1 χ[n−k,k−2](e) if k= (n−1)/2 χ[n−k,k−2](e)−χ[n−k−1,k−1](e) ifk=n/2.(2.38) By Eq. (2.14) the Hamiltonian for a clique Knis HKn= 2/summationdisplay (i,j)∈E(Kn)I−2/summationdisplay (i,j)∈E(Kn)Swapij (2.39) = 2/parenleftigg n 2/parenrightigg I−2/summationdisplay (i,j)∈E(Kn)ρ[n−k,k]((i j)), (2.40) and taking the trace of this expression gives Tr/bracketleftig H[n−k,k] Kn/bracketrightig = 2/parenleftigg n 2/parenrightigg χ[n−k,k](e)−2/summationdisplay

Question 139 multiple-choice

Algebraic geometry and quantum field theory intersect in the modeling of topological quantum computation, where the structure of symmetries and algebraic properties determines the behavior of quantum systems. The construction of Gröbner bases and the representation of the Braid group are key for understanding anyon statistics and the scalability of quantum algorithms. Which modification allows the reconstruction of a faithful Braid group representation in quantum computing models with indefinite metrics, after initial limitations due to twisted inner automorphisms and parity invariance? 1) Introducing time-reversal symmetry before associativity in the axiomatic framework 2) Applying quantum deformations to toric varieties along with a specific coordinate prescription 3) Utilizing classical coordinate systems on projective varieties without deformation 4) Replacing the cluster property with locality in the axiomatic sequence 5) Constructing Gröbner bases solely with lexicographic monomial ordering 6) Removing parity invariance from the Wightman axioms 7) Defining the Braid group via ideal quotients in order domains

✓ Correct Answer:

The correct answer is 2) Applying quantum deformations to toric varieties along with a specific coordinate prescription.

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Question 140 multiple-choice

In computational quantum chemistry and materials science, efficiently generating localized orbitals from large electronic structure calculations is critical for interpreting chemical bonding and reducing computational cost. Advanced algorithms often incorporate mathematical properties and physical intuition to optimize performance in these tasks. Which strategy most directly accelerates repeated orbital localization in large systems by enabling parallelism and focusing computation on regions of significant electron density? 1) Performing exhaustive global QR factorizations on the entire density matrix 2) Partitioning the column selection into local QRCP factorizations guided by electron density and using randomized sampling 3) Increasing the frequency of I/O operations to improve data throughput 4) Ignoring electron density and sampling columns uniformly across all regions 5) Selecting columns based solely on their numerical magnitude without regard to conditioning 6) Using only the original orthonormal basis for all calculations 7) Restricting computations to vacuum regions where electron density is minimal

✓ Correct Answer:

The correct answer is 2) Partitioning the column selection into local QRCP factorizations guided by electron density and using randomized sampling.

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regardless of the version used. Another key feature of the algorithm, especially for our modi cations later, is that because we are e ectively working with the spectral projector P;the method performs equivalently if a di erent orthonormal basis for the range of is used as input. In physics terminology, the SCDM procedure is gauge-invariant. Lastly, the key factor in forming a localized basis is the selection of a well conditioned subset of columns. Small changes to the selected columns, provided they remain nearly as well conditioned, may not signi cantly impact the overall quality of the basis. 3. The approximate column selection algorithm. When the SCDM proce- dure is used as a post-processing tool for a single atomic con guration, the computa- tional cost is usually a ordable. In fact in such a situation, the most time consuming part of the computation is often the I/O related to the matrices especially for systems of large sizes. However, when localized orbitals need to be calculated repeat- edly inside an electronic structure software package, such as in the context of hybrid functional calculations with geometry optimization or ab initio molecular dynamics simulations, the computational cost of SCDM can become relatively large. Here we present an algorithm that signi cantly accelerates the SCDM procedure. The core aspect of the SCDM procedure is the column selection procedure. Given a set of appropriate columns the requisite orthogonal transform to construct the 6 SCDM may be computed from the corresponding rows of , as seen in Algorithm 2. Here we present a two stage procedure for accelerating this selection of columns and hence the computation of . First, we construct a set of approximately localized orbitals that span the range of via a randomized method that requires only and the electron density ;though ifis not given it may be computed directly from without increasing the asymptotic computational complexity. We then use this approximately localized basis as the input for a procedure that re nes the selection of columns from which the localized basis is ultimately constructed. This is done by using the approximate locality to carefully partition the column selection process into a number of small, local, QRCP factorizations. Each small QRCP may be done in parallel, and operates on matrices of much smaller dimension than : 3.1. Approximate localization. The original SCDM procedure, through the QRCP, examines all Ncolumns of to decide which columns to use to construct Q. However, physical intuition suggests that it is often not necessary to visit all columns to nd good pivots. For instance, for a molecular system in vacuum, it is highly unlikely that a pivot comes from a column of the density matrix corresponding to the vacuum space away from the molecule. This inspires us to accelerate the column selection procedure by restricting the candidate columns. This is accomplished by generating O(nelogne) independent and identically dis- tributed (i.i.d.) random sample columns, using the normalized electron density as the probability distribution function (pdf). Indeed, if a column of

Question 141 multiple-choice

In computational group theory, the Hidden Subgroup Problem (HSP) and its generalization, the Hidden Symmetry Subgroup Problem (HSSP), play key roles, especially in the context of group actions and algorithmic reductions. Frobenius groups, with their distinct structure and permutation properties, are of particular interest for efficiently solving such problems. Which property of Frobenius groups enables a polynomial-time reduction from HSSP to HSP, thereby facilitating the efficient solution of certain computational problems like the Hidden Quadratic Polynomial Problem? 1) The kernel subgroup K is always abelian in Frobenius groups. 2) Every subgroup of a Frobenius group is cyclic. 3) Frobenius groups are always simple groups with no normal subgroups. 4) The action of the group on the kernel K is always regular. 5) The complement subgroup H is always unique up to isomorphism. 6) All elements outside the kernel K have order two. 7) The existence of an efficiently computable H-strong base in the set M.

✓ Correct Answer:

The correct answer is 7) The existence of an efficiently computable H-strong base in the set M..

📚 Reference Text:

ﬁnite group, Ma ﬁnite set, ◦a polynomial time computable action of GonM,a n dHa family of subgroups of G. If there exists an eﬃciently computable H-strong base in M,t h e n HSSP(G,M,◦,H)is polynomial time reducible to HSP(G,H). Together with Proposition 3.1, this result demonstrates that, in contrast to or- dinary bases, ﬁnding (and even understanding the existence of) strong bases can be quite diﬃcult. The results in the rest of the paper rely on constructing strong basesin two diﬀerent contexts. 4. The HSSP for Frobenius complements and the HQPP. In view of Proposition 3.4, we are interested in group actions for which there exist easily com-putable (and therefore also small) bases for some interesting families of subgroups. If in addition the related HSPis easy to solve, then we have eﬃciently solvable HSSPs. It turns out not only that Frobenius groups under some conditions have these prop- erties, but also that the HQPPcan be cast as one of these HSSPs. 4.1. Strong bases in Frobenius groups. AFrobenius group is a transitive permutation group acting on a ﬁnite set such that only the identity element has more than one ﬁxed point and some nontrivial element ﬁxes a point (see, for example, [23]). Let us recall here some notions and facts about these groups. Let Gbe a Frobenius group with action ◦ MonM. The identity element together with the elements of G that have no ﬁxed points form a normal subgroup K,t h eFrobenius kernel , for which we also have |K|=|M|. This latter fact and that Kis closed under conjugation are easy to prove. Surprisingly, all the known proofs for the statement that Kis a subgrouprequirerepresentationtheory. Asubgroup HofGisaFrobenius complement if it is the stabilizer Hmof some element m∈M. It is a subgroup complementary toK,t h a ti s , K∩H={1}andG=KH. Hence, the group Gis a semidirect product K/multicloserightHofKandH.W ed e ﬁ n et h eb i n a r yo p e r a t i o n ◦K:G×K→Kby g◦Kx=yhxh−1, whenx∈Kandg=yhwithy∈Kandh∈H. Itisastraightforwardcomputationto check that ◦Kis an action of GonK.O b s e r v et h a t Kacts on itself by multiplication from the left, while Hacts onKby conjugation. Furthermore, we can identify the action◦Mwith the action ◦Kvia the map φ:M→Kdeﬁned as follows. For any n∈M,t h e r ee x i s t s gn∈Gsuch that gn◦Mm=nsinceGis transitive. If gn=ynhn withyn∈Kandhn∈H, by deﬁnition we set φ(n)=yn.N o t et h a t φdepends on the choice of m, or, equivalently, on the choice of the complement H.O b s e r v ea l s o thatφ(n) can be characterized as the unique element ynofKwithyn◦Mm=n, and therefore φis a bijection. Then indeed for every g∈Gandn∈M,w eh a v e g◦Kφ(n)=φ(g◦Mn). From now on we will suppose without loss of generality that the action is ◦K, which we denote for simplicity by ◦. HIDDEN SYMMETRY SUBGROUP PROBLEMS 1995 Observethen that with respectto ◦, the Frobenius complement His the stabilizer ofe, the identity element of K. The orbits of Hare{e}and some

Question 142 multiple-choice

Isogeny-based cryptography relies on the hardness of finding isogenies between supersingular elliptic curves, but recent advances have revealed vulnerabilities in widely used protocols. Cryptographic schemes in this area often balance security and efficiency by carefully choosing which mathematical data to reveal during key exchange. Which cryptanalytic breakthrough demonstrated that SIDH protocols with balanced parameters and known endomorphism ring can be broken in polynomial time using superspecial abelian surfaces? 1) The development of group actions on supersingular elliptic curves (KMPW21) 2) The introduction of suborder representations for large prime degree isogenies (pSIDH) 3) The proposal of an isogeny representation based on torsion point images 4) The SCALLOP scheme using partial isogeny representations 5) The computation of the endomorphism ring of the codomain to break pSIDH 6) Robert's attack on SIDH with unknown endomorphism ring 7) The Castryck-Decru and Maino-Martindale polynomial-time attacks using superspecial abelian surfaces

✓ Correct Answer:

The correct answer is 7) The Castryck-Decru and Maino-Martindale polynomial-time attacks using superspecial abelian surfaces.

📚 Reference Text:

alreadyincluded parametersetswhichcouldhave been used in SIDH variants.Neverthelessnone ofthese attacksd irectly impacted SIDH where AandBare roughly the same size. Then in 2022 Castryck and De- cru [CD22] (and independently Maino and Martindale [ MM22]) vastly improved these using ingenious techniques (utilizing superspecial abelian surf aces) which break SIDH with known endomorphism ring in polynomial time even if AandB are balanced. Finally, Robert proposed a polynomial-time attack on S IDH with unknown endomorphism ring (furthermore, he only needs B2> Aas opposed toB >Ain other attacks). These attacks have shown that using smooth degree isogenies and providing torsion point information will potentially not lead to secure and eﬃcien t cryp- tographic constructions (in [ FMP23] some countermeasures are proposed, but the ones that are not broken are much less eﬃcient than the origina l SIDH con- struction). Thus, in order to navigate in the supersingular isogeny graph parties have to share some other kind of extra information. Alternative isogeny representations. In the pSIDH protocol introduced by Leroux [Ler22a], one reveals suborder representations for isogenies of large prime degreestobuildakeyexchange.Suborderrepresentationsarea particularkindof weak isogeny representations, i.e. some data to represent isogen ies together with an algorithm to eﬃciently evaluate these isogenies on any point up to a scalar. Prime degree isogenies were not really used before as one cannot wr ite down the isogeny itself (but one can compute its codomain with non-trivial tec hniques). More recently, a similartype ofsecret isogenywas used in the SCALL OPscheme [DFFK+23]. In SCALLOP, a partial isogeny representation is revealed to the attacker. From a cryptanalytic point of view, the unlimited amount of torsion inf or- mation provided by the isogeny representation revealed in pSIDH (a nd more generally, any isogeny representation) is very interesting. Howev er, when the kernel points are not deﬁned over a small extension, the known alg orithms do not apply and it is still unclear how to exploit the isogeny representat ion to recover the secret isogeny. 4 M. Chen, M. Imran, G.Ivanyos, P.Kutas, A.Leroux, C. Petit Lerouxstudied the case where a speciﬁc isogenyrepresentation( the suborder representation) is revealed, but we can generalize this setting to a ny isogeny rep- resentation. He showed that computing the endomorphism ring of t he codomain wouldmakepSIDH insecure, thereforemotivating Problem1.1in the p rime case. Morerecently,Robertintroducedyetanotherisogenyrepresen tationbasedon torsion point images and the recent SIDH attacks [ Rob22]. This representation could be used (for isogenies with large prime degrees) instead of the suborder representation to derive a key exchange protocol similar to pSIDH , and this protocol would be similarly aﬀected by our new results. A group action for SIDH and pSIDH In [KMPW21 ] the authors introduce a group action on a particular set of supersingular elliptic curves. Le tEbe a supersingular elliptic curve with endomorphism ring isomorphic to O. Then (O/NO)∗acts naturallyon the set of cyclicsubgroupsof EoforderN. Ifthere is a one-to-one correspondence between cyclic subgroups and N-isogenous curves, then one can look at this action as acting on a set of curves. This act ion was used to provide a

Question 143 multiple-choice

In the study of finite abelian groups and their automorphisms, matrix representations are a powerful computational tool, especially when analyzing group characters and transformations. These concepts are essential in quantum algorithms that leverage group structure for problem-solving efficiency. When representing automorphisms of finite abelian p-groups using matrices, which statement correctly describes the relationship established between character evaluations and matrix conjugation? 1) Character evaluations remain unchanged under any non-singular matrix transformation of group elements. 2) Character evaluations are always inversely related when automorphisms are represented by diagonal matrices. 3) Evaluating a character on a transformed group element always results in a zero value unless the automorphism is the identity. 4) The equivalence of character evaluations relies solely on the group being cyclic. 5) Evaluating the character χx on Φ(d) is equivalent to evaluating the conjugate character χˆΦ(x) on d via matrix conjugation. 6) The use of matrix representations eliminates the need for character theory in quantum algorithms. 7) Direct products of cyclic groups cannot be represented with matrix transformations for automorphisms.

✓ Correct Answer:

The correct answer is 5) Evaluating the character χx on Φ(d) is equivalent to evaluating the conjugate character χˆΦ(x) on d via matrix conjugation..

📚 Reference Text:

M. R¨ otteler and T. Beth. Polynomial-time solution to t he hidden subgroup problem for a class of non-abelian groups. arXiv:quant-ph/9812070. [29] K. Shoda. ¨UberdieAutomorphismen einer endlichen Abelschen Gruppe. Math. Ann. , 100:674– 686, 1928. [30] P. W. Shor. Algorithms for quantum computation: Discre te log and factoring. In Proc. 35th Annual IEEE Symposium on Foundations of Computer Science , pages 124–134, 1994. [31] J. Watrous. Quantum algorithms for solvable groups. In Proc. 33rd Annual ACM Symposium on Theory of Computing , pages 60–67, 2001. arXiv:quant-ph/0011023. [32] H. P. Yuen, R. S. Kennedy, and M. Lax. Optimum testing of m ultiple hypotheses in quantum detection theory. IEEE Trans. Inform. Theory , 21:125–134, 1975. 16 A Matrix representation of Φ(b) In this appendix, we show that Φ(b)andˆΦ(b)can be represented by (simply related) matrices, and furthermore, that χx(Φ(b)(d)) =χˆΦ(b)(x)(d). Lemma 7. LetAbe a ﬁnite abelian group, ϕ∈Aut(A),b∈Nwith the corresponding Φ :A→A deﬁned by Φ :=/summationtextb−1 i=0ϕi, andχ:A→Ca character of A. Then there exists a function ˆΦ :A→A such thatχx(Φ(d)) =χˆΦ(x)(d)for alld,x∈A. Proof.LetA∼=Ap1×···×Aprbetheelementary divisor decomposition of A, i.e., thedecomposition intop-groups where the piare distinct primes. Accordingly, let d= (d1,...,dr),x= (x1,...,x r), andχ(i) xi:Api→Csuch thatχx(d) =χ(1) x1(d1)···χ(r) xr(dr). It is known that any automorphism ϕ∈Aut(A) can be decomposed as ϕ= (ϕ1,...,ϕ r) withϕi∈Aut(Api) [29]. Similarly, we have the decomposition Φ( d) = (Φ 1(d1),...,Φr(dr)). Hence if we prove the lemma for p-groups, then this proves it for all ﬁnite abelian groups. Assume therefore that Ais ap-group,A∼=Zpe1×··· ×Zpekwithe1≥ ··· ≥ek. By [29], we can represent ϕ∈Aut(A) as a matrix transformation ( x1,...,x k)/ma√st≀→(x1,...,x k)µwhereµ∈Zk×k andpej−ei|µijfor alli>j. Consequently, we can also represent the transformation Φ b y a matrix Mwithpej−ei|Mijfor alli>j. Now deﬁne a conjugate matrix ˆMji:=pei−ejMijfor alli,j(note that all entries of ˆMare integers). For any character χx:A→C, we ﬁnd χx(Φ(d)) = exp/parenleftig 2πi/summationdisplay jidiMijxj/pej/parenrightig = exp/parenleftig 2πi/summationdisplay ijdixjˆMji/pei/parenrightig =χˆΦ(x)(d),(49) whereˆΦ :A→Ais the matrix transformation deﬁned by ( x1,...,x k)/ma√st≀→(x1,...,x k)ˆM. Note that if A=ZN(as in Section 5), then Φ can be represented by a single scalar M∈ZN, and hence ˆΦ = Φ. Also, if A=Zr p(as in Section 6), then M∈Zr×r pandˆM=MT. For notational simplicity, since we only need the matrix representation of ˆΦ (and not of Φ) in these two cases, we putµ→µTandM→MTthroughout the body of the paper. B Proof of Lemma 5 Proof.Forx∈(Zn p)k,w∈Zn p, andb∈Zk p, let Ξx w(b) = 0 denote the system of polynomial equations (48). We want to understand the typical behavior of ηx w:=|{b: Ξx w(b) = 0}|=/summationdisplay bδ[Ξx w(b) = 0] (50) for uniformly random x,w. Speciﬁcally, we want to show that it is typically close to it s mean, µ:=E x∈Ak,w∈A[ηx w] =pk−r(51) where we have used (14). To compute the variance, note that E x∈Ak,w∈A[(ηx w)2] =1 prk+r/summationdisplay x,w(ηx w)2(52) =1 prk+r/summationdisplay x,w/parenleftbigg/summationdisplay bδ[Ξx w(b) = 0]/parenrightbigg/parenleftbigg/summationdisplay cδ[Ξx w(c) = 0]/parenrightbigg (53) =1 prk+r/summationdisplay x,w/parenleftbigg/summationdisplay bδ[Ξx w(b) = 0]+/summationdisplay b/negationslash=cδ[Ξx w(b) = Ξx w(c) = 0]/parenrightbigg .(54) 17 The ﬁrst (diagonal) term is just the mean. For the second (oﬀ-d iagonal) term, note that the

Question 144 multiple-choice

In quantum information science, matrix product states (MPS) and matrix product operators (MPO) are essential tools for efficiently representing quantum states and operations, particularly in simulations of quantum circuits such as the quantum Fourier transform (QFT). The efficiency and accuracy of these representations depend critically on parameters like bond dimension and error bounds. For an MPS representation of the function f2(x) = sum of 20 cosines with n = 12, which of the following bond dimension sequences accurately describes how the required resources vary across the chain? 1) [2, 3, 5, 6, 7, 8, 10, 12, 7, 5, 2] 2) [2, 4, 7, 8, 8, 9, 10, 10, 8, 4, 2] 3) [2, 4, 6, 8, 9, 11, 13, 8, 4, 2, 2] 4) [2, 4, 6, 7, 8, 9, 11, 13, 8, 4, 2] 5) [2, 3, 6, 8, 10, 11, 12, 12, 9, 5, 2] 6) [3, 5, 7, 8, 9, 10, 12, 13, 9, 5, 3] 7) [2, 4, 5, 7, 8, 8, 9, 11, 7, 4, 2]

✓ Correct Answer:

The correct answer is 4) [2, 4, 6, 7, 8, 9, 11, 13, 8, 4, 2].

📚 Reference Text:

˜emax on both Dχ n+1,jand Dχ n+1,j+1; therefore ˜emaxonDχ n+1,jand Dχ n+1,j+1will also share the same bound, which makes it independent of j. For details, see the diagrammatic illus- tration in Fig. 9. Therefore, because of the decay of ˜σk j, the maximal error is also approximately O(e−χlog(χ/3)/√χ), and is thus at most O(ne−χlog(χ/3)/√χ)among the entire QFT MPO. APPENDIX K: FUNCTIONS USED FOR TIMING OF SUPERFAST FOURIER TRANSFORM For the timing of the superfast Fourier transform algorithm based on the MPO representation of the QFT, we used three example functions, all deﬁned on the domain 0≤x≤1. The ﬁrst function, labeled “one cosine” in Fig. 4,i s given by f1(x)=cos(2πx). (K1) It compresses exactly into an MPS with uniform bond dimensions all of size χm=2. The second function, labeled “20 cosines” in Fig. 4,i s given by f2(x)=20/summationdisplay j=1cos/bracketleftbigg 1.1(4j−2)/parenleftbigg x−1 2/parenrightbigg/bracketrightbigg . (K2) When this function was represented as an MPS with n=12, for example, the bond dimensions were [2, 4, 6, 7, 8, 9, 11, 13, 8, 4, 2]. The third function, labeled “one cosine + cusps” in Fig.4, is given by f3(x)=cos(2πx)+2e−3|x−0.4|+e−2|x−0.7| +2e−3|x−0.8|+e−2|x−0.9|. (K3) When this function was represented as an MPS with n=12, for example, the bond dimensions were [2, 4, 7, 8, 8, 9, 10, 10, 8, 4, 2]. [1] P. W. Shor, Polynomial-time algorithms for prime factor- ization and discrete logarithms on a quantum computer,SIAM J. Comput. 26, 1484 (1997). [2] A. Y. Kitaev, Quantum measurements and the Abelian sta- bilizer problem, Electron. Colloquium Comput. Complex. TR96 (1996). [3] P. Høyer, Conjugated operators in quantum algorithms, P h y s .R e v .A 59, 3280 (1999). 040318-29 JIELUN CHEN, STOUDENMIRE, and WHITE PRX QUANTUM 4,040318 (2023) [4] I. Kassal, J. D. Whitﬁeld, A. Perdomo-Ortiz, M.-H. Yung, and A. Aspuru-Guzik, Simulating chemistry using quantumcomputers, Annu. Rev. Phys. Chem. 62, 185 (2011). [5] A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear systems of equations, P h y s .R e v .L e t t . 103, 150502 (2009). [6] L. Ruiz-Perez and J. C. Garcia-Escartin, Quantum arith- metic with the quantum Fourier transform, Quantum Inf. Process. 16, 152 (2017). [7] M. A. Nielsen, C. M. Dawson, J. L. Dodd, A. Gilchrist, D .M o r t i m e r ,T .J .O s b o r n e ,M .J .B r e m n e r ,A .W .H a r r o w ,and A. Hines, Quantum dynamics as a physical resource, Phys. Rev. A 67, 052301 (2003). [8] J. Tyson, Operator-Schmidt decomposition of the quantum Fourier transform on C N1tensor CN2,J. Phys. A: Math. General 36, 6813 (2003). [9] K. Woolfe, C. Hill, and L. Hollenberg, Scaling and eﬃ- cient classical simulation of the quantum Fourier transform, Quantum Inf. Comput. 17, 1 (2014). [10] D. Slepian and H. O. Pollak, Prolate spheroidal wave func- tions, Fourier analysis and uncertainty—I, Bell Syst. Tech. J.40, 43 (1961). [11] H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions,

Question 145 multiple-choice

Quantum photonic systems rely on precise control of photons across multiple modes to perform computational and networking tasks. The combination of linear optical elements and specialized gates enables a wide range of quantum operations, though scalability and error mitigation remain key challenges. In a quantum optical system with n indistinguishable photons distributed over m modes, what is the exact dimension of the Hilbert space describing possible photon configurations? 1) n × m 2) mⁿ 3) C(n + m - 1, n) 4) n! × m! 5) 2ⁿ 6) C(m, n) 7) n + m

✓ Correct Answer:

The correct answer is 3) C(n + m - 1, n).

📚 Reference Text:

[8] to prove that augmenting the linear optical unitaries with any nontrivial SNAP gate is sufficient to achieve universality. These results are in Sec. IV. • In Sec. V, a brief overview of two future directions related to this work. First we discuss the applicability to larger systems of our analysis techniques and results regarding distinguishability errors. We propose a low-order approx- imation of the effect of distinguishability errors, as elaborated upon in Remark 5. We then consider the problem of 3 how many SNAP gates are required to promote linear optics to an approximate 2-design and provide some numerical results for small system sizes. II. DISTINGUISHABILITY AND LOSS ERRORS Loss and distinguishability are two of the most dominant contributions to noise and errors in optical settings, both of which are known to reduce the classical complexity of computational tasks such as Boson sampling [9, 10]. As such, it is important to be able simulate the effects of each in software. Our starting point is Fock basis simulation. Generally we will be interested in simulating an mmode system with a fixed number nof (for now) identical photons. As such the Hilbert space can be represented via the second quantization as H= span {|n1, . . . , n m⟩}P ini=n, with dimension dn,m:=n+m−1 n . The initial state is typically composed of nsingle photons in m≥nmodes, |1⟩⊗n|0⟩⊗(m−n). Evolution occurs via a linear optical network, which maps creation operators1 to a linear combination Ua† iU†=mX j=1ujia† j, (1) where uis an m×munitary matrix, called the transfer matrix. It is known that any linear optical transformation can be implemented, to arbitrary accuracy, by a sequence of O(m2)beamsplitters2[12]. The full state output of such a network can be computed in time O(ndn,m), and space O(dn,m)[5, 13], by multiplying the evolved creation operators: U|1⟩⊗n|0⟩⊗(m−n)=UnY i=1a† iU†|⃗0⟩=nY i=1 mX j=1ujia† j |⃗0⟩. (2) In the first step we write |1⟩=a†|0⟩, and use that any linear optical unitary acting on the vacuum is the identity, i.e. U†|⃗0⟩=|⃗0⟩. In the second step we insert resolutions of the identity I=U†Uafter each creation operator. In this work we will assume measurements are all of the photon-number-resolving (PNR) type, which, in the absence of error, counts the number of photons in each measured mode. That is, it is equivalent to performing a measurement in the Fock basis. In Secs. II A and II B, we will discuss how to modify such a simulation to include losses and distinguishability errors. This is followed by Sec. II C, in which we review work of Marshall [6] that gives a protocol for distilling less distinguishable photons. This protocol was discovered using the simulation techniques of Sec. II B. Then in Sec. II D, we consider the effect that distinguishability has on circuits for fusion, GHZ state generation, and related operations. The first several subsections give analytical results; Sec. II D 3 gives numerics for Bell state generation in the presence of distinguishability, using the techniques of Sec. II B. A. Simulation of Loss

Question 146 multiple-choice

Quantum algorithms have revolutionized certain computational tasks, particularly those involving algebraic structures and hidden patterns in functions over finite fields. Understanding the difference in complexity between classical and quantum approaches to identifying hidden polynomials is essential in computational algebra and cryptography. When identifying an m-variate polynomial of total degree greater than 2 hidden within a function over a finite field with d elements, which of the following statements best describes the quantum algorithm's advantage over classical algorithms? 1) The quantum algorithm requires exponentially more queries than the classical algorithm as d increases. 2) Both quantum and classical algorithms require Ω(d) queries for constant success probability. 3) Quantum algorithms do not offer any speedup for polynomials of degree greater than 2. 4) The classical algorithm is faster due to its ability to directly sample all polynomial coefficients. 5) Quantum algorithms cannot identify polynomials of degree greater than 2 with constant probability. 6) The quantum algorithm identifies hidden polynomials with constant probability in polylogarithmic time, whereas classical methods require at least Ω(√d) queries. 7) Classical algorithms outperform quantum algorithms when d is large and m > 1.

✓ Correct Answer:

The correct answer is 6) The quantum algorithm identifies hidden polynomials with constant probability in polylogarithmic time, whereas classical methods require at least Ω(√d) queries..

📚 Reference Text:

Question 147 multiple-choice

Quantum algorithms for hidden subgroup problems use group-theoretic methods to efficiently uncover hidden structure in functions defined over groups. The complexity and implementation of the quantum Fourier transform vary significantly between abelian and non-abelian groups due to differences in their representations. In the context of quantum algorithms for hidden subgroup problems, which of the following is a direct consequence of the structure of non-abelian groups when implementing the quantum Fourier transform? 1) The Fourier basis states are always equal to the standard basis states. 2) All irreducible representations are one-dimensional characters. 3) The quantum algorithm requires an exponential number of queries even for simple groups. 4) The irreducible representations used to construct the Fourier basis are higher-dimensional unitary matrices. 5) The Fourier transform over non-abelian groups can be implemented using only Hadamard gates. 6) The sum of the dimensions of irreducible representations equals the group order. 7) The quantum Fourier transform does not require orthogonality of basis states.

✓ Correct Answer:

The correct answer is 4) The irreducible representations used to construct the Fourier basis are higher-dimensional unitary matrices..

📚 Reference Text:

but to convert this into an explicit description of K(say an actual set of generators) we need to use further mathematical properties of Ge.g. properties of co-primality as illustrated in our examples. The above group-theoretic framework serves to generalize a nd extend the applicability of the quantum algorithm for periodicity determination. Fo r example Simon considered the following problem: suppose that we have a black box which com putes a function ffromn-bit strings ton-bit strings. It is also promised that the function is “two-t o-one” in the sense that there is a ﬁxed n-bit stringξsuch that f(x+ξ) =f(x) for all n-bit strings x. (18) (Here + denotes binary bitwise addition of nbit strings.) Our problem is to determine ξ. To see that this is just a generalized periodicity determina tion note that in the group ( Z2)n ofn-bit strings, every element satisﬁes x+x= 0. Hence eq. (18) states just that fis periodic on the group with periodicity subgroup K={0,ξ}. Thus to determine ξwe construct the 10 Fourier transform on the group of n-bit strings and apply the standard algorithm above. The relevant Hilbert space Hwith a basis labeled by n-bit strings is just a row of nqubits. The irreducible representations of the group ZN 2are the functions fx(y) = (−1)x1y1...(−1)xnyn wherex=x1...x nandy=y1...y nare n bit strings. Thus the Fourier transform may be easily seen [11] to be just the application of the 1-qubit Had amard transform: H=1√ 2/parenleftBigg 1 1 1−1/parenrightBigg to each of the nqubits. The resulting quantum algorithm for determining th e hidden subgroup then reproduces Simon’s original algorithm [6]. It determi nesξinO(n2) steps whereas it may be argued [6] that any classical algorithm must evaluate fat leastO(2n) times. 6 Non-abelian groups We will now consider the hidden subgroup problem in the situa tion whereGand the subgroup Kmay be non-abelian i.e. we have f:G→Xwhich is constant on the (left) cosets of Kin G. We now also write the group operation multiplicatively. As before our algorithm begins in the same way by producing the state |g0K/an}bracketri}htwhereg0has been chosen at random. The passage from abelian to non-abelian groups is accompanied b y various potential conceptual problems: (a) (Construction of non-abelian Fourier transform). For a belian groups the irreducible representations are always one dimensional (i.e. the funct ionsχlin eq. (17)) whereas for non-abelian groups they are functions χ:G→U(d) taking values in the set U(d) of all d×dunitary matrices for suitable values of d. According to a basic theorem of group representation theory [3], if d1...,d mare the dimensions of a complete set of irreducible unitary representations χ1,...,χ mthend2 1+...+d2 m=|G|. Let us write χi,jk(g) for the (j,k)th component of the unitary matrix χi(g). Thus asi,j,kvary we get |G|complex valued functions and as in eq. (17) we may deﬁne the |G|states: |χi,jk/an}bracketri}ht=1/radicalbig |G|/summationdisplay g∈Gχi,jk(g)|g/an}bracketri}ht. The orthogonality relations of irreducible representatio ns [3] guarantee that these are again orthonormal states, called the Fourier basis, and the non-a belian Fourier transform is deﬁned as the unitary operation that rotates this basis into standa rd

Question 148 multiple-choice

Quantum algorithms have shown significant speedups for certain algebraic problems, but their performance often depends on the underlying algebraic structure. The complexity of membership and discrete logarithm problems varies greatly between groups and semigroups. Which statement correctly characterizes the quantum query complexity of the constructive membership problem in general semigroups with k ≥ 2 generators? 1) It can always be solved with a polynomial number of quantum queries regardless of the semigroup structure. 2) It requires only logarithmic quantum queries in the size of the semigroup. 3) It requires exponentially many quantum queries for general semigroups when there are at least two generators. 4) It is efficiently solvable by Shor’s algorithm for any semigroup. 5) It becomes trivial if the semigroup is non-abelian. 6) It does not depend on the number of generators. 7) It is equivalent in difficulty to the discrete logarithm problem in abelian groups.

✓ Correct Answer:

The correct answer is 3) It requires exponentially many quantum queries for general semigroups when there are at least two generators..

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solve the shifted discrete logarithm problem in a black-box semigroup Swith only poly .logjSj/queries. As in the proof of Theorem 3.1, given a candidate value a, we can check whetherxDyga. If this check fails, we can conclude (with bounded error) that no solution exists. The dihedral hidden subgroup problem (DHSP) is apparently hard. Despite con- siderable effort (motivated by a close connection to lattice problems [17]), Kuper- berg’s algorithm remains the best known approach, and it is plausible that there might be no efﬁcient quantum algorithm. Note that the DHSP can be reduced to a quantum generalization of the constructive orbit membership problem, namely, Quantum computation of discrete logarithms in semigroups 411 orbit membership for a permutation action on pairwise orthogonal quantum states [7, Proposition 2.2]. Thus, intuitively, a solution of the shift problem for a (classi- cal) permutation action (such as in the shifted discrete logarithm problem) should exploit that the action is on classical states, unless it also solves the DHSP. In Section 5, we describe another variant of the discrete logarithm problem that is even harder than the shifted discrete logarithm problem, requiring exponentially many queries. We also show that our lower bound for that problem is nearly opti- mal using the algorithm of Lemma 4.1 as a subroutine. 5 Constructive semigroup membership Given an abelian semigroup generated by g1;:::;gkand a semigroup element x2 hg1;:::;gki, the constructive membership problem asks us to ﬁnd a1;:::;ak2 N0´¹0;1;2;:::ºwitha1CCak1such thatxDga1 1gak k. The notation g0 isimply indicates that no factor of giis present, so solutions with aiD0for some values of iare well deﬁned even though the semigroup need not have an identity element. This natural generalization of the discrete logarithm problem is easy for abelian groups (see for example [10, Theorem 5]). In that case, let ri´jhgiijfor alli2 ¹1;:::;kº,r´jhxij, andL´Zr1 ZrkZr. The values.r1;:::;rk;r/ can be computed efﬁciently by Shor’s order-ﬁnding algorithm [19]. Now consider the functionfWL!Gdeﬁned byf.a1;:::;ak;b/Dga1 1gak kxb. This func- tion hides the subgroup H´® .a1;:::;ak;b/2LWga1 1gak kDxb¯ L; so generators of Hcan be found in polynomial time [15]. To solve the construc- tive membership problem, it sufﬁces to ﬁnd the solutions with bD1modr. This corresponds to a system of linear Diophantine equations, so it can be solved clas- sically in polynomial time (see for example [18, Corollary 5.3b]). Here we show that the constructive membership problem in semigroups is con- siderably harder. Speciﬁcally, given a black-box semigroup S, we need expo- nentially many quantum queries (in log jSj) to solve the constructive membership problem with respect to k2generators. Theorem 5.1. For any ﬁxed k2N, there is a black-box semigroup Swithk generators for which at least .jSj1 21 2k/quantum queries are required to solve the constructive membership problem. Proof. For anyn2N, consider the abelian semigroup SD® ga1 1gak kWa1;:::;ak2N0;1a1CCakn¯ [¹0º 412 A. M. Childs and G. Ivanyos generated by g1;:::;gk, with the following multiplication rules: 0.ga1 1gak k/D0; .ga1 1gak k/.gb1 1gbk k/D´ ga1Cb1 1gakCbk kifPk iD1.aiCbi/n, 0 ifPk iD1.aiCbi/>n: Let´¹.a1;:::;ak1/2Nk1 0Wa1CCak1nº. We show that the problem of inverting a

Question 149 multiple-choice

Quantum information theory studies how information can be transmitted and processed using quantum systems, with special attention to the effects of channel noise and symmetries. Entanglement-assisted classical capacity quantifies the optimal rate for sending classical bits over quantum channels when unlimited entanglement is shared between sender and receiver. Which statement correctly describes the impact of a quantum Markov semigroup satisfying a modified logarithmic Sobolev inequality (MLSI) on the entanglement-assisted classical capacity of quantum channels with finite symmetry groups? 1) MLSI causes the capacity to remain constant over time regardless of channel evolution. 2) MLSI guarantees polynomial decay of the capacity upper bound as time increases. 3) MLSI leads to exponential decay of the relative entropy, resulting in exponentially tightening upper bounds for capacity over time. 4) MLSI only affects capacities in infinite-dimensional Hilbert spaces, not finite ones. 5) MLSI increases the channel capacity by enhancing entanglement generation. 6) MLSI eliminates the strong converse property for these channels. 7) MLSI ensures that lower bounds on capacity exceed upper bounds for all time.

✓ Correct Answer:

The correct answer is 3) MLSI leads to exponential decay of the relative entropy, resulting in exponentially tightening upper bounds for capacity over time..

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/uni27E9/uni27E8 /divides.alt0) on systemBand where /divides.alt0 /uni27E9/uni27E8 /divides.alt0stands for the rank-one orthogonal projection on the norm-one vector ∈H. A closed formula for the entanglement assisted capacity 1note that we are just rewriting the mutual information in terms of the relative entropy in (56).was obtained in [34]. There, they show that for a conditional expectation Efixwe have CEA(Efix)=log/parenleft.alt4m /summation.disp i=1n2 i/parenright.alt4: Using similar ideas as before we can estimate the entanglement assisted classical capacity using relative entropy transference. Theorem V .6 (Bounding the entanglement assisted classical capacity) .Let(Tt)t≥0be a transferred QMS. Then: log/parenleft.alt4m /summation.disp i=1n2 i/parenright.alt4≤CEA(Tt())≤log/parenleft.alt4m /summation.disp i=1n2 i/parenright.alt4+: Furthermore, if (St)t≥0satisﬁes an -MLSI and Gis ﬁnite, then: CEA(Tt)≤log/parenleft.alt4m /summation.disp i=1n2 i/parenright.alt4+e−2 tlog(/divides.alt0G/divides.alt0) Moreover, the upper bounds are in the strong converse sense. Proof. The proof is similar to the ones of Theorems V .1 and V .5 and Proposition V .13, and hence is omitted. D. Capacities from a modiﬁed logarithmic Sobolev inequality Similarly to the derivation of decoherence times, one expects to get tighter bounds on the various capacities by directly applying a quantum functional inequality, when the latter is known. As was the case with decoherence times, the decay of the capacities we obtain does not depend on particular properties of the representation at hand and will in general not be tight. For capacities, the right functional inequality to consider is the modiﬁed logarithmic Sobolev inequality (MLSI). To the best of our knowledge, this connection between a MLSI and capacity bounds cannot be found in the literature beyond primitive semigroups [46], so we establish it here for more general semigroups. Here, we still assume that (Tt)t≥0is a quantum Markov semigroup on B(H)that is symmetric with respect to the Hilbert Schmidt inner product. Then, instead of using the entropy comparison theorem (Theorem III.1), one can simply decompose the relative entropy between Tt(), ∈D(H), and any state ∈D(Nfix)as follows: Lemma V .7. For any∈D(H), and any∈D(Nfix), D(Tt()/parallel.alt1)=D(Efix()/parallel.alt1)+D(Tt()/parallel.alt1Efix()): 18 Proof. D(Tt()/parallel.alt1) =Tr(Tt()(ln(Tt())−ln)) =Tr(Tt()(ln(Tt())−ln(Efix()))) +Tr(Tt()(ln(Efix())−ln)) =D(Tt()/parallel.alt1Efix())+Tr(Efix()(ln(Efix())−ln)) =D(Tt()/parallel.alt1Efix())+D(Efix()/parallel.alt1); where in the third line we used that Efixis a conditional expectation with respect to the completely mixed state, so thatEfix=E† fixand for any ∈D(Nfix)∩D(H)+, ln()=Efix(ln()). We recall that, given a faithful quantum Markov semigroup (Tt=e−tL)t≥0, its decoherence-free modiﬁed logarithmic Sobolev constant 1(L)has been deﬁned in [27] as follows: 1(L)∶=inf ∈D+(H)Tr(L()(ln−lnEfix())) D(/parallel.alt1Efix()) (we recall our convention that Lis a positive semi-deﬁnite operator). It is the largest constant >0 such that the following decay in relative entropy occurs2: D(Tt()/parallel.alt1Efix())≤e− tD(/parallel.alt1Efix()): The classical logarithmic Sobolev inequality is then deﬁned in an analogous way. An extension of Theorem III.1 then gives the Theorem V .8. Let(Tt)t≥0be a transferred QMS from a ﬁnite group GonB(H),∈D(Nfix)and suppose that Sthas an MLS constant 1(L)>0. Then, for any state ∈D(H)andt≥0: D(Efix()/parallel.alt1)≤D(Tt()/parallel.alt1) (57) ≤D(Efix()/parallel.alt1)+e− 1(L)tlog(/divides.alt0G/divides.alt0) Proof. It follows from Theorem III.1 that D(Tt()/parallel.alt1)≤D(Efix()/parallel.alt1)+/integral.dispGktlog(kt)dG: Now note that /integral.dispGktlog(kt)dG=D(t/parallel.alt1G); wheregis an arbitrary element of the group and where dt=kt∗gdG. Remark also that t=St(g). Thus, as we assumed that Stsatisﬁes a MLSI, we conclude that D(t/parallel.alt1G)≤e− 1(L)tlog(/divides.alt0G/divides.alt0); (58) where in the last step we used the

Question 150 multiple-choice

Parity encoding schemes in quantum computing architectures offer alternative ways to represent logical qubits and perform universal gate operations. The Lechner-Hauke-Zoller (LHZ) architecture leverages parity qubits and data qubits to address hardware connectivity challenges and optimize quantum algorithm implementation. Within the LHZ architecture, which operation specifically requires chains of CNOT gates due to the structure of its corresponding logical operator, making its implementation more complex than single-qubit rotations? 1) Logical Y rotation (˜Ry) 2) Logical X rotation (˜Rx) 3) Logical Z rotation (˜Rz) 4) Physical Z rotation on a single data qubit 5) Measurement in the computational basis 6) Preparation of graph states 7) Application of a single Hadamard gate

✓ Correct Answer:

The correct answer is 2) Logical X rotation (˜Rx).

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zero or an even number oftimes. The square brackets around σ (l4) zindicate that a con- straint can contain either three or four qubits. The specialcase of the parity encoding involving all n(n−1)/2 two-body terms (parity qubits) for nlogical qubits is known as the LHZ architecture [ 12]. It is possible to extend this by including physical qubits representing single logical qubits such that ˜σ (i) z=σ(i) z. (3) We refer to these qubits as data qubits in the following. The presence of data qubits in the parity encoding ensures that thephysical qubits uniquely deﬁne the state of the logical qubits. 1 A variant of the LHZ architecture involving all data qubits hasrecently been shown to provide a universal gate set [ 11], based on the logical operators ˜R (i) x(α)=exp/parenleftBigg −iα 2σ(i) x/productdisplay jiσ(ij) x/parenrightBigg , (4) ˜R(i) z(α)=exp/parenleftBig −iα 2σ(i) z/parenrightBig =R(i) z(α), (5) C˜P(i,j) φ=R(i) z/parenleftbiggφ 2/parenrightbigg R(ij) z/parenleftbigg −φ 2/parenrightbigg R(j) z/parenleftbiggφ 2/parenrightbigg . (6) 1Note that two-body parity qubits alone determine the logical state only up to a global spin ﬂip. 2469-9926/2022/106(4)/042442(8) 042442-1 Published by the American Physical Society FELLNER, MESSINGER, ENDER, AND LECHNER PHYSICAL REVIEW A 106, 042442 (2022) Logical operators (with a tilde) act on the logical qubits deﬁned in the constraint-fulﬁlling subspace and commutewith all constraint operators. In contrast to that, physicaloperators (without a tilde) do not necessarily preserve theconstraint-fulﬁlling subspace. The ˜R xoperators require chains of controlled- NOT (CNOT ) gates due to the product of Pauli operators in the exponent (see, for example, Refs. [ 30–32]f o r further background on exponentiating products of Pauli ma-trices), while the other logical operators can be implementedwith physical single-qubit operations only. The set of physicalqubits that are involved in ˜R (i) x(i.e., the set of all physical qubits containing the logical index i) is referred to as the logical line associated with qubit ( i). In this work we study the implementation of several es- sential quantum algorithms in this scheme and ﬁnd that,depending on the algorithm, the parity scheme can show anadvantage in circuit depth or multiqubit gate count. In par-ticular, we focus on quantum algorithms essential for Shor’sfactoring algorithm [ 2], the quantum Fourier transform (QFT) [31], and the quantum addition algorithm based on the QFT [33], as well as the implementation of Grover’s diffusion operator [ 4]. Furthermore, we present a strategy to efﬁciently prepare graph states, which represent an important resourcefor measurement-based quantum computing [ 34]. II. COMMON GATES AND GATE SEQUENCES A. Arbitrary single-qubit gates Any single-qubit unitary Ucan be decomposed into rota- tions [ 31] U=Rz(α)Rx(β)Rz(γ), (7) with some angles α,β, andγ. We can thus construct any log- ical single-qubit gate using the operators deﬁned in Eqs. ( 4) and ( 5)a s ˜U=˜Rz(α)˜Rx(β)˜Rz(γ). (8) The two ˜Rzrotations can be easily implemented in the LHZ scheme with physical rotations on the corresponding dataqubits. The ˜R xrotation requires a chain of CNOT gates along the logical line and a physical Rxrotation on one of its qubits,

Question 151 multiple-choice

Quantum algorithms leverage specialized circuit operations to efficiently manipulate and measure properties of quantum states, particularly in problems involving symmetry and signal processing. Techniques such as the quantum Fourier transform (QFT), phase gates, and the Hadamard test are foundational for tasks like operator diagonalization and expectation value estimation. Which circuit technique enables efficient diagonalization of the cyclic permutation operator in quantum computing, resulting in circuit depth that does not scale with the exponent of the permutation? 1) Controlled-X gate sequences combined with classical shift operations 2) Grover diffusion operator implemented repeatedly 3) Use of Toffoli gates for iterative cyclic shifts 4) Variational quantum eigensolver with parameterized rotation gates 5) Classical fast Fourier transform applied before measurement 6) Quantum Fourier Transform (QFT) followed by phase gates 7) Quantum error correction codes integrated with permutation logic

✓ Correct Answer:

The correct answer is 6) Quantum Fourier Transform (QFT) followed by phase gates.

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1 with quantum computers, obtaining estimations of each entry of VR,VI, qR, and qIwith the Hadamard test [22], which are then stored in classical memory. We modify quantum circuit implementations of shift operations shown by Koch et al. [30] and quantum adder circuit by Draper [31] to implement the cyclic permutation operator Qand its powers with circuit depth independent to the power m, a contrast to the implementation of quantum circuits for solving general linear systems by Huang et al. [7]. Steps 5-8 of the algorithm remain implemented on classical computers. As shown in the following section, the powers of Qcan be implemented with a known quantum circuit with O(n2) quantum gates. Therefore, we can construct classical sketches of the circuit implementation of output states |um⟩=QmUb|0n⟩that correspond to the computed optimal parameters αm, using O(Tn2) memory in total to provide a classical sketch of the estimator ˜ xas a combination of quantum states. We now provide details of the circuit implementation as follows: Quantum circuit implementation. Recall from Section II that the eigendecomposition of the cyclic permutation matrix Qis given by Q=F−1ΛF, where Λ is the diagonal matrix with all eigenvalues of Qand the diagonalization matrix Fis the matrix representation of DFT. On quantum systems, for cases where n= log2(N),Fcan be implemented with O(n2) quantum logic gates using the quantum Fourier transform (QFT) [17]. For the more general case where n̸= log2N, one can use the arbitrary-size quantum Fourier transform proposed by Kitaev [32] which utilizes quantum phase estimation to construct DFT. As such implementations require fault-tolerance, we discuss only the standard QFT for our implementation, while details on the generalized case are provided in Appendix B. 7 (a) ...Λ =P(π) P(π 2) P(2π N) (b) ...Λ =P(mπ) P(mπ 2) P(2mπ N) (c) |0⟩H I/S H |0⟩ UbQFT Λm ... |0⟩ FIG. 2. Circuit implementation of quantum subroutines in our algorithm. The cyclic permutation matrix Qcan be diago- nalized by the DFT matrix Fsuch that Q=F−1ΛF. DFT can be implemented by QFT circuits. See Appendix B for the implementation of QFT for arbitrary N. The diagonal matrix Λ can be implemented as a tensor product of phase gates as shown in Panel 2a. Powers of Λ can be implemented with the same phase gates, by multiplied rotation angles as shown in Panel 2b. To retrieve the real and imaginary parts of ⟨b|Qm|b⟩, we use the Hadamard test (on the state QFT|b⟩), which requires a controlled version of Λm, i.e. controlled phase gates, as shown in Panel 2c. For the remainder of the discussion, we assume n= log2(N), while keeping in mind that cases where n̸= log2(N) are still implementable. Matrix Λ can be written as: Λ = diag ω0 Nω1 Nω2 N···ωN−1 N (14) We use the phase gate P, whose matrix form is P(θ) =1 0 0eiθ , (15) where the parameter θis the rotation angle. We note that ω0= 1. Hence Λ can be written as the tensor product of P(θ) gates as follows: Λ =1 0 0ωN/2

Question 152 multiple-choice

Quantum magnetism and many-body physics often require specialized computational techniques and careful handling of symmetries to explore exotic phases such as spin liquids and quantum paramagnets. Theoretical frameworks like Schur-Weyl duality and advanced variational ansatzes are essential for efficient simulation and understanding of these systems. Which property distinguishes the eSWAP ansatz from the Sn-CQA ansatz in preserving global SU(2) symmetry for spin systems, but not extending its universality to higher-dimensional (qudit) cases? 1) eSWAP supports adiabatic evolution at large circuit depth with global SU(d) symmetry 2) eSWAP universally preserves SU(2) symmetry in relevant sectors, but loses universality for qudits 3) eSWAP alternates between problem and mixer Hamiltonians, resembling QAOA 4) eSWAP is commutative and requires fixed circuit layout 5) eSWAP enables efficient initialization using the Young basis for all local dimensions 6) eSWAP guarantees ground state location in frustrated systems via the Marshall-Lieb-Mattis theorem 7) eSWAP performs exact diagonalization for large-scale quantum systems

✓ Correct Answer:

The correct answer is 2) eSWAP universally preserves SU(2) symmetry in relevant sectors, but loses universality for qudits.

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is known about the intermediate quantum paramagnetic phase—recent evidence of decon- ﬁned quantum criticality [ 84,85] sparked further interest in studying these regimes. Gaining physical insights in the intermediate quantum paramagnetic phase requires solving the problem of the ground-state sign structure the system approaches the phase transition. Recently, there were a number of numerical attempts to address the existence of the U(1)gapless spin-liquid phase, using recently the ten- sor networks [ 86], restricted Boltzmann machine (RBM) [87], convolutional neural network (CNN) [ 57,58,88], and graphical neural network (GNN) [ 89]—all yielding partial progress. As a signiﬁcant diﬀerence from the unfrustrated case, the Marshall-Lieb-Mattis theorem does not hold gen- erally and there is no guarantee that the ground state still lives at J=0 or equivalently λ=(n/2,n/2), which urges us to preserve the global SU(2) symmetry, which fur- ther gives us access to search in all inequivalent Snirreps decomposed from the system by Schur-Weyl duality. Global SU (2)symmetry and challenges in NQS ansätze Taking advantage of the global SU (2)symmetry, we address this problem in a diﬀerent way: we recast the Hamiltonian in Eq. (8)by the following identity: π((ij))=2ˆSi·ˆSj+1 2I,( 9 ) withˆSibeing further expanded as the half of standard Pauli operators {X,Y,Z}. Equation (9)was ﬁrst discov- ered by Heisenberg himself [ 90,91] (an elementary proof can be found in the Appendix) and more recently noted by Ref. [ 46] in analyzing the ground-state property of the 1D Heisenberg chain. As designed by products of expo- nentials of SWAPs (eSWAPs), the method proposed inRef. [ 46] truly preserves the global SU(2) symmetry. As a brief comparison with S n-CQA, eSWAP ansätze are uni- versal in relevant sectors given by the SU(2) symmetry. However, this property no longer holds for qudits with 020327-12 SPEEDING UP LEARNING QUANTUM... PRX QUANTUM 4,020327 (2023) FIG. 9. Scaling properties of small total spin irreps dimen- sion, respectively. The graph shows the scale: 2n/dimSλwith the partitions λ1=(n/2,n/2)(red),λ2=(n/2+1,n/2−1)(gray), λ3=(n/2+2,n/2−2)(orange). The orange dashed line is exp(−n/8)the exponential decay, since the plot starts at n=8. d≥3 (Sec. III C). As eSWAPs are noncommutative oper- ators in general, there are various ways to place them in a quantum circuit. A more suitable perspective to describe the ansatz might be sampling them as two-local SU(2) ran- dom circuits [ 45]. On the other hand, the Sn-CQA ansatz is designed by alternating exponentials of the problem and mixer Hamiltonians HP,HMjust like the framework of QAOA. Similar to QAOA, Sn-CQA at large pcorre- sponds to a form of adiabatic evolution with global SU (d) symmetry, which could hint a theoretically guaranteed performance as pis large (see Sec. VII in the Appendix). An immediate consequence of using Eq. (9)is that the resulting Heisenberg Hamiltonian can be expressed in the Young basis where every Snirrep is indexed by the total spin label j. Mapping to this basis can be done using the constant-depth circuit state initialization in Sec. IV B . Using our C[Sn] variational ansatz leads to a more eﬃ- cient algorithm by polynomially reducing the

Question 153 multiple-choice

Quantum computing offers novel approaches for solving structured linear systems, particularly those with circulant and banded matrix properties, which frequently arise in physics and engineering applications. Efficient algorithms for these systems can leverage matrix symmetries and specialized quantum operations to reduce computational resources. Which algorithmic strategy enables a reduction from exponential to linear quantum resource requirements when solving banded circulant linear systems with bandwidth parameter $ K $? 1) Using Grover’s search to optimize eigenvalue estimation 2) Encoding matrix entries directly into quantum amplitudes 3) Employing classical iterative refinement after quantum subroutines 4) Applying amplitude amplification to classical solutions 5) Decomposing banded circulant matrices into cyclic permutations and combining quantum states via convex optimization 6) Utilizing quantum phase estimation on non-circulant matrices 7) Implementing variational quantum eigensolvers without matrix decomposition

✓ Correct Answer:

The correct answer is 5) Decomposing banded circulant matrices into cyclic permutations and combining quantum states via convex optimization.

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Question 154 multiple-choice

Quantum algorithms for linear algebra and simulation frequently exploit circuit decompositions and measurement strategies to optimize resource efficiency. Understanding the implementation and analysis of quantum operators is essential for evaluating algorithmic scalability and error bounds. Which of the following statements best describes why the quantum circuit for implementing powers of the operator Λm achieves scalability regardless of the exponent m? 1) The circuit uses ancilla qubits to store the value of m, eliminating dependence on its size. 2) Λm is implemented via iterative Grover rotations, making the runtime independent of m. 3) The circuit employs amplitude amplification to compress the effect of m into a single operation. 4) The circuit relies on Hamiltonian simulation, where the time evolution parameter substitutes for m. 5) The operator Λm is realized using controlled-NOT gates whose number does not depend on m. 6) The implementation uses classical preprocessing to encode m into initial states, removing m from the circuit depth calculation. 7) The circuit’s depth is determined solely by the Quantum Fourier Transform, and the phase gate powers are set by multiplying phase angles, so the depth does not depend on m.

✓ Correct Answer:

The correct answer is 7) The circuit’s depth is determined solely by the Quantum Fourier Transform, and the phase gate powers are set by multiplying phase angles, so the depth does not depend on m..

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N ⊗ ··· ⊗ 1 0 0ω2 N ⊗ 1 0 0ωN =n−1O j=0P(θj),where θj=2π N·2j. (16) The implementation of Λ is then as illustrated in Figure 2a. We note that this is the non-controlled version of the quantum adder circuit proposed by Draper [31]. The number of gates for this implementation is equal to the qubit number n. We note that for m∈Z,Qmhas eigendecomposition F−1ΛmF. Further note that the powers of phase operations are phase operations with multiplied rotation angles, that is, P(θ)m=P(mθ). As Λ can be decomposed into a tensor product of phase gates, we can simply implement Λmwith the same gate, but with the angles multiplied by m, as shown in Figure 2b. Observe that the implemented quantum circuit is independent of mand only depends on the depth of the QFT implementation. Given the above implementation of powers of Q, to obtain the expectation value of ⟨b|Qm|b⟩, we prepare a quantum state|ψb⟩=QFT|b⟩. A Hadamard test is then performed on Λmto calculate the expectation values with respect to|ψb⟩as described in Appendix B, such that we obtain the values of Re ⟨ψb|Λm|ψb⟩and Im ⟨ψb|Λm|ψb⟩, which are equivalent to Re ⟨b|Qm|b⟩and Im ⟨b|Qm|b⟩. The full quantum circuit is shown in Figure 2c. Number of total measurements. As we conduct quantum measurements to obtain auxiliary system matrices Wand r, we do not gain the exact value of the entries in the matrices. Instead, what we obtain are estimations of the values whose accuracy depends on the number of measurements conducted. The total number of measurements required to achieve a close enough estimate of ˜ xis stated in the following proposition: Proposition 6 (Number of measurements needed) .Given K-banded circulant matrix C=PK ℓ=−KcℓQℓ∈ M N(C), hardware efficient implementation of Ubas shown in Assumption 5 and truncation threshold T. This defines auxiliary systems Wandrin Equation 12. Let the sum of absolute values of the coefficients of decomposed CbeB=PK ℓ=−K|cℓ|. We can find ˜α:{˜αm∈C,∀m∈[−T..T]}for estimator ˜x(α) =PT m=−TαmQmUb|0n⟩such that the following is satisfied ∥C˜x(˜α)− |b⟩∥2≤min α∈C2T+1∥C˜x(α)− |b⟩∥2+ε usingO(B4(K+T)T∥W∥∥W−1∥2(1 +∥W−1r∥2/ε)measurements via Hadamard test. 8 Algorithm 2 Classical subroutine for inner product estimation with sample and query access Input: Classical data structure Sbas shown in Assumption 8, Power of cyclic perturbation matrix m, Additive error ε, Failure rateδ Output: ε-close estimation of ⟨b, Qmb⟩with success rate 1 −δ 1:fori←0 to 6 log(2 /δ)do 2: ηi←0 3: forj←0 to 9 /ε2do 4: s←Sample (Sb) 5: q←Query (Sb, s−m(mod N)) 6: r←Query (Sb, s) 7: ηi←ηi+q/r 8: end for 9: ηi←ηiε2/9 10:end for 11:return Median (η) We note that the number of different expectation values needed is 4 K+ 4T+ 1∈ O(K+T) due to the high overlap of values required. So the number of total measurements for all entries on Wandrto have variance O(1/R) is O((K+T)B4R). From Proposition 12 of [7], we have R > CT ∥W∥∥W−1∥2(1 +∥W−1r∥2/ε). We now provide informal bounds of the above proposition. Based on discussions given by Huang et al. [7] on the upper bounds of ∥W∥,∥W−1∥and∥W−1r∥, we can see that the number

Question 155 multiple-choice

Quantum computing hardware is limited by the number of available qubits and susceptibility to errors, making efficient circuit designs crucial for practical applications. The Quantum Fourier Transform (QFT) is a foundational algorithm, often used in tasks such as factoring, but typically requires significant resources. Which property most directly enables an in-place quantum Fourier transform circuit to minimize hardware requirements while maintaining logarithmic depth? 1) Utilization of randomized measurements to reduce error rates 2) Addition of entangled ancilla qubits for error correction 3) Implementation of global gates acting on all qubits simultaneously 4) Encoding input states into higher-dimensional Hilbert spaces 5) Operating entirely without ancilla qubits and using only local gates in a 1D array 6) Repetition of QFT circuits with post-selection for accuracy 7) Reliance on adaptive classical feedback during computation

✓ Correct Answer:

The correct answer is 5) Operating entirely without ancilla qubits and using only local gates in a 1D array.

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Question 156 multiple-choice

Quantum error-correcting codes that are covariant with respect to unitary group actions are important for fault-tolerant quantum computing but face significant resource and error reduction limitations. Theoretical results explore the trade-offs between error rates, subsystem dimensions, and group representation properties when designing such codes. Which of the following best characterizes the lower bound result for SU(d_L)-covariant quantum codes regarding the scaling of physical subsystem dimension with logical system dimension for fixed n? 1) The physical subsystem dimension must grow exponentially with the logical system dimension. 2) The physical subsystem dimension can remain constant regardless of logical system size. 3) The error parameter can be made arbitrarily small without increasing subsystem dimension. 4) The scaling is always linear with respect to logical system dimension. 5) The lower bound only applies to codes without symmetry constraints. 6) The subsystem dimension decreases as logical system dimension increases. 7) The scaling is polynomial in the logical system dimension for all n.

✓ Correct Answer:

The correct answer is 1) The physical subsystem dimension must grow exponentially with the logical system dimension..

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entangled with alarge reference system. In such a situation, we require ϵ worst≲d−1 L. Bound (55)then asserts that, at constant n, the physical subsystem dimension must grow exponentially in the logical system dimension. From Theorem 6, we can further write ϵworst½SUðdLÞ-covariant code /C138 ≥1 2n1 max ilndiþO/C181 ndL/C19 ð56Þ (see proof in Supplemental Material, Sec. E. 2 [37]). The bound (56) is useful to determine the precision limit of a code that has a universal set of transversal gates. If weimagine that each physical subsystem is composed of m i¼log 2ðdiÞqubits lumped together, then the error parameter of the code scales at least inversely in the largestnumber of qubits m ithat are lumped together. If we consider, for instance, ten logical qubits ( dL¼210) that are encoded into nsystems consisting of ten qubits each, i.e.,di¼210, we obtain the rather prohibitive error param- eterϵworst≳0.14=n. In this example, we can improve this estimate to ϵworst≥0.5=nby using Eq. (54a) directly: If max idi¼dL, we must have ⌈½2nϵworst/C138−1⌉≤1, because any larger value would make the binomial coefficient too large to satisfy Eq. (54a) . B. Random constructions The bounds of Theorem 6 severely limit the error- correction capability of the unitary SUðdLÞ-covariant codes. We now show that it is possible to find good SUðdLÞ-covariant codes in regimes of large physical systems that are not excluded by Theorem 6. The constructions we present are randomized as well as asymptotic in the dimension of the physical subsystems.More precisely, we consider the encoding of one d L- dimensional Hilbert space HLin a physical space which is a tensor product of three Hilbert spaces HA¼HA1⊗ HA2⊗HA3. The encoding is done via an isometry VL→A, which is UðdLÞcovariant: For all U∈UðdLÞ, VU¼r1ðUÞ⊗r2ðUÞ⊗r3ðUÞV: ð57Þ Here, r1,r2, and r3are three irreps of UðdLÞ.O u r constructions are randomized in the following way: (i)Vis chosen randomly from all possible isometries satisfying the covariance condition (57);(ii) the irreps r1,r2, and r3are chosen randomly, or at least generically . In fact, we need only that the irreducible representation does not belong to a small subset of all possible irreducible representations. We use randomized constructions to prove the existence of UðdLÞ-covariant codes with a small error [measured by ϵe based on the (fixed-input) entanglement fidelity], as sum- marized in the following theorem. We refer to theSupplemental Material, Sec. F [37] for a complete proof. Theorem 7. Ford L≥4and every ϵ>0, there exists a UðdLÞ-covariant code with error ϵe≤ϵand physical dimensions di,i∈f1;2;3g, such that max ilndi≤dLðdL−1Þln/C181 ϵe/C19 þC2; ð58Þ for some C2which is only a function of dL. It is not clear how to compare the performance of our code given by Eq. (58) to our bounds of Theorem 6, because our nonconstructive proof does not specify the behavior of C2as a function of dL, which is given by details of the representation theory of UðdLÞ. It remains open whether the lower bound can be strengthened or theconstructions can be improved. Our proof technique does not immediately work for Uð2Þ-covariant codes, as it is harder to

Question 157 multiple-choice

Quantum error correction is fundamental for the reliable operation of quantum computers, which are vulnerable to noise and decoherence. Stabilizer codes represent a versatile class of quantum error-correcting codes enabling robust protection against certain types of quantum errors. Which quantum code is the smallest stabilizer code capable of correcting arbitrary single-qubit errors? 1) Nine-qubit Shor code 2) Five-qubit code 3) Seven-qubit Steane code 4) Ten-qubit Bacon-Shor code 5) Four-qubit code 6) Surface code 7) Color code

✓ Correct Answer:

The correct answer is 2) Five-qubit code.

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it was soon recognized that error-correction techniques could play an important role in a functioning quantum computer. Semi nal papers by Shor [64] and Steane [68,69] suggested a methodology, and a theory of quantum error-correcting codes was soon in place. In Chapter 4 we trace the development of this class of algorithms, beginning with a brief discussion of the physical problem of decoherence and discussing the early models of Shor and Steane. A "truly quantum" five- "qubit" code is discussed in detail, since it provides the paradigm for the "stabilizer" codes which are described subsequently. The relation of classical codes to stabilizer codes is recorded, and we conclude with a discussion of error correcting codes from the abstract Hilbert space perspective a la Knill and Laflamme [47]. There is much more to the subject of quantum computation and re lated fields than we discuss here. For example, in 1984 Charles Bennett and Gilles Brassard [10] introduced a key distribution system based on quantum mechanical principles. (For a novel extension of the ideas in [10], one which has a practical experimental realization, see Ekert [81].) Without going into detail, the goal is to enable sender and receiver (Alice and Bob) to securely construct a key sequence for encryption using the properties of photons. Thus, concurrently with quantum computers and quantum algorithms, the fields of quantum cryptography and quantum x Preface communication were developing. This quickly led to quantum information theory and to related complexity issues. The notation and results we present here provide a good foundation for further reading in these various subjects. A broad overview from the information-theoretic perspective is given in [70j, and readers interested in learning more about these various topics will find appropriate refer ences at the start of the bibliography section. Acknow ledgments An Introduction to Quantum Computing Algorithms reflects my own experience in learning the mathematics and theoretical physics required for the subject. I am most grateful to many colleagues who helped in that process, especially the core members of our Quantum Computation Seminar at the Baltimore County campus of the University of Maryland (UMBC) who comprised the (usually) patient audience for the initial presentation of parts of this book. Special thanks go to Rich Davis, Keith Miller, and Mort Rubin who read all or parts of the manuscript in its various incarnations. Readers who find additional errors are encouraged to notify me at pittenge@math.umbc.edu. I am also indebted to the many scholars who routinely made their research available on the Los Alamos web site quant-ph. The field has developed so quickly that an introduction of this nature would not have been possible otherwise. Ann Kostant at Birkhauser Boston was most supportive throughout the entire process, and indeed there would have been no book without her encouragement. Viet Ngo provided crucial support in preparing the various drawings, and I also received critical technical help in the prepara tion of the manuscript from Rouben Rostamian, Boris Alemi, and Esther M. Scheffel, colleagues at UMBC. That assistance

Question 158 multiple-choice

Quantum approximate counting is a technique used to estimate the number of solutions (marked items) in a dataset using quantum algorithms. These algorithms often leverage quantum speedups and can be implemented using different subroutines such as Grover iterations and the Quantum Fourier Transform (QFT). Which quantum algorithmic improvement allows for approximate counting and amplitude estimation using only Grover iterations, thereby eliminating the need for the Quantum Fourier Transform? 1) The use of quantum phase estimation based on QFT 2) A QFT-free algorithm that applies only Grover iterations for estimation 3) Classical probabilistic estimation with amplitude amplification 4) The integration of hidden subgroup algorithms 5) Quantum walk-based search algorithm 6) Variational quantum eigensolver for amplitude estimation 7) Quantum annealing with error correction

✓ Correct Answer:

The correct answer is 2) A QFT-free algorithm that applies only Grover iterations for estimation.

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Question 159 multiple-choice

In image processing, quantitative metrics are essential for evaluating the fidelity of processed images compared to their original versions. Two widely used metrics are Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM), which assess image quality from different perspectives. Which of the following statements best describes a fundamental difference between PSNR and SSIM when used to evaluate image quality? 1) PSNR measures color accuracy, while SSIM measures noise level. 2) PSNR is calculated using histograms, whereas SSIM uses edge detection. 3) PSNR directly models human perception, while SSIM is a purely mathematical metric. 4) PSNR is suitable only for grayscale images, but SSIM works for color images. 5) PSNR reports values in percentages, whereas SSIM reports values in decibels. 6) PSNR evaluates pixel-wise intensity differences, while SSIM incorporates luminance, contrast, and structural information. 7) PSNR quantifies average pixel-wise error, but SSIM assesses perceptual similarity by considering luminance, contrast, and structure.

✓ Correct Answer:

The correct answer is 7) PSNR quantifies average pixel-wise error, but SSIM assesses perceptual similarity by considering luminance, contrast, and structure..

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Question 160 multiple-choice

Heterotic string theory models often employ the four-dimensional free fermionic construction to investigate particle physics phenomena, including dark matter and solutions to the strong CP problem. These constructions yield hidden sector gauge groups and possible axion candidates which impact phenomenological predictions. In a heterotic string-derived model built using the free fermionic construction, which hidden sector feature most naturally facilitates self-interacting dark matter candidates? 1) The presence of a single U(1) gauge group 2) The existence of extra dimensions with compactified geometry 3) The inclusion of supersymmetry-breaking terms 4) The identification of light axion-like particles 5) The presence of four non-Abelian SU(2) gauge groups 6) The embedding of the Standard Model gauge group in E8 7) The appearance of massive scalar fields

✓ Correct Answer:

The correct answer is 5) The presence of four non-Abelian SU(2) gauge groups.

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Question 161 multiple-choice

Quantum algorithms for the hidden subgroup problem (HSP) play a central role in computational group theory and cryptanalysis, especially for non-Abelian groups. Advanced mathematical transforms, such as the Clebsch-Gordan transform, have been instrumental in designing efficient algorithms for these complex cases. Which mathematical procedure is crucial for enabling efficient quantum algorithms for the hidden subgroup problem over semidirect product groups like $ \mathbb{Z}_k^p \rtimes \mathbb{Z} $, particularly by allowing optimal measurements across multiple quantum registers? 1) Quantum error correction codes 2) Grover's search algorithm 3) Quantum phase estimation 4) Clebsch-Gordan transform 5) Classical random sampling 6) Quantum amplitude amplification 7) Bell state preparation

✓ Correct Answer:

The correct answer is 4) Clebsch-Gordan transform.

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in [19]. However, building upon the optimal m easurement approach, Bacon, Childs, andvanDam then appliedthe apparatusofoptimalme asurementsforthe HSP to the HSP for certain semidirect product groups of the for Zk p⋊Z, for a ﬁxed kand prime p[14]. Again it was discovered that the optimal measurement on multiple registers containing the hidden subgroup states required a measurement over multiple r egisters containing the hidden subgroup state (in fact over rregisters.) However this time the authors were able to ﬁnd an eﬃcient quantum algorithm implementing this measurement. Thus a non-trivial quantum algorithm for a non-Abelian hidden subgroup problem was dis covered which was optimally solved by a measurement across multiple copies of the hidden subgroup state. However, in spite of this success, there is much that remains myste rious about the eﬃcient quantum algorithm for the Heisenberg HSP. In particular, why is the re an eﬃcient algorithm for implementing the optimal measurement in this case? Is there any structure behind mea- surements across multiple copies of hidden subgroup states which c an be used to solve the hidden subgroup problem? In this paper we present partial answer s to these questions and highlight the role of an important transform over many registers co ntaining the hidden sub- group state, the Clebsch-Gordan transform, in providing an eﬃcie nt algorithm. We believe that this is an important insight, ﬁrst of all because it gives a new exp lanation for the eﬃcient pretty good measurement based algorithm of Bacon, Childs, and va n Dam. Further we be- lieve that our result highlights the important role of that Clebsch-Go rdan transform can play in quantum algorithms. Clebsch-Gordan transforms, like quantum F ourier transforms over ﬁnite groups, suﬀer from there not being a canonical choice for th e bases of registers output by these transforms. In this work we show that by a judicious choic e of this arbitrary basis, we can use Clebsch-Gordan transforms to solve a non-Abelian HSP. Thus beyond showing that Clebsch-Gordan transforms can be useful for solving HSPs w e can focus the search for eﬃcient HSP algorithm to an even smaller problem of understanding th e choose of basis for these transforms. 3 The Hidden Subgroup Conjugacy Problem In this section we present a variant of the hidden subgroup problem which we label the hidden subgroup conjugacy problem. Two subgroups H1⊂ GandH2⊂ Gare said to be conjugate to each other if there exists an element of g∈ Gsuch that H1={gh2g−1,∀h2∈ H2}. (4) 8How a Clebsch-Gordan Transform Helps to Solve ... The notion of conjugate subgroups forms an equivalence relations hip. Therefore we can classify all subgroups into distinct sets of subgroups which are all c onjugate to each other. The hidden subgroup conjugacy problem (HSCP) is exactly like the HS P, but instead of requiring that we correctly identify the hidden subgroup Hof a function, we only require than one correctly identify which set of conjugate subgroups Hbelongs to. Clearly solving the HSP in its original form will allow one to solve

Question 162 multiple-choice

Quantum algorithms have demonstrated significant advantages over classical algorithms for certain problems, particularly those involving group structures and hidden symmetries. The hidden subgroup problem (HSP) is a central framework in quantum computing that underlies famous algorithms for tasks such as factoring and period finding. Which step in the typical quantum algorithm for solving the hidden subgroup problem involves projecting the quantum state onto a coset superposition associated with the hidden subgroup by measuring part of the quantum register? 1) Applying the quantum Fourier transform to the entire quantum register 2) Preparing an initial state as a uniform superposition over group elements 3) Performing a classical post-processing of the measurement outcomes 4) Utilizing entanglement between group elements and function values via oracle queries 5) Implementing group-theoretic reductions to simplify the subgroup structure 6) Sampling from the Fourier basis before any measurements occur 7) Measuring the function register, which collapses the state to a coset superposition of the hidden subgroup

✓ Correct Answer:

The correct answer is 7) Measuring the function register, which collapses the state to a coset superposition of the hidden subgroup.

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revised form) December 20, 2006; published electronically August 24, 2007. Support for this work was provided by theCalifornia Institute of Technology’s Institute for Quantum Information (IQI), the Mathematical Sci-ences Research Institute (MSRI), the Institute for Advanced Study (IAS), NSF grants ITR-0220070,ITR-0220264, CCR-0093065, CCR-0049092, EIA-0218443, QuBIC-0218563, and CCF-0524613, theCharles Lee Powell Foundation, and the Bell Fund. http://www.siam.org/journals/sicomp/37-3/44717.html †University of New Mexico, Albuquerque, NM 87131, and Santa Fe Institute, Santa Fe, NM 87501 (moore@cs.unm.edu) ‡Dartmouth College, Hanover, NH 03755 (rockmore@cs.dartmouth.edu). §University of Connecticut, Storrs, CT 06269 (acr@cse.uconn.edu). ¶California Institute of Technology, Pasadena, CA 91125 (schulman@caltech.edu). 938 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE POWER OF STRONG FOURIER SAMPLING 939 subgroup problem is the problem of determining the subgroup Hfrom such a function. Algorithms for these problems typically adopt the approach detailed below, calledFourier sampling [3]. Step1. Prepare two registers, the ﬁrst in a uniform superposition over the ele- ments of a group Gand the second with the value zero, yielding the state ψ 1=1/radicalbig |G|/summationdisplay g∈G|g/angbracketright⊗|0/angbracketright. Step2. Calculate (or, if it is an oracle, query) the function fdeﬁned on Gand XOR it with the second register. This entangles the two registers and results in thestate ψ 2=1/radicalbig |G|/summationdisplay g∈G|g/angbracketright⊗|f(g)/angbracketright. Step3. Measure the second register. This produces a uniform superposition over one of f’s level sets, i.e., the set of group elements gfor which f(g) takes the measured value f0. As the level sets of fare the cosets of H, this puts the ﬁrst register in a uniform distribution over superpositions on one of those cosets, namely cH, where f(c)=f0for some f0. Moreover, it disentangles the two registers, resulting in the state ψ3⊗|f0/angbracketright, where ψ3is a so-called coset state , ψ3=|cH/angbracketright=1/radicalbig |H|/summationdisplay h∈H|ch/angbracketright. Alternately, since the value f0we observe has no bearing on the algorithm, we can use the formulation in which the environment, rather than the user, measures f.I n that case, tracing over fyields a mixed state with density matrix ρH=1 [G:H]/summationdisplay f0|ψ3/angbracketright/angbracketleftψ3|=1 |G|/summationdisplay c|cH/angbracketright/angbracketleftcH|, i.e., a classical mixture consisting of one pure state ψ3for each coset. Kuperberg refers to this as the coherent hidden subgroup problem [18]. Step4. Carry out the quantum Fourier transform on ψ3orρHand measure the result. For example, in Simon’s algorithm [26], the “ambient” group Gover which the Fourier transform is performed is Zn 2,fis an oracle with the promise that f(x)=f(x+ y) for some y, and H={0,y}is a subgroup of order 2. In Shor’s factoring algorithm [25]Gis the group Z∗ n, where nis the number we wish to factor, f(x)=rxmod n for a random r

Question 163 multiple-choice

In higher representation theory, 2-quantum groups and their categorified representations play a central role in understanding the algebraic and topological properties of knots and links. Explicit constructions often involve quiver Hecke algebras and their cyclotomic quotients, leading to new insights in knot homology and quantum invariants. Which property is crucial for ensuring that cyclotomic quiver Hecke algebras possess a non-degenerate bilinear form and duality, making them symmetric Frobenius algebras in the categorification of quantum knot invariants? 1) The existence of a symmetric Frobenius structure on the algebra 2) The presence of an infinite-dimensional center in the algebra 3) The absence of standardly stratified category structure 4) The failure of the double centralizer property for self-dual modules 5) The lack of equivalence between diagrammatic and algebraic constructions 6) The inability to categorify tensor products of irreducible representations 7) The degeneracy of Hom spaces in the associated 2-category

✓ Correct Answer:

The correct answer is 1) The existence of a symmetric Frobenius structure on the algebra.

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Question 164 multiple-choice

In quantum information theory and operator algebras, analyzing the structure of completely positive maps and their dilations is essential for understanding quantum channels, group actions, and fixed-point spaces. Tools such as Stinespring dilation, ternary rings of operators (TROs), and interpolation theory play a central role in studying norm inequalities and hypercontractivity properties. Which of the following statements correctly characterizes the role of Kosaki's interpolation theory in the context of quantum semigroups and operator algebras? 1) It provides a method for constructing group representations from operator spaces. 2) It establishes norm inequalities for noncommutative Lp spaces, enabling interpolation between different operator norm regimes relevant to hypercontractivity. 3) It proves the existence of fixed points for completely positive maps under group actions. 4) It classifies TROs based on their underlying Hilbert space structure. 5) It describes the entropic properties of quantum channels through ultracontractivity. 6) It relates tensor products of Hilbert spaces to dual operators in quantum Markov processes. 7) It transfers hypercontractivity properties from classical to quantum settings without restriction.

✓ Correct Answer:

The correct answer is 2) It establishes norm inequalities for noncommutative Lp spaces, enabling interpolation between different operator norm regimes relevant to hypercontractivity..

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( )=/integral.dispG (g)u(g)∗ dG(g); ∗ (')(g)=/uni27E8 ;u(g)'/uni27E9H: Denote byV∶H→H′⊗L2(G)the Stinespring dilation of the conditional expectation that is given by V =(g/uni21A6u(g) ), so that Efix()=(idB(H′)⊗EG)(VV∗) =/integral.dispGu(g)u(g)∗dG(g): Next, identify the space V(H)with a subspace of L2(G)r⊗hH′c≅B(L2(G);H′), so thatV( )= . Lemma A.2. V(H)is a TRO. Proof. Let ;'; ~ ;~'∈H. We compute /uni27E8~ ; ∗ '(~')/uni27E9H=/integral.dispG/uni27E8~ ;u(g)∗ /uni27E9 /uni27E8';u(g)~'/uni27E9dG(g) =Tr(/divides.alt0~'/uni27E9/uni27E8~ /divides.alt0Efix(/divides.alt0 /uni27E9/uni27E8'/divides.alt0)) =/uni27E8~ ;Efix(/divides.alt0 /uni27E9/uni27E8'/divides.alt0)~'/uni27E9H ≡/uni27E8~ ;~'/uni27E9H; (74) where∶=Efix(/divides.alt0 /uni27E9/uni27E8'/divides.alt0). Since this holds for all ~ ∈H, ∗ '(~')=~'. We also have that, since ∈u(G)′, for any ′∈H ∗ ′(~')(g)=/uni27E8 ′;u(g)~'/uni27E9H=/uni27E8∗ ′;u(g)~'/uni27E9H =∗ ∗ ′(~')(g): Hence∗ ′ ∗ '=∗ ∗ ′. SinceEfix(/divides.alt0 /uni27E9/uni27E8'/divides.alt0)∗ ′∈H, ∗ ∗ ′∈V(H)∗, so thatV(H)is a TRO. Next, for any ∈T2(H), deﬁne the following dual operators: ^ ∶/braceleft.alt4H⊗L2(G)→H′ /uni21A6∫Gu(g)∗ (g)dG(g); ^∗ ∶/braceleft.alt4H′→H⊗L2(G) /uni21A6(g/uni21A6 ∗u(g) ): By deﬁnition, the operators ^ span the space V(H)⊗h Hr. SinceV(H)is a TRO, as shown in Lemma A.2, V(H)⊗hHris also a TRO. Lemma A.3. Let1≤p≤∞of Hölder conjugate ^p, and Lk∶L2(G)→L2(G)denote left multiplication by kfork∶G→R+bounded measurable. Then, for any ∈T2(H)and positive operator ∈L(V(H))≡Nfix, /parallel.alt1−1 2^p^ (1H⊗L1 2 k) /parallel.alt12p≤/parallel.alt1k1 2/parallel.alt1L2p(G)/parallel.alt1−1 2^p^ /parallel.alt12p: (75) Proof. By Theorem 5.2 of [34], for any positive operator ∈L(V(H)⊗hHr)≡L(V(H))≡Nfix, the spaces ~Xp;∶=V(H)⊗hHrequipped with the Kosaki norm /parallel.alt1x/parallel.alt1̂Lp()form an interpolation scale. In particular, ~X2p;=[~X∞;~X2;]1 p. Next, observe that −1 2^ ∈ V(H)⊗hHr. Therefore, assuming that /parallel.alt1−1 2^ /parallel.alt1̂Lp()< 1, then there exists an analytic function ∶{z∈C∶0≤ Re(z)≤1}→~X∞+~X2;, of ﬁnite dimensional range such that(1/slash.leftp)=−1 2^ [61], /parallel.alt1(it)/parallel.alt1∞=/parallel.alt1(it)/parallel.alt1̂L∞()<1;and/parallel.alt1(1+it)/parallel.alt1̂L2()<1: 27 Let us deﬁne the analytic function x(z)∶= z/slash.left2(z)(1H⊗Lpz 2 k). We have that /parallel.alt1x(it)/parallel.alt1∞=/parallel.alt1it 2(it)(1H⊗Litp 2 k)/parallel.alt1∞=/parallel.alt1(it)/parallel.alt1∞<1: Next, since(1+it)∈V(H)⊗hHr, there is ′∈T2(H) such that(1+it)≡^ ′. Therefore, /parallel.alt1x(1+it)/parallel.alt12 2=/parallel.alt11 2(1+it)(1H⊗Lp 2 k)/parallel.alt12 2 =/parallel.alt11 2^ ′(1H⊗Lp 2 k)/parallel.alt12 2 =/integral.dispG/divides.alt0k(g)/divides.alt0p/parallel.alt11 2u(g)∗ ′/parallel.alt12 2dG(g) =/integral.dispG/divides.alt0k(g)/divides.alt0p/parallel.alt1u(g)∗1 2 ′/parallel.alt12 2dG(g) =/parallel.alt1k1 2/parallel.alt12p L2p(G)/parallel.alt11 2 ′/parallel.alt12 2 =/parallel.alt1k1 2/parallel.alt12p L2p(G)/parallel.alt11 2(1+it)/parallel.alt12 2

Question 165 multiple-choice

Quantum information experiments often compare simulated and experimental implementations of unitary operations, such as the Quantum Fourier Transform (QFT), to assess fidelity and identify sources of error. Kraus operators and Choi matrices are central tools for characterizing the physical validity and noise characteristics of quantum processes. When analyzing complete positivity in quantum processes using the Choi matrix, which factor most directly explains a lower positivity value in experimental results compared to simulations? 1) Use of non-optimal control sequences in experiments 2) Incomplete state tomography in simulations 3) Presence of non-Hermitian operators in phase space 4) Strong correspondence only for smaller Kraus operators 5) Additional physical errors from spectral fitting in experimental tomography 6) Absence of permutation structure in QFT implementation 7) Lack of singular value normalization in Kraus operator comparisons

✓ Correct Answer:

The correct answer is 5) Additional physical errors from spectral fitting in experimental tomography.

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into ac- count. All of the unitary operations performed during these simula tions were implemented using exactly the same strongly modulating control sequences tha t were used for the exper- iments, and complete state tomography was performed for each in put and output state. It should be noted, however, that these states were reconstruct ed directly from the observables in the simulated density matrices, without further simulating the spe ctra and ﬁtting them as required in experimental state tomography (which of course giv es rise to additional errors in the actual experiments). Figure 8 shows the correlation and att enuated correlation of the initial and ﬁnal states obtained from these simulations. Figure 9 plots the sorted Kraus operator amplitudes obtained from the simulated su- permatrix, along with those from the experimental supermatrix fo r comparison (cf. Fig. 7). The negative values plotted are actually the negative square roots of the corresponding Choi matrix eigenvalues, and are shown to illustrate that the experiment al supermatrix was sig- niﬁcantly further from being completely positive than was the simulat ed (the positivity of the simulated was 0 .86, as opposed to 0 .60 for the experimental). The most likely reason for this is the absence in the simulations of the additional errors exp ected from ﬁtting the spectra to extract the product operator amplitudes for state t omography. Also shown once again is the real part of the theoretical QFT unitary matrix (cf. Fig . 7), along with the real parts of the matrices of the largest Kraus operators from the ex perimental and simulated supermatrices. Finally, the corresponding best unitary approxima tions to the largest Kraus 19 operators, obtained by setting their singular values to unity, have their real parts shown on the bottom right-hand side of Fig. 9. It may be seen that there is a g ood correspondence between the largest Kraus operators as well as between their bes t unitary approximations, withacorrelationbetween thesimulation andtheexperiment of0 .90inbothcases. Asnoted previously, the smaller Kraus operators cannot be expected to co rrespond signiﬁcantly to one another. Since the Fourier transformation converts between the position a nd momentum bases, it has a very simple interpretation in phase space. Classically, a Fourie r transform rotates phase space by 90◦. In quantum phase space, the QFT superoperator has only one no nzero element equal to unity in each row and column, and so constitutes a p ermutation matrix [33]. This is shown in Fig. 10, along with the corresponding plots for the simulated and experimental supermatrices. One can see immediately from these p lots that there are errors in the QFT implementation. The phase space basis, however, consist s of operators that are neither Hermitian nor products of any underlying Hilbert space b asis, and which are not related in any simple way to the physical operators used to implem ent the QFT. Thus, although the action of the QFT is very easy to understand in the pha se space basis, it is very hard to

Question 166 multiple-choice

In quantum computing and cryptography, the generalized hidden shift problem is an extension of the hidden subgroup problem with crucial implications for algorithmic efficiency and security. Solution counts to related modular equations often rely on group theory, number theory, and properties of the Chinese Remainder Theorem. For a fixed vector b and modulus N, what is the number of solutions x in the group Zk_N to the equation b·x = 0 mod N? 1) N^k divided by the least common multiple of the entries of b and N 2) N^{k-1} times the greatest common divisor of the entries of b and N 3) N^{k-2} times the sum of the entries of b 4) N^{k-1} times the least common multiple of the entries of b and N 5) k times the greatest common divisor of the entries of b and N 6) N^{k} minus the product of the entries of b 7) N^{k-1} divided by the greatest common divisor of the entries of b and N

✓ Correct Answer:

The correct answer is 2) N^{k-1} times the greatest common divisor of the entries of b and N.

📚 Reference Text:

and the generalized hidden shift pro blem. AMC received support from the National Science Foundation unde r Grant No. EIA-0086038. WvD’s work was supported in part by the Advance d Re- search and Development Activity (ARDA). Appendix: Number of solutions of the matrix sum problem In this section, we prove Lemma 2. Before giving the proof, we need the following fact: Lemma 4.For any ﬁxed b, the number of solutions x∈Zk Nto the equation b·x= 0 mod NisNk−1gcd(b1,...,bk,N). Proof. First, consider the case where N=pris a prime power. Then gcd(b1,...,bk,pr) =psfor some s∈ {0,1,...,r}. In particular, there must be an index isuch that gcd( bi,pr) =ps, and hence bi=cpsfor some c∈ {1,...,p−1}. Without loss of generality, assume i= 1. Now we can rewrite the equation b·x= 0 ascpsx1+/summationtextk j=2bjxj= 0 mod pr, or equivalently, sincepsis a common divisor of all bj,cx1=−/summationtextk j=2b′ jxjmodpr−swhere b′ j=bj/ps. Because c∈Z× pr, for any ﬁxed ( x2,...,x k)∈Zk−1 pr, there are pssolutions x1= (/summationtextk j=2b′ jxj)/c+λpr−smodpr(one solution for for each λ∈ {0,...,ps−1}). Hence the total number of solutions ( x1,...,x k) is Nk−1ps, proving the lemma for the case N=pr. Now ifNis not a prime power, let N=pr1 1···prt tbe the factorization ofNinto powers of distinct primes, and let τ:ZN→Zpr1 1× ··· ×Zprt t be the ring isomorphism provided by the Chinese remainder th eorem: for x∈ZN,τ(x) = (xmodpr1 1,...,xmodprt t). Since τis a ring isomorphism, b·x= 0 mod Nif and only if b·τ(x)i= 0 mod pri ifor alli= 1,...,t. By QUANTUM ALGORITHM FOR A GENERALIZED HIDDEN SHIFT PROBLEM 13 the special case of the lemma for Na prime power, the number of solutions to theith equation is pri(k−1) igcd(b1,...,bk,pri i); hence the total number of solutions is/producttextt i=1pri(k−1) igcd(b1,...,bk,pri i) =Nk−1gcd(b1,...,bk,N) as claimed. /square Now we are ready to give the proof of Lemma 2: Lemma 2.For the generalized hidden shift problem with M=⌊N1/k⌋ withk≥3andNsuﬃciently large, Pr(1≤ηx w≤4)is lower bounded by a constant. Proof. The main idea of the proof is the same as in the proof of Lemma 5 of [3]: we show that the variance of ηx wis small, so that the num- ber of solutions of the matrix sum problem is typically close to its mean. Because we have M=⌊N1/k⌋, the mean value of ηx wisµ:=Ex,w[ηx w] = Mk/N= 1+O(1/N) asNgrows. The variance of the number of solutions b∈ {0,...,M−1}kof the equation b·x=wmodNfor uniformly random x∈Zk N,w∈ZNisσ2:= Ex,w[(ηx w)2]−µ2, and E x,w[(ηx w)2] =1 Nk+1/summationdisplay x∈Zk N,w∈ZN(ηx w)2(40) =1 Nk+1/summationdisplay x∈Zk N,w∈ZN/parenleftBigg/summationdisplay bδb·x,w/parenrightBigg/parenleftBigg/summationdisplay cδc·x,w/parenrightBigg (41) =1 Nk+1/summationdisplay x,w /summationdisplay bδb·x,w+/summationdisplay b/ne}ationslash=cδb·x,wδb·x,c·x  (42) (with the b,csummations over {0,...,M−1}k). The ﬁrst (diagonal) term is just the mean. To handle the second (oﬀ-diagonal) term, we c an write E x,w[(ηx w)2] =µ+1 Nk+1/summationdisplay b/ne}ationslash=c/summationdisplay x∈Zk Nδb·x,c·x/summationdisplay w∈ZNδb·x,w (43) =µ+1 Nk+1/summationdisplay b/ne}ationslash=c/summationdisplay x∈Zk Nδb·x,c·x (44) =µ+1 N2/summationdisplay b/ne}ationslash=cgcd(b1−c1,...,bk−ck,N) (45) ≤µ+1 N2/summationdisplay b/ne}ationslash=cgcd(b1−c1,...,bk−ck) (46) 14 ANDREW M. CHILDS AND WIM VAN DAM where the next to last step follows from Lemma 4. Now for any q∈ {0,...,M−1}, for aﬁxedvalueof ci, thereare 1+ ⌊(M−ci−1)/q⌋+⌊ci/q⌋ ≤

Question 167 multiple-choice

Quantum-enhanced magnetometry utilizes advanced quantum algorithms and superconducting qubits to surpass classical measurement limits for magnetic fields. Achieving high precision in parameter estimation is a central goal of quantum metrology. Which strategy enables simultaneous achievement of high measurement precision and a broad measurement range in quantum magnetometry using a single superconducting qubit? 1) Increasing the measurement duration without algorithmic modifications 2) Employing NOON states for entangled measurements 3) Using classical averaging to reduce shot-noise 4) Shortening the coherence time to expand measurement capabilities 5) Replacing transmon qubits with spin ensembles 6) Relying exclusively on environmental isolation techniques 7) Implementing phase estimation algorithms that resolve phase ambiguity

✓ Correct Answer:

The correct answer is 7) Implementing phase estimation algorithms that resolve phase ambiguity.

📚 Reference Text:

ARTICLE OPEN Quantum-enhanced magnetometry by phase estimation algorithms with a single arti ﬁcial atom S. Danilin1, A. V. Lebedev2,3, A. Vepsäläinen1, G. B. Lesovik3,4, G. Blatter2and G. S. Paraoanu1 Phase estimation algorithms are key protocols in quantum information processing. Besides applications in quantum computing, they can also be employed in metrology as they allow for fast extraction of information stored in the quantum state of a system. Here, we implement two suitably modi ﬁed phase estimation procedures, the Kitaev and the semiclassical Fourier-transform algorithms, using an arti ﬁcial atom realized with a superconducting transmon circuit. We demonstrate that both algorithms yield a ﬂux sensitivity exceeding the classical shot-noise limit of the device, allowing one to approach the Heisenberg limit. Our experiment paves the way for the use of superconducting qubits as metrological devices which are potentially able to outperform the bestexisting ﬂux sensors with a sensitivity enhanced by few orders of magnitude. npj Quantum Information (2018) 4:29 ; doi:10.1038/s41534-018-0078-y INTRODUCTION Phase estimation algorithms are building elements for many important quantum algorithms, 1such as Shor ’s factorization algorithm2,3or Lloyd ’s algorithm4for solving systems of linear equations. At the same time, phase estimation is a natural conceptin quantum metrology,5where one aims at evaluating an unknown parameter λthat typically enters into the Hamiltonian of a probe quantum system and de ﬁnes its energy states En(λ). In a standard (classical) measurement, the precision δλis restricted by the shot-noise limit δλ/1=ﬃﬃtp, where tis the measurement time. This, however, is not a fundamental limit: in principle, the ultimateattainable precision scales as δλ∝1/t, constrained only by the Heisenberg relation ΔE(λ)≥2πℏ/t, where ΔE=max n;mEn/C0Em ðÞ . The Heisenberg limit can be achieved with the help of entanglement resources, e.g., using NOON photon states in optics.6–8However, these states are dif ﬁcult to create in general and they typically have a short coherence time. Alternatively, onecan reach the Heisenberg limit without exploiting entanglement,by using the coherence of the wavefunction of a single quantum system as a dynamical resource. However, the uncontrollable interaction of the probe with the environment limits the timescale twhere the Heisenberg scaling can be attained by the probe ’s coherence time t~T 2. A further improvement then has to make use of an alternative measurement strategy with a precision following the standard quantum limit but with a better prefactor. The unknown parameter λcan be estimated from the phase ϕ =ΔE(λ)τ/ℏaccumulated by the system in the course of its evolution during the time τ~T2. The 2 π-periodicity of the phase limits the probe ’s measurement range Δλwhere λcan be unambiguously resolved within the narrow interval [ δλ]H=2πℏ/ (μT2), with μ≡∂ΔE/∂λdenoting the sensitivity of the probe ’s spectrum. Therefore, the improvement in the precision at larger T2 is concomitant with a proportional reduction of the measurement range Δλ. The use of phase estimation algorithms then allows toresolve the 2 πphase uncertainty and hence break this unfavorable trade-off between the measurement precision δλand the measurement range Δλ. Moreover, a metrological procedure based on a phase estimation algorithm is Heisenberg-limited: itattains

Question 168 multiple-choice

In quantum computing, the hidden subgroup problem (HSP) plays a crucial role in designing efficient algorithms for factoring, graph isomorphism, and cryptography. Understanding the sample complexity of quantum measurements is key to determining the resources required for solving variants of the HSP. When candidate hidden subgroups all have the same prime order $ p $, what is the sample complexity required for identifying the hidden subgroup among $ |H| $ candidates, and which quantum measurement strategy achieves the upper bound for this complexity? 1) Θ(|H| / p) samples, attained by the optimal measurement 2) Θ(p \cdot |H|) samples, attained by the standard basis measurement 3) Θ(|H|^p) samples, attained by the quantum Fourier transform 4) Θ(log p / log |H|) samples, attained by the majority vote strategy 5) Θ(log |H| / log p) samples, attained by a variant of the pretty good measurement (PGM) 6) Θ(p / |H|) samples, attained by projective measurement onto subgroup states 7) Θ(log |H| \cdot p) samples, attained by classical sampling

✓ Correct Answer:

The correct answer is 5) Θ(log |H| / log p) samples, attained by a variant of the pretty good measurement (PGM).

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Question 169 multiple-choice

In the representation theory of finite-dimensional algebras, Auslander-Reiten quivers and almost-split sequences are central tools for understanding the structure and classification of indecomposable modules. These concepts are especially important in the study of modular and quantum group algebras. Which statement correctly describes a property of almost-split sequences in the category of finite-dimensional modules over an Artin algebra? 1) They do not exist for any projective module. 2) Every module is the end term of an almost-split sequence. 3) For every non-projective indecomposable module, there is a unique almost-split sequence ending at that module. 4) The middle term of an almost-split sequence is always decomposable. 5) Almost-split sequences classify all projective modules. 6) Almost-split sequences never contain irreducible morphisms. 7) The existence of almost-split sequences depends on the choice of ground field.

✓ Correct Answer:

The correct answer is 3) For every non-projective indecomposable module, there is a unique almost-split sequence ending at that module..

📚 Reference Text:

there is an irreducible morphism N-+N'. Notice that on a fixed diagonal the dimension of the modules is constant; simples are located on the principal diagonal. If q is specialized to a primitive dih root of unity, the Auslander-Reiten quiver of J^(q)/Id is obtained by deleting the vertices located above the d — 1 diagonal. For kZn(q)/Id, the corresponding identi- fications shows that the Auslander-Reiten quiver lies on a cylinder. An almost-split sequence (see [3,4] ) ending at an indecomposable module N is a non-split exact sequence 0->M->Jf-»N-»0 such that any morphism/ Y -»N which is not a split surjection can be lifted to a morphism Y -> X. Such a sequence is unique and always exists in case AT is a non-projective indecomposable object Quiver Quantum Group 471 of a category of finitely generated modules over a finite dimensional algebra. In that case M is indecomposable and uniquely determined by N, it is called the Auslander-Reiten translate of N. The almost-split sequences in our situation are for a non-simple indecomposable module NίtU9 and ->i+l -N-jίvι, i ί+1 for a simple module St = Niti. We recall some properties of this category (see [12] ). The mesh relations arising from the Auslander-Reiten translate indicates that the squares of the quiver commute and that the composition of two arrows of any triangle over the principal diagonal is zero. The important fact is that these relations are sufficient in order to present the category of indecomposable modules, see for instance [12]. The following is an immediate consequence of the preceding discussion: Proposition 4.5 ([12]). Let NitU be a fixed indecomposable module. The indecompos- able modules X such that Hom^pf, Nittl)ή=0 are located in the A-region of the following diagram', the indecomposable modules Y such that Hom^(Nί>M? Y) ΦO are located in the B-region. Theorem 4.6. Let NiίU and NjtV be finite dimensional indecomposable J^- modules. The decomposition ofNitU u (x) NjtV) is a split basic semisimple K-algebra of K-dimension πύn{άίmκNitU9 dimκNjtV}. 5. Almost-Split Sequences Almost-split sequences are an important tool in understanding finite length mod- ule categories. In case of modular group algebras,

Question 170 multiple-choice

In quantum information theory, group symmetries and representation theory are powerful tools for constructing multipartite quantum states with desirable entanglement and mixing properties. Schur’s Lemma and the structure of irreducible representations play a key role in determining when subsystems of a quantum state are maximally mixed. Which statement most accurately describes when a subsystem's density matrix is guaranteed to be proportional to the identity operator in a multipartite quantum state constructed via group representation methods? 1) When the subsystem is acted on by a reducible representation of any group. 2) When the global state is invariant under a group action but the representation is non-unitary. 3) When the tensor product of representations contains only higher-dimensional irreducible components. 4) When the subgroup acts irreducibly on the subsystem and the global state is invariant under this action, ensuring maximal mixing by Schur’s Lemma. 5) When the subsystem is described by a pure state and the group contains the trivial representation. 6) When the group is abelian and all its projective representations are reducible. 7) When the density matrix is diagonal in the basis of the group’s irreducible representations, regardless of invariance.

✓ Correct Answer:

The correct answer is 4) When the subgroup acts irreducibly on the subsystem and the global state is invariant under this action, ensuring maximal mixing by Schur’s Lemma..

📚 Reference Text:

a phase, then the density matrix ρAis invariant under the action of H. LettingUhbe the representative of h∈HinU(dA), we then have that for all h,UhρAU† h=ρA, or [Uh,ρA] = 0. Since h→Uhis an irreducible representation ofH, Schur’s Lemma implies that ρAis a multiple of the identity operator.10 The theorem has an immediate application to multipart systems: Corollary 3.2. Consider a multipart quantum system with Hilbert space H=H1⊗···⊗H n upon which ˜K=U(d1)× ··· ×U(dn)acts. If a state |Ψ/an}b∇acket∇i}ht ∈ His invariant up to a phase under a subgroup H⊂˜K, then the density matrix for each subsystem Hα=⊗i∈αHi, upon whichHacts irreducibly is maximally mixed. This gives a way to construct states of multipart systems whose ele mentary subsystems are all maximally mixed: Corollary 3.3. Consider any group Hand any set of unitary irreducible representations Ri ofHwhose tensor product contains the trivial representation ( i.e. an invariant vector).11 10More directly, if ρAis not maximally mixed, then ρad A=ρA−1 dA1 1 does not vanish, and transforms in the adjoint representation of U(dA). We have seen that Uhcommutes with ρAand therefore with ρad A for allh. Acting on A,Uhtherefore does not mix subspaces with diﬀerent eigenvalues for ρad A. Sinceρad A is nonvanishing and traceless, it must have at least two diﬀerent eige nvalues, so there are proper invariant subspaces for the action of H, in contradiction with the assumption that Hacts irreducibly on A. 11The same result holds if the tensor product of Ricontains any one-dimensional representation, since the associated one-dimensional invariant subspace gives a state |Ψ/an}b∇acket∇i}htinvariant under Hup to a phase as required by Corollary 3.2. However, this generalization does not pro duce any new examples: if Ris the one-dimensional representation in the tensor product and R∗is the complex conjugate representation, then (R1⊗R∗,R2,...Rn) will be a set of irreducible representations whose tensor product contains the trivial representation, and the corresponding invariant state will be the state|Ψ/an}b∇acket∇i}ht. 13 Given an explicit representation of such an invariant as |0/an}b∇acket∇i}ht=/summationdisplay Ca1···an|a1/an}b∇acket∇i}ht⊗···⊗|an/an}b∇acket∇i}ht, (3.1) whereCa1···anare group-theoretic coeﬃcients describing how the trivial representation is em- bedded in the tensor product, the state |0/an}b∇acket∇i}ht, considered as a quantum state in Hilbert space H1⊗···⊗H nwill have all elementary subsystems maximally mixed. Furth er, any composite subsystem Hαfor which the tensor product of representations Riwithi∈αgives a single irreducible representation will also be maximally mixed. Remark 3.4. In some cases, it is useful to consider a variant of our constructio n based on a representation h/ma√sto−→u1(h)⊗···⊗un(h)∈U(d1···dn) (3.2) of a groupHfor which the individual maps h/ma√sto→ui(h)∈H(di) give only projective represen- tations ofH(i.e. where ui(h1)ui(h2) =ci(h1,h2)ui(h1h2) for some scalar function c(h1,h2)). If these projective representations are irreducible and if a state |Ψ/an}b∇acket∇i}htis invariant under the action ofHdeﬁned in this way, then essentially the same arguments as above sh ow that|Ψ/an}b∇acket∇i}ht will be LME. Here, we have the possibility that the group His abelian, since we can have irreducible projective representations of abelian groups with dimen sion greater than 1. It is possible to understand this construction as a case of the gene ral construction in

Question 171 multiple-choice

In photonic quantum information processing, the generation of high-fidelity entangled states such as Bell states is crucial, and various error models are used to analyze how imperfections in photon properties affect protocol performance. Simulations often focus on different types of distinguishability errors and employ metrics that account for practical post-selection procedures. Which error model specifically represents a scenario where every pair of photons shares a unique, orthogonal error state, commonly encountered with imperfect photon pair sources? 1) Orthogonal Bad Bits (OBB) 2) Orthogonal Bad Pairs (OBP) 3) Same Bad Bits (SBB) 4) Independent Phase Errors (IPE) 5) Collective Amplitude Damping (CAD) 6) Mode Mismatch Errors (MME) 7) Global Depolarizing Channel (GDC)

✓ Correct Answer:

The correct answer is 2) Orthogonal Bad Pairs (OBP).

📚 Reference Text:

” resulting in nontrivial contributions from terms that are normally “filtered out” by subsequent operations (as discussed in Lemma 2 and below). 3. Numerical distinguishability error analysis for Bell state generation In the previous section, we characterized the behavior of the n-GHZ state generation protocol, Protocol 2, under low-order distinguishability errors. We now use our simulation tools to numerically study the performance of the Bell state generation protocol (the special case n= 2) under all orders of distinguishability errors, using various performance metrics and error models. We note that Sparrow [20] has explicitly calculated the output state of this protocol (after dropping non-dual-rail terms and tracing out all internal degrees of freedom) as a function of each input photon’s internal state; further, Shaw et al. [25] have numerically compared the fidelity of this protocol with other BSG protocols. Here, our goal is to demonstrate the utility of our simulation paradigm for analyzing such protocols, and to examine the interactions between different error models and performance metrics. These techniques can be extended to larger protocols (especially via the simulation paradigm discussed in Sec. III below) and can easily incorporate additional types of errors. (Note that, as discussed in Remark 3, we neglect loss errors in this section as they will be filtered out by all the metrics we consider here.) We consider three different models for distinguishability errors. In the first two, as in (7) above, the ith photon’s internal state is independent and is modeled as ρ(ϵ) = (1 −ϵ)|ψ0⟩⟨ψ0|+ϵ|ψi⟩⟨ψi|, where ⟨ψ0|ψi⟩= 0fori >0. In the Orthogonal 11 Bad Bits (OBB) model [20], we have ⟨ψi|ψj⟩= 0fori̸=j. In the Same Bad Bits (SBB) model, we have only one error state:|ψ1⟩=|ψ2⟩=. . .. Finally, in the Orthogonal Bad Pairs (OBP) model (corresponding to the pair errors studied in Theorem 4), pairs of mutually indistinguishable photons are emitted. Each pair is independent of the others, with a unique error state. In other words, the ithpair has internal state of the form (1−ϵ′)|ψ0ψ0⟩⟨ψ0ψ0|+ϵ′|ψiψi⟩⟨ψiψi|, where ⟨ψi|ψj⟩= 0 fori̸=j. These mutually indistinguishable pairs are routed into adjacent modes (0,1)or(2,3)of the BSG circuit. In order to compare to the other models, we will take ϵ′= 1−(1−ϵ)2=ϵ(2−ϵ), so that in all three models, the probability of an ideal pair is (1−ϵ)2. We also consider four different metrics for benchmarking the quality of the four-mode state η=P jpj|ηj⟩⟨ηj|constructed by the BSG protocol (ideally a two-qubit dual-rail state). The first is the fidelity with the desired ideal Bell pair |B++⟩. (Here we use the notation that |Bs1s2⟩is the ideal Bell state with (X⊗X)|Bs1s2⟩=s1|Bs1s2⟩,(Z⊗Z)|Bs1s2⟩= s2|Bs1s2⟩.) The second is the post-selected fidelity : we drop all terms in the output state that do not correspond to a dual- rail state, renormalize, then calculate the fidelity with the ideal state |B++⟩. This is motivated by the fact that fusion and similar circuits tend to filter out non-dual-rail terms such as |11⟩,|20⟩, etc. (recall Lemma 2). For the final two metrics, we take this observation to its logical conclusion, asking the following question:

Question 172 multiple-choice

In post-quantum cryptography, the use of code-based encryption schemes such as McEliece and Niederreiter relies on the mathematical properties of error-correcting codes and their associated matrix structures. Security against quantum attacks often depends on the rank and automorphism group characteristics of these code-related matrices. Which property of a generator matrix constructed from a classical Goppa code is most critical for ensuring resistance to quantum Fourier sampling attacks in code-based cryptosystems? 1) Large field size q relative to code length n 2) Low minimal degree of the automorphism group 3) Sparse matrix structure with many zero entries 4) Full rank combined with an automorphism group of minimal degree at least n-2 and small size 5) Use of parity-check matrices instead of generator matrices 6) Dense matrix structure with repeated rows 7) Automorphism group growing exponentially with n

✓ Correct Answer:

The correct answer is 4) Full rank combined with an automorphism group of minimal degree at least n-2 and small size.

📚 Reference Text:

By the orbit-stabilizer formula, we have |Fix(N)|≤|GLk(Fq/lscript)| |Orb(N)|=(q/lscriptk−1)(q/lscriptk−q/lscript)...(q/lscriptk−q/lscript(k−1)) (q/lscriptk−1)(q/lscriptk−q/lscript)...(q/lscriptk−q/lscript(r−1)) =(q/lscriptk−q/lscriptr)(q/lscriptk−q/lscript(r+1))···(q/lscriptk−q/lscript(k−1))≤q/lscriptk(k−r). Corollary 1. Assume qk2≤n0.2nand the automorphism group Aut(M)has minimal degree Ω(n). Let r be the column rank of M. Then the subgroup K deﬁned in (5)has DK≤|Aut(M)|4q4/lscriptk(k−r)e−Ω(n). In particular, the subgroup K is indistinguishable if, further,|Aut(M)|≤eo(n)and r≥k−o(√ n)//lscript. The constraint qk2≤n0.2nimplies log |GLk(Fq)|=O(nlogn), so Alice only needs to ﬂipO(nlogn)bits to pick a random Sfrom GLk(Fq). Thus she needs only O(nlogn) coin ﬂips overall to generate her private key. Application to the McEliece cryptosystem. Consider a McEliece cryptosystem using aq-ary linear [n,k]-code C, with parameters satisfying qk2≤n0.2n. Since the automor- phism group of the code Cequals the automorphism group of its generator matrix, we can conclude that this McEliece cryptosystem resists the standard quantum Fourier sam- pling attack if the code Cis(i) well-scrambled , i.e., it has a generator matrix of rank at least k−o(√ n),a n di s (ii) well-permuted , i.e., its automorphism group has minimal degree at least Ω(n)and has size at most eo(n). Recall that in terms of security, the Niederreiter system using (n−k)×nparity check matrices over Fqof the same code C is equivalent to the McEliece system using the code C[10]. Application to Goppa codes. We would like to point out that if Mis a generator matrix of a rational Goppa code or a canonical parity check matrix of a classical Goppa code, it will give good bounds in Corollary 1. Speciﬁcally, we consider a matrix Mover a ﬁnite ﬁeld Fq/lscript⊃Fqof the following form: M=⎛ ⎜⎜⎜⎝v1f1(α1)v2f1(α2)···vnf1(αn) v1f2(α1)v2f2(α2)···vnf2(αn) ............ v1fk(α1)v2fk(α2)···vnfk(αn)⎞ ⎟⎟⎟⎠(7) 774 H. Dinh, C. Moore, and A. Russell where v1,..., vnare nonzero elements in the ﬁeld Fq/lscript,(α1,..., αn)is a list of distinct points in the projective line Fq/lscript∪{∞},a n d f1,..., fkareFq/lscript-linearly independent poly- nomials in Fq/lscript[X]of degree less than k(by convention, fi(∞)is the Xk−1-coefﬁcient of fi(X)). Note that such a matrix Mis a generator matrix of a rational Goppa [n,k]-code over the ﬁeld Fq/lscript, and is also a parity check matrix of a classical Goppa [n,≥n−/lscriptk]- code over Fq. To apply Corollary 1, we show the following properties of the matrix M: Lemma 3. T h em a t r i xMi nt h ef o r mo f (7)has full rank (i.e., its column rank equals k), and Aut(M)has minimal degree at least n −2, and|Aut(M)|≤q3/lscript. Proof. We can show that Mhas full rank directly by decomposing MasM=AV D , where A=(aij)is an k×kinvertible matrix with entry aijbeing the the Xj−1-coefﬁcient of polynomial fi(X);Vis ak×nVandermonde matrix with (i,j)-entry being αi−1 j;a n d Dis an n×ndiagonal matrix with viin the (i,i)-entry. Then the rank of Mequals the rank of the Vandermonde matrix V, which has full rank. Now we can view Mas a generator matrix of a rational Goppa [n,k]-code Rover the ﬁeldFq/lscript. Then we have Aut (M)⊂Aut(R),w h e r eA u t (R)is the automorphism group of the code R,t h a ti s ,A u t (R)=/braceleftBig P∈Sn|SMP =Mfor some S∈GLk(Fq/lscript)/bracerightBig .N o ww e can apply Stichtenoth’s Theorem [25] to

Question 173 multiple-choice

Finite W-algebras are associative algebras constructed from Lie algebras and play a key role in mathematical physics and representation theory. Their structure often involves central elements, explicit bases, and connections to symmetry algebras like sl2 and sl3. Which statement correctly describes the structure of a finite W-algebra generated by {H, E, F, G+, G0, G−, C} when its central elements are fixed to generic values? 1) It becomes isomorphic to the Lie algebra sl3. 2) It reduces to a single copy of the sl2 algebra. 3) The spin-1 representation decomposes into three one-dimensional representations. 4) The generators correspond to those of the Virasoro algebra. 5) It forms a direct sum with the u(1) algebra. 6) It becomes isomorphic to sl2⊕sl2. 7) The algebra loses all nontrivial commutators.

✓ Correct Answer:

The correct answer is 6) It becomes isomorphic to sl2⊕sl2..

📚 Reference Text:

2=e3, C= (1 24s2+¯h 2s) +1 2(e1f1+f1e1+e2f2+f2e2+e3f3+f3e3) +1 3(h2 1+h1h2+h2 2). (8.6) 55 InCwe again recognize the second casimir of sl3. It is a general feature of ﬁnite W algebras that they contain a central element C, whose Miura transform contains the second casimir of g0.Cis the ﬁnite counterpart of the energy momentum tensor that every inﬁnite Walgebra possesses. 8.2.4 = 2 + 2 A convenient basis to study this case is rata= r6 4+r10 2r5 2+r8 2r12r14 r7 2+r9 2−r6 4−r11 2r13r15 r1r2r6 4−r10 2r5 2−r8 2 r3r4r7 2−r9 2−r6 4+r11 2 . (8.7) Thesl2embedding is given by t+=t12+t15,t0=t10−t11andt−=1 2(t1+t4). The subalgebra g+is spanned by {t12,...,t 15},g0by{t5,...,t 11}andg−by{t1,...,t 4}. Thed1cohomology of Ω redis generated by ˆJ1,...,ˆJ7. Thed-closed representatives are W(ˆJa) =ˆJafora= 5,6,7, and W(ˆJ1) = ˆJ1−1 4ˆJ5ˆJ9+1 4ˆJ7ˆJ8+1 4ˆJ8ˆJ9+1 4ˆJ10ˆJ10+3¯h 4ˆJ10−¯h 4ˆJ11, W(ˆJ2) = ˆJ2+1 4ˆJ5ˆJ10+1 4ˆJ5ˆJ11−1 4ˆJ6ˆJ8+1 4ˆJ8ˆJ10−1 4ˆJ10ˆJ11+¯h 2ˆJ8, W(ˆJ3) = ˆJ3+1 4ˆJ6ˆJ9−1 4ˆJ7ˆJ10−1 4ˆJ7ˆJ11+1 4ˆJ9ˆJ10−1 4ˆJ9ˆJ11+¯h 2ˆJ9, W(ˆJ4) = ˆJ4+1 4ˆJ5ˆJ9−1 4ˆJ7ˆJ8+1 4ˆJ8ˆJ9+1 4ˆJ11ˆJ11+¯h 4ˆJ10−3¯h 4ˆJ11. (8.8) To display the properties of the algebra as clearly as possib le, we introduce a new basis of ﬁelds H=1 2W(ˆJ6), E=−W(ˆJ7), F=−W(ˆJ5), G+=W(ˆJ3), G0=W(ˆJ1)−W(ˆJ4), G−=−W(ˆJ2), C=W(ˆJ1) +W(ˆJ4) +1 8W(ˆJ6)W(ˆJ6) +1 4W(ˆJ5)W(ˆJ7) +1 4W(ˆJ7)W(ˆJ5) +¯h 4W(ˆJ6). (8.9) 56 Here,Cis the by now familiar central element, {E,H,F }form ansl2algebra and {G+,G0,G−}form a spin 1 representation with respect to this sl2algebra. The non- vanishing commutators are [E,F] = 2¯hH, [H,E] = ¯hE, [H,F] = −¯hF, [E,G0] = 2¯hG+, [E,G−] = ¯hG0, [F,G+] = ¯hG0, [F,G0] = 2¯hG−, [H,G+] = ¯hG+, [H,G−] = −¯hG−, [G0,G+] = ¯h(−CE+EH2+1 2EEF +1 2EFE)−2¯h3E, [G0,G−] = ¯h(−CF+FH2+1 2FFE +1 2FEF)−2¯h3F, [G+,G−] = ¯h(−CH+H3+1 2HEF +1 2HFE )−2¯h3H. (8.10) Sinceg0=sl2⊕sl2⊕u(1), the quantum Miura transformation expresses this algeb ra in term of generators {e1,h1,f1},{e2,h2,f2},sofg0. The relation between these generators and the ˆJaare:s=ˆJ10−ˆJ11,h1=1 2(ˆJ6+ˆJ10+ˆJ11),h2=1 2(ˆJ6−ˆJ10−ˆJ11), e1=1 2(ˆJ7+ˆJ9),e2=1 2(ˆJ7−ˆJ9),f1=1 2(ˆJ5+ˆJ8) andf2=1 2(ˆJ5−ˆJ8). The commutation relations for these are [ e1,f1] = ¯hh1, [h1,e1] = 2¯he1, [h1,f1] =−2¯hf1, and similar for {e2,h2,f2}. For the quantum Miura transformation one then ﬁnds H=1 2(h1+h2), E=−e1−e2, F=−f1−f2, G+=1 2e1h2−1 2e2h1+1 4s(e1−e2) + ¯h(e1−e2), G0=f1e2−f2e1+1 4s(h1−h2) + ¯h(h1−h2), G−=1 2f1h2−1 2f2h1−1 4s(f1−f2)−¯h(f1−f2), C= (1 8s2+ ¯hs) +1 2(e1f1+f1e1+e2f2+f2e2) +1 4(h2 1+h2 2). (8.11) The inﬁnite dimensional version of this algebra is one of the ‘covariantly coupled’ algebras that have been studied in [34]. The ﬁnite algebra (8 .10) is almost a Lie algebra. If we assign particular values to Cand to the second casimir C2= (H2+1 2EF+1 2FE) 57 of thesl2subalgebra spanned by {E,H,F }, then (8.10) reduces to a Lie algebra. For a generic choice of the values of CandC2this Lie algebra is isomorphic to sl2⊕sl2. An interesting question is, whether a similar phenomena occur s for diﬀerent covariantly coupled algebras. 8.3.4 = 3 + 1 The last nontrivial nonprincipal sl2embedding we consider is 44≃32⊕12. We choose yet another basis rata= r5 12−r6 3r8r11 r12 r4−r5 4r13 r14 r3 2−r9 2r10r5 12+r6 6+r7 2r15 r1r2r3 2+r9 2r5 12+r6 6−r7 2 , (8.12) in terms of which the sl2embedding is t+=t11+t15,t0=−3t6+t7andt−= 2t3. The subalgebra g+is

Question 174 multiple-choice

In quantum algorithms for the hidden subgroup problem, group representation theory and the quantum Fourier transform are crucial for analyzing and measuring hidden subgroup states. Understanding the role of irreducible representations and subgroup projectors is central to designing optimal measurement strategies. Which statement best describes the role of Schur’s lemma in the construction of measurement operators for distinguishing hidden subgroups in the quantum hidden subgroup problem? 1) It ensures all measurement operators are diagonal in the computational basis. 2) It guarantees measurement outcomes are uniformly distributed over all group elements. 3) It implies that summing subgroup projectors over all conjugate subgroups yields an operator proportional to the identity for each irreducible representation. 4) It establishes that characters of non-conjugate subgroups are always zero. 5) It requires that the sum of measurement operators equals the trace of the regular representation. 6) It proves that irreducible representations must be one-dimensional for abelian groups. 7) It determines the optimal coefficients for projectors by maximizing their commutator with the subgroup state.

✓ Correct Answer:

The correct answer is 3) It implies that summing subgroup projectors over all conjugate subgroups yields an operator proportional to the identity for each irreducible representation..

📚 Reference Text:

expressing the hidden subgroup state in terms of the right reg ular representation is helpful, it is even more helpful to examine this state in a diﬀerent basis. If we perform the q uantum Fourier transform over the ﬁnite group G, this transforms the basis of group elements {|g/an}bracketri}ht}to a basis with irreducible representation (irrep) labels along with row and column registers, {|µ,i,j/an}bracketri}ht}. In this basis the hidden subgroup state is ρH=/circleplusdisplay µ/bracketleftBigg Idµ⊗1 |G|/summationdisplay h∈HDµ(h)/bracketrightBigg , (14) whereµlabels the irreducible representations of G,Dµis theµth irrep, dµis the dimension of the µth irrep, and Idµ is thedµdimensional identity operator. Deﬁning the subgroup projectors PHfor theµth irrep, Pµ,H:=1 |H|/summationdisplay h∈HDµ(h), (15) we can express the hidden subgroup state as ρH=/circleplusdisplay µ/bracketleftbigg Idµ⊗|H| |G|Pµ,H/bracketrightbigg . (16) A ﬁnal useful symmetry relationship that we will use later comes abo ut when we sum the projectors Pµ,Hover all conjugate subgroups. A conjugate subgroup of His deﬁned as Hg:={ghg−1,h∈ H}forg∈ G. In particular we recall that/summationtext g∈GDµ(ghg−1) commutes with all Dµ(g′) and thus, via Schur’s lemma, it must be proportional to the identity matrix (this proportionality may be zero), /summationdisplay g∈GDµ(ghg−1) =rµ(h)Idµ, (17) whererµ(h) is the proportionality constant. Taking the trace of this equation allows us to calculate rµ(h): /summationdisplay g∈Gχµ(ghg−1) =|G|χµ(h) =rµ(h)dµ (18) whereχµ(g) := Tr[Dµ(g)] is the character of element gin irrepµ. Thus we obtain 1 |G|/summationdisplay g∈GPµ,Hg=1 |H|/summationdisplay h∈Hχµ(h) dµIdµ. (19) B. The Optimal Single Copy Measurement The optimal measurement criteria, Eq. (5), when all of the subgro ups are given with equal a prior probability is given by /summationdisplay H∈Sub(G)ρHEH=/summationdisplay H∈Sub(G)EHρH /summationdisplay H∈Sub(G)ρHEH≥ρH′,for allH′∈Sub(G), (20) 5 where this sum over His over all possible subgroups of G. Notice that if ρHandEHcommute, then the ﬁrst equation is satisﬁed. This will be true of the measurements we will consider, so the second equation will be the nontrivial condition. Our ansatz for the optimal measurement is a measurement of the f orm EH=/circleplusdisplay µ/bracketleftbig Idµ⊗cµ,HPµ,H/bracketrightbig . (21) wherecµ,Hare as of yet undetermined constants. The reason for beginning w ith this assumption is that it can easily be shown that the previous results of Ip [15] and Moore and Russell [18] can be written as measurements of this form. Notice that this choice of measurement implies that ρHandEHcommute so the ﬁrst optimal measurement criteria is automatically satisﬁed. Thus we must show that the second measu rement criteria is satisﬁed, while the condition that the measurement is valid also holds. The EHof our chosen form are a valid measurement if and only if cµ,H≥0 and, for all µ, /summationdisplay H∈Sub(G)cµ,HPµ,H=Idµ. (22) It will be convenient to break the sum over all subgroups that appe ars here down into a sum over conjugate subgroups. In particular two subgroups H1andH2are conjugate if there exists an element, gofG, such that H2={ghg−1,h∈ H1}. Whether two subgroups are conjugate to each other forms an e quivalence relation and thus we can partition the set of all subgroups of Ginto sets of conjugate subgroups. Let C(G) be the set of all of

Question 175 multiple-choice

Quantum algorithms often leverage both classical and quantum computational techniques to efficiently solve problems that are intractable for classical computers alone. The generalized hidden shift problem, which interpolates between dihedral and abelian cases as a parameter M varies, serves as a benchmark for such hybrid approaches and has connections to lattice problems and graph isomorphism. In the context of quantum algorithms for the generalized hidden shift problem, which choice most accurately describes why Lenstra’s algorithm for integer programming is effective in this setting with sufficiently large M? 1) Because it ensures polynomial-time performance regardless of the dimension size 2) Because it transforms the quantum problem into a purely classical one without entanglement 3) Because it enables solving non-linear integer equations in arbitrary dimensions 4) Because it guarantees exact solutions for all values of M and N 5) Because it provides exponential speedup for M = 2 6) Because it runs in polynomial time when the dimension is fixed, allowing efficient implementation of the quantum measurement step when M is large enough 7) Because it uses quantum state discrimination to bypass the need for integer programming

✓ Correct Answer:

The correct answer is 6) Because it runs in polynomial time when the dimension is fixed, allowing efficient implementation of the quantum measurement step when M is large enough.

📚 Reference Text:

with M≥Nǫfor any ﬁxedǫ >0can be solved in time poly(log N)on a quantum computer. Proof. Givenǫ, we will use k= max{⌈1/ǫ⌉,3}copies of the unknown quantum state (3). Because the generalized hidden shift pro blem on the full domain{0,...,M−1}×ZNcan be solved by solving the same problem on a reduced domain {0,...,M′−1} ×ZNwithM′≤M, it is suﬃcient to prove the theorem for a speciﬁc M≤Nǫ. We will do this for M=⌊N1/k⌋. By Lemma 2, there is a constant probability of having between 1 and 4 solutionstotherandommatrixsumequation b·x=wmodN, andhencethe success probability of the pgmis also aconstant (by Lemma1). Theeﬃcient approximate implementation of this measurement is describ ed in the second half of Section 3 in combination with the results on integer p rogramming that are described in the ﬁrst part of Section 4. /square In fact, the algorithm remains eﬃcient even if ǫdecreases (very) slowly withN. Lenstra’s algorithm for integer programming in dimension ktakes time 2O(k3)[20], so the generalized hidden shift problem can be solved e f- ﬁciently for M=NO(1/(loglogN)1/3). Indeed, a subsequent improvement by Kannan solves k-dimensional integer programming in time 2O(klogk)[16], which can be used to decrease Mslightly further. 5. Discussion We have applied the pgmapproach to the generalized hidden shift prob- lem, whichinterpolates fromthedihedral hsptotheabelian hspasMvaries from 2 to N. We found an eﬃcient quantum algorithm for this problem for anyM≥Nǫwithǫﬁxed (or decreasing very slowly with N). The algorithm works by solving the matrix sum problem using Lenstra’s algo rithm for in- teger programming in constant dimensions, thereby illustr ating (as in [3]) that nontrivial classical algorithms can beuseful for impl ementing entangled measurements to distinguish states obtained by Fourier sam pling. Our original motivation for studying this problem was the ob servation that a solution to the generalized hidden shift problem for s uﬃciently small Mcould lead to new algorithms for the unique shortest vector i n a lattice problem, just as Regev showed for the case M= 2 [24]. Unfortunately, 12 ANDREW M. CHILDS AND WIM VAN DAM M≥Nǫdoes not appear to be suﬃciently small to yield interesting l at- tice algorithms. Nevertheless, attempting to solve the gen eralized hidden shift problem for yet smaller Mmay be a promising path toward improved quantum algorithms for lattice problems. Indeed, for the ca seM= 2, Ku- perberg’s subexponential-time algorithm outperforms the algorithm given in this paper, so it seems likely that an improved algorithm c ould be found for values of Mintermediate between 2 and Nǫ. Another problem suggested by this work is the following gene ralization of graph isomorphism. Suppose that we are given a list of n-vertex graphs G0,...,G M−1, and are promised that either no two graphs are isomorphic, orGb=π(Gb+1) for some ﬁxed permutation π∈Snforb= 0,1,...,M−2. It would be interesting to show that this problem can be solve d eﬃciently even for very large M(where the graphs can be speciﬁed by a black box in the case where Mis superpolynomial in n). Acknowledgments We thank Oded Regev for for discussions about the relationsh ip be- tween lattice problems

Question 176 multiple-choice

Quantum k-designs are ensembles of unitary operations that replicate the statistical properties of completely random unitaries up to a specified moment k, and are essential for simulating randomness in quantum systems. Incorporating SU(d) symmetry into local quantum circuits introduces fundamental constraints and changes to the behavior of pseudorandomness and convergence. Which mathematical approach is particularly employed to analyze the representations of the symmetric group Sn in the context of SU(d)-symmetric quantum circuits and their convergence to k-designs? 1) Schur-Weyl duality 2) Fourier analysis on compact groups 3) Okounkov-Vershik approach 4) Lie algebra decomposition 5) Path integral formalism 6) Quantum error correction codes 7) Trotter-Suzuki expansion

✓ Correct Answer:

The correct answer is 3) Okounkov-Vershik approach.

📚 Reference Text:

Question 177 multiple-choice

Quantum error correction and symmetry play central roles in building robust quantum computing architectures and understanding the fundamental principles of quantum information. The interaction between code structure and physical symmetries determines the feasibility of fault-tolerant operations. Which statement correctly describes the implication of the Eastin-Knill theorem for quantum error-correcting codes with continuous symmetries? 1) Such codes can always implement arbitrary logical gates using transversal operations. 2) Quantum codes with continuous symmetries are immune to all types of noise. 3) Only codes defined on infinite-dimensional Hilbert spaces are constrained by the theorem. 4) The theorem allows universal transversal gates if the code corrects only bit-flip errors. 5) Discrete symmetries are subject to the same restriction as continuous symmetries under the theorem. 6) Quantum error-correcting codes with continuous symmetries cannot support a universal set of transversal gates in finite dimensions. 7) The theorem prohibits the use of covariant encoding maps in any code construction.

✓ Correct Answer:

The correct answer is 6) Quantum error-correcting codes with continuous symmetries cannot support a universal set of transversal gates in finite dimensions..

📚 Reference Text:

are known to form approximate quantum error- correcting codes [8,9], which are preserved under time evolution. Limits to sensitivity in quantum metrology arerelated to the degree of asymmetry of probe states, a notion formalized in the resource theory of asymmetry and refer- ence frames [10,11] . Thus, reference frame information can be protected against noise using quantum codes with suitable symmetry properties [12]. Furthermore, recent developments in quantum gravity have shown that theAdS/CFT correspondence can be viewed as a quantum error-correcting code which is expected to be compatible with the natural physical symmetries of the system, such astime-translation invariance [6,7,13 –16]. Finally, the Eastin- Knill theorem [17–19], which complicates the construction of fault-tolerant schemes for quantum computation by*Corresponding author. philippe.faist@fu-berlin.de †These authors contributed equally to this work. Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article ’s title, journal citation, and DOI.PHYSICAL REVIEW X 10,041018 (2020) 2160-3308 =20=10(4)=041018(31) 041018-1 Published by the American Physical Society forbidding quantum error-correcting codes from admitting a universal set of transversal gates [20–23], can be viewed as the statement that finite-dimensional quantum codes which correct erasure have no continuous symmetries [12]. Thus, there are loopholes to the Eastin-Knill theorem that are naturally exploited by holographic theories of quantum gravity. This article provides a detailed quantitative inves- tigation of those loopholes, critically evaluating their poten- tial for application to quantum fault tolerance (see also recent related results [24]). A continuous symmetry, as opposed to a discrete sym- metry, allows for infinitesimally small transformations that are arbitrarily close to the identity operation. Such symmetry transformations are generated by conserved operators calledcharges . For instance, consider a particle in three-dimen- sional space that we rotate about the Zaxis by an angle θ. Acting on the Hilbert space, this symmetry transformation isrepresented by a unitary U θthat is generated by the Z component of the Hermitian angular momentum operator Jz, i.e.,Uθ¼e−iJzθ. In general, we say that a quantum operation is“covariant ”with respect to a group Gif the operation commutes with the group action. Specifically, a unitarytransformation Vis covariant with respect to rotations about theZaxis if VandU θcommute up to a phase factor: VU θV−1U−1 θ¼eiαθ1, where αis a real number. If α¼0, this covariance property implies that Vconserves the physical quantity Jz; if the initial state jψiis an eigenstate ofJzwith eigenvalue m, then the transformed state Vjψi must also be an eigenstate of Jzwith the same eigenvalue. For nonzero α,Vadditively shifts all eigenvalues by α, preserving the difference of any two eigenvalues. When we say that a quantum code is covariant with respect to the group G, we mean that its encoding map is a G-covariant operation. More specifically, this map embeds a“logical ”code space in the tensor product of nphysical subsystems, and, as indicated in Fig. 1, a unitary group transformation acting on the code space can be realized as a tensor product of nunitary transformations,

Question 178 multiple-choice

The Hidden Subgroup Problem (HSP) plays a pivotal role in quantum computation, serving as the framework for quantum algorithms that outperform classical approaches in factoring, discrete logarithms, and related problems. Classifying the computational tractability of HSP across various algebraic structures has significant implications for both algorithm design and cryptography. Which of the following statements accurately describes the current state of quantum algorithms for the Hidden Subgroup Problem in different algebraic contexts? 1) Quantum computers have polynomial-time algorithms for HSP in all finite groups, regardless of their structure. 2) The generalized HSP on universal algebras is always classically tractable and never yields quantum speedup. 3) Quantum algorithms provide super-polynomial speedup for HSP only in the abelian case and never for other algebraic structures. 4) The abelian case of HSP is classically intractable and quantum intractable. 5) Quantum solutions exist for the general non-abelian case of HSP in polynomial time. 6) Quantum computers efficiently solve HSP in abelian groups, but no general polynomial-time quantum solution for non-abelian groups is known; moreover, certain universal algebras built from powers of 2-element systems allow super-polynomial quantum speedup. 7) Classical algorithms outperform quantum algorithms for HSP on powers of 2-element algebras in all cases.

✓ Correct Answer:

The correct answer is 6) Quantum computers efficiently solve HSP in abelian groups, but no general polynomial-time quantum solution for non-abelian groups is known; moreover, certain universal algebras built from powers of 2-element systems allow super-polynomial quantum speedup..

📚 Reference Text:

Question 179 multiple-choice

Quantum information theory leverages mathematical constructs like tensor networks and operator decompositions to analyze and simulate quantum circuits. Efficient classical simulation of certain circuits often depends on their entanglement properties and how well they can be represented using structures such as Matrix Product Operators (MPOs). Which property of the Quantum Fourier Transform (QFT) circuit most directly enables its efficient classical simulation via tensor networks like MPOs? 1) The low entanglement across circuit partitions, resulting in rapid decay of Schmidt coefficients 2) The presence of nonlocal operations in its gate sequence 3) The use of maximally entangled auxiliary systems in its implementation 4) The inherently exponential scaling of von Neumann entropy with qubit number 5) The requirement for large bond dimension in tensor network representation 6) The inability to compress its structure into an MPO with controlled error 7) The necessity to utilize classical FFT algorithms for simulation

✓ Correct Answer:

The correct answer is 1) The low entanglement across circuit partitions, resulting in rapid decay of Schmidt coefficients.

📚 Reference Text:

with the number of qubits, i.e., /Delta1Sj=/vextendsingle/vextendsingleSj(QP n|ψ/angbracketright)−Sj(|ψ/angbracketright)/vextendsingle/vextendsingle≤O(1), (37) where Sjis the von Neumann entropy calculated at parti- tion j. See Appendix Efor details. While this result does not directly translate to Schmidt coeﬃcients of QP n(or those of Qn), we note the oper- ator Schmidt decomposition can be reinterpreted as the Schmidt decomposition of the state QP n(|α/angbracketright⊗|β/angbracketright), where |α/angbracketrightis a maximally entangled state on system Aand an auxiliary RAwith dim (RA)=dim(A),a n d|β/angbracketrightis a max- imally entangled state on system Band an auxiliary RB with dim (RB)=dim(B). Therefore, the Schmidt coeﬃ- cients of QP narise from setting |ψ/angbracketright=|α/angbracketright⊗|β/angbracketrightin Eq. (37), which indicates that they must decay fast enough so they converge under the function Sjand thus /Delta1Sis a constant. Indeed, Theorem 1 is a stronger statement, which showsthat the Schmidt coeﬃcients decay exponentially fast, so /Delta1Sis not only bounded by a constant but also is extremely small. III. MATRIX PRODUCT OPERATOR FORM OF THE QFT The low-entanglement property of the QFT naturally raises the question of whether it is eﬃcient to simulate it classically. To answer this, we consider the tensor network simulations of the QFT in this section. We brieﬂy review the deﬁnition of tensor networks, tensor diagrams, and the MPO. Then we introduce a tensor network representation of the QFT’s circuit, and we show how to compress it into a bond-dimension- χMPO with O(ne−χlog(χ/3)/√χ)error inO(χ3n2)time. In Sec. IV, we apply this MPO in the classical setting and compare it with the FFT. A. Tensor networks and the matrix product operators We brieﬂy review the deﬁnition of tensor networks, ten- sor diagrams, and the MPO. In our context, a tensor is a d-dimensional array of complex numbers, with dbeing the order of the tensor or its number of indices. The contrac- tionof a set of tensors is the sum of all the possible values of indices shared by tensors; for example, Aγ α=/summationdisplay βBα βCγ β (38) represents contracting tensors Band C, where α,β,a n dγ are index variables and αandγareopen indices . Indices are put as either superscripts or subscripts, and in our con- text, both serve the same meaning. A tensor network is then a network of tensors being contracted in certain ways, with some (or none) of the indices left open. A tensor diagram is the diagrammatic representation of a tensor net- work following two basic rules: ﬁrstly, tensors are notated by shapes (e.g., squares, circles, or triangles), and tensor indices are notated by lines emanating from these shapes; secondly, connecting two index lines implies a contraction. For example, Eq. (38) can be represented by the diagram (39) The diagram represents the same tensor network if shapes are rearranged or rotated and lines are bent, as long ascontracted indices remain the same. Additionally, in our context, crossing lines do not intersect, i.e., one line can be thought of as out of the plane. To demonstrate the above points, the following two tensor diagrams are considered 040318-7 JIELUN CHEN, STOUDENMIRE, and WHITE PRX QUANTUM 4,040318

Question 180 multiple-choice

Quantum algorithms often rely on efficient techniques for simulating the evolution of physical systems, with Hamiltonian simulation playing a central role in applications ranging from quantum chemistry to control theory. Recent advancements leverage Fourier transform methods to improve both resource usage and algorithmic accessibility. Which innovation enables a significant reduction in quantum resources and circuit depth for differential equation algorithms by simplifying kernel function construction in Hamiltonian simulation? 1) Restricting kernel function requirements to the real axis using a Fourier transform-based formalism 2) Employing Chebyshev polynomial approximations for Hamiltonian evolution 3) Utilizing ancillary qubits to encode time-dependent potentials 4) Applying quantum phase estimation for unstable dynamics simulation 5) Incorporating variational principles into kernel design 6) Adopting Trotter-Suzuki decompositions for kernel optimization 7) Leveraging classical post-processing to reduce circuit depth

✓ Correct Answer:

The correct answer is 1) Restricting kernel function requirements to the real axis using a Fourier transform-based formalism.

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Question 181 multiple-choice

In quantum information theory, the study of symmetries and group representations is fundamental for understanding how quantum systems transform. Lie groups and their associated Lie algebras play a crucial role in modeling continuous symmetries, such as those found in spin systems and quantum operations. Which group is represented by 2x2 unitary matrices with determinant 1 and is fundamental in describing spin and qubit symmetries in quantum mechanics? 1) SU(2) 2) Z2 3) U(1) 4) GL(2, R) 5) SO(2) 6) SL(2, R) 7) O(2)

✓ Correct Answer:

The correct answer is 1) SU(2).

📚 Reference Text:

the Lie algebra as the unitary algebra su(2), and the Lie group as SU(2). (Note that the previous result also allows us to identify the group G=fU kjU2SU(2)gwithSU(2), for any value of k.) VI. REPRESENTATION THEORY FOR DISCRETE AND CONTINUOUS GROUPS In the previous sections we have taken a constructive approach where we started with a given dataset S= fi;yigN i=1, identiﬁed the symmetries that leave the labels invariant according to Deﬁnition 1, and found their ab- stract counterparts. This led us to prove that both the groupGb ipof the bit-ﬂip symmetry transformation and 8 the groupGSWAPof two qubit permutations have the ab- stract structure of Z2. Hence, at this point, we are ready to say that the bit-ﬂip and the qubit-swapping groups are bothrepresentations ofZ2. Similarly, we can say that Gprodfrom Eq. (11) is a representation ofSU(2). Representationtheorystudieshowgroupscanactonvec- tor spaces through linear transformations. In this section and those to come, we formally ﬂesh out the mathemati- cal basics of representation theory and Lie theory, provid- ing a rigorous footing for the examples earlier in the text and in Box 11 and Box 12. This is a standard topic with many excellent texts, so we opt instead here to instruct through examples rather than developing the theory com- pletely. Especially important examples will appear repeat- edly throughout the text, so we have added these in color- coded boxes to aid the reader. In writing this, we primarily reference Hall [51], periodically deferring to Serre [54], Ful- ton and Harris [53], and Nachtergaele and Sims [58]. We warn that most quantum scientists who come in close con- tact with Lie theory ﬁnd themselves inevitably enamored by the subject, and we encourage them to use these won- derful texts to explore further. In the following sections we will present some key mathe- maticaldeﬁnitionsthatwillbeusedthroughoutthisarticle. A. Groups: discrete and continuous First, we deﬁne a group. Deﬁnition 2. Agroupis a setGwith a binary operation :GG!Gobeying the following axioms: 1. Associativity: For all g;h;k2G,(gh)k=g(hk). 2. Identity element: There exists an identity element 12Gsuch that for every g2G,1g=g=g1. 3. Inverse element: For all g2G, there exists a g1 such thatgg1= 1 =g1g. We will commonly write gh=:gh. If for all g;h2G we have that gh=hg, we callGabelian, and if this does not hold, we call Gnonabelian . At this point, we refer the reader to Box 13 places at the end of this work, where we have listed some commonly appearing discrete groups and their key properties. Here we also note that there is a stark diﬀerence in theory be- tween ﬁnite and inﬁnite discrete groups: most of the rep- resentation theory we will develop will only apply to ﬁnite discrete groups. That being said, Deﬁnition 2 also includes continuous groups as well: for instance the general linear groupGL(n;C)(resp.GL(n;R)) of invertible nncom- plex (resp. real) matrices forms a group under the matrixmultiplication operation. Here, the identity matrix 1is the identity element. With the previous, we now formally deﬁne what a con- tinuousmatrix group is. Recall that a

Question 182 multiple-choice

Quantum phase estimation is a cornerstone algorithm in quantum computing, enabling access to spectral information and powering applications in simulation, machine learning, and statistical inference. Recent developments focus on improving phase estimation in coherent scenarios, where circuit complexity and resource demands are critical concerns. Which technique is specifically employed in recent coherent phase estimation algorithms to replace the Quantum Fourier Transform and enable bitwise estimation while reducing ancillary qubit requirements? 1) Grover's search 2) Hamiltonian simulation 3) Quantum walk operators 4) Variational quantum eigensolver 5) Quantum amplitude amplification 6) Block-encoding and singular value transformation 7) Quantum principal component analysis

✓ Correct Answer:

The correct answer is 6) Block-encoding and singular value transformation.

📚 Reference Text:

Faster Coherent Quantum Algorithms for Phase, Energy, and Amplitude Estimation Patrick Rall Quantum Information Center, University of Texas at Austin Weconsiderperformingphaseestimationunderthefollowingconditions: we are given only one copy of the input state, the input state does not have to be an eigenstate of the unitary, and the state must not be measured. Most quan- tum estimation algorithms make assumptions that make them unsuitable for this ‘coherent’ setting, leaving only the textbook approach. We present novel algorithms for phase, energy, and amplitude estimation that are both concep- tually and computationally simpler than the textbook method, featuring both a smaller query complexity and ancilla footprint. They do not require a quan- tum Fourier transform, and they do not require a quantum sorting network to compute the median of several estimates. Instead, they use block-encoding techniques to compute the estimate one bit at a time, performing all ampliﬁca- tion via singular value transformation. These improved subroutines accelerate the performance of quantum Metropolis sampling and quantum Bayesian in- ference. Phase estimation is one of the most widely used quantum subroutines, because it grants quantum computers two unique capabilities. First, it yields a quadratic speedup in the accuracy of Monte Carlo estimates [BHMT00, Mon15, HM18]. Achieving ‘Heisenberg lim- ited’ accuracy scaling has numerous applications in physics, chemistry, machine learning, and ﬁnance [Wright&20, ESP20, Rall20, An&20]. Second, it allows quantum computers to diagonalize unitaries in a certain restricted sense: if U=/summationtext je2πiλj|ψj/angbracketright/angbracketleftψj|then phase estimation performs the transformation /summationdisplay jαj|0n/angbracketright|ψj/angbracketright→/summationdisplay jαj|λj/angbracketright|ψj/angbracketright (1) where|λj/angbracketrightis ann-bit estimate of λj. This access to spectral information enables quantum speedupsforlinearalgebra[HHL08,CKS15], studyingphysicalsystems[Temme&09,YA11, Lemi&19, JKKA20], estimating partition functions [Mon15], and performing Bayesian in- ference [HW19, AHNTW20]. Phase estimation is also a very complicated algorithm. Textbook phase estimation [NC00] requires the Quantum Fourier Transform (QFT), a tool we commonly associate withtheexponentialspeedupofShor’salgorithm[Shor95]. However, phaseestimationonly delivers a quadratic speedup for estimation and the exponential speedup for linear algebra can sidestep the QFT [CKS15, GSLW18]. When applied to energies, phase estimation also requires Hamiltonian simulation, a quantum subroutine that is an entire subject of study in its own right: it requires recent innovations to apply optimally in a black-box setting [BCK15, LC1606, LC1610, LC17, GSLW18] and optimal Hamiltonian simulation for spe- ciﬁc systems is still being actively studied [SHC20, Cam20]. Furthermore, phase estimation Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 1arXiv:2103.09717v3 [quant-ph] 14 Oct 2021 demands median ampliﬁcation to guarantee an accurate answer. This can be challenging to implement coherently on a quantum computer [HNS01, Klauck02, Beals&12], because it requires many ancillae and a quantum sorting network. The probability with which phase estimation gives the correct answer (sometimes referred to as the ‘Fejer kernel’ [Rogg20]) is not always high enough to be ampliﬁed, which must be dealt with either by rounding or adaptivity [AA16]. The conceptual complexity of textbook phase estimation and the resulting computational overhead motivates a search for alternatives. Fortunately, a lot of simpliﬁcation is possible if we allow for ‘incoherent’ algorithms, where incoherence manifests via a variety of assumptions. We could be

Question 183 multiple-choice

In quantum information theory, universality refers to the ability of a set of quantum gates to approximate any unitary operation on a finite-dimensional Hilbert space. The structure of unitary groups and their subgroups underpins the requirements for universal quantum computation. Which of the following statements accurately describes the relationship between infinite closed subgroups of U(d) that form a continuous 2-design and universality in quantum computation for d ≥ 2? 1) Such subgroups generate only classical operations and cannot achieve universality. 2) They can form non-universal gate sets that approximate any state up to the first moment only. 3) These subgroups necessarily exclude all nontrivial elements outside SU(d). 4) They may fail to generate universal gate sets if they lack entangling gates. 5) These subgroups allow exact implementation of only diagonal unitary operations. 6) They can form 2-designs without being universal if restricted to real orthogonal matrices. 7) Any infinite closed subgroup of U(d) that forms a continuous 2-design must be universal, necessarily containing SU(d).

✓ Correct Answer:

The correct answer is 7) Any infinite closed subgroup of U(d) that forms a continuous 2-design must be universal, necessarily containing SU(d)..

📚 Reference Text:

gates. As we will see in Theorem 8, linear optics assisted by nearly any other unitary is sufficient to generate a universal gate set in U(H). In particular, we will show in Theorem 9 that linear optics with any nontrivial SNAP gate is sufficient for universality. We briefly remark on our choice of the SNAP gate as the “resourceful” gate on top of linear optical unitaries. For the single mode case this shows up naturally in cavity-QED systems where a cavity dispersively coupled to a qubit is used to perform quantum computation [32]. It was shown here that displacements on the infinite-dimensional Hilbert space of a single mode bosonic system along with SNAP gates is sufficient to perform universal quantum computation on any d-dimensional subspace of the bosonic mode. Namely, one can realize universal control of a single qudit via this approach. Our use of the SNAP gates is inspired by this result. For0≤k≤n,0≤s≤m−1, and θ∈R, define the SNAP gate Sk,s(θ) = exp( iθ|k⟩⟨k|s), (27) where the subscript indicates that the action is on the sth mode. Note that the SNAP gates are diagonal in the Fock basis. Of course, if θ≡0 mod 2 π, then Sk,s(0)is the identity operator. Note that if θ̸= 0 mod 2 π, then Sk,s(θ)̸∈Un,m. This is quite intuitive using the symmetric subspace perspective above. For example, recalling the case m=n= 2 from Remark 6, we see that S2,1(θ)has the following action: v0⊗v0=|2,0⟩ 7→eiθ|2,0⟩=eiθv0⊗v0 v1⊗v1=|0,2⟩ 7→ | 0,2⟩=v1⊗v1 1√ 2(v0⊗v1+v1⊗v0) =|1,1⟩ 7→ | 1,1⟩=1√ 2(v0⊗v1+v1⊗v0). This clearly cannot arise from an action of the form U⊗U. D. Result: Continuous 2-designs In this section, we observe that there are no non-universal infinite closed (i.e. continuous) 2-designs in U(V)ifdimV ≥2. This is essentially Theorem 1.1.6(2) in Ref. 7, and we closely follow the proof given there. This result was not stated or 17 proved in the context of unitary t-design theory, so for the sake of accessibility we choose to restate it here and prove it in Appendix B 3. The main idea of the proof is as follows. First, due to Proposition 3, G⊆U(V)is a2-design if and only if V ⊗ V decomposes into exactly 2irreducible representations for G. Second, following Ref. 7, one can prove that V ⊗V∗decomposes into exactly 2irreducible representations for Gif and only if SU(d)⊆G. (Note that V ⊗V∗∼=gl(V), which is very useful in the proof of this result.) To obtain the result, then, one needs only to show that V ⊗ V andV ⊗ V∗ decompose into the same number of irreducible G-representations. This is a much more general fact that holds whenever the finite-dimensional representations of Gare known to be completely reducible. Theorem 7. LetV∼=Cd, with d≥2. Let Gbe an infinite closed subgroup of U(d) =U(V). IfGis a unitary 2-design, thenSU(d)⊆G. In particular, any set of unitaries generating Gis a universal set. In the following section, we apply this result to observe that there are no non-universal extensions of linear optics to a 2-design. E. Results on linear optical t-designs

Question 184 multiple-choice

Quantum algorithms have significantly advanced the ability to efficiently solve group-theoretic problems, especially in settings where group elements are accessed through oracles and encoded as binary strings. The Hidden Subgroup Problem (HSP) is central to quantum computing, with solutions that extend to a variety of group structures relevant in cryptography and computational algebra. Which group property most directly enables efficient quantum algorithms for the Hidden Subgroup Problem by allowing quantum techniques to overcome classical algorithmic barriers related to factoring and membership testing? 1) Existence of a non-normal subgroup of large index 2) Presence of a simple non-Abelian factor group 3) A small commutator subgroup, making the group close to Abelian 4) The group being non-solvable and non-permutation 5) Absence of any cyclic subgroups 6) Lack of any elementary Abelian normal 2-subgroups 7) Group elements encoded as decimal integers

✓ Correct Answer:

The correct answer is 3) A small commutator subgroup, making the group close to Abelian.

📚 Reference Text:

will focus on the HSP, and we will show that it c an be solved in polynomial time in several black-box groups. In particular, we will pre sent eﬃcient quantum algorithms for this problem for groups with small commutator subgroup and f or groups having an elementary Abelian normal 2-subgroup of small index or with cyclic fact or group. Our basic ingredient will be a series of deep algorithmical results of Beals and Babai fro m classical computational group theory. Indeed, in [5] they have shown that, up to certain computatio nally diﬃcult subtasks – the so-called Abelian obstacles – such as factoring integers and construc tive membership test in Abelian groups many problems related to the structure of black-box groups, such as ﬁnding composition series, can be solved eﬃciently for groups without large composition fa ctors of Lie type, and in particular, for solvable groups. As quantum computers can factor integers a nd take discrete logarithms, and, more generally, perform the constructive membership test in Abe lian groups eﬃciently, one expects that a large part of the Beals–Babai algorithms can be eﬃciently i mplemented by quantum algorithms. 2 Indeed, the above results of Watrous partly fulﬁll this task , although his algorithms are not using the Beals–Babai algorithms. Here we will describe eﬃcient q uantum implementations of some of the Beals–Babai algorithms. It turns out, that beside pavin g the way for solving the HSP in the groups mentioned previously, these implementations give a lso almost “for free” eﬃcient solutions for ﬁnding hidden normalsubgroups in many cases, including solvable groups and perm utation groups. The rest of the paper is structured as follows. In Section 2 we review the necessary deﬁnitions about black-box groups in the quantum computing framework, and will summarize the most impor- tant results about Abelian and solvable groups. In Section 3 we state the result of Beals and Babai andCorollary 5 which makes explicit two hypotheses (disposability of orac les for order computing and for constructive membershiptest in elementary Abelian subgroups)underwhich thealgorithms have eﬃcient quantum implementations. Section 4 deals with these quantum implementations in the following cases: unique encoding ( Theorem 6 ), modulo a hidden normal subgroup ( Theorem 7) and modulo a normal subgroup given by generators in case of u nique encoding ( Theorem 10 ). As a consequence, we can derive the eﬃcient quantum solution for the normal HSP in solvable and permutation groups without any assumption on computability of noncommutative Fourier trans- forms(Theorem 8 ). Section 5 contains the eﬃcient algorithm for the HSP for gr oups with small commutator subgroup ( Theorem 11 ), and Section 6 for groups having an elementary Abelian normal 2-subgroup of small index or with cyclic factor group (Theorem 13 ). 2 Preliminaries In order to achieve suﬃciently general results we shall work in the context of black-box groups. We will suppose that the elements of the group Gare encoded by binary strings of length nfor some ﬁxed integer n, what we call the

Question 185 multiple-choice

In quantum information theory and quantum computing, understanding the properties of Hilbert spaces, norms, and efficient quantum circuit implementations is crucial for algorithm design and error analysis. Concepts such as Gram-Schmidt orthonormalization and measures of overlap between subspaces play a central role in studying distinguishability of quantum states. Which of the following statements accurately describes the effect of Gram-Schmidt orthonormalization on a set of n nearly orthogonal unit vectors in a Hilbert space with pairwise overlap δ? 1) Each orthonormalized vector will be exactly equal to its original vector if δ is nonzero. 2) The difference between each orthonormalized vector and its original is bounded by δ²·n for all δ. 3) Each orthonormalized vector remains close to its original, with the difference in norm bounded by a function proportional to δ√n. 4) Gram-Schmidt orthonormalization always creates vectors orthogonal to their originals, regardless of δ. 5) The maximum possible difference after orthonormalization is independent of the number of vectors n. 6) If δ is small, Gram-Schmidt orthonormalization causes the vectors to become nearly parallel. 7) The orthonormalized vectors coincide only if the original vectors are linearly dependent.

✓ Correct Answer:

The correct answer is 3) Each orthonormalized vector remains close to its original, with the difference in norm bounded by a function proportional to δ√n..

📚 Reference Text:

t∈TΦ(t), this means that we have to implement the following transformation: Λ :|x/an}bracketri}ht|a/an}bracketri}ht|b/an}bracketri}ht /ma√sto→/braceleftbigg|x/an}bracketri}ht|a/an}bracketri}ht|b−ax/an}bracketri}ht:ifa/ne}ationslash= 0, ωxb p|x/an}bracketri}ht|0/an}bracketri}ht|b/an}bracketri}ht:ifa= 0. Itisstraightforwardtoimplement Λusingclassicalefﬁcientcircuitsformodularadditionand multiplica- tion. Hence Λcanbeimplementedusing O(log3(p))quantumgates. Notethat QFTN∞= QFTZp⊗QFTZp andQFTG/N∞= QFTZp, both of which can be either implemented approximately [Kit 95] or exactly [MZ04]onaquantum computer using O(log2p)manyelementary quantum gates. Hence theclaimed com- plexity for computing aquantum Fourier transform for Hpfollows. 3 Random basesandFourier sampling 3.1 Nearly orthogonal vectors In this subsection, we state some results about sets of nearl y orthogonal unit vectors in a Hilbert space. We use/bardbl·/bardblto denote the ℓ2-norm of vectors as well as the ℓ2-induced operator norm of matrices. We use /bardblv/bardbl1 to denote the ℓ1-norm of a vector v. We let/bardblM/bardbltr= Tr√ M†Mdenote the trace norm of a matrix M. For subspacesV1,V2having trivial intersection, their overlapis deﬁned as ovlap(V1,V2) = max v1,v2|/an}bracketle{tv1|v2/an}bracketri}ht|, wherevirange over unit vectors in Vi. LetΠidenote the orthogonal projection operator onto Vi. It is easy tosee that ovlap(V1,V2) =/bardblΠ1Π2/bardbl. Proposition 2 LetV1,V2be subspaces of a Hilbert space having trivial intersection . Letσ2denote the totally mixed state in V2. LetV′ 2denote the orthogonal complement of V1inV1+V2andσ′ 2denote the totally mixed state in V′ 2. Letδ= ovlap(V1,V2). Then, /bardblσ2−σ′ 2/bardbltr≤2δ1/2(1−δ2)−1/4. Proof: Letd= dimV2anda1,...,adbeanorthonormal basis for V2. Leta′ 1,...,a′ dbetheGram-Schmidt orthonormalisation of a1,...,a dwith respect to V1. Hence,a′ 1,...,a′ dis an orthonormal basis for V′ 2. We will show that /bardbl|ai/an}bracketri}ht/an}bracketle{tai|−|a′ i/an}bracketri}ht/an}bracketle{ta′ i|/bardbltr≤2δ1/2(1−δ2)−1/4for all1≤i≤d. Sinceσ2=1 d/summationtextd i=1|ai/an}bracketri}ht/an}bracketle{tai| andσ′ 2=1 d/summationtextd i=1|a′ i/an}bracketri}ht/an}bracketle{ta′ i|, wewill get /bardblσ2−σ′ 2/bardbltr≤1 dd/summationdisplay i=1/bardbl|ai/an}bracketri}ht/an}bracketle{tai|−|a′ i/an}bracketri}ht/an}bracketle{ta′ i|/bardbltr≤2δ1/2(1−δ2)−1/4. 9 Fix somei,1≤i≤d. Letbi+cidenote the orthogonal projection of aionto the space spanned by V1 anda1,...,a i−1,wherebi∈V1andci∈span{a1,...,ai−1} ⊆V2. Then, 1≥ /bardblbi+ci/bardbl2≥ /bardblbi/bardbl2+/bardblci/bardbl2−2|/an}bracketle{tbi|ci/an}bracketri}ht| ≥ /bardblbi/bardbl2+/bardblci/bardbl2−2δ/bardblbi/bardbl/bardblci/bardbl = (1−δ2)/bardblbi/bardbl2+(δ/bardblbi/bardbl−/bardblci/bardbl)2 ≥(1−δ2)/bardblbi/bardbl2, i.e./bardblbi/bardbl ≤1√ 1−δ2. Now, /bardblbi+ci/bardbl2=/an}bracketle{tai|bi+ci/an}bracketri}ht=/an}bracketle{tai|bi/an}bracketri}ht+/an}bracketle{tai|ci/an}bracketri}ht=/an}bracketle{tai|bi/an}bracketri}ht ≤δ√ 1−δ2, i.e./bardblbi+ci/bardbl ≤δ1/2(1−δ2)−1/4. The third equality above follows from the fact that a1,...,a i−1,aiare pairwise orthogonal. Now /an}bracketle{tai|a′ i/an}bracketri}ht=/bardblai−bi−ci/bardbl=/radicalbig 1−/bardblbi+ci/bardbl2, and hence, /bardbl|ai/an}bracketri}ht/an}bracketle{tai|−|a′ i/an}bracketri}ht/an}bracketle{ta′ i|/bardbltr= 2/radicalBig 1−|/an}bracketle{tai|a′ i/an}bracketri}ht|2= 2/bardblbi+ci/bardbl ≤2δ1/2(1−δ2)−1/4. Thiscompletes the proof of the proposition. Proposition 3 Letv′ 1,...,v′ nbeunit vectors inaHilbert space. Let 0≤δ<1 2n. Suppose forall i,j,i/ne}ationslash=j, |/an}bracketle{tvi|vj/an}bracketri}ht| ≤δ. Letv1,...,vnbe unit vectors obtained by Gram-Schmidt orthonormalising v′ 1,...,v′ n. Then for anyi,1≤i≤n, /bardbl|vi/an}bracketri}ht/an}bracketle{tvi|−|v′ i/an}bracketri}ht/an}bracketle{tv′ i|/bardbltr<2√ 6·δ√n. Proof: Fix some i,1≤i < n. Letai+1=/summationtexti j=1αjv′ jbe the orthogonal projection of v′ i+1onto the subspace spanned by v′ 1,...,v′ i. Then for all k,1≤k≤i,/an}bracketle{tv′ k|v′ i+1−ai+1/an}bracketri}ht= 0i.e./an}bracketle{tv′ k|v′ i+1/an}bracketri}ht=/summationtexti j=1αj/an}bracketle{tv′ k|v′ j/an}bracketri}ht. Supposek,1≤k≤iis such that |αk|= max j:1≤j≤i|αj|. Then, δ≥ |/an}bracketle{tv′ k|v′ i+1/an}bracketri}ht| ≥ |αk||/an}bracketle{tv′ k|v′ k/an}bracketri}ht|−/summationdisplay j:1≤j≤i j/negationslash=k|αj||/an}bracketle{tv′ k|v′ j/an}bracketri}ht| ≥ |αk|−/summationdisplay j:1≤j≤i j/negationslash=k|αk||/an}bracketle{tv′ k|v′ j/an}bracketri}ht| ≥ |αk|(1−(i−1)δ) >|αk|·1 2, i.e.maxj:1≤j≤i|αj|<2δ. Now, /bardblai+1/bardbl2≤i/summationdisplay j=1|αj|2/bardblvj/bardbl2+/summationdisplay j,j′:1≤j,j′≤i j/negationslash=j′|αj||αj′||/an}bracketle{tvj|vj′/an}bracketri}ht| <4δ2n+4δ3n2 <4δ2n+2δ2n = 6δ2n. 10 Reasoning asat the end of theproof of Proposition 2,weget /bardbl|vi+1/an}bracketri}ht/an}bracketle{tvi+1|−|v′ i+1/an}bracketri}ht/an}bracketle{tv′ i+1|/bardbltr= 2/radicalBig 1−|/an}bracketle{tvi+1|v′ i+1/an}bracketri}ht|2 = 2/radicalBig 1−/bardblv′ i+1−ai+1/bardbl2 = 2/radicalbig 1−(1−/bardblai+1/bardbl2) = 2/bardblai+1/bardbl <2√ 6·δ√n. Thiscompletes the proof of theproposition. 3.2 Random orthonormal vectors Inthissubsection, westatesomefactsaboutrandomorthono rmal setsofvectorsin Cd. Onewayofgenerat- ing arandom unit vector in Cdis asfollows: Consider (y1,...,y2d)∈R2d,where each yiis independently chosen according to the one dimensional Gaussian distribut ion with mean 0and variance 1(i.e.yiis a real valued random variable with probability density funct ion1√ 2πexp(−y2/2)). Normalise to

Question 186 multiple-choice

In quantum computing, finding hidden structures such as symmetries or algebraic relations often relies on formulating problems in terms of group actions and partitions. The Hidden Symmetry Subgroup Problem (HSSP) extends these frameworks, allowing researchers to address broader classes of computational challenges. Which statement most accurately describes a critical distinction between the Hidden Subgroup Problem (HSP) and the Hidden Symmetry Subgroup Problem (HSSP) in the context of quantum query complexity and group actions? 1) HSP is always harder than HSSP for all group actions. 2) HSSP is restricted to abelian groups, whereas HSP is not. 3) HSSP always admits an efficient quantum algorithm regardless of group structure. 4) Certain instances of HSSP, such as Grover’s search, have exponential quantum query complexity, while HSP generally has polynomial complexity for similar group actions. 5) HSP and HSSP are equivalent in all computational scenarios. 6) HSSP is only defined for groups acting trivially on sets. 7) Quantum algorithms for HSSP do not benefit from reductions involving generalized bases.

✓ Correct Answer:

The correct answer is 4) Certain instances of HSSP, such as Grover’s search, have exponential quantum query complexity, while HSP generally has polynomial complexity for similar group actions..

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group Gacting on some ﬁnite set M, and we suppose that we have at our disposal a black-box function whose level sets deﬁne a partition of M.T h e object we would like to recover is the group of symmetries of this partition inside G, i.e., the largest subgroup whose orbits under the action coincide with the classes of the partition. We call this problem the hidden symmetry subgroup problem ( HSSP). It is easy to see that the HSPis a special case of the HSSPwhen the group acts on itself and the action corresponds to the group operation. But, for some actions, theHSSPis provably harder than any HSP. We show that Grover’s search can be cast as an HSSP, establishing that certain cases of the HSSPhave exponential quantum query complexity. This is in contrast to the HSP, which has polynomial quantum query complexity for all groups [16]. The potential of the HSSPlies mainly in the possibility of extending the HSP techniquestomoregeneralgroupactionsthatstilladmiteﬃcientquantumprocedures. We demonstrate the power ofthis new approachby designing and improvingquantum algorithms for several algebraic problems. To achieve this we reduce both the HQPP and the univariate HPGPto appropriate HSSPs for which we can give eﬃcient HIDDEN SYMMETRY SUBGROUP PROBLEMS 1989 quantum solutions in some interesting cases. Besides the construction of eﬃcient algorithms, the formulation of problems as HSSPcan also shed new light on their structure. For example, the apparent diﬃculty of the HQPPover prime ﬁelds might be explained by the equivalence of this problem to the HSPin the dihedral group, a connection discovered via their relations to the HSSP. It is also worth noting that the hidden shifted multiplicative character problem of van Dam, Hallgren, and Ip [17] is a version of the HSSPwith an additional promise on the input. To establish our algorithmic results, we ﬁrst concentrate on the question of whether the HSSPcan be reduced in some cases to the related HSPthat we ob- tain by forgetting about the action. We design a reduction scheme, which involves the generalization of bases known from the theory of permutation groups. We are able to show that when the action has an eﬃ ciently computable generalized base then theHSSPis indeed eﬃciently reducible to the related HSP(Proposition 3.4). Then we describe a probabilistic construction of such bases for a large class of Frobenius groups. Therefore, the above reduction applies to these groups (Theorem 4.4). These groups include among others a large variety of aﬃne groups, and the HSSPis eﬃ- ciently solvable for these groups by a quantum algorithm. We remark that in [18] it isproved that the HSSP(in a slightly diﬀerent formulation) can be solved eﬃciently for some of these aﬃne groups. The proof uses essentially the same reduction technique. Wethen establishseveralsurprisingco nnectionsbetweenhidden polynomialprob- lems and the HSSP. In fact, the HQPPturns out to be equivalent in a very strong sense to the HSSPover a related aﬃne group. Combined with the above reduction to the related HSP, we are able to give the ﬁrst ever quantum polynomial time solution for

Question 187 multiple-choice

In mathematical physics and quantum information theory, labeling basis states of irreducible representations (irreps) of the special unitary group SU(d) is essential for distinguishing states in symmetric quantum systems, especially when permutation symmetry is present. Sophisticated methods involving subgroup structure and group-theoretic operators are used to achieve unique labeling at higher dimensions. Which combination of quantum numbers and operators provides a complete and unique labeling of basis states for SU(d) irreducible representations in systems with permutation symmetry? 1) Only the weight components μ1 through μd−1 2) The partition labels μ plus the eigenvalues of SU(2) Cartan generators 3) Hypercharge and isospin quantum numbers for all d 4) The branching path αT and total spin-z operator 5) The set of μ1,..,μd−1 together with the eigenvalues of Casimir operators for nested SU(k) subgroups 6) The eigenvalues of the SU(d) Casimir operator alone 7) The coordinates of the weight diagram in the fundamental representation

✓ Correct Answer:

The correct answer is 5) The set of μ1,..,μd−1 together with the eigenvalues of Casimir operators for nested SU(k) subgroups.

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may wish to distinguish these isomorphic copies by permutation modules like Fig. 2(c). However, as mentioned in Sec. II, when d≥3 isomorphic copies of Snirreps can even be found from the same per- mutation module Mμ[50,59] and hence the superscript μ is no longer enough to identify μS. Interestingly, this problem can be solved in the SU (d) picture, in which αTtells us exactly the path to cou- ple SU (d)irreps sequentially. Our ﬁnal destination Wλis uniquely determined by the path, but we now need to label |αT;μS/angbracketrightas a state inWλ. When d=2,μSis simply taken as the spin- zcomponent m. When d≥3, we use classical results by Gelfand and Tsetlin [ 76] and Biedenharn [ 77]. We illustrate the main idea for d=3: consider weight dia- grams of SU (3)irreps on Fig 5. Each dot in a diagram stands for a basis vector of the irrep [ 49,78]. With αT being determined by YJM elements, we need only to iden- tify which dot |αT;μS/angbracketrightcorresponds to the weight diagram ofWλ. Diagrammatically, these dots have planar coordi- nates, which are rigorously called weights measured by the isospin I3and hypercharge Yoperator of SU (3)[49,78] just like measuring spin components by Szof SU(2). However, being diﬀerent from the SU (2)case, some weight vec- tors (dots) occupy the same positions. It is known as the branching rule for SU (3)that dots with the same horizon- tal coordinate form irreps of SU (2)⊂SU(3). For instance, two brown dots in Fig. 5form a spin-1 /2 irrep while the other four red dots from the same horizontal line form a spin-3 /2 irrep. Thus after measuring weights and posi- tions by I3,Y, we simply need to apply the SU (2)Casimir operator J2to discern dots occupying the same position. In Ref. [ 77], authors provide the recipe to ﬁnd these operators for a general SU (d)group. Roughly speaking, we FIG. 5. Typical weight diagrams of SU (3)irreps. Some weight vectors (dots) may occupy the same positions. 020327-9 HAN ZHENG et al. PRX QUANTUM 4,020327 (2023) ﬁrst employ d−1 operators μi, which span its Cartan sub- algebra, to label the weights of a given basis vector. Then we take Casimir operators of SU (2)⊂···⊂ SU(d−1) to distinguish dots that occupy the same position. Their eigenvalues, which we record as μS, form the so-called Gelfand-Tsetlin pattern [76], which is widely used to study SU(d)irreps. To construct a CQA model with Snsymme- try, we replace the YJM elements labeling any Sλbasis states by the the following ones labeling any Wλbasis states: μ1,...,μd−1,CSU(d−1) 1 ,...,CSU(d−1) d−2,...,CSU(2) 1=J2. The number of required operators to build a SU (d)-CQA ansatz is1 2d(d−1)—a constant for ﬁxed d, no matter how many qudits the system contains. However, Casimir operators of SU (d−1)are supported on d−1 qudits [see Eq.(4)and [ 78]]. Applying the same proof method from Theorem 1, SU (d)-CQA is thus made of 2 (d−1)-local Sn-symmetric unitaries. As a simple example, SU (2)-irrep basis states are uniquely determined by the summation Sn zof spin operator1 2Zion

Question 188 multiple-choice

Hidden sector models of dark matter propose new particles that interact through forces separate from the Standard Model, affecting cosmic structure formation and early universe evolution. The relic abundance and cosmological impact of such particles are tightly constrained by observational data. Which of the following best explains why relic hidden gluinos are restricted to contribute only a small fraction of the total dark matter density in the universe? 1) Their rapid decay into Standard Model particles leads to strong indirect detection constraints. 2) They possess electric charge, causing electromagnetic interactions that disrupt large-scale structure. 3) Their annihilation cross-section is too large, resulting in negligible relic abundance. 4) Their production mechanism requires extremely high reheating temperatures that are cosmologically disfavored. 5) They couple to visible photons, leaving distinctive imprints on the cosmic microwave background. 6) Their mass is significantly below the weak scale, making them incompatible with observed galactic dynamics. 7) Their strong kinetic coupling to a hidden gluon bath generates pressure that prevents clustering and suppresses structure formation, leading to stringent cosmological limits.

✓ Correct Answer:

The correct answer is 7) Their strong kinetic coupling to a hidden gluon bath generates pressure that prevents clustering and suppresses structure formation, leading to stringent cosmological limits..

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in the ground state [149{151], is much smaller than the Hubble rate at temperatures below the binding energy. Together, these two results imply that the relic gluinos remain kinetically coupled to the gluon bath. The pressure induced by the gluons will drive gluinos out of overdense regions and interfere with structure formation, analogous to the photon pressure felt by baryons before recombination. This is very di erent from the behaviour of standard collisionless cold dark matter, and implies the hidden gluinos can only be a small fraction of the total dark matter density. This fraction, can be constrained using observations of the CMB and galaxy surveys. A study along these lines was performed in Ref. [152], and their results suggest that the fraction fx= eGx= cdmmust be less than a few percent, depending on the temperature ratioTx=T'(cx=cv)1=4.14Hidden gluino interactions may also modify the distribution of dark matter on galactic scales [155]. E. Summary This supersymmetric hidden SU(N)xextension can produce a much smaller non-thermal LSP relic density than the MSSM, and has only invisible annihilation modes that are not constrained by indirect detection. However, the hidden gluino LSP remains in thermal contact with a bath of hidden gluons, and thus can only make up at most a few percent of the total dark matter density. Obtaining such small relic densities is non-trivial and leads to new challenges, as we will discuss here. From Fig. 6 we see that reducing the gauge coupling gxlowers the non-thermal hidden gluino density until TRHTfo, at which point the relic abundance becomes approximately constant in gx. At the same time, Eq. (56) shows that smaller values of gxalso suppress the decay rate of the lightest MSSM superpartner. If such decays happen after the onset on primordial nucleosynthesis, they can disrupt the abundances of light elements [156, 157]. The direct two-body decays 0 1!eGxGxare invisible. However, the operator of Eq. (55) also gives rise to the semi-visible three-body mode 0 1!h0eGxGxif it is kinematically allowed. The decay products of the Higgs boson will be signi cantly hadronic, and can 14A relic population of millicharged particles will have a similar e ect. This was considered in Refs. [153, 154], and a limit of fx.1% was obtained. 31 modify light-element abundances. The branching fraction of this three-body mode depends on the available phase space. Taking it to be Bh103and estimating the Higgsino yield as in Section III, we nd that Higgsino lifetimes below 0 1.1100 s are allowed [156]. This can occur for larger values of N,NF, oru, or smaller values of orF. Note that reducing Fbelowm'=2 is dangerous because it would lead to the production of stable massive F andPstates which would tend to overclose the Universe. An acceptable hidden gluino relic density with a suciently rapid MSSM decay can be obtained in this scenario, but only in a very restricted and optimistic region of parameters. For example, with rx= 3N=5,gx= 0:01,N= 2,NF= 3,u= 0:75,cx=cv= 1=9, and m'= 2m3=2= 2F= 100 TeV, we obtain eGx= cdm= 0:023 and

Question 189 multiple-choice

Quantum sensing leverages quantum algorithms and entanglement to achieve measurement precision beyond classical methods, particularly in spectroscopy and metrology. Techniques such as Quantum Phase Estimation Algorithm (QPEA) and Quantum Fourier Transform (QFT) have redefined limits for information acquisition and sensor performance. Which of the following statements best explains why implementing the Quantum Fourier Transform (QFT) in phase estimation protocols can reduce overall measurement time, despite its modest operational overhead? 1) QFT increases the coherence time of single-qubit sensors, allowing longer measurement intervals. 2) QFT eliminates all noise contributions from classical electronics, improving speed. 3) QFT enables classical signals to be digitized more rapidly by skipping quantum entanglement steps. 4) QFT allows measurement protocols to bypass the need for Fisher information analysis. 5) QFT optimizes the readout and correlation steps, reducing lengthy operations such as full register readout even though it adds a small time overhead. 6) QFT replaces adaptive phase estimation methods entirely, removing the need for multiple trials. 7) QFT guarantees perfect matching between theoretical and experimental results, which accelerates measurement cycles.

✓ Correct Answer:

The correct answer is 5) QFT optimizes the readout and correlation steps, reducing lengthy operations such as full register readout even though it adds a small time overhead..

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ADC. The dashed lines represent the canonical SNR =6.03 Nbits+C (db) of the classical ADC with noise and single (blue) and 16 trials of signal sampling. Solid read and green curves are quantum sensor with QPEA. Red - single trial of signal sampling, green —100 trials.V. Vorobyov et al. 5 Published in partnership with The University of New South Wales npj Quantum Information (2021) 124 Quantum limit in a number of used qubit resources (see SI S.I). In the same plot, we show the experimentally achieved values of the Fisher information for the case of a twelve-level register which we denote as nq=3.5. These values were estimated from the experiment shown in Fig. 2for the bare QFT†(Fig. 2b) and QFT† with sensing (Fig. 2e). While the experimental values show a slight deviation in the achievable information compared to the theoretically calculated case, they outperform the StandardQuantum Limit. DISCUSSION In our work we studied the application of QFT-based phase estimation algorithm to improve the performance of correlation measurement protocols by increasing its dynamic range for precision spectroscopic measurements. To categorize the applica-tion areas of such protocol we classify the sensing scenarios as following: B rmsγeT2>2πand BrmsγeT2≪2π. In the latter case, the sensing could not be further improved with phase estimationprotocols, since the signal can not saturate the full sensor rangeeven with single qubit register. In the ﬁrst case, when the single qubit sensor is saturated, the phase estimation algorithm enhances sensing performance of thesystem. We further divide the case of classical and quantum signal.With the classical signal, one can use both quantum and semicalssical phase estimation methods such as adaptive or semiclassical QFT, for better usage of available resourcescomprised of time and qubits. However, the case of a quantum signal is different. Though, there is work showing the possibility for single shot determination of quantum signals using adaptivesensing 28, it is subject for further research in application for correlation measurements in nanoscale NMR. The presented QFT- based approach, without intermediate state projection preserves entanglement throughout the protocol, and opens new ways tostudy the effects of quantum measurements in the saturatedsensor regime. In particular, it paves the way to new regimes of generalized measurements with multiqubit register and studies of quantum back-action. The result of Fig. 2e with sensing of the arti ﬁcialﬁeld show that the QFT brings additional time overhead on the order of 5% to the total operational time. It also brings additional complexity and necessity to have a nuclear spin register, in comparison to thereference 5, where a quantum phase estimation could be done on single spin. However, when it comes to the correlation protocol, the QFT allows to spare either correlation time between alliterations, or one readout time, thus ef ﬁciently reducing overall time. We note that addition of QFT adds roughly 300 μs compared to 15 ms of readout for full register or 8 ms of correlation time. The analysis of quantum digitizers requires further work, since various aspects of the quantum Shannon theorem like informationstorage, transfer

Question 190 multiple-choice

Quantum algorithms have profoundly impacted cryptography by efficiently solving problems believed to be hard for classical computers, such as factoring and discrete logarithms. The Hidden Subgroup Problem (HSP) serves as a foundational framework for these quantum breakthroughs and is actively researched in both discrete and continuous mathematical settings. Which of the following statements accurately describes a significant cryptanalytic implication of generalizing the Hidden Subgroup Problem (HSP) to finding full-rank lattices in continuous vector spaces? 1) It enables efficient quantum algorithms for classical block cipher analysis. 2) It allows quantum algorithms to break hash functions based on collision resistance. 3) It guarantees the security of elliptic curve cryptography against quantum attacks. 4) It provides efficient quantum algorithms for solving NP-complete problems in general. 5) It permits quantum factoring of arbitrary composite numbers regardless of size. 6) It leads to quantum algorithms capable of finding short vectors in ideal lattices, threatening lattice-based cryptography. 7) It results in quantum supremacy for all known cryptographic schemes.

✓ Correct Answer:

The correct answer is 6) It leads to quantum algorithms capable of finding short vectors in ideal lattices, threatening lattice-based cryptography..

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Title: On the Quantum Complexity of the Continuous Hidden Subgroup Problem Year: 2020 Paper ID: 75dcfbcab4333bbe991559f3dc41b93593c3d1e4 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/75dcfbcab4333bbe991559f3dc41b93593c3d1e4 Abstract: The Hidden Subgroup Problem (HSP) aims at capturing all problems that are susceptible to be solvable in quantum polynomial time following the blueprints of Shor’s celebrated algorithm. Successful solutions to this problems over various commutative groups allow to efficiently perform number-theoretic tasks such as factoring or finding discrete logarithms. The latest successful generalization (Eisenträger et al. STOC 2014) considers the problem of finding a full-rank lattice as the hidden subgroup of the continuous vector space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^m$$\end{document}Rm, even for large dimensions m. It unlocked new cryptanalytic algorithms (Biasse-Song SODA 2016, Cramer et al. EUROCRYPT 2016 and 2017), in particular to find mildly short vectors in ideal lattices. The cryptanalytic relevance of such a problem raises the question of a more refined and quantitative complexity analysis. In the light of the increasing physical difficulty of maintaining a large entanglement of qubits, the degree of concern may be different whether the above algorithm requires only linearly many qubits or a much larger polynomial amount of qubits. This is the question we start addressing with this work. We propose a detailed analysis of (a variation of) the aforementioned HSP algorithm, and conclude on its complexity as a function of all the relevant parameters. Our modular analysis is tailored to support the optimization of future specialization to cases of cryptanalytic interests. We suggest a few ideas in this direction.

Question 191 multiple-choice

Programmable quantum processors are specialized quantum devices that apply different quantum channels to inputs depending on the state of a program register. Limitations and possibilities for such processors are deeply influenced by quantum symmetry and representation theory. Which condition allows all group-covariant quantum channels to be exactly implemented using a finite-dimensional program register? 1) The symmetry group acts reducibly on the input space 2) The symmetry group acts irreducibly on the input space 3) The input space is a tensor product of invariant subspaces 4) Only abelian groups are considered for channel covariance 5) The program register contains a classical description of channels 6) The group-covariant channels are restricted to diagonal unitary operations 7) The quantum processor uses infinite-dimensional ancilla states

✓ Correct Answer:

The correct answer is 2) The symmetry group acts irreducibly on the input space.

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Question 192 multiple-choice

In quantum gravity research, connecting fundamental theories describing spacetime at the quantum level to the large-scale cosmological models is crucial for understanding the universe's origins and evolution. Group field theory (GFT) provides a framework from which classical cosmological dynamics, such as those of the FLRW model, can be derived through effective methods. Which methodological feature allows effective cosmological dynamics to be derived from group field theory without assuming a specific quantum state? 1) Calculation based solely on spin foam amplitudes 2) Direct focus on macroscopic observables using one-body operators 3) Imposing a fixed clock variable from the outset 4) Restricting to semiclassical coherent states 5) Ignoring quantum corrections by setting ℏ to zero 6) Employing only two-body interactions 7) Defining time evolution through external classical fields

✓ Correct Answer:

The correct answer is 2) Direct focus on macroscopic observables using one-body operators.

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Question 193 multiple-choice

In quantum computing, the efficient synthesis of circuits for multi-level systems (qudits) often relies on algebraic properties of operators and tensor products. The quantum Fourier transform (QFT), a foundational component for many algorithms, is generalized from qubits to qudits using d-dimensional gates and operators. Which linear algebra property is essential for decomposing a direct sum of diagonal operators into a sum and product of Kronecker products in the construction of the QFT circuit for qudits? 1) Commutativity of matrix multiplication 2) Invertibility of unitary matrices 3) Trace-preserving nature of quantum channels 4) Orthogonality of projectors 5) Hermiticity of diagonal operators 6) Symmetry of the Kronecker product 7) Unitarity of the Hadamard gate

✓ Correct Answer:

The correct answer is 4) Orthogonality of projectors.

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of the m atrixIk⊕Ωk⊕··· ⊕Ωd−1 kcan be stated as follows. Theorem 6. The diagonal operators Ik⊕Ωk⊕···⊕Ωd−1 k∈Cdk+1×dk+1from Theorem 5 admit the decomposition: Ik⊕Ωk⊕···⊕Ωd−1 k=k/productdisplay i=1d/summationdisplay ℓ=1Eℓ⊗Ii−1⊗Rℓ−1 i+1⊗Ik−i (36) withEℓ:=eℓe⊤ ℓ∈Cd×dandRi+1deﬁned in Deﬁnition 10. Proof.We start by splitting the direct sum of ddiagonal matrices as a sum of Kronecker products, which yiel ds Ik⊕Ωk⊕···⊕Ωd−1 k=E1⊗Ik+E2⊗Ωk+···+Ed⊗Ωd−1 k, (37) 14 where we used a generalization of (4). Next, we rewrite the ℓth term on the right-hand side in a redundant form based on the mixed product identity of Kronecker product (2) . Using Lemma 7 and the property Eℓ=E2 ℓ, we get Eℓ⊗Ωℓ−1 k=Eℓ⊗Rℓ−1 2⊗···⊗Rℓ−1 k+1, = (Eℓ⊗Rℓ−1 2⊗I2⊗···⊗I2)(Eℓ⊗I2⊗Rℓ−1 3⊗···⊗I2)···(Eℓ⊗I2⊗I2⊗···⊗Rℓ−1 k+1), =k/productdisplay i=1Eℓ⊗Ii−1⊗Rℓ−1 i+1⊗Ik−i. Hence, the sum of Kronecker products (37) is equal to the prod uct of sums, i.e., Ik⊕Ωk⊕···⊕Ωd−1 k=d/summationdisplay ℓ=1k/productdisplay i=1Eℓ⊗Ii−1⊗Rℓ−1 i+1⊗Ik−i=k/productdisplay i=1d/summationdisplay ℓ=1Eℓ⊗Ii−1⊗Rℓ−1 i+1⊗Ik−i because the mixed terms in the latter expression cancel out d ue to the property EiEj= 0 fori/ne}ationslash=j. The position of the Ri+1andEℓmatrices in the Kronecker product expression can again be sw apped. Lemma 8. The Kronecker product matrices in (36)satisfy d/summationdisplay ℓ=1Eℓ⊗Ii−1⊗Rℓ−1 i+1⊗Ik−i=d/summationdisplay ℓ=1Rℓ−1 i+1⊗Ii−1⊗Eℓ⊗Ik−i, (38) fori= 1,...,k. Proof.SplittingRi+1in itsE1,E2,...,E dcomponents, i.e., Ri+1=E1+ωdi+1E2+...+ωd−1 di+1Ed, yields d/summationdisplay ℓ=1Eℓ⊗Ii−1⊗Rℓ−1 i+1⊗Ik−i =E1⊗Ii−1⊗(E1+E2+···+Ed)⊗Ik−i+ E2⊗Ii−1⊗(E1+ωdi+1E2+···+ωd−1 di+1Ed)⊗Ik−i+ ···+ Ed⊗Ii−1⊗(E1+ωd−1 di+1E2+···+ω(d−1)2 di+1Ed)⊗Ik−i, =E1⊗Ii−1⊗E1⊗Ik−i+E2⊗Ii−1⊗E1⊗Ik−i+···+Ed⊗Ii−1⊗E1⊗Ik−i+ E1⊗Ii−1⊗E2⊗Ik−i+ωdi+1E2⊗Ii−1⊗E2⊗Ik−i+···+ωd−1 di+1Ed⊗Ii−1⊗E2⊗Ik−i+ ···+ E1⊗Ii−1⊗Ed⊗Ik−i+ωd−1 di+1E2⊗Ii−1⊗Ed⊗Ik−i+···+ω(d−1)2 di+1Ed⊗Ii−1⊗Ed⊗Ik−i, = (E1+E2+···+Ed)⊗Ii−1⊗E1⊗Ik−i+ (E1+ωdi+1E2+···+ωd−1 di+1Ed)⊗Ii−1⊗E2⊗Ik−i+ ···+ (E1+ωd−1 di+1E2+···+ω(d−1)2 di+1Ed)⊗Ii−1⊗Ed−1⊗Ik−i, =d/summationdisplay ℓ=1Rℓ−1 i+1⊗Ii−1⊗Eℓ⊗Ik−i, where we subsequently used the distributivity and scalar sh ift properties (1) of the Kronecker product, and combined the terms with the same matrix, E1,E2up toEd, in the (i+1)st position. 15 6.3 Quantum circuit for Radix-d QFT The generalization of a qubit for encoding a d-dimensional vector is a qudit. Whend= 3, it is called a qutrit. The implementation of the QFT on a quantum computer working with qudits is studied in [2, 13, 6]. In the current section, we show that our derivation of the QFT on a qubit syst em extends to qudit systems in a straightforward manner, and leads to a class of equivalent QFT circuits on a qu dit system. Single quditgates are unitary d×dmatrices and the state space of multi-quditsystems is ( Cd)⊗n. A controlled- Uoperation acting on two qudits is denoted with the same circu it symbol as for the qudit case, but the action on the target qudit is now dependent on the ddiﬀerent states of the control qudit: • U↔(E1⊗Id)+(E2⊗U)+···+(Ed⊗Ud−1) =:Uc, withEk:=eke⊤ k∈Cd×d, fork= 1,...,d. This deﬁnition is in agreement with the deﬁnition of a contr olled-NOT operation on a d-level quantum system, also called a SUM gate [9]. Theorem 5 gives the following quantum circuit for the Fourie r transform of dimension dn×dnacting onn qudits: q1Fd In−1⊕Ωn−1⊕···⊕Ωd−1 n−1··· × q2 Fd In−2⊕Ωn−2⊕···⊕Ωd−1 n−2··· × |ψn/an}⌊ra⌋ketri}ht/Fn=...... qn−1 ···Fd I1⊕Ω1⊕···⊕Ωd−1 1× qn ··· Fd× The structure of this circuit is exactly the same as in the qub it case. The diﬀerence is that the Hadamard gate His replaced by the FouriergateFdwhich applies the d-dimensional DFT matrix to a single qudit, and that the diagonal operators now consist of a direct sum of ddiagonal matrices.

Question 194 multiple-choice

In algebraic topology and group cohomology, the Morava K-theory provides a powerful invariant for classifying finite groups, especially those of a given order. Techniques such as spectral sequences and analysis of group structure are commonly employed to investigate the cohomological properties of these groups. Which property distinguishes groups G18 and G20 among groups of order 32 in terms of their subgroup structure and cohomological simplicity? 1) They are minimal nonabelian, having no proper nontrivial nonabelian subgroups. 2) They are split metacyclic, being extensions of cyclic groups. 3) Their centers are isomorphic to C2×C2 with quotients C3_2. 4) Their centers have order 4 with quotient D8. 5) They are classified as nonsplit metacyclic groups. 6) Their maximal abelian subgroups allow the use of permutation modules to prove goodness. 7) They require detailed spectral sequence analysis due to complex abelian subgroup structure.

✓ Correct Answer:

The correct answer is 1) They are minimal nonabelian, having no proper nontrivial nonabelian subgroups..

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result, provided one can check the condition on the E4–page of the Serre spectral sequence. One only has to do this for G16andG17, since G18andG20are minimal nonabelian, whereas G19,G21andG20QD16are split metacyclic. G16is a semidirect product .C4C4/ÌC2with presentation G16Dha;b;cja4Db4Dc2Da;bDb;cD1;cacDab2i: The centre Zisha2;bi. Consider the Serre spectral sequence of the central extension 1!Z!G16!V!1withE2K.n/.BZ/ H.VIF2/. Then K.n/.BZ/is generated by the Euler classes of the two linear characters ; deﬁned by.a2/D1, .b/D1, and.a2/D1,.b/Di, respectively. The character extends to a representationzofG16by settingz.a/Diandz.c/D1; therefore the Euler class ofis in the image of restriction and thus a permanent cycle. On the other hand, zWDe./is not a permanent cycle, but it sufﬁces to see that z2is. Let DIndG16 ha;bi./ where.a/D1and.b/Di. Then ResG16 Z./D2; whence Resg Z.e.//Dz2. Algebraic & Geometric Topology , Volume 11 (2011) 512 Björn Schuster Secondly, G17Dha;b;cja8Dc2Da;cDa;bD1;b2Da2;cbcDa4bi with centre ZDhaicyclic of order 8 and central quotient VC2C2. Consider as above the Serre spectral sequence of the central extension; this time K.n/.BZ/ is generated by the Euler class of a generator ofRZ. Clearlyextends to a representationzofha;ci. Inducingzup to G17and restricting to the centre yields 2, implying that e./2is a permanent cycle. 4.3 Groups 23–33 The groups G23–G33all have centre of order 4 with quotient D8– again a case for Lemma 2.3 and henceforth an easy exercise in representation theory. They also share the same Euler characteristic n;2.G/D1 216nC8n1 24n: In fact, one has G23DC2D16,G24DC2SD16,G25DC2Q16,G26DC4D16, G31DC4oC2,G33D.C2C2/oC2, and G27Dha;b;cja8Db2Dc2Da;bDb;cD1;cacDa1bi G28Dha;b;cja8Db2Da;bDb;cD1;c2Da4;c1acDa3bi G29Dha;bja8Db4D1;b1abDa1i G30Dha;bja8Db4D1;b1abDa3i G32Dha;b;cja8D1;bDc2;c4Da4; a;bDb;cD1;c1acDa3i: Of these latter groups, G29andG30are split metacyclic and G32is nonsplit metacyclic, so that leaves G27andG28. For both groups, aandbgenerate a maximal abelian subgroup AC8C2. As prescribed by Lemma 2.3, we consider the Serre spectral sequence of the central extension 1!ha4;bi! G!hxa;ci! 1 E2DH.D8IK.n/.BZ// with F2x1;w1;w2=.x2 1Cx1w1/ K.n/z1;z2=.z2n 1;z2n 2/: Herez1andz2are the Euler classes of 1;2corresponding to a4andb, respectively, while we keep the notation for H.D8IF2/from Section 2. SinceG;GDha2biC4, we have a one-dimensional representation ofGwith .b/D1(and .a/D .c/D1); this restricts to 2on the centre. Thus z2is a Algebraic & Geometric Topology , Volume 11 (2011) Morava K–theory of groups of order 32 513 permanent cycle. Now let ADha;biC8C2as above, and deﬁne 2RA by .a/Dexp.i=4/and.b/D1. Thencis either1(forG27) or3(forG28); in any case, ResG ZIndG A./DResA Z.Cc/D21; soz2 1is a permanent cycle, too. 4.4 Groups 34–37 Presentations of G34–G37are as follows: G34Dha;b;cja4Db4Dc2Da;bD1;cacDa1;cbcDb1i G35Dha;b;cja4Db4Da;bD1;c2Da2;cacDa1;cbcDb1i G36Dha;b;cja4Db4Dc2Db;cD1;a1baDb1;cacDa1i G37Dha;b;cja4Dc2Dd2Db;cD1;dDa;c;b2Da2;bab1Da1i All four groups have centre ZC2C2with quotient C3 2, and Euler characteristic n;2D1 216nC8n1 24n: G34andG35have the maximal abelian subgroup ADha;biC4C4, on which the quotient acts (diagonally) by inverting aandb. Since D8could be written as a semidirect product C4ÌC2with that action, Theorem 1.1 tells us that MWD zK.n/.BC4/is a permutation module for the automorphism inverting the generator of the group, thuszK.n/.BA/M Mis a permutation module, too. Theorem 1.1 again thus implies that G34andG35are both good. G36contains the maximal abelian subgroup ADhb;a2;ciC4C2C2. From the relations one reads off that zK.n/.BA/M N, where NDzK.n/.BC2C2) with the switch action, so this is again a permutation module; the situation is similar forG37and the maximal abelian subgroup ADhb;c;di. 4.5 Groups 38–41 Presentations of G38–G41are as follows: G38Dha;b;cja4Db2Dc4Da;bD1;cac1Dac2;cbc1Da2bi G39Dha;b;cja4Db4Dc2Da;bD1;cacDa3;cbcDa2b3i G40Dha;b;cja4Db4D1;c2Db2;

Question 195 multiple-choice

In quantum information processing, decoherence and incoherence are critical challenges that degrade the performance of quantum gates. Experimental techniques such as RF field compensation and Kraus operator analysis are used to assess and improve quantum control fidelity in platforms like NMR. Which technical strategy most directly increases the amplitude of the dominant Kraus operator to more closely approximate the ideal quantum gate implementation in a system affected by RF inhomogeneity? 1) Increasing sample temperature to reduce relaxation rates 2) Using longer pulse durations during quantum operations 3) Applying uncompensated control sequences 4) Reducing the number of product operator basis states 5) Ignoring RF field mapping during pulse design 6) Decreasing the strength of the RF field uniformly 7) Compensating control sequences to account for RF inhomogeneity

✓ Correct Answer:

The correct answer is 7) Compensating control sequences to account for RF inhomogeneity.

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incoherence leads to an apparent decoherence, as e videnced by a reduction in the eigenvalue magnitudes. The average eigenvalue reduction is 1.2% forSbothand 0.9% for S1,2 180xS1 90¯y. FIG. 4: The 33 point RF proﬁle shows how much of the sample sees what fract ion of the desired power (so 1 .0 on the horizontal axis is the desired power). This proﬁle was measu red experimentally and used for the simulations. FIG. 5: Measured Hadamard relaxation operator versus the product ope rator basis for the three carbons of alanine in the absence of RF ﬁelds, with rates color -coded as indicated by the legend. The labels on the axes are a short-hand for the produc t operators associated with with xandydirections on the three carbons, with e.g. X1X↔σ1 xσ3 x. The labels for each entry of the matrix are obtained by analogy with the produ ct rules for the Pauli operators; for example, the matrix entry in the row labeled 1XXand column labeled Y1Yis YXZ↔σ1 yσ2 xσ3 z[23]. See Table I for the numerical values of the various entries. FIG. 6: The various correlations (see Section II) for the 64 states of the product operator basis before and after applying the QFT, where those obtained fro m the uncompensated strongly modulating control sequences have been plotted against those obtained with the sequences designed to compensate for RF inhomogeneity [10]. To sp read the points out, the (co)ordinates plottedarelog10(1+(1−x)),althoughthe axeshave beenlabeled by theactual correlations 0 ≤x≤1. Circles ( /circlecopyrt) are used for the initial state correlations, while triangles (△) denote the ﬁnal state correlations and stars ( ⋆) indicate the ﬁnal state attenuated correlations. For the uncompensated sequences the average init ial state correlation is 0.93, the average ﬁnal state correlation is 0.64, and the average atten uated correlation is 0.37. For the compensated sequences, the average initial state corre lation is 0.96, the average ﬁnal state correlation is 0.82, and the gate ﬁdelity 0.64. Moreover, since a large majority of the points lie below the diagonal in the plot (dashed line), the corre lations with the 30 compensated pulses were in fact better in almost every case. FIG. 7: The bar graph on the left shows the amplitudes ak=/bardblAk/bardbl/√ 8 of experimental QFT Kraus operators, where the light grey portions of the bars ar e the amplitudes obtained from QPT done with the uncompensated control sequences, and t he dark grey bars those obtained from QPT done with the RF-compensated control sequen ces (see text). With the uncompensated control sequences the dominant Kraus operato rhad anamplitude au 1= 0.72, whereasthecompensatedhad ac 1= 0.86. ThesecondlargestKrausoperatorshadamplitudes ofau 2= 0.30 andac 2= 0.34, respectively, after which the amplitudes decreased much more slowly. The plots on the right (which are shaded by column number to a id viewing) compare the real part of the desired unitary operator UQFT(top center) to the real part of the largest Kraus operator A1obtained using the uncompensated and compensated control seq uences (left and right-hand plots on the second

Question 196 multiple-choice

In spectral graph theory and quantum computation, decomposing a graph’s Hamiltonian into sums of simpler components provides powerful tools for analyzing eigenvalues and symmetries. Tree clique decomposition is one hierarchical technique used to break down complex graphs into combinations of cliques. Which property uniquely distinguishes a tree clique decomposition from other graph decomposition methods that use parity over F2? 1) It requires all leaf nodes to be triangles. 2) It mandates that every vertex in the tree corresponds to a single edge. 3) It forces every subgraph in the decomposition to be regular. 4) It uses spectral partitioning to select subgraphs at each step. 5) It applies only to bipartite graphs. 6) It includes each edge in the decomposition if it appears an odd number of times. 7) It organizes subgraphs in a rooted tree where each vertex corresponds to a subgraph, with leaves being either fully connected or fully disconnected components.

✓ Correct Answer:

The correct answer is 7) It organizes subgraphs in a rooted tree where each vertex corresponds to a subgraph, with leaves being either fully connected or fully disconnected components..

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Since there is only one valid way to remove a box from a rectangular 2 ×n/2 Young tableaux, we obtain H[n/2,n/2] Kn−1=H[n/2,n/2−1] Kn−1=η[n/2,n/2−1]I=1 2(n2−4)I. (6.7) Combining Eq. (6.7) with Eq. (6.6) in Eq. (6.5) yields H[n/2,n/2] ⋆n= (n+ 2)I. 6.2 Graph Clique Decomposition and Simpler Hamiltonians In the previous example we showed how to compute the eigenvalues of the star graph Hamiltonian by writing it as a difference of cliques and then using Young’s Branching Rule. In that example, the cliques considered differed in size by one vertex. But the same techniques also apply generally to any graph whose corresponding Hamiltonian can be de- composed as an arbitrary signed sum of clique Hamiltonians. In this section, we provide an algorithm which computes such a sum for a given graph Gwhenever such a decomposition is possible. Then we show how to speed up the computation of some QMC Hamiltonian’s eigenvalues using this decomposition. While this algorithm is straightforward, we do not know of an occurrence of it elsewhere in the literature, and so for completeness prove correctness of the algorithm and runtime guarantees in this paper. We also note that a similar, but distinct, decomposition of a graphs is considered in [ BPR22 ,PKT22 ]. These papers investigate ways in which graphs can be decomposed into sums of cliques over F2 (i.e. decompositions where existence of an edge in the original graph corresponds to the parity of the number of cliques containing that edge). These decomposition can be applied more generally than the decomposition considered in this paper. 6.2.1 The Tree Clique Decomposition To keep track of how the QMC Hamiltonian associated with a graph Gdecomposes into a sum of signed cliques we introduce an object which we call the Tree Clique Decomposition ofG. This is introduced formally in the next definition. In that definition and throughout the remainder of the paper we assume that all trees discussed are rooted (or equivalently directed) trees. Definition 6.2 (Tree Clique Decomposition) .For any connected graph G, the tree clique decomposition of G, denoted T(G)consists of an m-vertex tree Tand set ofnconnected graphs{G(v1),...,G (vm)}, where each graph G(vi)is associated with a vertex viofTand the following properties hold: (1)For the root vertex v1ofT, we haveG(v1) =G. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 50 (2)For any vertex viofTwhich is not a leaf vertex, let c1,...,ckbe its children. Then we have G(vi)c=/uniondisplay j∈{1,2,...,k}G(cj), (6.8) where we understand the union of graphs to be their disjoint union. (3)For any leaf vertex vjofTwe have that G(vj)cis connected or G(vj)cis totally disconnected. Given a graph Git is reasonably straightforward to compute the tree clique decom- position of Gby iteratively taking complements of graphs and then breaking them into their connected components. We make this process formal in Algorithm 6. Algorithm 6: Tree Clique Decomposition Input : A graphGto be decomposed. Output : A tree clique decomposition T(G) ={T,{G(v1),...,G (vm)}} InitializeTas the singleton graph with vertex v1; activeTLeaves :={v1}; i:= 0 ; G(v1) :=G; while activeTLeaves

Question 197 multiple-choice

Siegel modular threefolds are complex algebraic varieties arising from quotients of Siegel upper half-space by arithmetic subgroups, and are pivotal in the study of moduli spaces of abelian varieties and related geometric structures. Their geometry and topology are deeply influenced by group actions, polarizations, and ramification loci. In the construction of the minimal Siegel modular threefold associated to the maximal extension Γ*t of the paramodular group, which geometric locus forms the divisorial part of the ramification in the finite map from the moduli space of (1, t)-polarized abelian surfaces to the modular threefold? 1) Heegner divisors 2) Shimura curves 3) Modular discriminant divisors 4) Satake boundary components 5) Fermat surfaces 6) Prym varieties 7) Humbert surfaces

✓ Correct Answer:

The correct answer is 7) Humbert surfaces.

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Question 198 multiple-choice

Variational quantum algorithms are increasingly utilized for solving problems in group theory, particularly those involving finite Abelian groups and hidden subgroup problems. Hybrid quantum-classical approaches employ parameterized quantum circuits and classical post-processing to enhance computational efficiency in these domains. Which of the following best describes the role of the parameterized quantum Fourier transform (QFT) in variational algorithms for hidden subgroup problems over finite Abelian groups structured as products of cyclic groups? 1) It ensures that all group operations are performed deterministically without the need for classical post-processing. 2) It restricts the algorithm to only work with groups that are cyclic and not their direct products. 3) It removes the necessity for bit-permutations when encoding group structures in the quantum circuit. 4) It replaces the need for a cost function by directly maximizing the fidelity of the reconstructed data. 5) It encodes group structure in non-tunable static parameters, preventing adaptation to subgroup configurations. 6) It enables data compression exclusively through purely quantum mechanisms, excluding classical bottlenecks. 7) It introduces tunable parameters that adapt the quantum circuit to different group structures and subgroup configurations, allowing extraction of group type and permutations for efficient hybrid quantum-classical computation.

✓ Correct Answer:

The correct answer is 7) It introduces tunable parameters that adapt the quantum circuit to different group structures and subgroup configurations, allowing extraction of group type and permutations for efficient hybrid quantum-classical computation..

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bit-permutations at the circuit level. Parameterized quantum circuits for variational HSP . Substituting the standard QFT module in an HSP circuit with the provided parametrized QFT circuits yields the circuit ansatz for the variational HSP, denoted as HSP ~θ, as follows: Corollary 1 .Parameterized quantum circuits for hidden subgroup problems . Upon suitable con ﬁguration of the parameters ~θ,t h ef o l l o w i n g parametrized quantum circuits can solve the HSP over any type of ﬁnite Abelian group of the form Z2m1×Z2m2×...×Z2mq, that is assigned by bit- permutations, ð16Þ where W~θis the parameterized unitary corresponding to bit-permutations andUf∣ii∣0i¼∣ii∣fðiÞ/C11 . Obtaining group structures from parameters . Given valid parameters ~θ, if the parametrized quantum circuits execute the quantum Fourier transform on Gσ, the group Gσcan be determined from ~θ. This involves two steps: ﬁrstly, extracting the group type using parameters in QFT, and secondly, deducing permutations from W~θparameters. With the estab- lished group structure, group operations between elements can be computed, revealing the hidden subgroup. Remarkably, these compu-tations are achievable using classical algorithms on classical devices. Hidden subgroup auto-encoder We now construct an auto-encoder that automatically ﬁnds the correct parameters of HSP ~θ, compressing the input data into a description of sand the group structure. We describe the autoencoder algorithm, the com- pression ratio, numerical experiments and discuss potential applications. The autoencoder algorithm . Our primary algorithm is outlined below, with comprehensive details available in the Methods section. We employ the quantum autoencoder design, consisting of an encoder Eand a decoder D. The quantum autoencoder bottleneck, designed to select speciﬁc features, generally transmits qua ntum states to the decoder. For our problem, we impose a classical bottleneck to ensure the classical output asthe compressed data. Remarkab ly, a classical decoder is suf ﬁcient to reconstruct the original data, consi stent with the compression being the difﬁcult direction of a one-way function and the decoding the easy direction. Unlike typical autoencoders character ized by arbitrary node connections, we enforce the distinct structure denoted HSP ~θas deﬁned in Eq. ( 16), to serve as our circuit ansatz. We run HSP ~θfor multiple times to solve a tentative hiddenhttps://doi.org/10.1038/s41534-024-00865-2 Article npj Quantum Information | (2024) 10:74 4 subgroup H. Then we use classical post-processing to generate a classical message σclassical which describes the hidden subgroup Hand the data values f(i) on different cosets ciH. This classical message serves as the compressed data, which is passed to the decoder to reconst ruct the data. The classical decoder determines the group structure using the message, and then for each index i, reconstructs ^fðiÞ¼fðimod HÞ,w h e r e fðimod HÞis the data value on some cosets ciHthat can be decoded from the classical message σclassical .T oe v a l u a t e the variational autoencoder ’s performance, a cost function is de ﬁned using Euclidean distance between the original data and the reconstructed data, Cð~θÞ¼X i2training setfðiÞ/C0 ^fðiÞhi2 ;ð17Þ where ^fimplicitly depends on ~θ. For computational ef ﬁciency, we

Question 199 multiple-choice

The McEliece cryptosystem is a public-key encryption scheme that leverages error-correcting codes, with binary Goppa codes being preferred for their robust security properties. Its key structure and resistance to quantum attacks make it a significant candidate in post-quantum cryptography. In the McEliece cryptosystem, which component's secrecy is essential because its exposure, along with the permutation matrix, would directly compromise the security of the private key? 1) The syndrome vector 2) The public key matrix 3) The code length parameter 4) The randomly chosen invertible matrix 5) The error vector used in encryption 6) The original generator matrix 7) The ciphertext produced after encryption

✓ Correct Answer:

The correct answer is 4) The randomly chosen invertible matrix.

📚 Reference Text:

cryptosystem is a public-key cryptosys- tem proposed by McEliece in 1978 [13], conventionally built over Goppa codes. A dual variant of the system, proposed by Niederreiter [16], can provide slightly improved efﬁciency with equivalent security [10]. This dual system can additionally be used to construct a digital signature scheme [2], a shortcoming of the original system. There are two basic types of attacks known against the McEliece-type cryptosys- tems: decoding attacks, and direct attacks o n the private key. The former appears chal- lenging, considering that the general decoding problem is NP-hard; indeed, historical conﬁdence in the security of the McEliece system relies on the idea that this hardness can be retained for scrambled version of speciﬁc codes. This same intuition applies to quantum attacks: NP-hard problems are believed to be intractable, in general, for quantum computers and no signiﬁcant quantum algorithmic developments appear to be directly relevant to these decoding problems. The latter—direct attacks on the key—can be successful on certain classes of linear codes, and is our focus. In a McEliece-type cryptosystem, the private key of a user Alice consists of three matrices: a k×nmatrix Mover a ﬁnite ﬁeld Fq/lscript,ak×kinvertible matrix Sover the ﬁeld Fq,a n da n n×npermu- tation matrix P. In the McEliece version, Mis a generator matrix of a q-ary[n,k]-linear code (hence, /lscript=1), while in Niederreiter’s dual system, Mis a parity check matrix of aq-ary linear code of length n. The matrices SandPare selected randomly. Alice’s public key consists of the matrix M∗=SMP . An adversary may attack the private key, attempting to recover the secret row “scrambler” Sand the secret permutation Pfrom M∗andM, assuming he already knew M.1As pointed out in [4], it crucial to keep Sand Psecret for the security of the McEliece system. The security of these McEliece-type system s have received considerable attention in the literature, often focusing on partic ular choices for the unde rlying codes. Various classes of Goppa codes have received the great est attention: along these lines, Sidel- nokov and Shestakov’s attack [23] can efﬁciently compute the matrices SandMPfrom the public matrix M∗=SMP if the underlying code is a generalized Reed-Solomon code.2While this attack can reveal the structure of an alternative code, it does not re- veal the secret permutation. An attack in wh ich the secret permutation is revealed was proposed by Loidreau and Sendrier [11], using the Support Sp litting Algor ithm [21]. However, this attack only works with a very limited subclass of classical binary Goppa codes, namely those with a binary Goppa polynomial. 1Recovering the secret scrambler and the secret permutation is different from the Code Equiv- alence problem. The former ﬁnds a transformation between two equivalent codes, while the latter decides whether two linear codes are equivalent. 2We remark that the class of generalized Reed-Solomon codes is essentially equal to the class of rational Goppa codes. McEliece and Niederreiter Cryptosystems 763 Although the McEliece-type crypt osystems are efﬁcient and still considered classi- cally secure, at least with classical binary Goppa codes [4],

Question 200 multiple-choice

In the study of infinite abelian p-groups, the structure of basic subgroups and their properties play a crucial role in determining automorphism groups and extension behaviors. The relationship between divisibility, thinness, and the presence of infinite direct factors can characterize when certain automorphism subgroups exist. For an abelian p-group A, which of the following is equivalent to the automorphism group Aut containing no subgroup isomorphic to the infinite cyclic group C_∞? 1) A has no cyclic direct factor of order p 2) Every basic subgroup of A is infinitely generated 3) The divisible part of A is trivial 4) A is a restricted direct product of infinite cyclic groups 5) A admits a decomposition into non-homocyclic components only 6) The divisible part D of A has finite rank and a basic subgroup L is thin 7) A is a direct sum of countably many copies of C_{p^k}

✓ Correct Answer:

The correct answer is 6) The divisible part D of A has finite rank and a basic subgroup L is thin.

📚 Reference Text:

p is prime and A and B are both /^-groups, initially not necessarily abelian. Our first result is straightforward, and we omit the proof. THEOREM 3.1. If A is a finite p-group and B a nilpotent p-group, then Jf(A, B). The second is also easy, but not quite so elementary: THEOREM 3.2. Let A be an infinite abelian p-group of finite exponent and B a nilpotent p-group. Then JV(A, B) if and only if B is finite. Proof. One way this follows is from the well-known theorem of G. Baumslag [1] stating that every extension of a nilpotent /?-group of finite exponent by a finite p-group is nilpotent. Conversely, suppose that ^V(A, B). As A is infinite abelian of finite exponent, it follows from Prufer's theorem that A has a direct factor isomorphic with cffi for some k and so (2.8) says that B must be finite, and the proof is complete. When A has infinite exponent, the situation is more complicated and results are governed by the structure of a basic subgroup of A. We say that a subgroup L of an abelian /7-group A is basic (see [3] ) if (i) L is a restricted direct product of cyclic groups, (ii) L is pure in A, (hi) AIL is divisible. Every /?-group has a basic subgroup, and all basic subgroups of a given group are isomorphic. Additional properties of basic subgroups that we shall need are: (iv) If OO L = n L H, https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press NILPOTENT EXTENSIONS 1029 where L n is the homocyclic component of exponent p n, then for each n, A = L, X L 2 X ... X L n X A* 9 where A* = (A"", L*> and L* = II L m. m>n (v) If Aj is a homomorphic image of an abelian /?-group A, a basic subgroup of A x is a homomorphic image of a basic subgroup of A. (vi) ^ = LAP. We shall need the following definition when we come to investigate AT(A, B) for divisible B. Definition 3.3. A direct product of cyclic ^-groups is said to be thin if the ranks of the homocyclic components are all finite. It turns out that our investigation of the Suprunenko problem for A of infinite exponent requires that we treat the cases of B divisible and B non-divisible separately. Recall that a divisible subgroup of a nilpotent /?-group is central, so that a divisible nilpotent /?-group is abelian. A crucial first step is to consider the case where B is the quasicyclic group CpOQ . THEOREM 3.4. Let A be an abelian p-group. Then the following statements are equivalent. (i) ^(A, C p0o); (ii) Aut A contains no subgroup isomorphic to Coo; (iii) the divisible part D of A has finite rank and a basic subgroup L of A is thin; (iv) A has no direct factor of the form K^ \ where K is non-trivial. Proof That (i) implies (ii) is easy;

Question 201 multiple-choice

Quantum phase estimation (QPE) is a foundational algorithm in quantum computing, pivotal for tasks such as eigenvalue determination and factoring. The reliability of QPE is challenged by noise inherent in quantum hardware, especially incoherent noise impacting qubit states. Which of the following statements correctly characterizes the scaling behavior of eigenvalue estimation error in quantum phase estimation when subjected to incoherent noise? 1) The standard deviation of the estimated eigenvalue decreases exponentially with increasing error probability per qubit. 2) The standard deviation is independent of both the number of qubits and the error probability per qubit. 3) For any error probability, the standard deviation grows quadratically with the number of qubits. 4) The standard deviation grows linearly with the error probability per qubit regardless of its magnitude. 5) For large error probabilities, the standard deviation decreases with the number of qubits. 6) The standard deviation is unaffected by the type of incoherent noise model applied. 7) The standard deviation grows exponentially with the error probability per qubit, but only linearly with the number of qubits for small error probabilities.

✓ Correct Answer:

The correct answer is 7) The standard deviation grows exponentially with the error probability per qubit, but only linearly with the number of qubits for small error probabilities..

📚 Reference Text:

Question 202 multiple-choice

Quantum doubles of finite groups, denoted D, play a crucial role in topological quantum computation and the efficient implementation of quantum Fourier and Clebsch-Gordan transforms. These techniques underlie algorithms for simulating topological invariants and analyzing computational hardness in quantum complexity theory. Which type of computational hardness is established for exact evaluation of link invariants derived from the quantum double of a finite group? 1) BQP-hard 2) #P-hard 3) NP-complete 4) SBP-hard 5) BPP-hard 6) PSPACE-complete 7) DQC1-hard

✓ Correct Answer:

The correct answer is 2) #P-hard.

📚 Reference Text:

register has a trivial irrep of Zp, then perform a QFT over Zqin the second register. This gives us a QFT over G. Since we can now perform eﬃcient QFTs over all the centraliz ers, we can perform eﬃcient QFT over D(G). For the Clebsch-Gordan transform, as the ﬁrst condition, we need to perform CG transforms over all centralizers and their intersections. Since all intersect ions and two of the centralizers are abelian, their CG transforms are eﬃcient. For the CG transform over G, we only have to consider the case of high dimensional irreps. For these irreps, the tensor product looks like (ρk⊗ρl)(a,b) =/summationdisplay s,tωa(kα−s+lα−t) p|s,t/a\}bracketri}ht/a\}bracketle{ts−b,t−b|. The Clebsch-Gordan transform, if k+l/\e}ati≀\slash= 0, is |s,t/a\}bracketri}ht−→|t−s,t/a\}bracketri}ht. This produces the state /summationdisplay s,tωa(kα−t+s+lα−t) p|s,t/a\}bracketri}ht/a\}bracketle{ts,t−b|=/summationdisplay s,tωaα−t(kαs+l) p|s,t/a\}bracketri}ht/a\}bracketle{ts,t−b|, which is a direct sum of qirrepsρkαs+lfors∈Zq. Now, ifk+l= 0, then again we ﬁrst perform the above transform to get/summationdisplay s,tωaα−t(kαs−k) p|s,t/a\}bracketri}ht/a\}bracketle{ts,t−b|. Notice that when s= 0, this is the regular representation of Zq. Therefore, we have to perform a conditional QFT (conditioned on s= 0) additionally. Then we would obtain a direct sum of the qone dimensional irreps andq−1 high dimensional irreps ρk(αs−1)fors∈Zq,s/\e}ati≀\slash= 0. This completes the CG transform over G. We can now perform QFT and CG transforms over centralizers and i ntersections of centralizers. To complete the conditions for CG transform over D(G), we need to block diagonalize ρkrestricted to intersections of centralizers i.e., block diagonalize ρkwhen restricted to ZpandZq. From the structure of ρk, it is easy to see that when restricted to Zp, it is already diagonal and when restricted to Zq, it is the regular representation of Zq. Therefore, a QFT over Zqwould diagonalize it. Thus we can perform eﬃcient QFT and CG transforms over D(G). 8 Conclusions In this paper, we gave an eﬃcient circuit for the quantum Four ier transform over D(G), the quantum double of a ﬁnite group Gand show how to apply it to D(Sn). We used this circuit to give eﬃcient algorithms for approximating link invariants. We then showed some hardnes s results for approximating and exact evaluation of link invariants arising from D(G). We showed that additive approximations of link invariant s arising from irreps of D(G) are BPP-hard and multiplicative approximations are SBP-hard and exact evaluations are # P- hard. We also gave an eﬃcient randomized algorithm to additi vely approximate the link invariants when the conjugacy class (or ﬂuxon) irrep is used. This shows that for the ﬂuxon irrep the problem is BPP-complete. We then gave an eﬃcient circuit for the Clebsch-Gordan trans form for ﬂuxon irreps of D(G) and, under certain conditions, for general irreps. We gave an example o f a quantum group, namely D(Zp⋊Z q) (which is powerful enough to do universal quantum computation) for wh ich we show how to perform the QFT and CG transforms. We also showed how to simulate topological quan tum computation inside exponentially large irreps of D(G) eﬃciently using the Fourier and Clebsch-Gordan transform s. 27 In our hardness results, we need that

Question 203 multiple-choice

In the study of abelian varieties and their associated theta functions, isogenies and morphisms between these varieties play a crucial role, often characterized by properties of matrices and maps between groups. Understanding how these structures interact and the significance of specific algebraic operations is fundamental in algebraic geometry. If two g x g integer matrices S and T satisfy S - T = nI for n ≠ 0, and there exist associated group maps λ₁ and λ₂ whose composition is multiplication by n, what is the degree of the induced graded homomorphisms between the corresponding rings of theta functions? 1) n 2) n + 1 3) 2n 4) n³ 5) n² 6) g · n 7) n!

✓ Correct Answer:

The correct answer is 5) n².

📚 Reference Text:

symmetric and (ord. J/^ ) is positive definite). Let S = (Sij) and T = (tiό) be g x g matrices over Z such that S-T = nl, n Φ 0 and let δi> = Π ^lfSJe It is readily seen that the matrix (6^) also satisfies the Riemann conditions. Attached to the matrix (J/^) are the period vectors Vi9 the group Γ, the graded ring R(j^tj) of theta functions, the abelian variety A and the map ψ\ Gg/Γ —> A*.; similarly attached to (δ^) we have TF^, Γ', R{bis), B and ^: GgjΓ -* β&. The following identities are obvious: (1) (2) Let λ2:Gg I.— !(χ)Gg be = (yi> = <*,] the[} My = Π Kih maps > Ug)= Π ^Ur7 defined where where• by: Vt = 11 */t/^ ΓT /y^y11 xjj PROPOSITION III.1.1. λλ maps Γ into Γ',λ2 maps Γf into Γ and the composition in either order is the map x —> xn<ί. Proof. The image of Vr under λλ is the vector whose i-th compo- nent is Uj^ΐ?J = UJ MY- But this is just the vector Πy ^ry Similar- ly λ2(Wr) = Y[j Vfri. The last assertion is obvious. For θe£> let ψί(X) = 0(Y?, , Y;) where Y4=Πi-x'5ii and - , Zg) where Z, = \[sXψ. https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press 92 HORACIO TAPIA-RECILLAS PROPOSITION III.1.2. If θ e Rm(bi3) then ψ, e Rmn^υ). If θ e Rm(s/ij) then ψ2 e Rmni{biό). Consequently θ —* ψ1 (resp. θ —> ψ2) gives a graded homomorphism of degree n2μλ: R(jbi3) —> R(stfiό) (resp. μ2: R(stfi3) )), and the composition {in either order) is the map an2: Θ(X) Proof. ψ^VrX) = Θ(Z?, , ZJ) where Zt = flj (^r^)* S and μf : B —> A. Proof. Since the composition (in either order) is the map #n2 which is finite (cf. Theorem 1.1.3), μ1 and μ2 are finite. So we get morphisms of varieties A —> B and B —* A which are readily seen to be group variety morphisms. From Proposition III.1.3 we get homomorphisms μf\ Ak-*Bk and μf : Bk—> AIύ. The composite map Ak—> Bk—> Ak is the map induced by ) which by Theorem 1.3.5 is multiplication by n2. §111.2. φ is bijective PROPOSITION III.2.1. There is a commutative diagram of maps: GJΓ-^+GJΓ'-^+GJΓ ψ μί * μt Λk > Bk > Λk where the λt are induced by the maps of Proposition III.1.1. Furthermore (λ2 o λ^)(x) = xn* and μf o μf is just multiplication by n2. Proof. The commutativity of the diagram follows in a straight- https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press ABELIAN VARIETIES 93 forward way from

Question 204 multiple-choice

In modern number theory, elliptic curves over the rational numbers and their connections to modular forms have profound implications for the understanding of Diophantine equations and arithmetic geometry. One of the greatest breakthroughs in this area was the proof of the modularity theorem. Which statement accurately describes the Modularity Theorem in relation to elliptic curves over the rational numbers? 1) Every elliptic curve over Q can be parametrized by a modular form of weight 1 and level N. 2) Every elliptic curve over Q can be parametrized by a modular form of weight 2 and level N, where N is the conductor. 3) Every modular curve X₀ can be mapped to any elliptic curve over Q with cyclic subgroup of order N. 4) The conductor N of an elliptic curve over Q is always equal to its rank. 5) The Modularity Theorem implies that all elliptic curves over Q have complex multiplication. 6) Only elliptic curves with trivial conductor are associated with modular forms. 7) Elliptic curves over Q are classified exclusively by their torsion subgroups.

✓ Correct Answer:

The correct answer is 2) Every elliptic curve over Q can be parametrized by a modular form of weight 2 and level N, where N is the conductor..

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Question 205 multiple-choice

In group theory, the structure of abelian groups and their extensions with other groups, such as p-groups, is central to understanding properties like nilpotency, commutator behavior, and classification. Certain conditions regarding divisibility, cardinality, and subgroup structure can determine whether specific properties hold within these group extensions. Which of the following statements about the property JV(A, B) in group extensions is correct? 1) JV(A, B) holds for every pair of abelian groups A and B, regardless of their cardinality or divisibility. 2) If B is non-divisible and A is uncountable with infinite exponent and cardinality greater than 2^ω, then JV(A, B) always holds. 3) The divisible part of A can have arbitrary rank and JV(A, B) will still be satisfied for any abelian B. 4) If B is infinite, non-divisible, and the cardinality of A exceeds 2^ω, then JV(A, B) does not hold. 5) JV(A, B) is only determined by the order of p in the group extension and not by the structure of A or B. 6) When B is divisible, the presence of torsion in A prevents JV(A, B) from holding. 7) JV(A, B) holds for all finite p-groups B and any choice of A.

✓ Correct Answer:

The correct answer is 4) If B is infinite, non-divisible, and the cardinality of A exceeds 2^ω, then JV(A, B) does not hold..

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4.2.1, there exists a positive integer k such that [A, b] p = 1 for all b in B. Thus [A,B] ^ Q k(A). With Ai denoting the i-th homocyclic factor of A, we have Qk(A) ^ (A x,A,..-,A 2k-x,Apk)\ that is, [A, B] ^ HA pk for a finite group H. With G = (A, B), the split extension, we have y,(G) Si [A, B] for some /. So the image of Y/(G) in G/A p is a finite/?-group, which means that [Y/CG), mB] ^ ^ for some m ^ 1. Thus y /+w+1 (G) = 1 since 5 fixes the elements of A p . Case 2. ^1 is not a direct product of cyclic groups. Then A/AS * is such a product, and from Case 1 we have that G = (A/A (l\ B) is nilpotent. Thus yc(G) = A^ ) for some c. Theorem 5.3 now gives that there exists a k ^ 0 such that [A, b]pk = 1 for all b in B. So yc(G) ^ A {X) ^ A pk and as B fixes all elements of A p , yf+i(G) = 1- If B is not reduced, its divisible part C is central and it is easy to see that https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press 1052 J. BUCKLEY AND J. WIEGOLD B/C is reduced. Furthermore (B/C) ab is still a direct product of a group of finite exponent and a divisible group. Since L is thin, C acts trivially on A by Theorem 3.4. Hence every homomorphism B —> Aut A factors through a map B/C -> Aut A. But -group. We have now classified all pairs A, B with B divisible for which JV(A, B). Further, when B is non-divisible we have classified all A, B with A countable or of finite exponent for which Jf{A, B). There remains the case where A is uncountable, of infinite exponent and B is non-divisible. From our previous work there are many obvious necessary conditions IOXJV(A, B), for example, restrictions on the structure of countable direct factors of A. Again because of Theorem 5.1 the divisible part of A must have rank at most p — 2 and one can restrict attention to the case where A is reduced. Let L be a basic subgroup of such an A and let B be infinite and non-divisible. Again from (2.8) we see that L must be thin and since, as is well-known, the cardinality \A\ of A is at most \L\ ° we conclude that \A\ ^ 2 °. In other words, if B is infinite, non-divisible and \A\ exceeds 2 °, then —* Jf{A, B). For the case where B is a finite, non-trivial /7-group recall that ^V(A, B) implies ^V(A, C ). Consider the special case where A is the torsion subgroup

Question 206 multiple-choice

High-performance computing platforms are increasingly used to simulate quantum algorithms, leveraging specialized hardware to accelerate complex operations such as the Quantum Fourier Transform (QFT). Optimizing how quantum logic gates are mapped onto heterogeneous architectures can dramatically impact simulation speed and efficiency. Which optimization specifically contributed to minimizing communication overhead and achieving a 13.44x speedup in a 28-qubit QFT simulation on a heterogeneous supercomputer architecture? 1) Increasing CPU clock speeds for faster serial execution 2) Duplicating probability amplitude data across all computing units 3) Employing Grover's algorithm for gate decomposition 4) Parallelizing only the initialization step of the simulation 5) Scheduling phase shift operations on the CPU instead of accelerators 6) Mapping both Hadamard and phase shift (R gate) operations onto the DCU accelerator using HIP programming 7) Reducing the number of quantum logic gates in the QFT circuit

✓ Correct Answer:

The correct answer is 6) Mapping both Hadamard and phase shift (R gate) operations onto the DCU accelerator using HIP programming.

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Question 207 multiple-choice

In modern algebraic geometry and number theory, the Grothendieck-Teichmueller group (GT) and the absolute Galois group of the rational numbers ($G_Q$) are central objects studying symmetries of algebraic structures and field extensions. The construction of non-abelian quotients and groupoids associated with these groups has deep implications for understanding the interaction between arithmetic and geometric phenomena. Which property holds for the natural homomorphism from the absolute Galois group $G_Q$ to the endomorphism group $GTSh(K, K)$ in the groupoid $GTSh$ when $K$ is an object in the dihedral poset $Dih$ corresponding to a dihedral group whose order is a power of 2? 1) The homomorphism is surjective. 2) The homomorphism is trivial. 3) The homomorphism factors through an abelian quotient. 4) The homomorphism is injective but not surjective. 5) The homomorphism has dense image but is not surjective. 6) The homomorphism is only defined for cyclic groups. 7) The homomorphism is bijective only when the group order is odd.

✓ Correct Answer:

The correct answer is 1) The homomorphism is surjective..

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Title: First examples of non-abelian quotients of the Grothendieck-Teichmueller group that receive surjective homomorphisms from the absolute Galois group of rational numbers Year: 2024 Paper ID: dd01dce3779ee54b75e6039b2da3d688c27e2718 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/dd01dce3779ee54b75e6039b2da3d688c27e2718 Abstract: Many challenging questions about the Grothendieck-Teichmueller group, $GT$, are motivated by the fact that this group receives the injective homomorphism (called the Ihara embedding) from the absolute Galois group, $G_Q$, of rational numbers. Although the question about the surjectivity of the Ihara embedding is a very challenging problem, in this paper, we construct a family of finite non-abelian quotients of $GT$ that receive surjective homomorphisms from $G_Q$. We also assemble these finite quotients into an infinite (non-abelian) profinite quotient of $GT$. We prove that the natural homomorphism from $G_Q$ to the resulting profinite group is also surjective. We give an explicit description of this profinite group. To achieve these goals, we used the groupoid $GTSh$ of $GT$-shadows for the gentle version of the Grothendieck-Teichmueller group. This groupoid was introduced in the recent paper by the second author and J. Guynee and the set $Ob(GTSh)$ of objects of $GTSh$ is a poset of certain finite index normal subgroups of the Artin braid group on 3 strands. We introduce a sub-poset $Dih$ of $Ob(GTSh)$ related to the family of dihedral groups and call it the dihedral poset. We show that each element $K$ of $Dih$ is the only object of its connected component in $GTSh$. Using the surjectivity of the cyclotomic character, we prove that, if the order of the dihedral group corresponding to $K$ is a power of 2, then the natural homomorphism from $G_Q$ to the finite group $GTSh(K, K)$ is surjective. We introduce the Lochak-Schneps conditions on morphisms of $GTSh$ and prove that each morphism of $GTSh$ with the target $K$ in $Dih$ satisfies the Lochak-Schneps conditions. Finally, we conjecture that the natural homomorphism from $G_Q$ to the finite group $GTSh(K, K)$ is surjective for every object $K$ of the dihedral poset.

Question 208 multiple-choice

Lie groups and their associated Lie algebras are central in understanding continuous symmetries in mathematics and physics. The structure of the Lie algebra is closely tied to the matrix properties defining the group and to the geometry of the group's manifold. Which property must hold for any element X in the Lie algebra of the special unitary group SU(d)? 1) X is a real symmetric matrix 2) X is a diagonalizable matrix with all positive eigenvalues 3) X is a skew-Hermitian, traceless d×d complex matrix 4) X has determinant equal to one 5) X is a real skew-symmetric matrix 6) X is a Hermitian matrix 7) X is a block diagonal matrix with zero trace

✓ Correct Answer:

The correct answer is 3) X is a skew-Hermitian, traceless d×d complex matrix.

📚 Reference Text:

see this in action). Let us present a diﬀerent way of working with tangent space using a computational trick: any Lie group element g2Gnearby 1can be Taylor expanded g=1+"X+O("2) whereX2g. Then matrix equations deﬁning the Lie groupGinduce equations deﬁning the Lie algebra g: for instance, if Gis a unitary group, gyg=1and so (1+"X+O("2))y(1+"X+O("2)) =1:(17) Equating"terms, we see that Xy=X, meaning the Lie algebra u(d)of a unitary group U(d)consists of skew- Hermitian matrices with the standard matrix commutator 7More generally, when Gis a Lie group, the Lie algebra is g=T1G withLiebracketgivenbythevectorﬁeldcommutatorofthevectors X;Ypushedforwardbyleft-multiplication: [X;Y ] = [LX;LY ]1for allX;Y2g. 16 [;]. The same method can be applied to ﬁnd that when G=O(d;R), then g=fX2Md(R) :XT+X= 0g, or whenG=GL(d;C), then g=gl(d;C) :=Md(C). In- cluding the condition “special” means that det(g) = 1, and since det etX =eTr[tX], we have the condition Tr[X] = 0 on the Lie algebra. For instance, one ﬁnds su(d) =fX2 u(d) : Tr[X] = 0g. The tangent space deﬁnition, depicted pictorially in Fig. 7 tells us quite a bit geometrically. For one, it im- mediately tells us that Gand ghave the same dimension as manifolds (note that any ﬁnite dimensional vector space is automatically a manifold, where its dimension as a man- ifold is its dimension as a real vector space). It also tells us that we can think of the Lie algebra gas the vector space of directional derivatives arising from paths in the Lie group G. Crucially, this means that when we have symmetries arising as representations of G, we can take derivatives and work on representations of the Lie algebra. In prac- tice it is often easier to work on the Lie algebra: whileG is generically not a vector space, gis, and so the tools of linear algebra are at our disposal . We need then to under- stand how to properly move between Gand gand their representations. The series of following deﬁnitions, propo- sitions, and theorems provides a bare-bones toolkit for this program which we will later use to construct equivariant maps as in Eq (3). Again, the beating heart of our treat- ment here is through the boxed examples and the ﬁgures: follow along with those to get a feel for how the toolkit works. The exponential map is central to moving between G and g. Recall a diﬀeomorphism is a smooth map with smoothinverse. The“local” conditionhereiscrucial: either injectivity or surjectivity may fail globally. Theorem 1 (Exponential map is a local diﬀeomorphism) . The exponential map e():g!Gis locally a diﬀeomor- phism nearby 02g, meaning there exists open regions Ugwith 02UandVGwith12Vsuch that e():U!Vis a diﬀeomorphism. Notice that by continuity, since the vector space gis con- nected, the exponential map can only map into the con- nected component of Gcontaining 1. This immediately knocks out the possibility of exp()being onto GwhenG is disconnected. A key example of this type are the orthog- onal groups O(d), which have 2 disconnected pieces (which can be seen by realizing that det :O(d)!f 1;1gis a continuous map and so the

Question 209 multiple-choice

The Hidden Kernel Problem (HKP) is a computational challenge rooted in universal algebra and complexity theory, closely related to the well-known Hidden Subgroup Problem in quantum computing. Understanding the query complexity of HKP for various algebraic structures sheds light on the boundaries of classical and quantum algorithmic efficiency. For n-dimensional powers of certain algebras, which statement accurately describes the lower bound for the number of queries any classical algorithm must make to solve the Hidden Kernel Problem? 1) The algorithm must make at least Ω(2ⁿ) oracle queries due to the exponential number of congruence classes. 2) The algorithm can solve the problem efficiently in polynomial time for all n. 3) The algorithm requires only O(n) queries because congruence classes are linearly bounded. 4) The algorithm can solve HKP with O(log n) queries using probabilistic methods. 5) The lower bound does not depend on the dimension n of the algebra. 6) The quantum query complexity for these algebras is always sub-exponential. 7) The number of necessary queries is determined solely by the number of generating operations, not by n.

✓ Correct Answer:

The correct answer is 1) The algorithm must make at least Ω(2ⁿ) oracle queries due to the exponential number of congruence classes..

📚 Reference Text:

InordertocompletetheproofofTheorem 5.1,whichwe opened the section with, we need only prove item 3: for allB/√re⌋edesequal A, the HKP for powers of Bclassically requires exponentially-manyoraclequeries.Theproofofthisiscon- tainedinthefollowinglemma,combinedwithObservation 3.1. Lemma5.6. LetBbea2-elementalgebrawith Clo(B)=A and letA=Bn. Any classical (resp. probabilistic) algorithm solvingHKP(A)must make Ω(2n)(resp.Ω(2n/2)) queries to the oracle.Proof.Theclone Ahastheoperation x↔/y.altasagenerating operation(seeFigure 1).Choosearandom z∈A\{0n}and deﬁneθztobethecongruencegeneratedbyrelating0nand z, θz≔Cg(0n,z). Observethatmodulo-2additionisdeﬁnablein A(andhence BandA) by x+/y.alt≔x↔ (0↔/y.alt). Claim.The congruence θzhas equivalence classes x/θz≔/braceleftbig x,x+z/bracerightbig . Proof of claim. Takeψto be the equivalence relation with the claimed equivalence classes. To prove that ψis a con- gruencewe needonlyshow that if aψ a′andbψ b′then (a↔b)ψ(a′↔b′). Bythedeﬁnitionof ψ,wehavethat a′=a+εzandb′=b+ τzfor someε,τ∈ {0,1}. The operation ↔is commutative andassociative,sousing“ +”asdeﬁnedbeforetheclaim,we have (w+x) ↔/y.alt=w↔0↔x↔/y.alt =w↔0↔/y.alt↔x=(w+/y.alt) ↔x. Thus(w+x) ↔/y.alt=(w+/y.alt) ↔x.Usingthisidentity twice, a′↔b′=(a+εz) ↔ (b+τz) =/parenleftbiga↔ (b+τz)/parenrightbig+εz=(a↔b)+(ε+τ)z. Addition isdeﬁnedmodulo2,so ε+τ∈ {0,1}andwe have (a↔a)ψ(a′↔b′), as desired. The congruence θzis minimaland contains ψ, soθz=ψ. ◦ Wewillnowshowthatit isexponentiallyhardforaclas- sical algorithm to distinguish between θzand the identity congruence 0for randomly chosen z. Letφbe the homo- morphism φ:A→A/θz, a/ma√sto→a/θz, so thatker (φ)=θz. Suppose that we have a classical procedure for solving HKP(A), and that this procedure evaluates φon a subset E={e1,...,eℓ} ⊆A. We are able to distinguish θzfrom0 if and only if |φ(E)|<|E|. By the claim above, for distinct ei,ej∈E,wehavethat φ(ei)=φ(ej)ifandonlyif ei=ej+z. The element zwas chosen randomly, so the probability of this occurring is 1 /(|A| −1). It follows that for ﬁxed j, the probability that ei=ej+zforsomei

Question 210 multiple-choice

Quantum-inspired classical algorithms have enabled efficient computation for large-scale matrix problems in theoretical physics and computational mathematics. One notable application is the detection of projectors and calculation of associated eigenvalues using randomized sampling methods. Which feature of l2-norm sampling in quantum-inspired randomized algorithms most directly allows the query complexity for approximating inner products to be independent of matrix dimension? 1) The sampling probability is proportional to the squared magnitude of vector entries, focusing queries on significant components. 2) All entries of the matrix are accessed in a uniform random order, ensuring dimension independence. 3) The matrix is decomposed into eigenvectors, allowing dimension-free estimation. 4) Queries are restricted to the diagonal entries of the matrix, reducing complexity. 5) An error correction code is applied to the sampled entries, removing dependence on size. 6) The algorithm exploits symmetry properties of the matrix, eliminating dimension effects. 7) The l1-norm is used for sampling, which naturally scales with dimension.

✓ Correct Answer:

The correct answer is 1) The sampling probability is proportional to the squared magnitude of vector entries, focusing queries on significant components..

📚 Reference Text:

the condition of validity of the super-gravity approximation in describing the physics of high dimension CFT operators. – 18 – JHEP05(2023)1915 Classical matrix algorithms for projector detection: randomised clas- sical algorithm In section 4 we described a classical algorithm which related by holography to the quantum detection of projectors in section 3. A more conventional approach to classical/quantum comparison of algorithms is to used quantum-inspired randomised classical algorithms [ 29]. In this section we will develop such an approach for the projector detection task. 5.1 Randomised classical algorithm We can compare classical and quantum algorithms for this problem of detecting projector states. In the classical problem, we are given a set of matrices of size D, one matrix corresponding to every projector PR. We can construct such matrices by taking the group algebra elements PR=dR n!/summationdisplay σχR(σ)σ (5.1) in a reducible representation of dimension D. We need a reducible representation which, in its decomposition into irreducibles, contain every irrep Rat least once. We will work with the regular representation. TASK 1: The task of ﬁnding the T2eigenvalue The input •We are given, as a black-box which we can query, a projector PRas a matrix DV(PR) of size DimV×DimV. •We are also given, as a black box, a matrix DV(T2). Algorithm •We query the DV(PR)black box and ask for a pair of numbers (i,j) 1≤i,j≤DimV with a non-zero entry (DV(PR))ij. •We consider the matrix equation (DV(T2)DV(PR))ij=/summationdisplay k(DV(T2))ik(DV(PR))kj=χR(T2) dR(DV(PR))ij(5.2) We query the i’th row{(DV(T2))ik:k∈{1,···,DimV}}of the matrix DV(T2)and thej’th column{(DV(PR))kj:k∈{1,···,DimV}}ofDV(PR). •The sum over kin the equation (5.2)is computing a scalar product of two vectors. The two vectors are real. We will do the calculation of this inner product using a randomized classical algorithm ( l2-norm sampling as in [ 29]). This will lead to a complexity which is independent of DimVand instead depends on the admissible error. This step is detailed below. •By taking the ratio of (DV(T2)DV(PR))ijover (DV(PR))ijwe can get the eigenvalue. – 19 – JHEP05(2023)191We now turn to the key computational step above is the multiplication of the i’th row ofDV(T2)with thej’th column (DV(PR)). This is the computation of a dot-product of two vectors in dimension dimV. We can immediately apply known results in quantum-inspired classical algorithms. We use the result in Proposition 4.2 of [ 29], which gives the number of queries required to calculate the inner product of two vectors with error/epsilon1||x||||y||, and with success probability (1−δ)). The number of queries needed is O(1 /epsilon12log(1 δ)). Here we are calculating (T2PR)ijwhich isχR(T2)/dRtimes (DV(PR))ij. We know the range of the normalized characters extends within the interval [−n2,n2]and we know they are integers. From this we extract an estimate for the biggest /epsilon1we can aﬀord. This will lead to query complexity which is polynomial in n, as we now show. Let us consider the application of Proposition 4.2 of [ 29] to projectors in the regular representation. Some important lemmas (to be proved in appendix A): Lemma 1./summationdisplay γ(Dreg γµ(PR))2=d2 R n!(5.3) Lemma 2./summationdisplay τ(Dreg στ(T2))2=n(n−1) 2(5.4) Useful to choose

Question 211 multiple-choice

Quantum algorithms for discrete logarithms and factoring leverage group-theoretic properties and quantum techniques like amplitude amplification to solve hard computational problems. The structure and order of the underlying group play a critical role in algorithmic efficiency and exactness. In the context of quantum discrete logarithm algorithms where the group order q is unknown and potentially composite, which method enables the algorithm to be compiled into a single quantum circuit without requiring knowledge of q's factorization? 1) Using only classical pre-processing to estimate the group order 2) Implementing a separate quantum circuit for each possible factor of q 3) Employing phase estimation to determine the order before running the algorithm 4) Designing all required quantum gates directly from q, ensuring no new special gates are introduced during recursive runs 5) Applying Grover's search algorithm to find the prime factors of q within the circuit 6) Limiting the algorithm to groups of prime order only 7) Utilizing classical post-processing to combine results from multiple quantum circuits

✓ Correct Answer:

The correct answer is 4) Designing all required quantum gates directly from q, ensuring no new special gates are introduced during recursive runs.

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qas a tensor product of a QFFT with odd order and a standard one with order a power of 2. Similarly, of course, we can generalise to QFF T’s over ﬁnite Abelian groups, not just cyclic ones. Also the discrete logarithm algorithm can be generalised to arbitrary orders q. Given the exact QFFT q, the algorithm will be successful whenever the ﬁrst number in the measured pair ( x,xamodq) is coprime to q. So the success probability is φ(q)/qwhereφ(q) is the Euler totient function. If we know the factorisation of q, this is easily calculated and so amplitude ampliﬁcation can be used to make the algorithm exact. 7 5.1.1 Factorisation of the order of the dlog not known In the following we give a more involved solution for the case when the f actori- sation of the order qis not known. It consists of O(logq) runs of (variants of) the dlog quantum circuit. What is important is, that these variants s till only use the special gates calculated (eﬃciently) at the beginning from q. In the ﬁrst run it is enough, as before, to use amplitude ampliﬁcation only to get rid of the case x= 0. We now measure a pair ( x,xamodq). Ifxis coprime toqwe can directly calculate aand are done. If gcd( x,q) =d >1, we still get some information about a, namely a′=amodq/d, and of course the factor d ofq. Now we have a=a′+a′′·q/d, where, in a standard way, a′′can be found by solving the dlog problem with ˜ α=αq/dand˜β=βα−a′. This dlog problem has smaller order, as ˜ αd=e, but we want to reuse the original quantum circuit for order q. If in this original circuit we simply replace α,βwith ˜α,˜β, we get (after the two QFFT’s): 1√ dd−1/summationdisplay k=0|k·q/d, a′′k·q/dmodq/an}⌊ra⌋k⌉tri}ht (Note that this is essentially the same as/summationtext k|k,a′′kmodd/an}⌊ra⌋k⌉tri}ht.) We want to avoid only the case k= 0, but in order not to introduce new “special” gates, we prefer to eliminate 3/4 of all states, such that one step of standa rd amplitude ampliﬁcation will lead to an exact solution. We can e.g. only retain the las t quarter of the values k= 0...d−1, although, if d is not divisible by 4, we will have to “partially tag” some states. (This can be done by appending a qubit in statec|0/an}⌊ra⌋k⌉tri}ht+s|1/an}⌊ra⌋k⌉tri}htwith|s|2= 1/4,1/2 or 3/4.) Now, like in the ﬁrst step, we will either directly get a′′, or will gain partial information on a′′, together with a factor of d. This can be iterated (at most O(logq) times) till the order of the dlog problem is small. Note that in our construction we have taken care not to introduce new “special” gates during the computation. This means that really the O(logq) quantum runs can be put together into one quantum circuit whose g ates can be computed from qalone (without knowing its factorisation). 5.2 No exact factorisation algorithm Let us also note that it is not clear how to make Shor’s integer factor isation algorithm exact with the

Question 212 multiple-choice

In quantum topology and modular representation theory, operators S and T arising from the quantum double of a finite-dimensional Hopf algebra play a crucial role in the construction of topological invariants and the representation of mapping class groups. Their relations encode important algebraic and topological information. Which property holds when the antipode operator S is restricted to the center Z(D) of the quantum double of a Hopf algebra D? 1) S acts as the identity on the center. 2) S becomes nilpotent of order 2 on the center. 3) S maps every element of the center to zero. 4) S exchanges the center with the commutator subalgebra. 5) S is involutive on the center, meaning S² acts as the identity. 6) S projects the center onto the space of primitive elements. 7) S acts as scalar multiplication by the central element v on the center.

✓ Correct Answer:

The correct answer is 5) S is involutive on the center, meaning S² acts as the identity..

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Sbe deﬁned as in (2.55) and T by multiplication with v. Then ST−1S=ν5TST (2.67) Proof :If we apply η◦S−1⊗1⊗Sfor some η∈D(A)∗to both sides of the equation in Lemma 3 we ﬁnd with τ(M) =S⊗S(M) that S◦f/parenleftBig λ⊳S◦f(η)/parenrightBig = (λ⊗S)/parenleftBig (S(f(η))⊗1)M/parenrightBig =η◦S−1⊗λ⊗S/parenleftBig (τ(M)⊗1)(1⊗M)/parenrightBig =/summationdisplay ijη/parenleftBig S−1(f(λ)′fifj)/parenrightBig ⊗S(ei)S(f(λ)′′)ej(2.68) Inserting into (2.68) the forms λ=µD⊳ρandη=µD⊳Afor some A,ρ∈D(A) and by using the deﬁnition (2.55) we ﬁnd: S◦Lρ◦S(A) =/summationdisplay ijµD/parenleftBig AS−1/parenleftBig/parenleftBig S−1(S(ρ))/parenrightBig′fifj/parenrightBig/parenrightBig S(ei)S/parenleftBig S−1(S(ρ))′′/parenrightBig ej =/summationdisplay ijµD/parenleftBig AS−1(fj)fiS−2(S(ρ)′′)/parenrightBig S2(ei)S(ρ)′ej (2.69) 24 HereLρis the left multiplication with ρ. The left hand side of the assertion (2.67) is now found by specializing ρ=v−1whereLv=T. In order to evaluate the right hand side of (2.69) we notice that Lemma 12 implies the following identities: ∆(S(v−1)) =ν5∆(v) =ν5v⊗vM−1=ν5v⊗vR−1τ(R−1) =ν5v⊗vS3⊗S2(R)1⊗S(τ(R)) =ν5/summationdisplay klvS3(ek)fl⊗vS2(fk)S(el) ReplacingS(v−1)′⊗S(v−1)′′in (2.69) by this expression yields the assertion: ST−1S=ν5/summationdisplay ijklµD/parenleftBig AS−1(fj)fivfkS−1(el)/parenrightBig vS2(ei)S3(ek)flej =ν5/summationdisplay jl/summationdisplay ikµD/parenleftBig vAS−1(fj)(fifk)S−1(el)/parenrightBig vS2(eiS(ek))flej by(2.13)=ν5/summationdisplay jlµD/parenleftBig vAS−1(fj)S−1(el)/parenrightBig vflej (2.70) We readily identify the last equation with the right hand side of (2.67). This completes the proof. ✷ TheSmatrix was originally deﬁnd as an element in the End-set of the coend of the representation category. As a map on D(A) it therefore intertwines the ad−- action ( see (2.60)) of the algebra on itself. (This property can also be inferred directly from Lemma 2.1.).) The same is true for multiplications with central elements as for exa mple forT. Hence the center Z(D(A)) - which is the invariance of the ad−- action - is an invariant subspace of both operators. It follows immediately from (2.59) tha t the restriction of Γ to the center is the usual antipode Sand thus involutive. We summarize these observations and the relations found in (2.60), (2.61), and (2.67) in the following theorem: Theorem 1 Suppose D(A)is the double of a ﬁnite dimensional Hopf algebra. Assume thatD(A)admits a balancing and let νandωbe as in Proposition 8 . Furthermore, let T be the multiplication with v, andSandΓbe deﬁned as in (2.55) and (2.59), respectively. Then 1. The generators deﬁne a projective representation of the m apping class group D:= πo(Diff(T,D))of torus maps ﬁxing a disk with the following relations: S2=ωΓ−1TΓ = ΓT (2.71) /parenleftBig ST/parenrightBig3=ν3Γ−2=ν3ad−(v) (2.72) 25 2. The mapsSandTstabilize the center Z(D(A)). The restrictions SandTsatisfy (ST)3=ν3S2=ωS±1TS=ST (2.73) whereSis the involutive map given by the restriction of the antipod e to the center. The relations in (2.73) show that SandTdeﬁne a projective representation of SL(2,Z). The normalization of the S-operation was deﬁned by the canonical normal- ization of µD. For the computation of topological invariants it is often more conv enient to have a normalization for which the operators are inverted if we inv ert the braided structure. For a given balancing kletS′andT′be the analogous operators deﬁned with respect to R′=τ(R−1). Then as u′= ˆuwe have thatT′=T−1is already correctly normalized. A computation similar to the one in Proposition 9 yields S′=ωS−1. Thus it is the matrix S.:=ν2S−1which inverts under inversion of the braided structure. For these generators we have the relations: S4.= 1 ( S.T)3=ν−3S2. (2.74) Comparing (2.74) to relations in [T] and [RT]

Question 213 multiple-choice

Quantum algorithms have shown remarkable efficiency in solving certain number-theoretic problems, sometimes outperforming classical approaches. Some of these problems involve quadratic residues and modular arithmetic, which are central concepts in computational number theory. Which feature distinguishes the Shifted Legendre Symbol Problem from many previously studied quantum algorithm problems such as factoring and discrete logarithm? 1) It can be solved efficiently by known classical algorithms. 2) It involves only linear algebraic techniques for its solution. 3) It is a variant of the standard Hidden Subgroup Problem framework. 4) It does not utilize quadratic residues or modular arithmetic. 5) Its complexity has been fully classified in classical computational theory. 6) It does not fit into the Hidden Subgroup Problem framework that underlies most famous quantum algorithms. 7) It relies solely on randomness rather than structured number-theoretic sequences.

✓ Correct Answer:

The correct answer is 6) It does not fit into the Hidden Subgroup Problem framework that underlies most famous quantum algorithms..

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Question 214 multiple-choice

Quantum phase estimation is a fundamental algorithm in quantum computing, used to determine the eigenphase of a unitary operator and is pivotal for applications such as quantum simulation and factorization. Advanced implementations can utilize qudits—quantum systems with d levels—instead of qubits, offering distinct computational advantages. Which statement best explains why using qudits (d-dimensional quantum systems) instead of qubits in quantum phase estimation algorithms can be advantageous when coherence time is limited? 1) Qudits allow the use of classical post-processing to eliminate the need for error correction. 2) Qudits increase the number of oracle queries required for each measurement cycle. 3) Qudits reduce the need for auxiliary registers by encoding all information in a single state. 4) Qudits enable measurement outcomes to be determined without the inverse Fourier transform. 5) Qudits guarantee zero statistical uncertainty from photon detection in experiments. 6) Qudits make it impossible to extract the global phase after computation. 7) Qudits provide log₂(d) greater computational capacity and accuracy, or log₂(d) fewer computational steps, decreasing total computation time exposed to decoherence.

✓ Correct Answer:

The correct answer is 7) Qudits provide log₂(d) greater computational capacity and accuracy, or log₂(d) fewer computational steps, decreasing total computation time exposed to decoherence..

📚 Reference Text:

parentheses of Fcand Fqrefer to ± 1 σuncertainty from photon statistics.NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-28767-x ARTICLE NATURE COMMUNICATIONS | (2022) 13:1166 | https://doi.org/10.1038/s41467-022-28767-x | www.nature.com/naturecommunications 5 coefﬁcients Ai. The output state of the xregister can be derived as ω/C0A0A1;A2; :::;An/C12/C12/C11 ,w h e r et h e A1;A2; :::;An/C12/C12/C11 state can be directly read out in its computational basis ( A0is lost as a global phase). From the experimental results in Fig. 4b, c, h, i one can therefore determine the multi-value function with A1={0, 1, 2, 3}, respectively, by a single query of the oracle. Details of the generalisation of the Deutsch ’s algorithms are provided in Supplementary Note 4. Benchmarking of d-ary phase estimation and order ﬁnding . Quantum phase estimation and order ﬁnding are two of the most featured quantum Fourier transform ralgorithms, that are essential to molecular simulation66and fast factorisation25. Kitaev ’s scalable implementation of both algorithms (in binary)26–29has been reported in several leading quantum platforms67–72. The remarkable idea is to replace the 2 nqubits bya single qubit in the auxiliary x-register, but at the expense of repeating m-sequences of single-qubit measurement and single- qubit feedforwarded operation, see quantum circuits in Fig. 5a. In Kitaev ’s phase estimation and order- ﬁnding algorithms, the computational capacity is determined by the number of n-qubits in the y-register, and the computational accuracy is determined by the number of m-sequences in the x-register. In this respect, one can see processing quantum algorithms with qudits results in nontrivial advantages: a log 2(d) larger computational capacity, and log 2(d) higher computational accuracy or log 2(d)-less com- putational steps to achieving the same precision, as shown in Supplementary Fig. 3b, which could be important to quantum computers with limited coherence time. In the quantum phase estimation, we aim to compute the eigenphase ϕof an unitary as Oψ/C12/C12/C11 =ei2πϕψ/C12/C12/C11 , given the eigenstate of ψ/C12/C12/C11 .T h ee i g e n p h a s eo f ϕcan be described in d-ary as 0. ϕ1ϕ2… ϕm−1ϕm,w h e r e mdenotes iterative steps determining the approx- imation accuracy26,29,a n de a c hd i to ft h ep h a s ei si n [0, 1,..., d−1]67,69,73.W et a k et h e s-th step as an example (see quantum circuit in Fig. 5a). We prepare an input state of 1ﬃﬃ dp∑d/C01 j¼0j/C12/C12/C11 ψ/C12/C12/C11 and perform the MVCU gate, that results in a state of1ﬃﬃ dp∑d/C01 j¼0eij2πð0:ϕsϕsþ1:::ϕmÞj/C12/C12/C11 ψ/C12/C12/C11 .T h e n ,t h e x-register qudit state is feed-forwardly rotated around the Pauli Zdbasis as diag[1 ;ei2πθs; :::;ei2πðd/C01Þθs], where the rotation angle θsof−0.0ϕs +1ϕs+2…ϕmis given by previous measurement outcomes. Remark- ably, implementing an inverse Fdin the x-register returns an output state as ϕs/C12/C12/C11 ¼sji( s e eS u p p l e m e n t a r yN o t e5 ) .M e a s u r i n gt h e x- register in the computational basis of sjitherefore allows the extraction of the s-th dit of the dit expansion.

Question 215 multiple-choice

The Hidden Subgroup Problem (HSP) is a foundational challenge in quantum computing, influencing the efficiency of quantum algorithms for tasks like factoring and discrete logarithms. Query complexity, which counts the number of times an algorithm accesses a function oracle, varies significantly depending on the group properties involved. Which of the following best describes the total query complexity of Algorithm 4 for solving the Hidden Subgroup Problem when the underlying group is abelian and possesses a subgroup of order n/m intersecting the hidden subgroup only at the identity? 1) O((log n)√(m/n)) 2) O((log m) n/m) 3) O((log m)√(n/m)) 4) O(√(κn/m) ln(κn/m)) 5) O(n log m) 6) O(√(n/m) + log m) 7) O((log m)√(m/n))

✓ Correct Answer:

The correct answer is 3) O((log m)√(n/m)).

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below: 1. if G is abelian, the algorithm has query complexity O(√ κn/m), and 2. if G is not abelian, the algorithm has query complexity O(/radicalbig (κn/m)ln(κn/m)). Proof. When H1={e}, the algorithm is identical to Algorithm 1, and its correctn ess follows by Theorem 5.2. Suppose H1is not the trivial subgroup. Since the algorithm reports a collision a,bonly when a−1b/\⌉}atio\slash∈H1, it gives the correct answer when H1=H. Suppose H1/\⌉}atio\slash=H. By hypothesis, Ghas a subgroup G0of order at least n/msuch that G0intersects H1only in the identity element. Then |G0|≤n/m1≤n/2; otherwise, we would have two distinct elements of G0in the same coset of H1, which implies that |G0∩H1|>1. If it does not ﬁnd a collision in earlier iterations, the algor ithm ﬁnds a subgroup G1with|G1|≥ n/mand G1∩H1={e}in an iteration with k≥k0≥0 (recall that k0:=⌈log2m1⌉−1≥0, since m1≥2). Let S1,S2be the generating pair for G1computed by the algorithm. Suppose G1H1=G. Consider any element h∈H\H1. We have h=g1h1for some g1∈G1 and h1∈H1. So g1=hh−1 1∈G1∩H, and g1/\⌉}atio\slash∈H1. We also have g1=abfor some a∈S1 and b∈S2, so f(a−1) = f(b). The algorithm queries the oracle fat the elements x−1and y, for allx∈S1and y∈S2, so it ﬁnds and returns a collision. Now suppose G1H1/\⌉}atio\slash=G, so that the algorithm also ﬁnds an element g∈G\(G1H1)in the same iteration. IfG1H=G, we have g=g1hfor some g∈G1and h∈H. Since g/\⌉}atio\slash∈G1H1,g1h/\⌉}atio\slash∈G1H1, and h/\⌉}atio\slash∈H1. Suppose g1=ab, with a∈S1and b∈S2. Then a−1g=bh, and f(a−1g) = f(b). The algorithm queries the oracle fat the elements x−1gand y, for all x∈S1and y∈S2, so it ﬁnds f(a−1g) = f(b), and returns a collision. IfG1H/\⌉}atio\slash=G, as in Lemma 5.1, there are two distinct elements g1,g2∈G1such that g−1 2g1∈H. Since G1∩H1={e}, and g1/\⌉}atio\slash=g2, we have g−1 2g1∈H\H1. We also have g−1 2g1=ab, for some a∈S1and b∈S2. The algorithm queries the oracle fat the elements x−1and y, for allx∈S1and y∈S2, so it ﬁnds f(a−1) = f(b), and returns a collision in this case as well. Assuming n/m≤|G0|≤κn/m≤nfor some κ≥1 with G0as in the statement of the theorem, the algorithm executes iterations with k≥ℓ, whereℓis such that κn/m∈/bracketleftbig n/2ℓ+1,n/(⌊2ℓ⌋+1)/bracketrightbig . So the query complexity of the algorithm follows by same kind of analysis as in Theorem 5.2. Thus, whenever there is a large enough subgroup G0that intersects with a proper subgroup H1 of the hidden subgroup Honly in the identity element, Algorithm 3 gives us an element hofH that is not in H1. Along with H1, the element hgenerates a strictly larger subgroup of H. As long as the condition above holds for all proper subgroups of H, we can repeat Algorithm 3 until we ﬁnd a set of generators for the hidden subgroup. This process i s described in Algorithm 4. Theorem 6.3. Algorithm 4 (Find-Subgroup) solves the Hidden Subgroup Pro blem over a group G and ﬁnds the hidden subgroup H when G has a subgroup G 0of order n /m such that|G0∩H|=1, where n and m are the orders of G and H, respectively. Moreover, the alg orithm makes • O((logm)√ n/m)queries when G is abelian, and • O((logm)/radicalbig

Question 216 multiple-choice

Quantum algorithms often leverage abstract mathematical structures to achieve computational speed-ups over classical approaches. Diagrammatic techniques provide alternative frameworks for understanding the underlying principles and correctness of these algorithms. Which feature of the diagrammatic approach to the abelian Hidden Subgroup Problem quantum algorithm enables its extension beyond finite-dimensional complex quantum theory? 1) Reliance on standard group-theoretic decompositions 2) Dependence on complex vector space representations 3) Use of strongly complementary observables to capture core quantum properties 4) Restriction to finite abelian groups only 5) Emphasis on classical simulation techniques 6) Requirement for explicit Fourier transform calculations 7) Focus on probability distributions over group elements

✓ Correct Answer:

The correct answer is 3) Use of strongly complementary observables to capture core quantum properties.

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Question 217 multiple-choice

In quantum computation, group-theoretical methods are essential for constructing efficient algorithms, particularly those involving representation theory and the quantum Fourier transform. The structure and properties of semidirect product groups like Zp ⋊ Zq play a crucial role in enabling universal quantum computation. Which one of the following conditions must be satisfied for the group D(Zp ⋊ Zq) to be sufficient for universal quantum computation using anyons with non-abelian statistics? 1) p and q are coprime and p divides (q−1) 2) Both p and q must be odd integers greater than 2 3) p divides q and p and q are square-free 4) q is a power of p and both are composite 5) p and q are both even and q divides (p+1) 6) p and q are prime numbers and q divides (p−1) 7) Both p and q must be prime and p divides (q−1)

✓ Correct Answer:

The correct answer is 6) p and q are prime numbers and q divides (p−1).

📚 Reference Text:

decomposed into irreps of Z(ghd). These multiplicities depend on ρ,σanddin general. 1. The ﬁrst step takes us from the basis |ha,v1,gb,v2/a\}bracketri}htto the basis|(ghd)k,t,v 1,v2/a\}bracketri}ht, wherev1andv2are vectors in the irrep spaces of ρandσ. This step is the same as in the previous case (for ﬂuxon irrep s) since we do not operate on the vectors v1andv2. The transversal is picked in the way described above. 2. The second step can be done if we know how to decompose any ir rep ofZ(g) andZ(hd) into irreps ofZ(g)∩Z(h)dand then perform Clebsch-Gordan transform over the group Z(g)∩Z(h)d. This step may be done for particular groups. 3. The third step can be done using the procedure described ab ove for decomposing induced irreps, since νis an irrep of Z(g)∩Z(hd) (the group Babove) and it is induced to Z(ghd) (the group Aabove). This gives an eﬃcient algorithm for the CG transform under ce rtain conditions. Next, we show how these conditions are satisﬁed for D(Zp⋊Z q) which is suﬃcient for universal quantum computation [ 28]. 7.2 Fourier and Clebsch-Gordan transforms over D(Zp⋊Z q) It is shown in [ 28], that the group G=Zp⋊ Z qcan be used to perform universal quantum computation whenpandqare prime and q|(p−1). Here we show how to perform the Fourier and Clebsch-Gorda n transforms over D(G). First, in order to fully describe the group, we have to pick the homomorphism from Zqto Aut(Zp). Such homomorphisms are characterized by elements αsuch thatαq= 0 modp. Having picked such an α, we see that the group multiplication in Gis (a1,b1)(a2,b2) = (a1+a2αb1,b1+b2). Notice that whenα= 1, we obtain the direct product of ZpandZq. We now describe its irreps. There are qone dimensional irreps and ( p−1)/q,qdimensional irreps. The one dimensional irreps are obtained as the extension of the t rivial irrep of ZptoGand tensored with each of theqirreps of Zq. Theqhigher dimensional irreps are obtained as induced represen tations from a non-trivial irrep ofZp. The irreps of Zpare characterized by kand are of the form exp(2 πika/p ) =ωka pfor a group elementa. All those kin the orbit of kαbforb∈Zqwill be induced to the same irrep of G. The induction can be given as ρk(a,b) =/summationdisplay s∈Zqωkaα−s p|s/a\}bracketri}ht/a\}bracketle{ts−b|. We now describe the centralizers of G. The centralizer of the identity element (0 ,0) isG. The centralizer of any element ( a,b), wherea/\e}ati≀\slash= 0 isZp. The centralizer of (0 ,b), whereb/\e}ati≀\slash= 0 isZq. 26 The Fourier transform is now easy to construct. Notice that s ince two of the centralizers are abelian, their QFT is eﬃcient. For the QFT over G, ﬁrst take the group basis |a,b/a\}bracketri}htand convert it into the basis |z,t/a\}bracketri}ht, wherez∈Zpandtis a transversal which can be picked to be an element of Zq. Now perform a QFT overZpin the ﬁrst register. Then conditioned on the value in the ﬁrs t register perform a second transform as follows. If the value in the ﬁrst register is a non-trivial irrep ofZp, then do nothing since we already have an irrep of G. If the ﬁrst

Question 218 multiple-choice

Quantum algorithms often exploit group-theoretic structures and specialized quantum operations to solve problems more efficiently than classical methods. One prominent example is solving the quantum hidden subgroup problem using state preparation and measurement techniques that leverage group properties. Which technique is specifically designed to amplify the probability of obtaining a quantum state suitable for parity measurement in dihedral hidden subgroup problems, thereby reducing the required runtime from exponential to subexponential? 1) Grover's search algorithm 2) Quantum phase estimation 3) Simon's algorithm 4) Amplitude amplification 5) Quantum walk 6) Kuperberg sieve 7) Shor's factoring algorithm

✓ Correct Answer:

The correct answer is 6) Kuperberg sieve.

📚 Reference Text:

ˆH=H=eH=h(b;1)i, and it sufﬁces to consider subgroups of the formh(b;1)i: 57 We have (b;1)2= (b;1)(b;1) = b+(1)b+2n2;2 = 2n2;0 : (4.72) Since 2n2;0 corresponds to j2=1, the QHSP reduces to ﬁnding a hiddenp1, and ulti- mately to determining b. 4.5.2 Coset Sample Generation Procedure LetjREG 1i,jREG 2idenote the top (input) register and the bottom (output) register, re- spectively. The algorithm starts with a coset sample generation procedure (CSG)that produces single-qubit states of a particular form, with bencoded in a phase factor. The routine CSG is given by the following sequence. 1. Create the uniform superposition over all elements of Q2n, i.e., the state jy0i=1p 2nå (x;y)2Q2njx;yi: (4.73) 2. Query the black-box function fto obtain the state jy1i=1p 2nå (x;y)2Q2njx;yijf(x;y)i: (4.74) 3. MeasurejREG 2ito collapse the state (4.74) to jy2i=1p 2(jx;0i+jx+b;1i): (4.75) 58 4. Apply the QFT†tojREG 1ito produce the state jy3i=1p 2n+12n1 å z=0 e2pizx 2njz;0i+e2piz(x+b) 2njz;1i =1p 2n2n1 å z=0e2pizx 2njzi 1p 2 j0i+e2pizb 2nj1i : (4.76) 5. MeasurejREG 1ito obtain the state jyzi=1p 2 j0i+e2pizb 2nj1i (4.77) for some zchosen uniformly at random from f0;1;:::;2n1g. We now exploit the form of jyzito ﬁnd the least signiﬁcant bit of b. Proposition 4.5.3. All the bits of b can be determined with n iterations of an algorithm that ﬁnds the least signiﬁcant bit of b. Proof. The group Q2ncontains two subgroups isomorphic to Q2n1, namely, G1=f(2a;0);(2a;1)ja2Z2n1g (4.78) G2=f(2a;0);(2a+1;1)ja2Z2n1g: (4.79) Ifbis even, then HG1, where His the hidden subgroup. On the other hand, if bis odd, then HG2. Once an algorithm determines the parity of b, the hiding function fis restricted to G1or G2, and the algorithm is called with a modiﬁed oracle. The algorithm outputs the least signiﬁcant bit of bwithin an appropriate subgroup, and this result corresponds to the second least signiﬁcant bit of binQ2n. Therefore, niterations of the algorithm determine all bits of b. 59 4.5.3 The Kuperberg Sieve The goal is to prepare the state jy2n1i=1p 2 j0i+e2pi(2n1)b 2nj1i! =1p 2 j0i+epibj1i =1p 2 j0i+(1)bj1i : (4.80) which is subsequently subjected to a measurement in the Hadamard basis ji=j0ij1ip 2(4.81) to extract the least signiﬁcant bit of b. The extraction works as follows. Proposition 4.5.4. The parity of b can be determined by measuring the state (4.80) in the Hadamard basis. Proof. The Hadamard gate Happlied to an arbitrary state jji=aj0i+bj1iis Hjji=aHj0i+bHj1i=aj0i+j1ip 2 +bj0ij1ip 2 =a+bp 2 j0i+abp 2 j1i: (4.82) Then application of Hadamard to the state (4.80) yields Hjy2n1i= 1+(1)b 2! j0i+ 1(1)b 2! j1i: (4.83) Ifbhas even parity, the measurement results in H1p 2(j0i+j1i) =j0i: (4.84) 60 If the parity of bis odd, the result is H1p 2(j0ij1i) =j1i: (4.85) However, the routine CSG produces the desired state (4.80) with an exponentially small probability of1 2n. The Kuperberg sieve resolves this difﬁculty by combining states of the form (4.77) to generate states of the same form but with preferred values of z. The Kuperberg sieve is essentially a sequence of m=pn1 routines such that the output of routine jis input to routine j+1 for

Question 219 multiple-choice

Quantum phase estimation algorithms, such as the Kitaev and Fourier approaches, are central to high-precision magnetometry, where the goal is to estimate magnetic flux with maximal accuracy even in the presence of environmental noise. The performance of these quantum algorithms is often compared to classical methods, especially in terms of their robustness and scaling behavior. In quantum-enhanced magnetometry, which property of the Kitaev and Fourier estimation algorithms most directly accounts for their superior stability and precision over classical methods when the system's passport function is irregular or affected by noise? 1) Their ability to eliminate measurement uncertainty entirely after a finite number of steps 2) Their reliance on entanglement to create superpositions of flux states 3) Their use of adaptive measurement delays to maximize sensitivity at all times 4) Their insensitivity to the choice of initial reference flux 5) Their capacity to perform measurements without environmental decoherence 6) Their strict dependence on monotonic and well-behaved passport functions 7) Their robustness against imperfections in the passport function, allowing precise estimation even when classical Bayesian inference fails

✓ Correct Answer:

The correct answer is 7) Their robustness against imperfections in the passport function, allowing precise estimation even when classical Bayesian inference fails.

📚 Reference Text:

the ﬁrstk=1,…, 4 steps of the Fourier (red panels) and Kitaev (blue curves) estimation algorithms. The magnetic ﬂux is measured with respect to a reference ﬂux (as explained in the Methods). The actual ﬂux value is shown by a thick black line in the ﬂux-step plane. The Kitaev algorithm starts at a zero delay τ=0 and a ﬁrst step returns a broad probability distribution with a single peak centered near the actual ﬂux value. During the run of the Kitaev algorithm this peak narrows down. The Fourier algorithm starts from the Ramsey measurement at large delay τ(s)=360 ns, with the ﬁrst step returning a probability distribution with six out of twelve ﬂux intervals assuming a non-vanishing value. Hence, this ﬁrst step selects half of the n=τ(s)/τ0~12 different ﬂux intervals ΔΦ mgiven by Δωm=Δω0+2πi/τs,m=0,…,n−1, where Δωm≡ω01(ΔΦ m)−ωdis the frequency interval corresponding to the ﬂux interval ΔΦ m, and determines the parity of the yet unknown index m∈[0,n−1] associated with the true ﬂux interval. In the second step, the Fourier algorithm proceeds to a shorter delay and rules out another half of the remaining six intervals. In the next two steps the algorithmdiscriminates between the remaining three alternatives and ends up with the correct ﬂux interval. The green line at the fourth step displays the probability distribution learned by the standard (classical) procedure during the same number of Ramsey measurements as was required by the quantum procedures. The distributions obtained at the step number 4 for the Kitaev and Fourier estimation algorithms and in the standard (classical) measurement are shown in the insetQuantum-enhanced magnetometry by phase estimation algorithms S Danilin et al. 5 Published in partnership with The University of New South Wales npj Quantum Information (2018) 29 exponent α, assuming a value α≈0.39 at short sensing times, while at large times αdecreases to a much smaller value ≈0.046. The scaling exponent 0.39 deviates from the shot noise exponent 1/2 due to the cases when the actual ﬂux value is located near the boundaries of the ﬂux interval Φ∈[Φ1,Φ161], where the passport function Pp(0,Φ) has an extremum and the scaling exponent for the standard procedure is reduced to 1/4, δΦ(t)∝t−1/4. As a result, the aggregated scaling exponent of Eq. ( 5) is reduced below 1/2. On the other hand, the crossover to α≈0.046 is a consequence of the irregularity of the passport function set by low-frequencynoise ﬂuctuations during the passport measurement. Indeed, at large sensing times, the standard procedure needs to distinguish ﬂuxes within a narrow interval where the passport function P p(τ= 0,Φ) has a non-regular and non-monotonic dependence on Φ.A s a result, the Bayesian learning procedure fails to converge to acorrect ﬂux value. In contrast, at a larger scale of Φ, the passport function is smooth and monotonic and the standard procedure behaves properly. These arguments are indeed con ﬁrmed by a numerical simulation with a regular passport function given by Eq.(1). Importantly, both our quantum metrological algorithms are more stable than the standard procedure with respect to passportimperfections

Question 220 multiple-choice

Quantum algorithms for order-finding leverage modular arithmetic and specialized quantum gates to solve number-theoretic problems exponentially faster than classical methods. Circuit efficiency and correct qubit register management are crucial for their successful implementation. Which operation is essential for extracting periodicity in quantum order-finding algorithms and is implemented through a unitary transformation analogous to the classical discrete Fourier transform? 1) Modular exponentiation 2) Hadamard transformation 3) Modular addition 4) Controlled modular multiplication 5) Quantum Fourier Transform 6) Modular multiplicative inverse calculation 7) Greatest common divisor computation

✓ Correct Answer:

The correct answer is 5) Quantum Fourier Transform.

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Question 221 multiple-choice

In algebraic topology, Morava K-theory is a family of extraordinary cohomology theories that provide rich insights into the cohomological structure of finite groups, particularly through spectral sequence computations. These computations often involve group extensions, differentials, restriction and transfer maps, and Steenrod-like operations. When analyzing the Lyndon–Hochschild–Serre spectral sequence for a group extension in Morava K-theory, what is the role of the permanent cycles in determining the structure of the cohomology ring? 1) Permanent cycles establish the action of the transfer map on subgroup cohomology. 2) Permanent cycles are used to compute the induced representation of the group on its classifying space. 3) Permanent cycles determine the decomposition of the module into free and torsion parts only. 4) Permanent cycles dictate which elements are killed by the d₃ differential exclusively. 5) Permanent cycles are responsible for the calculation of the E₁ page in the spectral sequence. 6) Permanent cycles identify generators of trivial summands that do not interact with differentials. 7) Permanent cycles are elements that survive to the E_∞ page and represent actual elements in the cohomology ring after all differentials have been accounted for.

✓ Correct Answer:

The correct answer is 7) Permanent cycles are elements that survive to the E_∞ page and represent actual elements in the cohomology ring after all differentials have been accounted for..

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Question 222 multiple-choice

Quantum computing offers significant advantages for specific mathematical operations, especially those involving Fourier transforms. However, encoding and extracting data from quantum states pose practical challenges in algorithm design. Which approach allows quantum algorithms to directly encode discrete Fourier coefficients for easier extraction and manipulation, but requires more qubits for fixed-point representation compared to amplitude encoding? 1) Classical Fast Fourier Transform using amplitude encoding 2) Quantum Fourier Transform with amplitude encoding 3) Quantum spectral methods for linear differential equations 4) No-cloning theorem-based state replication 5) Oracular input encoding with classical post-processing 6) Standard qubit amplitude encoding in quantum circuits 7) Quantum Fourier Transform in the Computational basis (QFTC)

✓ Correct Answer:

The correct answer is 7) Quantum Fourier Transform in the Computational basis (QFTC).

📚 Reference Text:

with complexity O log 2NðÞ/C0/C1 for prob- lem size Nhas exponential speed-up compared to the classical fast Fourier trans- form (complexity O NlogNðÞ ) and plays an important role in quantum computation as an essential part of many quantum algorithms. The exponential speed-up realized is due to superposition and quantum parallelism. However, in same quantum algo-rithms, the Fourier coefficients may be needed in the computational basis [21]. Here, the two different encoding methods are illustrated using the discrete Fourier Transform (DFT). The QFT performs the DFT in terms of amplitudes as, X N/C01 j¼0xjjji!XN/C01 k¼0ykkji (1) The QFT performs a DFT on a list of complex numbers, and the result is stored as amplitudes of a quantum state vector. In order to extract the individual Fouriercomponents, measurements need to be performed on the quantum state vector.Therefore, the QFT is not directly useful for determining the Fourier-transformedcoefficients of the input state. However, the QFT is widely used as a subroutine inlarger algorithms. In contrast to the amplitude encoding in Eq. (1), Zhou et al. [21] presented a quantum algorithm computing the Fourier transform in the computa- tional basis (termed QFTC). This quantum algorithm encodes Fourier coefficientswith fidelity 1 /C0δand digit accuracy εfor each Fourier coefficient. Its time com- plexity depends polynomially on log NðÞ, and linearly on 1 =δand 1 =ε. The QFTC, enables the Fourier-transformed coefficient to be encoded in the computational basis as follows, kji0ji!kjiy k/C12/C12/C11 (2) where ykcorresponds to the fixed-point binary representation of yk∈/C01, 1ðÞ using two ’s complement format. In the algorithm proposed by Zhou et al. [21], the input vector x!is provided by an oracle Oxsuch that, Ox0ji¼XN/C01 j¼0xjjji (3) which can be efficiently implemented if x!is efficiently computable or by using the qRAM that takes complexity log NðÞunder certain conditions [21]. Comparing Eq. (1) and Eq. (2), it is clear that encoding in the computational basis requires a number of additional qubits depending on the required fixed-point representation. 3Quantum Algorithms for Nonlinear Equations in Fluid Mechanics DOI: http:/ /dx.doi.org/10. 5772/intechopen.9 5023 3. Nonlinear problems on quantum computers An early work by Leyton and Osborne [20] introduced a quantum algorithm to solve nonlinear differential equations with an unfavorable complexity. Since then,very few works have considered quantum algorithms for nonlinear equations. Incontrast, algorithms for linear differential equations have continued to receivesignificant attention. As an example, advanced quantum spectral methods for dif-ferential equations were published recently by Childs and Liu [19]. A key contributing factor to the limited progress in algorithms for non-linear problems is the inherent linearity of quantum mechanics. For quantum algorithmsencoding information as amplitudes of a quantum state vector, nonlinear (prod-uct) terms cannot be obtained by multiplying these amplitudes by themselves, as aresult of the no-cloning theorem that prohibits the copying of an arbitrary quan-tum state. Furthermore, all quantum-gate operations (with the exception of mea-surements) in the quantum-circuit model used here need to be unitary andreversible. These requirements add further challenges to representing nonlinear terms when using the amplitude-based encoding approach. Specifically, in a

Question 223 multiple-choice

In quantum information theory, operators on tensor product spaces often exhibit interesting properties under permutations and averaging over the unitary group. The Flip (SWAP) operator plays a central role in simplifying calculations involving entanglement and random quantum states. Which operator identity directly connects the trace of a tensor product involving the Flip operator to the trace of the product of the original operators? 1) $\text{Tr}((A \otimes B)I) = \text{Tr}\text{Tr}$ 2) $\text{Tr}((A \otimes B)F) = \text{Tr}(AB)$ 3) $\text{Tr}((A \otimes B)F) = \text{Tr}\text{Tr}$ 4) $\text{Tr}((A \otimes B)I) = \text{Tr}(AB)$ 5) $\text{Tr}(AF) = \text{Tr}(A^2)$ 6) $\text{Tr}((A \otimes B)F) = \text{Tr}(A+B)$ 7) $\text{Tr}((A \otimes B)F) = \text{Tr}\text{Tr}$

✓ Correct Answer:

The correct answer is 2) $\text{Tr}((A \otimes B)F) = \text{Tr}(AB)$.

📚 Reference Text:

A=/summationdisplay π∈Sksgn(π)Vd(π). (39) 2The calculation of moments over the Haar measure is often referred to as the Weingarten calculus [57, 58]. 6 Now we show this linear combination is the zero operator, proving linear dependence. We consider the action of Aon an arbitrary product basis state |i1⟩⊗···⊗|ik⟩. Sincek>d , at least two tensor factors must have matching entry, i.e., there exist l̸=m∈[k] such that il=im. Due to anti-symmetrization, the output vector has to be the zero vector. In fact, we have: A|i1⟩⊗···⊗|ik⟩=AVd(τl,m)|i1⟩⊗···⊗|ik⟩ (40) =/summationdisplay π∈Sksgn(π)Vd(π)Vd(τl,m)|i1⟩⊗···⊗|ik⟩ (41) =/summationdisplay π∈Sksgn(π)Vd(πτl,m)|i1⟩⊗···⊗|ik⟩ (42) =/summationdisplay π∈Sksgn(πτ−1 l,m)Vd(π)|i1⟩⊗···⊗|ik⟩ (43) = sgn/parenleftig τ−1 l,m/parenrightig/summationdisplay π∈Sksgn(π)Vd(π)|i1⟩⊗···⊗|ik⟩ (44) =−A|i1⟩⊗···⊗|ik⟩. (45) ThereforeA|i1⟩⊗···⊗|ik⟩= 0 and so A= 0. In the following we define the identity permutation operator Iand the Flip operator F(also often referred to as SWAP operator) which are the permutation operators corresponding to the elements of the permutation group S2. Definition 12 (Identity and Flip operators) .The identity permutation operator Iis defined as the linear operator that leaves any tensor product state |ψ⟩⊗|ϕ⟩unchanged, that is: I(|ψ⟩⊗|ϕ⟩) =|ψ⟩⊗|ϕ⟩, for all|ψ⟩,|ϕ⟩∈Cd. (46) The Flip operator Fis defined as the linear operator that interchanges the order of the tensor factors of any product state |ψ⟩⊗|ϕ⟩, that is: F(|ψ⟩⊗|ϕ⟩) =|ϕ⟩⊗|ψ⟩, for all|ψ⟩,|ϕ⟩∈Cd. (47) Writing the Identity and the Flip in the computational basis, we have: I=d/summationdisplay i,j=1|i,j⟩⟨i,j|, F=d/summationdisplay i,j=1|i,j⟩⟨j,i|. (48) From this, it is evident that the Flip operator is Hermitian. Another key property of the Flip operator is the swap-trick , which states that for all operators A,B∈L(Cd), we have: Tr((A⊗B)F) = Tr(AB). (49) This property can be easily verified using the definition of the Flip operator. The swap-trick is particularly useful as it allows us to simplify calculations involving tensor products and permuta- tions, and it will be used extensively in the subsequent sections. We now present a corollary of Theorem 10, which plays a crucial role in many calculations involving Haar integrals in the context of quantum information, as we will see in section 9. In the following corollary we assume d>1(which makes sense for an n-qubit system, since d= 2n>1). Corollary 13 (First and second moment) .GivenO∈L/parenleftbig Cd/parenrightbig , we have: E U∼µH/bracketleftbig UOU†/bracketrightbig =Tr(O) dI. (50) GivenO∈L((Cd)⊗2), we have: E U∼µH/bracketleftbig U⊗2OU†⊗2/bracketrightbig =cI,OI+cF,OF, (51) where: cI,O=Tr(O)−d−1Tr(FO) d2−1andcF,O=Tr(FO)−d−1Tr(O) d2−1. (52) 7 Proof. According to Theorem (10), we have that the first-order moment operator is proportional to the identity operator (the only permutation operator associated with the permutation group of one element S1), i.e. E U∼µH/bracketleftbig UOU†/bracketrightbig =cII, (53) withcI∈C. Taking the trace to both sides, we deduce that cI=Tr(O) d. Moreover, using Theorem (10), we can see that that the second-order moment operator is a linear combination of the two permutation operators associated with the permutation group S2 which are the identity Iand the flip Foperator: E U∼µH/bracketleftbig U⊗2OU†⊗2/bracketrightbig =cI,OI+cF,OF, (54) withcI,O,cF,O∈C. To find these numbers, we left-multiply both sides by Iand take the trace, which gives us: Tr(O) =cI,Od2+cF,Od, (55) where we used the fact that Tr( I) =d2and Tr( F) =d. Similarly, by left-multiplying by Fand taking

Question 224 multiple-choice

In the study of integrable systems and algebraic geometry, τ-functions, Schur polynomials, and determinant formulas play a central role in representing solutions and encoding geometric structures such as Grassmannians. Understanding how these concepts interrelate is crucial for exploring the connection between symmetry and solvable models. Which mathematical structure acts as the generating function for local Grassmannian coordinates, encapsulates solutions to integrable hierarchies like KP and Toda, and is typically represented as a determinant involving Schur polynomials and matrix elements? 1) The Plücker relations 2) The fundamental representation 3) The H-matrix 4) The τ-function 5) The time variables si, ¯si 6) The Schur polynomial basis 7) The KP hierarchy

✓ Correct Answer:

The correct answer is 4) The τ-function.

📚 Reference Text:

in F(k)⊗F(k′)is: Ψ(k) [i1...ikΨ(k′) ik+1]i′ 1...i′ k′−1. (65) Operation Γ can be now rewritten in terms of matrix elements g(k)/parenleftiggi1...ik j1...jk/parenrightigg ≡ /an}bracketle{tΨi1...ik|g|Ψj1...jk/an}bracketri}ht= det 1≤a,b≤kgia jb(66) as follows: g(k)/parenleftiggi1...ik [j1...jk/parenrightigg g(k′)/parenleftiggi′ 1...i′ k jk+1]j′ 1...j′ k′−1/parenrightigg = =g(k+1)/parenleftiggi1...ik[i′ k′ j1...jk+1/parenrightigg g(k′−1)/parenleftiggi′ 1...i′ k′−1] j′ 1...j′ k′−1/parenrightigg (67) This is the explicit expression for eq.(9) in the case of fundamental r epresentations, and it is certainly identically true for any g(k)of the form (66).10 10To see this directly it is enough to rewrite the l.h.s. of (67) as g(k)/parenleftbiggi1...ik [j1...jk/parenrightbigg g[i′ k jk+1]g(k′−1)/parenleftbiggi′ 1...i′ k−1] j′ 1...j′ k−1/parenrightbigg 19 Let us note that one can easily construct from the minors (66) loca l coordinates in the Grassmannian [12]. Then, these coordinates satisfy a set of (bilinea r) Plucker relations [12], which are nothing but deﬁning equations of the Grassmannian consis ting of allk-dimensional vector subspaces of an n-dimensional vector space. Parametrizing determinants (66) by t ime variables (see (74)), one get a set of bilinear diﬀerential equations on the generating function of these Plucker coordinates, which is just τ-function [13]. Now let us introduce time-variables and rewrite (67) in terms of τ-functions. We shall denote time variables through si,¯si,i= 1,...,rin order to emphasize their diﬀerence from generictα,¯tαlabeled by all the positive roots αofG. Note that in order to have a closed system of equations we need to introduce all the rtimessifor allF(k)(thoughτ(k)actually depends only on kindependent combinations of these). Since the highest weight of representation F(k)is identiﬁed as |0/an}bracketri}htF(k)=|Ψ(k) 1...k/an}bracketri}ht, (68) we have: τ(k)(s,¯s|g) =/an}bracketle{tΨ(k) 1...k|exp/parenleftigg/summationdisplay isiRk(Ti +)/parenrightigg gexp/parenleftigg/summationdisplay i¯siRk(Ti −)/parenrightigg |Ψ(k) 1...k/an}bracketri}ht.(69) Now, exp/parenleftigg/summationdisplay isiRk(Ti)/parenrightigg = exp/parenleftigg Rk/parenleftigg/summationdisplay isiTi/parenrightigg/parenrightigg = =/parenleftigg exp/parenleftigg/summationdisplay isiTi/parenrightigg/parenrightigg⊗k = /summationdisplay jPj(s)Tj ⊗k ,(70) where we used the deﬁnition of the Schur polynomials exp/parenleftigg/summationdisplay isizi/parenrightigg =/summationdisplay jPj(s)zj, (71) their essential property being: ∂siPj(s) = (∂s1)iPj(s) =Pj−i(s). (72) (expansion of the determinant g(k′)in the ﬁrst column) and now the ﬁrst two factors can be composed in to g(k+1)(expansion of the determinant g(k+1)in the ﬁrst row), thus giving the r.h.s. of (67). 20 Because of (70), we can rewrite the r.h.s. of (69) as τ(k)(s,¯s|g) = =/summationdisplay i1,...,ik j1,...,jkPi1(s)...Pik(s)/an}bracketle{tΨ(k) 1+i1,2+i2,...,k+ik|g|Ψ(k) 1+j1,2+j2,...,k+jk/an}bracketri}htPj1(¯s)...Pjk(¯s) = = det 1≤α,β≤kHα β(s,¯s),(73) where Hα β(s,¯s) =/summationdisplay i,jPi−α(s)gi jPj−β(¯s). (74) This formula can be considered as including inﬁnitely many times siand ¯si, and it is only due to the ﬁniteness of matrix gi j∈SL(n) thatH-matrix is additionally constrained /parenleftigg∂ ∂s1/parenrightiggn Hα β= 0, ... ∂ ∂siHα β= 0,fori≥n.(75) The characteristic property of Hα βis that it satisﬁes the following “shift” relations (see (72)): ∂ ∂siHα β=Hα+i β,∂ ∂¯siHα β=Hα β+i. (76) Expressions (73), (74) and (76) are, of course, familiar from the theory of KP and Toda hierarchies (see [2, 3] and references therein). Coming back to bilinear relation (67), it can be easily rewritten in terms ofH-matrix: it is enough to convolute them with Schur polynomials. For the sake of co nvenience let us denote H/parenleftigα1...αk β1...βk/parenrightig = det1≤a,b≤kHαa βb. In accordance with this notation τ(k)=H/parenleftig1...k 1...k/parenrightig , while bilinear equation turns into: H/parenleftiggα1...αk

Question 225 multiple-choice

In the study of quantum planes and braided spaces, classical mathematical operations such as integration and Fourier transformation are redefined to accommodate noncommutative and braided algebraic structures. These generalizations are essential for applications in quantum calculus, noncommutative geometry, and quantum group theory. In the algebraic framework for integration and Fourier transformation on braided groups, which operation is equivalent to applying the braided Fourier transform twice to a function? 1) The braided coaddition 2) The Jackson q-integral 3) The braided derivative 4) The braided-antipode 5) The bosonic scaling map 6) The q-Gaussian projection 7) The ordinary inversion

✓ Correct Answer:

The correct answer is 4) The braided-antipode.

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Question 226 multiple-choice

In quantum computing and information theory, random vectors, orthonormal sets, and representation theory play key roles in algorithms such as quantum Fourier sampling and solutions to the hidden subgroup problem. In the context of constructing a random orthonormal set of vectors in $ \mathbb{C}^d $, which statement accurately describes the probabilistic behavior of the average probability vector formed by the squared magnitudes of coordinates when the set is sufficiently large? 1) The average probability vector is typically concentrated on a subset of coordinates. 2) The average probability vector is always sparse regardless of the number of vectors. 3) The average probability vector becomes more uneven as the dimension increases. 4) The average probability vector is dominated by the largest coordinate in every set. 5) The average probability vector shows significant deviation from uniformity as more vectors are added. 6) The average probability vector rapidly converges to a distribution with zeros in some coordinates. 7) The average probability vector tends to be close to uniform given a sufficiently large orthonormal set.

✓ Correct Answer:

The correct answer is 7) The average probability vector tends to be close to uniform given a sufficiently large orthonormal set..

📚 Reference Text:

get the unit vector(x1,...,x 2d), wherexi=yi√ y2 1+···+y2 2d(note that any yi= 0with zero probability). We thus get a random unit vector in R2d. Identifying a pair of real numbers with a single complex num ber, we get a ran- dom unit vector (z1,...,zd)inCd. To generate a random orthonormal ordered set {v1,...,vm}of vectors inCd,wecanﬁrstsample munitvectors {v′ 1,...,v′ m}inCdandthendoGram-Schmidtorthonormalisation on them to get {v1,...,vm}(note that withprobability 1,{v′ 1,...,v′ m}arelinearly independent). The following fact can be proved by combining Theorem 14.3.2 and Proposition 14.3.3 of [Mat02, Chapter 14] and using the concavity of the square-root funct ion. Fact 2Lett>0, and|v/an}bracketri}ht,|w/an}bracketri}htindependent random unit vectors in Cd. Then, Pr/bracketleftbigg |/an}bracketle{tv|w/an}bracketri}ht|>t+10√ d/bracketrightbigg ≤2exp(−t2d). We will require the following upper and lower bounds on the ta ils of the chi-square distribution (the chi-square distribution with ddegrees of freedom is the sum of squares of dindependent Gaussians with mean0and variance 1). The upper bound can be proved via Chernoff-style argument s on the moment generating function of the chi-square distribution. The lo wer bound follows, for example, from the central limittheoreminprobability theory. Onecanalsogiveadire ctproofofthelowerboundusingtheprobability density function of the chi-square distribution and estima ting it via Stirling’s approximation of the gamma function. Fact 3Let(X1,...,X d)beindependent random variables such that Xiisone-dimensional Gaussian with mean0and variance 1. LetX2=X2 1+···+X2 d. Let0≤ǫ <1/2. There exists a universal constant γ >0such that 1.Pr[|X2−d|>dǫ]<2exp(−dǫ2/6), 2.Pr[X2>d+√ d]>γ,Pr[X2γ. Thefollowing result follows easily from Fact3. Asimilar re sult appears as Lemma2in [MRRS04]. 11 Fact 4LetV={a1,...,ap}be a random orthonormal set of pvectors in Cd. Letai jdenote thejth coordinate of vector ai. Deﬁne the d-dimensional probability vector Sas follows:Sj=1 p/summationtextp i=1|ai j|2. Let 0≤ǫ<1/2. Supposep= Ω(ǫ−2logd). LetUdenote the uniform probability distribution on {1,...,d}. Then, with probability at least 1−exp(−Ω(ǫ2p))over thechoice of V,/bardblS−U/bardbl1≤ǫ. WewillalsoneedthefollowingChernoffupperboundsonthet ailofthesumof dindependentidentically distributed binary random variables. Fact 5([AS00,Cor. A.7,Theorem A.13]) Let(X1,...,X d)beindependentbinaryrandomvariablessuch thatPr[Xi= 1] =p. LetX=X1+···+Xd. Let0≤ǫ<1/2. Then, 1.Pr/bracketleftbig/vextendsingle/vextendsingleX d−p/vextendsingle/vextendsingle>ǫ/bracketrightbig <2exp(−2ǫ2d), 2.Pr[X

Question 227 multiple-choice

Quantum algebras and supergroups play an essential role in mathematical physics, particularly in the study of integrable systems and representation theory. The interplay between Hecke-type symmetries, quantum groups, and tensor categories informs the structure and actions of quantum algebras on vector spaces. Which statement accurately describes a property of the Hilbert-Poincaré series of the R-exterior algebra associated with a Hecke symmetry R on a superspace V of super-dimension (m|n)? 1) It is always a polynomial with degree equal to the sum m+n. 2) It is given by a product of two polynomials with degrees m and n. 3) It is expressed as a ratio of two polynomials whose degrees reflect the bosonic (m) and fermionic (n) dimensions. 4) It coincides with the Hilbert series of the symmetric algebra on V. 5) It vanishes when m equals n. 6) It cannot be computed due to the noncommutativity of the algebra. 7) It is independent of the Hecke symmetry R chosen.

✓ Correct Answer:

The correct answer is 3) It is expressed as a ratio of two polynomials whose degrees reflect the bosonic (m) and fermionic (n) dimensions..

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Question 228 multiple-choice

In quantum computing, representation theory of groups like the symmetric group Sn underpins advanced algorithms for state preparation and symmetry exploitation. Concepts such as induced representations, multiplicity spaces, and basis transformations are essential in constructing efficient quantum algorithms for high-dimensional systems. Which procedure allows quantum algorithms to reorganize multiplicity spaces so that the basis vectors transform trivially under a subgroup YT, thereby facilitating block diagonalization of the representation and isolation of the trivial irrep component? 1) Schur-Weyl duality 2) Quantum phase kickback 3) Generalized Phase Estimation (GPE) 4) Quantum Fourier Sampling 5) Grover’s search algorithm 6) Tensor product decomposition 7) Quantum error correction

✓ Correct Answer:

The correct answer is 3) Generalized Phase Estimation (GPE).

📚 Reference Text:

re-write these vectors in a way that re ects the structure of the induced representation. In other words, we choose a transversal and think of elements of the transversal |t/angbracketrightas basis vectors. Now, we can use the algorithm to construct Fourier transforms over induced representations. For clarity, we make the steps involved more explicit here. First, take an ancilla of size |YT|(more precisely, m=⌈log|YT|⌉qubits) with all the qubits in the state|0/angbracketright. Here|YT|stands for the number of elements in the subgroup YT. The state|0/angbracketrightmis taken to be the label of the trivial irrep of YT. Then perform the inverse Fourier transform over YTto obtain the equal superposition over all the elements of YT. This can now be thought of as a subspace of C(Sn)spanned by equal superpositions over elements of cosets of YTinSn. On this space, we can perform the quantum Fourier transform over Sn. This produces the basis |λ,i,j/angbracketright withiandjlabeled by SYTs. The set of irreps λthat appear in the decomposition are the ones that dominate the Young diagram corre- sponding to YT. Similarly, the multiplicity space, although embedded inside a space of dimension dλ, does not have support on all the vectors. It will be supported only on Kλµmany vectors. This space is the multiplicity of the trivial representation of the subgroup YTwhenλis restricted to YT. 11 While the algorithm above does an essential block diagonalization, we would ideally like to have the multiplicity index label basis vectors of the right subspace rather than have it label the basis of a larger space. In order to do this, we need to change the basis inside multiplicity spaces to correspond to the trivial space under the action of the subgroup YT. This follows from Frobenius reciprocity as mentioned earlier. It turns out this is also important to get to the Gelfand-Tsetlin basis in the dual Schur transform that we construct in the next section. To perform this base change in the multiplicity spaces, we can use generalized phase estimation (GPE) [18]. GPE is a generalization of Kitaev's phase estimation technique [28]. In this technique, one can block diagonalize any representation ρ(g)if one can perform ρ(g)withgas control. The main reason why we need the GPE is to organize the multiplicity space into a basis that consists of vectors that transform trivially under YT. We can use this primitive to re-organize the multiplicity space. The action of YTandSnin the multiplicity space is the so called right regular representation R. This acts on Young tableau according to the Young orthonormal representation since for the right action of Sn, the multiplicity space is an irrep. The result of performing GPE on the basis vector12 34(say) forYT=S2×S2 would be to stabilize it. In other words, performing (12) or(34) does not a ect the vector. This would then correspond to the SSYT11 22. Some vectors do not appear in the multiplicity space of the trivial irrep of YTin GPE. For example, the vector13 24would be taken to zero by the action of YT. These statements are

Question 229 multiple-choice

In the context of representation theory and quantum information, the spectral properties of permutation-invariant Hamiltonians play a crucial role in understanding the behavior of quantum systems modeled by graphs. Swap matrices are linear operators representing transpositions and satisfy various algebraic relations fundamental to quantum and combinatorial algorithms. Which closed-form expression gives the maximal eigenvalue η[n−k,k] of the Hamiltonian associated with the complete graph Kn when analyzed via the [n−k,k] irreducible representation of the symmetric group Sn? 1) η[n−k,k] = n(n−1) − k² 2) η[n−k,k] = k(n−k) 3) η[n−k,k] = 2k(n+1) − 2k² 4) η[n−k,k] = n² − 2k(n−k) 5) η[n−k,k] = (n+1)² − k² 6) η[n−k,k] = n(n+1) − 2k² 7) η[n−k,k] = k(n+1) − k²

✓ Correct Answer:

The correct answer is 3) η[n−k,k] = 2k(n+1) − 2k².

📚 Reference Text:

(i,j)∈E(G)χ[n−k,k]((i j)) (2.41) = 2/parenleftigg n 2/parenrightigg /parenleftbigχ[n−k,k](e)−χ[n−k,k]((1 2))/parenrightbig. (2.42) But we also have Tr/bracketleftig H[n−k,k] Kn/bracketrightig = Tr/bracketleftig η[n−k,k]I/bracketrightig =η[n−k,k]χ[n−k,k](e) (2.43) by definition of the trace. Putting these together gives η[n−k,k]=2/parenleftbign 2/parenrightbig(χ[n−k,k](e)−χ[n−k,k]((1 2))) χ[n−k,k](e)= 2/parenleftigg n 2/parenrightigg −2/hatwideη[n−k,k] (2.44) Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 17 where /hatwideη[n−k,k]:=/summationtext (i,j)∈E(Kn)χ[n−k,k]((i j)) χ[n−k,k](e)=/parenleftigg n 2/parenrightigg χ[n−k,k]((1 2)) χ[n−k,k](e). (2.45) Finally, combining Eqs. (2.36), (2.38) and (2.44) gives a closed form expression for η[n−k,k]. We do that below and simplify, assuming that n>5 and treating several cases separately. Casek= 1: /hatwideη[n−k,k]=/parenleftigg n 2/parenrightigg χ[n−k−2,k](e)/χ[n−k,k](e) =/parenleftbiggn(n−1) 2/parenrightbigg/parenleftigg (n−2k−1)(n−2)[k−1] k!/parenrightigg/parenleftigg k! (n−2k+ 1)n[k−1]/parenrightigg =n(n−3) 2 Casek= 2: /hatwideη[n−k,k]=/parenleftigg n 2/parenrightigg (χ[n−k−2,k](e) +χ[n−k,k−2](e))/χ[n−k,k](e) =/parenleftbiggn(n−1) 2/parenrightbigg/parenleftigg (n−2k−1)(n−2)[k−1] k!/parenrightigg/parenleftigg k! (n−2k+ 1)n[k−1]/parenrightigg +/parenleftbiggn(n−1) 2/parenrightbigg/parenleftbiggk! (n−2k+ 1)(n)[k−1]/parenrightbigg =(n−1)(n−5)(n−2) 2(n−3)+(n−1) (n−3) =n(n−5) 2+ 2 Case 3≤k≤(n−1)/2: /hatwideη[n−k,k]=/parenleftigg n 2/parenrightigg (χ[n−k−2,k](e) +χ[n−k,k−2](e))/χ[n−k,k](e) =/parenleftbiggn(n−1) 2/parenrightbigg/parenleftigg (n−2k−1)(n−2)[k−1] k!/parenrightigg/parenleftbiggk! (n−2k+ 1)(n)[k−1]/parenrightbigg +/parenleftbiggn(n−1) 2/parenrightbigg/parenleftigg (n−2k+ 3)(n−2)[k−3] (k−2)!/parenrightigg/parenleftbiggk! (n−2k+ 1)(n)[k−1]/parenrightbigg =(n−2k−1)(n−k+ 1)(n−k) 2(n−2k+ 1)+(n−2k+ 3)k(k−1) 2(n−2k+ 1) =n(n−1) 2+k2−k(n+ 1) Casek= (n−1)/2: /hatwideη[n−k,k]=/parenleftigg n 2/parenrightigg χ[n−k,k−2](e)/χ[n−k,k](e) Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 18 =/parenleftbiggn(n−1) 2/parenrightbigg/parenleftigg (n−2k+ 3)(n−2)[k−3] (k−2)!/parenrightigg/parenleftbiggk! (n−2k+ 1)(n)[k−1]/parenrightbigg =(n−2k+ 3)k(k−1) 2(n−2k+ 1) =(n−1)(n−3) 4 Casek=n/2: /hatwideη[n−k,k]=/parenleftigg n 2/parenrightigg (χ[n−k,k−2](e)−χ[n−k−1,k−1](e))/χ[n−k,k](e) =/parenleftbiggn(n−1) 2/parenrightbigg/parenleftigg (n−2k+ 3)(n−2)[k−3] (k−2)!/parenrightigg/parenleftigg k! (n−2k+ 1)n[k−1]/parenrightigg −/parenleftbiggn(n−1) 2/parenrightbigg/parenleftigg (n−2k+ 1)(n−2)[k−2] (k−1)!/parenrightigg/parenleftigg k! (n−2k+ 1)n[k−1]/parenrightigg =(n−2k+ 3)k(k−1) 2(n−2k+ 1)−(n−k+ 1)k 2 =3n(n−2) 8−n(n+ 2) 8 =n(n−4) 4 Straightforward calculation then shows that all of these cases are consistent with the formula /hatwideη[n−k,k]=/parenleftigg n 2/parenrightigg +k2−k(n+ 1) (2.46) from the 3≤k≤(n−1)/2 case. Hence η[n−k,k]= 2k(n+ 1)−2k2and we are done. Lemma 2.12 gives us a straightforward way to compute the max eigenvalue (and in fact, alleigenvalues) of the Hamiltonian HKn. In Section 6.1 we show how, with a bit more work, we can also use this lemma to compute the eigenvalues of graphs other than the complete graph. 2.6 Identities Satisfied by Swap Matrices A key tool that will be developed in the subsequent section and then used in the remainder of this paper is a fully algebraic characterization of the swap matrices. In preparation for this, we observe some simple algebraic facts about the swap matrices. Since the transpositions (i j)are generators of Sn, it follows that Mswap n is generated by the swap matrices Swapij. The latter must satisfy the relations Swap2 ij=I, (2.47a) SwapijSwapjk= SwapikSwapij, (2.47b) SwapijSwapkl= SwapklSwapij, (2.47c) for alli,j,k,l distinct, arising from the relations satisfied by the transpositions (ij). One can verify using Eq. (2.13) that, in addition to the above relations, the swap matrices satisfy SwapijSwapjk+ SwapjkSwapij= Swapij+ Swapjk+ Swapik−I, (△) Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 19 which is in general not true for transpositions (ij),(j k)∈C[Sn]. For instance, under the signature representation π∝⇕⊣√∫⊔≀→sgn (π)we have sgn ((i j)) sgn ((j k))+sgn ((j k)) sgn ((i j))= 2andsgn ((i j)) + sgn ((j k)) + sgn ((i k))−1 =−4. Remark 2.13. The commutation relations Eq. (2.47c) are (somewhat surprisingly) implied by Eqs. (2.47a), (2.47b) and ( △). To prove this, note Eq. (2.50)

Question 230 multiple-choice

In abstract algebra and topology, pseudocompact group topologies are studied to understand when algebraic structures can be given topologies with bounded continuous real-valued functions. The interplay between group decompositions, cardinalities, and set-theoretic assumptions is central to characterizing these topologies for Abelian groups. Which of the following is a necessary and sufficient condition for an Abelian torsion group to admit a pseudocompact group topology under certain set-theoretic assumptions? 1) The group contains an infinite cyclic subgroup and is locally compact. 2) Each element of the group has order two and the group is finite. 3) The group is a direct sum of uncountably many free Abelian groups of rank one. 4) The group is divisible and its cardinality is a singular cardinal. 5) The group is of bounded order and each p-component admits a pseudocompact topology. 6) The group is simple and non-Abelian with finite exponent. 7) Each subgroup satisfies the descending chain condition and the group is compact.

✓ Correct Answer:

The correct answer is 5) The group is of bounded order and each p-component admits a pseudocompact topology..

📚 Reference Text:

then, noticing that G' and G" satisfy the cofinal admissibility condition, we use the inductive hypothesis to find pseudocompact group topologies on G' and G". The product topology on G = G' χ G" is then pseudocompact, s required. α Now using (M) we can rewrite 3.17, casting condition (b) only in the language of algebra and cardinal numbers. 3.18. Theorem. Assume (M). Let p E P and let G be an Abelian p-group. Then the following conditions are equivalent. (a) G admits a pseudocompact group topology. (b) G has the form G = 0"= ί 0y . Z (pn) with rl<··· yk and (log log y^ < yt. Proof. This is simply 3.17, incorporating 3.8(b). D 3.19. Theorem. Let G be an Abelian torsion group, and let G = 0pePG(/?) be the decomposition ofG into itsp-components. Then the following conditions are equivalent. (a) G admits a pseudocompact group topology; (b) G is ofbounded order, and each ofthe groups G (p) admits a pseudocompact group topology; (c) G is ofbounded order, and each ofthe groups G (p) satisfies the cofinal admissibility condition. Proof. Since (b) «> (c) is given by 3.17, and (b) => (a) since the product of pseudocompact groups is pseudocompact, it is enough to prove (a) => (b). Since G is ofbounded order by 1.3, we have G = 0JL i G^.). For l ^x takes G onto G (/?,·), so G (/?,·) is pseudocompact in the group topology inherited from G. G The combination of 3.18 and 3.19 yields this definitive formulation. Bereitgestellt von | Technische Informationsbibliothek Hannover Angemeldet Heruntergeladen am | 27.03.18 14:50 334 W. W. Comfort, D. Remus 3.20. Theorem. Assume (M). For an Abelian torsion group G, thefollowing conditions are equivalent. (a) G admits a pseudocompact group topology. (b) G has theform G = @JL , 0?4 , 0^, Z (j>y·*) with rM < · · · < rjtltjfor \ y^k and <7i. D In the interest of completeness, we give the analogue of 3.20 under the stronger hypothesis SCH. 3.21 . Theorem. Assume SCH. For an Abelian torsion group G, thefollowing conditions are equivalent. (a) G admits a pseudocompact group topology.

Question 231 multiple-choice

Quantum computing architectures are evolving to use multi-level quantum systems known as qudits, which can enhance computational efficiency compared to traditional qubit-based systems. Advances in silicon-photonic integrated circuits have enabled the robust and programmable manipulation of these multi-level quantum states. Which advantage does a qudit-based quantum processor using silicon-photonic integrated circuits provide over conventional qubit-based architectures? 1) Increased information density and parallelism due to multiple accessible quantum levels per element 2) Inherent immunity to all decoherence mechanisms in quantum information processing 3) Guaranteed error-free quantum operations without the need for error correction 4) Universal compatibility with all existing classical computing hardware 5) Ability to simulate exclusively two-level quantum systems with higher speed 6) Automatic scalability to unlimited qudit numbers without fabrication constraints 7) Elimination of the need for initialization and measurement in quantum algorithms

✓ Correct Answer:

The correct answer is 1) Increased information density and parallelism due to multiple accessible quantum levels per element.

📚 Reference Text:

ARTICLE A programmable qudit-based quantum processor Yulin Chi1,12, Jieshan Huang1,12, Zhanchuan Zhang1,12, Jun Mao1, Zinan Zhou1, Xiaojiong Chen1, Chonghao Zhai1, Jueming Bao1, Tianxiang Dai1, Huihong Yuan1,2, Ming Zhang3, Daoxin Dai3, Bo Tang4, Yan Yang4, Zhihua Li4, Yunhong Ding5,6, Leif K. Oxenløwe5,6, Mark G. Thompson7, Jeremy L. O ’Brien8, Yan Li1,9,10,11, Qihuang Gong1,2,9,10,11& Jianwei Wang1,2,9,10,11 ✉ Controlling and programming quantum devices to process quantum information by the unit of quantum dit, i.e., qudit, provides the possibilities for noise-resilient quantum commu-nications, delicate quantum molecular simulations, and ef ﬁcient quantum computations, showing great potential to enhance the capabilities of qubit-based quantum technologies.Here, we report a programmable qudit-based quantum processor in silicon-photonic inte-grated circuits and demonstrate its enhancement of quantum computational parallelism. Theprocessor monolithically integrates all the key functionalities and capabilities of initialisation,manipulation, and measurement of the two quantum quart (ququart) states and multi-valuequantum-controlled logic gates with high-level ﬁdelities. By reprogramming the con ﬁguration of the processor, we implemented the most basic quantum Fourier transform algorithms, allin quaternary, to benchmark the enhancement of quantum parallelism using qudits, whichinclude generalised Deutsch-Jozsa and Bernstein-Vazirani algorithms, quaternary phaseestimation and fast factorization algorithms. The monolithic integration and high program-mability have allowed the implementations of more than one million high- ﬁdelity prepara- tions, operations and projections of qudit states in the processor. Our work shows anintegrated photonic quantum technology for qudit-based quantum computing with enhancedcapacity, accuracy, and ef ﬁciency, which could lead to the acceleration of building a large- scale quantum computer.https://doi.org/10.1038/s41467-022-28767-x OPEN 1State Key Laboratory for Mesoscopic Physics, School of Physics, Peking University, 100871 Beijing, China.2Beijing Academy of Quantum Information Sciences, 100193 Beijing, China.3State Key Laboratory for Modern Optical Instrumentation, College of Optical Science and Engineering, Ningbo Research Institute, International Research Center for Advanced Photonics, Zhejiang University, 310058 Hangzhou, China.4Institute of Microelectronics, Chinese Academy of Sciences, 100029 Beijing, China.5Department of Photonics Engineering, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark. 6Center for Silicon Photonics for Optical Communication (SPOC), Technical University of Denmark, 2800 Kgs. Lyngby, Denmark.7Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering, University of Bristol, BS8 1FD Bristol, Un ited Kingdom.8Department of Physics, The University of Western Australia, Perth 6009, Australia.9Frontiers Science Center for Nano-optoelectronics & Collaborative Innovation Center of Quantum Matter, Peking University, 100871 Beijing, China.10Collaborative Innovation Center of Extreme Optics, Shanxi University, 030006 Taiyuan, Shanxi, China.11Peking University Yangtze Delta Institute of Optoelectronics, Nantong 226010 Jiangsu, China.12These authors contributed equally: Yulin Chi, Jieshan Huang and Zhanchuan Zhang.✉email: jww@pku.edu.cn NATURE COMMUNICATIONS | (2022) 13:1166 | https://doi.org/10.1038/s41467-022-28767-x | www.nature.com/naturecommunications 11234567890():,; Natural quantum matters store rich multidimensional quantum information in a superposition of more than two electronic or mechanical modes. Engineering arti ﬁcial multilevel quantum devices to mimic nature may allow funda-mental innovations and technological advances. Recently, though the state-of-the-art qubit-based quantum technologies have demon- strated revolutionary milestones, e.g., loophole-free Bell tests 1,2, satellite-relayed quantum communications3,4and quantum compu- tational advantages5,6, qudit-based quantum technologies might be able to further enhance quantum capab ilities as they are intrinsically consistent with our natural quantu

Question 232 multiple-choice

Matrix valued orthogonal polynomials play a significant role in group representation theory and are closely linked to the analysis of stochastic processes such as quasi-birth-and-death (QBD) chains. Stochastic block Jacobi matrices are used to model transition probabilities in these complex systems. Which property distinguishes a stochastic block Jacobi matrix in the modeling of nonhomogeneous QBD processes with multiple phases? 1) Its entries are always symmetric and strictly positive. 2) It is diagonalizable with real eigenvalues only. 3) Each block corresponds to an irreducible representation of a Lie group. 4) The sum of entries in each column equals one. 5) It represents a time-homogeneous Markov process. 6) All off-diagonal blocks must be zero. 7) The sum of entries in each row equals one and all entries are non-negative.

✓ Correct Answer:

The correct answer is 7) The sum of entries in each row equals one and all entries are non-negative..

📚 Reference Text:

Question 233 multiple-choice

Quantum channel capacities determine the maximum rates at which information—either classical or quantum—can be reliably transmitted, with various bounds and inequalities playing a crucial role in quantum information theory. Two-way classical communication and advanced bounding techniques help clarify ultimate limits of secure quantum transmission. Which statement accurately reflects the relationship and upper bounds for the two-way private and quantum capacities of a transferred quantum Markov semigroup Tt(ε) as established by strong converse results? 1) Both two-way capacities are always strictly less than max_k log(n_k) for any ε 2) The two-way quantum capacity can exceed the two-way private capacity in all cases 3) The upper bound for two-way capacities is given by max_k log(n_k) plus twice ε 4) Both two-way private and quantum capacities satisfy max_k log(n_k) ≤ P↔(Tt(ε)), Q↔(Tt(ε)) ≤ max_k log(n_k) + ε 5) Strong converse bounds only apply when the channel is entanglement-breaking 6) The lower bound for two-way capacities is zero for all quantum Markov semigroups 7) The capacities do not depend on the dimension of fixed-point subspaces

✓ Correct Answer:

The correct answer is 4) Both two-way private and quantum capacities satisfy max_k log(n_k) ≤ P↔(Tt(ε)), Q↔(Tt(ε)) ≤ max_k log(n_k) + ε.

📚 Reference Text:

without resorting to interpolation as remarked after Theorem III.1, we still believe that these inequalities are of interest. This is due to the fact that it is usually nontrivial to derive them without noting that the underlying quantum channels arise as transferred semigroups. Moreover, one can directly use an entropy decay estimate instead of the bound (52) from the modiﬁed logarithmic Sobolev inequality for the underlying classical Markov kernel (kt)t≥0. All of our bounds will be based on comparing the capacity of the semigroup at time tto that of its limit ast→∞and using transference techniques to estimate how close the channel is to its limit. Note, however, that our estimates will be tighter than working directly with continuity bounds for capacities. To the best of our knowledge, known continuity bounds for various capacities such as the ones of [37], allow to conclude that the capacity of two channels with output dimension ddiffer by a factor of order log(d)if they are-close in diamond norm. However, in the case of transference we will be able to obtain that the capacities are close under similar assumptions. A. (Two way) Private and Quantum Capacity and Entanglement Breaking Times The private capacity quantiﬁes the rate at which classical information that is secret to the environment can be reliably transmitted by a given quantum channel in the asymptotic limit of many channel uses. For a quantum channel T, the private capacity is denoted by P(T). Similarly, the quantum capacity of a quantumchannel quantiﬁes at which rate quantum information can be reliably transmitted with a quantum channel, and is denoted by Q(T). Note that we always have Q(T)≤P(T)and thus, any upper bound on the private capacity extends to a bound on the quantum capacity. Moreover, we may also consider variations of these capacities in which we also allow for unlimited classical communication between the sender and the receiver of the output of the quantum channel. These are usually called the two-way private and quantum capacities and we will denote them by P↔(T)and Q↔(T), respectively. Clearly, we have P(T)≤P↔(T). We refer to e.g. [38] for a precise deﬁnition of these quantities. Using the techniques above, we can derive strong converses on the two-way quantum and private capacities based on the results of [38] and mixing time estimates. More speciﬁcally, in [38] the authors show that for a quantum channel T∶B(H)→B(H)the quantity Emax(T) (53) =sup ∈D(H⊗H)inf ∈D(H⊗H);∈SEPD∞(T⊗id()/divides.alt0/divides.alt0) is a strong converse upper bound on the two-way private and quantum capacities of T. Here SEP stands for the convex subset of D(H⊗H)of separable states, that is, SEP ={/summation.disp ipiA i⊗B i;pi≥0;/summation.disp ipi=1;A i;B i∈D(H)}: We then have: Theorem V .1 (Bounding the two-way quantum and private capacity) .Let(Tt)t≥0be a transferred QMS. Then: max klog(nk)≤ P↔(Tt());Q↔(Tt())≤max klog(nk)+ Moreover, the upper bound is a strong converse bound. Proof. We have the following chain of inequalities: Emax(Tt) =sup ∈D(H⊗H)inf ∈D(H⊗H);∈SEPD∞(Tt⊗id()/parallel.alt1) ≤sup ∈D(H⊗H)inf ∈D(Nfix)⊗D(H) ∈SEPD∞(Tt⊗id()/parallel.alt1) ≤sup ∈D(H⊗H)inf ∈D(Nfix)⊗D(H) ∈SEPD∞(Efix⊗id()/parallel.alt1)+ =max klog(nk)+; where in the ﬁrst inequality we used the fact that restricting the inﬁmum

Question 234 multiple-choice

Topological phases of matter in two spatial dimensions are central to both condensed matter physics and quantum information theory, offering robust properties useful for fault-tolerant quantum computation. Lattice models such as the Kitaev quantum double model serve as foundational examples in the study of topological quantum order and quantum error correction. Which property is rigorously established for all finite groups in Kitaev's quantum double models, ensuring that states with locally zero energy density are locally indistinguishable and supporting fault-tolerant quantum computation? 1) The existence of a unique ground state for each lattice size 2) The ability to encode information exclusively using local observables 3) Spontaneous symmetry breaking leading to magnetic order 4) Quasi-particle excitations that always obey Abelian statistics 5) Ground state degeneracy that depends only on local Hamiltonian terms 6) Satisfaction of TQO-1 and TQO-2 conditions, guaranteeing quantum error correction with macroscopic distance 7) Universal scalability of the model to arbitrary spatial dimensions

✓ Correct Answer:

The correct answer is 6) Satisfaction of TQO-1 and TQO-2 conditions, guaranteeing quantum error correction with macroscopic distance.

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. . . . . . . . . . . . . . . . . . . . . . . 14 4 Conclusion 21 Shawn X. Cui: cui177@purdue.edu Dawei Ding: dding@stanford.edu Xizhi Han: hanxzh@stanford.edu Geoﬀrey Penington: geoﬀp@stanford.edu Daniel Ranard: dranard@stanford.edu Brandon C. Rayhaun: brayhaun@stanford.edu Zhou Shangnan: snzhou@stanford.edu Accepted in Quantum2020-09-17, click title to verify. Published under CC-BY 4.0. 1arXiv:1908.02829v3 [quant-ph] 22 Sep 2020 A The “irrep basis” and the non-Abelian Fourier transform 21 B Appendix: Entanglement entropy in gauge theory and the algebraic formalism 22 1 Introduction Topological phases of matter in two spatial dimensions are gapped quantum liquids that exhibit exotic properties such as stable ground state degeneracy, stable long-range entanglement, existence of quasi-particle excitations, (possibly) non-Abelian exchange statistics, etc. These phases are characterized by a new type of order, topological quantum order (TQO), that is beyond the conventional Landau theory of spontaneous symmetry breaking and local order parameters.1An important application of topological phases of matter is in topological quantum computing [11, 13], where information is encoded in non-local degrees of freedom and processed by manipulating quasi- particle excitations. A large class of TQOs can be realized as lattice models in quantum spin systems where the Hamiltonian is given as a sum of pairwise commuting and geometrically local projectors. Examples of such constructions include the Levin-Wen string-net lattice models [16] and Kitaev’s quantum doublemodels[13]. In[4,5],theauthorsgaveamathematicallyrigorousproofofgapstabilityunder weak perturbations for quantum spin Hamiltonians satisfying two physically plausible conditions, TQO-1 and TQO-2. Roughly, TQO-1 states that the ground state space is a quantum error correcting code with a macroscopic distance, and TQO-2 means that the local ground state space coincides with the global one. See [4, 5] or §2.2 for a formal deﬁnition. It is widely believed that Kitaev’s quantum double models satisfy TQO-1 and TQO-2. This would rigorously justify the inherit fault-tolerance of Kitaev’s models for topological quantum computing. Previously, it has been shown that Kitaev’s model for ﬁnite Abelian groups satisfy the TQO conditions. See [1, 2], although some of the results in the above mentioned references may be stated in a diﬀerent form. See also [9] for some relevant results. But it is not clear if the methods in those references can be generalized to the case of non-Abelian groups which are signiﬁcantly more complicated than the Abelian case. In this paper, we provide a proof of the TQO conditions that work for all ﬁnite groups. In fact, we prove a stronger property of Kitaev’s model that simultaneously implies TQO-1 and TQO-2. Our result can be informally stated as: States with locally zero energy density are locally indistinguishable. See Theorem 3.1 for a formal statement. Anothermotivationforthecurrentworkistoconstructquantumerrorcorrectingcodesinlattice models. While the toric code and its variant surface code have been studied extensively as error correcting codes, the corresponding study for the case of non-Abelian groups seems insuﬃcient. We have shown in this paper that all non-Abelian Kitaev’s models are quantum error correcting codes with macroscopic distance. It will be interesting to see if there are non-Abelian models that

Question 235 multiple-choice

In group-based cryptography, non-commutative groups are increasingly studied for constructing secure key exchange protocols. Certain properties, such as the structure of commutator subgroups and the complexity of conjugacy problems, play critical roles in the cryptographic strength of these systems. Which property of Miller-Moreno groups makes them particularly attractive for non-commutative cryptographic key exchange protocols? 1) Every proper subgroup is abelian, leading to a computationally difficult conjugacy problem 2) All elements have order two, simplifying key generation 3) The group is always commutative, ensuring easy implementation 4) Their Sylow 2-subgroups are always cyclic 5) They are simple groups, lacking nontrivial normal subgroups 6) The group always has a prime order, maximizing security 7) All subgroups are normal, making the conjugacy problem trivial

✓ Correct Answer:

The correct answer is 1) Every proper subgroup is abelian, leading to a computationally difficult conjugacy problem.

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Question 236 multiple-choice

Quantum algorithms are often expected to outperform classical ones, especially for certain mathematical problems. However, the efficiency of quantum algorithms for large-scale linear algebra tasks can be fundamentally limited by time-space tradeoffs. For binary matrix multiplication of two n × n matrices on a quantum computer using T queries and S qubits, which lower bound on T best reflects the optimal time-space tradeoff proven for strong quantum query algorithms? 1) T = Ω(n²/S) 2) T = Ω(n³/√S) 3) T = Ω(n^{2.5}/S^{1/2}) 4) T = Ω(n²/√S) 5) T = Ω(n³/S) 6) T = Ω(n^{2.5}/S^{1/4}) 7) T = Ω(n⁴/S)

✓ Correct Answer:

The correct answer is 2) T = Ω(n³/√S).

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Question 237 multiple-choice

In quantum information theory, invariant polynomials are crucial tools for classifying the entanglement properties of multi-qubit systems under transformations such as SLOCC (stochastic local operations and classical communication). The study of degree-4 invariant polynomials for systems with several qubits is fundamental for distinguishing between different entanglement classes efficiently. Which mathematical approach is most directly employed to systematically construct the complete basis of degree-4 invariant polynomials for a system of 7 qubits under local symmetry group actions? 1) Representation theory combined with algebraic combinatorics 2) Numerical optimization of entanglement measures 3) Experimental tomography of quantum states 4) Differential geometry of state manifolds 5) Perturbative expansions in quantum field theory 6) Topological classification via homology groups 7) Spectral analysis using Fourier transforms

✓ Correct Answer:

The correct answer is 1) Representation theory combined with algebraic combinatorics.

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Question 238 multiple-choice

Quantum algorithms offer significant speedups for certain computational problems by leveraging properties of quantum mechanics, especially in the context of group theory and oracle models. The Hidden Subgroup Problem (HSP) is a central challenge in quantum computing with implications for cryptography and complexity theory. Which statement correctly characterizes the advancement in query complexity for the Hidden Subgroup Problem on arbitrary finite groups using quantum algorithms? 1) Quantum algorithms for HSP require an exponential number of oracle queries for non-Abelian groups. 2) Classical algorithms and quantum algorithms have equivalent query complexity for all finite groups in HSP. 3) Efficient quantum algorithms for HSP with bounded error exist only for Abelian groups. 4) The quantum query complexity for HSP on arbitrary finite groups is independent of the group size. 5) Polynomial-time quantum algorithms for HSP on general finite groups have been discovered. 6) Quantum algorithms can identify hidden subgroups in any finite group using a polynomial (in log|G|) number of oracle queries, even though the overall time complexity remains exponential. 7) Query complexity is the primary bottleneck preventing efficient solutions to HSP on non-Abelian groups.

✓ Correct Answer:

The correct answer is 6) Quantum algorithms can identify hidden subgroups in any finite group using a polynomial (in log|G|) number of oracle queries, even though the overall time complexity remains exponential..

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arXiv:quant-ph/0401083v1 14 Jan 2004Thequantumquerycomplexityofthehidden subgroupproblemis polynomial MarkEttinger LosAlamos National Laboratory∗PeterHøyer Universityof Calgary† EmanuelKnill Los Alamos National Laboratory∗ January 12,2004 Abstract We present a quantum algorithm which identiﬁes with certain ty a hidden sub- group of an arbitrary ﬁnite group Gin only a polynomial (in log|G|) number of calls to the oracle. This is exponentially better than the be st classical algorithm. Howeverourquantumalgorithmrequiresexponentialtime,a sintheclassicalcase. Our algorithm utilizes a new technique for constructing err or-free algorithms for non-decision problems onquantum computers. 1 Introduction LetGbe a ﬁnite group, written multiplicatively with identity 1G. A function fonG (with arbitrary range) is said to be H-periodic iffis constant on the left cosets of a subgroup HofG. Iffalso takes distinct valueson distinct cosets we say fisstrictly H-periodic and we call Hthehidden subgroup off. Thehidden subgroup problem (HSP) is stated as follows: Given a description of Gand a function fonGthat is promised to be strictly H-periodic for some subgroup H/lessorequalslantG, ﬁnd a generating set forH. Letrdenote the number of distinct subgroups of G. Fix any ordering of the r subgroups (K1,K2,...,K r)satisfying that |Kµ| ≥ |Kµ+1|for all1≤µ < r. In the HSP, we are searching for a generating set for one out of rcandidate subgroups. Let N=|G|denote the order of G. We consider n= log|G|to be the input size. Since any subgroup of Gis generated by a set of at most nelements of G, the number rof distinctsubgroupsof Gis2O(n2). We assume the function fis given as an oracle so that the only way we can gain knowledgeabout fisbyaskingforitsvalueonelementsof G. Formally,onaquantum ∗Email:{ettinger,knill }@lanl.gov . †Email:hoyer@cpsc.ucalgary.ca . 1 computer, the oracle is a unitary operator Of, that maps |g/an}brack⌉⊔ri}h⊔|0/an}brack⌉⊔ri}h⊔to|g/an}brack⌉⊔ri}h⊔|f(g)/an}brack⌉⊔ri}h⊔for all g∈G. We assume without loss of generality that algorithms for th e HSP always outputasubsetof H. Supposeinsteadthatanalgorithmoutputs X/n⌉}a⊔ionslash⊆H. Thenwecan ﬁnd the intersection of XwithHby evaluating fon each element x∈Xand only keepingxiff(x) =f(1G). Thisrequiresat most |X|+1evaluationsof f. If the group GisAbelian, then it is possible to solve the HSP in polynomial time with bounded error on a quantum computer. That is, we can efﬁc iently ﬁnd a subset X⊆Hthatgenerates Hwithprobabilityatleast2 3. Thisresultfollowsfromthework ofSimon[1],Shor[2]andKitaev[3]. It ispossibletoimprov ethe successprobability tooneforAbeliangroupsofsmoothorder[4](agroupisof c-smoothorder ifallprime factorsof |G|areatmost (log|G|)cforsomeconstant c). Fornon-Abelian groups,our knowledgeismuchmorelimited[5,6,7,8,9,10]. The efﬁcient HSP algorithm for Abelian groupsof smooth orde r implies that only apolynomial(in log|G|)numberofcallstotheoraclearenecessarytoidentify Hwith certainty. The main result of this paper is that this more lim ited result holds for all groups of ﬁnite order. That is, there exists a quantum algori thm that determines H usinga polynomialnumberofcallsto theoracle. Theorem1 There exists a quantum algorithm that, given a ﬁnite group Gand an oraclefonGpromised to be strictly H-periodic for some subgroup H/lessorequalslantG, calls the oracle O(log4|G|)times and outputs a generating set for H. The algorithm fails with probability exponentially small in log|G|. The algorithm can be made exact in anymodelallowingarbitraryone-qubitgates. An important consequence of this result is that it rules out m ost known methods for proving super-polynomial lower bounds on the total comp lexity of bounded-error quantum algorithmsfor the HSP. Most of these methods boundt he query complexity, including the recent

Question 239 multiple-choice

The Quantum Fourier Transform (QFT) is a fundamental operation in quantum computing, enabling efficient solutions to problems in cryptography, simulation, and machine learning. Optimizing the implementation of QFT is crucial for the practical use of large-scale quantum algorithms on fault-tolerant hardware. Which technique enables an O(n log(n)) T-count complexity for Quantum Fourier Transform implementations while maintaining high accuracy for applications such as factoring 2048-digit RSA keys? 1) Using only Clifford gates and omitting all non-Clifford operations 2) Applying quantum error correction codes without circuit modifications 3) Employing global entangling gates instead of local rotations 4) Replacing QFT with Grover's search algorithm for periodicity 5) Utilizing measurements and feedforward with a phase gradient quantum state 6) Increasing the number of small-angle rotation gates in the circuit 7) Implementing QFT via adiabatic quantum computing exclusively

✓ Correct Answer:

The correct answer is 5) Utilizing measurements and feedforward with a phase gradient quantum state.

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ARTICLE OPEN Approximate quantum Fourier transform with O(nlog(n)) T gates Yunseong Nam1✉, Yuan Su2✉and Dmitri Maslov3✉ The ability to implement the Quantum Fourier Transform (QFT) ef ﬁciently on a quantum computer facilitates the advantages offered by a variety of fundamental quantum algorithms, such as those for integer factoring, computing discrete logarithm over Abelian groups, solving systems of linear equations, and phase estimation, to name a few. The standard fault-tolerantimplementation of an n-qubit unitary QFT approximates the desired transformation by removing small-angle controlled rotations and synthesizing the remaining ones into Clifford +T gates, incurring the T-count complexity of Oðnlog 2ðnÞÞ. In this paper, we show how to obtain approximate QFT with the T-count of OðnlogðnÞÞ. For brevity, the above ﬁgures omit the dependence on the approximation error ε, assuming the error is ﬁxed. Our approach relies on quantum circuits with measurements and feedforward, and on reusing a special quantum state that induces the phase gradient transformation. We report asymptotic analysis as well as concrete circuits, demonstrating signi ﬁcant advantages in both theory and practice. npj Quantum Information (2020) 6:26 ; https://doi.org/10.1038/s41534-020-0257-5 INTRODUCTION Quantum Fourier Transform (QFT) is one of the most important operations in quantum computing. It can extract the periodicity encoded in the amplitudes of a quantum state, which is employedby an ef ﬁcient algorithm for integer number factoring, widely known as Shor ’s algorithm 1. Shor ’s integer factoring algorithm can be generalized (while still relying on the QFT) into a polynomial-time algorithm for the discrete logarithm problem over Abelian groups 1. The importance of the above is witnessed through the threat such algorithms pose to modern public-key cryptosystems,such as the RSA or the ECC. Using the QFT as a subroutine, the eigenphase of a black-box unitary can be estimated up to an arbitrary precision 2, which may be used to estimate quantum amplitudes3,4, simulate quantum chemistry/dynamics5,ﬁnd the ground state/energy of a Hamiltonian6, compute Hessian to optimize molecular geometry7, exponentiate unitaries8, construct fractional powers of the QFT using constantly many copies of the controlled-QFT8,9, extract features of the solution of linear systems10, and more. QFT has also been used in quantum arithmetics11,12and quantum cryptography13. QFT can be implemented approximately by removing all rotation gates with angles smaller than a certain threshold value, resulting in the Approximate QFT (AQFT). In practice, it was shown that it suf ﬁces to apply AQFT with ~5.3 × 104controlled rotation gates to factor 2048-digit numbers (re ﬂecting the de facto key size for today ’s standard14) with a high expected algorithmic accuracy (≳99.992%)15. AQFT has been studied extensively in the literature. The robustness of the quantum computer equipped with the AQFT was investigated in detail16–20. A study of the optimal level of the approximation of the AQFT in the presence of certain errorsmay be found in ref.21. Implementation of the QFT and its approximate version over restricted architectures was addressed in refs22,23.A ne f ﬁcient approximate implementation of the AQFT that harnesses certain quantum hardware features was also investigated24.Quantum information is fragile, and it

Question 240 multiple-choice

In mathematics and quantum information theory, tensors and moment polytopes are central objects, deeply connected to group symmetries, quantum entanglement, and computational complexity. Recent advances in algorithmic techniques have expanded the limits of explicit moment polytope computation for high-dimensional tensors. Which statement best describes a significant achievement of the latest algorithmic approach for computing moment polytopes of tensors in multipartite spaces? 1) It enables exact computation of moment polytopes for all tensor products involving spaces of dimension six or higher. 2) It is limited to computing moment polytopes only for bipartite tensor spaces such as $\mathbb{C}^2\otimes\mathbb{C}^2$. 3) It exclusively computes entanglement polytopes for one-dimensional vector spaces. 4) It provides partial results for tensor products in spaces with odd dimensions greater than five, but with low certainty. 5) It computes moment polytopes for tensors in $\mathbb{C}^4\otimes\mathbb{C}^4\otimes\mathbb{C}^4$ with full certainty, but fails for three-dimensional cases. 6) It only analyzes tensors relevant to classical signal processing, excluding quantum applications. 7) It achieves full certainty for tensors in $\mathbb{C}^3\otimes\mathbb{C}^3\otimes\mathbb{C}^3$ and high probability for $\mathbb{C}^4\otimes\mathbb{C}^4\otimes\mathbb{C}^4$.

✓ Correct Answer:

The correct answer is 7) It achieves full certainty for tensors in $\mathbb{C}^3\otimes\mathbb{C}^3\otimes\mathbb{C}^3$ and high probability for $\mathbb{C}^4\otimes\mathbb{C}^4\otimes\mathbb{C}^4$..

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Title: Computing moment polytopes -- with a focus on tensors, entanglement and matrix multiplication Year: 2025 Paper ID: 14c822f063a1c34dd9e6e99c94e82d85646e37cb Source: semantic-scholar URL: https://www.semanticscholar.org/paper/14c822f063a1c34dd9e6e99c94e82d85646e37cb Abstract: Tensors are fundamental in mathematics, computer science, and physics. Their study through algebraic geometry and representation theory has proved very fruitful in the context of algebraic complexity theory and quantum information. In particular, moment polytopes have been understood to play a key role. In quantum information, moment polytopes (also known as entanglement polytopes) provide a framework for the single-particle quantum marginal problem and offer a geometric characterization of entanglement. In algebraic complexity, they underpin quantum functionals that capture asymptotic tensor relations. More recently, moment polytopes have also become foundational to the emerging field of scaling algorithms in computer science and optimization. Despite their fundamental role and interest from many angles, much is still unknown about these polytopes, and in particular for tensors beyond $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2$ and $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2$ only sporadically have they been computed. We give a new algorithm for computing moment polytopes of tensors (and in fact moment polytopes for the general class of reductive algebraic groups) based on a mathematical description by Franz (J. Lie Theory 2002). This algorithm enables us to compute moment polytopes of tensors of dimension an order of magnitude larger than previous methods, allowing us to compute with certainty, for the first time, all moment polytopes of tensors in $\mathbb{C}^3\otimes\mathbb{C}^3\otimes\mathbb{C}^3$, and with high probability those in $\mathbb{C}^4\otimes\mathbb{C}^4\otimes\mathbb{C}^4$ (which includes the $2\times 2$ matrix multiplication tensor). We discuss how these explicit moment polytopes have led to several new theoretical directions and results.

Question 241 multiple-choice

In extensions of the Minimal Supersymmetric Standard Model, hidden-sector neutralinos are considered as dark matter candidates whose relic abundance must be carefully tuned to match cosmological observations. The interplay between hidden-sector parameters and annihilation channels critically determines their viability and detectability. Which of the following statements best explains why direct detection experiments are generally insensitive to hidden-sector neutralino dark matter? 1) The hidden-sector neutralino decays rapidly before reaching detectors. 2) The neutralino annihilates primarily into visible-sector particles, evading nuclear interactions. 3) The neutralino is a scalar particle with no couplings to nuclei. 4) The neutralino is a Majorana fermion with highly suppressed couplings to nuclei, minimizing observable scattering events. 5) The neutralino’s mass is always below the detection threshold of current experiments. 6) The hidden-sector neutralino interacts only gravitationally, precluding direct detection. 7) The relic abundance of the neutralino is too low for detectable interactions.

✓ Correct Answer:

The correct answer is 4) The neutralino is a Majorana fermion with highly suppressed couplings to nuclei, minimizing observable scattering events..

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density as described in Eq. (10). The relevant annihilation modes of the LSP are x 1 x 1!hx 1hx 1; Xhx 1; XX . Computing the corresponding annihilation rates using the method of Ref. [90] near TTRH, we nd that the XX nal state typically dominates provided it is open, as we will assume here. Using these rates, we compute the relic abundance of x 1by numerically solving the system of equations presented in Section II. In doing so, the decays of the MSSM LSP and all hidden states are treated as being prompt. Before presenting our numerical results, it is instructive to examine the parametric de- pendence of the approximate solution of Eq. (10). Writing 0=Mx; (29) 14 0.511.522.533.5 0.1 1m3/2 [100 T e V] µ/prime/MxΩχh2andmχx 1forgx= 0.1 30 10 5 110 3 1 0.3 0.12 0.06FIG. 2. Contours of the hidden neutralino x 1mass in GeV (dashed grey) and moduli-generated relic abundance h2(solid red) as a function of of 0=Mxandm3=2. The moduli parameters are taken to be m'=m3=2,c= 1, andN = 1, with the hidden-sector parameters as described in the text. and focusing on a hidden Higgsino-like LSP with 1, we obtain g =gxandr = 2 in Eq. (16). Thus, smaller values of andgxare expected to produce decreased x 1relic abundances. The results of a full numerical analysis are illustrated in Fig. 2, where we show the contours of the nal x 1abundance (solid red) and DM mass (dashed grey) in the m3=2 plane form'=m3=2,c= 1, andN = 1. The range of m3=2considered corresponds toM22[100;1000] GeV, and the hidden sector parameters are taken to be gx= 0:1, tan= 10, andmx= 0:2 GeV,mAx= 10 GeV. The shape of the abundance contours in Fig. 2 is in agreement with the scaling predicted by Eq. (16). We also see that =0=Mx<1 is typically needed to avoid creating too much dark matter, and this implies some degree of ne tuning for hidden-sector symmetry breaking. Larger values of are allowed when the moduli decay parameter cis greater than unity, since this leads to a higher reheating temperature and more ecient reannihilation. D. Constraints from Indirect Detection While this extension of the MSSM can yield an acceptable hidden neutralino relic density from moduli reheating, it is also constrained by indirect detection (ID) searches for DM.9 The pair annihilation of hidden neutralinos can produce continuum photons at tree level 9Constraints from direct detection are not relevant; the x 1LSP is a Majorana fermion, and scatters o nuclei mainly through a suppressed Higgs mixing coupling [86]. 15 from cascades induced by x 1 x 1!XX withX!ff, as well as photon lines at loop level through kinetic mixing with the photon and the Z0. These signals have been searched for by a number of gamma-ray telescopes, and limits have been placed on the corresponding gamma- ray uxes. We examine here the constraints on the x 1state from observations of the galactic centre (GC) gamma ray continuum by the Fermi Large Area Telescope (Fermi-LAT) [91], as well as

Question 242 multiple-choice

Gröbner bases are fundamental computational tools in algebra used for ideal membership testing, solving polynomial equations, and analyzing quotient algebras. Their structure and computation depend critically on properties like monomial order and admissibility. Which property of monomial orders is essential to guarantee the termination of Gröbner basis algorithms such as Buchberger’s? 1) Compatibility with addition 2) Well-ordering of monomials (admissibility) 3) Existence of multiplicative identity 4) Invariance under variable permutation 5) Distributivity over multiplication 6) Commutativity of monomials 7) Homogeneity of polynomials

✓ Correct Answer:

The correct answer is 2) Well-ordering of monomials (admissibility).

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Question 243 multiple-choice

Quantum state discrimination is a fundamental problem in quantum information theory, particularly relevant for optimizing measurements in communication systems. Symmetries among sets of quantum states can be exploited to improve both theoretical understanding and computational efficiency of minimum probability of error (MPE) measurements. Which approach enables the efficient computation of minimum probability of error measurements for compound geometrically uniform (CGU) quantum state sets by leveraging group symmetries? 1) Using tensor product decompositions of Hilbert spaces 2) Applying Bayesian inference to measurement outcomes 3) Solving large-scale semi-definite programs numerically 4) Utilizing perturbation theory for state overlap estimation 5) Reducing the problem to simultaneous equations via group representation theory 6) Approximating states with orthogonal projections 7) Implementing Monte Carlo simulations for error probabilities

✓ Correct Answer:

The correct answer is 5) Reducing the problem to simultaneous equations via group representation theory.

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Title: Optimal measurements for symmetric quantum states with applications to optical communication Year: 2015 Paper ID: 0534920de17b073f71d2cfbe3dad973415d09245 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0534920de17b073f71d2cfbe3dad973415d09245 Abstract: The minimum probability of error (MPE) measurement discriminates between a set of candidate quantum states with the minimum average error probability allowed by quantum mechanics. Conditions for a measurement to be MPE were derived by Yuen, Kennedy and Lax (YKL). MPE measurements have been found for states that form a single orbit under a group action, i.e., there is a transitive group action on the states in the set. For such state sets, termed geometrically uniform (GU) by Forney, it was shown that the `pretty good measurement' (PGM) attains the MPE. Even so, evaluating the actual probability of error (and other performance metrics) attained by the PGM on a GU set involves inverting large matrices, and is not easy in general. Our first contribution is a formula for the MPE and conditional probabilities of GU sets, using group representation theory. Next, we consider sets of pure states that have multiple orbits under the group action. Such states are termed compound geometrically uniform (CGU). MPE measurements for general CGU sets are not known. In this paper, we show how our representation-theoretic description of optimal measurements for GU sets naturally generalizes to the CGU case. We show how to compute the MPE measurement for CGU sets by reducing the problem to solving a few simultaneous equations. The number of equations depends on the sizes of the multiplicity space of irreducible representations. For many common group representations (such as those of several practical good linear codes), this is much more tractable than solving large semi-definite programs---which is what is needed to solve the YKL conditions numerically for arbitrary state sets. We show how to evaluate MPE measurements for CGU states for some examples relevant to quantum-limited classical optical communication.

Question 244 multiple-choice

In survival analysis, researchers often face challenges in estimating how treatments affect outcomes over time, especially when data is incomplete due to right-censoring and unobserved confounding. Advanced statistical methods seek to improve individualized predictions and causal inference in such scenarios. Which methodological innovation enables estimation of heterogeneous treatment effects in survival data with both heavy right-censoring and unobserved confounding from instrumental variables, without requiring direct modeling of the censoring mechanism? 1) Marginal structural models with time-dependent weights 2) Accelerated failure time models using parametric imputation 3) Causal survival forests with inverse-censoring weighting 4) Recursively imputed survival trees through non-parametric multiple imputation 5) Cox proportional hazards models with frailty terms 6) Random forests trained only on uncensored observations 7) Propensity score matching with censored outcomes

✓ Correct Answer:

The correct answer is 4) Recursively imputed survival trees through non-parametric multiple imputation.

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Question 245 multiple-choice

In quantum information theory, understanding the entanglement structure of multi-qubit states is essential for analyzing quantum systems and developing efficient algorithms. Recent advances connect entanglement testing to group-theoretic methods and quantum algorithm techniques. Which approach allows efficient identification of the specific bipartition where an n-qubit pure state is unentangled by formulating the problem as a hidden subgroup search using Abelian group action and Fourier sampling, generalizing Simon's algorithm? 1) State Hidden Subgroup Problem (StateHSP) formulation with Abelian group action and Fourier sampling 2) Quantum phase estimation using non-Abelian group symmetry 3) Classical brute-force search over all bipartitions using separability tests 4) Variational quantum eigensolver with entanglement witness measurements 5) Grover's search algorithm applied to tensor product decompositions 6) Quantum walk algorithms on entanglement graphs 7) Shor's algorithm applied to state factorization

✓ Correct Answer:

The correct answer is 1) State Hidden Subgroup Problem (StateHSP) formulation with Abelian group action and Fourier sampling.

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Question 246 multiple-choice

In quantum information theory, representation theory and group symmetries play a crucial role in the construction and analysis of quantum error-correcting codes. Properties of coefficients arising from these symmetries can strongly influence the performance and robustness of the codes. Which property of Littlewood-Richardson coefficients is essential for ensuring that random covariant quantum error-correcting codes exhibit reliable asymptotic error correction performance? 1) Smoothness of the coefficients as representation labels vary 2) Maximization of the coefficients for all tensor products 3) Vanishing of the coefficients for large system sizes 4) Symmetry of the coefficients under subsystem exchange 5) Sparsity of the coefficients in finite dimensions 6) Non-negativity of the coefficients in all cases 7) Periodicity of the coefficients with respect to subsystem count

✓ Correct Answer:

The correct answer is 1) Smoothness of the coefficients as representation labels vary.

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bound the fluctua- tions of the fidelity of recovery when the logical Hilbert FIG. 5. Smoothness of the Littlewood-Richardson coefficients, required for our proof that random covariant codes can asymp- totically correct against errors. A Littlewood-Richardson coef-ficient c λμνis the coefficient that counts the degeneracy of the UðdLÞirrep labeled by the Young diagram λin the tensor product of two other irreps labeled by μand ν. The Littlewood- Richardson coefficients are nonzero in a convex cone in thespace of ðμ;ν;λÞ. This cone —orchamber complex —is divided into several smaller convex cones —or chambers —in which c λμνis a polynomial of μ,ν, and λ. Hence, a generic choice of irreps on which we choose a random code has corresponding coefficientsthat are smooth, which we show implies good asymptoticperformance of the code.PHILIPPE FAIST et al. PHYS. REV . X 10,041018 (2020) 041018-18 space is too small. For Uð3Þ-covariant codes, our methods lead to codes with a slightly different scaling from Eq. (58). In fact, for the Uð3Þcase, one can provide randomized and nonasymptotic constructions (which work for known finite physical dimensions) using the explicit formulas for theLittlewood-Richardson coefficients [63]. These construc- tions are not included in the present paper, as there is little specific interest in the d L¼3case. The proof of Theorem 7 is technical and relies on the representation theory of the unitary group (cf. Supple- mental Material, Sec. F [37]for details). The proof starts by connecting the average fidelity recovery of erasure of afixed subsystem to the smoothness of the Littlewood- Richardson coefficients . Littlewood-Richardson coeffi- cients are representation theory quantities that count thedegeneracy of a particular irrep of Uðd LÞin the tensor product of two other irreps, and their smoothness followsfrom modern results in representation theory of the unitarygroup [63] (Fig. 5). C. Generalized W-state encoding Here, we consider another example of an approximate quantum error-correcting code, covariant with respect tothe full unitary group on the logical system. It is based ontheWstate and achieves an arbitrarily small ϵ worst in the limit of a large number of subsystems, n→∞. The logical system Lof dimension dLis encoded into a physical system composed of ncopies of a ðdLþ1Þ-dimensional space, where each subsystem is a copy of the logical systemwith an additional basis vector j⊥i. The encoding is jψi L→1ﬃﬃﬃnpðjψ;⊥;…;⊥iþj⊥;ψ;⊥;…i þ/C1/C1/C1þj ⊥;…;⊥;ψiÞ: ð59Þ Any logical unitary Ucan be carried out on the encoded state transversally by applying the unitary U⊗U⊗ /C1/C1/C1⊗U, where we let Uact trivially on the extra state j⊥i. Remarkably, aside from being UðdLÞcovariant, this trivial code is also effective against random erasures. Intuitively, there is only a probability 1=nthat the logical state jψiis stored on the subsystem the environment chose to access; that is, the environment is unlikely to learnanything about the logical state. This effectiveness can beformalized with a direct application of our criterion(Proposition 4). Given a basis jxi LofL, the reduced state ρx 1on any single physical subsystem of the code word (59) corresponding to jxiLis ρx 1¼1 njxihxjþ/C18 1−1 n/C19 j⊥ih⊥j; ð60Þ and, thus, F2ðρx 1;j⊥ih⊥jÞ

Question 247 multiple-choice

Quantum computing relies on manipulating qubits using unitary transformations, with operations built from elementary gates represented in matrix form. Understanding the roles and effects of these gates is foundational for designing and analyzing quantum algorithms. Which single-qubit gate is essential for creating equal superpositions of basis states, enabling quantum parallelism and the generation of entanglement in quantum circuits? 1) Identity gate 2) Quantum NOT gate 3) Hadamard gate 4) Phase shift gate 5) Pauli-Y gate 6) Pauli-Z gate 7) T gate

✓ Correct Answer:

The correct answer is 3) Hadamard gate.

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Usuch that U(jyi jfi) =jyi jyi: (2.22) By linearity of U U(ajyi jfi) =a(U(jyi jfi)) =ajyi jyi: (2.23) Letjci=ajyi. Then from Eq (2.22) it follows that U(jci jfi) =jci jci=ajyi ajyi =a2jyi jyi (2.24) which contradicts Eq (2.23). 2.3 Quantum Gates To perform a quantum computation, it is necessary to: 1. Prepare the quantum computer in a well-deﬁned initial state, for example, j00:::0i. 2. Manipulate the quantum-computer wave function via unitary transformations. 3. Perform a measurement in the computational basis at the end of an algorithm. The evolution of an n-qubit system is described by a unitary matrix of degree 2n, but any 11 such matrix can be decomposed into a product of unitary operations on one or two qubits, and these operations are the elementary quantum gates of the quantum circuit model of computation. It follows from linearity that the action of a gate is completely speciﬁed by its action on the com- putational basisfj0i;j1ig. 2.3.1 Single-qubit gates A quantum gate acting on a single qubit is represented by a 2 2 unitary matrix. The simplest single-qubit transformation is the identity Iwhose action on the computational basis is deﬁned by I:j0i7!j 0i;j1i7!j 1i (2.25) with the matrix representation I=j0ih0j+j1ih1j=0 B@1 0 0 11 CA (2.26) The quantum NOT gate implements the mapping X:j0i7!j 1i;j1i7!j 0i (2.27) and is represented by the matrix X=j1ih0j+j0ih1j=0 B@0 1 1 01 CA=sx: (2.28) Similarly, Y:j0i7!j 1i;j1i7!j 0i (2.29) 12 is represented by Y=j0ih1jj1ih0j=0 B@01 1 01 CA=isy; (2.30) and Z:j0i7!j 0i;j1i7!j 1i (2.31) has a representation Z=j0ih0jj1ih1j=0 B@1 0 011 CA=sz: (2.32) The matrices sx;sy;szare Pauli matrices used in quantum mechanics to describe the spin of a spin-1/2 particle in three spatial dimensions. TheZgate is a special case (with q=p) of the phase shift gate Pthat alters the relative phase of the amplitudes of a qubit: P:j0i7!j 0i;j1i7!eiqj1i (2.33) P=j0ih0j+eiqj1ih1j=0 B@1 0 0eiq1 CA (2.34) Other special cases of interest include the Sgate with q=p 2whose matrix is given by S=0 B@1 0 0i1 CA (2.35) and the Tgate with q=p 4: T=0 B@1 0 0eip 41 CA (2.36) 13 A variation of the phase gate is the discrete phase gate Rkgiven by Rk=0 B@1 0 0e2pi 2k1 CA (2.37) A gate used to create superposition states is the Hadamard gate Hspeciﬁed by H=1p 2(j0ih0j+j0ih1j+j1ih0jj1ih1j): (2.38) The action of Hon the computational basis is given by H:8 >>< >>:j0i7!j0i+j1ip 2 j1i7!j0ij1ip 2(2.39) with a representation H=1p 20 B@1 1 111 CA (2.40) In the quantum circuit notation, a qubit is denoted by a single line, called a quantum wire, which represents a qubit that is not evolving, i.e., a qubit acted upon by the identity operator I. If a qubit is initialized into a particular state, the state is written on the left of the appropriate quantum wire, and if a qubit is to be subjected to a particular unitary transformation, a corresponding gate symbol is placed along the quantum wire. For example, the process of initializing a qubit to a state j0i,

Question 248 multiple-choice

The classification of semi-direct product groups in abstract algebra relies on understanding group homomorphisms and the structure of multiplicative groups modulo prime powers. These concepts are crucial for applications in computational group theory and quantum algorithms. Given primes p and q, and an integer r ≥ 1, how many distinct non-trivial elements of order q exist in the multiplicative group Z*_{p^r} when r > 1 and q = p ≠ 2? 1) Exactly r distinct elements 2) Exactly p^r − 1 distinct elements 3) Exactly q distinct elements 4) Exactly one unique element 5) Exactly p − 1 distinct elements 6) Exactly two elements when p is odd 7) Zero elements if r > 2

✓ Correct Answer:

The correct answer is 5) Exactly p − 1 distinct elements.

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permutation groups and m atrix groups. However, we stress that the converse is not generally true: eﬃcie nt algorithms for a group problem can use information about the structure of the group that are not availab le in the black-box context. It is known that the HSP over an Abelian group input as a (quantum) b lack-box group with unique encoding can be solved in polynomial time by a quantum computer [19]. Notice that the same problem is open when the black-box group has not unique encoding. 3 A Classiﬁcation of Semi-direct Product Groups 3.1 Number of possibilities for φ For given nandq, how many possibilities are there for φdeﬁning a semi-direct product group Zn⋊φZq? Theconditionthat φshouldbe ahomomorphismimpliesthat φ(1)(1)q≡1 modn. Deﬁning φ(1)(1) satisfying this condition is actually necessary and suﬃcient to deﬁne completely φ. Notice that the case φ(1)(1) = 1 is a trivial possibility that leads to the direct product Zn×Zq. By considering the usual decomposition Zn∼=Zp1e1×···×Zpkek, (1) we can determine the number of possibilities for Zn⋊φZqby determining the number of possible φin the deﬁnition of the groups Zpei i⋊φZq. Therefore, it is suﬃcient to study only the case of nbeing a power of a prime number. Finding the number of acceptable deﬁnitions for φthus reduces to ﬁnding elements of order qinZ∗ nwithna prime power. In this paper, we will consider only the case qprime, which gives a clear classiﬁcation into ﬁve classes of semi-direct product groups. Proposition 4. Letpandqbe two prime numbers, and ran integer such that r≥1. The only cases where there exist non-trivial elements αof order qinZ∗ prare the following three cases. (i)q|p−1. There are exactly q−1distinct possibilities for α. (ii)r >1, q=p/ne}ationslash= 2. There are exactly p−1distinct possibilities: α=tpr−1+1for0< t < p. (iii)r >1, q=p= 2. Ifr >2then there are exactly three distinct possibilities: 2r−1+ 1,2r−1−1and 2r−1. Ifr= 2then there is only only possibility: α= 3. Proof.First, we consider the case p/ne}ationslash= 2. Recall that the group Z∗ pris a cyclic group. Let ube a primitive element of Z∗ pr. Thenαcan be written as ukfor some kless than the order of u. Since the order of uis pr−1(p−1),pr−1(p−1) divides kq. As 1≤k < pr−1(p−1) and we assume qis prime, qmust be por any prime that divides p−1. Ifq=p,kmust be of the form lpr−2(p−1) where l∈ {1,···,p−1}, so the number of non-trivial possibilities for αisp−1. In fact, it can be checked that the order of α=tpr−1+1 isp, for every 1≤t≤p−1: these p−1 values of αare thus the exact solutions. Else if qis a prime that divides p−1,kmust be lpr−1p−1 qwherel∈ {1,···,q−1}. Next, we consider the case p= 2. Assume r >2 (the case r= 2 is trivial: one unique solution, α= 3). As the order of group Z∗ 2ris 2r−1, the prime qmust be 2. Since α∈Z∗ 2ris odd, we denote αas 2kl+1 for k∈ {1,···,r−1}and odd l. From the condition α2= 2k+1l(2k−1l+ 1) +1 ≡1 mod 2r, we get k= 1 or 2r|2k+1. We thus obtain three cases: the case k= 1 and l= 2r−2−1

Question 249 multiple-choice

Quantum computing leverages group-theoretic problems to achieve algorithmic speedups, particularly in areas such as subgroup identification and isomorphism testing. Recursive methods and quantum superpositions are central to extending these techniques to more complex group structures. Which of the following statements accurately describes a unique property of the Translating Coset problem in finite solvable groups that enables recursive quantum algorithm design? 1) It requires injective hiding functions for identifying subgroup structure. 2) It can only be solved efficiently in non-abelian groups. 3) It is equivalent to the standard Hidden Subgroup Problem for all group types. 4) It does not involve quantum group actions on orthogonal quantum states. 5) Its solution set is never a coset of any subgroup. 6) It is well-suited for classical algorithms due to lack of superposition requirements. 7) It possesses a self-reducibility property allowing reduction to instances in factor groups and normal subgroups.

✓ Correct Answer:

The correct answer is 7) It possesses a self-reducibility property allowing reduction to instances in factor groups and normal subgroups..

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3.5 is that Hidden Subgroup inZn p/multicloserightZ2is eﬃciently solvable by a quantum algorithm. To solve Hidden Translation in other groups (which include abelian groups of constant exponent), we embark in a radically new direction whose basic idea is self- reducibility .S i n c e Hidden Translation is not well-suited for this self-reducibility based approach, we deﬁne a new paradigmatic group problem. Notice that there is anatural combination of Hidden Translation withHidden Subgroup .T h i si st h e version of Hidden Translation where the functions f 0andf1are not necessarily injective, but they are certain subgroup hiding functions. Indeed, if f1hides a sub- groupHandf0(x)=f1(xu)f o rs o m e u∈Gand for every x∈G, then the set of all such elements uform a right coset of H. (In the context of graph isomorphisms, the correspondingproblemwouldbedetermining allthe bijections betweenthe vertexsets which are isomorphisms. This set is a coset of the automorphism group of one of the graphs.) The self-reducibility will be based on “averaging” over normal subgroups sothat we actually get a problem over the factor group. We will give an averaging pro- cedure which results in quantum superpositions. Therefore our new problem, called Translating Coset , is a combination of Hidden Translation andHidden Sub- groupwhere we have quantum states as input. 1Translating Coset also involves quantum group actions, that is, groups acting on a ﬁnite set of mutually orthogonalquantum states. Given two such states, |φ 0/angbracketrightand|φ1/angbracketright,t h e Translating Coset problem consists of ﬁnding their translating coset , which is deﬁned to be the stabilizer subgroup of |φ1/angbracketrightand a group element that maps |φ1/angbracketrightto|φ0/angbracketright. It turns out that with a slight modiﬁcation, our algorithm of Theorem 3.5 also works for Translating Coset inZn pwhenever many copies of the input states 1In the preliminary version [15] of the present paper, the problem Translating Coset was called Orbit Coset . This was due to the fact that the problem is actually a constructive version of testing membership in orbits of permutation groups. 4 FRIEDL, IVANYOS, MAGNIEZ, SANTHA, AND SEN are given. Moreover, we show that Translating Coset has the following self- reducibility property in any ﬁnite solvable group G: it is reducible to instances ofTranslating Coset inG/NandNfor any normal subgroup NG(Theo- rem 4.11). This is the ﬁrst general self-reducibility result for a problem subsum-ingHidden Subgroup . The proof of the result involves a new technique which is based upon constructing the uniform superposition of the orbit of a given quan- tum state ( Orbit Superposition ). The importance of generating speciﬁc super- positions for solving important algorithmic problems has been observed before; see, for instance, the paper of Aharonov and Ta-Shma [3]. For example, generating theuniform superposition of all graphs isomorphic to a given graph, which in fact is an instance of the Orbit Superposition problem of the symmetric group S nact- ing on an n-vertex graph, would allow us to solve the graph isomorphism problem. We show how Orbit Superposition is related to Translating Coset (Theo- rem 4.10). The self-reducibility of

Question 250 multiple-choice

In quantum algorithms for lattice problems, understanding the properties of lattices, sublattices, and their associated functions is crucial for distinguishing algebraic structures and reconstructing hidden subgroups. The analysis often involves inner products, Lipschitz constants, and the relationship between a lattice and its reciprocal. In the context of quantum algorithms addressing the hidden subgroup problem for continuous groups, under which condition is the inner product between functions associated to two shifted lattices notably large, and what algebraic structure does this condition correspond to? 1) When the lattices are orthogonal and the shift is arbitrary 2) When the shifted lattices have maximal determinant difference 3) When the sublattice is trivial and the functions are constant 4) When the shift corresponds to a vector outside the lattice’s span 5) When the lattices are related by a scaling transformation only 6) When the shift corresponds to a unit of norm 1 in the unit group of a number field 7) When the shifted lattices have disjoint supports in continuous space

✓ Correct Answer:

The correct answer is 6) When the shift corresponds to a unit of norm 1 in the unit group of a number field.

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Lemma 5.5. Suppose we are given lattices LandL/prime, and sublatticesI⊆LandI/prime⊆L/prime. Assume there is a 1-1 corre- spondenceh:I→I/primes.t., for any (x,x/prime)∈˜L×˜L/prime, 299 1. if (x,x/prime)∈C:={(u,v )∈I×I/prime:v=h(u)}, then /bardblx−x/prime/bardbl ≤ε, 2. otherwise /bardblx−x/prime/bardbl ≥2ν√n. IfI/subsetnoteqlLandI/prime/subsetnoteqlL/prime, then for any n≥5,/angbracketleftf(L)|f (L/prime)/angbracketright ≤ 3/4if in addition s≥4πn3max{λn(L),λ n(L/prime)}. Lemma 5.6. Letν/lessorequalslantλ 2√nands/greaterorequalslantcn(√n/λ)n−1dfor a certain constant c, and let us restrict the encoding L/mapsto→ |f(L)/angbracketright to lattices with d(L)/lessorequalslantdandλ1(L)/greaterorequalslantλ. On such lattices,fhas a Lipschitz constant a=√πns 4ν+ 1. Next we can choose the parameters to show that the hid- den subgroup property holds. Since the inner product is at most a constant <1, a tensor product of ncopies will reduce the inner product to be exponentially small. For simplicity, we state the theorem only for the free part of the unit group LogO∗≤Rs+t−1, and for elements of norm 1. Theorem 5.7. Lets= 22n√nD,ν= 1/(4n(s√n)2n). Lett1,t2∈Rn. Letγ= (γ 1,...,γ n)bet2−t1−u, where u is the unit closest to t2−t1in Log O∗. Then the function fisa-Lipschitz with constant a=√πn s 4ν+ 1. The inner product /angbracketleftet1O|et2O/angbracketrightis at most 3/4 if for some i, we either have ln` 1−(s√n)n−12ν√n´ ≥γiorγi≥ ln` 1 + (s√n)n−12ν√n´ . The idea is as follows: ﬁx an ideal et1O. We want to bound its inner product with ideals et2O. First we show that when the two lattices have points inside the ball of ra- diuss√nwhere they overlap exactly, then their inner prod- uct must be small (unless the two lattices coincide, i.e. when t2−t1is a unit). Then we show that this inner product does not get bigger when we perturb one of the lattices slightly. Finally, we show that when two lattices are not close to hav- ing an exact point of intersection, then their inner product is small as well. Finally, it can be shown that a good approximation of these states can be computed when evaluated on rational numbers. First an approximate basis is computed using the algorithm in the last section, and then the superposition is created over the points. 6. THE HIDDEN SUBGROUP PROBLEM ON A CONTINUOUS GROUP The HSP algorithm for Rmcan be thought of in the usual structure, however the analysis is diﬃcult, and we combine phase estimation and our new continuous deﬁnition of the HSP to get it to work. The analysis works in continuous space and is discretized in a general way to derive the al- gorithm. The algorithm creates a superposition of points in Rmwith a suﬃciently broad wavefunction w. We cannot measure in the Fourier basis of the continuous group, but we show that phase estimation can be used to approximate this to measure a point uof the reciprocal lattice L∗with the probability distribution qu=1d(L)2Z (Rm/L)2/angbracketleftf(x/prime)|f(x)/angbracketrighte2πi/angbracketleftx−x/prime,u/angbracketrightdxdx/prime.(6.1)This distribution is not close to uniform but we are able to show that the probability of staying in any sublattice is bounded. For rounding, the samples deviate from the lattice points by, roughly, the inverse width of the wavefunction w. We show that the reconstruction of a lattice from an approximate generating set can be done using an

Question 251 multiple-choice

In computational group theory, analyzing finite p-groups of nilpotency class 2 often involves studying their automorphisms and using refined polycyclic presentations for efficient computation. The structure and encoding of such groups enable polynomial-time algorithms for critical subgroup computations. For a finite p-group of exponent p and nilpotency class 2, which automorphism property holds for every j in Z_p and every element z in the derived subgroup G′? 1) φ_j(z) = z^{j} 2) φ_j(z) = z^{j^2} 3) φ_j(z) = z^{p} 4) φ_j(z) = z^{j+j^2} 5) φ_j(z) = z^{j-1} 6) φ_j(z) = z^{j^3} 7) φ_j(z) = z^{j^2+1}

✓ Correct Answer:

The correct answer is 2) φ_j(z) = z^{j^2}.

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p.In those groups we will indentify G′andZd p.Thus, for two elements zandz′ofG′, the product zz′is just z⊕z′where⊕denotes the coordinate-wise addition modulo p. IfGis a such a group then |G|=pm+d. The elements of Gcan be encoded by binary strings of length O((m+d)logp), and an eﬃcient algorithm on input Ghas to be polynomial in m,dand logp. Forj= 1,...,p−1, we consider on generators the maps xitoxj i. It turns out that these maps extend to automorphisms φjofG. We also deﬁne the map φ0by letting φ0(g) = 1, for every g∈G. Proposition 1. LetGbe a p-group of exponent pand of nilpotency class 2. Then the mappings φjhave the following properties: 1.∀j∈Zp,∀z∈G′, φj(z) =zj2, 2.∀g∈G,∃zg∈G′,∀j∈Zp, φj(g) =gjzj−j2 g. Proof.The ﬁrst statement is trivial when j= 0. Otherwise, observe that for every j∈Z∗ p,and for every g∈G, there exists z∈G′such that φj(g) =gjzsinceG/G′is abelian. To prove the ﬁrst statement, letz= [g1,g2]. Then by this remark, there exist z1andz2inG′such that φj([g1,g2]) = [gj 1z1,gj 2z2]. By repeated applications of Fact 2 this is easily seen to be ([ g1,g2])j2. We now turn to the second statement. Let j0be a ﬁxed primitive element of Z∗ p. Thenφj0(g) =gj0s, for somes∈G′. Setzg=s(j0−j2 0)−1, we have φj0(g) =gj0zj0−j2 0g. Letk=gzg, thenφj0(k) =gj0zj0−j2 0gzj2 0g=kj0. Therefore, for all j∈Zp, we have φj(k) =kjandφj(g) =φj(k)φj(z−1 g) =gjzj gz−j2 g. 4 Clearly, for every g∈G, the element zgwhose existence is stated in the second part of Proposition 1 is unique. From now on, let zgdenote this unique element. 3 Classical reductions in groups of constant nilpotency cla ss In order to present the reduction methods in a suﬃciently general way, in this section we assume that our groups are presented in terms of so-called reﬁned polycyclic presentations [9]. Such a presentation of a ﬁnite solvable group Gis based on a sequence G=G1⊲...⊲Gs+1={1},where for each 1 ≤i≤sthe subgroup Gi+1is a normal subgroup of Giand the factor group Gi/Gi+1is cyclic of prime order ri. For each i≤san elementgi∈Gi\Gi+1is chosen. Then gri i∈Gi+1. Every element gofGcan be uniquely represented as a product of the form ge1 1···gess, called the normal word for g, where 0 ≤ei< ri. In the abstract presentation the generators are g1,...,g s,and for each index 1 ≤i≤s,the following relations are included: •gri i=ui, whereui=gai,i+1 i+1···gai,s sis the normal word for gri∈Gi+1, •g−1 igjgi=wijfor every j > i, wherewij=gbi,j,i+1 i+1···gbi,j,ssis the normal word for g−1 igjgi∈Gi+1. Using a quantum implementation [11] of an algorithm of Beals and Babai [2], reﬁned polycyclic presentation for a solvable black box group can be computed in polynomial time. We a ssume that elements of Gare encoded by normal words and there is a polynomial time algorithm in log |G|, the so called collection procedure , which computes normal words representing products. This is the case for nilpotent groups of constant class [10]. If there is an eﬃcient collection procedure then reﬁned polycyclic presentations for subgroups and factor groups can be obtained in polynomial time [9]. Also, the major notable subgroups including Sylow subgroups, the center and the

Question 252 multiple-choice

Quantum computing is an emerging field that holds promise for simulating fundamental physical theories, particularly those involving complex gauge symmetries. Non-Abelian lattice gauge theories, such as those based on the SU(2) group, are of particular interest for understanding the strong force in particle physics. Which statement accurately describes the significance of using SU(2) gauge symmetry in quantum simulations of lattice gauge theories with dynamical matter? 1) SU(2) gauge symmetry is Abelian and primarily used to model electromagnetic interactions. 2) SU(2) gauge symmetry is non-Abelian and serves as a foundational step toward simulating quantum chromodynamics (QCD), enabling calculations relevant to hadron physics. 3) SU(2) gauge symmetry allows for exact classical simulation of all time evolution processes without encountering the sign problem. 4) SU(2) gauge symmetry is exclusively used for modeling scalar field theories without fermionic matter. 5) SU(2) gauge symmetry is less relevant than U(1) for studying strong interactions among quarks. 6) SU(2) gauge symmetry restricts quantum simulations to static properties, excluding real-time dynamics of hadrons. 7) SU(2) gauge symmetry cannot be implemented on current quantum hardware due to its complexity.

✓ Correct Answer:

The correct answer is 2) SU(2) gauge symmetry is non-Abelian and serves as a foundational step toward simulating quantum chromodynamics (QCD), enabling calculations relevant to hadron physics..

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Question 253 multiple-choice

Quantum phase estimation is a key algorithm for extracting eigenphases of unitary operators, with careful management of measurement error and amplification techniques. The rounding promise and median amplification are essential for ensuring high-probability correct outcomes in the presence of noise and precision constraints. In quantum phase estimation with α > 1/2 and the rounding promise, which of the following statements is correct regarding the error probability when employing median amplification with M repetitions? 1) The error probability increases linearly with M and cannot be bounded by a chosen constant. 2) The error probability decreases exponentially with M, and can be bounded by a chosen constant δ_med by selecting M = ⎡log(δ_med⁻¹)/(2η²)⎤. 3) The error probability remains fixed at 1/2 regardless of amplification. 4) The error probability decreases polynomially with M, independent of η. 5) The error probability depends only on α and is unaffected by the number of repetitions M. 6) The error probability can be increased above 1/2 by increasing M. 7) The error probability is minimized when M = 1 due to the rounding promise.

✓ Correct Answer:

The correct answer is 2) The error probability decreases exponentially with M, and can be bounded by a chosen constant δ_med by selecting M = ⎡log(δ_med⁻¹)/(2η²)⎤..

📚 Reference Text:

construction to work, but observing this will be useful for the α≤1/2case. However, it is necessary to observe that when α>1/2we are guaranteed that ηis positive. Then, from the rounding promise and the deﬁnition of λ(x) j, we are guaranteed that whenx=ﬂoor(2nλj)then|β(λ(x) j)|2≥1/2 +η. Now we perform a standard median ampliﬁcation analysis. The probability of a particular estimate xbeing ‘correct’ is ≥ 1/2 +η, so the probability of being incorrect is ≤1/2−η. The only way that a median ofMestimates can be incorrect is if more than half of the estimates are incorrect. Let X be the random variable counting the number of incorrect estimates. Then we can invoke the Chernoﬀ-Hoeﬀding theorem: Pr[median incorrect ]≤Pr[X≥M/2] (57) ≤Pr[X≥M/2 +E(X)−(1/2−η)M] (58) ≤exp/parenleftBig −2Mη2/parenrightBig (59) We can bound the above by δmed, a constant we will ﬁx later, if we select M:=/ceilingleftbigg log(δ−1 med) 2η2/ceilingrightbigg as we did above. Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 13 0.0 0.25 0.5 1.0 (x) j 0.0 8 π2 γ(1/2) = η 0.5 1.0 Lower Bounds on Phase Estimation Probabilities ((x) j) ((x) j 1) ((x) j) + ((x) j 1) α≈10%2 0Figure 1: Sketch of γ(λ(x) j), a lower bound on the magnitude of the amplitude of phase estimation |β(λ(x) j)|2. Recall that the rounding promise guarantees that λj/negationslash∈/bracketleftbigx 2n,x 2n+α 2n/bracketrightbig for allx, implying thatλ(x) j/negationslash∈[1−α 2,1+α 2]as indicated in the shaded region. From this sketch we draw two conclusions necessary for our proof. First, when α > 1/2thenγ(λ(x) j)is guaranteed to be strictly greater than 1/2, soη > 0(in fact,η > η 0). Second, even when no rounding promise applies, the sum of the probabilities of two adjacent bins is at least 8/π2, which we use to deﬁne the ampliﬁcation threshold η0. Also visible in this ﬁgure is the impossibility of reducing αfurther than≈10%without rounding: several values of λ(x) jnear 1/2have probabilities less than 1/2and cannot be ampliﬁed. We have demonstrated that pj,x≥1−δmedwheneverx=ﬂoor(2nλj). Now all that remains is a bound on the error in diamond norm. We begin by bounding the spectral norm. Let pj:=pj,ﬂoor(2nλj)and observe that 1≥pj≥1−δmed. Taking the diﬀerence between equations (47) and (48) we obtain: /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(√pj−1)eiϕj|ﬂoor(λj2n)/angbracketright|garj/angbracketright|ψj/angbracketright+/summationdisplay x,x/negationslash=ﬂoor(λj2n)eiϕj,x√pj,x|x/angbracketright|garj,x/angbracketright|ψj/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(60) =/radicalBigg |(√pj−1)eiϕj|2+/summationdisplay x,x/negationslash=ﬂoor(λj2n)/vextendsingle/vextendsingleeiϕj,x√pj,x/vextendsingle/vextendsingle2(61) =/radicalBigg (√pj−1)2+/summationdisplay x,x/negationslash=ﬂoor(λj2n)pjk (62) ≤/radicalBig (1−/radicalbig 1−δmed)2+δmed (63) ≤/radicalBig 2−2/radicalbig 1−δmed−δmed+δmed (64) ≤/radicalBig 2−2/radicalbig 1−δmed (65) Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 14 Now we apply Lemma 4 to bound the diamond norm, and since the error is ∼√δmedto leading order, we construct a bound that holds whenever√δmed≤1: 2/radicalBig 2−2/radicalbig 1−δmed≤2.5/radicalbig δmed (66) If we select δmed:=δ2/6.25then the diamond norm is bounded by δ. Having completed the α>1/2case, we proceed to the α≤1/2case. As stated above, the construction we just gave actually also works when α≤1/2. However, the asymptotic dependence on αis like∼α−2which is not optimal. Looking again at Figure 1, we see that the sum of the probability of two adjacent bins is at least 8/π2, even when λ(x) jis in a region disallowed by the rounding promise. Therefore, if we

Question 254 multiple-choice

In quantum information theory, various norms are used to quantify the closeness of distributions over quantum operations to the uniform (Haar) measure, which is crucial for randomness and security tasks. Approximate k-designs are central for simulating random quantum processes and benchmarking quantum circuits. If a distribution of quantum operations is a diamond ε-approximate k-design, which of the following statements about its relation to TPE (Tensor Product Expander) k-designs is correct? 1) It is also a TPE k-design with approximation error exactly ε. 2) It is also a TPE k-design with approximation error at most εdk/2. 3) It is also a TPE k-design with approximation error at most ε/dk. 4) It is also a TPE k-design with approximation error at most ε². 5) It is also a TPE k-design with approximation error at most ε/2. 6) It is also a TPE k-design with approximation error at most 2ε. 7) It is also a TPE k-design with approximation error at most εk/d.

✓ Correct Answer:

The correct answer is 2) It is also a TPE k-design with approximation error at most εdk/2..

📚 Reference Text:

qubits nifλ=O(1). We will now define the notion of diamond ε-approximate k-design. 4Note that the submultiplicativity inequality is saturated if E V∼ν/bracketleftbig V⊗k,k/bracketrightbig is Hermitian (or more generally normal). 27 Definition 41 (Diamondε-approximate k-design) .We say that νis a diamond ε-approximate k-design if and only if /vextenddouble/vextenddouble/vextenddoubleM(k) ν−M(k) µH/vextenddouble/vextenddouble/vextenddouble ♢≤ε, (201) where the diamond norm of a superoperator Φ :L(CD)→L(CD)can be defined as ∥Φ∥♢:= sup X̸=0∥Φ⊗I(X)∥1 ∥X∥1, (202) whereI:L(CD)→L(CD)is the identity channel. It is worth mentioning that the diamond norm distance between two quantum channels has an important operational interpretation, as it is closely related to the one-shot distinguishability probability between the two channels [75]. In the following Proposition, we will see the relation between the Diamond approximate design definition and the TPE one previously introduced. Proposition 42 (Diamond vs TPE) . •We have: /vextenddouble/vextenddouble/vextenddoubleM(k) ν−M(k) µH/vextenddouble/vextenddouble/vextenddouble ♢≤dk/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE V∼ν/bracketleftbig V⊗k,k/bracketrightbig −E U∼µH/bracketleftbig U⊗k,k/bracketrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble ∞. (203) Thus, ifνis a TPEε-approximate k-design, then νis a diamond εdk-approximate k-design. •Conversely, we have: /vextenddouble/vextenddouble/vextenddouble/vextenddoubleE V∼ν/bracketleftbig V⊗k,k/bracketrightbig −E U∼µH/bracketleftbig U⊗k,k/bracketrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble ∞≤dk/2/vextenddouble/vextenddouble/vextenddoubleM(k) ν−M(k) µH/vextenddouble/vextenddouble/vextenddouble ♢. (204) Hence, ifνis a diamond ε-approximate k-design, then νis a TPEεdk/2-approximate k- design. Proof. Let Φ :=M(k) ν−M(k) µH. We have: ∥Φ∥♢= sup X̸=0∥Φ⊗I(X)∥1 ∥X∥1≤√ d2ksup X̸=0∥Φ⊗I(X)∥2 ∥X∥1≤dksup X̸=0∥Φ⊗I(X)∥2 ∥X∥2, (205) where we used that ∥A∥1≤√ D∥A∥2and∥A∥2≤∥A∥1forA∈L(CD). Now we observe that: sup X̸=0∥Φ⊗I(X)∥2 ∥X∥2= sup X:∥X∥2=1∥vec(Φ⊗I)|X⟩⟩∥2=∥vec(Φ⊗I)∥∞, (206) where in the first equality we used that the Hilbert-Schmidt norm of a matrix is the 2-norm of the vectorized matrix and that vec (Φ ⊗I(X)) = vec(Φ⊗I)|X⟩⟩(see Eq.(105)), while in the third equality we used that ∥A|v⟩∥2=/radicalbig ⟨v|A†A|v⟩and that the supremum of this quantity over all the vectors|v⟩that are normalized with respect to the 2-norm is achieved by the largest eigenvalue of A†A, which coincides with the largest singular value of A. Now we have: /vextenddouble/vextenddoublevec/parenleftbig U⊗k(·)U†⊗k⊗I/parenrightbig/vextenddouble/vextenddouble ∞=/vextenddouble/vextenddoubleU⊗k⊗I⊗k⊗U∗⊗k⊗I⊗k/vextenddouble/vextenddouble ∞=/vextenddouble/vextenddoubleU⊗k⊗U∗⊗k⊗I⊗k⊗I⊗k/vextenddouble/vextenddouble ∞, (207) where in the last step we permuted the tensor products using the p-norm property/vextenddouble/vextenddoubleUAU†/vextenddouble/vextenddouble p= ∥A∥pwithUunitary matrix. Using this we can write: /vextenddouble/vextenddouble/vextenddoubleM(k) ν−M(k) µH/vextenddouble/vextenddouble/vextenddouble ♢≤dk/vextenddouble/vextenddouble/vextenddoublevec/parenleftig M(k) ν⊗I/parenrightig −vec/parenleftig M(k) µH⊗I/parenrightig/vextenddouble/vextenddouble/vextenddouble ∞(208) =dk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg E V∼ν/bracketleftbig V⊗k,k/bracketrightbig −E U∼µH/bracketleftbig U⊗k,k/bracketrightbig/parenrightbigg ⊗I⊗k⊗I⊗k/vextenddouble/vextenddouble/vextenddouble/vextenddouble ∞(209) =dk/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE V∼ν/bracketleftbig V⊗k,k/bracketrightbig −E U∼µH/bracketleftbig U⊗k,k/bracketrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble ∞, (210) 28 where in the last step we used the p-norm property∥A⊗B∥p=∥A∥p∥B∥p. Hence the first claim follows. Now for the second claim, reviewing in reverse what we have shown for the first claim, we have: /vextenddouble/vextenddouble/vextenddouble/vextenddoubleE V∼ν/bracketleftbig V⊗k,k/bracketrightbig −E U∼µH/bracketleftbig U⊗k,k/bracketrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble ∞= sup X̸=0∥Φ⊗I(X)∥2 ∥X∥2. (211) By using that∥A∥2≤∥A∥1, that∥A∥1≤√ D∥A∥2forA∈L(CD) and the definition of diamond norm, we have: /vextenddouble/vextenddouble/vextenddouble/vextenddoubleE V∼ν/bracketleftbig V⊗k,k/bracketrightbig −E U∼µH/bracketleftbig U⊗k,k/bracketrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble ∞≤√ dk/vextenddouble/vextenddouble/vextenddoubleM(k) ν−M(k) µH/vextenddouble/vextenddouble/vextenddouble ♢. (212) Motivated by Proposition 34, we introduce an alternative notion of approximation based on the frame potential. Notably, the frame potential may be easier to estimate numerically than the other approximation notions previously introduced (if approached naively by their definition), as it involves computing the trace of matrices in L(Cd), rather than working with matrices defined in tensor product spaces. Specifically, we define a frame potential ε-approximate k-design as follows: Definition 43 (Frame potential ε-approximate k-design) .We say that νis a frame potential ε-approximate

Question 255 multiple-choice

In the study of group rings and modules, chain conditions play a crucial role in understanding the structure and classification of algebraic systems such as groups and their representations. The relationship between ideals in group rings and the absence of infinite strictly ascending chains of submodules is central to many finiteness results in group theory. Which statement best explains why an infinite strictly ascending chain of normal subgroups in a quotient group F/Uk is impossible under certain module-theoretic conditions? 1) Every normal subgroup in F/Uk is automatically maximal, preventing any ascending chains. 2) The group F/Uk is finite, so infinite chains cannot exist. 3) The module M3/M3Tk-1 satisfies a finiteness condition prohibiting infinite strictly ascending chains of submodules, and normal subgroups correspond to such submodules via homomorphisms. 4) All ideals in the group ring Z are principal, making ascending chains finite. 5) The endomorphism group acts trivially on every module, collapsing all subgroups. 6) The ideal Tk-1 equals the zero ideal, so all quotients are trivial. 7) Ascending chains of normal subgroups are always allowed in non-abelian groups.

✓ Correct Answer:

The correct answer is 3) The module M3/M3Tk-1 satisfies a finiteness condition prohibiting infinite strictly ascending chains of submodules, and normal subgroups correspond to such submodules via homomorphisms..

📚 Reference Text:

K. Gupta and A. N. Krasil'nikov [26] Since [v t ,v2,v3]- V is a product of elements of the form (33) and their inverses, the group Uk/ V is generated by the elements of the form (34) [x h,xh,xhy • V if el), where / is the two-sided ideal in Z(A) generated by the elements («i, u2, M 3) • • • (H 3*_5, M 3t_4, M 3t_3) (u, € A). Note that / = Tk~x (recall that T is the ideal in 1(A) generated by all elements (MI , w2, «3), where M, e 1{A)). Indeed, obviously / c Tk~l. On the other hand, since W(M,, M 2, M 3) = (UiM 2, M 3, V) - (« 2Ml, M 3, V) + («,, U 2, Uj)V for all ui, v e A, each element Ul(«l, M 2, U 3)V2 • • • Vk-i(u 3k-.s, M 3t_4, U 3k.3)vk (M,-, VJ G A) can be rewritten in the form (M'P «'2, K'J) • • • (M' 3J1_5, u' 3k_4, u' 3k^)v' («;, u' € A) so 7*-1 c /. Define a Z(A)-linear mapping a of M3 onto yi(F)/ V by (ji,J2,J3)a = [xh,xh,xh]- V. Clearly, (pa = cup for every $ e . Let ^ be the natural homomorphism of y3{F)/ V onto y 3(F)/U k. Since £/*/ V is a Z(A)-submodule in y}(F)/V closed under all mappings 0 e , y3(F)/ U k is a right Z(A)-module with mappings (p e 4> acting on it in such a way that $<$> = y3(F)/f/ t. Then /^ = (p/x for all 0 € . Since (U\,J2,J3)f)fi< if € /) is of the form (34), A/ 3T*-1 c ker(^) so one can define a 2(A)-linear homomorphism/I of M3/M3T*~' onto y 3(F)/ Uk by (w + M3Tk~l)JI = mix for all m e M 3. Clearly, 7I = 0/Z for all 0 e . Note that if N is a normal subgroup of FI Uk contained in y3(F)/ Uk then N is a Z(A)-submodule in y3(F)/ I/t so N/I"1 is a Z(A)-submodule in M 3/M3 T~l and if N is closed under all 0 e O then so is NJL"1. Now Proposition 2 follows immediately from Corollary 4. Indeed, if N1CJV2C- https://doi.org/10.1017/S1446788700002056 Published online by Cambridge University Press [27] Solution of a problem of Plotkin and Vovsi 355 is an infinite strictly ascending chain of normal -subgroups of F/ U k contained in K(F)/U k then Wi/I"' C N2JI-1 c • • • is an infinite strictly ascending chain of -submodules in M-$/M-$Tk~l. This contra- dicts Corollary 4 and completes the proof of Proposition 2. References [1] R. M. Bryant and M. F. Newman, 'Some finitely based varieties of groups', Proc. London Math. Soc. (3) 28 (1974), 237-252. [2] D. E. Cohen, 'On the laws of a metabelian variety', J. Algebra 5 (1967), 267-273. [3] C. K. Gupta and A. N. Krasil'nikov, 'Some non-finitely based varieties of groups and group representations',

Question 256 multiple-choice

Quantum optimization problems often require specialized hierarchies of relaxations to approximate or compute quantities such as the maximum eigenvalue of Hamiltonians. Algebraic methods involving swap operators can provide efficient tools for manipulating quantum subsystems and capturing fundamental quantum properties. Which hierarchy achieves exactness for the Quantum Max Cut problem at level ⌈n/2⌉ on n-vertex graphs by leveraging algebraic manipulation of swap operators instead of large matrix representations? 1) Classical Sum of Squares hierarchy 2) Quantum Lasserre hierarchy over Pauli operators 3) Matrix Product States hierarchy 4) Semidefinite Programming hierarchy with local constraints 5) Tensor Network Relaxation hierarchy 6) Non-commutative Lasserre hierarchy using Clifford algebra 7) Swap algebra-based hierarchy with polynomial identities

✓ Correct Answer:

The correct answer is 7) Swap algebra-based hierarchy with polynomial identities.

📚 Reference Text:

a subset of degree-2 polynomials sufficient to recover the Quantum Max Cut objective. Assuming the Unique Games Conjecture along with a technical conjecture known as the vector valued Borel’s inequality, the authors of [HNP+22] showed the second level of the quantum Lasserre hierarchy (and therefore also approximation algorithms building off the second level) could strictly outperform the first level. Indeed, the second level of the Lasserre hierarchy has been shown to be exact for certain graphs such as the unweighted star, and capture physically meaningful prop- 1There is some inconsistency in the literature between dand 2 dwhen discussing levels of the SoS hierarchy. Here we use the convention that level dof this hierarchy involves optimizing over linear functionals which are positive on squares of degree dpolynomials; note these squares may include terms of degree up to 2d. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 5 erties of quantum states such as monogamy of entanglement inequalities [PT21]. These properties have been used to improve the analysis of rounding algorithms [PT21, PT22]. 1.1 Results The first major contribution of this paper is to extend non-commutative Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of relaxations to the problem of computing the maximum eigenvalue of the QMC Hamiltonian. This hierarchy is a non-commutative sum-of-squares hierarchy (ncSoS) which captures the algebraic structure arising from the swap operators Swapijwhich define the QMC Hamiltonian objective. A critical ingredient in our construction of the SDP hierarchy is the ability to ma- nipulate polynomials in the swap operators and enforce polynomial identities. Towards this end, we give a fully algebraic charcterization (i.e., finite presentation) of the swap operators. We note that this presentation follows quickly as a special case of results in representation theory, cf. [Pro07, §6.1 Theorem]2but provide a standalone proof for the sake of completeness. This characterization is necessary for the use of “dimension free” computer algebraic techniques (e.g. Gr¨ obner bases) for manipulating polynomials in swap operators. That is to say, we do not need to use the matrix representations of the swap operators which could be exponentially large. We develop swap algebra theory including replacement rules useful for identifying polynomial identities in the swaps. These results may be of independent interest, beyond their use in constructing higher levels of the SDP hierarchy in swaps. This new hierarchy of relaxations to Quantum Max Cut is distinct from, and may sometimes outperform, the usual quantum Lasserre relaxations over Paulis. We show that this hierarchy is exact on an n-vertex graph at the level ⌈n 2⌉, no higher levels are needed – a consequence of the previously discussed identities for simplifying swap polynomials. For instance, the first level of this hierarchy enforces relations among the QMC terms which are not enforced by the first level of the quantum Lasserre relaxation [PT21]. More- over, tests on small instances indicate that our hierarchy can provide tighter bounds to Quantum Max Cut compared to the level-1 and level-2 quantum Lasserre relaxations over Paulis.

Question 257 multiple-choice

Quantum toroidal algebras are advanced algebraic structures with deep connections to representation theory and mathematical physics, often studied for their rich symmetry properties. The extended double affine braid group plays a significant role in inducing automorphisms and dualities within these algebras. Which of the following precisely describes the symmetry induced in simply laced quantum toroidal algebras by the action of the extended double affine braid group? 1) It permutes the highest weight modules without affecting central elements. 2) It fixes all central elements and intertwines only the root subspaces. 3) It exchanges vertical subalgebras with certain Hecke subalgebras. 4) It induces automorphisms that switch only the Cartan subalgebra generators. 5) It leaves the horizontal subalgebra invariant and acts trivially on the vertical subalgebra. 6) It permutes all simple roots while leaving all automorphism classes unchanged. 7) It yields automorphisms and anti-involutions that exchange horizontal and vertical subalgebras and invert central elements.

✓ Correct Answer:

The correct answer is 7) It yields automorphisms and anti-involutions that exchange horizontal and vertical subalgebras and invert central elements..

📚 Reference Text:

Title: Automorphisms of Quantum Toroidal Algebras from an Action of the Extended Double Affine Braid Group Year: 2023 Paper ID: 0b32ec3c6508540dada7c10ae7a0dc0581857478 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0b32ec3c6508540dada7c10ae7a0dc0581857478 Abstract: We first construct an action of the extended double affine braid group B¨\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {\ddot{B}}$$\end{document} on the quantum toroidal algebra Uq(gtor)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{q}(\mathfrak {g}_{\textrm{tor}})$$\end{document} in untwisted and twisted types. As a crucial step in the proof, we obtain a finite Drinfeld new style presentation for a broad class of quantum affinizations. In the simply laced cases, using our action and certain involutions of B¨\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {\ddot{B}}$$\end{document} we produce automorphisms and anti-involutions of Uq(gtor)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{q}(\mathfrak {g}_{\textrm{tor}})$$\end{document} which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements C and k0a0⋯knan\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{0}^{a_{0}}\dots k_{n}^{a_{n}}$$\end{document} up to inverse. This can be viewed as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups used by Cherednik to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures. Our work generalises existing results in type A due to Miki which have been instrumental in the study of the structure and representation theory of Uq(sln+1,tor)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{q}(\mathfrak {sl}_{n+1,\textrm{tor}})$$\end{document}.

Question 258 multiple-choice

In cryptography and computational complexity, the difficulty of guessing a secret variable from a probability distribution is quantified using measures like expected guess complexity and entropy. Quantum algorithms, such as Grover's algorithm, offer speedups for search problems, impacting the security analysis of cryptographic systems. For an n-dimensional discrete Gaussian distribution, which of the following best describes the classical expected guess complexity G in terms of entropy H? 1) G is approximately n·H 2) G is exactly 2^H 3) G is bounded above by H^n 4) G is roughly n·2^H 5) G is approximately e^n·H 6) G is approximately (2√e)^n·2^H 7) G is at most (√2e)^n·H

✓ Correct Answer:

The correct answer is 6) G is approximately (2√e)^n·2^H.

📚 Reference Text:

problem arises when we must ﬁnd the secret senumin the dual attack by asking the question “is the secret equal to ˜senum?”. LetNbe the number of guesses used in the guessing strategy that minimises E[N]. It can be shown that the best strategy is to try values of Xin decreasing order of probability. Without loss of generality, we can identify the possible values of XwithNin such a way that p0≥p1≥p2≥ ···where pi= Pr[X=i]. The expected number of guesses of the optimal strategy is therefore G(X) =/summationdisplay ii·pi. It is well-known that a lower bound onG(X) is given by the entropy of X. More precisely, Massey showed in [Mas94] that G(X)≥1 4·2H(X)+1 provided that H(X)≥2 bits, where Hdenotes Shannon’s entropy (i.e. in base 2). On the other hand, the same work shows that it is not, in genera l, possible to boundG(X) in terms of H(X) only. In Lemma 4, we heuristically show thatG(X)≈(2√e)n·2H(X)whenXis distributed according to a n-dimensional discrete Gaussian. In this work,we areinterestedin the quantum complexity ofguessing.Monta- naro showed [Mon11] that the expected number of guesses in this c ase becomes, Gqc(X) =/summationdisplay i√ i·pi and that this is the best possible. However, [Mon11] only deals with t he case where the oracle (which answers the question “is Xequal tox?”) always returns the correct answer. Using Lemma 1, a variant of Grover search th at can handle two-sided errors, we extend this algorithm to deal with bounded er ror oracles. Lemma 3. LetXbe a random variable taking values in some (eﬀectively de- scribable) set E. Assume that there is an eﬃciently computable bijective fun ction σ:N→Esuch that for all i/lessorequalslantj,PrX[X=σ(i)]/greaterorequalslantPrX[X=σ(j)],i.e.σorders Eby non-increasing probability according to X. Let(Ox)x∈Ebe a collection of oracles such that for any x∈E,Ox(x)returns1with probability at least 9/10 and for all y/n⌉}a⊔ionslash=x,Ox(y)returns0with probability at least 9/10. Then there is a quantum algorithm A, with quantum oracle access to σandOxsuch that for allx∈E,Aσ,Ox() =xwith constant probability, and EX[T(X)] =O(Gqc(X)),EX[Q(X)] =O(Gqc(X)), 14 whereT(x)is the running time complexity of Aσ,Ox(), andQ(x)its query com- plexity. Proof.See Appendix A. We now study the guessing complexity of a n-dimensional discrete Gaussian, its modular version and also relate these quantities to the entropy. The reason why we also study the (non-modular) Gaussian is that it is not clear ho w to order the elements by decreasing probability in the modular case, wh ereas it is easy in the non-modular one. Therefore we study the discrete Ga ussian ﬁrst and then show that its guessing complexity is an upper bound on the g uessing complexity of the modular Gaussian. Lemma 4. Letn/greaterorequalslant4. Then 1/lessorequalslantρσ(Zn) (σ√ 2π)n/lessorequalslantcoth(π2σ2), G(DZn,σ)/lessorequalslant2nρσ(Zn) 1−e−1/2σ2, and −n8π2σ2e−2π2σ2 log(2)coth( π2σ2)/lessorequalslantH(DZn,σ)−1 2+log(σ√ 2π) log(2)n/lessorequalslantnlog2(coth(π2σ2)). Furthermore, if σ/greaterorequalslant4/radicalig 2 27π2≈0.294then Gqc(DZn,σ)/lessorequalslant7 6·/parenleftbigg3 2/parenrightbigg3n/4/radicalbig ρσ(Zn) (1−e−1/3σ2)3/2. Furthermore, G(DZn,σ)/lessorequalslant1 1−e−1/2σ2/parenleftbigg2a(σ)√e/parenrightbiggn ·2H(DZn,σ), Gqc(DZn,σ)/lessorequalslant7 6·1 (1−e−1/3σ2)3/2/parenleftbigg27a(σ)2 8e/parenrightbiggn/4 ·2H(DZn,σ)/2 wherea(σ) =e8π2σ2e−2π2σ2tanh(π2σ2)is plotted on Figure 1 and a(σ) = 1+8π2σ2e−2π2σ2+o(σ2e−π2σ2), σ→ ∞. Finally, for all nandσ, G(DZnq,σ)/lessorequalslant2G(DZn,σ), Gqc(DZnq,σ)/lessorequalslant3 2Gqc(DZn,σ). Proof.See Appendix B. In Lemma 4, we have identiﬁed upper bounds which we believe to be tigh t up to polynomial factors

Question 259 multiple-choice

In quantum information processing, superoperators are used to model the evolution of quantum states, with their properties analyzed in various computational bases to benchmark quantum devices and error mitigation strategies. The product operator basis is particularly valuable for interpreting physical observables in experiments such as NMR and quantum computing. Which statement correctly identifies a unique advantage of representing supermatrices in the product operator basis for quantum benchmarking? 1) It ensures all eigenvalues are strictly real and positive. 2) It guarantees every quantum operation is represented as a diagonal matrix. 3) It allows direct visualization of quantum states in phase space. 4) It makes all matrix elements real, facilitating physical interpretation of observable expectation values and identification of fixed points. 5) It eliminates the need for error correction in experimental data. 6) It causes the correlation coefficient between simulated and experimental results to always be exactly 1. 7) It prevents coherent errors from affecting the superoperator’s eigenvalue spectrum.

✓ Correct Answer:

The correct answer is 4) It makes all matrix elements real, facilitating physical interpretation of observable expectation values and identification of fixed points..

📚 Reference Text:

theoretical (left), simulated (middle) and ex perimental (right) supermatrices in the computational (Zeeman) basis, wher e the theoretical superop- erator is given in terms of the unitary operator by UQFT th⊗UQFT th. The correlations and attenuated correlations among these matrices may be found in Tab le II. Although the pat- tern of the theoretical superoperator’s matrix elements is visible in both the simulated and the experimental supermatrix, it is still not at all obvious from this v antage point what errors have occurred during implementation or how they can be cor rected. FIG. 12: The theoretical (left), simulated (middle) and experimental super matrices in the product operator basis, wherein all the elements are necessarily r eal (see text; the scale has been reduced by a factor of 2 to create more contrast). The cor respondence between the rows / columns of this supermatrix and the individual product opera tors is the same as that given by the labels on the vertical axis of the bar graph in Fig. 8. Beca use all the entries of the corresponding “real” density matrix are the expectation va lues of observables which transform nicely under rotations [23], the physical interpretation is somewhat easier in this basis. This is manifest, for example, in the appearance of several ﬁ xed points with a 1 on the diagonal and zeros elsewhere in the same row (see text). FIG. 13: Theoretical (left), simulated (middle) and experimental (right) su permatrices in 32 the phase space basis after correcting the latter two by a unitary supermatrix designed to bring them as close as possible to the theoretical (as described in th e text). The levels of background noise are a great deal less than in the corresponding u ncorrected supermatrices (Fig. 10). FIG. 14: As may be seen from row 14 of the supermatrices shown in Fig. 12, th e operator σ1 xσ3 zis a ﬁxed point of the QFT. The upper bar graph displays the corresp onding row of the theoretical (dark grey), simulated (medium grey) and simulate d with coherent errors corrected (light grey) supermatrix elements versus the product operator basis, whereas the lower bar graph displays these same statistics for the experimenta l supermatrix. In contrast to the phase space and Zeeman basis (not shown), there is a signiﬁc ant correlation between the experimental and simulated values; speciﬁcally, the simulated-t o-experimental correla- tion coeﬃcients were 0 .80 and 0.94 before and after correction, respectively. In the present case these correlations are due almost entirely to the single large dia gonal value, but similar correlations were obtained for all other pairs of rows; the averag e correlations were 0 .78 and 0.94 before and after correction, respectively. FIG. 15: Plot of the eigenvalues of simulated QFT superoperators in the comp lex plane, speciﬁcally, the results of simulations with only coherent errors (blu e “/circlecopyrt”), the results in- cluding relaxation using the relaxation superoperator shown in Fig. 5 and Table I (red “ ∗”), the results including incoherent errors

Question 260 multiple-choice

In computational algebraic geometry and representation theory, algorithms for testing membership in moment polytopes frequently utilize properties of highest-weight vectors and randomized sampling techniques. The effectiveness of such algorithms depends on results that provide explicit degree bounds and efficient methods for choosing generic points. Which result enables explicit, computable upper bounds on the degree of highest-weight vectors required for algorithmic membership testing in moment polytopes? 1) Schwartz-Zippel lemma 2) Effective Mumford’s Theorem (with Derksen’s degree bounds) 3) Grothendieck’s finite generation theorem 4) NP-hardness of highest-weight vector existence 5) Rational convex polytope property 6) Randomized construction using random matrices 7) Algorithmic potential function approach

✓ Correct Answer:

The correct answer is 2) Effective Mumford’s Theorem (with Derksen’s degree bounds).

📚 Reference Text:

allRpRp1q;:::;RpdqqPB. We give a proof of a reﬁnement of Theorem 2.4 in Section 2.5. Theorem 2.4 alone does not appear to give an eﬃcient way of characterizing the moment polytopesinceitprovidesnoboundonthedegree knorarecipeforﬁndingapoint Xs.t.PpXq0. In fact, it is known that even deciding the existence of highest-weight vectors is NP-hard [ IMW17]. Our algorithm does not solve the membership problem via the dual description provided by Mumford’stheorem. Instead, suitablehighestweight vectorswillfeature aspotential functionsin the analysis of our algorithm! WewillneverthelessrequireamoreeﬀectiveunderstandingofTheorem2.4. Thiswillbethe concern of the remainderof thissection. Westart by observing that the algebra ofhighest weight vectorswithhighestweightamultipleof pisﬁnitelygenerated. Thus,Theorem2.4canbemade more eﬀective by bounding the degree of the highest weight vectors that need to be considered. This is achieved by our next result, which relies on recent degree bounds by Derksen [Der01]: Proposition 2.5 (Eﬀective Mumford’s Theorem) .LetpPPpn1;:::;ndqand`¡0an integer such that`pis integral. If pPpXq, then there exists an integer m¡0and a highest weight vector PPHWVmpCrVsp`mqqof degree`m¤K, where K: `dmaxd i1nidmaxd i1n2 i; such thatPpXq0for someXPX. Proof.ByLemma2.3,anyhighestweightvectorcanbewrittenas PpXqQ Xb`bv ,whereQ isaG-invariantpolynomialon WSym`pVqbV. Thus,byTheorem2.4, pRpXqifandonlyif Q Xb`bv 0forallXPXandallnonconstanthomogeneouspolynomials QPCrWsG. By deﬁnition, the latter means that Xb`bvis in the null cone of the G-action onWor, equivalently, oftheactionofthesubgroup ~GSLpn1q SLpndq./two.superior⁴Thelatterhasﬁnitekernel,soDerksen’s degree bound from [ Der01, Proposition 2.1] is applicable. It shows that the null cone is already deﬁned by G-invariant polynomials of degree m¤Ht~mA~m¤Hd°d i1nipiq 1 °d i1pn2 i1q ¤pH`dqdH2{`; /two.superior⁴Since the action is scale invariant for each GLpniq. 20 whereH:maxd i1ni,A:°d i1nipiq 1,~m:°d i1pn2 i1q, andt:°d i1n2 i. Since degpPq `degpQq, we obtain the desired degree bound. InSection2.4,wewillproveboundsontheevaluationofhighestweightvectors(Proposition2.9). Proposition 2.5 shows that we only need to consider ﬁnitely many highest weight vectors to test whetherpPpXq(e.g., a basis of the space of all highest weight vectors of degree ¤K). However, westillneedtotestif PpXq0forsomeXPX. Howcanweﬁndsuchan X? Clearly,PpXq0 for someXPXiﬀPpXq0for generic XPX, so: Corollary 2.6 ([Bri87]).pPpXqif and only ifpPpXqfor genericXPX. In fact, pXqpXqforgeneric XPX, sinceboth arerationalconvex polytopes. Thiscan be seen from the nontrivial fact that the algebra of allhighest weight vectors is also ﬁnitely generated [Gro73]. To make Corollary 2.6 eﬀective, we need a way to select generic elements in X. This may be doneusingtheSchwartz-Zippellemma–providedwehaveasuitableparametrizationof X. We will show how to do this in full generality in Section 6.2. We will also show there that if piqis degenerate i.e. piq jpiq kforjk, we can use signiﬁcantly fewer (sometimes even zero!) random bits to generate X. For now, we only carry this out for XGX. This is critical for our analysis of Item 1 in Algorithm 1. Corollary 2.7 (Good starting points) .LetXPTenpn0;n1;:::;ndq. SupposepPpXqand`¡0such that:`pisintegral. Let Mpnqdenotethespaceof nnmatrices. Chooseall maxd i1n2 ientriesofthe d-tuple of matrices ApAp1q;:::;ApdqqPMpn1qMpndqindependently and uniformly at random fromt1;:::;Mu, with M2dK; K : `dmaxd i1nidmaxd i1n2 i: Set ZAX: Ap1qbbApdq X: Then, there exists a highest weight vector PPHWVmpCrVsp`mqqof degree 0 `m¤Ksuch that, with probability at least 1{2,PpAXq0. Proof.Set:`p. According to Proposition 2.5, there exists a highest weight vector PP HWVmpCrVsp`mqqof degree 0 `m¤Ksuch thatPpYq0for someYPXGX. Then QpAq:PpAXqisnotequaltothezeropolynomialon M:Mpn1qMpndq. Ifthiswere not the case, then Pwould vanish in the Zariski dense (in X) setGX, contradicting PpYq0. Its degree is no larger than dK, so the

Question 261 multiple-choice

In quantum gravity research, group field theory (GFT) provides a framework for understanding how spacetime and cosmological metrics may emerge from more fundamental quantum structures. Recent approaches use effective metrics reconstructed from GFT states to analyze cosmological perturbations and their implications for early universe physics. Which feature distinguishes squeezed GFT perturbative modes from oscillating modes in analyses of effective spacetime metrics for cosmological perturbations? 1) Squeezed modes always correspond to classical fields with Lorentzian signature. 2) Oscillating modes exhibit non-classical behavior and yield Euclidean signature equations. 3) Squeezed modes are associated with enhanced gauge invariance compared to oscillating modes. 4) Oscillating modes generate inhomogeneities that match general relativity in all regimes. 5) Squeezed modes lead to the dominance of vector perturbations in cosmological models. 6) Oscillating modes are incompatible with any effective metric reconstruction from GFT. 7) Squeezed modes yield equations with Euclidean signature and distinct late-time behavior, while oscillating modes have Lorentzian signature and different dynamics from general relativity.

✓ Correct Answer:

The correct answer is 7) Squeezed modes yield equations with Euclidean signature and distinct late-time behavior, while oscillating modes have Lorentzian signature and different dynamics from general relativity..

📚 Reference Text:

Title: Cosmological scalar perturbations for a metric reconstructed from group field theory Year: 2025 Paper ID: 7222cdb328143f36b0c40474005c587d86d9fbe6 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/7222cdb328143f36b0c40474005c587d86d9fbe6 Abstract: While homogeneous cosmologies have long been studied in the group field theory (GFT) approach to quantum gravity, including a quantum description of cosmological perturbations is highly non-trivial. Here we apply a recent proposal for reconstructing an effective spacetime metric in GFT to the case of a metric with small inhomogeneities over a homogeneous background. We detail the procedure and give general expressions for cosmological scalar perturbations defined in terms of the GFT energy-momentum tensor. These include all the scalar components of standard perturbation theory and hence can be used to define gauge-invariant quantities. This is a major advantage of the effective metric approach compared to previous GFT studies limited to volume perturbations. We compute these perturbations explicitly for a particular Fock coherent state. While it was previously shown that such a state can be interpreted as an approximately flat homogeneous cosmology at late times, here we find that, in a very simple example, inhomogeneities do not follow the dynamics of general relativity in the semiclassical regime. More specifically, restricting ourselves to a specific coherent state in a simple (free) GFT, we study two types of perturbative GFT modes, squeezed and oscillating modes. For squeezed modes we find perturbation equations with Euclidean signature and a late-time limit that differs from general relativistic perturbation equations. Oscillating modes satisfy different dynamical equations that also differ from those of general relativity, but show a Lorentzian signature. Considering that our results were obtained within a number of simplifying assumptions and arguably the simplest possible example, we discuss how going beyond these assumptions could lead to a more desirable phenomenology. Overall, our analysis should be understood as a first step in understanding cosmological perturbations within the effective GFT metric.

Question 262 multiple-choice

In the study of group representations, particularly for the group SU(2), mathematical constructions such as direct sums, tensor products, and dual representations are central to understanding symmetries in quantum mechanics and particle physics. These techniques help classify and analyze the structure of composite and fundamental systems. Which specific representation of SU(2) corresponds to the antisymmetric square of the fundamental representation on C², and what is its dimension? 1) The tensor product representation with dimension 4 2) The adjoint representation with dimension 3 3) The symmetric square with dimension 3 4) The fundamental representation with dimension 2 5) The regular representation with dimension equal to the group's order 6) The singlet representation (Alt²(C²)) with dimension 1 7) The trivial representation with dimension 2

✓ Correct Answer:

The correct answer is 6) The singlet representation (Alt²(C²)) with dimension 1.

📚 Reference Text:

let us consider Gto beSU(2). Then, take R1(g)be the representation Sym (C2)from Example 5 (of- ten called the spin-1 representation), and R2(g)be the rep- resentation Alt2(C2)from the same example (often called the spin-0 representation, or singlet). Their direct sum is of the form (R1R1)(g) =0 BB@0 0 0 0 0 01 CCA:(23) We have also already seen through Example 5 that the tensor product of two representations is also a representa- tion. Moreover, from Example 8 we also saw how Leibniz’s rule for diﬀerentiation induces the following form of tensor representations of a Lie algebra g. Deﬁnition 10. LetR1;R2be representations of a Lie groupGacting on vector spaces V1;V2. Thetensor rep- resentation R1 R2ofGonV1 V2is (R1 R2)(g) =R1(g) R2(g): Ifr1;r2are representations of a Lie algebra gacting on V1;V2, then the tensorrepresentation r1 r2ofgonV1 V2 is (r1 r2)(X) =r1(X) 1+1 r2(X): Here, let us consider again the case of G=SU(2). Recalling from Example 5 that in the symmetric- antisymmetric basis we have Ug Ug=0 BB@0 0 0 0 0 01 CCA: (24) The fact that there exists a basis where Ug Ugin Eq. (24) has the exact block diagonal form as that in Eq. (23) indi- cates that these two representations are isomorphic (Deﬁ- nition16). Moreimportantly, itunravelsafundamentalas- pect of representation theory: given a representation (such as the tensor one), it is extremely useful to express it as a direct sum of smaller representations. For our previous example, we now know that the tensor representation of SU(2)can be decomposed as a direct sum of two “smaller”representations, the spin- 1and the spin- 1=2. We will re- turn to this in Example 10. The previous examples show how to build larger repre- sentations from smaller ones (though direct sums or prod- ucts). There is another key construction which allows us to build new representations from old, but this one keeps the dimension the same, a commonly useful feature for change of representation tasks. Recall that the dual space of a ﬁnite dimensional vector spaceV, where we think of Vas spanned by column vec- tors, is the space of row vectors V, i.e. linear functionals fV!FgwithF2fR;Cg. Deﬁnition 11. LetRbe a representation of Gacting on a ﬁnite-dimensional vector space V. Then the dual repre- sentationRis the representation of Gacting onVgiven by R(g) := [R(g1)]T;for allg2G; whereTdenotes the transpose. If ris a representation of g onV, then the dual representation rofgonVis given by r(X) :=r(X)T;for allg2G: Note that this is also called the contragredient represen- tation. The dual representation and tensor representation con- structions immediately yield natural representations of G on linear maps :V!W(which includes operators and quantum channels) and more generally tensors with mixed covariance and contravariance. This occurs because we can think of the space of linear maps f :V!Wgas the ten- sorproduct V W, whereabasisforthisspaceisgivenby, e.g.,jwihvjwhere we havejvi2V;jwi2Wandvdenotes the complex conjugate of v. In this case, given representa- tionsR,SonV,W, respectively, the natural action on is given by R Sfor anyg2Gis ((R S)(g)) (v) =Sg(R1 gv);wherev2V; which, when RandSare unitary representations (Deﬁni- tion 15)

Question 263 multiple-choice

In the study of algebraic combinatorics and group theory, permutation groups often act on sets constructed from vector spaces over finite fields, with specific attention to regular and transitive actions and associated algebraic structures. The interplay between symmetric algebras, group commutators, and bilinear forms is crucial for analyzing automorphism groups and the Cayley Isomorphism (CI) property. Which of the following statements correctly describes the role of the symmetric bilinear form $\varphi$ defined by the Kronecker delta on a vector space $V$ over a finite field $\mathbb{F}_p$, with respect to the analysis of group actions and annihilators? 1) It defines a norm on $V$ and allows for the construction of orthogonal complements in the symmetric algebra. 2) It provides a way to classify all subgroups of the automorphism group of $V$ via their eigenvalues. 3) It ensures that every subgroup of the group $G$ acts by scalar multiplication on $V$. 4) It determines the dual basis in the symmetric algebra and relates to the filtration by degree. 5) It is used to construct the upper central series of the group $U$ based on the commutator identities. 6) It allows identification of the symmetric algebra with the exterior algebra of $V$. 7) It enables the characterization of annihilators of elements in $V$ with respect to group actions, ensuring non-degeneracy and facilitating the analysis of orbits and stabilizers.

✓ Correct Answer:

The correct answer is 7) It enables the characterization of annihilators of elements in $V$ with respect to group actions, ensuring non-degeneracy and facilitating the analysis of orbits and stabilizers..

📚 Reference Text:

tion of Aon/Sigma1(i.e. the stabilizer of the set {0W}×V). It remains to prove that Lacts regularly on {0W}×V. The group Lcontains {0W}×Vand so Lacts transitively on each set of the form {w}×V.L e tgbe inL. Assume that (0,0)g=(0,0).L e tvbe inVand let us prove that (0,v)g=(0,v). Note that ((0,v),( 0,0))∈/Gamma1(0,v)and so ((0,v),( 0,0))g=((0,v1),(0,0))is an element of /Gamma1(0,v)=Cay(W×V,(0,v)),s o , v1=v. This proves that Ais a subgroup of U. 3 The construction Since the elementary Abelian 2-group of rank 6 is known not to be a CI-group, see [7], we assume that p≥3. We strongly use the well-known structure of the upper central series of the group U,s e e[ 4]. LetVbe the Galois ﬁeld Fpn, where n=2p−1, and let ε1,...,ε nbe an Fp-basis ofV. Also, let Wbe an Fp-vector space of dimension nwith basis e1,...,e nand dual basis e∗ 1,...,e∗ n.L e tSbe the symmetric V-algebra on e∗ 1,...,e∗ nand natural ﬁltration S=⊕m∈NSm(recall that Smis spanned by the monomials e∗j1 i1⊗···⊗ e∗jk ikof degree m). LetMmbe the V-subspace of Smspanned by the monomials e∗j1 i1⊗···⊗ e∗jk ikof degree msuch that j1,...,j k≤p−1. Let π:S→B=VW be the valuation map and Zi=π(Mi). For instance π(e∗j1 i1⊗···⊗ e∗jk ik)(w) is the element of Vdeﬁned by (e∗ i1(w))j1···(e∗ ik(w))jkfor any w∈W. It is known that B∩ξm(U)=⊕i≤m−1Ziand that π|Miis injective. Without loss of generality we identify e∗j1 i1⊗···⊗ e∗jk ikinMmwith its image under π. LetXbe the set {1,...,n }andXi={A⊆X||A|=i}fori=0,...,p .I fAis a subset of X, then we denote by e∗ Athe element of Bdeﬁned by ⊗i∈Ae∗ i. Similarly εA denotes/summationtext i∈Aεi∈V.W el e t εandedenote εXand/summationtext i∈Xei, respectively. Consider f=/summationdisplay A∈XpεX\Ae∗ A∈B. J Algebr Comb (2007) 26: 343–355 349 Deﬁne L=[W,f].I fw1,w2∈W, then [w1+w2,f]=[w1,f]w2+[w2,f].T h i s shows that G=/angbracketleftW,L/angbracketright=WL. Since [G,f]=[W,f]⊆G,w eh a v et h a t fnormal- izesG. LetCbe a subset of Xof size p−i. Deﬁne gC=/summationdisplay B∈Xi,C∩B=∅εX\(B∪C)e∗ B. Lemma 5 The group Lis generated by the set {gC|C∈Xi,1≤i≤p}andL0is generated by {gC|C∈Xi,1≤i≤p−1}. Proof Ifi∈A, then(e∗ A)ei(w)=/producttext a∈Ae∗ a(w−ei)=e∗ A(w)−e∗ A\{i}(w)and so (†)[ei,e∗ A]=/braceleftbigg0i f i/∈A, e∗ A\{i}ifi∈A. Using (†)we get [ei,f]=/summationdisplay B∈Xp−1,i /∈BεX\(B∪{i})e∗ B=g{i}∈Lfor any i∈X. More generally we have [ei,gC]=/braceleftbigg0i f i∈C, gC∪{i}ifi/∈C. The group Lis generated by the left-normed commutators [f,ei1,...,e ik]fork≥1 and therefore Lis generated by {gC|C∈Xi,1≤i≤p}. Our claim on L0is just an easy remark. /square Corollary 1 Gcontains ξ1(U). Proof By Lemma 5we have that L∩ξ1(U)is generated by {gC|C∈Xp}={εX\C| C∈Xp}. As usual we are identifying the elements of Vwith the elements of ξ1(U). We leave the reader to show that {εX\C|C∈Xp}spansV. /square By Corollary 1we have that U⊇G⊇Wξ1(U)and so we can apply Section 2to G. Letϕbe the non-degenerate bilinear symmetric form on Vdeﬁned by ϕ(εi,εj)= δij, where δijis the Kronecker delta. Lemma 6 H(ei)=ε⊥ ifor any i∈X. Proof We have that gC(ei)=0 for any C∈Xp−j,j≥2. Therefore, by Lemma 5, H(ei)=/angbracketleftgC(ei)|C∈Xp−1/angbracketright=/angbracketleftεX\(C∪{i})|i/∈C,C∈Xp−1/angbracketright, while it is an easy exercise in linear algebra to prove that this vector space is ε⊥ i. /square Lemma 7 Ifi/negationslash=j,thenH(ei+ej)=(εi−εj)⊥. 350 J Algebr Comb (2007) 26: 343–355 Proof Arguing is the same way as in Lemma

Question 264 multiple-choice

Quantum hidden subgroup algorithms leverage group theory and the quantum Fourier transform to efficiently solve problems that are intractable for classical computers, such as factoring and finding hidden symmetries. Understanding how these algorithms extract subgroup information is key to appreciating their exponential speed-up. In the quantum hidden subgroup algorithm for a finite Abelian group G, which step enables the extraction of information about the hidden subgroup K by mapping coset superpositions to states labeled by irreducible characters? 1) Applying the quantum Fourier transform to the measured coset superposition 2) Measuring the first register immediately after state preparation 3) Constructing a superposition over all elements of G before evaluating the function f 4) Evaluating the function f on each element of G individually using classical computation 5) Randomly sampling elements from G after measurement 6) Discarding the second register without measurement 7) Preparing shift-invariant states without using representation theory

✓ Correct Answer:

The correct answer is 1) Applying the quantum Fourier transform to the measured coset superposition.

📚 Reference Text:

|G|)) where |G|is the size of Gand the evaluation of fon an input counts as one computational step. (Note that we ma y easily determine Kin timeO(poly( |G|)) by simply evaluating and examining all the values of f). Further discussion of this time constraint will be given in t he next section. We begin by constructing the state |f/an}bracketri}ht=1/radicalBig |G|/summationdisplay g∈G|g/an}bracketri}ht|f(g)/an}bracketri}ht and read the second register. Assuming that fis suitably non-degenerate – in the sense thatf(g1) =f(g2) iﬀg1g−1 2∈Ki.e. thatfis one-to-one within each period – we will obtain in the ﬁrst register |ψ(g0)/an}bracketri}ht=1/radicalBig |K|/summationdisplay k∈K|g0k/an}bracketri}ht (10) 9 corresponding to seeing f(g0) in the second register and g0has been chosen at random. Examples. In Simon’s algorithm Gis the additive group BnandKis the cyclic subgroup {0,ξ}generated by ξ. In Shor’s algorithm Gis the additive group Zqand Kis the cyclic subgroup {0,r,2r,...}generated by r. In each case Kis speciﬁed by giving its generator. The state (10) is obtained in step 3 of t he algorithm. ✷ Remark. The construction leading to the state (10) applies in a more g eneral context than just a function on a group. Suppose we have any mathemati cal objectFwith an action of the group Gon it: g:F→gF such that (g1g2)F=g1(g2F). The symmetry group of Fis the subgroup K={k∈G:kF=F}. By constructing/summationtext g|g/an}bracketri}ht, applying it to a suitable state description |F/an}bracketri}htofFand reading the second register we obtain the state/summationtext k|g0k/an}bracketri}htas in eq. (10). ✷ In eq. (10) we have an equal superposition of labels correspo nding to a randomly chosen coset of KinG. NowGis the disjoint union of all the cosets so that if we read the label in eq. (10) we will see a random element chosen equip robably from all of G yielding no information at all about K. The Fourier transform will provide a way of eliminating g0from the labels which may then provide direct information ab outK. We ﬁrst construct a basis |χi/an}bracketri}htof states which are shift invariant in the sense: g|χi/an}bracketri}ht=eφi(g)|χi/an}bracketri}htfor allg∈G Such states are guaranteed to exist since the shift operatio nsgare unitary and they all commute. Next note that the state in eq. (10) may be writte n as ag0-shifted state: /summationdisplay k∈K|g0k/an}bracketri}ht=g0 /summationdisplay k∈K|k/an}bracketri}ht  Hence if we write this state in the basis {|χi/an}bracketri}ht,i= 1,...,|G|}then/summationtext k|k/an}bracketri}htand/summationtext k|g0k/an}bracketri}ht will contain the same pattern of labels, determined by the su bgroupKonly. The Fourier transform is simply deﬁned to be the unitary operati on which transforms the shift-invariant basis into the standard basis. After apply ing it to eq. (10) we may read the shift-invariant basis label by reading in the stand ard basis. This explains the essential role of the Fourier transform in step 4 of the al gorithms. The shift-invariant states |χi/an}bracketri}htare constructed using some basic group representa- tion theory [12]. Consider any (nonzero) complex valued fun ction on the group χ:G→ C which respects the group operation in the sense that χ(g1g2) =χ(g1)χ(g2) (11) 10 For Abelian groups these are the irreducible representatio ns [12] ofG. By listing the valuesχmay

Question 265 multiple-choice

Quantum walks are advanced quantum computational models that utilize quantum interference and superposition, offering improvements over classical random walks for tasks such as search and simulation. Efficient circuit design is essential for executing these algorithms on current quantum devices with limited resources. Which of the following statements best describes how the quantum Fourier transform (QFT) contributes to the efficiency of quantum walk algorithms on Noisy Intermediate-Scale Quantum (NISQ) devices? 1) It increases the circuit depth by introducing additional entangling gates. 2) It replaces the need for quantum error correction during quantum walks. 3) It enables exponential scaling of circuit size with respect to the number of qubits. 4) It limits the applicability of quantum walks to only search algorithms. 5) It implements the measurement process more efficiently in quantum circuits. 6) It facilitates a resource-efficient shift operation by diagonalizing circulant matrices, resulting in quadratic circuit size and linear depth. 7) It eliminates noise and decoherence in quantum hardware entirely.

✓ Correct Answer:

The correct answer is 6) It facilitates a resource-efficient shift operation by diagonalizing circulant matrices, resulting in quadratic circuit size and linear depth..

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arXiv:1912.00978v3 [quant-ph] 13 Jan 2020EFFICIENT AND SCALABLE QUANTUM WALK ALGORITHMS VIA THE QUANTUM FOURIER TRANSFORM ASIF SHAKEEL Abstract. Quantum walks (QWs) are of interest as examples of uniquely quantu m behav- ior and are applicable in a variety of quantum search and simulation mod els. Implementing QWs on quantum devices is useful from both points of view. We descr ibe a prototype one-dimensional discrete time QW algorithm that economizes resour ces required in its im- plementation. Our algorithm needs only a single shift (increment) ope ration. It also allows complete ﬂexibility in choosing the shift circuit, a resource intensive p art of QW implemen- tations. We implement the shift using the quantum Fourier transfor m (QFT), yielding, to date, the most eﬃcient and scalable, quadratic size, linear depth cir cuit for the basic QW. This is desirable for Noisy Intermediate-Scale Quantum (NISQ) devic es, in which fewer computations implies faster execution and reduced eﬀects of noise and decoherence. As the QFT diagonalizes unitary circulant matrices, we generalize the shift in the basic QW to introduce spatial convolutions in the QW. We demonstrate our basic QW algorithm using the QFT based shift by running it on publicly accessible IBM quantum co mputers. 1.Introduction Quantum walks (QWs) exhibit properties that are fundamentally qua ntum mechanical, serving to demonstrate in their most basic form, walk patterns tha t are unachievable by classical random walks. In their sophisticated conﬁgurations, the y appear in applications in areas as widely apart as quantum simulation models [1,2] and quantu m search algo- rithms [3]. QWs have been experimentally realized on diﬀerent physical systems [4,5]. The reader is referred to [6] for a comprehensive review of QWs. With th e public availability of quantum computers, quantum algorithms can be tested and their r esource requirements and performance examined. In the current era of Noisy Intermediate -Scale Quantum (NISQ) [7] devices, decoherenceandnoisemayseverely degradetheperfor manceofaquantumalgorithm implemented on a device, negating any gains to be had from a using a qu antum algorithm over a classical one. As a consequence, the issue of resource use in implementation deserves consideration. In this paper we develop a computational descriptio n of a one-dimensional discrete time QW that turns into a ﬂexible, scalable and highly eﬃcient im plementation in terms of circuit size and depth.1It also illustrates that partial optimizations of an algorithm can be done mathematically prior to circuit implementation on a quantu m device A basic discrete time QW is speciﬁed by a particle walking on a lattice, at e ach time step moving from its current siteposition to another in its vicinity, and scattering through self- interaction . We will, for this paper, consider a one dimensional walk on a ﬁnite lattic e. The state of the QW is speciﬁed by its position and velocity. Taking the latt ice to be of size N, E-mail address :asif.shakeel@gmail.com . 1The circuit size is the number of at most two-qubit gates used in the c ircuit. The circuit depth is the longest path

Question 266 multiple-choice

In the representation theory of finite groups of Lie type, the group SL(2, F_{2n}) consists of 2x2 matrices with determinant 1 over a finite field. Understanding the character values of irreducible representations on specific conjugacy classes is fundamental in this domain. Which irreducible representation of SL(2, F_{2n}) has degree q and yields a character value of 0 when evaluated at the standard unipotent element $h = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$? 1) Irreducible representation θ_k 2) Irreducible representation χ_j 3) Trivial representation 11 4) Steinberg representation ψ 5) Irreducible representation η 6) Symmetric square representation 7) Regular representation

✓ Correct Answer:

The correct answer is 4) Steinberg representation ψ.

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SL(2,F2n)are conjugate tothe element h=/parenleftbigg1 1 0 1/parenrightbigg ∈SL(2,Fq), andthereisatotalnumberof q2−1ofsuchinvolutions. Thecharactersandtheirvalueson haresummarized inthe following table. Irrep name Parameters Number of irreps Degree Character val ue onh 11 — 1 1 1 ψ — 1 q 0 θkk= 1,2,...,q 2q 2q−1 −1 χjj= 1,2,...,q−2 2q−2 2q+1 1 25

Question 267 multiple-choice

Quantum algorithms for hidden subgroup problems are central to advances in cryptanalysis, group theory, and scalable quantum computation. Recent developments have focused on distributed quantum systems and refining algorithms for resource efficiency and exactness. Which feature distinguishes a recent distributed quantum algorithm for the generalized Simon’s problem, making it more physically feasible for implementation? 1) Utilization of classical random sampling methods 2) Reliance on post-quantum cryptographic assumptions 3) Reduction in the number of qubits required in the oracle 4) Elimination of quantum entanglement between processors 5) Replacement of amplitude amplification with Grover’s search 6) Focus solely on Abelian group structures 7) Use of error-correcting codes to achieve polynomial speedup

✓ Correct Answer:

The correct answer is 3) Reduction in the number of qubits required in the oracle.

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Title: Exact distributed quantum algorithm for generalized Simon’s problem Year: 2023 Paper ID: 8fec004a336e83783c5954a2d4cad605ded5165e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/8fec004a336e83783c5954a2d4cad605ded5165e Abstract: Simon’s problem is one of the most important problems demonstrating the power of quantum algorithms, as it greatly inspired the proposal of Shor’s algorithm. The generalized Simon’s problem is a natural extension of Simon’s problem and also a special hidden subgroup problem: Given a function f:{0,1}n→{0,1}m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:\{0,1\}^n \rightarrow \{0,1\}^m$$\end{document}, it is promised that there exists a hidden subgroup S≤Z2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\le \mathbb {Z}_2^n$$\end{document} of rank k such that for any x,y∈{0,1}n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x, y\in {\{0, 1\}}^n$$\end{document}, f(x)=f(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x) = f(y)$$\end{document} iff x⊕y∈S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \oplus y \in S$$\end{document}. The goal of generalized Simon’s problem is to find the hidden subgroup S. In this paper, we present two key contributions. Firstly, we characterize the structure of the generalized Simon’s problem in distributed scenario and introduce a corresponding distributed quantum algorithm. Secondly, we refine the algorithm to ensure exactness due to the application of quantum amplitude amplification technique. Our algorithm offers exponential speedup compared to the distributed classical algorithm. When contrasted with the quantum algorithm for the generalized Simon’s problem, our algorithm’s oracle requires fewer qubits, thus making it easier to be physically implemented. Particularly, the exact distributed quantum algorithm we develop for the generalized Simon’s problem outperforms the best previously proposed distributed quantum algorithm for Simon’s problem in terms of generalizability and exactness.

Question 268 multiple-choice

Quantum computing relies on both mathematical abstractions and physical implementations to perform computations using qubits and quantum gates. Accurate representation and verification of quantum algorithms are essential for practical and theoretical advancements in the field. Which of the following best describes the role of the Solovay-Kitaev algorithm in quantum computing? 1) It guarantees error-free transmission of quantum states over optical fibers. 2) It provides a method for measuring the spin of individual qubits. 3) It decomposes classical binary numbers into quantum states. 4) It verifies the correctness of quantum algorithms using formal proof assistants. 5) It simulates the behavior of beam splitters in quantum optics experiments. 6) It enables symbolic computation of Dirac notation for multi-qubit systems. 7) It efficiently approximates arbitrary unitary operations with a finite set of universal quantum gates.

✓ Correct Answer:

The correct answer is 7) It efficiently approximates arbitrary unitary operations with a finite set of universal quantum gates..

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instance the binary representation for the number 17 is: 1710→  2) 17 2) 8 2) 4 2) 2 2) 1/arrowtp1 0 0 0 1  10001 2 or for instance: 11001.01b= 25.2510 (25) 0.001b= 0.12510 (26) 10.1101b= 2.8125 10 In our case, in the Dirac notation, a 3-qubit state may be represented as follows: |Ψ/angbracketright=1√ 8(|000/angbracketright+|001/angbracketright+...+|111/angbracketright) =1√ 8(|1/angbracketright+|2/angbracketright+...|7/angbracketright) (27) 3 FORMAL VERIFICATION OF THE QFT ALGORITHM In this section, we shall brie y examine approaches for the formal veri cation of quantum algorithms. Quantum program accuracy is critical, and mathematicians have traditionally used theorem proving techniques to formalize and verify conjectures and theorems [8]. Quantum information scientists will face a common dilemma [15] [9]) as quantum com- puting develops from theory to practice: how can they be con dent that their code performs what they wish it to do? For example, the authors propose "a novel framework for modelling and evaluating quantum protocols and their implementations using the proof assistant Coq" in formalization of Quantum Protocols using Coq [15].This quantum programming diculty presents an opportunity for formal techniques [9]. The Coq proof assistant is a mathematical programming language that is primarily used to ensure that algorithms are accurate [15].It's a general-purpose environment for creating mathematical proofs that's been enhanced by a number of authors [7] [1] [8] [9] to support quantum information processing proofs. Formal approaches can be used to verify that the code implementing a quantum algorithm accomplishes what it should for all potential inputs and con gurations in advance. In this study, we limit ourselves to a simpler, symbolic approach for representing and reasoning about the quantum QFT circuit, which we then validate using the Pythonic SymPy package [16]. Symbolic computing is used to manipulate symbols. and solve mathematical equations and formulas in order to get results that are mathematically correct. This method is based on a symbolic depiction of quantum states and quantum gates using symbols , rather than numbers. 9 SciPost Physics Codebases Submission 3.1 Symbolic representation of universal quantum gates A quantum operator can be any unitary matrix (/parenleftBig U†U/parenrightBig =I) of size 2nand in the same way as traditional logic gates are referred to as operators, they are also referred to as gates. The Hadamard and the phase operator are not only unitary but also universal [4]. Here, we break down the Fourier transform into a set of universal gates made up of CNOT and one- qubit gates. This decomposition (5) allows us to understand it's optimal realization on a real quantum computer. For instance, the Solovay-Kitaev is an algorithm that can approximate any single transformation by breaking it down into the basic gates that make it up [16]. "Any QC algorithm can be represented as a composition of Walsh-Hadamard transforms and associated conditional phase shifts" [17] From the quantum optics or hardware perspective, His realized by a half-silvered mirror (beam splitter) and P(ω)represents a phase shifter, as in a standard Mach-Zehnder inter- ferometer [18] [19] [17]. In [20] the authors implemented the

Question 269 multiple-choice

In abstract algebra, the study of group rings over free groups explores the structure of ideals generated by specific elements such as commutators and their products. This area is significant in understanding how complex algebraic objects can be simplified using properties of group theory and ring theory. Which of the following most accurately describes how the two-sided ideal T in the integral group ring of a free group is generated, according to results on simplifying ideal generators? 1) By elements of the form (commutator products) minus the identity, specifically expressions like [a_i, a_j][a_k, a_l] - 1 2) By all monomials of degree greater than or equal to k in the group ring 3) By the set of all group elements raised to even powers minus the identity 4) By arbitrary products of group generators and their inverses without consideration of commutators 5) By sums of symmetric group elements and their inverses 6) By elements representing the center of the group ring only 7) By differences of individual generators of the free group minus the identity

✓ Correct Answer:

The correct answer is 1) By elements of the form (commutator products) minus the identity, specifically expressions like [a_i, a_j][a_k, a_l] - 1.

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= («,, M 2, U3U3') = M 3(M1? M 2, MJ) + (Ml, M 2, U3)Uj € 5". If M2 = u'2u2' then /=(«!, M 2M'2', M 3) = (M 2(H,, M 2') + (Mi, «i)j4', M 3) = M 2(«i, M 2, M 3) + (M 2, M 3)(M,, M 2) + (Ui, M 2)(M 2, M 3) + (M n M 2, M 3)M2' G ^'. (If M2, M3 are of degree 1 but u\ = u,\u'l where u\, u" are monomials of degree at least 1 then the proof is quite similar). Thus, under the inductive assumption each element of the form (15) and of degree k is contained in &'. This completes the proof of Lemma 6. • https://doi.org/10.1017/S1446788700002056 Published online by Cambridge University Press [15] Solution of a problem of Plotkin and Vovsi 343 Let A = F/yiiF), I(A) the group ring of A. Suppose that a, = *,y 3(F) (i g N) so that {a, | / g N) is a free generating set of A. Let T be the two-sided ideal in 2(A) generated by all elements of the form (ui, u 2, u3) (u\, u 2, K3 g 2(A)). LEMMA 7. T is generated (as an ideal in Z(A)) by all elements of the form (16) ([a,,, ah] - 1)(K, au] - 1) + {[a h,ait\ - l){[a h, ah] - l), where i\, i 2, h, U £ N. PROOF. By Lemma 6, T is the two-sided ideal generated by all elements (17) «', fl£, a£) (i lf i2, ij € N, £l, e2, £3 e {-1, 1}) and (18) «,a-)( flJ,<) + «,a-)«,<) O'i, i2,h, U G N, £1,£2,£3,e 4 € {-1, 1}). Let T\ be the two-sided ideal in Z(A) generated by all elements (19) «;,<2)(flJ"« - 1) and (22) ([a,-, a,]"1 - l) = -[ fl<1 aj-'dfl,-, a,\ - 1) so https://doi.org/10.1017/S1446788700002056 Published online by Cambridge University Press 344 C. K.Gupta and A. N. Krasil'nikov [16] for all iu i2, h 6 N, eit e2, £3 € (—1,1} and some g € A, e e {—1,1} which depend on I'I, i2, h and eu e2, £3. Therefore, the elements (19) and (20) generate the same ideal (which is the ideal 7*0. Further, if i k =j, = q for some k,l,(\

Question 270 multiple-choice

Code-based cryptography relies on the hardness of mathematical problems involving error-correcting codes to ensure security, especially in the face of quantum computing advancements. Understanding the interplay between classical and quantum algorithmic limitations is essential for evaluating the robustness of cryptosystems like McEliece and Sidelnikov. Which statement best explains why current quantum algorithms are unable to efficiently solve the Code Equivalence problem for Goppa codes in McEliece-type cryptosystems? 1) Quantum algorithms are inherently slower than classical algorithms for all code equivalence problems. 2) The corresponding Hidden Subgroup Problem for Goppa codes requires highly entangled quantum measurements, which existing quantum algorithms based on Fourier sampling cannot perform. 3) Classical algorithms have been proven to be faster than quantum algorithms for permutation-based code equivalence. 4) Goppa codes cannot be represented as linear codes, making quantum algorithms inapplicable. 5) The hardness of factoring integers is directly tied to the security of Goppa codes, limiting quantum applicability. 6) Reed-Muller codes, not Goppa codes, are the basis for quantum-resistant cryptosystems. 7) Quantum algorithms cannot process permutation groups due to their abelian structure.

✓ Correct Answer:

The correct answer is 2) The corresponding Hidden Subgroup Problem for Goppa codes requires highly entangled quantum measurements, which existing quantum algorithms based on Fourier sampling cannot perform..

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Title: Quantum Fourier sampling, Code Equivalence, and the quantum security of the McEliece and Sidelnikov cryptosystems Year: 2011 Paper ID: 38fc33b60af60b7b013d7d18e924e2c90bbca603 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/38fc33b60af60b7b013d7d18e924e2c90bbca603 Abstract: The Code Equivalence problem is that of determining whether two given linear codes are equivalent to each other up to a permutation of the coordinates. This problem has a direct reduction to a nonabelian hidden subgroup problem (HSP), suggesting a possible quantum algorithm analogous to Shor's algorithms for factoring or discrete log. However, we recently showed that in many cases of interest---including Goppa codes---solving this case of the HSP requires rich, entangled measurements. Thus, solving these cases of Code Equivalence via Fourier sampling appears to be out of reach of current families of quantum algorithms. Code equivalence is directly related to the security of McEliece-type cryptosystems in the case where the private code is known to the adversary. However, for many codes the support splitting algorithm of Sendrier provides a classical attack in this case. We revisit the claims of our previous article in the light of these classical attacks, and discuss the particular case of the Sidelnikov cryptosystem, which is based on Reed-Muller codes.

Question 271 multiple-choice

In computational algebra, determining explicit isomorphisms between rings, especially matrix rings over modular integers, is important for applications in coding theory and cryptography. Efficient algorithms often utilize structural properties and the factorization of the modulus. When constructing an explicit isomorphism between a ring A isomorphic to M₂(Z/NZ) and the matrix ring M₂(Z/NZ), which technique is crucial for reducing the problem when N can be factored as N = ab with coprime integers a and b? 1) Applying the Euclidean algorithm to find the greatest common divisor of a and b 2) Using group cohomology to classify all ring extensions 3) Decomposing the problem into M₂(Z/aZ) and M₂(Z/bZ) via the Chinese Remainder Theorem 4) Constructing a minimal polynomial for each matrix in M₂(Z/NZ) 5) Employing spectral decomposition to diagonalize matrices over Z/NZ 6) Utilizing the Jordan canonical form for matrices over Z/NZ 7) Transforming the ring into a direct sum of cyclic groups

✓ Correct Answer:

The correct answer is 3) Decomposing the problem into M₂(Z/aZ) and M₂(Z/bZ) via the Chinese Remainder Theorem.

📚 Reference Text:

a multiplication table of the basis elements. Such a representation is c alled a structure constant representation. R´ onyai [ R´ on90] proposed a polynomial-time algorithm for this problem when Nis prime. The next lemma generalizes the algorithm to arbitrary Nwhose factorization is known. 16 M. Chen, M. Imran, G.Ivanyos, P.Kutas, A.Leroux, C. Petit Proposition 4.1. LetAbe a ring isomorphic to M2(Z/NZ)given by a struc- ture constant representation. Suppose that factorization ofNis known. Then there exists a polynomial-time algorithm that computes an e xplicit isomorphism betweenAandM2(Z/NZ). Proof.First we reduce the problem to the case where Nis prime power. Let N=abwhereaandbare coprime. Then A/aAis isomorphic M2(Z/aZ) and A/bAis isomorphic to M2(Z/bZ). SinceM2(Z/aZ)×M2(Z/bZ) is isomorphic to M2(Z/NZ), knowing an explicit isomorphism between A/aandM2(Z/aZ) and an explicit isomorphism between A/bAandM2(Z/bZ) is enough to recover the isomorphism between AandM2(Z/NZ). Using this procedure iteratively (using the fact that the factorization of Nis known) one can reduce to the case where N=qkwhereqis some prime number. Now suppose that Ais isomorphic to M2(Z/qkZ). Observe that A/qAis isomorphic to M2(Z/qZ). One can compute a non-trivial idempotent in A/qA using R´ onyai’s algorithm [ R´ on90], let that be e0. Now one has that e2 0−e0∈qA ande0ande0−1 are not in qA. Our goal is to ﬁnd an element ewhich is an idempotent of A. We will perform an iteration which starts with e0and in theith step we return an element eifor whiche2 i−ei∈qi+1A. Suppose we have an element ei−1for whiche2 i−1−ei−1∈qiA. Now we are looking for anf∈Asuch that ( ei−1+fqi)2−(ei−1+fqi)∈qi+1A. This is clearly equivalent to ( e2 i−1−ei−1) + (2ei−1−1)fqi∈qi+1A. Now dividing by qilet (e2 i−1−ei−1)/q=Ei−1. We need an fsuch that that Ei−1+(2ei−1−1)f∈qA. Observe that since ( e2 i−1−ei−1)∈qAone has that (2 ei−1−1)2−1∈qA, hence 2ei−1reduces to an invertible matrix in A/qA. Hence any choice for which f≡ −Ei−1(2ei−1−1)−1(modqA) is suﬃcient and we can set ei=ei−1+fqi. Thenek−1will be a non-trivial idempotent of A. Non-triviality follows from the fact that every eiis congruent to e0moduloqA. SinceAis a 2×2 matrix ring, Aeis isomorphic to ( Z/qkZ)2as anA-module. Then the left action of AonIprovides an explicit isomorphism between Aand M2(Z/qZ). We could not ﬁnd a reference for this fact, so we present a quick simple proof. Let/parenleftbigga b c(1−a)/parenrightbigg be an idempotent matrix. We can assume that it has the above form as eis not congruent to 0 or the identity matrix modulo qA. We also have that a(1−a) =bc. Now the following is a generating set (as an abelian group) of Ae: /parenleftbiggab 0 0/parenrightbigg ,/parenleftbigg0 0 ab/parenrightbigg ,/parenleftbiggc(1−a) 0 0/parenrightbigg ,/parenleftbigg0 0 c(1−a)/parenrightbigg . One has that either aor 1−ais invertible in Z/qkZ, one may suppose that a is invertible (the calculation is the same in the other case). Then it is cle ar that every element of the form/parenleftbigg αα(b/a) β β(b/a)/parenrightbigg Title Suppressed Due to Excessive Length 17 is in the left ideal for any α,β∈Z/qkZ. We show that every element of Aeis of this form. This follows from the fact that every element of the form /parenleftbiggγcγ(1−a) δc

Question 272 multiple-choice

Quantum computing protocols vary in their use of digital and analog techniques, impacting the scalability and fidelity of algorithms such as the Quantum Fourier Transform (QFT). Hardware connectivity and error sources play a critical role in determining overall computational performance. Which protocol is considered optimal for implementing Quantum Fourier Transform on multi-qubit systems, offering superior fidelity and scalability particularly when system size increases? 1) bDAQC (best Digital-Analog Quantum Computing) 2) DQC (fully Digital Quantum Computing) 3) Quantum Error Mitigation Protocol 4) Classical Fourier Transform Simulation 5) Quantum Annealing Protocol 6) Analog-Only Quantum Computing 7) Surface Code Error Correction

✓ Correct Answer:

The correct answer is 1) bDAQC (best Digital-Analog Quantum Computing).

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shows that the bDAQC protocol is the best option if one wants to implement the Q FT on a system built up from several qubits. (c) Fidelity evolution with growing errors. We want to show how the ﬁdelity behaves if the errors we have estimated are slightly different. We have computed the Q FT of state |ψ 0/angbracketright=sinπ 4|Wn/angbracketright+cosπ 4|GHZ n/angbracketright for a system of three, ﬁve, six, and seven qubits. The ﬁdelity of the two DAQC protocols is better than the ﬁdelity obtained with the DQC, no matter what the errors are. Note that, in the DQC protocol, the error of the two-qubit gates dominates in the total error. Similarly, the ﬁdelityin the DAQC is mainly affected by the errors in the analog blocks, to the DQC case in which the error of the two-qubit gates dominates in the total error, the ﬁdelity in the DAQC is mainlyaffected by the errors in the analog blocks. The results ofthe simulations shown in Fig. 4should not be understood as the expected ﬁdelity arising from an experiment sinceit will strongly depend on the fabrication, architecture, andmaterials, among other factors. The actual result is that thedistance between the ﬁdelities of the bDAQC and the DQCare always remarkable in favor of the bDAQC. Even thoughwe have considered a sensible choice for the noise model ofthe DAQC implementation, this model will not be accurateuntil comparing it against experimental data. We would like to point out that, although the connectivity of most quantum systems is not ATA, with the remarkableexception of trapped ions and NMR, we have chosen it forthe sake of simplicity. It can be shown that, using the DAQCparadigm, one can simulate a N-qubit ATA Ising Hamiltonian making use of, at most, 1 2N(N−1) nearest-neighbor (NN) Hamiltonians. This quadratic overhead holds in the worst-casescenario, but it can be substantially reduced if there existssome pattern in our couplings (see the Appendix). However,this overhead (or worse) also holds for the DQC paradigm. Consequently, the values of the ﬁdelities could change in thecase of considering NN Hamiltonians, but the comparisonshown between DQC and DAQC is fair, and the conclusionrelated to the better performance of the bDAQC with respectto the DQC still holds. V . CONCLUSIONS AND PERSPECTIVES We have shown that the DAQC paradigm can enhance the depth of the implementation of a quantum algorithm. Inparticular, we have provided a digital-analog algorithm for theQFT, a ubiquitous quantum subroutine, which is a relevant part of several quantum algorithms. Improving the ﬁdelity ofthe implementation of Q FT, consequently, enhances applica- bility of other quantum algorithms, such as Shor’s algorithmfor prime number factorization or the HHL algorithm forsolving linear systems of equations. The main problem of the digital approach for the Q FTi n a real NISQ chip is that its ﬁdelity decays fast when scalingup since the depth of the algorithm grows quadratically with 013012-6 DIGITAL-ANALOG QUANTUM ALGORITHM FOR THE … PHYSICAL REVIEW RESEARCH 2, 013012 (2020) the number of qubits. Although there exist QEM techniques to reduce the

Question 273 multiple-choice

Extensions of gauge symmetries in theoretical physics can lead to novel hidden sectors and alter fundamental particle interactions, particularly in the context of dark matter and early universe cosmology. Innovative models may propose unconventional charge types and field dynamics with observable and cosmological implications. Which of the following is a predicted consequence of introducing imaginary charges and extended abelian gauge symmetry in the context of dark matter theory? 1) Like imaginary charges attract, and unlike imaginary charges repel, reversing the usual electromagnetic interaction pattern. 2) The Lagrangian for the hidden sector must be an identical copy added to the visible sector's Lagrangian. 3) All electromagnetic fields, including those from imaginary charges, have strictly positive energy density. 4) Imaginary charges always produce effects indistinguishable from ordinary electric charges in experiments. 5) The weak energy condition cannot be violated by any electromagnetic fields, standard or dark. 6) Dark matter consisting of imaginary charges would have identical distribution and interaction properties as cold, collisionless dark matter. 7) Only standard U(1) gauge symmetry can account for dark photon phenomena in cosmology.

✓ Correct Answer:

The correct answer is 1) Like imaginary charges attract, and unlike imaginary charges repel, reversing the usual electromagnetic interaction pattern..

📚 Reference Text:

Title: Generalization of Abelian Gauge Symmetry, Dark Matter and Cosmological Expansion Year: 2014 Paper ID: 0e0138b062d07fd6963e2f8bd487717f2569836e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0e0138b062d07fd6963e2f8bd487717f2569836e Abstract: A commutative generalization of the U(1) gauge symmetry group is proposed. The two-parametric family of two-connected abelian Lie groups is obtained. The necessity of existence of so-called imaginary charges and electromagnetic fields with negative energy density (dark photons) is derived. The possibilities when the overall Lagrangian represents a sum or difference of two identical Lagrangians for the visible and hidden sectors (i.e. copies of unbroken U(1)) are ruled out by the extended symmetry. The distinction between the two types of fields resides in the fact that for one of them current and electromagnetic kinetic terms in Lagrangians are identical in sign, whereas for another type these terms are opposite in sign. As a consequence, and in contrast to the common case, like imaginary charges attract and unlike charges repel. Some cosmological issues of the proposed hypothesis are discussed. Particles carrying imaginary charges are proposed as one of the components of dark matter. Such a matter would be imaginarily charged on a large scale for the reason that dark atoms carry non-compensated charges. It leads to important predictions for matter distribution, interaction and other physical properties being different from what is observed in dominant dark matter component in the standard model. These effects of imaginary charges depend on their density and could be distinguished in future observations. Dark electromagnetic fields can play crucial dynamical role in the very early universe as they may dominate in the past and violate weak energy condition which provides physical grounds for bouncing cosmological scenarios pouring a light on the problem of origin of the expanding matter flow.

Question 274 multiple-choice

Gastric cancer is a biologically diverse malignancy with multiple modifiable risk factors. Understanding these risks and the disease’s heterogeneity is important for targeted prevention and research. Which of the following statements best explains why subgroup analyses are considered crucial for clarifying associations between diabetes and gastric cancer risk? 1) Subgroup analyses reduce the prevalence of H. pylori infection in study populations. 2) Subgroup analyses allow researchers to exclude confounding factors such as medication use. 3) Subgroup analyses account for the heterogeneity of gastric cancer subtypes, which may have distinct risk profiles and responses to risk factors. 4) Subgroup analyses increase the statistical power by enlarging sample sizes. 5) Subgroup analyses eliminate the effect of genetic variability in patients. 6) Subgroup analyses ensure only patients with advanced gastric cancer are studied. 7) Subgroup analyses focus solely on environmental exposures rather than biological mechanisms.

✓ Correct Answer:

The correct answer is 3) Subgroup analyses account for the heterogeneity of gastric cancer subtypes, which may have distinct risk profiles and responses to risk factors..

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has an important clinical meaning because authors provided a clue to other modifiable risk factors of gastric cancer besides H. pylori, smoking and alcohol consumption. Given that main hypothesized mechanisms for the carcinogenic effect of diabetes underlies insulin resistance with secondary hyperinsulinemia, the generally null association between a history of diabetes and the risk of gastric cancer in the previous literature is somewhat surprising. Because gastric cancer has been considered more heterogeneous diseases compared to colon cancer, it may be important to discern between the subgroups in investigations of gastric cancers. For the future per-

Question 275 multiple-choice

In variational quantum simulation of spin systems, symmetry-adapted neural network ansätze and advanced optimizers are used to accurately approximate ground states while maintaining physical interpretability. Careful selection of model parameters, activation functions, and wave function postprocessing plays a crucial role in achieving robust and precise results. Which of the following practices directly contributes to the compatibility of a trial wave function with quantum hardware and leverages time-reversal symmetry in spin systems? 1) Using complex-valued neural network parameters 2) Increasing the circuit depth by adding more alternating layers 3) Employing U(1) symmetry in the optimizer 4) Initializing variational parameters randomly 5) Applying the Hamiltonian shift to ensure positive semidefiniteness 6) Restricting to the real, normalized part of the wave function (Re(ψ) = ψ + ψ*) 7) Selecting a learning rate of 0.02 for all models

✓ Correct Answer:

The correct answer is 6) Restricting to the real, normalized part of the wave function (Re(ψ) = ψ + ψ*).

📚 Reference Text:

comparison. Note that we implement no explicit global SU(2) or U(1)symmetry for these benchmark algorithms. For numerical simula- tion of Sn-CQA, we perform random initialization of the parameters. We ﬁnd that the random initialization already returns the energy, which is within roughly 10−2precision within ED ground-state energy and nonoscillating descents around the ED ground-state energy comparing with that ofGCNN and RBM. This is likely due to the fact that we use theSn-Fourier space activation with real-valued trial wave functions with explicit SU(2) symmetry. We record the optimized energy for the Sn-CQA ansätze every ﬁve iter- ations, and we set the number of alternating layers p=4 for the 3 ×4 lattice and p=6 for the 12-spin kagome lat- tice. In the implementation, we shift the Hamiltonian to /tildewideH(λ)=H(λ)+m1dλto ensure /tildewideH(λ)is positive semidef- initeness in the Snirrep speciﬁcity by the partition λ= (λ1,λ2)with the total spin label j=(λ1−λ2)/2 (Sec. II), where mis the total number of transpositions. We take only the real (normalized) part of the wave function Re (ψ)= ψ+ψ∗. This can be seen as a postprocessing step for the realization of Sn-CQA on a quantum computer. We use the Nesterov-accelerated Adam [ 96] for the Sn-CQA optimization with hyperparameters: betas =[0.99, 0.999]. We also utilize NetKet’s exact diagonalization (ED) result 020327-14 SPEEDING UP LEARNING QUANTUM... PRX QUANTUM 4,020327 (2023) for the comparison with the exact ground-state energy. The ground state is additionally calculated in the Young (Schur) basis by diagonalizing the Heisenberg Hamilto- nian H(λ)in the irrep λwhere the ground state lives. It is worth mentioning that our optimized ground state is strictly real valued and has explicitly SU(2) symmetry, oﬀering the missing yet essential physical interpretation. We provide code and Jupiter notebook in open access on Github in python. The numerical simulations are run in the CPU platform with the 9th-Gen 1.4-GHz Intel Core i5 processors. 3×4rectangular lattice In frustrated region of J2/J1=0.5,J2/J1=0.8, we ﬁnd that the ground state lives entirely in the total spin 0 irrep, corresponding to the partition λ=(4, 4). In the case of J2=0.5, we report that the Sn-CQA ansätze are able to smoothly converge to the ground state, with error to the exact ground-state energy 9.1049 e−5.F o r J2=0.8, the Sn- CQA returns 5.0587 e−4precision to the ED ground-state energy. We notice that the Sn-CQA seem always to con- verge to the ground state with reasonable good accuracy without the issue of trapping in local minima, regardless of initialization (random initialization from Gaussian is used). The learning rate used here is 0.01. For the GCNN layers in both J2values we set the feature dimensions of hidden layers (8, 8, 8, 8 )and 1024 samples with the learning rate set for 0.02. For the RBM model, we ﬁx the learning rate 0.02 with 1024 samples. 12-spin kagome lattice We ﬁnd by comparing with the ED result that the ground state of 12-spin kagome lattice lives in the total spin-2 irrep, corresponding to partition [8, 4] in J2=0, which suggests it to be ﬁvefold degenerate.

Question 276 multiple-choice

Loop Quantum Gravity and spin-foam models utilize advanced group theoretical concepts to describe quantum properties of spacetime. Lie groups such as SU(2) and SL(2,C) play a central role in formulating their underlying mathematical structures. Which mathematical tool is essential for decomposing tensor products of group representations, thereby enabling the combination of quantum states in Loop Quantum Gravity? 1) Gauge fixing 2) Path integration 3) Differential geometry 4) Symplectic reduction 5) Fiber bundle theory 6) Renormalization group analysis 7) Recoupling theory

✓ Correct Answer:

The correct answer is 7) Recoupling theory.

📚 Reference Text:

Title: A primer of group theory for Loop Quantum Gravity and spin-foams Year: 2019 Paper ID: f47a4ff99a1997545f21a6e1dcfc987e5344db2b Source: semantic-scholar URL: https://www.semanticscholar.org/paper/f47a4ff99a1997545f21a6e1dcfc987e5344db2b Abstract: Calculations in Loop Quantum Gravity (LQG) and spin-foams theory rely heavily on group theory of SU(2) and SL2(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SL_{2}({\mathbb {C}})$$\end{document}. Even though many monographs exist devoted to this theory, the different tools needed (e.g. representation theory, harmonic analysis, recoupling theory...) are often dispersed in different books, with different conventions and notations. This was the initial motivation for the compilation of the present document. Generally, these notes can serve three main purposes: a concise introduction for students to the essential mathematical tools of LQG, a convenient compendium for researchers, a translational hub between the conventions of the main references. These notes are aimed both at physicists, caring about their tools being mathematically well grounded, and at mathematicians, curious about how some of their familiar abstract structures can reveal the beauty of quantum gravity.

Question 277 multiple-choice

In algebraic geometry, the study of abelian varieties involves analyzing their group structure, morphisms, and properties such as projectivity, dimension, and behavior under field extensions. The construction of abelian varieties often relies on graded rings and various maps that preserve group operations. Which property is essential to show that the scheme A = Proj is an abelian variety of dimension g over a field k? 1) A admits a non-trivial unipotent subgroup 2) A is projective, reduced, irreducible, of finite type, and has dimension g 3) A has a dense open subscheme isomorphic to affine space 4) A is a fiber product of elliptic curves 5) The coordinate ring R is local and Artinian 6) A is a disjoint union of group schemes of lower dimension 7) The global sections of A form a non-commutative ring

✓ Correct Answer:

The correct answer is 2) A is projective, reduced, irreducible, of finite type, and has dimension g.

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into V. An obvious but tedious calculation shows that the two induced maps Γ(V) —* Γ(U) coin- cide and we apply Lemma 1.3.2 (for a more detailed proof of a similar result see Theorem 1.3.5). Finally, (4) of Theorem 1.3.3 shows that m: Ak x Afc —> Ak takes (Px,Py) to P^: i.e. that x—>Px is a homomor- phism. THEOREM 1.3.5. For each n > 0 ίfeβ map a*: A-> A induced by an is just group scheme multiplication by n (which we will denote by nA). Proof, Since R is integral over an(R), we get a morphism of schemes α*: A -> A. We show first that if θ and θr are in R'm then the pull-back of θ'/θeΓ((A x A),) under α* x id is θ'(Xn,X)/θ(Xn,X), at least on some principal open subset U of Aθ(XniZ). To see this, take ψ Φ 0 in i2m. Since R'm = Rm®Rm9 direct calcula- tion shows that the pull-back of ^/ψ(Z)ψ(Y) under α* x id. is ^(Z72,^) /Λ/Γ(ZW)^(Z). Since a similar formula holds for the pull-back of 6>7ψ(Z)ψ(Y), we get our result where V is defined by ψ(Zn)ψ(Z). The theorem can now be proved by induction on n. n = 1 is obvious. in + ΐ)A is the composite map πλ o β* o (^ x id.) :A->AχA-*AχA->A. Fix G ΦO in βj and suppose FeRm. Then F/Gw in Γ(AG) pulls back to F(Z)G(Γ)W/G(Z)WG(Γ)W under πλ and this pulls back to F(XY)G(XY~1)m IG(XY)mG(XY-ι)m under β*. By induction, (n^ x id) = (a* X id). If we apply the result of the paragraph above with ψ = G2m, we conclude that the pull-back of F/Gm under (n + 1)A = πx o i8* o (α* X id.) is F(ZW+1)/G(ZW+1)TO over the affine subset of A defined by G(Xn+ί)G(Xn-ι)G(Xn)G(X). The theorem then follows from Lemma 1.3.2 applied to the maps an+1 and in + 1)^. https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press ABELIAN VARIETIES 77 THEOREM 1.3.6. The scheme A = Proj. (R) is an abelian variety of dimension g over k. Proof. From Theorem 1.3.4, A has the structure of commutative group scheme over fc. Since R is a finitely generated fc-algebra and an integral domain, A is of finite type, reduced and irreducible. If L is a finite extension of k, let R(L) be the graded L-algebra corresponding to the matrix (st^ over the field L. Then # (x)fc L ~ R(L) and is a domain. Hence, A remains reduced and irreducible under finite extensions of k, and since it is projective, it is an abelian variety. Since dim. Rm = (2m)g for all m > 0, A has dimension g. II In this part we show that the map φ: Gg/Γ —* Ak defined in §1.3 is an isomorphism provided the elements off the main diagonal of the matrix (J/^ ) are units in the valuation ring Θ. Throughout part II ive make this assumption on the J^I/S. Note that qt — s/ti e Jί because of positive definiteness. §Π.l.

Question 278 multiple-choice

Quantum algorithms have dramatically reshaped the computational landscape for hard mathematical problems in cryptography, particularly through breakthroughs in the discrete logarithm problem (DLP) and the hidden subgroup problem (HSP). Understanding their complexity and underlying algebraic structures is essential for evaluating their impact on cryptographic protocols. Which statement correctly identifies a quantum algorithmic complexity or property directly associated with solving the abelian hidden subgroup problem (HSP) in finite groups? 1) Quantum algorithms require Ω(|G|) queries to solve the abelian HSP. 2) The abelian HSP is solved by classical methods in polynomial time for all finite groups. 3) Quantum solutions to the abelian HSP are only applicable to non-abelian groups. 4) The quantum Fourier transform provides no advantage in solving the abelian HSP. 5) Quantum algorithms for the abelian HSP do not utilize measurements of quantum registers. 6) Classical algorithms solve the abelian HSP with O(log|G|) queries. 7) Quantum algorithms solve the abelian HSP with O(log|G|) queries, providing exponential speedup over classical approaches.

✓ Correct Answer:

The correct answer is 7) Quantum algorithms solve the abelian HSP with O(log|G|) queries, providing exponential speedup over classical approaches..

📚 Reference Text:

Childs [ 46] described an effective quantum algorithm for computing discrete logarithms over semi- groups. Recently, further generalized Shor’s quantum DLP algorithms are proposed for different algebraic structures [ 47–50]. With the identity g r=y, if we deﬁne a binary function f:Zp−1×Zp−1→Zp as follows: (a,b)/mapsto→gaybmod p, 123 178 Page 12 of 26 J. Suo et al. |0 Fp−1⊗Fp−1UfFp−1⊗Fp−1|0 |0 Fig. 4 The circuit of discrete logarithmic problems [ 51].Fp−1is the Fourier transform over Zp−1 then ftakes(r,−1)as its period, considering that f(a+r,b−1)=f(a,b)(∀a,b∈Zp−1). Therefore, the aforementioned idea for ﬁnding period can be used to solve the discrete logarithms problems. Now, suppose a quantum circuit Uffor implementing |a/angbracketright|b/angbracketright|0/angbracketright→| a/angbracketright|b/angbracketright|gaybmod p/angbracketright is at hand, then Shor’s discrete logarithm algorithm is depicted in Fig. 4. A modiﬁed version of Shor’s quantum DLP algorithm, due to Wang [ 51], consists of the following six steps: •Step 1 Initialize three quantum registers |ϕ0/angbracketright=| 0/angbracketright|0/angbracketright|0/angbracketright •Step 2 Apply the Fp−1⊗Fp−1on the ﬁrst two registers and get the superposition |ϕ1/angbracketright=1 p−1p−2/summationdisplay a=0p−2/summationdisplay b=0|a/angbracketright|b/angbracketright|0/angbracketright •Step 3 Perform Ufin the third register |ϕ2/angbracketright=1 p−1p−2/summationdisplay a=0p−2/summationdisplay b=0|a/angbracketright|b/angbracketright|gaybmod p/angbracketright •Step 4 Measure and then discard the third register, leading that the ﬁrst two registers collapse to |ϕ3/angbracketright=1√p−1p−2/summationdisplay λ=0|a0+λr/angbracketright|b0−λ/angbracketright. •Step 5 Apply QFT to the ﬁrst two registers, and we get |ϕ4/angbracketright=1√p−1p−1/summationdisplay u=0e2πi(a0+b0r)u p−1|u/angbracketright|ru/angbracketright 123 Quantum algorithms for typical hard problems… Page 13 of 26 178 Table 4 The complexity of quantum DLP algorithms Authors Year Time complexity Space complexity Shor et al. [ 3] 1994 O(n3) O(n) Proos et al. [ 42] 2003 O(n2) – Ekera et al. [ 49] 2019 – O(n 2) The comparison of time and space complexities is in table •Step 6 Measure the ﬁrst two registers to get |u0/angbracketright|ru0/angbracketright, and then, derive rby r=ru0u−1 0mod(p−1)(assuming that gcd (u0,p−1)=1 with high probability). The complexities of Shor’s quantum DLP algorithm and some related algorithms are collected in Table 4. 6 Quantum algorithms for abelian hidden subgroup problems Let Hbe the subgroup of group G,Sbe any set and f:G→Sa function that distinguishes cosets of H, i.e., ∀g1,g2∈G,f(g1)=f(g2)⇔g1H=g2H. The hidden subgroup problem (HSP) is to ﬁnd the subgroup Husing f.T os o l v e this problem classically, /Omega1(|G|)queries on fare required, while it is solvable on a quantum computer using merely O(log|G|)f-queries. In 1995, Kitaev [ 52] gave a polynomial quantum algorithms to solve the abelian stabilizer problem (ASP) and prove that the integer factorization and discrete loga- rithm problems can be solved as special cases. In 1995, Dan and Lipton [ 53] ﬁrst built the relationship between quantum algorithm and HSP and designed a quan-tum algorithm to solve the hidden linear function. In 1997, Brassard and Hoyer [ 54] extended the Simon’s problem to HSP . In 1998, Jozsa [ 55] gave a uniﬁed description of Deustch–Jozsa’s algorithm, Simon’s algorithm and Shor’s algorithm in the form ofHSP . Subsequently, Mosca [ 56,57] and Jozsa [ 58] introduced the more general abelian HSP and gave quantum Fourier transform to solve it. Abelian HSP mainly focuses on ﬁnite abelian groups, and related

Question 279 multiple-choice

Supersymmetric models propose new particles that could explain dark matter, but their cosmological implications depend critically on particle properties and early universe dynamics. String-theory-inspired scenarios often involve moduli fields affecting the relic abundance of dark matter candidates. Which scenario most effectively eliminates the cosmological overproduction of the lightest supersymmetric particle (LSP) dark matter without requiring fine-tuned extensions to the Minimal Supersymmetric Standard Model (MSSM)? 1) Introducing extra hidden U(1) gauge symmetries 2) Enhancing thermal freeze-out via additional Higgs doublets 3) Tuning modulus decay rates to match nucleosynthesis constraints 4) Including axion-like particles as alternative dark matter candidates 5) Increasing the number of hidden sector generations 6) Allowing R-parity violation or making all superpartners and moduli very heavy 7) Lowering the gluino mass to suppress dark matter interactions

✓ Correct Answer:

The correct answer is 6) Allowing R-parity violation or making all superpartners and moduli very heavy.

📚 Reference Text:

a potential tension with primordial nucleosynthesis from late MSSM decays, and leads to a negligibly small hidden con nement scale. In contrast to the two previous extensions, light scalar superpartners are not required and this mechanism can work in the context of mini-split supersymmetry [38{43]. While this theory is not constrained by standard indirect detection searches, the coupling of the hidden gluino to a bath of hidden gluons leads to non-standard DM dynamics that require the hidden gluino density to be only a few percent of the total DM density. It is very dicult to obtain relic densities this small in this scenario. Our main conclusion is that it is challenging to avoid producing too much LSP dark matter in the course of string-motivated moduli reheating. For seemingly generic modulus parameters, the relic density in the MSSM is either too large or at odds with limits from indirect detection. This may be a hint that the properties of moduli (in our vacuum at least) di er from the general expectations discussed above [6, 18]. Alternatively, this could be an indication of new light physics beyond the MSSM. We have considered three examples of the latter possibility in this paper and have shown that they can produce a stable LSP abundance that is consistent with current observations and limits. Even so, these three extensions all lead to a signi cant complication of the MSSM and require a somewhat fortuitous conspiracy of parameters for them to succeed. A more direct solution might be the absence of a stable LSP through R-parity violation, or simply the absence of light superpartners and very large m'm3=2. ACKNOWLEDGEMENTS We thank Matthew Reece, Kris Sigurdson, Scott Watson, and Kathryn Zurek for helpful discussions. This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). [1] G. D. Coughlan, W. Fischler, E. W. Kolb, S. Raby and G. G. Ross, Phys. Lett. B 131, 59 (1983). [2] J. R. Ellis, D. V. Nanopoulos and M. Quiros, Phys. Lett. B 174, 176 (1986). [3] T. Banks, D. B. Kaplan and A. E. Nelson, Phys. Rev. D 49, 779 (1994) [hep-ph/9308292]. [4] J. Polchinski, \String theory. Vol. 1: An introduction to the bosonic string," Cambridge, UK: Univ. Pr. (1998) 402 p [5] J. Polchinski, \String theory. Vol. 2: Superstring theory and beyond," Cambridge, UK: Univ. Pr. (1998) 531 p 33 [6] B. S. Acharya, G. Kane and P. Kumar, Int. J. Mod. Phys. A 27, 1230012 (2012) [arXiv:1204.2795 [hep-ph]]. [7] B. de Carlos, J. A. Casas, F. Quevedo and E. Roulet, Phys. Lett. B 318, 447 (1993) [hep- ph/9308325]. [8] F. Quevedo, Class. Quant. Grav. 19, 5721 (2002) [hep-th/0210292]. [9] B. S. Acharya, P. Kumar, K. Bobkov, G. Kane, J. Shao and S. Watson, JHEP 0806 , 064 (2008) [arXiv:0804.0863 [hep-ph]]. [10] M. Dine, L. Randall and S. D. Thomas, Phys. Rev. Lett. 75, 398 (1995) [hep-ph/9503303]. [11] R. Durrer and J. Hasenkamp, Phys. Rev. D 84, 064027 (2011) [arXiv:1105.5283 [gr-qc]]. [12] A. L. Erickcek and

Question 280 multiple-choice

Quantum algorithms often rely on techniques from group theory to efficiently solve problems such as factoring and discrete logarithms. The Hidden Subgroup Problem (HSP) is fundamental in this area, with recent advances leveraging representation theory to improve solution strategies. Which query method maximizes the probability of correctly identifying the hidden subgroup in single-query algorithms for finite abelian groups under a uniform prior? 1) Character query 2) Standard equal superposition query with 〈0〉 in the response register 3) Grover’s search query 4) Oracle randomization query 5) Hadamard query 6) Coset state query 7) Conjugate subgroup query

✓ Correct Answer:

The correct answer is 1) Character query.

📚 Reference Text:

Title: An improved query for the hidden subbroup problem Year: 2011 Paper ID: a28e28d23b1440140e685b765e95ea6528bc7475 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a28e28d23b1440140e685b765e95ea6528bc7475 Abstract: The Hidden Subgroup Problem (HSP) is at the forefront of problems in quantum algorithms. In this paper, we introduce a new query, the character query, generalizing the well-known phase kickback trick that was first used successfully to efficiently solve Deutsch's problem. An equal superposition query with 〈0〉 in the response register is typically used in the "standard method" of single-query algorithms for the HSP. The proposed character query improves over this query by maximizing the success probability of subgroup identification under a uniform prior, for the HSP in which the oracle functions take values in a finite abelian group. We apply our results to the case when the subgroups are drawn from a set of conjugate subgroups and obtain a success probability greater than that found by Moore and Russell.

Question 281 multiple-choice

In quantum information processing, the physical validity of quantum operations is guaranteed by certain mathematical properties, such as complete positivity, which can be analyzed using matrices derived from quantum channels. Nuclear Magnetic Resonance (NMR) quantum computers implement these operations using electromagnetic pulses, with experimental fidelity affected by system and control parameters. Which of the following statements about completely positive superoperators and their Choi matrices is correct? 1) The sum of all Choi matrix eigenvalues must be zero for a completely positive superoperator. 2) Negative Choi matrix eigenvalues indicate complete positivity of the superoperator. 3) The ratio of positive to total Choi matrix eigenvalues exceeds one for a completely positive superoperator. 4) A superoperator is completely positive if and only if all eigenvalues of its Choi matrix are non-negative. 5) Complete positivity implies that the Choi matrix is traceless. 6) The presence of any positive eigenvalue in the Choi matrix guarantees complete positivity. 7) Complete positivity is unrelated to the eigenvalues of the Choi matrix.

✓ Correct Answer:

The correct answer is 4) A superoperator is completely positive if and only if all eigenvalues of its Choi matrix are non-negative..

📚 Reference Text:

This is described for the e xperimental results in the Section VIII. In order to quantify the extent to which a supermatrix violates the completely positive requirement, we deﬁne the positivity as the ratio of the sum of eigen values of the Choi matrix corresponding to the supermatrix over the sum of positive e igenvalues of the said Choi matrix. For a completely positive superoperator the positivity is equal to 1, while the presence of negative Choi eigenvalues causes the positivity to be le ss than 1. IV. THE SPIN SYSTEM AND THE EXPERIMENTS The experiments were implemented on a three-qubit NMR quantum inf ormation proces- sor [28, 29]. The three qubits used were the three carbon spins in mo lecules of13C-enriched alanine in an aqueous solution. The internal Hamiltonian of this system has the form Hs=π3/summationdisplay i=1νi Cσi C,z+π 23/summationdisplay j>i=1Ji,j C,C/vector σi C·/vector σj C, (19) whereνi Care the Larmor frequencies of the spins and Ji,j C,Care the strengths of the couplings between them, both in frequency units. In our indexing scheme, a s uperscript 1 labels the carbonyl carbon of alanine, 2 the alpha carbon and 3 the methyl ca rbon. A separate time- dependent external Hamiltonian, shown in Eqn. (11), must be adde d whenever an RF pulse is applied to rotate qubits. In the 7 Tesla magnet used for the exper iments, the resonant frequency of carbon-13 is approximately 75.468 MHz. Frequency c hanges, also known as chemical shifts, among the spins introduce diﬀerences ν2 C−ν1 C= 9456.5 Hz,ν3 C−ν2 C= 2594.3 Hz andν3 C−ν1 C= 12050.8 Hz. The coupling constants between the three spins are J1,2 C,C= 54.2 Hz, J2,3 C,C= 35.1 Hz, and J1,3 C,C=−1.2 Hz. In the absence of RF, the T1relaxation times 10 of the three spins are all longer than 1.5 s, while the T2relaxation times are longer than 400 ms (see Table I for exact numbers). To rotate qubits we used the strongly modulating pulses introduced in Refs. [3, 10]. The ﬁrst generation of strongly modulating pulses were designed to per form the desired propaga- tor while refocussing the spins’ known internal Hamiltonian [3]; the s econd generation was designed to also compensate for RF ﬁeld inhomogeneity [10]. This was d one by including the RF inhomogeneity proﬁle in the simulations and, hence, in the target f unction minimized to design the compensated pulses. Pulses designed in this way perfo rmed near-optimally over the range of RF powers experienced by the ensemble of spins. While the simulated peak ﬁdelities were lower than with the uncompensated pulses, the a ctual performance of the compensated pulses in the spectrometer was greatly enhance d. The method of implementing the QFT via NMR is the same as in our previou s QFT implementations [8, 22], with the added beneﬁt of the RF pulses that c ompensated for RF-inhomogeneity. The pulse sequences for the Hadamard and con ditional phase gates are derived from an idempotent or correlation operator description of the propagators [30]. The

Question 282 multiple-choice

In quantum machine learning, symmetry properties play a crucial role in designing neural network architectures that process data consistently under group actions. The concept of equivariance ensures that network operations remain compatible with underlying symmetries present in the data or task. Which of the following sets of generators guarantees that a two-qubit quantum neural network layer is equivariant with respect to the SWAP symmetry, treating both qubits identically under exchange? 1) $\{X \otimes Y, Y \otimes X, Z \otimes X, X \otimes Z\}$ 2) $\{X \otimes 11, Y \otimes 11, Z \otimes 11, 11 \otimes X, 11 \otimes Y, 11 \otimes Z\}$ 3) $\{X \otimes Y - Y \otimes X, Y \otimes Z - Z \otimes Y, Z \otimes X - X \otimes Z\}$ 4) $\{X \otimes Z + Z \otimes X, Y \otimes Y + Z \otimes Z, X \otimes X - Y \otimes Y\}$ 5) $\{X \otimes X + Y \otimes Y, Z \otimes Z, 11 \otimes 11\}$ 6) $\{X \otimes 11 + 11 \otimes X, Y \otimes 11 + 11 \otimes Y, Z \otimes 11 + 11 \otimes Z, X \otimes X, Y \otimes Y, Z \otimes Z\}$ 7) $\{X \otimes X - Y \otimes Y, Z \otimes Z - X \otimes X, Y \otimes Z + Z \otimes Y\}$

✓ Correct Answer:

The correct answer is 6) $\{X \otimes 11 + 11 \otimes X, Y \otimes 11 + 11 \otimes Y, Z \otimes 11 + 11 \otimes Z, X \otimes X, Y \otimes Y, Z \otimes Z\}$.

📚 Reference Text:

zero (to ﬁrst order) if [H;Ug] = 0. That is, the quantum neural network W()will be equivariant if its generatorHcommutes with all the representations of the group elements. In fact, one can check that [H;Ug] = 0;8g2G; (22) is enough to guarantee that all remaining higher orders terms will also commute with Ug[27]. Not surprisingly, it Figure 8. Example of equivariant quantum neural net- work.Here we consider the task in Fig. 3 of classifying real valued data x= (x1;x2)living in a two-dimensional plane. We use tools from the Lie group / Lie algebra correspondence to build equivariant quantum neural networks to the symmetry group of the task, GSWAP =f11;SWAPg. is easier to solve [H;Ug] = 0at the algebra level than to solve [W();Ug]atthegrouplevel. Herewefurtherremark that if G is a Lie group, then we can map to the Lie algebra and ﬁnd all operators such that [H;X ] = 0for allX2g. This exempliﬁes how working at the algebra level of the layer is easier. Note that these results can be generalized to a quantum neural network with multiple layers. Here, W() =Q leilHl, whereW()will be equivariant if each layerisequivariant. Thus, werequire [Hl;Ug] = 0,8Hland 8g2G. As an example, consider the QML task of Fig. (3) of classifying real valued data x= (x1;x2)living in a two- dimensionalplane. Here,thedataisencodedintotwoqubit states, and the symmetry group is Z2whose representation acting on two qubits is GSWAP =f11;SWAPgof Eq. (9). From the previous result we have that a quantum neural network will be equivariant if its generators commutes with the identity operator (always trivially true), and with the SWAP operator. The reader can verify that any generator Hinspan(fX 11+11 X;Y 11+11 Y;Z 11+11 Z;X X;Y Y;Z Zg), satisﬁes [H;SWAP] = 0 (below we will explicitly show how this set can be found, we also refer the readers to Refs. [25, 27, 30, 62]). In Fig. 8 we show an example of a quantum neural network that is constructed by exponentiation of these generators, and thus, that is equivariant to GSWAP. D. Representation Theory: the fundamentals Our working examples the entire time have been repre- sentations of a few key groups. It is high time we listed key deﬁnitions to provide formal structure to the pictures they paint. First, let us tackle a conceptually diﬀerent aspect of representation theory: Given a representation of a symme- try group. How can I build a new representation? Answer- ing this question can be both conceptually rich, but also 20 Example 8: Tensor rep of su(2)on 2 qubits Returntothetensorrepresentationof SU(2)ontwo qubitsV= (C2) 2, which we recall decomposes as V=Sym2(C2)Alt2(C2). Just as in the previous example, we study the induced representation on the Lie algebra by writing Ug() =eiHwithH2 su(2). Then using Leibniz rule for tensor products, d d(Ug() Ug()) =0=d d eiH eiH =0 =i HeiH eiH +eiH HeiH =0 =i(H 1+1 H): So, the Lie group representation g7!Ug Uy g2 GL(V)induces the Lie algebra representation H7! H 1+1 H2gl(V). Again, as demanded by Theorem 2, this representation respects the de- composition

Question 283 multiple-choice

Quantum algorithms excel at solving certain group-theoretic problems, particularly the hidden subgroup problem (HSP), with techniques like the quantum Fourier transform and entangled measurements. Advances in combining quantum and classical algorithms have broadened the scope of tractable problems, including generalized hidden shift problems and applications of integer programming. Which algorithm enables efficient classical solution of the matrix sum problem involved in the generalized hidden shift problem for fixed dimension k? 1) Shor's algorithm 2) Grover's algorithm 3) Lenstra's algorithm 4) Berlekamp-Massey algorithm 5) Lattice basis reduction algorithm 6) Babai's nearest plane algorithm 7) Elliptic curve method

✓ Correct Answer:

The correct answer is 3) Lenstra's algorithm.

📚 Reference Text:

cases, the matrix sum problem can be solved, giving an eﬃcient quantum algorithm for the co rresponding hidden subgroup problem [3]. However, since the average cas e subset sum problem appears to be diﬃcult, this approach has not yielded an improved algorithm for the dihedral hsp. In this article, we show how the pgmapproach provides an eﬃcient quantum algorithm for a problem that interpolates between t he abelian and dihedral hidden subgroup problems. The dihedral hspis equivalent to the hidden shift problem , in which the goal is to determine a hidden shift s∈ZN given two injective functions f0,f1satisfying f0(x) =f1(x+s). Instead of only two such functions, we consider Msuch functions, each one shifted from the previous by a ﬁxed hidden shift s. IfM=N, this problem is an instance of the abelian hsponZN×ZNwith the hidden subgroup /an}⌊ra⌋k⌉tl⌉{t(1,s)/an}⌊ra⌋k⌉tri}ht, which can be solved eﬃciently using abelian Fourier samplin g. Even the caseM=Nis classically intractable, and the problem only becomes mo re diﬃcult for smaller M. In particular, for M≪N, abelian Fourier sampling fails to determine the hidden shift. Using the pgmapproach, we give, for any ﬁxed integer k≥3, an eﬃcient quantum algorithm that solves this problem for M=⌊N1/k⌋. The algorithm works by implementing a joint measurement on kcopies of certain quantum states that encode the hidden shift. Because for each M≥M′the generalized hidden shift problem on {0,...,M−1}×ZNcan be reduced to the generalized hidden shift problem on{0,...,M′−1}×ZN, for any ﬁxed ǫ >0, this gives an eﬃcient quantum algorithm for all M≥Nǫ. By applying the general pgmapproach developed in [3], we ﬁnd that the matrix sum problem corresponding to the generalized hid den shift prob- lem is the following: given uniformly random x∈Zk Nandw∈ZN, ﬁnd b∈ {0,...,M−1}ksuch that b·xmodN=w. As this is a linear Dio- phantine equation with convex constraints, it is an instanc e of integer pro- gramming, and can be solved using Hendrik Lenstra’s algorit hm for that problem [20], which is eﬃcient as long as the dimension kis constant. Thus QUANTUM ALGORITHM FOR A GENERALIZED HIDDEN SHIFT PROBLEM 3 our algorithm for the generalized hidden shift problem reit erates a theme of [3]: by combining abelian quantum Fourier transforms with n ontrivial clas- sical (or quantum) algorithms, one can ﬁnd eﬃcient quantum a lgorithms for hsp-like problems via the implementation of entangled quantum measure- ments. This result is encouraging since entangled measurem ents are known to be necessary for some hidden subgroup problems—in partic ular, for the symmetric group [22]. The remainder of this article is organized as follows. In Sec tion 2, we describe the generalized hidden shift problem in detail and explain how it can be viewed as a quantum state distinguishability problem . In Section 3, we review the pretty good measurement approach to such probl ems, prove that this approach solves the generalized hidden shift prob lem when the number of states is k≥ ⌈1/ǫ⌉, and explain how it can be implemented by solvinganappropriatematrixsumproblem. InSection4, wee xplainhowthe matrix sumproblemcan besolved eﬃciently (for constant k) usingLenstra’s algorithm for

Question 284 multiple-choice

Quantum error-correcting codes play a vital role in protecting information in physical systems that exhibit symmetries, such as charge conservation. Advanced constructions use continuous degrees of freedom and principles from statistical mechanics to enhance error correction and robustness against noise. Which statement correctly describes the behavior of thermodynamic quantum codes as the number of subsystems n approaches infinity? 1) The global state becomes completely mixed, and local subsystems lose all thermal properties. 2) The local temperature of subsystems diverges, leading to increased distinguishability between code words. 3) The worst-case error for erasure scenarios grows exponentially with n. 4) Code words become perfectly distinguishable by local measurements. 5) The code loses all covariance under symmetry operations for large n. 6) The local reduced states become pure, and temperature differences increase. 7) The temperature difference between code words vanishes, leading to local indistinguishability and imperfect correctability of erasure.

✓ Correct Answer:

The correct answer is 7) The temperature difference between code words vanishes, leading to local indistinguishability and imperfect correctability of erasure..

📚 Reference Text:

states, which can be done viaa sharp or a smooth cutoff as in the ½½3;1;2/C138/C138 Zcode. We perform the latter using a cyclically symmetric Gaussian envelope with spread w, prepending exp ½−ð1=4w2Þðj2þ k2þl2þm2þn2Þ/C138to the tensor Tð∞Þ jklmn in Eq. (43), and then normalizing the code words. The resulting code is covariant with respect to a Uð1Þsymmetry generated by the total physical charge TA¼P5 i¼1TAi, analogous to the three-rotor code (40). With the addition of the envelope, the resulting tensor becomes approximately perfect. This rotor version can be stacked to form an approximately error- correcting Uð1Þ-covariant holographic code in the same way as the qubit perfect tensors are connected in Ref. [7]. One can apply the certification criteria to this code to yield the same scaling as for the three-rotor code (41) for the model of a single erasure (see Supplemental Material,Sec. D. 2 [37] for details): ϵ worstðN1erasure∘EðwÞÞ≲1ﬃﬃﬃﬃﬃﬃﬃﬃ 160ph w: ð46Þ However, this code is capable of correcting any single- subsystem error, so it can correct for known erasure of anytwo subsystems. Calculating the bound for the noisechannel Nconsisting of erasure of any two sites with equal probability yields the same scaling: ϵ worstðN2erasures∘EðwÞÞ≲1ﬃﬃﬃﬃﬃ 60ph w: ð47Þ The larger coefficient is sensible, since a code approx- imately correcting at most two erasures should be better atcorrecting only one. In both cases, there are additional corrections of the order of Oðhe −cw2Þforc> 0arising from a detailed application of our criterion. C. Thermodynamic codes for n→∞ We now investigate a class of covariant codes in the limit where the number of subsystems ngrows large. We exploit the codes developed in Ref. [8], relevant for quantum computing with atomic ensembles [54]. For these codes, the basis vectors for the code space can be chosen to be energy eigenstates of a many-body system,with the property that the reduced state on a subsystemappears to be thermal with a nonzero temperature; wetherefore call them thermodynamic codes . This thermal behavior of local subsystems is expected for closed quantum systems that satisfy the eigenstate thermalizationhypothesis [55] or dynamical typicality [56,57] (states exhibiting many-body localization or the eigenstate ther-malization hypothesis have been prepared experimentally[58]). Energy eigenstates with slightly different values of the total energy also have slightly different values of thePHILIPPE FAIST et al. PHYS. REV . X 10,041018 (2020) 041018-14 locally measurable temperature; thus, the identity of a code word is imperfectly hidden from a local observer, and, therefore, erasure of a subsystem is imperfectly correctable. Consider a many-body system, such as a one- dimensional spin chain, and pick out two global energylevels jEi AandjE0iAin the middle of the spectrum, with a given energy difference ΔE¼E0−E. Assume, in the spirit of the eigenstate thermalization hypothesis, that the reduced states of both jEiAand jE0iAon each individual system Aiare approximately thermal. The corresponding temperature scales as T∝E=n, since the temperature is an intensive thermodynamic variable. Then, the temperaturedifference vanishes for n→∞, and the resulting reduced thermal states for these two states are very close. Intuitively, this result means that

Question 285 multiple-choice

In quantum hypothesis testing, the ability to distinguish between different quantum states depends heavily on the choice of measurement scheme and the structure of the state space, especially as system size increases. Representation theory and information-theoretic quantities such as entropy play critical roles in optimizing state discrimination protocols. Which property accurately characterizes the limitation of permutation-invariant Positive Operator-Valued Measures (POVMs) for distinguishing entangled from separable states in high-dimensional quantum systems? 1) They guarantee exact identification of all entangled states for large n. 2) Their effectiveness fundamentally decreases, making it impossible to reliably distinguish entangled from separable states as n grows. 3) They perform optimally only when measuring pure states in low dimensions. 4) Their success rate improves exponentially with the number of subsystems. 5) They yield tight bounds only when irreducible subspaces are ignored. 6) Their distinguishability power is independent of system size or dimensionality. 7) They can be made optimal by ignoring entropy-related terms.

✓ Correct Answer:

The correct answer is 2) Their effectiveness fundamentally decreases, making it impossible to reliably distinguish entangled from separable states as n grows..

📚 Reference Text:

by replacing εwithεn=√ 1α/radicalBig 1 nlog(ν)+d2+√n nlog(2n). Remark 3. By linearity of the trace, this implies that we can distingui sh the hypotheses conv({ρsn}sn∈Sn) andσ⊗nat the same rate. The proof of Theorem 4 reveals an interesting Lemma, which is a gene ralization of the well-known concentration theorem for spectral estimation [44]. We state it he re exclusively for sake of completeness: Lemma 2. Letρsn:=⊗n i=1ρsifor a ﬁnite set {ρs}s∈Sof states. Let ¯ρ:=/summationtext s1 nN(s|sn)ρsand let λ∈Yd,n. Then tr{Pλρsn} ≤(2n)|S|+d22−n(D(¯λ/bardblspec(¯ρ)). (36) Our last result is of negative character: 8 Theorem 5. Letn∈N,ε≥0and a permutation invariant operator A∈ B((Cd)⊗n)which is bounded by0≤An≤ /BD⊗nbe given. Then min ρ∈S(Cd)tr{Anρ⊗n} ≥1−ε⇒An≥(1−ε·(2dn)4d2) /BD⊗n(37) and, by applying this result to ( /BD⊗n−A)instead of A: max ρ∈S(Cd)tr{Aρ⊗n} ≤ε⇒An≤ε·(2dn)4d2/BD⊗n. (38) Remark 4. A look at the proof of this theorem reveals that the decomposi tionVλ=⊕fVf,λis still not suﬃciently ﬁne for proving the theorem, indicating that the re is some more relevant information to be gained by using POVMs that employ projections onto irreduci ble subspaces rather than the (generally reducible) ones that we employ. An even stronger method of ty pes is thus still a possibility. An operational implication of Theorem 5 result is that any su ch sequence ((An, /BD⊗n−An))n∈Nof POVMs is useless for distinguishing between separable states and entangled states on (Cd)⊗n, for large values of n: In the light of our Theorem 4, and since S(Cd) = conv( S(Cd)), we see that any measurement scheme that successfully distinguishes between the hypotheses “s ome ﬁxed entangled state” versus “any separable state” has probability of successfully detecting the secon d hypothesis converging to one only polynomially fast, or it cannot be permutation invariant. We arrive at our two main results by exploiting the simple observation t hat every frequency typical subspace Vfcarries a representation of the symmetric group Snand can therefore be decomposed into irreducible subspaces of Sn. V Proofs Proof of Theorem 2. This proof is split into several subsections. We start with the error exponent. Proof that we have the correct error exponent in Theorem 2. For a projection Pf,λwe have - including the case that td−k,...,td= 0 for some k >0 through the convention 2−∞= 0 and the case that dim( Fλ)>0 but tr{Pf,λ}= 0 that can occur if Vfdoes not carry any of the representations corresponding to λ- tr{Pf,λσ⊗n} ≤mλ,ndim(Fλ)2n/summationtextd i=11 nfilog(ti)(39) ≤mλ,n2n(H(λ)+/summationtextd i=11 nfilogti). (40) If, for some s∈S, both/bardbl¯λ−rs/bardbl ≤εand/bardblf−˜rs/bardbl ≤ε, then equation (39) leads to tr{Pf,λσ⊗n} ≤mλ,n2n(H(rs)+ε|log(ε/d)|+/summationtextd i=1˜rs(i)logti+dε·maxi|logti|). (41) But for an orthonormal eigenbasis {es i}d i=1ofρswe have ˜ rs(i) =/summationtext j|/an}bracketle{t˜ei,es j/an}bracketri}ht|2rs(j) and so H(r)+d/summationdisplay i=1˜rs(i)logti=−D(ρs/bardblσ), (42) and for the projection Pnthis means that tr{Pnσ⊗n}=/summationdisplay (f,λ)∈Λεtr{Pf,λσ⊗n} (43) ≤(2n)d2max (f,λ)∈Λεtr{Pf,λσ⊗n} (44) ≤(2n)d2sup s∈S2−n(D(ρs/bardblσ)−ε|log(ε/d)|−dε·maxi|logti|)(45) = 2−n(infs∈SD(ρs/bardblσ)−d2 nlog(2n)−ε|log(ε/d)|−dε·maxi|logti|)(46) = 2−n(infs∈SD(ρs/bardblσ)−Θ(n,ε,d,σ)), (47) 9 which is what we wanted to show. Proof that we can identify every state in our given set with hi gh probability. For every s∈S, deﬁne the projections Pn,s:=/summationtext f:/bardbl¯f−˜rs/bardbl≤ε/summationtext λ:/bardbl¯λ−rs/bardbl≤εPf,λ. These satisfy the lower bound ∀s∈S, P n≥Pn,s. (48) Another inequality we will need is an upper bound that is

Question 286 multiple-choice

In quantum chemistry, constructing accurate wavefunctions for many-electron systems requires careful consideration of both spin and spatial symmetries, often utilizing mathematical tools from group theory. Advanced methods seek to optimize these aspects while maintaining physically meaningful interpretations and compliance with fundamental theorems. Which characteristic uniquely distinguishes the Spin-Coupling Optimized Generalized-Independent Particle (SOGI) method from the Generalized-Independent Particle (GI) method in the construction of many-electron wavefunctions? 1) It uses only localized atomic orbitals for all electrons. 2) It excludes the use of self-consistent field approximations. 3) It relies exclusively on delocalized molecular orbitals. 4) It enforces antisymmetry through direct determinant construction without projection operators. 5) It disregards spin coupling for electrons in its formulation. 6) It restricts its wavefunctions to those violating the Pauli principle. 7) It systematically removes arbitrariness in the choice of spin functions by optimizing spin-coupling using orthogonal Wigner projection operators based on Young tableaux.

✓ Correct Answer:

The correct answer is 7) It systematically removes arbitrariness in the choice of spin functions by optimizing spin-coupling using orthogonal Wigner projection operators based on Young tableaux..

📚 Reference Text:

:2:=4.0°, f: 2:=10°. Distance is in atomic units: the standard spin functions. The exact wavefunctions can again be written as -qrexaet = G;I'Li.L,exaetX• (4) We now consider approximate wavefunctions of the form (5) and select i, L, and q,i,L,product such that the energy is ~he lowes~ possi~le, which leads to a new set of coupled mtegro-drfferential equations analogous to (3), R,.,.iLc/>k=EI!/>k k=l, •• ·N (6) f?r the best orbitals. Since this approach is a generaliza tion of the GI method, which eliminates the arbitrari ness in choice of spin function and yet retains the independent particle interpretation, we will refer to it as the spin-coupling optimized GI or SOGI method. We will show that the SOGI wavefunction referred to in (5) as 'lFarL, is the best possible wa~efunction which can be interpreted in terms of N spatial orbitals as if an electron were moving in each orbital in the self-consistent field due to the electrons in the N -1 other orbitals. In addition, we will show that the SOGI wavefunction has all the general properties of GI wave fun.ctions; e.g., the virial, Hellmann-Feynman, Bril loum, and Koopmans theorems still apply5 (for a fixed spin function). We also show that the SOGI method can be considered as the synthesis of the valence bond and Hartree--Fock methods. 6 W.:A. Goddard rn1 J. Chern. Phys. 48, 5337 (1968). _Iz: Sec. II we derive the general equations for deter mrmng. the SOGI orbitals a~d consider some aspects of spatial symmetry restrictiOns. In order to demon ~trate various aspects of the SOGI method we report Ill Sec. III the results of SOGI calculations on some three-and four-electron atoms and molecules. II. THE SPIN-COUPLING OPTIMIZED Gl METHOD A. The SOGI Equations The G;11 operators were defined in Paper equation I by the Gl= ",.., O,·"w--il .L.... ~ n.,.. ' r1. ' (7) r where Ori" and Wriii are orthogonal Wigner projection operators3 based on Young's orthogonal representation of SN, and t>.r~ is the parity of the permutation ;\. . which changes the ith standard Young tableau S;" in;~ the rth standard Young tableau Sr". For a spin-inde pendent Hamiltonian H the energy is given by Eq. ( 15) of Paper I E= (G;~'x I HI G;"x)/ (G;~'x I G;"x) =( I HI o;;~•)/( I 0;/') (8) which results from the fact that the orthogonal units obey the following equationsi.3•4: and (0;/'t I 2) = (I l Oj/'z), (wn;n"XI I x2) = (xt I w;;.n;"xz). The orthogonal units are defined byl,3,4 Ori''= (j~'/N!) L Uri(•l"f, TN (9a) (9b) (lOa) (lOb) (11a) (11b) where the U~'(T) are orthogonal matrices which yield the JLth irreducible representation of SN. Picking a set of representative matrices is equivalent to picking a set of orthogonal (not necessarily normalized) basis vectors in the !~'-dimensional space of spin functions. A convenient way of constructing the U~''s is described in Paper I and involves the use of Young tableaux. The arbitrariness in the GI method results from choos ing our one

Question 287 multiple-choice

Noether's problem in field theory explores whether the fixed field of a finite group acting on a rational function field is itself rational, with significant implications for Galois theory and algebraic geometry. Recent advances involve using cohomological invariants to determine rationality or stable rationality of these fixed fields. For a finite group $G$ with a normal subgroup $N$ such that $G/N \simeq C_{2^n}$ for $n \geq 3$, and a field $k$ of characteristic zero where the extension $k(\zeta_{2^n})/k$ is not cyclic, which statement accurately describes the rationality of the fixed field $k$? 1) $k$ is always rational over $k$ for any such group $G$ 2) $k$ is stably rational only if $G$ is abelian 3) The rationality of $k$ depends solely on the characteristic of $k$ 4) $k$ is not stably rational or retract rational over $k$ under these conditions 5) $k$ is stably rational if $G/N$ is non-cyclic 6) $k$ is always rational if $k$ contains all roots of unity 7) The existence of a normal subgroup $N$ guarantees rationality of $k$

✓ Correct Answer:

The correct answer is 4) $k$ is not stably rational or retract rational over $k$ under these conditions.

📚 Reference Text:

Title: An application of cohomological invariants Year: 2019 Paper ID: fc9de7bc38d0535878667025d6663595a570b78e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/fc9de7bc38d0535878667025d6663595a570b78e Abstract: Let $G$ be a finite group, $k$ be a field and $G\to GL(V_{\rm reg})$ be the regular representation of $G$ over $k$. Then $G$ acts naturally on the rational function field $k(V_{\rm reg})$ by $k$-automorphisms. Define $k(G)$ to be the fixed field $k(V_{\rm reg})^G$. Noether's problem asks whether $k(G)$ is rational (resp. stably rational) over $k$. When $k=\bQ$ and $G$ contains a normal subgroup $N$ with $G/H\simeq C_8$ (the cyclic group of order $8$), Jack Sonn proves that $\bQ(G)$ is not stably rational over $\bQ$, which is a non-abelian extension of a theorem of Endo-Miyata, Voskresenskii, Lenstra and Saltman for the abelian Noether's problem $\bQ(C_8)$. Using the method of cohomological invariants, we are able to generalize Sonn's theorem as follows. Theorem. Let $G$ be a finite group and $N$ $\lhd$ $G$ such that $G/N\simeq C_{2^n}$ with $n\geq 3$. If $k$ is a field satisfying that ${\rm char}\,k=0$ and $k(\zeta_{2^n})/k$ is not a cyclic extension where $\zeta_{2^n}$ is a primitive $2^n$-th root of unity, then $k(G)$ is not stably rational (resp. not retract rational) over $k$. \end{abstract}

Question 288 multiple-choice

Quantum max-cut problems employ swap algebra and semidefinite programming to optimize quantum Hamiltonians, leveraging advanced mathematical tools such as pseudomoment matrices and Gröbner bases. These approaches have demonstrated surprising accuracy for small graphs, notably surpassing classical relaxation techniques. In quantum max-cut optimization using swap algebra, which step enables the conversion of degree-4 swap monomial expressions into a form suitable for semidefinite programming? 1) Symmetrizing the swap operators across all vertices 2) Diagonalizing the quantum max-cut Hamiltonian directly 3) Restricting the basis to only degree-2 swap monomials 4) Introducing new variables for each degree-4 monomial to populate the pseudomoment matrix 5) Applying the standard commutative Gröbner basis algorithm to the swap operators 6) Maximizing the trace of the swap basis matrix 7) Replacing all swap operators with classical adjacency matrices

✓ Correct Answer:

The correct answer is 4) Introducing new variables for each degree-4 monomial to populate the pseudomoment matrix.

📚 Reference Text:

Swap algebra We continue our discussion of relaxation by describing the second relaxation. This requires sets of monomials B3,B4(to be described later in Appendices B.1 and B.2 which are degree 3 and 4 analogs of B2. In order to build the second relaxation, we replace V2(n) by the vector ⃗B2consisting of the swap basis B2determined in Proposition 4.12. The corresponding symbolic pseu- domoment matrix ⃗B2⃗B∗ 2will contain products of up to four swaps. To form the SDPs in Eq. (4.5) or Eq. (4.10) we need to understand the linear space spanned by all products of up to four Swap matrices, a topic we describe in some detail below in Appendix B. Algorithm 1: 2nd relaxation of the quantum max-cut Input : GraphG= (V,E) onnvertices Output : Solution to the 2nd relaxation of quantum max-cut for hG. Alternately, “upper bound on the minimum eigenvalue of the QMC Hamiltonian” Form⃗B2; Form the symbolic pseudomoment matrix M2:=⃗B2⃗B∗ 2; Express each entry in M2as a linear combination of elements of B4(see Appendix B.2) to obtain M′ 2; Replace each distinct term appearing in M′ 2with a new (scalar) variable; call the resulting matrix M′′ 2; Solve the SDPν2(hG) = sup{⟨M′′ 2,ΓG⟩|M′′ 2⪰0,(M′′ 2)1,1= 1}, (4.22) where ΓGis any matrix satisfying hG=⃗B∗ 2ΓG⃗B2; returnν2(hG) 4.3.4 Swap relaxation behaves well in experiments We implemented Algorithm 1 in Mathematica (using the out-of-the-box semidefinite optimization module in Mathematica) on all graphs with ≤8 vertices: Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 39 Proposition 4.14. Forn≤8the 2nd relaxed value of an nvertex Quantum Max Cut Hamilton with uniform edge weights is up to the tolerance of 10−7equal to the max eigenvalue of that Hamiltonian. It would be interesting to find the smallest graph on which the second relaxation is not exact. Remark 4.15. Proposition 4.14 is surprising when compared to classical (commutative) max cut relaxation which performs much worse. For example, the second Lasserre relaxation is not exact for the 5 cycle. Also, while we do not give comparative statistics, the first classical relaxation is worse than the one for swaps, e.g., even the triangle is classically not first Lasserre exact. 4.3.5 Segue Since, clearly, it is not feasible to find linear algebra bases for linear spaces spanned by products of d>4 swaps by hand, we turn to Gr¨ obner bases in next. 5 Gr ¨obner Bases This section gives a few observations about Gr¨ obner Bases (GB) with an eye toward computation. 5.1 Our Gr ¨obner Basis Set Up Our presentation is aimed at readers who have a basic familiarity with GBs; a standard reference in the commutative setting is [ CLO15 ], while [ Mor86 ,Gre00 ,Xiu12 ] describe the appropriate nc analogs. Noncommutative GBs have properties similar to those of commutative GB with the dramatic exception that a noncommutative GB might not be finite. However, we start with some conceptual motivation to provide a little context for readers who are unfamiliar with GBs. 5.1.1 Motivation for why we need Gr ¨obner generators

Question 289 multiple-choice

Quantum algorithms have shown remarkable efficiency for certain problems framed as hidden subgroup problems (HSP), especially in abelian groups. However, for non-abelian group problems such as graph isomorphism, extracting useful information often requires complex operations on quantum states. What is the minimum number of coset states on which entangled quantum measurements must be performed to extract meaningful information for solving the graph isomorphism problem using the HSP approach? 1) Ω(n log n) 2) O(1) 3) O(log n) 4) Θ(n) 5) Ω(n^2) 6) O(n) 7) Θ(1)

✓ Correct Answer:

The correct answer is 1) Ω(n log n).

📚 Reference Text:

Title: Limitations of quantum coset states for graph isomorphism Year: 2005 Paper ID: 15261eada627f7c8cf3db8f0dc0821e2e7f6dc63 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/15261eada627f7c8cf3db8f0dc0821e2e7f6dc63 Abstract: It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard it is to use these states to solve graph isomorphism. It was recently shown by Moore, Russell, and Schulman [30] that only an exponentially small amount of information is available from one, or a pair of coset states. A potential source of power to exploit are entangled quantum measurements that act jointly on many states at once. We show that entangled quantum measurements on at least Ω(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound. This may be viewed as a negative result because highly entangled measurements seem hard to implement in general. Our main theorem is very general and also rules out using joint measurements on few coset states for some other groups, such as GL(n,Fpm) and Gn where G is finite and satisfies a suitable property.

Question 290 multiple-choice

Quantum computing has progressed rapidly in recent years, with public cloud platforms enabling remote access to quantum processors for experiments and research. This increased accessibility has fostered innovation in key quantum information protocols and technologies. Which of the following protocols is essential for mitigating the high error rates found in early-stage quantum hardware and for scalable quantum computation? 1) Quantum Error Correction 2) Quantum Supremacy 3) Quantum Teleportation 4) Quantum Key Distribution 5) Quantum Annealing 6) Quantum Machine Learning 7) Quantum Random Number Generation

✓ Correct Answer:

The correct answer is 1) Quantum Error Correction.

📚 Reference Text:

Title: Performing Quantum Computing Experiments in the Cloud Year: 2016 Paper ID: 41a29af06c54f2419cda1fe3dd3d633804b522a7 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/41a29af06c54f2419cda1fe3dd3d633804b522a7 Abstract: Quantum computing technology has reached a second renaissance in the past five years. Increased interest from both the private and public sector combined with extraordinary theoretical and experimental progress has solidified this technology as a major advancement in the 21st century. As anticipated by many, the first realisation of quantum computing technology would occur over the cloud, with users logging onto dedicated hardware over the classical internet. Recently IBM has released the {\em Quantum Experience} which allows users to access a five qubit quantum processor. In this paper we take advantage of this online availability of actual quantum hardware and present four quantum information experiments that have never been demonstrated before. We utilise the IBM chip to realise protocols in Quantum Error Correction, Quantum Arithmetic, Quantum graph theory and Fault-tolerant quantum computation, by accessing the device remotely through the cloud. While the results are subject to significant noise, the correct results are returned from the chip. This demonstrates the power of experimental groups opening up their technology to a wider audience and will hopefully allow for the next stage development in quantum information technology.

Question 291 multiple-choice

In quantum computing, amplitude estimation is a key technique for estimating the probability of specific measurement outcomes, often used in algorithms for finance, chemistry, and optimization. Non-destructive amplitude estimation enables repeated use of quantum states without disturbing them, improving resource efficiency. Which of the following statements accurately describes the resource usage for non-destructive amplitude estimation using controlled applications of unitary reflections in an ancilla register of n qubits, targeting an error tolerance δ? 1) It requires O(n log(1/δ)) controlled applications of unitary reflections. 2) The procedure uses O(log(n) log(1/δ)) controlled operations. 3) It requires O(2ⁿ) controlled applications, independent of δ. 4) The resource usage is O(2ⁿ log(1/δ)) controlled applications of the relevant unitary reflections. 5) The number of required operations scales as O(δ log(2ⁿ)). 6) It uses O(log(1/δ) / 2ⁿ) controlled operations. 7) The complexity is O(2ⁿ + δ) controlled applications.

✓ Correct Answer:

The correct answer is 4) The resource usage is O(2ⁿ log(1/δ)) controlled applications of the relevant unitary reflections..

📚 Reference Text:

approximation of√xcould be constructed from Corollary 66 of [GSLW18]. We leave a careful error analysis of a direct estimate of ato future work. Corollary 20. Non-destructive amplitude estimation. SayΠis a projector and RΠ is a unitary that reﬂects about this projector: RΠ:= 2Π−I (213) Let|Ψ/angbracketrightbe some quantum state such that a:=|Π|Ψ/angbracketright|. LetM:=ﬂoor(a22n). Then for any positive integer nand anyδ > 0there exists a quantum channel that implements δ-approximately in diamond norm the map: |0n/angbracketright/angbracketleft0n|⊗|Ψ/angbracketright/angbracketleftΨ|→(p|M/angbracketright/angbracketleftM|+ (1−p)|M−1mod 2n/angbracketright/angbracketleftM−1mod 2n|)⊗|Ψ/angbracketright/angbracketleftΨ| (214) for some probability p(i.e., there is some probability that instead of obtaining ﬂoor (a22n) we obtain ﬂoor (a22n)−1). This channel uses O/parenleftbig2nlog(δ−1)/parenrightbigcontrolled applications of RΠandR|Ψ/angbracketright:= 2|Ψ/angbracketright/angbracketleftΨ|−I. Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 44 Proof.We use linear combinations of unitaries to make block-encodings of Πand|Ψ/angbracketright/angbracketleftΨ|. Π =I+RΠ 2,|Ψ/angbracketright/angbracketleftΨ|=I+R|Ψ/angbracketright 2(215) Then, we multiply these projectors together to make a block-encoding of A: A:=|Ψ/angbracketright/angbracketleftΨ|·Π·|Ψ/angbracketright/angbracketleftΨ|=a2|Ψ/angbracketright/angbracketleftΨ| (216) Now we simply invoke the algorithm described in Corollary 16 for the case when no rounding promise is present, and apply it to the input state |Ψ/angbracketright. Since|Ψ/angbracketrightis an eigenstate ofA, we are guaranteed to measure either ﬂoor (a22n−1)or ﬂoor (a22n−1)−1mod 2n. Observe that the random error only occurs for particular values of a2, which is where the rounding promise is violated. We can ignore the rounding promise precisely because the input state|Ψ/angbracketrightis an eigenstate of the Hamiltonian: an incorrect estimate does not damage the input state. 6 Acknowledgments The author thanks Scott Aaronson, Andrew Tan, Pawel Wocjan, William Zeng, Yosi Atia, András Gilyén, Philip Jensen, Lasse Bjørn Kristensen and Alán Aspuru-Guzik for helpful conversations. This work was supported by Scott Aaronson’s Vannevar Bush Faculty Fellowship from the US Department of Defense. References [WT21] Pawel Wocjan, Kristan Temme Szegedy Walk Unitaries for Quantum Maps arXiv:2107.07365 (2021) [MRTC21] John M. Martyn, Zane M. Rossi, Andrew K. Tan, Isaac L. Chuang A Grand Uniﬁcation of Quantum Algorithms arXiv:2105.02859 (2021) [LT21] Lin Lin, Yu Tong Heisenberg-limited ground state energy estimation for early fault-tolerant quantum computers arXiv:2102.11340 (2021) [Cam20] Earl T. Campbell Early fault-tolerant simulations of the Hubbard model arXiv:2012.09238 (2020) [SHC20] Yuan Su, Hsin-Yuan Huang, Earl T. Campbell Nearly tight Trotterization of interacting electrons arXiv:2012.09194 Quantum 5, 495 (2020) [ESP20] Alexander Engel, Graeme Smith, Scott E. Parker A framework for applying quan- tum computation to nonlinear dynamical systems arXiv:2012.06681 Physics of Plasmas 28, 062305 (2020) [An&20] DongAn, NoahLinden, Jin-PengLiu, AshleyMontanaro, ChangpengShao, Jiasu WangQuantum-accelerated multilevel Monte Carlo methods for stochastic diﬀerential equations in mathematical ﬁnance arXiv:2012.06283 Quantum 5, 481 (2020) [Chuang20] Isaac Chuang. Grand uniﬁcation of quantum algorithms. Seminar presentation at IQC Waterloo. (2020) [Wright&20] Lewis Wright, Fergus Barratt, James Dborin, George H. Booth, Andrew G. GreenAutomatic Post-selection by Ancillae Thermalisation arXiv:2010.04173 Phys. Rev. Research 3, 033151 (2020) Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 45 [AHNTW20] Srinivasan Arunachalam, Vojtech Havlicek, Giacomo Nannicini, Kristan Temme, Pawel Wocjan Simpler (classical) and faster (quantum) algorithms for Gibbs partition functions arXiv:2009.11270 (2020) [GST20] András Gilyén, Zhao Song, Ewin Tang An improved quantum-inspired algorithm for linear regression arXiv:2009.07268 (2020)

Question 292 multiple-choice

In quantum algorithms addressing the Hidden Subgroup Problem, states are often analyzed in the Fourier basis, where group-theoretic structures such as irreducible representations and subgroup projectors play a crucial role. Understanding the optimal quantum measurements for distinguishing hidden subgroups relies on principles from representation theory and quantum measurement theory. When constructing optimal Positive Operator-Valued Measures (POVMs) in the Fourier basis to distinguish hidden subgroup states of a finite group, which validity condition must be satisfied by the coefficients $ c_{\mu, H} $ in the measurement operators $ E_H $? 1) They must be strictly positive and equal for all subgroups. 2) They must be real numbers summing to zero within each irrep. 3) They must form a unitary matrix across all irreps and subgroups. 4) They must vanish for non-trivial conjugacy classes. 5) They must be non-negative and, when summed over all subgroups, equal the identity operator within each irrep. 6) They must be proportional to the character values of the group elements. 7) They must be invariant under subgroup intersections.

✓ Correct Answer:

The correct answer is 5) They must be non-negative and, when summed over all subgroups, equal the identity operator within each irrep..

📚 Reference Text:

expressing the hidden subgroup state in terms of the right reg ular representation is helpful, it is even more helpful to examine this state in a diﬀerent basis. If we perform the q uantum Fourier transform over the ﬁnite group G, this transforms the basis of group elements {|g/an}bracketri}ht}to a basis with irreducible representation (irrep) labels along with row and column registers, {|µ,i,j/an}bracketri}ht}. In this basis the hidden subgroup state is ρH=/circleplusdisplay µ/bracketleftBigg Idµ⊗1 |G|/summationdisplay h∈HDµ(h)/bracketrightBigg , (14) whereµlabels the irreducible representations of G,Dµis theµth irrep, dµis the dimension of the µth irrep, and Idµ is thedµdimensional identity operator. Deﬁning the subgroup projectors PHfor theµth irrep, Pµ,H:=1 |H|/summationdisplay h∈HDµ(h), (15) we can express the hidden subgroup state as ρH=/circleplusdisplay µ/bracketleftbigg Idµ⊗|H| |G|Pµ,H/bracketrightbigg . (16) A ﬁnal useful symmetry relationship that we will use later comes abo ut when we sum the projectors Pµ,Hover all conjugate subgroups. A conjugate subgroup of His deﬁned as Hg:={ghg−1,h∈ H}forg∈ G. In particular we recall that/summationtext g∈GDµ(ghg−1) commutes with all Dµ(g′) and thus, via Schur’s lemma, it must be proportional to the identity matrix (this proportionality may be zero), /summationdisplay g∈GDµ(ghg−1) =rµ(h)Idµ, (17) whererµ(h) is the proportionality constant. Taking the trace of this equation allows us to calculate rµ(h): /summationdisplay g∈Gχµ(ghg−1) =|G|χµ(h) =rµ(h)dµ (18) whereχµ(g) := Tr[Dµ(g)] is the character of element gin irrepµ. Thus we obtain 1 |G|/summationdisplay g∈GPµ,Hg=1 |H|/summationdisplay h∈Hχµ(h) dµIdµ. (19) B. The Optimal Single Copy Measurement The optimal measurement criteria, Eq. (5), when all of the subgro ups are given with equal a prior probability is given by /summationdisplay H∈Sub(G)ρHEH=/summationdisplay H∈Sub(G)EHρH /summationdisplay H∈Sub(G)ρHEH≥ρH′,for allH′∈Sub(G), (20) 5 where this sum over His over all possible subgroups of G. Notice that if ρHandEHcommute, then the ﬁrst equation is satisﬁed. This will be true of the measurements we will consider, so the second equation will be the nontrivial condition. Our ansatz for the optimal measurement is a measurement of the f orm EH=/circleplusdisplay µ/bracketleftbig Idµ⊗cµ,HPµ,H/bracketrightbig . (21) wherecµ,Hare as of yet undetermined constants. The reason for beginning w ith this assumption is that it can easily be shown that the previous results of Ip [15] and Moore and Russell [18] can be written as measurements of this form. Notice that this choice of measurement implies that ρHandEHcommute so the ﬁrst optimal measurement criteria is automatically satisﬁed. Thus we must show that the second measu rement criteria is satisﬁed, while the condition that the measurement is valid also holds. The EHof our chosen form are a valid measurement if and only if cµ,H≥0 and, for all µ, /summationdisplay H∈Sub(G)cµ,HPµ,H=Idµ. (22) It will be convenient to break the sum over all subgroups that appe ars here down into a sum over conjugate subgroups. In particular two subgroups H1andH2are conjugate if there exists an element, gofG, such that H2={ghg−1,h∈ H1}. Whether two subgroups are conjugate to each other forms an e quivalence relation and thus we can partition the set of all subgroups of Ginto sets of conjugate subgroups. Let C(G) be the set of all of

Question 293 multiple-choice

In the study of abelian varieties over non-archimedean fields, the structure and properties of the ring of theta functions play a central role, especially in the context of projective embeddings and the analysis of group morphisms. Techniques such as induction, decomposition theorems, and isogeny arguments are vital for understanding the relationships between algebraic objects in this setting. Which concept is essential for proving that the projective spectrum A = Proj. R is an abelian variety of dimension g over a complete non-archimedean field k, given the ring R constructed from Laurent series over k? 1) The existence of a unique maximal ideal in R 2) The construction of R via Laurent series and the use of parity decomposition for theta functions 3) The requirement that all matrix coefficients are algebraic integers 4) The surjectivity of the valuation function ord restricted to the unit group U 5) The irreducibility of the valuation ring Θ 6) The rationality of ord. s/tJ for all elements in R 7) The compactness of the group Gg under the discrete topology

✓ Correct Answer:

The correct answer is 2) The construction of R via Laurent series and the use of parity decomposition for theta functions.

📚 Reference Text:

onϊ. In § II. 3 we use an implicit function type argument to show that ψ is 1-1 and that all unit points are in the image of φ. The proof that any PeAk is in φ(Gg) is by induction on the cardinality of S(P). The key steps in the induction are an addition formula on A, (Theorem II. 4.6), and the "decomposition theorem", (Theorem II. 6.6), whose proof depends on the study of the zeroes of a certain Laurent series ΘP. We return to the general case in part III. Using the diagonal case and an isogeny argument we show that φ is bijective, assuming only that each ord. s/tJ is rational. This mild restriction is unnecessary as Gerritzen's result show, but we have been unable to avoid it. Throughout this paper we use the following notation: k is a field complete with respect to a non-trivial, non-archimedean valuation, ord: fc* —> Reals is the associated order function, Θ, Jί and k are the valu- ation ring, maximal ideal and residue class field of the valuation. U is the unit group of Θ and Gg is the product of g copies of fc*. Part I is concerned with the definition and basic properties of the ring of theta functions R. It contains a proof that A = Proj. R is an abelian variety of dimension g over k. Most of this material can be found in the first three chapters of McCabe [2], but our arguments are somewhat simpler. §1.1. The ring of theta-functions A Laurent series over k is a formal sum ΣIeZg s^jX1, J/Z e k, which converges for all (x19 , xg) e Gg. (we shall use standard multivariable notation throughout. If / = (ix, , ig) then X1 means \[3 X*/). The Laurent series form a fe-algebra =£?. The subring of ££ consisting of https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press ABELIAN VARIETIES 67 series with J/7 e Θ is called «£?„,. S£ is a domain, and if an element of ££ vanishes on all of G^, each J/7 = 0. Suppose n>l and (r) = (r1? ,τg) e (Z/nZ)*7. An element 2] ^ iX1 of «£? is said to have n-parity (r) if j/7 = 0 unless each ij reduces to r3 mod. n. Let «£?(r) denote the sub- space of elements of Jδf having n-parity (r). Then we get a decomposi- tion £f = Θ(r) <^(r) the "n-parity decomposition of J2"\ Let (J/^) be a ^ x (jί symmetric matrix with entries in &* such that the associated matrix (ord. stfί3) is positive definite. Let Vj = (^!, ',s/jg), qό = J/^. If m > 0, Rm(s/tJ) (or just #J will denote the set of elements Θe& which satisfy the following func- tional relation: (P) θ(VjX) - qjmXj2mθ(X) j = 1,2, , g . Note that if 0(X) = 2] 6jZ7 € Rm and 7^ = []ί=i ^% it follows from the relation (P) that the &/s satisfy: = &/+2m5j 3 = 1,2,

Question 294 multiple-choice

In quantum information theory, the structure of quantum operations and the representation of random quantum states are central topics. The use of mathematical tools like permutation operators, Haar measure, vectorization, and Choi matrix formalism provides deep insights into symmetries and properties of quantum channels. Which property guarantees that the average of a function over all quantum states, when sampled according to the Haar measure, is independent of the initial state chosen? 1) The bijection property of vectorization 2) The uniform weighting of permutation coefficients in symmetric subspaces 3) The positivity of the Choi matrix for completely positive maps 4) The invariance of the Haar measure under right multiplication by unitaries 5) The ABC-rule for manipulating tensor products of operators 6) The singular value decomposition of superoperators 7) The preservation of the Hilbert-Schmidt inner product during vectorization

✓ Correct Answer:

The correct answer is 4) The invariance of the Haar measure under right multiplication by unitaries.

📚 Reference Text:

π∈SkcπVd(σ−1π) =/summationdisplay π∈SkcσπVd(π). (98) Therefore, we have: M(k) µH/parenleftig (|ϕ⟩⟨ϕ|)⊗k/parenrightig =/summationdisplay π∈SkcσπVd(π). (99) Assumingd > 1, the previous equation implies that all the coefficients must be the same, i.e., cσ=cIfor allσ∈Sk(note that this can be shown without using linear independence of the 13 permutation matrices, but just using that the permutation matrices are all different linear operators ford>13). Hence, we have: M(k) µH/parenleftig (|ϕ⟩⟨ϕ|)⊗k/parenrightig =cI/parenleftigg/summationdisplay π∈SkVd(π)/parenrightigg =cIk!P(d,k) sym. (100) Thus, taking the trace, we get: cI=1 k! Tr/parenleftig P(d,k) sym/parenrightig=1 k!1/parenleftbigk+d−1 k/parenrightbig. (101) Definition 23 (Haar measure on states) .Given a state|ϕ⟩inCd, we denote E |ψ⟩∼µH/bracketleftig |ψ⟩⟨ψ|⊗k/bracketrightig :=E U∼µH/bracketleftig U⊗k|ϕ⟩⟨ϕ|⊗kU†⊗k/bracketrightig . (102) Note that the right invariance of the Haar measure implies that the definition of E |ψ⟩∼µH/bracketleftig |ψ⟩⟨ψ|⊗k/bracketrightig does not depend on the choice of |ϕ⟩. Moreover, due to Theorem 22, we have: E |ψ⟩∼µH/bracketleftig |ψ⟩⟨ψ|⊗k/bracketrightig =P(d,k) sym Tr/parenleftig P(d,k) sym/parenrightig. (103) 5 Vectorization formalism We define the linear operator vec :L/parenleftbig Cd/parenrightbig →(Cd)⊗2as follows: for all i,j∈[d], we set vec(|i⟩⟨j|):= |i⟩⊗|j⟩. This means that if A∈ L/parenleftbig Cd/parenrightbig is written as A=/summationtextd i,j=1Ai,j|i⟩⟨j|, then vec(A) =/summationtextd i,j=1Ai,j|i⟩⊗|j⟩. This justifies the name “vectorization”, as the linear map vec(·)sends the d×dmatrixAto thed2-dimensional vector vec(A). Importantly, vec(·)is a bijection, meaning that for all|v⟩∈(Cd)⊗2there exists a unique A∈L/parenleftbig Cd/parenrightbig such that|v⟩= vec(A)(in fact, one can takeA=/summationtextd i,j=1⟨i,j|v⟩|i⟩⟨j|) and vice versa. It is often convenient to express vec(A) in the form vec(A) =A⊗I|Ω⟩, where|Ω⟩:= vec(I) =/summationtextd i=1|i,i⟩is the vectorization of the identity matrix, i.e., the non-normalized maximally entangled state. To simplify the notation, we can use the shorthand |A⟩⟩:= vec(A). Another useful property is that the canonical inner product between two vectorized operators AandBis equal to their Hilbert-Schmidt scalar product, i.e., ⟨⟨A|B⟩⟩=⟨A,B⟩HS, where⟨⟨A|B⟩⟩:= vec(A)†vec(B)is the canonical inner product. Using the so-called transpose -trickA⊗I|Ω⟩=I⊗AT|Ω⟩, we have the ABC-rule : |ABC⟩⟩=A⊗CT|B⟩⟩, (104) for allA,B,C∈L/parenleftbig Cd/parenrightbig . Consider a linear superoperator Φ:L(Cd)→L(Cd). It is worth noting that the map |Φ(X)⟩⟩ is linear with respect to |X⟩⟩for all|X⟩⟩∈(Cd)⊗2. This linearity implies the existence of a matrix vec(Φ)∈L((Cd)⊗2)associated with this linear transformation. In other words, we can express the action of Φon an operator Xas follows: |Φ(X)⟩⟩= vec(Φ)|X⟩⟩for allX∈L(Cd). (105) It is important to mention that for all linear superoperators Φ :L(Cd)→L(Cd)there exists a set of matrices {Ai}d2 i=1,{Bi}d2 i=1∈L(Cd)such that Φ(X) =/summationtextd2 i=1AiXB† i. To see why, observe that: Φ(X) = Tr 2/parenleftbig I⊗XTρΦ/parenrightbig , (106) 3To see it explicitly, one can choose the permutation σ, which fixes all indices except for two, which are inter- changed. 14 whereρΦ:= Φ⊗I(|Ω⟩⟨Ω|) =/summationtextd i,j=1Φ(|i⟩⟨j|)⊗|i⟩⟨j|is the so-called Choi matrix of Φ,I:L(Cd)→ L(Cd)is the identity channel and Tr2indicates the partial trace with respect to the second tensor space. Now, performing a singular value decomposition (SVD) on ρΦ, we have: ρΦ=d2/summationdisplay i=1λi|ui⟩⟨vi|=d2/summationdisplay i=1|vec(Ai)⟩⟨vec(Bi)|=d2/summationdisplay i=1Ai⊗I|Ω⟩⟨Ω|B† i⊗I, (107) whereλi≥0,|ui⟩and|vi⟩are respectively the singular values and the column vectors associated to the SVD unitaries, and we defined Ai:=√λivec−1(|ui⟩)andBi:=√λivec−1(|vi⟩). Substituting this expression back into (106) , we obtain the desired claim. (Note that if Φis completely positive, then its Choi matrix is positive, so we can perform

Question 295 multiple-choice

In computational materials science, localized representations of electronic states, such as Wannier functions, are essential for analyzing chemical bonding and constructing electronic structure models. The efficiency and robustness of algorithms for orbital localization are critical when dealing with large molecular systems. Which of the following statements most accurately describes a key advantage of a two-stage orbital localization algorithm over traditional Wannier90 in large-scale electronic structure calculations? 1) It produces orbitals with significantly lower spread than Wannier90 in all tested systems. 2) It eliminates the need for overlap matrix calculations entirely, reducing memory requirements. 3) It always converges faster than any algorithm that uses parallel computing resources. 4) It achieves comparable orbital localization to Wannier90 but with substantially reduced sensitivity to the choice of initial guess and much faster runtime. 5) It guarantees exact eigenfunctions of the Kohn-Sham Hamiltonian for all molecular systems. 6) It requires manual tuning of parameters for each new system to avoid failure. 7) It can only be applied to systems with fewer than 100 molecules.

✓ Correct Answer:

The correct answer is 4) It achieves comparable orbital localization to Wannier90 but with substantially reduced sensitivity to the choice of initial guess and much faster runtime..

📚 Reference Text:

spreads of 0.47, 0.48, 0.55, and 0.55 A2, which matched the results when starting with a Wannier90 18 Table 4.2: Runtime for localization algorithms as applied 256 water molecules. Operation time (s) Matrix-matrix multiplication Q 47.831 Randomized version, Algorithm 3 14.024 Re nement step, Algorithm 4 78.361 Total cost of our two stage algorithm 140.22 Original algorithm, Algorithm 1 4496.1 Table 4.3: Final spread of orbitals for 256 water molecules and time to solution. Algorithm spread A2 time to solution (s) Algorithm 1 589.91 4496.1 Randomized version, Algorithm 3 636.60 14.024 Two stage procedure using Algorithms 3 and 4 589.97 140.22 Wannier90 550.20 1715.2 generated random initial guess. S p r e a d 8A20.45 0.5 0.55 0.6 0.65 0.7number of orbitals 0102030 Two stage algorithm S p r e a d 8A20.45 0.5 0.55 0.6 0.65 0.7number of orbitals 0102030 Wannier90 converged Fig. 4.7: Histogram of localized orbitals for 256 water molecules computed by two di erent algorithms based on the Wannier90 spread functional. (top) output of the re nement algorithm applied to the output of the randomized algorithm, and (bottom) converged local minimum for Wannier90. 19 Admittedly, the time to solution for Wannier90 may depend strongly on the initial guess, e.g., in this experiment the initial spread for Wannier90 was 5339 A2 and convergence took 20 iterations. However, a poor initial guess could result in Wannier90 failing to converge, converging to a worse local minimum, or taking longer to converge. Our algorithms are direct and have no such dependence on an initial guess. In this experiment, each iteration of Wannier90 took roughly 85 seconds. So even two iterations costs more than our two stage algorithm and we have omitted the cost of computing the overlap matrices based on , which Wannier90 requires as input and we computed using Quantum ESPRESSO on 256 processors. Collectively, these results make our algorithm an attractive alternative to Wannier90. If a local minimum of the objective is desired, the output from Algorithms 3 or 4 may be used as an algorithmically constructed initial guess for Wannier90. Remark 4.1. Seeding Wannier90 with our two stage algorithm convergence took 16 iterations and we appear to arrive at an equivalent local minima. For this system, it would appear the area around the local minimum is quite at and the default absolute convergence criteria is rather stringent. In fact, after one iteration starting with our algorithm Wannier90 yields localized functions whose spread is withing 0:4%of the converged value. To get to within 1%starting from the random initial guess takes ve iterations. 5. Conclusion. We have presented a two stage algorithm to accelerate the com- putation of the SCDM for nding localized representation of the Kohn-Sham invariant subspace. We rst utilize an algorithm based on random sampling to approximately localize the basis and then perform a subsequent re nement step. This method can achieve computational gains of over an order of magnitude for systems of relatively large sizes. Furthermore, the orbitals computed are qualitatively and quantitatively similar to

Question 296 multiple-choice

Quantum computing leverages sophisticated techniques in state preparation and measurement to solve hidden subgroup problems, which are central to important algorithms like factoring and period finding. The dihedral hidden subgroup problem (DHSP) offers unique challenges due to its connection with non-abelian groups and classical hard problems such as subset sum. Which quantum measurement strategy is proven to be optimal for distinguishing dihedral hidden subgroup states when multiple copies are available, as determined by the Holevo-Yuen-Kennedy-Lax theorem? 1) Helstrom measurement 2) Von Neumann projective measurement 3) Pretty good measurement (PGM) 4) Swap test 5) Quantum Fourier transform measurement 6) Maximum likelihood measurement 7) Bell basis measurement

✓ Correct Answer:

The correct answer is 3) Pretty good measurement (PGM).

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approach produces a random state |t,x/angbracketrightwithtuniformly sampled from Z2 andxuniformly sampled from ZN. Thus the hidden subgroup state when the hidden subgroup is H=C1is simply the maximally mixed state (11) ρ{e}=1 N/summationdisplay t∈Z2/summationdisplay x∈ZN|t,x/angbracketright/angbracketleftt,x|=I2N 2N whereI2Nis the 2N-dimensional identity matrix. Our goal is to determine dgivenkcopies of the state ρd. It will be helpful to write the state in a way that begins to reveal the connection to the subset 8 DAVE BACON, ANDREW M. CHILDS, AND WIM VAN DAM sum problem. Note that we can write (12) ρd=1 2N/summationdisplay b,c∈Z2/summationdisplay x∈ZNω(b−c)xd|b,x/angbracketright/angbracketleftc,x|. Therefore ρ⊗k d=1 (2N)k/summationdisplay b,c∈Zk 2/summationdisplay x∈Zk Nω[(b−c)·x]d|b,x/angbracketright/angbracketleftc,x| (13) =1 (2N)k/summationdisplay x∈Zk N/summationdisplay p,q∈ZNωd(p−q)/radicalbigηxpηxq|Sx p,x/angbracketright/angbracketleftSx q,x| (14) where |Sx r/angbracketrightis the (normalized) uniform superposition over subsets of x∈Zk N that sum to r∈ZN, |Sx r/angbracketright:=1√ηxr/summationdisplay b∈Sxr|b/angbracketright (15) withSx r:={b∈Zk 2:b·x=r}denoting the set of bit strings corresponding to subsets of xthat sum to r, andηx r:=|Sx r|denoting the number of such subsets. If ηx r= 0, then no such state can be deﬁned, and we use the convention |Sx r/angbracketright= 0. When the subgroup is trivial, kcopies of the hidden subgroup state are simply kcopies of the maximally mixed state, ρ⊗k {e}=I(2N)k/(2N)k. 4.The optimal measurement In this section, we present the optimal measurement for distinguishing the hidden subgroup states ρ⊗k d. The measurement will have Noutcomes, one for each possible value of d, and will be optimal in the sense that the prob- ability of obtaining the correct outcome will be as large as possible. Recall that a general quantum measurement, a positive operator-valued measure (povm ), is speciﬁed by a set of positive operators {Ej},Ej>0, that sum to the identity, i.e.,/summationtext jEj=I. Given a density matrix ρ, the probability of obtaining the outcome jis trEjρ. Ip was the ﬁrst to consider optimal measurements for hidden subgroup problems [32]. In particular, he found the optimal measurement for the abelian hidden subgroup problem when the hidden subgroups are all given with equal a priori probabilities, thereby showing that the methods devel- oped to solve the factoring problem are optimal. Ip also derived the optimal measurement for the dihedral hidden subgroup problem given a single copy of the hidden subgroup state. As we shall see, this measurement fails to eﬃciently identify the order two subgroups. Since we know that there exists a measurement for solving the dhsp usingO(logN) copies of the hidden subgroup states, it is of interest to understand the optimal measurement givenk/greatermuch1 copies. OPTIMAL MEASUREMENTS FOR THE DIHEDRAL HSP 9 The optimal measurement turns out to be the pretty good measurement (pgm) [25] (also known as the square root measurement orleast squares measurement [11]). To prove that the pgm is optimal, we will use the following theorem: Theorem 1 (Holevo [28], Yuen-Kennedy-Lax [52]) .Given an ensemble of quantum states ρiwith a priori probabilities pi, the measurement with povm elementsEjmaximizes the probability of successfully identifying the state if and only if /summationdisplay ipiρiEi=/summationdisplay ipiEiρi (16) and/summationdisplay ipiρiEi≥pjρj∀j. (17) This condition follows most easily from noting that the maximization prob- lem

Question 297 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) in non-abelian groups rely on measurement processes and group-theoretic properties to distinguish hidden subgroups. Efficient distinguishability often depends on the structure and representation theory of groups such as affine and q-hedral groups. For a prime p and a divisor q of p−1, which of the following statements correctly describes the sample complexity required to distinguish between p different conjugate subgroups in the affine group Ap, given a bounded below total variation distance between quantum states corresponding to distinct cosets? 1) The number of samples required grows linearly with p. 2) The number of samples required is constant, independent of p. 3) The number of samples required is O(log p). 4) The number of samples required grows quadratically with p. 5) The number of samples required is O(p). 6) The number of samples required is O(√p). 7) The number of samples required is exponential in p.

✓ Correct Answer:

The correct answer is 3) The number of samples required is O(log p)..

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has rank ( p−1)/q, and the total probability that we observe ρis (p−1)/p=1−1/pas before. Then these four partial measurements determine k, remove the eﬀect of the coset, and determine that the row has one of two values, uoru+ 1. Up to an overall phase we can write this as a two-dimensional vector: 1√ 2/parenleftbiggωkaub p ωkau+1b p/parenrightbigg . Our only goal in doing this is to create a one-qubit state where the relative phase between the two basis vectors depends on the conjugate b. Moreover, the relative phase is multiplied by the irrep label k, which is uniformly random. As a result, the typical angle between the states corresponding to any two distinct cosets b, b/primewill be Ω(1), and a simple measurement yields a constant variation distance between them. To make this precise, apply the Hadamard transform 1√ 2/parenleftbigg11 1−1/parenrightbigg and measure according to the resulting basis. The probability that we observe the ﬁrst or second basis vector is then cos2θand sin2θ, respectively, where θ=(kau(a− 1)bπ)/p. Now when we observe a q-dimensional representation ρkofNk, the observed label kis uniformly distributed over Z∗ p/Zq. Moreover, when we perform the POVM in step (iv) above, the uwe observe is uniformly distributed over Zq. It follows that the coeﬃcient m=kau(u−1) is uniformly distributed over Z∗ p. For any two distinct b,b/prime, the total variation distance is then 1 2(p−1)/summationdisplay m∈Z∗p/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos 2πmb p−cos2πmb/prime p/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsinglesin 2πmb p−sin2πmb/prime p/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg . Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 950 MOORE, ROCKMORE, RUSSELL, AND SCHULMAN This we rewrite 1 p−1/summationdisplay m∈Z∗ p/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos 2πmb p−cos2πmb/prime p/vextendsingle/vextendsingle/vextendsingle/vextendsingle =1 2(p−1)/summationdisplay m∈Zp/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos2πmb p−cos2πmb/prime p/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≥1 4(p−1)/summationdisplay m∈Zp/parenleftbigg cos2πmb p−cos2πmb/prime p/parenrightbigg2 =p 4(p−1)>1 4. (Adding the m= 0 term contributes zero to the sum in the second line. In the third line we use the facts that |x|≤x2/2 for all |x|≤2, the average of cos2xis 1/2, and the two cosines have zero inner product.) Since the total variation distance between any two distinct conjugates is bounded below by a constant, we can distinguish between the pdiﬀerent conjugates with only O(logp) = poly( n) samples. Thus, hidden conjugates in Apare information- theoretically reconstructible, completing the proof. By embedding the q-hedral groups in Apas in Theorem 2, we can generalize Theorem 3 to the q-hedral groups ( qnot necessarily prime) as follows. Let pbe prime, letqbe a divisor of p−1, and let q/primebe a divisor of q. The q-hedral group Zq/multicloseleftZ p has a unique normal subgroup K∼=Zq/prime/multicloseleftZ p, in which there is a maximal subgroup Hq/primeof order q/prime. Moreover, all nonnormal subgroups of order q/primeare conjugates of Hq/prime. Theorem 4. Ifp, q, q/prime, and Hq/primeare as above, then the hidden conjugates of Hq/prime inZq/multicloseleftZ pare information-theoretically reconstructible. We now wish to information-theoretically reconstruct all subgroups of the aﬃne andq-hedral groups. We can do this by using the fact that, except for normal sub- groups, there is a unique conjugacy class of each order q/prime, namely the conjugates of Hq/prime. Thus if we can determine the order of H, this determines

Question 298 multiple-choice

Understanding phase transitions in strongly coupled gauge theories is important for modeling phenomena like dynamical symmetry breaking and mass generation. Lattice simulations often utilize special fermion formulations and exploit properties of gauge groups to study these effects. Which feature of the SU(2) gauge theory with $N_f=4$ Dirac fermions allows lattice simulations to avoid the sign problem, enabling efficient Monte Carlo analysis? 1) The use of continuous chiral symmetry in the lattice setup 2) The implementation of Wilson fermions for flavor representation 3) The abelian nature of the SU(2) gauge group 4) The presence of massless Goldstone bosons in the theory 5) The strong coupling limit inducing confinement 6) The pseudoreal property of the SU(2) fundamental representation 7) The direct coupling of gauge fields to the four-fermion interaction

✓ Correct Answer:

The correct answer is 6) The pseudoreal property of the SU(2) fundamental representation.

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Title: Non-Abelian gauged Nambu–Jona-Lasinio models on the lattice Year: 2011 Paper ID: 8c365d3e6cb93cef13f81699f63ffc15c67b9a32 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/8c365d3e6cb93cef13f81699f63ffc15c67b9a32 Abstract: We use Monte Carlo simulation to probe the phase structure of a SU(2) gauge theory containing $N_f$ Dirac fermion flavors transforming in the fundamental representation of the group and interacting through an additional four fermion term. Pairs of physical flavors are implemented using the two tastes present in a reduced staggered fermion formulation of the theory. The resultant lattice theory is invariant under a set of shift symmetries which correspond to a discrete subgroup of the continuum chiral-flavor symmetry. The pseudoreal character of the representation guarantees that the theory has no sign problem. For the case of $N_f=4$ we observe a crossover in the behavior of the chiral condensate for strong four fermi coupling associated with the generation of a dynamical mass for the fermions. At weak gauge coupling this crossover is consistent with the usual continuous phase transition seen in the pure (ungauged) NJL model. However, if the gauge coupling is strong enough to cause confinement we observe a much more rapid crossover in the chiral condensate consistent with a first order phase transition

Question 299 multiple-choice

Topological Quantum Field Theories (TQFTs) intertwine algebraic structures such as Hopf algebras and quantum groups with the topology of surfaces through modular group actions and invariants. A key aspect is the representation theory of mapping class groups, which often depends on the properties of the S-matrix and associated modularity conditions. In the setting of quantum groups and TQFTs, which approach restores the invertibility of the S-matrix and modularity for representation spaces when degeneracy arises due to objects braiding trivially at specific roots of unity? 1) Extending the category to include non-representable objects 2) Applying a universal central extension of SL(2,Z) 3) Introducing additional handlebody decompositions 4) Modifying the ribbon graph structure on surfaces 5) Increasing the genus of the underlying surfaces 6) Restricting to the subcategory of semisimple objects 7) Employing the double construction for Hopf algebras to obtain simple formulae for S and its inverse

✓ Correct Answer:

The correct answer is 7) Employing the double construction for Hopf algebras to obtain simple formulae for S and its inverse.

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to the corresponding union of handlebodies. If the Miare invertible morphisms in the cobordism category we obtain projective representations of the modular groups. For details see [T]. The main result of the ﬁrst chapter is the relation of th e phase cto intrinsic invariants of the Hopf algebra A. Inordertodiscussthemodularityconditionwerecallhowthe Smatrixcanbeobtained from the [RT]-construction for standard TQFT’s with S=∅. The cobordism describing the action ofSonT′is a 3-manifold whose boundary is ∂M=T′/coproducttext S1T′, the closed surface of genus two, and can thus be considered a cobordism Σ 2→∅. In [RT] the vectorspace associated to Σ 2is/circleplustext ijHom(i⊗i∨⊗j⊗j∨,1). The linear form assigned to Σ2→∅is found by computing the the invariant in S3of the ribbon graph embedded in the outside of Σ 2. On a vector fits value is 1coev⊗coev−−−−−−→ i⊗i∨⊗j⊗j∨1⊗ǫ2⊗1−−−−−→ i⊗i∨⊗j⊗j∨f−−−→1 6 In the description of an extended TQFT we have to consider this as t he matrix element ofS ∈End(T′)∼=/circleplustext ijHom(j⊗j∨, i⊗i∨). Thus on a summand we have S:j⊗j∨coev⊗1−−−→i⊗i∨⊗j⊗j∨1⊗ǫ2⊗1−−−−−→ i⊗i∨⊗j⊗j∨1⊗cj−−−→i⊗i∨ wherecjis proportional to j⊗j∨ǫ−→j∨⊗jev−→1. The generalization of this formula to non semisimple categories is described by [Ly] and will be reviewed in the next chapter. The matrix elements of the restriction of StoV(T) =Hom(1,XT′) =/circleplustext jHom(1,j⊗ j∨)∼=kIare given bySij=tri⊗j(ǫ(i,j)ǫ(j,i)) where tris the usual trace of a balanced category. A priori the operations SandTdeﬁned for a general semisimple Codo not yield a projective representation of SL(2,Z) unless we impose one further condition. This is the rather specialized “modularity condition” introduced in [T] asserting that theS- matrix is invertible. In case this condition is violated we may still apply Atiyah’s p rescription and reduce the space kIby the projection P=SS−where the matrix S− ij=Sij∨is assigned by the [RT]- prescription to the inverse cobordism. For example ifCois a symmetric category the Smatrix is of rank one so the SL(2,Z) representation is one dimensional. A degeneracy problem occurs qu ite generally ifI contains a subsetIo, of irreducible objects, which braid trivially, i.e., ǫ(k,j)ǫ(j,k) = 1 for allk∈Io, j∈I. In caseIois a subgroup of invertibles {σ}we have for the natural action of its elements on kIthatSσ=S. HenceSandTcan be deﬁned on the orbit spaceim(/summationtext σ∈I≀σ), where we can hope for the modularity condition to hold. This situation occurs for the semisimpliﬁed representation categor ies of quantum groups at certain roots of unity. The example we will come back to in t he last chap- ter isUq(sl2) where q1/2is anl= 2m+1- th root of unity. We have |I|= 2mand the 2m-th representation braids trivially and is invertible of order two. The truncated theory yields an m-dimensional representation of SL(2,Z). The problem of degeneracy is resolved in a very natural way in the un iversal picture forR(A) by choosingAto be a double constructed algebra. In this situation we ﬁnd very simple formulae for Sand its inverse. Survey of Contents and Summary of Results: In Chapter 2 we deﬁne and study the action of operators generat ing the mapping class groupD:=πo(Diff(T,D)) on the double D(A) of a ﬁnite dimensional Hopf algebra. We start in Section 1 with a review of

Question 300 multiple-choice

In canonical approaches to quantum gravity, models employing Abelian gauge groups such as U(1)^3 offer a tractable setting to rigorously study the quantisation of gravitational degrees of freedom. These frameworks also assist in testing mathematical techniques for constraint implementation and path integral formulations. Which feature distinguishes the U(1)^3 quantum gravity model as particularly useful for developing spin foam techniques and exact quantisation methods? 1) Its Abelian gauge group structure, which simplifies calculations and representations 2) Its description of only one physical polarisation, reducing complexity 3) Its lack of general covariance, avoiding diffeomorphism constraints 4) Its degenerate metric configurations being the focus of quantisation 5) Its presence of persistent quantum anomalies in the algebra representation 6) Its reliance exclusively on SU(2) symmetry for constraint solutions 7) Its phase space independent structure functions in the hypersurface deformation algebra

✓ Correct Answer:

The correct answer is 1) Its Abelian gauge group structure, which simplifies calculations and representations.

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Title: Exact quantisation of U(1)3 quantum gravity via exponentiation of the hypersurface deformation algebroid Year: 2022 Paper ID: a0e9474775d48af0ce0a30f1145b8abcd96f7432 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a0e9474775d48af0ce0a30f1145b8abcd96f7432 Abstract: The U(1)3 model for 3+1 Euclidian signature general relativity (GR) is an interacting, generally covariant field theory with two physical polarisations that shares many features of Lorentzian GR. In particular, it displays a non-trivial realisation of the hypersurface deformation algebroid with non-trivial, i.e. phase space dependent structure functions rather than structure constants. In this paper we show that the model admits an exact quantisation. The quantisation rests on the observation that for this model and in the chosen representation of the canonical commutation relations the density unity hypersurface algebra can be exponentiated on non-degenerate states. These are states that represent a non-degenerate quantum metric and from a classical perspective are the relevant states on which the hypersurface algebra is representable. The representation of the algebra is exact, with no ambiguities involved and anomaly free. The quantum constraints can be exactly solved using groupoid averaging and the solutions admit a Hilbert space structure that agrees with the quantisation of a recently found reduced phase space formulation. Using the also recently found covariant action for that model, we start a path integral or spin foam formulation which, due to the Abelian character of the gauge group, is much simpler than for Lorentzian signature GR and provides an ideal testing ground for general spin foam models. The solution of U(1)3 quantum gravity communicated in this paper motivates an entirely new approach to the implementation of the Hamiltonian constraint in quantum gravity.

Question 301 multiple-choice

Quaternions are a mathematical extension of complex numbers, essential in representing rotations in three-dimensional space and fundamental in abstract algebra. Their algebraic structure includes unique properties not shared by complex numbers or real numbers. Which statement accurately describes a property that differentiates quaternions from both complex numbers and triplets constructed with three imaginary units? 1) Quaternions form a commutative ring under multiplication, just like complex numbers. 2) Quaternions have only two distinct square roots of -1, similar to complex numbers. 3) The algebra of quaternions is associative but not closed under multiplication. 4) All nonzero quaternions lack multiplicative inverses, so division is undefined. 5) Triplets with three imaginary units can form a division algebra over the real numbers. 6) Quaternions are the first example of a noncommutative division ring, with infinitely many square roots of -1 corresponding to unit pure quaternions. 7) Quaternions cannot be used to encode spatial rotations in three dimensions.

✓ Correct Answer:

The correct answer is 6) Quaternions are the first example of a noncommutative division ring, with infinitely many square roots of -1 corresponding to unit pure quaternions..

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sacriﬁce is not enough. 37 For an algebra to be closed under multiplication, a product of triplets must produce another triplet, but (a1+ib1+jc1)(a2+ib2+jc2) = ( a1a2b1b2c1c2) +i(a1b2+b1a2)+j(c1a2+a1c2) (4.3) +i j(b1c2c1b2): One way to see the impossibility of this feat is to realize that the following must hold in a closed algebra: i j=a+bi+c j (4.4) fora;b;c2R. Multiplication of Eq (4.4) by iyields aib+ci j+j=0: (4.5) Substitution of Eq (4.4) into Eq (4.5) produces aib+c(a+bi+c j)+j=aib+ca+cbi+c2j+j = (cab)+(a+cb)i+ c2+1 j (4.6) =0 from which a contradiction immediately ensues as c2=1 (4.7) and hence c=2R. 38 The breakthrough for Hamilton came on that fateful October day when, in his own words [37]: I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i, j, k; exactly such as I have used them ever since. A rare act of mathematical vandalism followed: Hamilton pulled out his pocket knife and carved the equations on a side of the bridge. His carving did not survive, but a plaque commemorating the momentous occasion (Figure 4.1) was later placed on the bridge. Figure 4.1: Quaternion Plaque on Broom Bridge in Dublin Hamilton accomplished his extension of complex numbers by leaping from two dimensions straight into four. The quaternion was born as a quadruplet q=a+ib+jc+kd (4.8) where i2=j2=k2=1 and a;b;c;d2R. 39 The norm of a quaternion N(q)is N(q) = ( a+ib+jc+kd)(aibjckd) = a2+b2+c2+d2 bd(ik+ki)cd(jk+k j) (4.9) =a2+b2+c2+d2 if ik=ki (4.10) and jk=k j: (4.11) A complete set of multiplication rules for quaternions is then produced by setting i j=k (4.12) and the algebra is closed: (a1+ib1+jc1+kd1)(a2+ib2+jc2+kd2) =a1a2+ia1b2+ja1c2+ka1d2 +ib1a2+i2b1b2+i jb1c2+ikb1d2 +jc1a2+jic1b2+j2c1c2+jkc1d2 +kd1a2+kid1b2+k jd 1c2+k2d1d2 = (a1a2b1b2c1c2d1d2) (4.13) +i(a1b2+b1a2+c1d2d1c2) +j(a1c2b1d2+c1a2+d1b2) +k(a1d2+b1c2c1b2+d1a2): 40 Moreover, if q6=0, then N(q) =q¯q=¯qq6=0 . The inverse of a quaternion is given by q1=¯q N(q)(4.14) Hamilton produced the ﬁrst example of a noncommutative division ring, on which J.J. Silvester remarked [38]: In Quaternions the example has been given of Algebra released from the yoke of the commutative principle of multiplication - an emancipation somewhat akin to Lobachevsky’s of Geometry from Euclid’s noted empirical axiom. Proposition 4.1.1. There are inﬁnitely many square roots of 1in the algebra of Hamilton’s quaternions. Proof. Letqbe a pure quaternion, i.e., q=ia+jb+kcwith a;b;c2Rand consider the equation q2=1: (4.15) We have q2= (ia+jb+kc)(ia+jb+kc) =a2+i jab+ikac+jibab2+jkbc+kica+k jcbc2(4.16) = a2+b2+c2 =1 Thus, any pure quaternion of norm 1 is a solution to Eq (4.15), and there are inﬁnitely many such quaternions since they correspond to points on the imaginary 2-sphere. 41 4.1.2 The Struggle Hamilton spent much of the remaining twenty-two years of his life on the study of quater- nions, publishing 109 papers on the subject. His classic book Lectures on Quaternions appeared in 1853, but his most extensive treatise Elements of Quaternions came from beyond the grave, pub- lished in 1866, shortly after Hamilton’s death. He noted the connection between the quaternions and dimensions of time and space [32]: Time is said to have only one dimension, and space to

Question 302 multiple-choice

Quantum algorithms for hidden shift and hidden subgroup problems utilize advanced measurement techniques to efficiently distinguish between quantum states associated with algebraic structures. The pretty good measurement (PGM) is a key tool in these algorithms, closely related to the matrix sum problem and the probabilistic structure of solution spaces. In the context of quantum algorithms leveraging the pretty good measurement (PGM) for generalized hidden shift problems, which parameter regime marks the transition from typically having no solutions to the matrix sum problem to typically having many solutions, and thereby directly impacts the probability of successful identification of the hidden shift? 1) When k ≫ N/M and M ≪ N 2) When k ≪ logM/logN and M ≫ N 3) When k is constant and N is exponentially larger than M 4) When k is less than the square root of N and M is arbitrary 5) When k ≈ loglogN and M ≈ logN 6) When k ≈ N/M^k and M ≈ N 7) When k ≪ logN/logM yields almost no solutions, but larger k yields many solutions

✓ Correct Answer:

The correct answer is 7) When k ≪ logN/logM yields almost no solutions, but larger k yields many solutions.

📚 Reference Text:

ZM×ZNfollowed by a computational basis measurement also fails forM≪N. (Note that poly(log N) such classical samples information theoretically determine the answer even for M= 2 [9], but it is not known how to process this data eﬃciently.) Instead, wewilluse k >1statesandapplytheprettygoodmeasurement. To see the connection to the matrix sum problem, it is helpful to write these states in a diﬀerent basis. Fourier transforming the second r egister over ZN, we ﬁnd ˜ρ⊗k s=1 (MN)k/summationdisplay x∈Zk N/summationdisplay b,c∈{0,...,M−1}kω(b·x−c·x)s|b,x/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tc,x|.(6) QUANTUM ALGORITHM FOR A GENERALIZED HIDDEN SHIFT PROBLEM 5 Now introduce the states |Sx w/an}⌊ra⌋k⌉tri}ht:=1√ηxw/summationdisplay b∈Sxw|b/an}⌊ra⌋k⌉tri}ht (7) which are uniform superpositions over the solutions of the m atrix sum prob- lem, Sx w:={b∈ {0,...,M−1}k:b·x=wmodN}. (8) Here the number of solutions is ηx w:=|Sx w|. If there are no solutions (i.e., if ηx w= 0), then we deﬁne |Sx w/an}⌊ra⌋k⌉tri}ht:= 0. In terms of these states, we can rewrite (6) as ˜ρ⊗k s=1 (MN)k/summationdisplay x∈Zk N/summationdisplay w,v∈ZNω(w−v)s/radicalbig ηxwηxv|Sx w,x/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tSx v,x|.(9) Given such a state, we would like to determine the value of s. 3. Pretty good measurement approach In this section, we review the pgmapproach to distinguishing hidden subgroup states [2,3] as applied to the generalized hidden s hift states (9). Thepretty good measurement (also known as the square root measure- ment or least squares measurement) is a measurement that oft en does well at distinguishing the members of an ensemble of quantum stat es [13] (and in fact is sometimes optimal in a certain sense). For an ensem ble of states {σj}with equal a priori probabilities, the pretty good measurem ent is the povmwith elements Ej:= Σ−1/2σjΣ−1/2(10) where Σ:=/summationdisplay jσj (11) and where the inverse is taken over the support of the ensembl e. For the states (9), the pgmnormalization matrix is Σ =N (MN)k/summationdisplay x∈Zk N/summationdisplay w∈ZNηx w|Sx w,x/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tSx w,x|, (12) givingpovmelements Ej=1 N/summationdisplay x∈Zk N/summationdisplay w,v∈ZNω(w−v)j|Sx w,x/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tSx v,x|. (13) The probability of successfully identifying the hidden shi ftsis indepen- dent ofs, and is given by Pr(success) := trEs˜ρ⊗k s (14) =1 MkNk+1/summationdisplay x∈Zk N/parenleftbigg/summationdisplay w∈ZN/radicalbig ηxw/parenrightbigg2 . (15) 6 ANDREW M. CHILDS AND WIM VAN DAM Usingthisexpression, wecanshowthatthesuccessprobabil ityisappreciable when the matrix sum problem is likely to have a solution. Spec iﬁcally, we have Lemma 1 (Lemma 2 of [3]) .IfPr(ηx w≥α)≥βfor uniformly random x∈Zk Nandw∈ZN(i.e., if most instances of the matrix sum problem have many solutions), then αβ2N/Mk≤Pr(success) ≤Mk/N. Proof. For the upper bound, we have Pr(success) ≤M Nk+1/summationdisplay x∈Zk N/parenleftbigg/summationdisplay w∈ZNηx w/parenrightbigg2 =Mk N(16) since the η’s are integers and/summationtext w∈ZNηx w=Mkfor anyx. For the lower bound, we have Pr(success) ≥N Mk/parenleftbigg1 Nk+1/summationdisplay x∈Zk N/summationdisplay w∈ZN/radicalbig ηxw/parenrightbigg2 (17) by Cauchy’s inequality applied to (15). Now 1 Nk+1/summationdisplay x∈Zk N/summationdisplay w∈ZN/radicalbig ηxw≥√αPr(ηx w≥α), (18) so by the hypothesis, Pr(success) ≥αβ2N/Mkas claimed. /square For uniformly random x∈Zk Nandw∈ZN, the expected number of matrix sum solutions is µ:=E x,w[ηx w] =Mk N, (19) where we have again used the fact that/summationtext wηx w=Mkfor anyx. Thus, we expect the matrix sum problem to typically have no solutio ns fork≪ logN/logM, many solutions for

Question 303 multiple-choice

In quantum chemistry, the independent particle interpretation (IPI) allows electrons in a many-electron system to be modeled as if each moves independently in an averaged field created by all other electrons. Several electronic structure methods meet specific mathematical and physical criteria necessary for the IPI, while others do not. Which electronic structure wavefunction fulfills the requirements of having no more than N spatial orbitals for N electrons, orbital eigenfunctions of a proper Hamiltonian including the average electronic field, and mean-field effects resulting from the variational principle, thereby supporting the independent particle interpretation? 1) Configuration Interaction (CI) wavefunction 2) Multiconfiguration Self-Consistent Field (MC-SCF) wavefunction 3) Extended Hartree-Fock with spatial projection operators 4) Valence Bond (VB) wavefunction 5) Hartree-Fock (HF) wavefunction 6) Valence Bond (VB) wavefunction with fixed orbitals 7) Generalized Independent particle (GI) wavefunction with spatial projection operators

✓ Correct Answer:

The correct answer is 5) Hartree-Fock (HF) wavefunction.

📚 Reference Text:

must be no more than N different spatial orbit als since there are only N electrons. (ii) Each spatial orbital must be an eigenfunction of an operator equiva lent to the Hamiltonian for an electron moving in the field due to the nuclei and in some average field due to electrons in the other N -1 orbitals. (iii) This aver age field in (ii) can be nonlocal but it must be obtained directly from applying the variational principle to the energy. As discussed elsewhere2 the Hartree-Fock, UHF, and GI wavefunctions satisfy these criteria and can be given an independent particle interpretation (IPI). However configuration-interaction ( CI) and multiconfiguration SCF6 (MC-SCF) wavefunctions do not satisfy (ii) and usually not (i) and cannot be given the IPI. In addition extended Hartree-Fock methods which use spatial projection operators 7 do not necessarily satisfy (i) [e.g., for H2 such a wave function7 might involve two u orbitals, four 1r orbitals (two 7rz and two 7ru), four 8 orbitals, etc.] and thus cannot be given the above IPI. In the valence bond (VB) wavefunction,8 (i) is satisfied but the orbitals are not functionally optimized so that (ii) and (iii) are not satisfied and the VB wavefunction cannot be given the IPI. However we have shown9 that the G 1 wavefunction corresponds to a generalization of the VB wavefunction in which all orthogonality and double occupation constraints are removed and the orbitals are functionally optimized, and thus the G 1 method is the direct generalization of the VB method which does lead to the IPI. The criteria in (i)-(iii) are sufficient to require that the many-electron wavefunction be expressible as a product of N spatial orbitals and a suitable spin func tion with this perhaps operated on with an operator which does not change the form of the orbitals (if they were changed the space spanned by the orbitals would in general enlarge to dimensions larger than N). Thus the operator may involve permutations but not spatial projection operators. If all the orbitals in are allowed to be different the operator must take care of both the spin symmetry and Pauli principle. The most general operator which does this and nothing more is 0.70 0.60 0.50 -; _g 0.40 § 0o.3o 0.20 0.10 0 L: c,G;i', i FrG. 3. Spin density at the nucleus, Q(O), as a function of repre sentation parameter Z for the 22S state of Li. s G. Das and A. C. Wahl, J. Chern. Phys. 44, 87 (1966); A. C. Wahl, P. J. Bertoncini, G. Das, and T. L. Gilbert, Intern. J. Quant. Chern. IS, 123 (1967). 7 E. R. Davidson and L. L. Jones, J. Chern. Phys. 37, 2966 (1962); C. F. Bunge, Phys. Rev. 154, 70 (1967). s H. Eyring, J. Walter, and G. E. Kimball, Quantum Chemistry (John Wiley & Sons, Inc., New York, 1944), p. 218. v W. A. Goddard m, Phys. Rev. 169, 120 (1968). Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license

Question 304 multiple-choice

In quantum computing and representation theory, efficient decomposition of tensor product representations is essential for algorithms involving groups with complex symmetry, such as quantum doubles. The Clebsch-Gordan transform and quantum Fourier transforms (QFTs) over centralizers and their intersections play a crucial role in this process. Which of the following conditions is both necessary and sufficient for performing an efficient Clebsch-Gordan transform over the quantum double D for all irreducible representations, including non-fluxon types? 1) Efficient QFTs over the group algebra C[G] and its full group of automorphisms 2) Efficient block diagonalization of all irreducible representations of G restricted to abelian subgroups 3) Existence of a classical algorithm for enumerating all conjugacy classes of G 4) Efficient QFTs over G and its center Z only 5) Efficient decomposition of tensor products using only the regular representation of G 6) Efficient QFT and Clebsch-Gordan transforms over G without regard to subgroups 7) Efficient QFT and Clebsch-Gordan transforms over Z(h) and Z(g)∩Z(h) for all g, h in G, along with block diagonalization of irreducible representations of centralizers restricted to their intersections

✓ Correct Answer:

The correct answer is 7) Efficient QFT and Clebsch-Gordan transforms over Z(h) and Z(g)∩Z(h) for all g, h in G, along with block diagonalization of irreducible representations of centralizers restricted to their intersections.

📚 Reference Text:

Recall that for each element tinTd, we have that gt(hd)t=ghdand that each such tgives a distinct pair ( gt,(hd)t). Therefore, we can determine tafter ordering the pairs. This gives us the ﬁrst transformation. It is also clear from this that the co-a lgebra action is consistent across the both sides. Indeed, the action of any c∗on a state of the form |t1,v1/a\}bracketri}ht⊗|t2,v2/a\}bracketri}htis c∗(|t1,v1/a\}bracketri}ht⊗|t2,v2/a\}bracketri}ht) =/summationdisplay h2h1=ch∗ 1|t1,v1/a\}bracketri}ht⊗h∗ 2|t2,v2/a\}bracketri}ht=δ{t2t1=c}|t1,v1/a\}bracketri}ht⊗|t2,v2/a\}bracketri}ht. Under the Clebsch-Gordan transform, this state is taken to |t2t1,v3/a\}bracketri}ht, wherev3is a state determined by v1 andv2. This state is given by the second transformation described below. Therefore, the action of c∗is c∗|t2t1,v3/a\}bracketri}ht=δ{t2t1=c}|t2t1,v3/a\}bracketri}ht. This shows that the coalgebra action is consistent with this decomposition. For the second transformation, we need to block diagonalize the induced representation of the trivial irrep ofZ(g)∩Z(hd) toZ(ghd). We show how to do this if we can perform eﬃcient QFT over both groups. Suppose that ρis an irrep of a group Bwhich is a subgroup of the group Aand suppose that we can perform a QFT over AandB, then we show how to use these QFTs to give an eﬃcient way to blo ck diagonalize the induced representation ρ↑A. We ﬁrst embed the induced representation into C[A] in the following way. The induced representation consists of vectors of the form |t,v/a\}bracketri}ht, wheretis an element of the transversal of Bin Aandvis a vector in the irrep space ρ. We embed|v/a\}bracketri}htintoC[B] by taking an ancilla of size |B|/dρ, where dρis the dimension of ρ. This is possible if we know how to perform the QFT over C[B]. Now that we have an embedding into C[A], we can perform the QFT over Aand obtain a basis |ρ,i,j/a\}bracketri}htwhere only some irreps ρappear. Since we know which ones do not appear at all, we can la bel the irreps of Asuch that the state is|0/a\}bracketri}ht⊗|ρ,i,j/a\}bracketri}htwhere the ﬁrst register is of size |H|/dρ. We can now discard this register and we obtain the required block diagonalization. Using this technique when ρis a trivial representation of B=Z(g)∩Z(hd) andA=Z(ghd) gives us a way to decompose the induced representation from the trivial representation of a subgroup. This gives us the second transformation and compl etes the Clebsch-Gordan transform for ﬂuxon irreps. We have thus shown that if we can perform an eﬃcient QF T overZ(g) and overZ(g)∩Z(h) for all g,h∈G, then we can perform an eﬃcient Clebsch-Gordan transform ov erD(G). 25 7.1.2 General irreps For the more general irreps, we show that one can perform eﬃci ent Clebsch-Gordan transform if we can 1. perform QFT and CG transforms over Z(h) andZ(g)∩Z(h) for allg,h∈Gand, 2. block diagonalize irreps of centralizers restricted to i ntersections of centralizers. The procedure can be split into three steps in the following w ay. ([h],ρ)⊗([g],σ)∼=/circleplusdisplay d/parenleftig/parenleftbig ρ↓Z(g)∩Z(hd)⊗σ↓Z(g)∩Z(hd)/parenrightbig ↑Z(ghd)/parenrightig ↑G ∼=/circleplusdisplay d  /circleplusdisplay ν∈/hatwiderZ(g)∩Z(hd)nνν ↑Z(ghd) ↑G ∼=/circleplusdisplay d/circleplusdisplay µ∈/hatwiderZ(ghd)mµ(µ↑G). (7.3) In the above, nνis the multiplicity of ν(an irrep of Z(g)∩Z(h)) in the tensor product decomposition of ρ andσrestricted to Z(g)∩Z(hd) andmµis the multiplicity of µ(an irrep of Z(ghd)) whenνis induced to Z(ghd) and

Question 305 multiple-choice

In quantum information theory, group representation methods are often employed to construct highly entangled states in multi-qubit systems. These techniques are essential for understanding the relationship between symmetry, entanglement, and the structure of the Hilbert space. In a multi-qubit quantum system where a symmetry group acts irreducibly on each qubit and the tensor product of subsystem representations contains the trivial representation, what is the dimension of the physical Hilbert space? 1) The order of the symmetry group 2) The total number of qubits 3) The number of nontrivial representations in the tensor product 4) The number of times the trivial representation appears in the tensor product 5) The rank of the group’s character table 6) The product of the dimensions of all subsystem representations 7) The number of conjugacy classes of the group

✓ Correct Answer:

The correct answer is 4) The number of times the trivial representation appears in the tensor product.

📚 Reference Text:

subsystem (call these “small” subsets) are maximally mixed. Some pr evious discussions 19 (1/2,0) (1/2,0) (1/2,0)(1/2,0) (0,1/2)(0,1/2)(0,1/2) (0,1/2) (1/2,1/2) (1/2,1/2) (1/2,1/2)(1/2,1/2) R R{(0,0),(0,1),(1,0),(1,1)} Figure 5: Tensor network depiction of an eight qubit state with each spin and each nearest neighbor pair of spins in a maximally mixed state. of these states, including complete existence criteria for qubit sys tems may be found in [GB98, HS00, Sc04, HCRLL12, FFPP08, HGS17]. According to Corollary 3.3, we can obtain a maximally entangled state if we ﬁnd a group H with representations Riwhose tensor product contains the identity representation and f or which the tensor product of any small subset of representations is irreducible. As an example, consider the six-qubit state described in Appendix A.1 of [PYHP15]. Here, we have a representation of ( Z2)6toU(d1···dn) of the form described in Remark 3.4, where the six generators are represented as S1=X⊗Z⊗Z⊗X⊗I⊗I S2=I⊗X⊗Z⊗Z⊗X⊗I S3=X⊗I⊗X⊗Z⊗Z⊗I, S4=Z⊗X⊗I⊗X⊗Z⊗I S5=X⊗X⊗X⊗X⊗X⊗X S6=Z⊗Z⊗Z⊗Z⊗Z⊗Z (3.12) with I=/parenleftbigg 1 0 0 1/parenrightbigg X=/parenleftbigg 0 1 1 0/parenrightbigg Z=/parenleftbigg 1 0 0−1/parenrightbigg . (3.13) In this case, the corresponding representations of ( Z2)6on the individual subsystems are projective. Their tensor product includes the trivial representa tion: it is not hard to check directly that there is a state with |Ψ/an}b∇acket∇i}ht=|Ψ/an}b∇acket∇i}ht. Also, it is not hard to check that the indi- vidual representations are irreducible and the product of any two or three of these is also irreducible.13Thus, all subsystems of one, two, or three qubits are maximally ent angled. To understand this example in the language of Corollary 3.3, we can choo se a cover group ˆH to be the subgroup of Kgenerated by elements S′ iwithS′ 1= (X,Z,Z,X,I,I ) and so forth. This is a central extension of Hof order 64 ×32 = 2048 with 32 central elements of the form (±I,±I,±I,±I,±I,±I) with an even number of minus signs. 13For example, in each case, we can verify that any matrix commuting w ith the representatives of each generator of ( Z2)6is a multiple of the identity matrix. 20 3.2 Quantum systems with only LME states An interesting point (suggested by Eliot Hijano) is that we can descr ibe quantum mechan- ical systems for which all physical states are LME. Given any multipa rtite system with a Hamiltonian Hinvariant under the global symmetry group Hacting irreducibly in each ele- mentary subsystem (andpossibly insome composite subsystems), we simply gaugethegroup H. This restricts us to physical states which are invariant under H, so by our Corollary 3.3, any subsystem upon which Hacts irreducibly will be maximally mixed. In this case, local observables cannot be used to distinguish the possible physical sta tes. Forthese theories, the dimension ofthe Hilbert space of physical s tates is exactly the number of trivial representations appearing in the tensor product of rep resentations Riassociated to the elementary subsystems. Given a particular graph structur e for the associated tensor network (as in ﬁgure 5), we can choose basis elements correspond ing to the diﬀerent ways of labeling the edges with representations such that

Question 306 multiple-choice

Topological quantum computing leverages algebraic and geometric structures, such as group representations and character varieties, to encode and manipulate quantum information in robust ways. The interplay between SL(2,C), SU(2), and special algebraic surfaces is central to the mathematical formulation of qubits in this paradigm. Which property distinguishes representations with character κd, where d < 4 and |x|, |y|, |z| ≤ 2, in the context of topological quantum computing using link complements in S³? 1) They correspond to classical bits through abelian representations. 2) They generate trivial topological invariants in all cases. 3) They are associated exclusively with the topology of the two-sphere. 4) They produce non-unique fixed points in hyperbolic space H³. 5) They fix a unique point in H³ and are conjugate to SU(2) representations, corresponding to qubits. 6) They only arise in links with no irreducible character variety components. 7) They are equivalent to representations of the symmetric group S₃.

✓ Correct Answer:

The correct answer is 5) They fix a unique point in H³ and are conjugate to SU(2) representations, corresponding to qubits..

📚 Reference Text:

Title: Fricke Topological Qubits Year: 2022 Paper ID: db4481d095261faaf0134806f6acc6c46f5655e4 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/db4481d095261faaf0134806f6acc6c46f5655e4 Abstract: We recently proposed that topological quantum computing might be based on SL(2,C) representations of the fundamental group π1(S3\K) for the complement of a link K in the three-sphere. The restriction to links whose associated SL(2,C) character variety V contains a Fricke surface κd=xyz−x2−y2−z2+d is desirable due to the connection of Fricke spaces to elementary topology. Taking K as the Hopf link L2a1, one of the three arithmetic two-bridge links (the Whitehead link 512, the Berge link 622 or the double-eight link 632) or the link 732, the V for those links contains the reducible component κ4, the so-called Cayley cubic. In addition, the V for the latter two links contains the irreducible component κ3, or κ2, respectively. Taking ρ to be a representation with character κd (d<4), with |x|,|y|,|z|≤2, then ρ(π1) fixes a unique point in the hyperbolic space H3 and is a conjugate to a SU(2) representation (a qubit). Even though details on the physical implementation remain open, more generally, we show that topological quantum computing may be developed from the point of view of three-bridge links, the topology of the four-punctured sphere and Painlevé VI equation. The 0-surgery on the three circles of the Borromean rings L6a4 is taken as an example.

Question 307 multiple-choice

In electronic structure calculations, constructing a localized basis from delocalized orbitals is essential for computational efficiency and physical interpretation. The SCDM algorithm utilizes numerical linear algebra techniques to achieve this goal. Which statement best describes the principal advantage of performing QR with column pivoting (QRCP) on a smaller matrix $ U $ with orthogonal columns, instead of directly on the larger density matrix $ P $ in the context of constructing a localized basis? 1) It improves the accuracy of the resulting localized orbitals by reducing overlap between them. 2) It ensures that the selected columns are always spatially adjacent, increasing localization. 3) It eliminates the need for orthogonalization in the final basis construction. 4) It results in a basis with higher symmetry than that obtained from direct QRCP on $ P $. 5) It allows the algorithm to handle non-Hermitian matrices without modification. 6) It drastically reduces computational cost from $ O(N^3) $ to $ O(N n_e^2) $ while yielding an equivalent selection of well-conditioned columns. 7) It guarantees uniqueness of the resulting basis up to permutations of the columns.

✓ Correct Answer:

The correct answer is 6) It drastically reduces computational cost from $ O(N^3) $ to $ O(N n_e^2) $ while yielding an equivalent selection of well-conditioned columns..

📚 Reference Text:

well conditioned as possible. In particular, if we let Cdenote the columns selected by the rstnecolumns of then A:;Cshould be a well conditioned set of necolumns of A: In our setting, this means we would ideally compute an QRCP factorization of the matrix Pto identify newell conditioned columns from which we may construct a localized basis. However, this would be highly costly since Pis a large matrix of sizeN. Fortunately, we may equivalently compute the set Cby computing a QRCP factorization of or, in fact, any matrix Uwith orthogonal columns such that P=UU:More speci cally, we compute (2.4) =QR1R2 ; and the rst necolumns of encode C: The SCDM procedure to construct an orthonormal set of localized basis elements, denoted ifori= 1;:::;n e, and collected as columns of the matrix is presented in its simplest form in Algorithm 1. In such form the algorithm requires knowledge Algorithm 1 The SCDM algorithm Given: the Kohn-Sham orbitals 1:Compute a column pivoted QR of , =QR1R2 2:Compute = Qor, alternatively, =R1R2 Output: a localized basis for the Kohn-Sham subspace of the orthogonal factor from the QRCP. However, an alternative description simply requires the column selection C:We may equivalently write the SCDM algorithm as in Algorithm 2. Note that in Algorithm 2, the cost of the QR factorization for the matrix ( C;:)is onlyO(n3 e). 5 Algorithm 2 An alternative version of the SCDM algorithm Given: the Kohn-Sham orbitals 1:ComputeCassociated with a column pivoted QR of 2:Compute the QR factorization ( C;:)=QR 3:Compute = Q Output: a localized basis for the Kohn-Sham subspace Remark 2.1. There are various equivalent ways to construct the SCDM algo- rithm. While the simple presentation here di ers slightly from the original presenta- tion [10] the two are mathematically equivalent up to a choice of sign for the columns of. The original presentation corresponds to the computation of a QR factoriza- tion of ( C;:)via a Cholesky factorization of ( C;:) ( C;:):QR factorizations are not unique, there is always ambiguity up to diagonal matrix with entries on the unit circle. However, such an ambiguity does not have any a ect on the localization. This second interpretation allows us to brie y explain why this algorithm con- structs localized orbitals. Let Dbe a diagonal matrix with 1 on the diagonal such thatDRhas positive diagonal entries. This means that Q= ( C;:)R1D whereR1is a Cholesky factor of ( C;:) ( C;:)1:Importantly, ( C;:) ( C;:)= PC;Cand, therefore, exhibits o diagonal decay so long as PC;Cis well conditioned. This property is then inherited by R1[4]. Finally, since PC;:= ( C;:) =PC;:R1D we may conclude that it is well localized | PC;:is well localized and R1Ddoes not destroy that locality. Importantly, here we see that all the factorizations we are performing can be thought of as involving sub-matrices of P. The overall computational cost of the algorithm is O(Nn2 e);and practically the cost is dominated by the single QRCP factorization

Question 308 multiple-choice

In quantum algorithms for hidden subgroup problems, distinguishing between conjugate subgroups often relies on analyzing the statistical properties of measurement outcomes in high-dimensional representations of non-Abelian groups. Probabilistic concentration phenomena and representation theory play a crucial role in understanding the limitations of these algorithms. Which of the following conditions ensures that the total variation distance between the probability distribution Pb(v) for measuring basis vector v and the uniform distribution is exponentially small, making it information-theoretically hard to distinguish conjugate subgroups without an exponential number of measurements? 1) The subgroup order q is greater than the group order p. 2) The rank of the projection operator rkπ is less than ε₁. 3) The representation ρ has dimension less than p. 4) The projection operator π is not related to the subgroup structure. 5) The probability of observing representation ρ is less than 1/p. 6) The subgroup order q is approximately equal to p. 7) The subgroup order q satisfies q < p^(1–ε₁), resulting in rkπ > p/ε₁ and δ < p^(–β).

✓ Correct Answer:

The correct answer is 7) The subgroup order q satisfies q < p^(1–ε₁), resulting in rkπ > p/ε₁ and δ < p^(–β)..

📚 Reference Text:

In particular, there exists β>0, depending only on /epsilon1, such that /summationdisplay v|Pb(v)−Pb/prime(v)|δr d/bracketrightBig <4e−rδ2/48. Proof. We use an argument similar to [10]. We can think of a random d- dimensional complex vector vas a random 2 d-dimensional real vector of the same length, and we can think of this in turn as vi=wi/summationtext2d i=1w2 i, where the wiare independent Gaussian variables with zero mean and unit variance. By choosing a basis in which πprojects onto the ﬁrst r(complex) components of v, we have |π·v|2 2=/summationtext2r i=1w2 i/summationtext2d i=1w2 i=r d(1/2r)/summationtext2r i=1w2 i (1/2d)/summationtext2d i=1w2 i. Now we use the following Chernoﬀ bound, which can be derived from the moment generating function. For any t,w eh a v e Pr/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBigg 1 tt/summationdisplay i=1w2 i/parenrightBigg −1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>/epsilon1/bracketrightBigg <2/bracketleftBig (1 +/epsilon1) 1/2e−/epsilon1/2/bracketrightBigt . For|/epsilon1|<1/2, we have ln(1 + /epsilon1)/epsilon1/bracketrightBigg <2e −t/epsilon12/6. Now, for any a, b,i f|a/b−1|>δ, where δ<2, then either |a−1|>δ /4o r |b−1|>δ /4. Taking the union bound over these events where a=( 1/2r)/summationtext2r i=1w2 i andb=( 1/2d)/summationtext2d i=1w2 i, setting /epsilon1=δ/4 and t=2r≤2din (5) gives the stated bound. Setting d=dρandr=rkπ, Lemma 2 and the union bound imply that, for any constant A>√ 48, if (6) δ=A/radicalbigg logdρ rkπ, then, with high probability, for all dρbasis vectors vwe have /vextendsingle/vextendsingle/vextendsingle/vextendsingleP b(v)−1 dρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle<δ dρ. Summing over all v, this implies that the /lscript1distance between Pb(v) and the uniform Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE POWER OF STRONG FOURIER SAMPLING 953 distribution is at most δ. Now recall that rkπ=(p−1)/q.I f qp/epsilon1, and (6) gives δ

Question 309 multiple-choice

Indirect detection searches for dark matter rely heavily on the assumed density profile of the Milky Way, which influences expected gamma-ray signals from potential dark matter annihilation or decay. The choice of density profile affects the strength of exclusion limits set by experiments like Fermi-LAT and COMPTEL. Which dark matter density profile typically leads to the strongest indirect detection constraints due to its steep central density, increasing the predicted gamma-ray signal from the galactic center? 1) Isothermal profile 2) Uniform density profile 3) Cored profile 4) Contracted profile 5) Burkert profile 6) Plummer profile 7) King profile

✓ Correct Answer:

The correct answer is 4) Contracted profile.

📚 Reference Text:

m'=2m3=2) (right). The green shaded region is excluded by Fermi GC observations and the blue shaded region is excluded by COMPTEL. Both exclusions assume an Einasto galactic DM pro le. The thick solid and thin dotted contours correspond to the exclusions assuming the NFW and cored pro les, respectively. The green and blue dashed lines show the boundaries of the stronger exclusion obtained assuming a contracted pro le with = 1:4. Above and to the right of the solid red line, the hidden LSP density is larger than the observed DM density. The solid and dash-dotted orange lines shows the exclusion from deviations in the CMB for f= 0:2 andf= 1, respectively, with the excluded areas above and to the right of the lines. Note that the entire c= 1 parameter space is excluded by the CMB constraint for f= 1. The gray shaded region at the bottom has a hidden vector mass mx<20 MeV that is excluded by xed-target experiments. to sub-GeV hidden sector masses that could potentially be probed in current and planned precision searches [88]. E. Summary With optimistic but reasonable choices for the moduli parameters, a light hidden sector neutralino LSP produced in moduli reheating can be consistent with current DM searches. Even so, the scenario is tightly constrained by indirect detection and CMB measurements. The challenge here is precisely the same as in the MSSM: to avoid overproducing the neutralino LSP during moduli reheating, the annihilation rate must be large relative to the standard thermal value hvi31026cm3=s, and such an enhanced rate is strongly constrained by indirect DM searches. To avoid these bounds while not creating too much dark matter, the annihilation rate must be large enough that the LSP relic abundance is only a small fraction of the total DM density. 18 TheU(1)xhidden sector does slightly better than the MSSM in this regard for two reasons. First, the hidden gauge coupling can be taken small (as can =0=Mx), which helps to reduce the LSP relic abundance as suggested by Eq. (16). And second, the hidden LSP can be much lighter than an MSSM wino or Higgsino, leading to smaller photon yields below the primary sensitivity of Fermi-LAT. The strongest constraints for such light masses come from COMPTEL, which are less stringent than those from Fermi. Since the large late- time hidden neutralino annihilation rate is the primary hindrance to realizing this set-up, one might consider analogous scenarios in which the CMB and indirect detection signatures are suppressed; we address this possibility in the following section. Before moving on, let us also comment on the spectrum in the hidden sector. To avoid a large ne tuning, the hidden scalar soft terms must be relatively small, on the same order or less than the hidden gaugino mass. Given the large values of m3=2considered, the scalar soft masses must be sequestered from supersymmetry breaking. They must also receive new contributions beyond minimal AMSB, and the b0bilinear soft term must not be too much larger than ( 0)2.

Question 310 multiple-choice

In group theory and quantum computing, the structure of finite groups greatly influences the efficiency of algorithms for problems such as the Hidden Subgroup Problem. Concepts like derived series, smoothness, and group decompositions have significant computational implications. Which statement best describes a smoothly solvable group in the context of group-theoretic algorithms? 1) It is a group whose order is a prime number and whose elements commute. 2) It is any group that can be decomposed into a direct product of simple non-abelian groups. 3) It is a group whose derived length is infinite and whose exponents are unbounded. 4) It is an abelian group with no proper nontrivial subgroups. 5) It is a solvable group whose derived series consists entirely of factors that are (e, s)-smooth abelian groups, with constant derived length and bounded exponents. 6) It is a group for which the quantum Fourier transform cannot be efficiently implemented. 7) It is a group with a trivial center and no abelian quotient groups.

✓ Correct Answer:

The correct answer is 5) It is a solvable group whose derived series consists entirely of factors that are (e, s)-smooth abelian groups, with constant derived length and bounded exponents..

📚 Reference Text:

G0G1 ···Gmsuch that G=G0,Gm={1G}and the factors Gi/Gi+1(i=0,1,...,m−1) are abelian. A natural way of constructing a subnormal series of the solvable group Gis to consider its derived subgroups. For any group H, let us ﬁrst deﬁne and denote thecommutator subgroup H/primeofHbyH/prime=/angbracketleft{h−1k−1hk:h,k∈H}/angbracketright. Then the derived subgroups G(i)(i=0,1,2,...) are deﬁned by induction: G(0)=G;a n dt h e 6 FRIEDL, IVANYOS, MAGNIEZ, SANTHA, AND SEN (i+ 1)th derived subgroup G(i+1)is deﬁned as the commutator ( G(i))/primeofG(i).A l l the subgroups G(i)are normal subgroups of G(j)for 0≤j

Question 311 multiple-choice

Quantum Phase Estimation (QPE) is a foundational algorithm in quantum computing, enabling extraction of eigenphases for applications such as factoring and simulating quantum systems. Practical implementation on noisy intermediate-scale quantum (NISQ) devices requires consideration of hardware limitations and error mitigation strategies. Which circuit-level modification most directly improves phase estimation accuracy on NISQ hardware by reducing cumulative error from controlled gates in the standard QPE algorithm? 1) Replacing controlled phase rotation gates with unitary rotation gates and removing ancillary control qubits 2) Increasing the number of ancilla qubits for redundancy 3) Applying additional layers of controlled-NOT gates before measurement 4) Doubling the number of shots for each algorithm run 5) Implementing error correction codes for readout only 6) Extending coherence time by reducing gate speed 7) Using post-selection to discard runs with readout errors

✓ Correct Answer:

The correct answer is 1) Replacing controlled phase rotation gates with unitary rotation gates and removing ancillary control qubits.

📚 Reference Text:

each QPE algorithm, we ran the default 1 ,024 shots for both simulator and the IBMQX4. First, we implemented Kitaev’s algorithm to ﬁnd the phase ϕon both the Qiskit Aer simulator and IBMQX4 quantum computer. Fig.8 shows that the estimated ˆϕvalues of simulator results are almost the same as the original ϕvalues. The estimated ϕvalues from the IBM hardware platform are slightly different than the original ϕdue to the lack of full quantum computing error correction capabilities today. Nevertheless, we can estimate the correct binary values of bit positions by converting the estimated ϕvalues. Because the noise can be attributed to various factors among differe nt quantum computers, it is critical to ﬁnd the hardware error rates in order to increase the accuracy of the ϕestimation in Kitaev’s algorithm. The accuracy can be increased by adjust ing proper error rates for each quantum computer during the computation process from the estimated ϕinto the binary bit position. 6 q0:|0/angbracketright H • H•• ✌✌ q1:|0/angbracketright H • • −π 2H•• ✌✌ q2:|0/angbracketright H • • −π 4• −π 2H•✌✌ q3:|0/angbracketright H• • −π 4• −π 2H✌✌ q4:|0/angbracketright X• 11π 8• 11π 4• 11π 2• 11π✌✌ Fig. 13: Arbitrary constant precision QPE gate with 1 ancill ary qubit Arbitrary Constant Precision Quantum Phase Estimation 00000 00010 00100 00110 01000 01010 01100 01110 10000 10010 10100 10110 11000 11010 11100 11110 11111 Phase Phi = 0.x1x2x3x4 (Original Phi = 0.1011)00.010.020.030.040.050.060.070.080.09The probability of phase PhiIBM QX Fig. 14: Arbitrary constant precision QPE on IBMQX4 Second, we implemented iterative quantum phase estimation algorithm (IQPEA) to ﬁnd the phase ϕon both the Qiskit Aer simulator and IBMQX4 quantum computer. Fig.7 shows that the probability of ﬁnding ϕvalue from the simulation results are exactly the same as the original ϕ. The experiment results are slightly different than the original ϕbut we can estimate the correct binary values the same way as Kitaev’s algorithm . Third, we implemented QPE algorithms using the inverse quantum Fourier transform technique to ﬁnd the phase ϕon both the Qiskit Aer simulator and the IBMQX4 quantum computer. Fig.10 only shows the probability of ﬁnding ϕ values from the IBMQX4 experiments because the probability of ﬁnding ϕvalue from the simulation results are exactly the same as the original ϕ. However, the highest probability of the phase ϕfrom the experimental results is when the ϕis 0.01100 instead of the correct ϕ=0.1011. The main reasons for these inaccurate results are caused by the lack of error correction capabilities, short longitu dinal, and transverse coherence time for qubits and ancillary qubi ts respectively. Moreover, as described in Fig.9, the number o f controlled phase rotation gates on qubits can increase the readout errors. To solve this problem and increase the accuracy of exper- imental results, we remove the ancillary control qubit and replace the unnecessary controlled-rotation gates with un itary rotation gates for each qubit as described in Fig.11. Our experimental results Fig.12 shows that our solution can ﬁndq0:|0/angbracketright H U1(11π) H•• ✌✌ q1:|0/angbracketright H U1(11π 2) •

Question 312 multiple-choice

Quantum algorithms for hidden subgroup problems are a central concern in quantum computing, with important implications for cryptography and computational complexity. Non-Abelian groups introduce significant challenges compared to Abelian cases, requiring advanced mathematical tools to design efficient quantum algorithms. Which mathematical technique has been shown to enable efficient quantum algorithms for the Heisenberg hidden subgroup problem by leveraging group symmetries, while failing to provide similar breakthroughs for unitary groups and graph isomorphism? 1) Quantum Fourier transform over Abelian groups 2) Grover’s amplitude amplification 3) Classical brute-force search 4) Quantum phase estimation 5) Shor’s factoring algorithm 6) Clebsch-Gordan transform 7) Quantum error correction codes

✓ Correct Answer:

The correct answer is 6) Clebsch-Gordan transform.

📚 Reference Text:

was an eﬃcient quantumalgorithmforthenon-Abelianversionofthe problemwhichlie satthe heartofShor’s algorithm, the hidden subgroup problem[6, 7]. Thus motivated, a gre at deal of eﬀort has been expended in the last few years attempting to solve the non-Abelian h idden subgroup problem eﬃciently on a quantum computer. Towards this end, a series of eﬃc ient quantum algorithms for certain non-Abelian hidden subgroups have been developed[8, 9 , 10, 11, 12, 13, 14, 15]. At the same time a series of negative results towards the standard approach to solving the hidden subgroup problem on a quantum computer have also appeare d[8, 9, 16, 17, 18, 19, 20]. Viewed pessimistically, these results cast doubt on whether quantu m computers can be used to eﬃciently solve non-Abelian hidden subgroup problems. An alterna tive optimistic view is also possible. In this view what these results show is that any eﬃcient quantum algorithm for the non-Abelian hidden subgroup problem (in what is known as the sta ndard method) must have a particular form. Speciﬁcally such algorithms must perform qu antum circuits across many separate quantum queries to a hidden subgroup oracle. If we are to view these results in a positive manner, then this tells us that what is needed, if we are go ing to eﬃciently solve non-Abelian hidden subgroup problems, are new quantum transfor ms which can act across many such queries. In this paper we provide some evidence in favor of this optimistic view. Recently, Ba- con, Childs, and van Dam[14] have shown that quantum computers can eﬃciently solve the hidden subgroup problem for certain semidirect product groups. O ne such group which admits an eﬃcient quantum algorithm is the Heisenberg group, Z2 p⋊ Zp. The algorithm of Bacon, Childs, and van Dam[14] was discovered using the framewo rk of pretty good measurements[21, 14, 22, 15, 19]. This yields a particular algorithm f or solving the Heisen- berg hidden subgroup problem which is optimal and which is related to t he solution of certain algebraic equations. Here we show that the structure of this quan tum algorithm can be de- rived almost solely from symmetry arguments. These symmetry arg uments give rise to a transform, the Clebsch-Gordan transform over the Heisenberg group, which can be used to help eﬃciently solve the Heisenberg hidden subgroup problem. Previo usly a Clebsch-Gordan transform was used by Kuperberg to ﬁnd a subexponential algorit hm for the dihedral hidden subgroup problem[23, 24]. Clebsch-Gordan transforms over the u nitary group[25, 26] and a certain form of measurement were demonstrated to not help in solv ing a hidden subgroup problem by Childs, Harrow, and Wocjan[27]. Further Moore, Russell, and Sniady have re- cently shown that a certain form of Clebsch-Gordan transform us ed to perform a quantum algorithm which mimics Kuperberg’s algorithm cannot be used to obtain even a subexponen- tial time algorithm for the hidden subgroup problem relevant to grap h isomorphism[28]. Here we show that these negative results can be overcome, at least for the Heisenberg group, by performing a Clebsch-Gordan

Question 313 multiple-choice

Quantum error-correcting codes are essential for protecting quantum information from noise and loss, especially in systems governed by physical symmetries. When such codes are required to be covariant under continuous symmetries, their ability to correct certain errors can be fundamentally constrained. Which statement best explains why a quantum code covariant under a continuous symmetry cannot perfectly protect logical information against erasure errors at known locations? 1) The environment cannot distinguish logical charge states because symmetry generators commute with all observables. 2) The code's logical states are mapped to physical states with unrelated symmetry generator eigenvalues, leading to random charge leakage. 3) The local nature of symmetry generators allows the environment to learn about the logical charge by accessing erased subsystems, causing information to leak. 4) Entanglement fidelity is maximized when symmetry generators act globally and not locally. 5) Quantum codes with discrete symmetries always perfectly protect against erasure errors regardless of locality. 6) The trace distance between environment output states is always zero for covariant codes. 7) Covariant codes can avoid information leakage by encoding all logical states with identical charge eigenvalues.

✓ Correct Answer:

The correct answer is 3) The local nature of symmetry generators allows the environment to learn about the logical charge by accessing erased subsystems, causing information to leak..

📚 Reference Text:

poorly a code necessarily performs against erasures atknown locations, given that the code must be covariant with respect to a continuous symmetry. For the sake of clarity, we first present a simplified version of our general bound.Consider an encoding map jxi L→jψxiAwith respect to some basis fjxiLg, which we may represent by an isometry VL→A¼P xjψxiAhxjL. Denote the corresponding encod- ing channel by EL→Að·Þ¼Vð·ÞV†. Pick any generator TLfrom the Lie algebra of the symmetry acting on L. Let Tibe the corresponding generator acting on the ith physical subsystem Ai, with the total generator on Abeing TA¼P iTi. As Hermitian matrices, these are quantum mechanical observables whose eigenvalues we may think of as abstract “charges. ”(These charges might correspond to the component of angular momentum in a given direction, the number of particles, or some other physical quantity.) Crucially, since the code Eis covariant, a logical charge eigenstate jtiLmust be encoded into a code word jψtiAwhich is an eigenvector of TAwith the same eigenvalue t, up to a constant offset ν. Let us assume without loss of generality ν¼0, because this offset can be absorbed in TAby shifting Ti→Ti−ðν=nÞ1. Assume the environment erases a subsystem ichosen at random with probability qi¼1=n. Then, the environment gets the information represented by the complementary channel (14). That is, if the original state is jxiL, then the environment gets the state ρx i¼trAnAiðjψxihψxjLÞon sub- system iwith probability 1=n. Yet, because the charge observable is local, the environment can gain information about the expectation value of the charge. Indeed, foranyjxi L, trðTLjxihxjLÞ¼trðTAjψxihψxjAÞ¼X itrðTiρx iÞ;ð22Þ where the first equality holds because the code is covariant and the second because the charge is local. Hence, if wedefine the observable Z C0E¼nP ijiihijC0⊗Tion the environment systems, we have trðTLjxihxjLÞ¼tr½ZC0EdN∘EðjxihxjLÞ/C138; ð23Þmaking it clear that the environment can measure the average charge using the information it has available. Surely, if the charge expectation value leaks to the environment, then the code must be bad. However, the accuracy of the code is measured in terms of an entangle- ment fidelity (worst-case or fixed input) to the identitychannel. Hence, it still remains to relate the accuracy of thecode to the environment ’s ability to access the code word ’s total charge. On one hand, we observe that the difference in expectation value of Z C0Eon the environment can be translated into a distinguishability of code words in termsof the trace distance. More precisely, in our case, considertwo logical charge eigenstates jϕ /C6iLcorresponding to the maximum and minimum eigenvalues of TL. Then, we may exploit the characterization of the trace distance asδðρ;σÞ¼max Qtr½Qðρ−σÞ/C138=kQk∞for any states ρ,σand where the optimization is over all Hermitian operators Q,t o see that δ½dN∘Eðjϕ−ihϕ−jLÞ;dN∘EðjϕþihϕþjLÞ/C138≥ΔTL 2kZC0Ek∞;ð24Þ whereΔTLis the spectral range of TL, i.e., the difference between the maximum and minimum eigenvalue of TL.W e assume here for simplicity that the maximum and minimumeigenvalues of T iare equal in magnitude, such that ΔTi¼2kTik∞; hence, 2kZC0Ek∞¼2nmax ikTik∞¼ nmax iΔTi. On the other hand, if the environment ’s states are distinguishable for different

Question 314 multiple-choice

Quantum algorithms have shown significant advantages over classical methods for certain group-theoretic problems, especially those framed as hidden subgroup problems (HSPs). The intersection of quantum measurement strategies and classical algorithmic tools has led to new breakthroughs for previously hard instances. Which approach enables an efficient quantum algorithm for the generalized hidden shift problem when the number of shifted functions M equals ⌊N^(1/k)⌋ for any fixed integer k ≥ 3, and involves joint measurement on k copies of quantum states? 1) Pretty good measurement (PGM) 2) Classical abelian Fourier sampling 3) Lenstra’s algorithm alone 4) Shor’s algorithm 5) Dihedral group representation theory 6) Subset sum reduction 7) Grover’s search algorithm

✓ Correct Answer:

The correct answer is 1) Pretty good measurement (PGM).

📚 Reference Text:

cases, the matrix sum problem can be solved, giving an eﬃcient quantum algorithm for the co rresponding hidden subgroup problem [3]. However, since the average cas e subset sum problem appears to be diﬃcult, this approach has not yielded an improved algorithm for the dihedral hsp. In this article, we show how the pgmapproach provides an eﬃcient quantum algorithm for a problem that interpolates between t he abelian and dihedral hidden subgroup problems. The dihedral hspis equivalent to the hidden shift problem , in which the goal is to determine a hidden shift s∈ZN given two injective functions f0,f1satisfying f0(x) =f1(x+s). Instead of only two such functions, we consider Msuch functions, each one shifted from the previous by a ﬁxed hidden shift s. IfM=N, this problem is an instance of the abelian hsponZN×ZNwith the hidden subgroup /an}⌊ra⌋k⌉tl⌉{t(1,s)/an}⌊ra⌋k⌉tri}ht, which can be solved eﬃciently using abelian Fourier samplin g. Even the caseM=Nis classically intractable, and the problem only becomes mo re diﬃcult for smaller M. In particular, for M≪N, abelian Fourier sampling fails to determine the hidden shift. Using the pgmapproach, we give, for any ﬁxed integer k≥3, an eﬃcient quantum algorithm that solves this problem for M=⌊N1/k⌋. The algorithm works by implementing a joint measurement on kcopies of certain quantum states that encode the hidden shift. Because for each M≥M′the generalized hidden shift problem on {0,...,M−1}×ZNcan be reduced to the generalized hidden shift problem on{0,...,M′−1}×ZN, for any ﬁxed ǫ >0, this gives an eﬃcient quantum algorithm for all M≥Nǫ. By applying the general pgmapproach developed in [3], we ﬁnd that the matrix sum problem corresponding to the generalized hid den shift prob- lem is the following: given uniformly random x∈Zk Nandw∈ZN, ﬁnd b∈ {0,...,M−1}ksuch that b·xmodN=w. As this is a linear Dio- phantine equation with convex constraints, it is an instanc e of integer pro- gramming, and can be solved using Hendrik Lenstra’s algorit hm for that problem [20], which is eﬃcient as long as the dimension kis constant. Thus QUANTUM ALGORITHM FOR A GENERALIZED HIDDEN SHIFT PROBLEM 3 our algorithm for the generalized hidden shift problem reit erates a theme of [3]: by combining abelian quantum Fourier transforms with n ontrivial clas- sical (or quantum) algorithms, one can ﬁnd eﬃcient quantum a lgorithms for hsp-like problems via the implementation of entangled quantum measure- ments. This result is encouraging since entangled measurem ents are known to be necessary for some hidden subgroup problems—in partic ular, for the symmetric group [22]. The remainder of this article is organized as follows. In Sec tion 2, we describe the generalized hidden shift problem in detail and explain how it can be viewed as a quantum state distinguishability problem . In Section 3, we review the pretty good measurement approach to such probl ems, prove that this approach solves the generalized hidden shift prob lem when the number of states is k≥ ⌈1/ǫ⌉, and explain how it can be implemented by solvinganappropriatematrixsumproblem. InSection4, wee xplainhowthe matrix sumproblemcan besolved eﬃciently (for constant k) usingLenstra’s algorithm for

Question 315 multiple-choice

In modular representation theory, simple algebraic groups over algebraically closed fields of characteristic p ≥ h, where h is the Coxeter number, exhibit distinctive structural and combinatorial properties. Tools such as the Jantzen sum formula and its recursion play key roles in analyzing the filtration and decomposition of Weyl modules with p-regular highest weights. Which statement accurately describes a major consequence of applying a recursion formula to the Jantzen sum formula for Weyl modules with p-regular highest weights? 1) It allows efficient computation of the Jantzen sum formula, leading to practical bounds on the length of the Jantzen filtration for these modules. 2) It proves that all Weyl modules are simple over fields of characteristic p < h. 3) It shows that the filtration length is always equal for modules in arbitrary non-adjacent alcoves. 4) It establishes that the Jantzen sum formula is identical to Andersen’s sum formula for all representations. 5) It demonstrates that the Coxeter number has no relevance in modular representation theory. 6) It confirms that tilting modules cannot be filtered using Andersen’s sum formula. 7) It provides a direct classification of all simple algebraic groups without reference to their representations.

✓ Correct Answer:

The correct answer is 1) It allows efficient computation of the Jantzen sum formula, leading to practical bounds on the length of the Jantzen filtration for these modules..

📚 Reference Text:

Title: Coxeter combinatorics for sum formulas in the representation theory of algebraic groups Year: 2021 Paper ID: 1bef5a3c0406ad4924e998c1a0b94e09293d0cd7 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/1bef5a3c0406ad4924e998c1a0b94e09293d0cd7 Abstract: Let G G be a simple algebraic group over an algebraically closed field F \mathbb {F} of characteristic p ≥ h p \geq h , the Coxeter number of G G . We observe an easy ‘recursion formula’ for computing the Jantzen sum formula of a Weyl module with p p -regular highest weight. We also discuss a ‘duality formula’ that relates the Jantzen sum formula to Andersen’s sum formula for tilting filtrations and we give two different representation theoretic explanations of the recursion formula. As a corollary, we also obtain an upper bound on the length of the Jantzen filtration of a Weyl module with p p -regular highest weight in terms of the length of the Jantzen filtration of a Weyl module with highest weight in an adjacent alcove.

Question 316 multiple-choice

In lattice gauge theory, the construction of physical states and operators relies on group representation theory and algebraic techniques. Quantum simulations of such theories often utilize methods from tensor algebra and symmetry analysis. Which condition must be satisfied at each lattice site to ensure local gauge invariance when constructing gauge-invariant states using irreducible representations and Clebsch-Gordan coefficients? 1) The combination of irreducible representation labels must yield nonzero Clebsch-Gordan (3J) symbols at each site. 2) All link multiplicities must be set to zero at the boundaries. 3) Only symmetric tensor products of representations are allowed. 4) Each link must have identical representation labels at both ends. 5) The sum of all representation labels across the lattice must be even. 6) Clebsch-Gordan coefficients must vanish for all site combinations. 7) Only Abelian group representations can be used for gauge invariance.

✓ Correct Answer:

The correct answer is 1) The combination of irreducible representation labels must yield nonzero Clebsch-Gordan (3J) symbols at each site..

📚 Reference Text:

and the CG coeﬃcients for D3andD4which were listed in Appendix A, one can produce a gauge invariant state on the ladder by taking the ten- sor product of the analogous expression at each site. This is represented by the expres- Frommetal. EPJQuantumTechnology (2023) 10:31 Page17of19 sion |ψ/angbracketright=/circlemultiplydisplay s|00/angbracketrights (56) =/summationdisplay {mi,1}···/summationdisplay {mi,ns}/parenleftbigg/circlemultiplydisplay /lscript|j/lscriptmL/lscriptmR/lscript/angbracketright/parenrightbigg/productdisplay s/parenleftBigg j/lscript1,sj/lscript2,sj/lscript3,s m1,sm2,sm3,s/parenrightBigg (–)f(j/lscript1,s,j/lscript2,s,j/lscript3,s;m3,s), wherethe {mi,s}arethelocalmultiplicitiesatsite softhelink /lscriptinrepresentation j/lscript,andin theproductoverlinks,eachlinkhasmultiplicities mL/lscriptandmR/lscriptassociatedwiththe“left” and “right” ends of the link. One can thus label the states of the Hilbert space by a set of Nlinks=3Nintegers characterizing the irreps, where Nis the number of plaquettes, with the condition that at each site there is a valid tuple with nonzero 3J symbol. From here,one has everything one needs in order to enumerate the states in the physical Hilbert space. AppendixC: Plaquettecontribution Here we provide the details for the matrix element of a plaquette in the physical Hilbert space.Thisderivationissimilartothatof[ 9]forthecaseof SU(2).Theapplicationofthe plaquettein( 13)onthearbitraryphysicalstatelistedin( 56)isgivenby ˆP/vectorx|ψ/angbracketright=/summationdisplay n1,...,n4/summationdisplay {mLi}/summationdisplay {mRi}ˆU˜l1;n1n2|j˜l1mL˜l1mR˜l1/angbracketright (57) ⊗ˆU˜l2;n2n3|j˜l2mL˜l2mR˜l2/angbracketright⊗ˆU˜l3;n4n3|j˜l3mL˜l3mR˜l3/angbracketright ⊗ˆU˜l4;n1n4|j˜l4mL˜l4mR˜l4/angbracketright/circlemultiplydisplay i/∈L/vectorx|jimLimRi/angbracketright ×/productdisplay s/parenleftBigg j/lscript1,sj/lscript2,sj/lscript3,s m1,sm2,sm3,s/parenrightBigg (–)f(j/lscript1,s,j/lscript2,s,j/lscript3,s;m3,s), (58) wherethesumsover mLiandmRirepresent3 Nindividualsumswhoserangesdependon the conﬁguration of the state |ψ/angbracketright. Using the Clebsch-Gordan series, we can obtain the resultofeachlinkoperatoractingonanarbitrarylinkstate.Applyingthistoeachtermin(58)yields ˆP /vectorx|ψ/angbracketright=/summationdisplay n1,...,n4/summationdisplay {mLi}/summationdisplay {mRi}/productdisplay s/parenleftBigg j/lscript1,sj/lscript2,sj/lscript3,s m1,sm2,sm3,s/parenrightBigg (–)f(j/lscript1,s,j/lscript2,s,j/lscript3,s;m3,s)/circlemultiplydisplay i/∈L/vectorx|jimLimRi/angbracketright ⊗/summationdisplay J˜l1/summationdisplay ML˜l1/summationdisplay MR˜l1/radicalBigg dim(j˜l1) dim(J˜l1)/angbracketleft2n1j˜l1mL˜l1|J˜l1ML˜l1/angbracketright/angbracketleftJ˜l1MR˜l1|2n2j˜l1mR˜l1/angbracketright ×|J˜l1ML˜l1MR˜l1/angbracketright ⊗/summationdisplay J˜l2/summationdisplay ML˜l2/summationdisplay MR˜l2/radicalBigg dim(j˜l2) dim(J˜l2)/angbracketleft2n2j˜l2mL˜l2|J˜l2ML˜l2/angbracketright/angbracketleftJ˜l2MR˜l2|2n3j˜l2mR˜l2/angbracketright ×|J˜l2ML˜l2MR˜l2/angbracketright Frommetal. EPJQuantumTechnology (2023) 10:31 Page18of19 ⊗/summationdisplay J˜l3/summationdisplay ML˜l3/summationdisplay MR˜l3/radicalBigg dim(j˜l3) dim(J˜l3)/angbracketleft2n4j˜l3mL˜l3|J˜l3ML˜l3/angbracketright/angbracketleftJ˜l3MR˜l3|2n3j˜l3mR˜l3/angbracketright ×|J˜l3ML˜l3MR˜l3/angbracketright ⊗/summationdisplay J˜l4/summationdisplay ML˜l4/summationdisplay MR˜l4/radicalBigg dim(j˜l4) dim(J˜l4)/angbracketleft2n1j˜l4mL˜l4|J˜l4ML˜l4/angbracketright/angbracketleftJ˜l4MR˜l4|2n4j˜l4mR˜l4/angbracketright ×|J˜l4ML˜l4MR˜l4/angbracketright. (59) We now take the inner product of ( 59)w i t has e c o n dg e n e r i cs t a t e |ψ/prime/angbracketrightwhich satisﬁes Gauss’law.Wenotethatstatesintherepresentationbasisareorthonormal /angbracketleftbig j/primem/primen/prime|jmn/angbracketrightbig =δjj/primeδmm/primeδnn/prime, (60) andthusweobtaintheresultinEq.( 14). Acknowledgements TheauthorswouldliketothankBrendanPankovichforusefuldiscussionsandMichaelSpannowskyforcommentsonthemanuscript. Funding OpenAccessfundingenabledandorganizedbyProjektDEAL.TheworkissupportedbytheDeutscheForschungsgemeinschaft(DFG,GermanResearchFoundation)throughthegrantCRC-TR211“Strong-interactionmatterunderextremeconditions”–projectnumber315477589–TRR211andbytheStateofHessewithintheResearchClusterELEMENTS(ProjectID500/10.006). Abbreviations QCD,QuantumChromodynamics;QED,QuantumElectrodynamics;API,Applicationprogramminginterface. Availabilityofdataandmaterials Simulationdataareavailablefromthecorrespondingauthoruponreasonablerequest. Declarations EthicsapprovalandconsenttoparticipateNotapplicable. Consentforpublication Notapplicable. Competinginterests Theauthorsdeclarenocompetinginterests. Authorcontributions Allauthorscontributedequallyintheconceptionphase,CWandMFcarriedoutthecomputationsandsimulations.Allauthorsjointlydraftedandreviewedthemanuscriptandapprovedthesubmission. Received:2December2022 Accepted:14August2023 References 1. 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Question 317 multiple-choice

In the representation theory of compact Lie groups, the interplay between root systems, Weyl groups, and lattices underpins the structure of weight spaces and their associated symmetries. These algebraic and geometric objects are foundational in both mathematical physics and modern geometry. Which statement describes the precise relationship between the volume of the coroot lattice ΛR and the fundamental alcove A for the affine Weyl group action on the Cartan subalgebra t? 1) The volume of ΛR is equal to the volume of the Weyl chamber F0. 2) The volume of the fundamental alcove A is always larger than the volume of ΛR. 3) The volume of ΛR is equal to the number of coroots times the volume of A. 4) The volume of the fundamental alcove A equals the determinant of the Cartan matrix. 5) The volume of ΛR is the product of the volumes of all Weyl chambers in t. 6) The volume of ΛR equals the order of the Weyl group |W| times the volume of the alcove A. 7) The volume of ΛR and that of the alcove A are always unrelated for different Lie groups.

✓ Correct Answer:

The correct answer is 6) The volume of ΛR equals the order of the Weyl group |W| times the volume of the alcove A..

📚 Reference Text:

ΛR⊂tis theZ–span of the coroots. This gives the identiﬁcation T=t/ΛR. (74) 19 For every integer k≥1we deﬁne a k-scaled inner product (·,·)k≡k(·,·)1. We use it to deﬁne the dual of ΛRwith respect to (·,·)k Λw (k)≡/braceleftbig γ∈tsuch that (γ,λ)k∈Z, for everyλ∈ΛR/bracerightbig . ΛRis integral with respect to (·,·)k, meaning that ΛR⊂Λw (k)for allk. In the special casek= 1,Λw= Λw (1)is called the weights lattice . We will denote by Z(k)the ﬁnite abelian group Z(k)≡Λw (k)/ΛR. (75) For any full rank lattice Λwe write VolkΛfor the volume of a fundamental domain of the action of Λontwith respect to the inner product (·,·)k. For a ﬁnite set C, |C|is its cardinality. We have Vol1Λw (k)=Vol1Λw kn=/parenleftbig Vol1ΛRkn/parenrightbig−1,|Z(k)|=Vol1ΛR Vol1Λw (k)=|ΛR|2kn(76) The Weyl group Wis the group generated by the reﬂections sαthrough the hyperplane orthogonal to α, i.e. sα(v) =v−2(α,v) (α,α). (77) A fundamental domain for the action of Wontis called a Weyl Chamber . After a choice of a set of simple roots ∆is ﬁxed, we can identify a preferred chamber F0, called fundamental Weyl chamber as F0≡ {x∈t:(x,α)1>0, for allα∈∆}. (78) In this way tis decomposed into |W|Weyl chambers and Wacts simply transitively on the set of them, i.e. t=/uniondisplay w∈Ww(F0). (79) The space L2(t,⟪·,·⟫t)is deﬁned with respect to the following inner product ⟪f,g⟫t≡/integraldisplay tf(x)g(x)dvolk(x), (80) wheredvolkis the measure on tinduced by (·,·)k. Anf∈L2(t)can be projected orthogonally into the subspace of W–invariant (resp. W–anti-invariant ) functions L2 +(t)(resp.L2 −(t)) as follows P+(f)(x) =1 |W|/summationdisplay w∈Wf(w(x)) (inariant) (81) P−(f)(x) =1 |W|/summationdisplay w∈Wdet(w)f(w(x)) (anti–inariant) (82) wheredet(w)is the same as the sign character. In particular, both L2 +(t)andL2 −(t) are isomorphic to L2(F0)by restriction to a Weyl chamber. Theaﬃne Weyl group Wais the semidirect product ΛR⋊W. Itsfundamental alcoveA⊂tis the set A≡ {x∈t:0<(x,α)1<1,for allα∈∆+}, (83) 20 andAis a fundamental domain for the action of Waont. Denote by ΩR⊂tthe fundamental domain of ΛRcontainingA. We have Vol1ΛR= Vol1(ΩR) =|W|Vol1(A). (84) We deﬁne the following two sets of indexes Ik≡Λw (k)∩A, Ik≡Λw (k)∩A. (85) The setIkcan be thought as a fundamental domain for the action of WonZ(k) or the action of WaonΛw (k). By scaling by kwe get a bijection between Ikand the weights in the fundamental k-alcove (compare with [Je] equation A.16 and section A.5). The space of functions x:Z(k)−→Cis a ﬁnite dimensional vector space of dimen- sion|Z(k)|. A basis is given by the functions δγ,γ∈ Z(k)deﬁned as δγ(γ0) =δ/parenleftbig γ−γ0mod ΛR/parenrightbig . (86) The subspaces C(k)andC(k)of, respectively, W–invariant and W–anti-invariant functions are then spanned, respectively, by the following two basis C(k)= SpanC/an}b∇acketle{teγ/an}b∇acket∇i}htγ∈IkC(k)= SpanC/an}b∇acketle{t˜eγ/an}b∇acket∇i}htγ∈Ik (87) eγ≡1/radicalbig |W|/summationdisplay w∈Wδw(γ), for everyγ∈Ik, (88) ˜eγ≡1/radicalbig |W|/summationdisplay w∈Wdet(w)δw(γ), for everyγ∈Ik. (89) We use the sets IkandIkto index the basises {eγ}and{˜eγ}respectively. These basises are orthonormal with respect to the product ⟪x,y⟫Z(k)≡/summationdisplay γ∈Zx(γ)y(γ). (90) References [A1] J.E. Andersen, Geometric Quantization of Symplectic Manifolds with respe ct to reducible non-negative polarizations . Commun. in Math. Phys. 183(1997) 401–421. [A2] J.E Andersen, Deformation quantization and geometric quantization of ab elian moduli spaces , Comm. of Math. Phys. 255, (2005) 727–745. [A3] J.E. Andersen, Asymptotic faithfulness of

Question 318 multiple-choice

In computational quantum chemistry, constructing efficient and accurate basis sets is essential for simulating electronic structures of molecules and materials. Algorithms often exploit the spatial localization of orbitals to reduce computational cost and improve scalability. Which parameter in an overlap-based localized basis selection algorithm determines whether the overlap between two orbitals is considered substantial, and is independent of the system geometry? 1) The magnitude threshold ε for orbital overlap 2) The total number of orbitals in the system 3) The spatial separation between orbital centers 4) The dimensionality of the simulation domain 5) The symmetry group of the molecule or material 6) The maximum allowed number of candidate columns 7) The type of matrix factorization used in basis selection

✓ Correct Answer:

The correct answer is 1) The magnitude threshold ε for orbital overlap.

📚 Reference Text:

only exerts in uence on, i.e.signi cantly overlaps with, nearby localized orbitals. Hence, we may re ne the selected columns locally. This means that for each orbital we may simply gure out which orbitals it substantially overlaps with and compute a QRCP on just those orbitals (rows) ofewhile simultaneously omitting columns with small norm over those rows. This 9 process will yield a small number of selected columns that we add to a list of potential candidate columns for the nal localization. However, because we repeat this process for each localized orbital we might have more than netotal candidate columns by a small multiplicative factor. Therefore, we perform one nal column pivoted QR factorization on these candidate columns to select the nal set C. In principle, while the localized orbitals get small on large portions of the domain, they are not actually zero. Hence, for any given we say two orbitals substantially overlap if there is any spatial point where they both have relative magnitude greater than. We nd in practice that choosing close to the relative value at which one considers an orbital to have vanished suces, though taking !0 one recovers the original SCDM algorithm. Importantly, this parameter is completely independent of the geometry of the problem under consideration and thus does not require detailed knowledge of the system to set. We detail our complete algorithm for computing the orthogonalized SCDM from approximately localized input in Algorithm 4. The only parameter is how small a column is for it to be ignored. At a high level, the goal is simply to generate additional candidate columns by allowing each orbital to interact with its immediate neighbors. Provided that each orbital is only close to a small number of others, and that the approximate localization is sucient this procedure may be performed quickly. Algorithm 4 Re ning an approximately localized collection of basis vectors Given: approximately localized Kohn-Sham orbitals e and column tolerance 1:Ji=n j2f1;:::;Ngjje ij>max kje i(rk)jo fori= 1;:::;n e 2:fori= 1;:::;n edo 3: SetRi=fj2f1;:::;n egjJ i\Jj6=;g 4: SetLi=[ j2RiJj 5: Compute a column pivoted QR factorization of e Ri;Liand denote the pivot columnsCi 6:end for 7:SeteC=[iCi 8:Compute the column pivoted QR factorization eeC;: =QR 9:Form the localized basis = eQ To illustrate the behavior of this algorithm, we sketch the behavior of Algorithm 4 in two cases. In one case, after the approximate localization, we have two sets of orbitals whose support sets after truncation are disjoint. This is shown in Figure 3.1. Here, simply computing two independent QRCP factorizations is actually equivalent to computing the QRCP of the entire matrix. As we see Algorithm 4 partitions the orbitals into two sets and then only considers the columns with signi cant norm over the orbital set. In the second case, illustrated in Figure 3.2, we have a chain of orbitals whose support set after truncation forms a connected region in the spatial domain. In this situation we do not actually replicate the computation of

Question 319 multiple-choice

In computational chemistry, the accurate modeling of molecules and their interactions relies heavily on molecular mechanics force fields. These force fields are often parameterized to reproduce quantum mechanical properties and are evaluated using a range of quantitative metrics and benchmark comparisons. Which of the following most directly demonstrates a force field's ability to account for chemical diversity when modeling molecules with similar functional groups? 1) Using a single set of force constants for all carbon-halogen bonds regardless of halogen identity 2) Reporting only average vibrational frequencies for a series of molecules 3) Achieving rapid convergence of parameter optimization cycles 4) Matching simulated bulk densities to experimental values for a single compound 5) Assigning different force constants and equilibrium bond lengths to C–Cl bonds depending on their chemical environment 6) Utilizing B3LYP-D quantum calculations solely for intramolecular energy profiles 7) Applying the same equilibrium bond length to all C–Br bonds in various compounds

✓ Correct Answer:

The correct answer is 5) Assigning different force constants and equilibrium bond lengths to C–Cl bonds depending on their chemical environment.

📚 Reference Text:

ear to indicate a rather reliable matching between the QM and MM descriptions. A very similar accuracy was achieved along the whole benchmark set, as conﬁrmed by Table 3, which reports for each 18 0.20.40.60.81NM proj. coeff. 0 5 10 15 20 25 NM0100020003000ν (cm-1) QM QMD-FF -180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180 δ (degrees)05101520∆E (kJ/mol)QM QMD-FF Figure 4: Joyceparameterization results achieved for I2(2-iodo-propane). Top panel: overlap of QM and QMD-FF normal modes; middle panel: vibrational frequ encies computed at QM level and by means of the Joyceparameterized QMD-FF; bottom panel: QM and QMD-FF torsiona l energy proﬁle along the principal dihedral angle. compoundσJ(Joycestandard deviation22,33) andσν(the standard deviation computed from the squared diﬀerences between the vibrational freque ncies computed at QM level and the ones computed with the QMD-FF). The ﬁnal intramolecular parameters obtained for all molecu les are reported in detail in Table 4 and in Tables B-E. By looking at Table 4, where the intramolecular har- monic parameters are reported for the ﬁrst four investigate d molecules, the QMD-FF capability to speciﬁcally account for chemical diversity, even within a class of rather sim- ilar molecules, clearly appears. For instance, the C-Cl bon d, which takes place in the C1-C4compounds within diﬀerent chemical surroundings, is descr ibed byJoycewith diﬀerent force constants ksand equilibrium positions R 0, respectively spanning a range between 962 - 1624 kJ/mol ˚A−2and 1.75 and 1.83 ˚A. On the same foot, the Carbon - Halogen bond is consistently described along the series, bei ng (see Supporting Informa- tion for details) ∼1000 kJ/mol ˚A−2, 800 kJ/mol ˚A−2and 500 kJ/mol ˚A−2the average 19 moleculeσJ(kJ/mol) σν(cm−1) C1 0.016 55 C2 0.016 46 C3 0.011 59 C4 0.018 60 N1 0.005 66 N2 0.010 29 B1 0.020 40 B2 0.016 53 B3 0.036 55 B4 0.028 46 I1 0.025 42 I2 0.057 52 Table3: σJandσν(seetextfordeﬁnitions)standarddeviationsobtainedfor eachinvestigatedmolecules byJoyceintramolecular parameterizations. force constants found for Cl, Br and I, respectively, and ∼1.80˚A, 2.00˚A and 2.20 ˚A the equilibrium bond lengths. 3.3.2 Intermolecular Parameterization Once all the intramolecular QMD-FFs were prepared, the next step consisted in obtaining the parameters for the intermolecular description. As previ ously detailed, this was done through the Pickypackage, optimizing the QMD-FF dimer IPES against the refer ence one, sampled through B3LYP-D QM calculations, following the results of Section 3.2. Table 5 contains the ﬁnal values of selected quantities moni tored during the parame- terization cycles. By looking at the third and fourth column , it appears that all parame- terizations successfully ended within 5 to 8 cycles, reachi ng the selected ∆P threshold of 0.5 kJ/mol. The bulk densities ( ρMC F) achieved in the last parameterization cycle through theMCsimulationsandreportedinTable5cover abroadrange ofvalues, from952kg/m3 (chloroform at 273K) to 2549 kg/m3(dibromomethane at 298K). This can be interpreted as a ﬁrst indication that the QMD-FFs are able to maintain the capability of the parent QM description in accounting for chemical diversity. In par ticular, diﬀerent densities were predicted by the QMD-FF also for compounds with

Question 320 multiple-choice

Entanglement entropy in quantum circuits is closely tied to the nature of interactions between qubits and the structure of the circuit. The quantum Fourier transform (QFT) is a fundamental operation in quantum computing with unique entanglement properties. Which property of the QFT circuit ensures that its entanglement entropy remains bounded even as the number of qubits increases? 1) The presence of Hadamard gates that create uniform superpositions 2) Exponentially decaying Z-Z interactions implemented by controlled phase gates 3) Use of long-range interactions that scale as 1/r with distance 4) The application of SWAP gates after every controlled gate 5) Volume law scaling of entanglement due to nonlocal operations 6) Constant interaction strength between all qubits 7) Removal of all phase gates, leaving only single-qubit operations

✓ Correct Answer:

The correct answer is 2) Exponentially decaying Z-Z interactions implemented by controlled phase gates.

📚 Reference Text:

matrix is a doubly stochastic linear map, which directly implies majorization [ 32]. Therefore, Theorem 1 is proved. C. Relation to the area law In this subsection, we develop a more intuitive expla- nation for the QFT’s low entanglement. Speciﬁcally, we show that the QFT has the same entanglement structure as the time evolution of a Hamiltonian with exponential decay of interaction, which is pseudolocal. The dynamics under such interactions are known to obey a variant of the area law [ 33,34], which bounds by a constant how much the entanglement entropy of a system can be changed. We ﬁrst claim that removing Hadamard (H)gates from Qndoes not change its entanglement structure. To see how, recall the equation introduced in Sec. II B, Sch j(Qn)=Sch j(/Omega1n,j). (32) One observes that since /Omega1n,jcontains only controlled phase gates, the Hgates do not play any role in Qn’s Schmidt coeﬃcients. More generally, if we denote the operator obtained from removing Hgates in QnasQP n, we can show that any reasonable bipartite entanglement measure is identical between QnandQP n; see Appendix Efor details. Therefore, the entanglement study of QP ncan be directly applied to QP n. Note this is entirely due to the QFT circuit’s special structure, and generally single-qubit gates sand- wiched by multiqubit gates will change the entanglement structure of the operator. Interestingly, QP ncan be physically interpreted as the time-evolution operator of an exponentially decaying Hamiltonian over a time interval of t=π. To see how, we ﬁrst express QP nas a product of phase gates: QP n=n−1/productdisplay l=1n/productdisplay m=l+1Pl,m/parenleftBigπ 2m−l/parenrightBig . (33) 040318-6 QUANTUM FOURIER TRANSFORM... PRX QUANTUM 4,040318 (2023) One can expand Pl,m(θ)in the subspace spanned by qubits landmas Pl,m(θ)=exp(iθ|1/angbracketright/angbracketleft1|l⊗|1/angbracketright/angbracketleft1|m), (34) where |1/angbracketright/angbracketleft1|lis the projection to state |1/angbracketrighton qubit l.QP n can then be expressed as QP n=exp/parenleftBigg iπn−1/summationdisplay l=1n/summationdisplay m=l+1/parenleftbigg1 2/parenrightbiggm−l |1/angbracketright/angbracketleft1|l⊗|1/angbracketright/angbracketleft1|m/parenrightBigg , (35) which arises from time evolving by πtime units under the Hamiltonian HP n=−n−1/summationdisplay l=1n/summationdisplay m=l+1/parenleftbigg1 2/parenrightbiggm−l |1/angbracketright/angbracketleft1|l⊗|1/angbracketright/angbracketleft1|m. (36) Note that the projection operator |1/angbracketright/angbracketleft1|is equivalent to the Pauli Zoperator shifted and rescaled, so the Hamiltonian HP ncan also be physically interpreted as Z-Zinteraction on each pair of sites with exponentially decaying corre- lations. Such interaction is considered as short range, and the sum of interactions across any partition converges as the system size increases, and thus so does the entangle- ment entropy. Speciﬁcally, Gong et al. [34] proved that for a quantum state on a D-dimensional ﬁnite or inﬁnite lattice, if it evolves in time under a Hamiltonian with inter- actions decaying faster than 1 /rD+1, where ris the distance between two sites, the entanglement entropy of the state with respect to a subregion cannot change faster than a rate proportional to the size of the subregion’s boundary. For the case of QP non a 1D qubit chain, the interaction decays as 1/2r, which is exponentially faster than 1 /r2. Therefore, the entanglement rate is bounded by a very small constant, and the amount of entanglement generated through QP non any state |ψ/angbracketrightdoes not grow

Question 321 multiple-choice

In the study of abelian p-groups, concepts such as element height, direct sum decompositions, fully invariant subgroups, and the p-adic topology are crucial for understanding their structure and classification. The properties of these groups and their endomorphism rings have foundational significance in algebra. Which of the following statements accurately describes a key property of fully invariant subgroups of abelian p-groups with no elements of infinite height? 1) They are always equal to the basic subgroup of the group. 2) They coincide with the set of elements of order p in the group. 3) They must be direct summands of the group. 4) They are uniquely determined by the p-adic valuation of the group. 5) They are closed in the p-adic topology on the group. 6) They are finite whenever the group is countable. 7) They are precisely the subgroups invariant under all automorphisms but not necessarily under all endomorphisms.

✓ Correct Answer:

The correct answer is 5) They are closed in the p-adic topology on the group..

📚 Reference Text:

hull so the statement follows from Theorem 3.2 and Proposition 2.1. We remark that this will be generalized in the next section. https://doi.org/10.4153/CJM-1972-056-6 Published online by Cambridge University Press 620 G. D. POOLE AND J. D. REID 4. Direct sums of finite groups. In this section we show that direct sums of finite groups are quasi-injective over their endomorphism rings. In particular this covers the cases of bounded ^-groups as well as countable ^-groups without elements of infinite height. By Corollary 2.4 we may assume the groups in question are ^-groups. The result we are after will follow from more general considerations in which we only assume that our groups have no elements of infinite height. Let G be a ^-group with G 1 = 0, with fully invariant subgroup H, and with basic subgroup B. We want to estimate the size of Hom 5(i7, G), where E is the endomorphism ring of G. Write B = Yl B n, where B n = Y[b na] is a direct sum of cyclic groups of order p n, and G n = p nG + Yk> nBk. Then G = B 1 0... 0 5» ®G n for each n, so H = B1r\H ®... ®B nC\H ®G nr\H. Our immeidate object is to show that for /Ç Hom#(i7, G), we have / (Bj (^H) ^BjHH for each j. Since Bj = £[&*«], we have Bjr\H = £[&*] r\ H = Z[p njabja\ for some integers n jai 0 ^ n ja < j. For each pair of indices a, /3, [bja] and [bjp] are isomorphic summands of G so there is an endomorphism p of G such that p{bja) = bjp. Then p-Hjp = p(p nHjp) G [bjP] H H = \jr*b jp] so that tija ^ ritf. By symmetry, n ja = n^ and we conclude that Bû C\ H = pniBjy ftj being the common value of the n ja. Any / G Hom E(H, G) commutes in particular with the projections of G onto its direct summands. Therefore/ (Bj P\ H) S Bj for every j. By the re marks above, pj-n'Gj H H = 0 so f (Bj H H) ^ B j[pj~nq. But B['-nq = pnŒj = Bjr\H and f (Bj r\ H) ^ Bj C\ H as we wished to show. Now let x Ç H and write x = xi + . . . + x n + y n, Xj 6 Bj, y n £ G w. Then f(x) = £ /(*y) + f(y n) with /(*,) £BjC\H and /(y n) G G„. Since this is true of each n, at least f(x) belongs to the closure of H in the £-adic topology on G. However, from Kaplansky's characterization ot the fully invariant subgroups of ^-groups without elements of infinite height [3], it is easy to see that such subgroups are closed. Thus f(H) ^ H and we have proved PROPOSITION 4.1. Let G be a p-group with no elements of infinite

Question 322 multiple-choice

Quantum sensing leverages fragile quantum states to detect minute environmental changes, and integrating complex quantum algorithms can enhance sensor performance. The quantum Fourier transform (QFT) is a powerful tool in this domain, especially when implemented in systems with multiple qubits. In a quantum sensor employing a nitrogen-vacancy (NV) center electron spin and multiple nuclear spins, what is a key advantage of implementing the quantum Fourier transform (QFT) on the nuclear spins during correlation spectroscopy? 1) It enables quantum error correction for the electron spin state. 2) It allows measurement of spin relaxation times without decoherence. 3) It enhances the sensor’s resistance to thermal noise. 4) It increases the maximum achievable magnetic field strength. 5) It enables direct readout of the electron spin in real time. 6) It allows demultiplexing of overlapping nuclear magnetic resonance (NMR) signals, thereby increasing precision and dynamic range. 7) It eliminates the need for entanglement between nuclear spins.

✓ Correct Answer:

The correct answer is 6) It allows demultiplexing of overlapping nuclear magnetic resonance (NMR) signals, thereby increasing precision and dynamic range..

📚 Reference Text:

Title: Quantum Fourier transform for nanoscale quantum sensing Year: 2020 Paper ID: 55ac46a150e42209159f38bc3f47953e0565950a Source: semantic-scholar URL: https://www.semanticscholar.org/paper/55ac46a150e42209159f38bc3f47953e0565950a Abstract: The quantum Fourier transformation (QFT) is a key building block for a whole wealth of quantum algorithms. Despite its proven efficiency, only a few proof-of-principle demonstrations have been reported. Here we utilize QFT to enhance the performance of a quantum sensor. We implement the QFT algorithm in a hybrid quantum register consisting of a nitrogen-vacancy (NV) center electron spin and three nuclear spins. The QFT runs on the nuclear spins and serves to process the sensor—i.e., the NV electron spin signal. Specifically, we show the application of QFT for correlation spectroscopy, where the long correlation time benefits the use of the QFT in gaining maximum precision and dynamic range at the same time. We further point out the ability for demultiplexing the nuclear magnetic resonance (NMR) signals using QFT and demonstrate precision scaling with the number of used qubits. Our results mark the application of a complex quantum algorithm in sensing which is of particular interest for high dynamic range quantum sensing and nanoscale NMR spectroscopy experiments.

Question 323 multiple-choice

In algebraic graph theory, the CI-group property is a significant concept relating group structure to Cayley graphs and their automorphism groups. Elementary Abelian p-groups play a central role in ongoing research on the classification of CI-groups. Which of the following statements accurately describes a key criterion for when an elementary Abelian p-group fails to be a CI-group? 1) The group has rank less than or equal to p. 2) The group contains only cyclic subgroups of order p. 3) The group admits a nontrivial automorphism group. 4) The group has rank greater than or equal to 4p−2. 5) The group’s Cayley digraphs are always vertex-transitive. 6) The group is isomorphic to $(\mathbb{Z}_p)^2$. 7) The group’s regular subgroups are always conjugate.

✓ Correct Answer:

The correct answer is 4) The group has rank greater than or equal to 4p−2..

📚 Reference Text:

J Algebr Comb (2007) 26: 343–355 DOI 10.1007/s10801-007-0059-2 Elementary Abelian p-groups of rank greater than or equal to 4 p−2 are not CI-groups Pablo Spiga Received: 27 July 2006 / Accepted: 17 January 2007 / Published online: 17 April 2007 © Springer Science+Business Media, LLC 2007 Abstract In this paper we prove that an elementary Abelian p-group of rank 4 p−2 is not a CI(2)-group, i.e. there exists a 2-closed transitive permutation group con- taining two non-conjugate regular elementary Abelian p-subgroups of rank 4 p−2, see Hirasaka and Muzychuk (J. Comb. Theory Ser. A 94(2), 339–362, 2001 ). It was shown in Hirasaka and Muzychuk (loc cit) and Muzychuk (Discrete Math. 264(1–3), 167–185, 2003 ) that this is related to the problem of determining whether an elemen- tary Abelian p-group of rank nis a CI-group. As a strengthening of this result we prove that an elementary Abelian p-group E of rank greater or equal to 4 p−2 is not a CI-group, i.e. there exist two isomorphic Cayley digraphs over Ewhose corresponding connection sets are not conjugate in AutE. Keywords Cayley graph ·CI-group ·Schur ring ·2-closure 1 Introduction LetHbe a group and Sa subset of H. The Cayley digraph of Hwith con- nection set S, denoted Cay (H,S) , is the graph with vertex set Hand edge set {(h1,h2)|h1h−1 2∈S}. Two Cayley digraphs Cay (H,S) and Cay (H,T ) are said to beCayley isomorphic if there exists an element g∈AutHsuch that Sg=T. A sub- setSof a group His said to be a CI-subset (or Cayley isomorphic subset) if for each T⊆Hthe digraphs Cay (H,S) and Cay (H,T ) are isomorphic if and only if they are Cayley isomorphic. Finally, a group His said to be a CI-group if each subset of This research was supported by a fellowship from the Paciﬁc Institute for the Mathematical Sciences. P. Spiga (/letter) Department of Mathematics and Computer Science , University of Lethbridge, 4401 University Drive, Lethbridge, AB, Canada e-mail: spiga@cs.uleth.ca 344 J Algebr Comb (2007) 26: 343–355 His a CI-subset. We refer the interested reader to the survey article [ 5] for details, examples and the main results on CI-groups. We remark that the classiﬁcation of CI- groups is not known and at the time of this writing it seems that the classiﬁcation problem strongly depends on whether an elementary Abelian p-group of rank nis a CI-group, see [ 5]. It was proved in [ 1] thatS⊆His a CI-subset if and only if any regular subgroup of Aut (Cay(H,S)) isomorphic to His conjugate to Hin Aut(Cay(H,S)) . This result is the starting point of most of the results on CI-groups. In [3] the authors proved that if Vis a regular elementary Abelian p-subgroup of rankn(n≤4,p>2) of a 2-closed permutation group G, then any regular subgroup WofGisomorphic to Vis conjugate to VinG. In particular, by the former para- graph, as the automorphism group of a digraph is a 2-closed group we have that an elementary Abelian p-group of rank less than or equal to 4

Question 324 multiple-choice

Finite p-groups, whose orders are powers of a prime, exhibit intricate subgroup and automorphism structures central to group theory. Key problems involve the interplay of subgroup normality, regularity, and automorphism group sizes within such groups. In the study of p-groups with maximal abelian subgroups, which statement holds true for p-groups with p > 2 regarding the normality and regularity of these subgroups? 1) Maximal abelian subgroups are always nonnormal and the group is never regular. 2) All subgroups are abelian and regularity does not depend on the prime p. 3) There are no maximal abelian subgroups when p > 2. 4) Maximal abelian subgroups are never normal when p > 2. 5) If all maximal abelian subgroups are normal in a nonabelian p-group with p > 2, then the group is regular. 6) Regularity only occurs for p = 2 and not for greater primes. 7) Nonabelian p-groups with maximal abelian subgroups are always irregular for p > 2.

✓ Correct Answer:

The correct answer is 5) If all maximal abelian subgroups are normal in a nonabelian p-group with p > 2, then the group is regular..

📚 Reference Text:

Title: Further research problems and theorems on prime power groups Year: 2019 Paper ID: 8a8561a00fc11e893783c83e8bcfbf7bd72432e8 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/8a8561a00fc11e893783c83e8bcfbf7bd72432e8 Abstract: 4101. Describe the p-groups all of whose subgroups of index p, k ∈ {2, 3, 4}, are normal (three problems). Consider in detail the groups of exponent p. 4102. Study the nonabelian p-groups G all of whose maximal abelian subgroups are normal (any two elements of G generate a subgroup of class ≤ 2 so our group is regular if p > 2, by Theorem 7.1(b) in [B1]). Consider in detail the case p = 2. 4103. Find the maximal possible order of the automorphism groups of the groups of maximal class of order p. 4104. Study the non-Dedekindian p-groups covered by nonnormal subgroups. 4105. Study the p-groups G in which the intersection of any two nonincident subgroups, say A and B, of equal order (of different orders) is normal (i) either in A or in B, (ii) in 〈A,B〉. 4106. Study the p-groups G all of whose nonabelian subgroups of equal order are isomorphic (permutable). Consider in detail the case when exp(G) = p.

Question 325 multiple-choice

In computational mathematics and quantum algorithms, efficiently approximating solutions to large linear systems is crucial, especially when the system matrix has special structure such as banded circulant properties. Quantum and quantum-inspired algorithms can accelerate certain steps in these methods by exploiting data access models and structural features of the matrices involved. Which step in the approximate solution of a linear system with a K-banded circulant matrix is most directly targeted for acceleration by quantum and quantum-inspired algorithms due to its computational bottleneck in high dimensions? 1) Constructing the cyclic permutation matrix 2) Choosing the truncation threshold for the matrix polynomial 3) Normalizing the Ansatz vectors 4) Ensuring periodic boundary conditions 5) Initializing kernel parameters in machine learning applications 6) Solving the classical convex optimization for the estimator weights 7) Computing overlaps (inner products) between high-dimensional components

✓ Correct Answer:

The correct answer is 7) Computing overlaps (inner products) between high-dimensional components.

📚 Reference Text:

(MSE) loss of ˜ x, or∥C˜x−b∥2, is close to the MSE loss of x∗, or∥Cx∗−b∥2. More precisely, given additive error tolerance ε, we aim to find ˜ xsuch that LMSE(˜x) :=∥C˜x−b∥2≤ ∥Cx∗−b∥2+ε= min x∈CN∥Cx−b∥2+ε= min x∈CNLMSE(x) +ε. (4) High-level intuition. We follow the ideas introduced in [7] and divide the above problem into two parts. First, suppose we are given an Ansatz set of normalized vectors {um}M m=1, where um∈CN, such that estimator ˜ xcan be constructed by finding a weighted combination of the Ansatz set where ˜x(α) =MX m=1αmum. (5) We then define the following problem: 4 C IQ2QK Q-2 Q-KQ Q-1 ⋯ ⋯Decompose Broaden b Q2b QTb Q-2b Q-Tb Q b Q-1b Q3b Q-3b x̃ ⋯ ⋯Σ1. Ansatz set construction 2. Ansatz overlap computation 3. Classical convex optimization 〈,〉 = Σ 〈 , 〉 Cx̃ CQmb Cx̃ CQmb 〈,〉 = Σ 〈 , 〉 b b CQmb V q V ɑ ɑ†q†ɑ -2 Re | 〉 b QmQ-m||b||2Hadamard test Sample and query access b1b2b3b4 b1b2b3b4 Q b= 1 2 4ɑmQmb =CΣ=cℓBanded circulant matrix T = O(K·κC log κC/ε)Performance guarantee Cx̃Qℓ 31 2 4 3 |b1|2+|b2|2 |b3|2+|b4|2 |b1|2|b2|2|b3|2|b4|2b = FIG. 1. Here we provide an illustrated overview of our method to solve K-banded circulant linear systems. Note the fact that Ccan be written as a matrix polynomial in terms of the cyclic permutation matrix Qgiven in Equation 1 with the powers being bounded between −KandK. In step (1), we set a truncation threshold T≥Kto create a matrix polynomial with powers of Qfrom−TtoTthat serves as an approximation to C−1. To find the optimal coefficients of the matrix polynomial α, in step (2), we calculate the overlaps between individual components CQmbas well as the overlap between CQmbandb. Such overlaps can be computed with either the Hadamard test for the hybrid quantum-classical algorithm or sample and query access for the quantum-inspired algorithm. Lastly, in step (3) we perform classical convex optimization to find the optimal coefficients α. Definition 3 (CirculantOptim ).We are given a K-banded circulant matrix C∈ M N(C)with the decomposition C=PK ℓ=−KcℓQℓ, where K∈ O(poly log N), as well as a vector b∈CN. An Ansatz set {um}M m=1, and error tolerance ζ >0are also provided. The problem is to find {ˆαm}M m=1,αm∈Csuch that CMX m=1ˆαmum−b 2 ≤min α∈CM CMX m=1αmum−b 2 +ζ. By expanding the MSE loss term, we see that the above problem can be solved by convex optimization methods given that bounded error approximations of inner products of ⟨Cuk, Cum⟩and⟨b, Cum⟩are computed. This serves as the bottleneck to this problem as the computation of inner products may scale linearly with the dimension of the vector, a costly operation when dealing with high dimensional data. However, in different formulations of the problem where vector band the Ansatz set are given as quantum circuits or given with sample and query access, the problem variants may have a poly log Nalgorithm. We discuss the implementations of these methods in Sections IV and V, while the hardness of the variant problems is

Question 326 multiple-choice

Quantum computing relies on specialized gates and algorithms to leverage superposition and entanglement for computational tasks. The Quantum Fourier Transform (QFT) is a cornerstone in many quantum algorithms, including those for phase estimation and factoring. When implementing the Quantum Fourier Transform (QFT) on a single qubit, which operation is performed, and what is its inverse? 1) Application of the phase shift gate; inverse is the Pauli-X gate 2) Controlled-NOT followed by a Hadamard; inverse is the phase shift gate 3) Hadamard gate; inverse is also the Hadamard gate 4) Pauli-Y gate; inverse is the Pauli-Z gate 5) Toffoli gate; inverse is the Pauli-X gate 6) Phase shift followed by controlled phase gate; inverse is the controlled-NOT gate 7) Swap gate; inverse is the swap gate

✓ Correct Answer:

The correct answer is 3) Hadamard gate; inverse is also the Hadamard gate.

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QFT on a three bit nuclear magnetic resonance (NMR) quantum computer. Realizing the QFT using NMR works quite similarly [17]. In the factorization equation [19] each factor is of the form |0/angbracketright+eiω|1/angbracketright. This can be implemented using the application of two successive unitary operators. First, the Hadamard (34), which is one of most commonly used quantum gates: (i)H=1√ 2 1 1 1−1 (ii)Ψ H(iii) (H) Figure 3 The unitary gate for the 1-qubit Hadamard. The diagram is a commonly used [4] schematic representation of the gate Hacting on a qubit in state|Ψ/angbracketright. (i) The Hadamard matrix (ii) The Hadamard gate (iii) The hadamard symbol. Applying the Hadamard on the quantum state |0/angbracketrightfor instance, results in H|0/angbracketright=1√ 2(|0/angbracketright+|1/angbracketright). The Hadamard has an intuitive interpretation around the Bloch sphere [4]: If the qubit is initialised in the ground state |0/angbracketrightand then transformed via a Hadamard operator, it's state is transformed from |0/angbracketrightto an equal superposition of 0 and 1 and thus rotates the quantum state vector from the north pole of Bloch's sphere to the equator. In the Fourier basis, this alters our computational basis states from |0/angbracketrightand|1/angbracketrightto|+/angbracketrightand|−/angbracketright. and the shift operator: P(ω) = 1 0 0eiω  (28) together forming the following unitary: P(ω)[H|0/angbracketright] =1√ 2/parenleftBig |0/angbracketright+eiω|1/angbracketright/parenrightBig (29) Therefore, the QFT can be realized using universal quantum gates [4]. In the literature, there are several libraries for generating circuits from their respective unitary representation [9] and vice-versa. SymPy [16] is one such library which encompasses algebra, calculus and a base module for symbolic quantum computing in the Dirac [4] notation. 10 SciPost Physics Codebases Submission "SymPy's quantum module now supports Hilbert spaces, states, spin systems, bras, kets, inner/outer products, operators, commutators, anticommutators, dag- gers, etc. in their most abstract form." [16] Code listing (4) depicts a typical usage, involving the sympy.physics.quantum quantum package and the tensor product Ψ = (H0|φ/angbracketright)⊗(H0|ψ/angbracketright). 1from sympy import ∗ 2from sympy.physics.quantum import TensorProduct 3 4phi, psi = Ket(’phi’), Ket(’psi’) 5Uﬁnal=(TensorProduct(qapply(H(0) ∗phi),qapply(H(0) ∗psi))) 6Uﬁnal Figure 4 A typical usage involving the sympy.physics.quantum quan- tum package. The tensor product TensorProduct (psi1,psi2)results in (H0|φ/angbracketright)⊗(H0|ψ/angbracketright)which is a symbolic rather than a numerical represen- tation. Symbolically , Sympy would represent the expression exactly the same way as a human would in the Dirac notation. From here stems the relative advantage of being able to use the same ”language” both in proofs and in code. 3.2 Numerical examples: manually applying the QFT In this section, we present the results of manually applying the QFT's to a variety of quantum states. We repeat the exact same calculations later in sec (3.3), except for using SymPy programmatically, to show that they are identical. 3.2.1 Numerical examples: 1-qubit QFT First, we note that the Hadamard matrix Hisequivalent to the 1-qubit QFT and it's inverse, that is, the QFT is reduced to applying a Hadamard gate: F21=H=1√ 2 1 1 1−1 =H−1=F−1 21 (30) Also note that: ω=e2πi/N=e2πi/2=eiπ=−1 (31) 1. ForN= 21= 2. QFT|x/angbracketright=1√ 2/summationtextN−1 y=0e2πixy/N|y/angbracketright =1√ 2/bracketleftBig e2πix·0/2|0/angbracketright+e2πix·1/2|1/angbracketright/bracketrightBig =1√ 2/bracketleftbig|0/angbracketright+eπix|1/angbracketright/bracketrightbig(32) 11 SciPost Physics Codebases Submission

Question 327 multiple-choice

Efficient memory management and algorithmic design are crucial in large-scale electronic structure calculations, where orbital localization and matrix operations can significantly impact computational performance. Understanding how truncation thresholds and locality measures affect these algorithms is essential for ensuring both accuracy and scalability. Which approach allows measurement of orbital locality to remain independent of atomic geometry in quantum chemistry localization algorithms? 1) Using Gaussian basis sets for all orbitals 2) Calculating the geometric spread of each orbital 3) Counting nonzero matrix entries based on a fixed threshold value 4) Minimizing the total energy with respect to orbital positions 5) Applying periodic boundary conditions universally 6) Computing the overlap integrals only for adjacent atoms 7) Utilizing only orthonormalized orbitals for all systems

✓ Correct Answer:

The correct answer is 3) Counting nonzero matrix entries based on a fixed threshold value.

📚 Reference Text:

ecient. In fact, for practical purposes their memory cost is bounded by the cost of storing plus a few work arrays of size N: Besides the storage for ;all of the matrices we operate on cost at most O(n3 elogne) to store and may be discarded after use. Furthermore, the QR factorizations and matrix multiplication may be done in place excepting a few work arrays of length N: 4. Numerical examples. To demonstrate the performance of our method we use three examples that capture the di erent facets of our algorithm. The rst exam- ple is the dissociation of a BH 3NH3molecule, and the second example is the alkane chain. We select these two examples not because they are computationally expensive, 12 but that they clearly demonstrates that our approximate localization algorithm is ef- fective in two very di erent regimes. In particular, the e ectiveness of the algorithm is independent of whether localized orbitals form one single group or multiple discon- nected groups. Our third example is a large supercell with 256 water molecules. We demonstrate the performance gains over the existing SCDM method and provide a comparison with Wannier90 [31]. In all of the examples here we assume we have access to the electron density from the electronic structure calculation, and therefore exclude its computation from the timings of our randomized method. We use 3 nelognesamples in the randomized algorithm, corresponding to = 1=3;for all of the experiments. Furthermore, to more clearly illustrate the advantages of our method, we separately report timings for computing the orthogonal transform Qthat localizes the orbitals and subsequent com- putation of the localized orbitals by a single matrix product. Here we only consider the orthogonalized SCDM, as discussed in this paper. For the re nement algorithm we set the relative truncation threshold at = 5102and we observe that this is sucient for our new algorithm to closely match the results of the existing algorithm. Finally, in all of the experiments here we use 2 :5102as the relative truncation threshold of the localized orbitals when counting the fraction of entries that are non- zero. This measure of \locality" has the advantage of not depending on the geometry of the physical system. However, for completeness we also provide spread computa- tions in the nal example. Prior work validates the expected exponential decay of the orbitals by varying the truncation threshold [10]. All numerical results shown were run on a quad-socket Intel Xeon E5-4640 pro- cessor clocked at 2.4 GHz with 1.5 TB of RAM and our algorithms were implemented in MATLAB R2015a. Our code is sequential and the only multi-threading is in the LAPACK and BLAS routines called by MATLAB. The storage cost of all the algo- rithms presented here is O(neN);which is also the cost to store in memory. In our largest example a copy of cost roughly 60 GB to store. The Kohn-Sham orbitals were computed using Quantum ESPRESSO [16] and VMD [22] was used for plotting orbitals and molecules in the

Question 328 multiple-choice

Variational Quantum Computing (VQC) is a hybrid computational paradigm that leverages both quantum circuits and classical optimization algorithms to solve complex problems, including nonlinear partial differential equations. Its efficiency and scalability depend on how the quantum circuit architecture relates to the underlying problem characteristics. Which of the following conditions is critical for VQC to achieve potential exponential speed-up over classical algorithms in solving large-scale nonlinear problems? 1) The quantum circuit must use only linear operators throughout its layers. 2) The number of qubits should scale with the size of the classical problem instance. 3) Classical optimization steps must be performed after every single quantum gate operation. 4) The circuit depth and number of parameters should scale with the number of qubits, not with the classical problem size. 5) Nonlinear operations must be fully implemented on the quantum hardware without any classical assistance. 6) Measurement and re-initialization of quantum states must be performed repeatedly during computation. 7) All amplitude products in the quantum state must be computed using non-unitary operators.

✓ Correct Answer:

The correct answer is 4) The circuit depth and number of parameters should scale with the number of qubits, not with the classical problem size..

📚 Reference Text:

nor- malized quantum state vector all amplitudes in the vector are ≤1 (unless only a single amplitude is non-zero), therefore an operator performing products of theamplitudes cannot be unitary since the resulting quantum state vector will nolonger have a unit norm. One possible way around these problems associated with nonlinear terms would be a hybrid quantum-classical approach where the nonlinear products are computedon a classical computer. However, due to the complexity introduced by measuring the quantum state (needed before each transfer of information to the classical computer) and the cost of (re-)initialization of the quantum computer with theresult of these products, this is not a promising line of development. It is highlyunlikely to lead to a quantum speed-up. Recently, Variational Quantum Computing(VQC) was introduced as an effective hybrid classical-quantum approach [22, 23],firstly for applications in quantum chemistry and more recently for a wider range oflinear and nonlinear problems [24]. The VQC approach constructs the requiredsolution from a layered network, as illustrated in Figure 1 . As shown in Figure 1(a) , multiple layers are used (4 in the illustration), each taking as input multiple qubits (6 in example shown). Using depth 5 in the example, the quantum circuits defined Figure 1. Illustration of the Variational Quantum computing (VQC) approach (adapted from Lubasch et al. [24]). 4Quantum Computing and Communications byUλðÞinvolve 13 two-qubit gates as shown in Figure 1(b) . Each of these gates has a parameter λi∈1, 13½/C138 associated with it. A classical computer is used to create optimized parameters λemploying an iterative approach that takes the measured state of the ancilla qubit as input. A further key part of the approach is the problem-specific Quantum Nonlinear Processing Unit (QNPU). Recently, Lubasch et al. [24]published an example for the QNPU for the nonlinear Burgers equation. In applica-tions of the VQC approach, the efficiency strongly depends on the choice of thenumber of parameters λused in UλðÞ. The work by Lubasch et al. [24] showed that exponential speed-up is only possible if the depth of UλðÞscales with the number of qubits and not with the overall problem size. It is clear that the proposed VQC approach is an important development toward QC applications to nonlinear prob- lems. It therefore constitutes a leading candidate for applications to fluid dynamics.However, it is also clear that further investigation is needed to further assess itssuitability for a range of applications. 4. Nonlinear governing equations in fluid mechanics The Navier –Stokes equations for an incompressible, Newtonian fluid can be written as, ∂U ∂tþU/C1∇xU¼/C01 ρ∇xpþνΔU;∇x/C1U¼0 (4) where U,p,ρand νare the velocity, pressure, density and kinematic viscosity, respectively. xdenotes the coordinate in space. The second term on the right-hand side of Eq. (4) is the nonlinear convection term that poses a key challenge to devel- oping efficient quantum algorithms for the Navier –Stokes equations. Efficient quan- tum algorithms for linear convection equations discretized on regular Cartesianmeshes with periodic boundary conditions have been devised in recent years [6].When studying numerical methods for the Navier –Stokes equations, it is often useful

Question 329 multiple-choice

Quantum algorithms for the dihedral hidden subgroup problem (DHSP) are central to advances in cryptography and lattice-based computational problems. Recent approaches combine techniques from classical lattice theory with quantum state manipulation to improve efficiency and scalability. Which innovation in a quantum algorithm for DHSP directly leverages the SV (shortest vector) algorithm to achieve subexponential time scaling for moderate problem sizes? 1) Adapting the SV (shortest vector) algorithm to construct quantum measurement bases 2) Using Grover's search algorithm for hidden subgroup identification 3) Employing Shor's algorithm to factor the order of the dihedral group 4) Implementing a phase estimation routine for slope determination 5) Utilizing the quantum Fourier transform to reveal subgroup structure 6) Applying amplitude amplification to enhance measurement probability 7) Designing error-correcting codes for quantum state robustness

✓ Correct Answer:

The correct answer is 1) Adapting the SV (shortest vector) algorithm to construct quantum measurement bases.

📚 Reference Text:

Title: A quantum algorithm for the dihedral hidden subgroup problem based on algorithm SV Year: 2013 Paper ID: a16be1bcbee829f89e7c02fb80fb7952c40b0156 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a16be1bcbee829f89e7c02fb80fb7952c40b0156 Abstract: To accelerate the algorithms for the dihedral hidden subgroup problem, we present a new algorithm based on algorithm SV(shortest vector). A subroutine is given to get a transition quantum state by constructing a phase filter function, then the measurement basis are derived based on the technique for solving low density subset problem. Finally, the parity of slope is revealed by the measurements on the transition quantum state. This algorithm takes O(n) quantum space and O(n^2) classical space, which is superior to existing algorithms, for a relatively small n(n<6400),it takes (n^0.5)*(log(max aij))^3 computation time, which is superior to 2^(O(n^0.5)).

Question 330 multiple-choice

Simulating time evolution and spectrum of quantum many-body systems on finite lattices is a central challenge in computational physics, often requiring advanced mathematical techniques and optimization algorithms. Non-Abelian lattice gauge theories, particularly those with discrete gauge groups, pose unique difficulties due to the rapid growth of the Hilbert space and complex coupling structures. Which mathematical tool is essential for constructing the Hilbert space basis in non-Abelian lattice gauge theories, especially when decomposing quantum states and coupling representations for systems with discrete gauge groups such as D3 and D4? 1) Pauli matrices 2) Fourier transforms 3) Clebsch-Gordan coefficients 4) Laplace operators 5) Gram-Schmidt orthogonalization 6) Legendre polynomials 7) Taylor series expansions

✓ Correct Answer:

The correct answer is 3) Clebsch-Gordan coefficients.

📚 Reference Text:

penaltyterm,and ˆC0selectsapredeterminedinitialstate.Byconstruction, ˆCishermitian. OnecanshowthattheminimumofthefunctionalEq.( 20)correspondstotheexacttime- evolved state at each step. Thus, as a result of a single optimization problem one obtains thefulltime-evolutionofamany-bodyquantumstateoveraﬁnitetimeinterval.Fromthis functionalonecannowobtaintheQUBOmatrix.Aspreviouslydiscussedforthecaseof ﬁnding the ground state of our Hamiltonian, this is what the quantum annealer requires as input. Our discussion closely follows the derivation of [ 10]. Using a variational state |ψt/angbracketright|t/angbracketrightateachtimestep t,thefunctionalinEq.( 20)becomes L=/summationdisplay αβa∗ αLαβaβ, (22) where the expansion parameters aαare complex, the indices in the sum run over all NtNconfvalues,and Lαβarethematrixelementsofthefunctional.Thetermsintheabove sum can be written in terms of the real and imaginary parts of both Lαβandaα.U s i n g theﬁxed-pointrepresentationforboththerealandimaginarypartsof aα,oneobtainsthe followingQUBOmatrix Qα,i;β,j=⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩2 i+j–2K–2z(–1)δiK+δjK/RfracturLαβ+2δαβδij2i–K–z(–1)δiK ×/summationtext γ(/Rfractura(z) γ/RfracturLγβ+/Ifractura(z) γ/IfracturLγβ), 1≤i,j≤K, –2i+j/prime–2K–2z(–1)δiK+δj/primeK/IfracturLαβ,1≤i,j/prime≤K, 2i/prime+j–2K–2z(–1)δi/primeK+δjK/IfracturLαβ,1≤i/prime,j≤K, 2i/prime+j/prime–2K–2z(–1)δi/primeK+δj/primeK/RfracturLαβ+2δαβδi/primej/prime2i/prime–K–z(–1)δi/primeK ×/summationtext γ(/Ifractura(z) γ/RfracturLγβ–/Rfractura(z) γ/IfracturLγβ), 1≤i/prime,j/prime≤K,(23) where the Latin indices now run from 1 to 2 Kto allow for Kbits in representing both the real and imaginary parts of the variational parameters and the primed Latin indices are shifted by K. The dimension of this QUBO matrix is 2 KNtNconf×2KNtNconf,w h i c h is signiﬁcantly larger than the one used to determine the eigenstates of the Hamiltonian. This problem size is too large to be fully embedded on current quantum annealers and wereverttolocalsimulatedannealing3tosolveit.Theresultsofourtimeevolutionsim- ulationsaredisplayedinFig. 4wherewehavetimeevolvedthetrivialvacuum.Shownare t h ee x p e c t a t i o nv a l u eo ft h em a g n e t i cp a r t , ˆHB, as well as the probability that the trivial vacuumpersists, |/angbracketleft0|U(t)|0/angbracketright|2.Onecanseegoodagreementwiththeexactresults. 4 Conclusionandoutlook We have constructed the Hamiltonian formulation of non-Abelian lattice gauge theories fordiscretegaugegroups Dn.Fortheconcreteexamples D3,D4weworkedouttheKogut- 3Simulationparametersare K=3,Nt=3,zmax=11andnruns=8000. Frommetal. EPJQuantumTechnology (2023) 10:31 Page11of19 Figure4 Resultsfortimeevolutionusingsimulatedannealingat g2 H=0.75withsimulationparametersgiven inthetext.TheredandyellowpointsrepresenttheexpectationvalueofthemagneticHamiltonianinthe time-evolvedtrivialvacuumstateasafunctionoftimefor D3andD4.Theblueandblackpointsrepresentthe probabilityamplitudeforthetrivialvacuumstatetopersistasafunctionoftime.Thelinesrepresenttheexact resultforeachcase Susskind Hamiltonian as well as the representations of the corresponding Hilbert space basis states in terms of Clebsch-Gordan coeﬃcients. In principle, this construction can be generalized to larger gauge groups and higher dimensionality. By observing Eq. ( 55), it is clear that a full 2D lattice, which has a coordination number of four, would involvecouplingoneadditionallinkpersitetoyieldarepresentation j I(correspondingtothead- ditionofthreeangularmomentain SU(2)).Thenaturalcoeﬃcientsarisinginthiscontext wouldthenbetheWigner6J-symbols,alsoknownastheRacahcoeﬃcients[ 24].Forthree dimensions,thisconstructioncaninprincipleberepeated.However,inthatcaseitseemsadvisabletoworkpurelywithCGcoeﬃcients.Asanexample,wehaveshownhowtomapthese simple lattice gauge theories onto a quantum annealer. In doing so, we have beenable to compute the spectrum as well as time evolution for both D 3andD4on small lat- tices.Theseproof-of-principleresultsareofcourseaﬀectedbyﬁnite-sizeeﬀectsandgoing to larger system sizes will have signiﬁcant impact on observables, such as the spectrum of the theory. However, these are accessible using traditional Markov chain Monte Carlomethods(seee.g.[ 14]). Themainobstacletosimulatebothlargergroupsaswellaslargerlatticesinourapproach i st h es c a l i n go ft h ep h y s i c a lH i l b e r ts p a c ew i t hi n c r e a s i n gl a t t i c ea n dg r o u ps i z e .I tt h u smay be helpful to look in other directions in order to utilize the power of the quantumannealer. One idea to get around the barrier of rapidly expanding Hilbert spaces for the case of continuousgroupsistoperformatruncationinthenumberofallowedirrepsaswasdoneinearlierstudies of SU(2) and SU(3) intheHamiltonianformulation[ 4,5,9].Thiscould also be done for D n,w h e r en>4.

Question 331 multiple-choice

In quantum algorithms for nonabelian groups, the choice of basis and type of Fourier transform profoundly affects the ability to reconstruct hidden subgroup structures, with implications for computational efficiency and cryptographic applications. The affine group Ap serves as a key example where representation theory and subgroup-adapted methods are crucial. Which of the following statements most accurately explains why the abelian Fourier transform fails to distinguish maximal subgroup conjugates in the affine group Ap? 1) It lacks the ability to compute character probabilities for nonabelian groups. 2) The abelian Fourier transform always produces zero probability for nontrivial subgroups. 3) It treats all subgroups as if they were normal, resulting in identical measurement outcomes. 4) Its measurement outcomes depend only on the order of the subgroup, not its conjugacy class. 5) It collapses all group elements to their identity, erasing subgroup distinctions. 6) The character probabilities do not depend on the parameter labeling different conjugate subgroups, making these conjugates indistinguishable. 7) It applies only to groups of prime order, so cannot distinguish conjugates in larger groups.

✓ Correct Answer:

The correct answer is 6) The character probabilities do not depend on the parameter labeling different conjugate subgroups, making these conjugates indistinguishable..

📚 Reference Text:

cuts across these blocks, mixing diﬀerent“frequencies” ρ kand canceling out the useful information. To be more precise, a random basis is not adapted to the subgroup structure of Ap; it does not “know” that ρdecomposes into a direct sum of the ρkwhen restricted to Nq. Second, it is worth noting that for the values of qfor which we have an algorithm for full (as opposed to information-theoretic) reconstruction, namely q=p/polylog( p), a random basis works as well, since the /lscript1distance δbecomes 1 /polylog( p). Based on the strong evidence from representation theory that some bases are much betterfor computation than others, we conjecture that, for some families of groups, adaptedbases allow full reconstruction while random bases do not; but this remains an openquestion. Third, while we focused above on distinguishing conjugate subgroups from each other, in fact our proof shows that if q

Question 332 multiple-choice

In computational molecular spectroscopy, accurately modeling vibrational frequencies of molecules in solution is essential for interpreting experimental IR spectra and understanding local environmental effects. Advanced methods incorporate quantum mechanical and electronic structural details to predict spectral features in complex systems. Which statement best describes a key advantage of the nuclear quantum vibrational perturbation (QVP) method over traditional electrostatics-based spectroscopic mapping approaches? 1) QVP uses only classical mechanics to simulate molecular vibrations, neglecting quantum effects. 2) QVP disregards solvent fluctuations when calculating vibrational energy shifts. 3) QVP relies exclusively on harmonic oscillator models for all vibrational modes. 4) QVP incorporates both nuclear quantum effects and electronic structural details, providing more accurate vibrational frequency predictions in fluctuating environments. 5) QVP eliminates the need for perturbation theory in spectral simulations. 6) QVP does not account for anharmonicity in vibrational motions. 7) QVP is unable to simulate vibrational spectra in aqueous solutions.

✓ Correct Answer:

The correct answer is 4) QVP incorporates both nuclear quantum effects and electronic structural details, providing more accurate vibrational frequency predictions in fluctuating environments..

📚 Reference Text:

Perturbation Approach for Computing Infrared Spectra of the Local Mode of Probe Molecules Rui-Jie Xue†, Adam Grofe‡, He Yin†, Zexing Qu†, Jiali Gao*,†,‡,iD, and Hui Li*,† †Institute of Theoretical Chemistry, Jilin University, 2519 Jiefang Road, Changchun 130023, People’s Republic of China ‡Department of Chemistry and Supercomputing Institute, University of Minnesota, 207 Pleasant Street, SE, Minneapolis, Minnesota 55455, United States Abstract Linear and two-dimensional infrared (IR) spectroscopy of site-specific probe molecules provides an opportunity to gain a molecular-level understanding of the local hydrogen-bonding network, conformational dynamics, and long-range electrostatic interactions in condensed-phase and biological systems. A challenge in computation is to determine the time-dependent vibrational frequencies that incorporate explicitly both nuclear quantum effects of vibrational motions and an electronic structural representation of the potential energy surface. In this paper, a nuclear quantum vibrational perturbation (QVP) method is described for efficiently determining the instantaneous vibrational frequency of a chromophore in molecular dynamics simulations. Computational efficiency is achieved through the use of (a) discrete variable representation of the vibrational wave functions, (b) a perturbation theory to evaluate the vibrational energy shifts due to solvent dynamic fluctuations, and (c) a combined QM/MM potential for the systems. It was found that first-order perturbation is sufficiently accurate, enabling time-dependent vibrational frequencies to be obtained on the fly in molecular dynamics. The QVP method is illustrated in the mode-specific linear and 2D-IR spectra of the H–Cl stretching frequency in the HCl–water clusters and the carbonyl stretching vibration of acetone in aqueous solution. To further reduce computational cost, a hybrid strategy was proposed, and it was found that the computed vibrational spectral peak position and line shape are in agreement with experimental results. In addition, it was found that anharmonicity is significant in the H–Cl stretching mode, and hydrogen-bonding interactions further enhance anharmonic effects. The present QVP method complements other computational approaches, including path integral-based molecular dynamics, and represents a major improvement over the electrostatics-based spectroscopic mapping procedures. *Corresponding Authors gao@jialigao.org. Prof_huili@jlu.edu.cn. ORCID Jiali Gao: 0000-0003-0106-7154 ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.6b00733 . The convergent test of HCl stretch vibration within HClH 2O complex, the perturbation approach test, ab initio molecular simulation setup, the response function of rephrasing and non-rephrasing processes (PDF) The authors declare no competing financial interest. HHS Public Access Author manuscript J Chem Theory Comput . Author manuscript; available in PMC 2018 February 01. Published in final edited form as: J Chem Theory Comput . 2017 January 10; 13(1): 191–201. doi:10.1021/acs.jctc.6b00733. Author Manuscript Author Manuscript Author Manuscript Author Manuscript Graphical Abstract 1. INTRODUCTION Molecular vibrations and infrared spectroscopy of small chromophore probes embedded in molecular clusters, solutions, and biomacromolecules have been widely used to investigate solute–solvent interactions, hydrogen-bonding dynamics, and the local electric fields in enzymes.1–9 Vibrational frequency shifts are especially sensitive to the local electrostatic environment, which can be easily and accurately measured. Thus, the study of vibrational frequency shifts is an efficient way to analyze the local structure and

Question 333 multiple-choice

Quantum algorithms often utilize group-theoretic Fourier analysis, but the complexity differs significantly depending on whether the underlying group is abelian or non-abelian. Understanding how subgroup identification and measurement strategies vary between these cases is crucial for analyzing algorithmic efficiency and feasibility. Which statement accurately characterizes the differences in subgroup identification between finite abelian and non-abelian groups in the context of quantum algorithms? 1) In both abelian and non-abelian groups, subgroup identification is computationally efficient and always feasible. 2) Non-abelian groups allow their subgroups to be described compactly using generators, similar to abelian groups. 3) For abelian groups, subgroup identification may be undecidable due to the complexity of their subgroup structure. 4) Abelian groups permit efficient subgroup identification using generators, while non-abelian group subgroup identification is often computationally hard or uncomputable. 5) Quantum algorithms output a complete list of subgroup elements for any finite group, regardless of group type. 6) Shift invariance ensures that subgroup identification is straightforward in all group types. 7) Both abelian and non-abelian groups have subgroup enumeration algorithms that run in polynomial time.

✓ Correct Answer:

The correct answer is 4) Abelian groups permit efficient subgroup identification using generators, while non-abelian group subgroup identification is often computationally hard or uncomputable..

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position. In the abelian case, jandktake only the value 1 and may be omitted. The Fourier basis may be grouped into msubsets of sizes d2 1,...,d2 maccording to the value of iand we may consider the associated incomplete von Neumann measurement which distinguishes on ly the various representations. We will denote this incomplete measurement by Mrepand it will be important later (cf (d) below). (b) (Eﬃcient implementation of non-abelian FT). For the eﬃc iency of our quantum al- gorithms it is important that FT be implementable in poly(lo g|G|) computational steps. In the abelian case this was a consequence of the FFT formalism. Fortunately this formalism extends to the non-abelian case too [12] requiring only that the group contains a suitable tower of subgroups. For the standard FFT on Z2nthis tower is H0⊂H1⊂...⊂ Z2nwhere Hkis the subgroup of multiples of 2n−kinZ2n. A fundamental non-abelian group is the 11 permutation group G=Pnonnsymbols. Pncontains the tower P1⊂ P2⊂...⊂ Pnand its FT has been shown to be eﬃciently implementable [13]. (c) (Description of subgroups). Our quantum algorithm shou ld provide distinguishable outputs for diﬀerent possible subgroups K. In that case we say that the subgroup has been information-theoretically determined. However in ge neral it may still be a diﬃcult computational task to identify the actual subgroup from the output result. For ﬁnite abelian groups a fundamental structure theorem [4] asserts that any such group is isomorphic to a direct product of groups of the form Zn. In this case any subgroup Kwill have a simple poly(log |G|) sized description given by a list of generators, which we ca n require as the output of the algorithm. For non-abelian groups the classiﬁcation of possibilities is not so simple. For example even the problem of deciding whether or not two se ts of generators and relations give isomorphic groups, is known to be uncomputable! [4]. Fu rthermore it is not appropriate to ask for a list of all elements of Kas this may be of size O(|G|) i.e. exponentially large in log |G|. We may circumvent these diﬃculties of description by askin g for less – instead of characterising Kper se , we may for example ask that the algorithm outputs a randomly chosen element of Kor determines whether or not some chosen property of a subgro up holds for the hidden subgroup. (d) (Shift invariance). In the preceding section we used the existence of the shift invariant basis|χl/an}bracketri}htto give some intuitive insight into why FT is useful for abeli an hidden subgroups. It provided a means of eliminating the eﬀects of a randomly ch oseng0in the state |g0+K/an}bracketri}ht. The existence of a shift invariant basis relies on the commut ing of the shift operators U(g) and this is a consequence of the abelian-ness of G. In the non-abelian case such a basis will not exist. However a restricted form of shift invariance still s urvives because of the multiplicative property of representations: χi(g1g2) =χi(g1)χi(g2) (where the RHS is multiplication of di×diunitary matrices). If we

Question 334 multiple-choice

Quantum algorithms have transformed the field of computational algebra by enabling efficient methods for analyzing finite algebraic structures. One prominent approach utilizes the Hidden Subgroup Problem (HSP) framework to locate important substructures such as nuclei and centers within finite algebras. Which advantage does quantum reformulation of searching for substructures in finite algebras via the Hidden Subgroup Problem (HSP) over Abelian groups most directly provide compared to classical algorithms? 1) It eliminates the need for circuit construction. 2) It always produces exact solutions without error. 3) It restricts analysis to only non-Abelian groups. 4) It bypasses the requirement for knowledge of automorphisms. 5) It enables only exponential scaling with algebra dimension. 6) It reduces computational complexity from exponential to polynomial in the dimension of the algebra. 7) It prevents applications in cryptography and coding theory.

✓ Correct Answer:

The correct answer is 6) It reduces computational complexity from exponential to polynomial in the dimension of the algebra..

📚 Reference Text:

Title: Efficient quantum algorithms to find substructures on finite algebras Year: 2023 Paper ID: 2ae3c31ade9208c502801d99ce1952109e3635f1 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/2ae3c31ade9208c502801d99ce1952109e3635f1 Abstract: When classifying a collection of finite algebras (for instance, in the computational classification of finite semifields), an important task is the determination of substructures such as the right, middle, and left nuclei, the nucleus, and the center. Finding these structures may become computationally expensive when there is no additional information about the algebra properties. In this paper, we introduce quantum algorithms than solve this task efficiently, by formulating it as an instance of the Hidden Subgroup Problem (HSP) {over Abelian groups}. We give detailed constructions of the quantum circuits involved in the process and prove that the overall (quantum) complexity of our algorithm is polynomial in the dimension of the algebra, while a similar approach with classical computers would require an exponential number of queries to the HSP function

Question 335 multiple-choice

Fourier analysis plays a crucial role in quantum algorithms for lattice problems and hidden subgroup identification, particularly in the study of quantum states and their probability distributions in dual spaces. Understanding how wavefunctions transform and how their measurement probabilities are interpreted is essential for applications in quantum cryptography. Which statement best describes the relationship between the peaks in the Fourier-transformed wavefunction and the dual lattice when a quantum algorithm analyzes a hidden subgroup problem modulo a lattice L? 1) The peaks occur only at random positions in the dual space, independent of the lattice structure. 2) The peaks are uniformly distributed throughout the entire Fourier domain, unrelated to the dual lattice. 3) The peaks correspond to zeros of the original wavefunction in real space. 4) The peaks are determined solely by the normalization condition of the quantum state. 5) The peaks appear at points in the dual lattice L*, reflecting the underlying symmetry and periodicity of the lattice. 6) The peaks indicate eigenvalues of the Hamiltonian governing the quantum system. 7) The peaks are smoothed out due to convolution with a Gaussian, eliminating sharp features.

✓ Correct Answer:

The correct answer is 5) The peaks appear at points in the dual lattice L*, reflecting the underlying symmetry and periodicity of the lattice..

📚 Reference Text:

Fourier transform of ψδ andψ, respectively: pδ(y) =˙bψδ(y)˛˛bψδ(y)¸ withbψδ=FδZmψ; p(y) =˙bψ(y)˛˛bψ(y)¸ withbψ=FRnψ. (6.8) It can be shown that pδis close top. Let us now focus on the distribution p(y) =˙bψ(y)˛˛bψ(y)¸ . We have ψ=wf,bψ=bw∗(FRmf), wherebw=FRmw. Sincefis a hidden subgroup oracle, we may regard it as a function on Rm/Land deﬁnebf=FRm/Lf. That is, bfu=Z Rm/Le2πi/angbracketleftx,u/angbracketrightf(x)dx foru∈L∗, f(x) =1 d(L)X u∈L∗e−2πi/angbracketleftx,u/angbracketrightbfu. (6.9) 301 The Fourier transform over Rmis (FRmf)(y) =Z Rme2πi/angbracketleftx,y/angbracketright 1 d(L)X u∈L∗e−2πi/angbracketleftx,u/angbracketrightbfu! dx =1 d(L)X u∈L∗bfuδ(y−u). It follows that bψ(y) =` bw∗(FRmf)´ (y) =1 d(L)X u∈L∗bfubw(y−u),(6.10) p(y) =1 d(L)2X u,u/prime∈L∗˙bfu/prime˛˛bfu¸ bw(y−u)bw(y−u/prime).(6.11) The last equation is complicated, but to have a good ap- proximation it is enough to keep the terms with u=u/prime. Let us consider the quantum state FRm|ψ/angbracketrightwhose wavefunction is given by Eq. (6.10). It consists of identically shaped peaks at the points u∈L∗. Each peak has a weight qu=˙bfu˛˛bfu¸ d(L)2=1 d(L)2Z e2πi/angbracketleftx−x/prime,u/angbracketright˙ f(x/prime)˛˛f(x)¸ dxdx/prime, where the integral is over (Rm/L)2. The numbers qucan be interpreted as probabilities because they are nonnegative and add up to 1. Indeed, let us normalize fto make a function of unit norm, g(x) =d(L)−1/2f(x). Then X u∈L∗qu=1 d(L)X u∈L∗˙ bgu˛˛bgu¸ =˙ bg˛˛bg¸ =/angbracketleftg|g/angbracketright= 1. 7. REFERENCES [Ban93] W. Banaszczyk. New bounds in some transference theorems in the geometry of numbers. Mathematische Annalen, 296(1):625–635, 1993. [BK93] Johannes Buchmann and Volker Kessler. Computing a reduced lattice basis from a generating system, 1993. Preprint, August 4, 1993. [BP87] Johannes Buchmann and Michael Pohst. Computing a lattice basis from a system of generating vectors. In Eurocal’87 , volume 378 ofLNCS, pages 54–63. Springer-Verlag, June 1987. [BV11] Zvika Brakerski and Vinod Vaikuntanathan. Fully homomorphic encryption from ring-LWE and security for key dependent messages. In Advances in cryptology—CRYPTO 2011 , volume 6841 of LNCS, pages 505–524. Springer, 2011. [Coh93] Henri Cohen. A course in computational algebraic number theory. Springer-Verlag New York, Inc., New York, NY, USA, 1993. [FIM+03] Katalin Friedl, Gabor Ivanyos, Frederic Magniez, Miklos Santha, and Pranab Sen. Hidden translation and orbit coset in quantum computing. In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, San Diego, CA, 9–11June 2003. [GH11] C. Gentry and S. Halevi. Implementing gentry’s fully-homomorphic encryption scheme. Eurocrypt 2011 , pages 132–150, 2011.[GKP01] Daniel Gottesman, Alexei Kitaev, and John Preskill. Encoding a qubit in an oscillator. Phys. Rev. A , 64:012310, Jun 2001. [GVL96] Gene H. Golub and Charles F. Van Loan. Matrix Computations . Johns Hopkins University Press, Baltimore, MD, 3rd edition, 1996. [Hal05] Sean Hallgren. Fast quantum algorithms for computing the unit group and class group of a number ﬁeld. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pages 468–474, 2005. [Hal07] Sean Hallgren. Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem. Journal of the ACM, 54(1):1–19, 2007. [HMR+10] Sean Hallgren, Cristopher Moore, Martin R¨otteler, Alexander Russell, and Pranab Sen. Limitations of quantum coset states for graph isomorphism. J. ACM, 57:34:1–34:33, November 2010. [Kit95] Alexei Kitaev. Quantum measurements and the abelian stabilizer problem, 1995. quant-ph/9511026. [KW08] Alexei Kitaev and William A. Webb.

Question 336 multiple-choice

Quantum groups are algebraic structures that generalize symmetries found in quantum mechanics, and their properties often depend on parameters such as roots of unity. Hopf algebras associated with quivers play a fundamental role in categorical algebra and the representation theory of associative algebras. For which specific conditions is the Hopf algebra constructed from the cyclic quiver Zn and a root of unity q quasitriangular? 1) Only when n is odd and q = 1 2) Only when n = 2 and q is any root of unity 3) When q is a primitive root of unity of order n 4) When n is divisible by 3 and q = -1 5) Only when n is a prime number and q = 1 6) When n is even and q = -1 7) When n is any integer and q is a nontrivial root of unity

✓ Correct Answer:

The correct answer is 6) When n is even and q = -1.

📚 Reference Text:

Archive ouverte UNIGE https://archive-ouverte.unige.ch Article scientifique Article 1993 Published version Open Access This is the published version of the publication, made available in accordance with the publisher’s policy. A quiver quantum group Cibils, Claude Henri How to cite CIBILS, Claude Henri. A quiver quantum group. In: Communications in Mathematical Physics , 1993, vol. 157, n° 3, p. 459–477. doi: 10.1007/bf02096879 This publication URL: https://archive-ouverte.unige.ch/unige:12842 Publication DOI: 10.1007/bf02096879 © This document is protected by copyright. Please refer to copyright holder(s) for terms of use. Commun. Math. Phys. 157, 459-177 (1993) Communications IΠ Mathematical Physics© Springer-Verlag 1993 A Quiver Quantum Group Claude Cibils* ** Department of Mathematics, Brandeis University, Waltham, MA 02154, USA Received: 12 June 1992/in revised form: 29 March 1993 Abstract: We construct quantum groups at a root of unity and we describe their monoidal module category using techniques from the representation theory of finite dimensional associative algebras. 1. Introduction In the representation theory of finite dimensional algebras, important techniques as quivers, almost-split sequences and irreducible morphisms have been developed these last two decades. Our purpose is to use these methods in order to study finite dimensional Hopf algebras which are neither commutative nor cocommutative together with their monoidal category of modules. Hopf algebras of this sort produce non-trivial braided categories through the representations of the quasi- triangular Drinfeld's double ([8]) or through the "center construction" ([10,13,18,15]); according to Drinfeld [8] or Manin [19] these Hopf algebras are considered as quantum groups or rather better as functions on some hypothetic quantum group. Their representation theory through the double constructions has applications in various parts of physics and mathematics: one can use them to obtain solutions of the quantum Yang-Baxter equation or topological invariants (see for instance [24, 25,17]). Quivers with relations allow to present finite dimensional associative algebras in a useful way, see [11]. If the algebra is of finite representation type, its module category can be presented using the Auslander-Reiten quiver of the category of indecomposable modules and relations given by the almost-split sequences. In this note we construct a family of infinite dimensional Hopf algebras over an arbitrary field k. To each integer n ^ 2 and each nth root of unity q in k we attach a Hopf algebra Hn(q) which is the path algebra of the cyclic quiver Zn. These Hopf * Supported by the FNSRS (Swiss National Foundation) ** Current address: Section de Mathematiques, Universite de Geneve, C.P. 240, CH-1211 Geneve 24, Switzerland. Email: cibils@cgeugell.bitnet 460 C. Cibils algebras are neither commutative nor cocommutative, except for q = 1 when they are cocommutative. Let d be the precise order of q; we will show that Hn(q) has a finite dimensional quotient Λn(q) of dimension nd with antipode of order 2d. Those quantum groups are not quasitriangular unless n is even and q = — 1. If the characteristic of k does not divide n and if in addition q is primitive then the Hopf algebras An(q) are isomorphic to the Hopf algebras obtained by Taft

Question 337 multiple-choice

Quantum branching programs, such as quantum Ordered Binary Decision Diagrams (OBDDs), harness quantum superposition and interference to process Boolean functions with high efficiency. These techniques offer exponential improvements in space and computational resources compared to classical models for certain functions. Which function is efficiently computed by quantum OBDDs using the fingerprinting technique and demonstrates an exponential advantage in width over deterministic OBDDs? 1) Permutation Matrix (PERMn) 2) Boolean satisfiability (SAT) 3) Graph coloring 4) Subset sum 5) Maximum clique 6) Shortest path 7) Hamiltonian cycle

✓ Correct Answer:

The correct answer is 1) Permutation Matrix (PERMn).

📚 Reference Text:

Next, upon input symbol 0 identity transformation Iis applied. But if the value of xjis 1, then the state of the last qubit is transformed by the operator Ri,j, rotating it by the angle proportional to cj. Moreover, the rotation is done in each of tsubspaces with the corresponding amplitude 1 /√t. Such a parallelism is implemented by the controlled operators Ci(Ri,j), which transform the states |i/an}⌊ra⌋k⌉tri}ht|·/an}⌊ra⌋k⌉tri}htinto |i/an}⌊ra⌋k⌉tri}htRi,j|·/an}⌊ra⌋k⌉tri}ht, and leave others unchanged. For instance, having read the i nput symbol x1=1, the system would evolve into state |ψ2/an}⌊ra⌋k⌉tri}ht=1√tt ∑ i=1Ci(Ri,1)|i/an}⌊ra⌋k⌉tri}ht|0/an}⌊ra⌋k⌉tri}ht=1√tt ∑ i=1|i/an}⌊ra⌋k⌉tri}htRi,1|0/an}⌊ra⌋k⌉tri}ht =1√tt ∑ i=1|i/an}⌊ra⌋k⌉tri}ht/parenleftBig cos2πkic1 m|0/an}⌊ra⌋k⌉tri}ht+sin2πkic1 m|1/an}⌊ra⌋k⌉tri}ht/parenrightBig. 6 Algorithms for Quantum Branching Programs Thus, after having read the input σthe amplitudes would “collect” the sum ∑n j=1cjσj |ψ3/an}⌊ra⌋k⌉tri}ht=1√tt ∑ i=1|i/an}⌊ra⌋k⌉tri}ht/parenleftBig cos2πki∑n j=1cjσj m|0/an}⌊ra⌋k⌉tri}ht+sin2πki∑n j=1cjσj m|1/an}⌊ra⌋k⌉tri}ht/parenrightBig . At the next step weperform the rotations by the angle4πkic0 mabout the ˆ yaxis of the Bloch sphere for eachi∈{1,...,t}. Therefore, the state of the system would be |ψ4/an}⌊ra⌋k⌉tri}ht=1√tt ∑ i=1|i/an}⌊ra⌋k⌉tri}ht/parenleftBig cos2πkig(σ) m|0/an}⌊ra⌋k⌉tri}ht+sin2πkig(σ) m|1/an}⌊ra⌋k⌉tri}ht/parenrightBig . Applying H⊗logt⊗Iweobtain thestate |ψ5/an}⌊ra⌋k⌉tri}ht=/parenleftbigg 1 tt ∑ i=1cos2πkig(σ) m/parenrightbigg |0/an}⌊ra⌋k⌉tri}ht⊗logt|0/an}⌊ra⌋k⌉tri}ht+ +γ|0/an}⌊ra⌋k⌉tri}ht⊗logt|1/an}⌊ra⌋k⌉tri}ht+t ∑ i=2|i/an}⌊ra⌋k⌉tri}ht(αi|0/an}⌊ra⌋k⌉tri}ht+βi|1/an}⌊ra⌋k⌉tri}ht), whereγ,αi, andβiare someunimportant amplitudes. Theinput σisacceptedifthemeasurementoutcomeis |0/an}⌊ra⌋k⌉tri}ht⊗logt|0/an}⌊ra⌋k⌉tri}ht. Clearly,theacceptingprobability is Praccept(σ)=1 t2/parenleftBigg t ∑ i=1cos2πkig(σ) 2n/parenrightBigg2 . Iff(σ)=1 theng(σ)=0 and the program accepts σwith probability 1. Otherwise, the choice of the setK={k1,...,kt}guarantees that Praccept(σ)=1 t2/parenleftBigg t ∑ i=1cos2πkig(σ) 2n/parenrightBigg2 <ε. Thus,fcan becomputed by a q-qubit quantum OBDD,where q=log2t=O(loglogm). The width of the program is 2q=O(logm). The following functions have the aforementioned linear pol ynomials and thus are effectively com- puted via the ﬁngerprinting technique. MODmThe function MODmtests whether the number of 1’s in the input is 0 modulo m. The linear polynomial over Zmfor this function is n ∑ i=1xi. Thelowerbound forthewidthofdeterministic OBDDscomputi ng thisfunction is Ω(m)[13]. Thus, our method provides an exponential advantage of quantum OBDDov er any deterministic one. EQnThe function EQn, which tests the equality of two n-bit binary strings, has the following polyno- mial over Z2n n ∑ i=1xi2i−1−n ∑ i=1yi2i−1. F.Ablayev & A.Vasiliev 7 Palindrome n(x1,...,xn)This function tests the symmetry of the input, i.e. whether x1x2...x⌊n/2⌋= xnxn−1...x⌈n/2⌉+1or not. Thepolynomial over Z2⌊n/2⌋is ⌊n/2⌋ ∑ i=1xi2i−1−n ∑ i=⌈n/2⌉xi2n−i. PERMnThePermutation Matrix test function ( PERMn) is deﬁned on n2variables xij(1≤i,j≤n). It testswhether theinput matrixcontains exactlyone1ineach rowandeachcolumn. Hereisapolynomial overZ(n+1)2n n ∑ i=1n ∑ j=1xij/parenleftbig (n+1)i−1+(n+1)n+j−1/parenrightbig −2n ∑ i=1(n+1)i−1. Note, that this function cannot beeffectively computed by a deterministic OBDD–the lower bound isΩ(2nn−5/2)regardlessofthevariableordering[13]. Thewidthofthebe stknownprobabilistic OBDD, computingthisfunctionwithone-sidederror, is O(n4logn)[13]. Ouralgorithm hasthewidth O(nlogn). Since the lower bound Ω(n−logn)follows from Theorem 1, our algorithm isalmost optimal. The following table provides the comparison of the width of q uantum and deterministic OBDDsfor the aforementioned functions. OBDD QOBDD MODm Ω(m) O(logm) EQn 2Ω(n)O(n) Palindrome n2Ω(n)O(n) PERMnΩ(2nn−5/2)O(nlogn) 4 Generalized Approach The ﬁngerprinting technique described in the previous sect ion allows us to test a single property of the inputencodedbyacharacteristicpolynomial. Usingthesam eideaswecantesttheconjunctionofseveral conditions encodedbyagroupofcharacteristic polynomial s whichwecalla characteristic ofafunction. Deﬁnition 5. Wecall a set χm fof polynomials over Zmacharacteristic of aBoolean function f if for all polynomials g ∈χm fand allσ∈{0,1}nit holds that g (σ)=0iffσ∈f−1(1). Wesay that acharacteristic is linearif all of

Question 338 multiple-choice

Programmable quantum algorithms can be implemented using advanced photonic technologies, such as metasurfaces composed of geometric-phase metalens arrays. These platforms utilize spatial light modulation and precise photon detection to perform unitary operations on quantum states for tasks like search and signal processing. Which feature enables a static metasurface platform to switch between different quantum algorithms without physically altering its hardware configuration? 1) Adjusting the geometric parameters of each metalens in real-time 2) Changing the polarization state of output photons using external filters 3) Reprogramming the unitary matrix encoded in the metasurface 4) Dynamically reshaping the metasurface using microelectromechanical actuators 5) Selectively exciting different sets of metalenses and observing specific output directions 6) Varying the refractive index of the substrate with thermal control 7) Incorporating active gain materials to amplify specific output modes

✓ Correct Answer:

The correct answer is 5) Selectively exciting different sets of metalenses and observing specific output directions.

📚 Reference Text:

[ 36]–[38] and to solve differential equations[ 39],[40]. In this work, we have developed a platform with a tailor-made geometric-phase metalens array on a meta- surface to execute programmable quantum algorithms, enabling the implementation of unitary transformationsin a single step. By employing a spatial light modulator(SLM) and a single-photon camera for the generation anddetection of input and output states, we have performedprogrammable quantum algorithms and operations in theoptical domain. In this case, our work demonstrates pro-grammability on a static metasurface by using a spatiallightmodulator(SLM)toselectivelyilluminatedifferentsetsof metalenses and a single photon camera to selectively observe at different exit directions. Similar approaches using static metasurfaces have been used in generatingdynamic holograms by modulating incident polarization,orbitalangularmomentum,andspatialproﬁle[ 41]–[43].It allowsustoimplementGrover’ssearch(GS)andthequan-tum Fourier transform (QFT). GS algorithm is a quantumsearch algorithm to ﬁnd a unique marked element in anunstructureddatabasethatofferaquadraticspeedupoverthe classical search algorithm [ 44]. We have chosen the GS algorithm for its practical signiﬁcance, as fast search- ing facilitates solving difficult computational problems forfuture extension [ 22]. Likewise, QFT, a quantum equiva- lent of the discrete Fourier transform, serves as a funda-mental algorithmic primitive that enables the implemen-tation of several other quantum algorithms, with the mostnotable being Shor’s algorithm [ 45]. To substantiate ourmethodology, we conducted comprehensive simulations and experiments employing both classical and quantum lightsources.Theinherentprogrammabilityofourstrategy heraldsaneweraintheminiaturizationofquantumoptical circuits, presenting an economical and integrated avenue forquantuminformationprocessing. 2 Results and discussion 2.1 Computing scheme using metalens array for programmable quantum algorithms and operations Ourgoalistoimplementaspeciﬁcquantumalgorithmfor n-qubits, which can be summarized as a transformation: |j⟩→∑2n−1 i=0Uij|i⟩,where Uisaunitarymatrix,usingameta- surface. The ‘ket’ notation represents the n-qubit basis for eithertheinputoroutputstate,withtheinteger jorirang- ing from 0 to 2n−1. Our metasurface approach is shown in Figure1(a) .Aspatiallightmodulator(SLM)isutilizedtopre- pare the input state∑2n−1 j=0𝜓(in) j|j⟩, where input basis {|j⟩} representsauniformelectric-ﬁeldnormallyincidentonthe lens located at Rj,a n d𝜓(in) jrepresents the complex ﬁeld coefficient of the corresponding basis. The SLM operates in vertical polarization, which is subsequently convertedto left-circular polarization (LCP) before illuminating the metasurface. Our metasurface comprises 9 metalenses, of whichfourareemployedtoperformaspeciﬁc2-qubit( n = 2)unitaryoperation.Eachmetalensat Rjdiffractstheinput mode|j⟩intoadiscretesetofoutputmodes {|i⟩},withthe correspondingexitdirection kianddesignedcomplexmod- ulation Uij.Thebasisoftheoutputstate {|i⟩}correspondsto anexitbeamcenteredat kionthefar-ﬁeldofthemetasur- face. The different colors, physically all at the same wave-length,illustratedifferentoutputdirections;forinstance,a greenbeamcorrespondstothenormalexitdirection,while acyanbeamcorrespondstoa10 ◦deﬂectiontotheleft.Inour case,eachmetalensconvertsLCPtoright-circularpolariza- tion(RCP)usingacustomcomplextransmissionamplitudeandphaseproﬁlegivenbytheexpression: t j(𝛒)=2n−1∑ i=0Uijexp(iki⋅𝛒), (1) where 𝛒denotes the transverse position measured from the center of the entire metasurface. Consequently, light from different metalenses with the same output directioninterferes in the far-ﬁeld regime, resulting in the output mode|i⟩inRCP.Byﬁttingtheinterferencepatterncaptured by a single-photon camera, we can obtain the output ﬁeld R.S.Tanuwijayaetal.:Metasurfaceforprogrammablequantumalgorithms — 929 Figure 1:Computingmetasurfaceforprogrammablequantumalgorithms.(a)Overallschematicofthecomputingmetasurfaceforimplementing two-qubitunitaryquantumoperations.Aspatiallightmodulator(SLM)selectivelyexcitesthemetalensarrayandpreparesthetwo-qubitinputstat e ||𝜓(in)⟩bymodulatingthecomplexincidentE-fieldforeachselectedmetalensat R0toR3.Eachmetalensdiffractsthebeamintomultipleoutput directions,withamplitudeandphasemodulatedbythe U-matrix.Theinterferencepatternsofthediffractedbeamsinthesamedirectionareprojected ontotheFourierplane,wheretheintensityprofilearound k0tok3yieldstheoutputstate ||𝜓(out)⟩=̂U||𝜓(in)⟩.(b)Complextransmissionamplitudeand phaseprofileofthemetalensarray,encodingbothGrover’sSearch(GS)andQuantumFourierTransform(QFT).(c)Far-fieldintensityprofilewhenon ly thecenterlensisexcitedtoimplementacolumnoftheunitary Umatrix.Thecolorshowsdifferentamplitudeandphasemodulationfordifferent outputdirections(allofthesamewavelength).Programmabilityisachievedbyselectingthemetalensexcitation( R0toR3)andtheprojected interferencepatterns( k0tok3).Forexample,selectingtheredsquarein(b)and(c)enablesGS,whileselectingthebluesquareenablesQFT. (d)Thecorrespondingmatricesforthetwoprogrammedquantumoperators, ̂UGSand̂UQFT(normalizationfactoromittedforbrevity). amplitude|||𝜓(out) i|||.Theentireunitaryoperationcanbesum- marized as 𝜓(out) i=∑2n−1 j=0Uij𝜓(in) jin terms of the complex ﬁeldcoefficient. Our scheme incorporates programmability using a static metasurface with shared resources for implement- ing different quantum algorithm choices through selective illumination. The complex input ﬁeld coefficient 𝜓(in) jon a speciﬁc metalens at Rjis prepared by the correspond- ing region of a (phase) SLM at normal incidence. A cer- tain fraction of these pixels is deactivated by introducing a phase gradient, diverting the light at an angle that does not fall

Question 339 multiple-choice

Elliptic Calabi-Yau manifolds play a pivotal role in algebraic geometry and string theory, particularly due to their applications in compactification models and their rich deformation theory. The persistence of elliptic fibration structures under deformation is governed by intricate geometric and cohomological conditions. Which condition guarantees that all small deformations of a smooth elliptic Calabi-Yau manifold retain the elliptic Calabi-Yau property? 1) Non-vanishing first Chern class 2) Positivity of the third Todd class 3) Vanishing of the second cohomology group of the structure sheaf, H²(X, O_X) 4) Existence of a global holomorphic 3-form 5) Nontrivial fundamental group 6) Ampleness of the canonical bundle 7) Presence of continuous automorphisms

✓ Correct Answer:

The correct answer is 3) Vanishing of the second cohomology group of the structure sheaf, H²(X, O_X).

📚 Reference Text:

Title: Deformations of elliptic Calabi-Yau manifolds Year: 2012 Paper ID: fb3bf5bf2a5b5f619192181ffbe46e590ce8bb64 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/fb3bf5bf2a5b5f619192181ffbe46e590ce8bb64 Abstract: Abstract We investigate deformations and characterizations of elliptic Calabi-Yau varieties, building on earlier works of Wilson and Oguiso. We show that if the second cohomology of the structure sheaf vanishes, then every deformation is again elliptic. More generally, all non-elliptic deformations derive from abelian varieties or K3 surfaces. We also give a numerical characterization of elliptic Calabi-Yau varieties under some positivity assumptions on the second Todd class. These results lead to a series of conjectures on fibered Calabi-Yau varieties. To Robert Lazarsfeld on the occasion of his sixtieth birthday The aim of this paper is to answer some questions about Calabi-Yau manifolds that were raised during the workshop String Theory for Mathematicians , which was held at the Simons Center for Geometry and Physics. F -theory posits that the “hidden dimensions” constitute a Calabi-Yau 4-fold X that has an elliptic structure with a section. That is, there are morphisms g : X → B whose general fibers are elliptic curves and σ: B → X such that g ∘ σ = 1 B (see [Vaf96, Don98]). In his lecture, Donagi asked the following: Question 1 Is every small deformation of an elliptic Calabi-Yau manifold also an elliptic Calabi-Yau manifold? Question 2 Is there a good numerical characterization of elliptic Calabi-Yau manifolds? Clearly, an answer to Question 2 should give a solution of Question 1. The answers to these problems are quite sensitive to which variant of the definition of Calabi-Yau manifolds one uses. For instance, a general deformation of the product of an Abelian variety and of an elliptic curve has no elliptic fiber space structure and every elliptic K3 surface has non-elliptic deformations. We prove in Section 5 that these are essentially the only such examples, even for singular Calabi-Yau varieties (Theorem 31). In the smooth case, the answer is especially simple. Theorem 3 Let X be an elliptic Calabi-Yau manifold such that H 2 (X, O X ) = 0. Then every small deformation of X is also an elliptic Calabi-Yau manifold .

Question 340 multiple-choice

Quantum computing algorithms such as Quantum Fourier Transform (QFT) and quantum fingerprinting are increasingly adapted for real-world quantum devices, which often feature complex qubit connectivity graphs. Efficient circuit construction on these devices requires specialized methods to handle restricted two-qubit gate interactions and minimize error-prone operations. Which statement best describes the advantage of a generic circuit construction method for quantum algorithms on devices with arbitrary qubit connection graphs? 1) It enables quantum circuits to run exclusively on linear nearest neighbor architectures. 2) It guarantees the absolute minimum number of CNOT gates for every possible connectivity. 3) It restricts circuit implementation to only superconducting qubit devices. 4) It eliminates the need for two-qubit gates in quantum algorithms. 5) It tailors quantum algorithms solely for idealized hardware with full qubit connectivity. 6) It allows algorithm adaptation and circuit compilation for any hardware topology, improving portability and scalability despite a slightly higher CNOT gate count. 7) It optimizes quantum circuits exclusively for photonic quantum computers.

✓ Correct Answer:

The correct answer is 6) It allows algorithm adaptation and circuit compilation for any hardware topology, improving portability and scalability despite a slightly higher CNOT gate count..

📚 Reference Text:

Title: Implementation of Quantum Fourier Transform and Quantum Hashing for a Quantum Device with Arbitrary Qubits Connection Graphs Year: 2025 Paper ID: 420867117cb908f657e9f7e72c9c1187db06e654 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/420867117cb908f657e9f7e72c9c1187db06e654 Abstract: In the paper, we consider quantum circuits for Quantum fingerprinting (quantum hashing) and quantum Fourier transform (QFT) algorithms. Quantum fingerprinting (quantum hashing) is a well-known technique for comparing large objects using small images. The QFT algorithm is a very popular technique used in many algorithms. We present a generic method for constructing quantum circuits for these algorithms for quantum devices with restrictions. Many quantum devices (for example, based on superconductors) have restrictions on applying two-qubit gates. The restrictions are presented by a qubits connection graph. Typically, researchers consider only the linear nearest neighbor (LNN) architecture, but current devices have more complex graphs. We present a method for arbitrary connected graphs that minimizes the number of CNOT gates in the circuit. The heuristic version of the method is fast enough and works with $O(n^5)$ time complexity, where $n$ is the number of qubits. The certain version of the algorithm has an exponential time complexity that is $O(n^22^n)$. We compare quantum circuits built by our algorithm with quantum circuits optimized for specific graphs that are Linear-nearest-neighbor (LNN) architecture, ``sun'' (a cycle with tails, presented by 16-qubit IBMQ device) and ``two joint suns'' (two joint cycles with tails, presented by 27-qubit IBMQ device). Our generic method gives similar results with little bit more CNOT gates. At the same time, our method allows us to construct a circuit for arbitrary connected graphs.

Question 341 multiple-choice

Quantum theory relies on the mathematical structure of Hilbert spaces and is deeply influenced by the symmetries of spacetime, such as those described by the Poincaré group. The interplay between symmetry groups and operator algebras determines which types of Hilbert spaces are physically viable for formulating quantum systems. Which condition ensures that a real Hilbert space equipped with a unitary representation of the Poincaré group acquires a unique, Poincaré-invariant complex structure that commutes with all observables? 1) The existence of non-trivial quaternionic subspaces 2) The requirement that all observables are bounded operators 3) The irreducibility of the representation only under spatial rotations 4) The absence of time-translation symmetry 5) The non-negativity of the squared-mass operator 6) The maximal continuity of the Hilbert space topology 7) The decomposition of the Hilbert space into purely real components

✓ Correct Answer:

The correct answer is 5) The non-negativity of the squared-mass operator.

📚 Reference Text:

Title: Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from Poincar\'e symmetry Year: 2016 Paper ID: b57a37837b18703589d870e15cb99263b3346471 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b57a37837b18703589d870e15cb99263b3346471 Abstract: As established by Soler, Quantum Theories may be formulated in real, complex or quaternionic Hilbert spaces only. Stuckelberg provided physical reasons for ruling out real Hilbert spaces relying on Heisenberg principle. Focusing on this issue from another viewpoint, we argue that there is a fundamental reason why elementary quantum systems are not described in real Hilbert spaces: their symmetry group. We consider an elementary relativistic system within Wigner's approach defined as a locally-faithful irreducible continuous unitary representation of the Poincare group in a real Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincare invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation. All that leads to a physically equivalent formulation in a complex Hilbert space. Differently from what happens in the real picture, here all selfadjoint operators are observables in accordance with Soler's thesis, and the standard quantum version of Noether theorem holds. We next focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them and making our model physically more general. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions and we adopt a notion of continuity referred to the states. Also in this case, the final result proves that there exist a unique (up to sign) Poincare invariant complex structure making the theory complex and completely fitting into Soler's picture. This complex structure reveals a nice interplay of Poincare symmetry and the classification of the commutant of irreducible real von Neumann algebras.

Question 342 multiple-choice

In particle physics, the "flavour problem" concerns the inability of existing theories to predict the observed patterns of fermion masses and mixing angles. Extensions to the Standard Model often incorporate new symmetries and particles to address these shortcomings. Which of the following modifications specifically introduces right-handed neutrino singlets to generate neutrino masses and accommodate observed neutrino oscillations? 1) Inclusion of supersymmetric partners for all Standard Model particles 2) Addition of a scalar singlet field coupled to leptons 3) Imposing an abelian discrete symmetry on the lepton sector 4) Increasing the number of quark generations beyond three 5) Introducing extra charged leptons with vector-like couplings 6) Adding three right-handed neutrino singlets to the particle content 7) Extending the gauge group to include an extra U(1) factor

✓ Correct Answer:

The correct answer is 6) Adding three right-handed neutrino singlets to the particle content.

📚 Reference Text:

arXiv:1604.08072v2 [hep-ph] 17 Jun 2016The discrete family symmetries as the possible solution to t he ﬂavour problem Bartosz Dziewit,1,∗Jacek Holeczek,1,†Monika Richter,1,‡ Sebastian Zajac,2,§and Marek Zralek1,¶ 1Institute of Physics, University of Silesia Uniwersytecka 4, Katowice, Poland 2Faculty of Mathematics and Natural Studies Cardinal Stefan Wyszynski University in Warsaw Dewajtis 5, Warszawa, Poland In order to explain the fermions’ masses and mixing paramete rs appearing in the lepton sector of the Standard Model, one proposes the ext ension of its sym- metry. A discrete, non-abelian subgroup of U(3) is added to the gauge group SU(3)C×SU(2)L×U(1)Y. Apart from that, one assumes the existence of one extra Higgs doublet. This article focuses mainly on the math ematical theorems and computational techniques which brought us to the results. I. INTRODUCTION The ﬂavour problem constitutes one of the most serious disadvant ages of the Standard Model [1]. It doesn’t allow us to predict theoretically neither the part icles’ masses nor the parameters describing UPMNSmixing matrix (Pontecorvo-Maki-Nakagawa-Sakata matrix). This fact clearly points to some extension of the present-day theo ry. In our approach, one adds some new, non-abelian discrete symmet ryGFto the Standard Model’s gauge group. This step can be partially justiﬁed bearing in mind the enormous success of the tribimaximal mixing [2] which could be explained by such a group. Since ∗Electronic address: bartosz.dziewit@us.edu.pl †Electronic address: jacek.holeczek@us.edu.pl ‡Electronic address: mrichter@us.edu.pl §Electronic address: s.zajac@uksw.edu.pl ¶Electronic address: marek.zralek@us.edu.pl 2 2004 many similar ideas have already been widely studied, but gave no r easonable results. For this reason, we have decided to go one step further and widen t he scalar sector as well. To our knowledge, not so many attempts have been ventured in this direction, so far. Only one article concerning such a model (with 3 Higgs doublets) has been found in the literature [3]. This fact motivated us to investigate the model with two Higgs dou blets with a lot of possible choices for group GF[4]. A. The main concept To start with, it is necessary to introduce the Yukawa part of the c onsidered model: LY=−/summationdisplay α,β=e,µ,τ/summationdisplay j=1,2[(hl j)αβ¯LαLΦjlβR+(hν j)αβ¯LαL˜ΦjνβR], (1) where •e,µ,τ- three diﬀerent ﬂavors of charged leptons and neutrinos, •LαL= ναL lαL - the lepton doublet composed of left-handed charged lepton and n eu- trino ﬁeld of given ﬂavor α, •lβR,νβR- the right-handed charged lepton and neutrino ﬁeld of given ﬂavor β, •Φ1,2= Φ+ 1,2 Φ0 1,2 - two Higgs doublets, •˜Φ1,2=iσ2Φ∗ 1,2, •hl 1,2,hν 1,2- Yukawa couplings. We assumed here the minimal extension of the Standard Model: to ge t the masses for neutrinos, three right-handed neutrino singlets have been added . In principle, four Yukawa couplings hl,ν 1,2occurring in the model are arbitrary 3- dimensional matrices. This lack of any restrictions causes that the masses of fermions and mixing parameters are treated as free parameters within the Stan dard Model. In order to solve this problem, one can try to constrain the allowed m asses by imposing some additional symmetry GFon the Lagrangian. In simple terms, we expect Eq.(1) to be

Question 343 multiple-choice

Quantum programming languages and frameworks are evolving to support both the efficient implementation and formal verification of critical subroutines called quantum oracles, which are central to major algorithms such as Shor’s factoring and Grover’s search. These frameworks must address challenges including resource optimization, correctness assurance, and integration with classical operations. Which approach enables a quantum programming framework to optimize oracle implementation by reducing qubit usage while formally verifying correctness, through a pipeline involving an imperative language, quantum Fourier transform-based assembly, and a general-purpose quantum assembly output? 1) Utilizing reversible classical gates without formal verification 2) Employing unverified circuit synthesis with entanglement-based optimizations 3) Compiling from a verified imperative language to a QFT-based assembly language and then to general-purpose quantum assembly, leveraging formal proofs for correctness 4) Relying solely on randomized property-based testing for error detection 5) Direct translation of classical functions to quantum circuits without resource analysis 6) Implementing quantum oracles using only state-of-the-art hardware error correction 7) Generating oracles with a probabilistic programming language lacking formal guarantees

✓ Correct Answer:

The correct answer is 3) Compiling from a verified imperative language to a QFT-based assembly language and then to general-purpose quantum assembly, leveraging formal proofs for correctness.

📚 Reference Text:

Title: Verified compilation of Quantum oracles Year: 2021 Paper ID: f11b22e64bef2cccd1efc128b7f5b61ad6611765 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/f11b22e64bef2cccd1efc128b7f5b61ad6611765 Abstract: Quantum algorithms often apply classical operations, such as arithmetic or predicate checks, over a quantum superposition of classical data; these so-called oracles are often the largest components of a quantum program. To ease the construction of efficient, correct oracle functions, this paper presents VQO, a high-assurance framework implemented with the Coq proof assistant. The core of VQO is OQASM, the oracle quantum assembly language. OQASM operations move qubits between two different bases via the quantum Fourier transform, thus admitting important optimizations, but without inducing entanglement and the exponential blowup that comes with it. OQASM’s design enabled us to prove correct VQO’s compilers—from a simple imperative language called OQIMP to OQASM, and from OQASM to SQIR, a general-purpose quantum assembly language—and allowed us to efficiently test properties of OQASM programs using the QuickChick property-based testing framework. We have used VQO to implement a variety of arithmetic and geometric operators that are building blocks for important oracles, including those used in Shor’s and Grover’s algorithms. We found that VQO’s QFT-based arithmetic oracles require fewer qubits, sometimes substantially fewer, than those constructed using “classical” gates; VQO’s versions of the latter were nevertheless on par with or better than (in terms of both qubit and gate counts) oracles produced by Quipper, a state-of-the-art but unverified quantum programming platform.

Question 344 multiple-choice

Quantum algorithms have revolutionized the approach to solving certain cryptographic problems, notably the discrete logarithm problem, which underpins the security of many classical cryptosystems. Key techniques include quantum Fourier transform, amplitude amplification, and quantum eigenvalue estimation. In the context of quantum algorithms for the discrete logarithm problem in a finite cyclic group of order p, which method can be used to make the algorithm's success probability uniform and independent of specific problem instances, and what challenge remains with this approach? 1) Increasing the number of quantum registers and applying repeated measurements; computational complexity increases exponentially 2) Using Grover's search algorithm to amplify amplitude; requires classical preprocessing 3) Applying post-processing error correction after measurement; demands additional error syndrome extraction 4) Randomizing the group element β by multiplying with α^r for random r; efficient computation of the averaged success probability remains nontrivial 5) Reducing the group order to a prime factor; introduces a risk of information loss 6) Applying a classical hash function to β before quantum processing; may destroy group structure 7) Encoding β in a higher-dimensional Hilbert space; complicates state preparation

✓ Correct Answer:

The correct answer is 4) Randomizing the group element β by multiplying with α^r for random r; efficient computation of the averaged success probability remains nontrivial.

📚 Reference Text:

αp=e. Then another elementβof the group is given and we want to know which power of αit is; that is, the integer afor which β=αa. This is also written as a= logαβ. In the quantum solution (see Shor [11]), we prepare two registers, each in a uniform amplitude superposition of pbasis states: 1 pp−1/summationdisplay x=0p−1/summationdisplay y=0|x,y/an}⌊ra⌋k⌉tri}ht. 4 Then we compute the function αxβyin an additional register and measure it. This will leave the two registers in a superposition of the form/summationtext y|x0−a·y,y/an}⌊ra⌋k⌉tri}ht where all arithmetic operations are understood to be modulo p,x0is random andyruns over 0 ...p−1. By Fourier transforming each register with a QFFT p we get a similar state but without the oﬀset x0, namely an equally weighted superposition of all states of the form |x,a·x/an}⌊ra⌋k⌉tri}htwithx= 0...p−1. A measure- ment will now allow to compute ain all cases except when x= 0. Thus we have the known and instance independent success probability of 1 −1/p,which allows to easily make the algorithm exact by using (exact) amplitude ampliﬁca tion. 3.1 Alternatively: directly uniformising dlog Actually one can directly make the success probability of the dlog algo rithm instance independent. Thus one uses the usual algorithm with a QFF T2n, but replacesβwithβ·αrwhererisagainchosenuniformlyatrandomfrom0 ...p−1. We have noted this approach a while ago, but were not able to show th at the (now averaged) success probability can be computed eﬃciently, th us it is not clear whether the circuit for a given pcan be computed eﬃciently. 4 Eigenvalue estimation In our case we want to estimate the eigenvalue of |Ψx/an}⌊ra⌋k⌉tri}htunder the (unitary) cyclic shift operator Uwhich acts on computational basis states as: |x/an}⌊ra⌋k⌉tri}ht → |(x+1) mod p/an}⌊ra⌋k⌉tri}ht. For eigenvalue estimation we need to do large powers of U, which in this case is easy. Namely we ﬁrst prepare an auxiliary n-qubit register in a uniform amplitude superposition of all its N= 2nbasis states. (We will chooseNto be larger than p, see below.) Then we do: 1√ NN−1/summationdisplay y=0|y/an}⌊ra⌋k⌉tri}ht |Ψx/an}⌊ra⌋k⌉tri}ht →1√ NN−1/summationdisplay y=0|y/an}⌊ra⌋k⌉tri}htUy|Ψx/an}⌊ra⌋k⌉tri}ht=1√ NN−1/summationdisplay y=0e−2πixy p|y/an}⌊ra⌋k⌉tri}ht |Ψx/an}⌊ra⌋k⌉tri}ht. where we used that the eigenvalue of |Ψx/an}⌊ra⌋k⌉tri}htunderUise−2πix/p. Note that the operation we have to do is simply a modular addition on computational b asis states, thus |a,b/an}⌊ra⌋k⌉tri}ht → |a,(a+b) modp/an}⌊ra⌋k⌉tri}ht. After a Fourier transform of size 2non the auxiliary register, the probability of measuring ywould be given by: py=f2(y−xN/p) where f(z) =sin(πz) Nsin(πz/N). We illustrate the function f(z) withN→ ∞in ﬁgure 1. It is peaked around z= 0 so that after measuring some ywe would guess for the number we want to ﬁndx≈y·p/N. The choice with the highest probability of obtaining the correctxwould be to simply round y·p/Nto the closest integer. Partially to simplify notation, here we round up to the next integer, thus our guess is x′=⌈y·p/N⌉. (For us the loss of some success probability does not matter, at least not as long as it is at least 1/4.) 5 00.51 ±5 ±4 ±3 ±2 ±1 1 2 3 4 5x Figure 1: The functionsin(πx) πx. Because we have N > p, it is

Question 345 multiple-choice

Quantum photonic circuits are evaluated using various fidelity metrics and error models, which influence the generation and manipulation of quantum states. Understanding the mathematical structure of quantum states in infinite-dimensional Hilbert spaces is essential for analyzing these systems. Which statement accurately describes the fundamental relationship between the Fock basis and the coherent state basis in the Hilbert space of a single-mode harmonic oscillator? 1) Both Fock and coherent state bases are countable and orthonormal. 2) A unitary transformation connects the Fock and coherent state bases. 3) Fock states are always superpositions of other Fock states. 4) Coherent states are eigenstates of the number operator. 5) The Fock basis is overcomplete and uncountable. 6) Coherent states (except the vacuum) are superpositions of Fock states, while Fock states (for n > 0) are superpositions of coherent states. 7) The inner product of two Fock states depends exponentially on their separation.

✓ Correct Answer:

The correct answer is 6) Coherent states (except the vacuum) are superpositions of Fock states, while Fock states (for n > 0) are superpositions of coherent states..

📚 Reference Text:

First, we see that by all metrics, OBB errors lead to worse state generation than SBB errors. This fits the intuition that more distinguishability, even between photons that are already non-ideal, reduces the overall performance of photonic circuits. The comparison between the OBP and other models, however, is less clear-cut. The OBP model leads to higher standard fidelity than both the OBB and SBB models for small ϵ,0< ϵ⪅0.083. However, the OBP model in fact exhibits the lowest post-selected fidelity for reasonable values ofϵ,0< ϵ⪅0.186; for larger ϵ, OBB is worse than OBP. This difference in performance is because the OBP model does not lead to non-dual-rail terms and thus gives the same values for the standard and post-selected fidelity (compare the appropriate functions in Appendix A 3). Thus, post-selection improves the fidelity in the OBB and SBB models but does not affect the OBP model. For the fusion fidelity, we see similar behavior as for the post-selected fidelity, but the intersection between the OBB and OBP curves occurs much earlier, ϵ≈0.060. Finally, we note that the OBB model leads to more expected stabilizer errors than the OBP model for all ϵ. Thus we see that error models and performance metrics can interact nontrivially, and it is important to consider a variety of metrics when comparing the performance of two or more linear optical protocols. 12 Figure 7: Plot of the BSG circuit’s performance under various metrics, comparing 3 different error models. III. COHERENT-RANK FRAMEWORK We summarize the coherent-rank framework introduced in Ref. 5. There are two canonical bases of interest for the separable, infinite-dimensional Hilbert space of a simple harmonic oscillator, H∞∼=L2(R): the Fock basis, gener- ated by the eigenstates of the number operator, B={|n⟩}∞ n=0where ˆn|n⟩=n|n⟩. And the coherent state basis, eB={|α⟩}α∈C, which are eigenstates of the annihilation operator, ˆa|α⟩=α|α⟩. Recall that coherent states are de- fined via the action of the displacement operator on the vacuum state |α⟩:=ˆD(α)|0⟩=e−|α|2/2P∞ n=0αn √ n!|n⟩, where ˆD(α) =eαˆa†−¯αˆa=e−|α|2/2eαˆa†e−¯αˆais the displacement unitary with α∈C. The two bases interact in an interesting way: apart from the vacuum state |0⟩, no other Fock state is a coherent state and vice-versa. As a result, every coherent state |α⟩, α̸= 0 is a superposition of Fock states and every Fock state |n⟩, n > 0 is a superposition of coherent states. Given two bases in a Hilbert space, it is natural to seek a unitary (or isometry) that connects the two bases. However in this case, no such transformation exists. This is because the coherent states form an uncountable, overcomplete basis for H∞, with the following resolution of the identity,1 πR α∈Cdα|α⟩⟨α|=I. In contrast, the Fock states form a countable orthonormal basis with,∞P n=0|n⟩⟨n|=I. The inner product between two coherent states is⟨α|β⟩=e−1 2|α−β|2e−iIm(α¯β). This relation makes it evident that if two coherent states are “far away” from each other, namely, |α−β| ≫1, then they are approximately orthogonal. The above implies a Fock basis representation of a state, such as the maximum N-photon single-mode state |ψ⟩=PN n=0cn|n⟩,

Question 346 multiple-choice

Quantum algorithms often rely on precise quantum state preparation techniques, such as those used for constructing Fourier states and amplifying desired quantum outcomes. Strategies like amplitude amplification and phase estimation are central to increasing the probability of measuring specific solution states. In the context of amplitude amplification within quantum algorithms, which adjustment to the operator ensures that a single iteration can yield the desired state with certainty, given an initial success probability of exactly 1/4? 1) Setting both amplification phases to zero 2) Applying entangling gates after every iteration 3) Increasing the number of qubits used for the initial state 4) Modifying the operator so that its success probability is exactly 1/4 5) Replacing SO(2) rotations with controlled-NOT gates 6) Performing eigenvalue estimation after amplitude amplification 7) Using a projector with randomized phase shifts

✓ Correct Answer:

The correct answer is 4) Modifying the operator so that its success probability is exactly 1/4.

📚 Reference Text:

to this state and can thus again be left away. The ﬁrst step constructs the Fourier state |Ψx/an}⌊ra⌋k⌉tri}htfor a given x. This can be done exactly by ﬁrst obtaining the “uniform amplitude” superpositio n|Ψ0/an}⌊ra⌋k⌉tri}htof the ﬁrst pbasis states of a register and then “rephasing” it: |x,0/an}⌊ra⌋k⌉tri}ht → | x,Ψ0/an}⌊ra⌋k⌉tri}ht → | x,Ψx/an}⌊ra⌋k⌉tri}ht. (1) As pointed out by Kitaev, |Ψ0/an}⌊ra⌋k⌉tri}htcan be obtained from |0/an}⌊ra⌋k⌉tri}htby a sequence of SO(2) rotations applied to each qubit in order from high to low signiﬁcance, w hereby the rotation angle has to be controlled by the previously touched qu bits. The rephasing then simply consists of a rephasing on each qubit, propor tional to x and the place value of the qubit. The second step of Kitaev’s construction is the reverse of |Ψx,0/an}⌊ra⌋k⌉tri}ht → | Ψx,x/an}⌊ra⌋k⌉tri}ht. This is done through a technique known as “eigenvalue estimation” (s ee also the article by Cleve et al. [3]), which details how to ﬁnd the eigenvalue of an unknown eigenstate of some unitary U. We will describe this in more detail later. Here we only need to note that although this operation is not e xact, it leaves the eigenstate |Ψx/an}⌊ra⌋k⌉tri}htunchanged. Thus it does: |Ψx,0/an}⌊ra⌋k⌉tri}ht → | Ψx/an}⌊ra⌋k⌉tri}ht/summationdisplay x′cx,x′|x′,gx,x′/an}⌊ra⌋k⌉tri}ht (2) 2 where on the right hand side the superposition should be dominated b y the term withx′=x, such that a measurement would yield xwith good probability. We also included some (unwanted) “garbage” gx,x′which may be produced along with the eigenvalue. 2.1 Using amplitude ampliﬁcation We now use “amplitude ampliﬁcation” [1] to eliminate all but the desired term |x,gx,x/an}⌊ra⌋k⌉tri}ht. WegivehereaquickreviewofthisgeneralisationofGrover’salgorith m. We are given a unitary operator Awhich, when applied to the initial state |0/an}⌊ra⌋k⌉tri}ht, gives an output state which has some component in a “good” subspa ce. Thus the probability |PgoodA|0/an}⌊ra⌋k⌉tri}ht|2is not too small, where Pgoodis the projector onto the good subspace. The amplitude of the good component can be inc reased through the following procedure /bracketleftbig A(1+(eiφ−1)|0/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{t0|)A−1(1+(eiϕ−1)Pgood)/bracketrightbigTA|0/an}⌊ra⌋k⌉tri}ht where the sequence of operations in the brackets is repeated Ttimes, depending on the “success probability” of the “algorithm” Aalone. As in Grover’s algo- rithm, the fastest increase is achieved when both phases are chos enφ=ϕ=π. The algorithm can be analysed by noting that the state alwaysremain s in a sub- space spanned by the state we are seeking PgoodA|0/an}⌊ra⌋k⌉tri}htand by (1 −Pgood)A|0/an}⌊ra⌋k⌉tri}ht. Usually an integer number of iterations will not lead exactly to the des ired state and so we need to chose diﬀerent (non-optimal) phases, either in all steps or only in the last one or two. In our case we will leave the phases at their stan- dard settings, but will modify Aso that its success probability is reduced to 1/4 where a single iteration leads exactly to the desired state. The operator Awill be given by eq. 2, where the state |Ψx/an}⌊ra⌋k⌉tri}htwill have to be added as a “spectator” that is not changed. 2.1.1 “Recognising” the correct solution Apart from the “heuristic” algorithm A, amplitude ampliﬁcation requires a way to“recognise”the goodstates. Moreprecisely, weneed awaytoa pply the phase

Question 347 multiple-choice

Quantum algorithms for hidden subgroup problems are central to the development of efficient solutions for problems such as factoring and discrete logarithms. The dihedral hidden subgroup problem (DHSP) serves as a benchmark for understanding the limits of quantum computation, especially for non-abelian groups. Which statement accurately describes the sample complexity threshold for successfully identifying the hidden subgroup in the dihedral hidden subgroup problem using the pretty good measurement (PGM)? 1) Success probability is high for any nonzero number of copies of the hidden subgroup state. 2) At least Ω(√N) quantum samples are required for reliable success. 3) The threshold is determined by the ratio k / N, where k is the number of copies and N is the group order. 4) The probability of success remains constant regardless of the number of samples taken. 5) The number of required quantum samples scales linearly with the group order. 6) Ω(log |G|) copies are necessary, and the success probability exhibits a sharp threshold at ν = k / log₂ N, with low success for ν < 1 and high success for ν > 1. 7) Reliable identification is possible with only one copy of the hidden subgroup state.

✓ Correct Answer:

The correct answer is 6) Ω(log |G|) copies are necessary, and the success probability exhibits a sharp threshold at ν = k / log₂ N, with low success for ν < 1 and high success for ν > 1..

📚 Reference Text:

OPTIMAL MEASUREMENTS FOR THE DIHEDRAL HIDDEN SUBGROUP PROBLEM DAVE BACON, ANDREW M. CHILDS, AND WIM VAN DAM Abstract. We consider the dihedral hidden subgroup problem as the problem of distinguishing hidden subgroup states. We show that the optimal measurement for solving this problem is the so-called pretty good measurement. We then prove that the success probability of this measurement exhibits a sharp threshold as a function of the density ν=k/log2N, where kis the number of copies of the hidden subgroup state and 2 Nis the order of the dihedral group. In particular, for ν <1 the optimal measurement (and hence any measurement) identiﬁes the hidden subgroup with a probability that is exponentially small in logN, while for ν > 1 the optimal measurement identiﬁes the hidden subgroup with a probability of order unity. Thus the dihedral group provides an example of a group Gfor which Ω(log |G|) hidden subgroup states are necessary to solve the hidden subgroup problem. We also consider the optimal measurement for determining a single bit of the answer, and show that it exhibits the same threshold. Finally, we con- sider implementing the optimal measurement by a quantum circuit, and thereby establish further connections between the dihedral hidden sub- group problem and average case subset sum problems. In particular, we show that an eﬃcient quantum algorithm for a restricted version of the optimal measurement would imply an eﬃcient quantum algorithm for the subset sum problem, and conversely, that the ability to quantum sample from subset sum solutions allows one to implement the optimal measurement. 1.Introduction Quantum computers promise to solve certain problems asymptotically faster than their classical counterparts. In particular, Shor’s discovery of an eﬃcient quantum algorithm for factoring [49]—a cryptographically signiﬁ- cant task for which no eﬃcient classical algorithm is known—has motivated considerable investigation into the potential algorithmic uses of quantum computers. Along with its predecessors [3,8,9,50], Shor’s algorithm can be viewed as a solution to one of a large class of problems known as hidden sub- group problems [4,30], several of which also have applications to interesting computational problems for which no eﬃcient classical algorithm is known. Thus the broader question arises: under what circumstances can the hidden subgroup problem ( hsp) be solved eﬃciently by a quantum computer? 2000 Mathematics Subject Classiﬁcation. 81P68, 68Q17, 81P15. Key words and phrases. quantum algorithms, optimal measurements, combinatorics. 2 DAVE BACON, ANDREW M. CHILDS, AND WIM VAN DAM Encouragingly, the quantum query complexity of the general hspis poly- nomial: for any group G, only poly(log |G|) quantum queries of the function that hides a subgroup are suﬃcient to solve the problem [12–15, 30]. Thus lower bounds showing that the hspis intractable are unlikely to be forth- coming. However, the processing of the queries could take exponential time, so it remains a challenge to ﬁnd algorithms that are eﬃcient in terms of the number of elementary operations. Following Shor’s discovery, there has been considerable progress in show- ing that quantum computers can eﬃciently solve the hsp for particular groups

Question 348 multiple-choice

Quantum channels introduce various types of noise to qubit systems, affecting the distribution and reliability of measured outcomes in quantum circuits. The standard deviation of measurement outcomes provides insight into the transition from ordered to maximally mixed states under increasing noise. Which quantum noise channel is characterized by causing the most rapid exponential increase in the standard deviation of output distribution as noise level rises, thus demonstrating the greatest sensitivity to noise among common channels? 1) Bit flip channel 2) Phase flip channel 3) Bit-phase flip channel 4) Amplitude damping channel 5) Depolarizing channel 6) Phase damping channel 7) Generalized amplitude damping channel

✓ Correct Answer:

The correct answer is 5) Depolarizing channel.

📚 Reference Text:

phase ﬂip channel bit-phase ﬂip channel 0.0000 0.0025 0.0050 0.0075 0.0100 p0.00.10.20.30.4∆θθactual = 0.96875 Nruns= 10000 f(p) =k1+k2e−k3p depolarizing channel bit ﬂip channel phase ﬂip channel bit-phase ﬂip channel Figure 3: Best fit plot of ∆ θvs.pforn= 5 qubits. The model paramters are provided in Table 1. To characterize the approach of the circuit output towards the maximally mixed state, we computed the standard deviation ∆ θof the measured output and presented them in Fig. 2. The figure shows that ∆ θstarts small and approaches a saturation value when pincreases. This gives us a better measure of the approach to maximally mixed state even when the actual θcan be close to 0 .5. Since the results in Fig. 2 for ∆ θshow an exponential increase, we fitted ∆ θto the following model, when pis small, ∆θ(p) =k1+k2e−k3p,0≤p≤0.01, (1) for all four channels and obtained parameters of the curve that best models the data with more than 98% accuracy. The parameters are provided in Table 1. The table shows that ∆ θhas strong exponential dependence upon p. The plot of standard deviation ∆ θvs. noise level psuch that 0 < p < 0.01 is illustrated in Fig. 3. The results in Table 1 indicate that the depolarization channel has the strongest exponential dependence on noise. Furthermore, ∆ θapproaches k1≈0.3 when actual θis either close to 0 or 1 but k1≈0.2 when θwas 0 .5 when p= 0.01. However, when p= 0.05, ∆ θapproaches 0 .3 for all values of θas can be seen from Fig. 2. This can also be expected from the fact that the standard deviation of uniformly distributed data from 0 to 1 is about 0.3 because a maximally mixed state at the output will lead to a uniform distribution of all possible values of θ∈ {0,1}. Table 1: The optimal set of parameters for the function ∆ θ(p) =k1+k2e−k3pthat best fits the standard deviation ∆θfor 0≤p≤0.01 with more than 98% accuracy. (a)θactual = 0.03125 Noise Channel k1 k2 k3 Depolarizing 0.32 -0.28 654 Bit flip 0.31 -0.27 317 Phase flip 0.31 -0.27 448 Bit-phase flip 0.30 -0.26 413(b)θactual = 0.5 k1 k2 k3 0.24 -0.21 338 0.18 -0.16 238 0.22 -0.19 293 0.22 -0.19 267(c)θactual = 0.96875 k1 k2 k3 0.35 -0.31 927 0.38 -0.33 460 0.33 -0.28 448 0.34 -0.29 508 5 6 7 8 9 10 11 n0.00.20.40.60.81.0¯θθactual = 0.03125 Nruns= 10000 p= 0.1%depolarizing channel bit ﬂip channel phase ﬂip channel bit-phase ﬂip channel 5 6 7 8 9 10 11 n0.00.20.40.60.81.0∆θθactual = 0.03125 Nruns= 10000 p= 0.1%depolarizing channel bit ﬂip channel phase ﬂip channel bit-phase ﬂip channel 5 6 7 8 9 10 11 n0.00.20.40.60.81.0¯θ θactual = 0.96875 Nruns= 10000 p= 0.1%depolarizing channel bit ﬂip channel phase ﬂip channel bit-phase ﬂip channel 5 6 7 8 9 10 11 n0.00.20.40.60.81.0∆θθactual = 0.96875 Nruns= 10000 p= 0.1%depolarizing channel bit ﬂip channel phase ﬂip channel bit-phase ﬂip channel Figure 4: Average value θand standard deviation ∆ θas function of number of qubits nfor fixed pand for

Question 349 multiple-choice

Quantum algorithms for integer factorization rely on advanced number-theoretic and quantum computing concepts, including modular arithmetic, group theory, and efficient quantum transforms. Shor's algorithm, in particular, exploits the periodicity in modular exponentiation and the efficiency of the Quantum Fourier Transform (QFT). Which one of the following statements correctly describes a key advantage of implementing the Quantum Fourier Transform (QFT) over the group Z₂ⁿ in quantum circuits for order-finding algorithms? 1) The QFT over Z₂ⁿ requires exponential time in the number of qubits due to complex gate operations. 2) The QFT over Z₂ⁿ can only be performed using Toffoli gates, making it impractical for large-scale quantum systems. 3) The QFT over Z₂ⁿ does not preserve orthonormality of basis states, limiting its usefulness in quantum algorithms. 4) The QFT over Z₂ⁿ outputs only classical values, preventing the use of quantum superposition. 5) The QFT over Z₂ⁿ fails to exploit periodicity, making it unsuitable for factoring algorithms. 6) The QFT over Z₂ⁿ can be efficiently implemented in O(n²) time using Hadamard and controlled phase gates, providing exponential speedup over classical Fourier transforms for large n. 7) The QFT over Z₂ⁿ requires post-selection measurements, increasing the risk of decoherence in practical quantum computations.

✓ Correct Answer:

The correct answer is 6) The QFT over Z₂ⁿ can be efficiently implemented in O(n²) time using Hadamard and controlled phase gates, providing exponential speedup over classical Fourier transforms for large n..

📚 Reference Text:

integer asuch that 1

Question 350 multiple-choice

Quantum walks are an important algorithmic paradigm in quantum computing, with applications in simulation, search, and transport. The use of highly symmetric matrices and efficient quantum operations enables scalable implementations of quantum walks on multi-qubit systems. Which property of circulant matrices is crucial for enabling efficient quantum walk implementations using the Quantum Fourier Transform, especially on multi-qubit systems? 1) Circulant matrices are diagonalizable in the Fourier basis, allowing efficient application via QFT and phase gates. 2) Circulant matrices always commute with all coin operators, simplifying circuit decomposition. 3) The entries of circulant matrices are restricted to binary values, enabling efficient classical simulation. 4) Circulant matrices guarantee perfect state transfer between all positions after one time step. 5) The eigenvalues of circulant matrices are always real, facilitating noise-resistant quantum operations. 6) Circulant matrices can only represent nearest-neighbor interactions, limiting their use in long-range coupling. 7) Every circulant matrix is equivalent to a permutation matrix, allowing trivial implementation in quantum circuits.

✓ Correct Answer:

The correct answer is 1) Circulant matrices are diagonalizable in the Fourier basis, allowing efficient application via QFT and phase gates..

📚 Reference Text:

any unitary circulant matrix (see section 4 .8 of [9]), i.e., any convolution matrix. Recall that an N×Ncirculant matrix is a special type of Toeplitz matrix of the form C= c0c1c2···cN−1 cN−1c0c1...... cN−2cN−1......c2 ............c1 c1···cN−2cN−1c0 . Instead of the dynamics based on shift X, we would like to accomplish the dynamics of QW based on unitary circulant matrices. In this scheme, the state at a site after a time step would depend on the state of the walk at the neighboring sites befor e the step through a unitary spatial convolution more general than a shift. Let circulan t matricesC,C′determine the dynamics in that the propagation matrix, in the joint velocity-po sition3basis, is σ=/bracketleftbigg C0 0C′/bracketrightbigg . (14) The QW evolution in this setting proceeds as before by scattering th e velocity by a unitary matrixSfollowed by the propagation above. The evolution of the QW is: T=σˆS, whereˆSis as in eq. (1). Let C=F−1ΛF=F†ΛF where Λ = eiθ00 0 ···0 0eiθ1......... 0 0eiθ20 0 ............0 0 0 ···0eiθN−1 , andθj,j∈ {0,...,N−1}, are real. Similarly we have the decomposition C′=F†Λ′F, where Λ′is a diagonal unitary matrix. We can write σin eq. (14) as σ=/bracketleftbigg F†0 0F†/bracketrightbigg/bracketleftbigg Λ0 0Λ′/bracketrightbigg/bracketleftbigg F0 0F/bracketrightbigg . 3We realize that velocityis no longer an appropriate name for this form of steering. 10 EFFICIENT AND SCALABLE QUANTUM WALK ALGORITHMS We adopt the following notation for |v/an}⌊ra⌋ketri}ht-controlled switch between matrices MandM′as Cv(M,M′) =/bracketleftbigg M0 0M′/bracketrightbigg . To be consistent with the previous section, we deﬁne Cv(M) :=Cv(IN,M), andC¯v(M) := Cv(M,IN). In the above notation, σin Eq. (14) may be written σ= (Iv⊗F†)Cv(Λ,Λ′)(Iv⊗F), (15) whereIvis the identity operator on the velocity space. Insofar as Λ ,Λ′are eﬃciently imple- mentable, and controllable by |v/an}⌊ra⌋ketri}htto switch between Λ and Λ′, we can eﬃciently implement a QW with its evolution based on C,C′using the QFT. Generally, if we can implement both Cv(Λ,IN) =C¯v(Λ) andCv(IN,Λ′) =Cv(Λ′), we can obtain Cv(Λ,Λ′) as Cv(Λ,Λ′) =Cv(Λ,IN)Cv(IN,Λ′) =C¯v(Λ)Cv(Λ′) =Cv(Λ′)C¯v(Λ). The last equality is there as the controlled operations may be applied in either order. Let us re-examine the basic QW as an illustrative example of the setup of t his section in a multi-qubit system and then consider the general case. 3.1.Basic QW in a multi-qubit system revisited. For the basic QW, C=Xand C′=X⊤, with Λ = Ω from eq. (13) and Λ′=Ω, the element-wise complex conjugate of Ω, Ω =Rn⊗Rn−1⊗Rn−2⊗···⊗R1, whereRjare as in eq. (12), Rj=/bracketleftbigg1 0 0e2πi/2j/bracketrightbigg , and Rj=/bracketleftbigg1 0 0e−2πi/2j/bracketrightbigg . We start with the standard controlled rotations (with |v/an}⌊ra⌋ketri}htas the control), Cv j(Rj) =/bracketleftbiggI20 0Rj/bracketrightbigg , and C¯v j(Rj) =/bracketleftbigg Rj0 0I2/bracketrightbigg . Note that C¯v j(Rj) = (X⊗I2)Cv j(Rj)(X⊗I2) (16) Using these, we can construct C¯v(Ω) =n/productdisplay j=1C¯v j(Rj), and Cv(Ω) =n/productdisplay j=1Cv j(Rj), EFFICIENT AND SCALABLE QUANTUM WALK ALGORITHMS 11 and thus Cv(Ω,Ω) =C¯v(Ω)Cv(Ω) Substituting in (15), we get the propagation σin the setup of this section, σ= (Iv⊗F†)Cv(Ω,Ω)(Iv⊗F). This form of σgives us another algorithm for the basic QW. 3.2.Multi-qubit system: the general case. Let us turn to the general

Question 351 multiple-choice

The XXZ model is a fundamental quantum spin chain with anisotropic interactions, notable for its integrability and rich symmetry structure. Quantum group symmetries, such as $U_q(\slt)$, play a crucial role in simplifying computations and understanding local quantum correlations in lattice systems. Which mathematical framework is especially effective for calculating exact expectation values of $U_q(\slt)$-invariant local operators in quantum integrable spin chains of the XXZ type? 1) Bethe ansatz 2) Matrix product states 3) Fermionic basis 4) Tensor network renormalization 5) Quantum Monte Carlo 6) Transfer matrix method 7) Density functional theory

✓ Correct Answer:

The correct answer is 3) Fermionic basis.

📚 Reference Text:

Title: Exact Density Matrix for Quantum Group Invariant Sector of XXZ Model Year: 2018 Paper ID: 929723af36872334119adc9dc57c40227ca52347 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/929723af36872334119adc9dc57c40227ca52347 Abstract: Using the fermionic basis we obtain the expectation values of all $U_q(\slt)$-invariant local operators on 8 sites for the anisotropic six-vertex model on a cylinder with generic Matsubara data. In the case when the $U_q(\slt)$ symmetry is not broken this computation is equivalent to finding the entire density matrix up to 8 sites. As application, we compute the entanglement entropy without and with temperature, and compare the results with CFT predictions.

Question 352 multiple-choice

In quantum computing, the design of variational ansätze that respect symmetries such as SU(d) and the symmetric group Sn is critical for efficient algorithms. Representation theory plays a central role in constructing quantum circuits that can exploit these symmetries. Which statement accurately describes a key property of the Sn-equivariant convolutional quantum alternating ansatz (Sn-CQA) for n-qudit systems with SU(d) symmetry? 1) It requires two-local SU(d)-symmetric unitaries to achieve universality for all d ≥ 2. 2) It cannot generate arbitrary unitaries within Sn-irrep blocks for n-qudit systems. 3) It is only applicable to systems with d=2 and fails for higher-dimensional qudits. 4) It relies exclusively on SWAP gates and eSWAPs for circuit construction. 5) It does not utilize any concepts from representation theory in its design. 6) It achieves restricted universality by generating any unitary within an Sn-irrep block for n-qudit systems, leveraging SU(d)-symmetric Hamiltonians. 7) It avoids symmetry considerations and uses purely random variational parameters.

✓ Correct Answer:

The correct answer is 6) It achieves restricted universality by generating any unitary within an Sn-irrep block for n-qudit systems, leveraging SU(d)-symmetric Hamiltonians..

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the superexponential cost to compute Fourier coeﬃcients over Sn. Finding variational ansätze respecting continuous rotation symmetry is desirable because it not only helps to gain important physical insights about the system but also leads to more eﬃcient simulation algorithms [ 37,38]. Motivated by the class of problems with a global SU(d)symmetry, in Sec. III, we construct what we call the variational Sn-equivariant convolutional quan- tum alternating ansätze ( Sn-CQA), which are products ofalternating exponentials of certain Hamiltonians admitting SU(d)symmetry. This is a concrete example of the PQC+ framework and may also be thought as a special case of QAOA with SU (d)symmetry. Using the Okounkov- Vershik approach [39]t oSnrepresentation theory as well as other classical results from the theory of Lie group and Lie algebra [ 40,41] we prove that Sn-CQA generates any unitary matrix in each given Sn-irrep block decomposed from an n-qudit system, hence it acts as a restricted uni- versal variational model for problems that possess global SU(d)symmetry (Theorem 1). Consequently, it can be applied to a wide array of machine learning and optimiza- tion tasks that exhibit global SU (d)symmetry or require explicit computation of high-dimensional Snirreps, pre- senting a quantum superexponential speedup. Our proof techniques are of independent interest and we provide two more applications. It is shown in Refs. [ 42,43] that QAOA ansätze generated by simple local Hamiltonians are universal in the common sense. Forgetting the imposed symmetry, we use our techniques to derive the univer- sality for a diﬀerent but related class of QAOA ansätze with a richer set of mixer Hamiltonians (Theorem 2). In addition, we ﬁnd a four-local S n-CQA model, which is uni- versal to build any SU (d)symmetric quantum circuits up to phase factors (Theorem 3) with awareness of the fact thattwo-local SU(d)-symmetric unitaries cannot fulﬁll the task when d≥3[44,45]. Consequently, when compared with other SU (d)-symmetric ansätze, products of expo- nentials of SWAPs (eSWAPs) proposed in Ref. [ 46] admit the restricted universality with SU (d)symmetry only when d=2 and the CQA model is universal in general cases when restricted to any one of Snirreps. In Sec. IVwe explore more details about Schur-Weyl duality on qudit systems. To be speciﬁc, talking about Sn or SU(d)irrep blocks in a qudit system requires using the Schur basis , instead of the computational basis. The Schur basis can be constructed by either SU (d)Clebsch-Gordon decomposition [ 30,47,48]o rb yt h e Snbranching rule [49–51], which yields two ways to label the basis elements by either SU (d)Casimir operators or by the so-called YJM elements used in the Okounkov-Vershik approach. We rigorously demonstrate the equivalence of these label- ing schemes (Theorem 4), ﬁrst conjectured in Harrow’s thesis [ 47], (see also discussions by Bacon, Chuang, and Harrow [ 52] and Krovi [ 51]). As a result, we ﬁnd a state- initialization method, using constant-depth qudit circuits, to produce linear combinations of Schur-basis vectors, which may be preferred in NISQ devices rather than imple- menting a quantum Schur transform (QST). We show

Question 353 multiple-choice

In group theory, torsion-free abelian groups are studied for their structural properties and classification challenges, especially as their rank increases. Understanding the classification of these groups requires knowledge of invariants and their behavior in different ranks. Which of the following statements accurately describes the main reason why torsion-free abelian groups of rank two and higher are much harder to classify than those of rank one? 1) Rank two and higher groups are always finite, making their classification less intuitive. 2) Higher rank torsion-free abelian groups always possess torsion elements, complicating their structure. 3) The classification for rank one groups fails because their elements are not linearly independent. 4) Invariants used for rank one groups cannot distinguish all isomorphism classes in higher ranks due to increased internal complexity. 5) All torsion-free abelian groups of any rank can be classified solely by their order. 6) Rank two and higher groups cannot be embedded in rational vector spaces. 7) No system of complete invariants exists for distinguishing all torsion-free abelian groups of rank two or higher, due to their richer and more intricate internal structure.

✓ Correct Answer:

The correct answer is 7) No system of complete invariants exists for distinguishing all torsion-free abelian groups of rank two or higher, due to their richer and more intricate internal structure..

📚 Reference Text:

Title: On the complexity of the classification problem for torsion-free abelian groups of rank two Year: 2002 Paper ID: 37206444009560ba1614b1eaad7121e80b386529 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/37206444009560ba1614b1eaad7121e80b386529 Abstract: This paper is a contribution to the project [9], [8], [1], [13] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n~>2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space Qn which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank 1 <~r<~n can be naturally identified with the set S(Q n) of all non-trivial additive subgroups of Qn. In 1937, Baer [3] solved the classification problem for the class S(Q) of rank-one groups as follows. Let P be the set of primes. If G is a torsion-free abelian group and Or then the p-height of x is defined to be

Question 354 multiple-choice

Quantum circuit optimization is a critical challenge in the development of practical quantum computing applications, especially given hardware limitations and the complexity of quantum gate arrangements. Various algorithmic strategies have been developed to automate the search for efficient quantum circuit designs. Which optimization method is specifically designed to balance exploration and exploitation when searching large, complex spaces for optimal quantum circuit architectures? 1) Genetic algorithms 2) Gradient descent 3) Gibbs sampling 4) Reinforcement learning 5) Monte Carlo Tree Search (MCTS) 6) Simulated annealing 7) Neural network predictors

✓ Correct Answer:

The correct answer is 5) Monte Carlo Tree Search (MCTS).

📚 Reference Text:

Monte Carlo Graph Search for Quantum Circuit Optimization Bodo Rosenhahn Institute for Information Processing (tnt/L3S), Leibniz Universit¨ at Hannover, Germany Tobias J. Osborne Institute of Theoretical Physics and L3S, Leibniz Universit¨ at Hannover, Appelstrasse 2, 30167 Hannover, Germany (Dated: July 17, 2023) The building blocks of quantum algorithms and software are quantum gates, with the appropriate combination of quantum gates leading to a desired quantum circuit. Deep expert knowledge is nec- essary to discover effective combinations of quantum gates to achieve a desired quantum algorithm for solving a specific task. This is especially challenging for quantum machine learning and signal processing. For example, it is not trivial to design a quantum Fourier transform from scratch. This work proposes a quantum architecture search algorithm which is based on a Monte Carlo graph search and measures of importance sampling. It is applicable to the optimization of gate order, both for discrete gates, as well as gates containing continuous variables. Several numerical experi- ments demonstrate the applicability of the proposed method for the automatic discovery of quantum circuits. Quantum computing has brought about a paradigm shift in information processing and promises breakthroughs in the solution of indus- trial use cases [1–3], physics [4, 5], medicine [6], chemistry [7], biology [8], robotics [9], general pattern recognition [10], machine learning [11– 13], and much more. These opportunities are ameliorated, however, by the significant chal- lenges arising in the discovery and application of quantum software. This is because the de- sign of quantum algorithms still requires expert knowledge. Further, since quantum devices are still small and costly, it is essential to optimize resources, in particular, gate counts. Thus the discovery and optimization of quantum circuits is of significant near-term relevance. Methods for the automated search for opti- mal quantum circuits have been investigated in the literature, and the term Quantum Ar- chitecture Search (QAS) has been adopted to describe this body of research. The name is borrowed and adapted from Neural Architec- ture Search (NAS) [14, 15], which is devoted to the study and hyperparameter tuning of neu- ral networks. Recent works on QAS are often specific to a problem setup, e.g., it has been applied to quantum circuit structure learning [16] for finding the ground states of LithiumHydride and the Heisenberg model in simula- tion, as well as for finding the ground state of a Hydrogen gas. Many QAS-variants are fo- cussed on discrete optimization and exploit op- timization strategies for non-differentiable opti- mization criteria. Here variants of Gibbs sam- pling [17], evolutional approaches [18], genetic algorithms [19, 20], neural-network based pre- dictors [21], variants with noise-aware circuit learning [22], and the optimization of approx- imate solutions [23] have been suggested. A re- cent survey on QAS can be found in [24]. Going beyond discrete optimisation it is also possible to exploit gradient-descent based optimization schemes [25, 26] or reinforcement learning [27] for QAS. The recent work [28] proposes a Monte Carlo Tree Search (MCTS) based on a multi-armed bandit formulation. This paper is closest

Question 355 multiple-choice

In quantum lattice gauge theories, defining entanglement entropy is complicated by the non-factorization of the gauge-invariant Hilbert space, leading to multiple inequivalent definitions. Topological entanglement entropy is used as a robust probe of topological order in such systems. Which of the following statements accurately describes the relationship between the "extended Hilbert space" (ordinary) and "algebraic" (electric center) definitions of entanglement entropy in lattice gauge theories, and their implications for calculating topological entanglement entropy? 1) Only the algebraic definition yields the correct value for the topological entanglement entropy, while the extended Hilbert space definition fails due to overcounting. 2) The extended Hilbert space definition always provides a lower value for entanglement entropy than the algebraic definition for any region. 3) The two definitions are equivalent in all respects and do not differ even by boundary terms. 4) The extended Hilbert space definition ignores gauge invariance, leading to an incorrect topological entanglement entropy in all cases. 5) The algebraic definition includes contributions from non-gauge-invariant states, which are excluded in the extended Hilbert space definition. 6) Both definitions yield different values for the topological entanglement entropy due to their treatment of boundary vertices. 7) Both the extended Hilbert space and algebraic definitions differ by a term proportional to the boundary length, but agree on the universal value of the topological entanglement entropy.

✓ Correct Answer:

The correct answer is 7) Both the extended Hilbert space and algebraic definitions differ by a term proportional to the boundary length, but agree on the universal value of the topological entanglement entropy..

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inequivalent ways to deﬁne the entanglement entropy of a region. The ambiguity arises because when considering a regionA(consisting of some edges), the gauge-invariant Hilbert space Hphysdoes not factorize into a tensor product of factors associated with Aand the complement ¯A, unlike the full Hilbert space H. For a discussion of deﬁning entanglement entropy in gauge theories, see [24, 25]. Also see Appendix B for an introduction. Because there are multiple deﬁnitions for the full entanglement entropy in gauge theory, we can ask whether these deﬁnitions lead to the same answer for the topological entanglement entropy. We will discuss three deﬁnitions in particular (introduced in the paragaph below equation (53)), including the “ordinary” deﬁnition which is calculated using the non-gauge-invariant/extended Hilbert space. We will ﬁnd that two of these deﬁnitions — the extended Hilbert space deﬁnition and the so-called “algebraic” deﬁnition — diﬀer by a term proportional to L, hence yielding the same answer for the term Stopo. This calculation corrects a comment made by [24] and repeated elsewhere [17, 27], in the context of lattice gauge theories, stating that only one of these two deﬁnitions correctly reproduces the topological entanglement entropy.12(The quantum double model coincides with the weakly coupled limit of a lattice gauge theory.) We begin by considering reduced density matrices of general states in the gauge-invariant Hilbert space Hphys. After examining the structure of the density matrix for a gauge-invariant state, we will be able to describe various possible deﬁnitions of entanglement entropy when the system is considered as a gauge theory. 11Deﬁning the L→∞limit requires a family of regions of increasing size. However, on the lattice, the boundaries of these regions will have “sharp” features like corners, and these sharp features make the limit sensitive to the choice of region. 12We agree with the explicit calculations in the seminal work [24], but we rectify their incorrect suggestion that the algebraic deﬁnition of entanglement entropy does not accurately capture the topological entanglement entropy, e.g. when they state “Dropping some or all of these classical contributions would therefore not yield the correct result.” Note that in [24], they refer to the algebraic deﬁnition of entanglement entropy as the “electric centre” deﬁnition. Accepted in Quantum2020-09-17, click title to verify. Published under CC-BY 4.0. 15 Consider a rectangular region A(a subset of edges) and the reduced state ρAonAfor some |ψ/angbracketright∈Hphys. The region is rectangular for simplicity and includes the edges on its boundary, e.g. like regionAin Figure 1. Consider the boundary vertices v∈∂A, and letL=|∂A|be the number of such vertices. Then for each boundary vertex v, consider the gauge transformation Av(g)atvbut restricted to the edges from vthat lie inA. That is,Av(g)multiplies by g(org−1, according to orientation) for each edge that lies in Aof the boundary vertex v. We call this the “boundary gauge transformation” at v. For each boundary vertex v, there is a separate action of Gon the Hilbert space HAof region A, where HAis the associated tensor factor in the “extended” (non-gauge-invariant) Hilbert space. The aforementioned actions commute. We

Question 356 multiple-choice

Dynamical quantum groups extend the theory of quantum groups by incorporating dependencies on additional parameters, often leading to rich algebraic structures such as bialgebroids. The quantum dynamical Yang-Baxter equation (QDYBE) plays a critical role in the representation theory of these structures. Which property or condition on the function $ R: T \to \text{End}(V \otimes V) $ is essential for the existence of non-trivial dynamical representations of the $ H $-bialgebroid $ \overline{A}_R $? 1) $ R $ must factor through a commutative subalgebra of $ \text{End}(V \otimes V) $. 2) $ R $ must be constant with respect to the dynamical parameter. 3) $ R $ must be invertible for every value in $ T $. 4) $ R $ must satisfy the ordinary (non-dynamical) Yang-Baxter equation. 5) $ R $ must be a solution to a classical Lie algebra cocycle condition. 6) $ R $ must satisfy the quantum dynamical Yang-Baxter equation (QDYBE). 7) $ R $ must commute with all elements of $ MT $.

✓ Correct Answer:

The correct answer is 6) $ R $ must satisfy the quantum dynamical Yang-Baxter equation (QDYBE)..

📚 Reference Text:

bialgebroids. If Ais anH-bialgebroid, and WandUare two dynamical rep- resentations of A, then we endow the H-module W⊗Uwith the structure of a dynamicalrepresentationvia πW⊗U(x) =θWU◦(πW⊗πU)◦∆(x). Iff:W1→W2 andg:U1→U2are homomorphisms of dynamical representations, then so is f¯⊗g:W1⊗U1→W2⊗U2. Thus, dynamicalrepresentationsof Aformamonoidal category Rep( A), whose identity object is the trivial representation. Remark. IfAis anH-Hopf algebroid and Vis a dynamical representation, then one can deﬁne the left and right dual dynamical representat ions∗Vand V∗. We will not discuss this notion here and refer the reader to [EV2]. 7.3. The H-bialgebroid associated to a function R:T→End(V⊗V). Now, following [EV2], let us deﬁne an H-bialgebroid ¯ARassociated to a mero- morphic, zero weight function R:T→End(V⊗V), where Vis a ﬁnite di- mensional diagonalizable H-module (we assume that R(λ) is nondegenerate for generic λ). This is the dynamical analogue of the Faddeev-Reshetikhin- Takhtajan-Sklyaninconstructionofabialgebrafromanelement R∈End(V⊗V), whereVis a vector space. By deﬁnition, the algebra ¯ARis generated by two copies of MT(embedded as subalgebras) and matrix elements of the operator L∈End(V)⊗¯AR. We denote the elements of the ﬁrst copy of MTbyf(λ1) and of the second copy 20 byf(λ2), where f∈MT. We denote by Lαβthe weight components of L with respect to the natural T-bigrading on End( V), so that L= (Lαβ), where Lαβ∈HomC(V[β],V[α])⊗¯AR. Introduce the moment maps for ¯ARbyµl(f) =f(λ1),µr(f) =f(λ2), and deﬁne the weight decomposition by f(λ1),f(λ2)∈(¯AR)00, L αβ∈HomC(V[β],V[α])⊗(¯AR)αβ. The deﬁning relations for ¯ARare: f(λ1)Lαβ=Lαβf(λ1+α);f(λ2)Lαβ=Lαβf(λ2+β);[f(λ1),g(λ2)] = 0; and the dynamical Yang-Baxter relation R12(λ1)L13L23=:L23L13R12(λ2) :. (7.5) Here the :: sign means that the matrix elements of Lshould be put on the right of the matrix elements of R. Thus, if {va}is a homogeneous basis of V, and L=/summationtextEab⊗Lab,R(λ)(va⊗vb) =/summationtextRab cd(λ)vc⊗vd, then (7.5) has the form /summationdisplay Rxy ac(λ1)LxbLyd=/summationdisplay Rbd xy(λ2)LcyLax, where we sum over repeated indices. Deﬁne the coproduct on ¯AR, ∆ :¯AR→¯AR˜⊗¯AR, and the counit of ¯ARby ∆(L) =L12L13,ǫ(Lαβ) =δαβIdV[α]⊗T−1 α, where Id V[α]:V[α]→V[α] is the identity operator. Proposition 7.1 ([EV2]). (¯AR,∆,ǫ)is anH-bialgebroid. Example. Suppose that Ris the basic trigonometric solution of the QDYBE (see Section 6). Then the deﬁning relations for ¯ARlook like f(λ1)Lbc=Lbcf(λ1+ωb), f(λ2)Lbc=Lbcf(λ2+ωc), LasLat=αst(λ2) 1−βts(λ2)LatLas,s/\e}atio\slash=t, LbsLas=αab(λ1) 1−βab(λ1)LasLbs,a/\e}atio\slash=b, αab(λ1)LasLbt−αst(λ2)LbtLas= (βts(λ2)−βab(λ1))LbsLat,a/\e}atio\slash=b,s/\e}atio\slash=t, whereαab(λ) =qλa−λb+1−1 qλa−λb−1,βab(λ) =q−1 qλa−λb−1. We note that we don’t need any special properties of R(like the dynami- cal Yang-Baxter equation or Hecke condition) to deﬁne the H-bialgebroid ¯AR. However, if we take a ”randomly chosen” function R, theH-bialgebroid ¯AR will most likely have rather bad properties; i.e. it will be rather small an d will not have interesting dynamical representations. The simplest way to ensure the existence of at least one interesting dynamical representation is t o require that Rsatisﬁes the QDYBE. This is so because of the following proposition. 21 If (W,πW) is a dynamical representation of an H-algebra A, we denote π0 W: A→Hom(W,W⊗MT) the map deﬁned by π0 W(x)w=πW(x)w,w∈W(the diﬀerence operator πW(x) restricted to the constant functions). It is clear that πWis completely determined by π0 W. Proposition 7.2. IfRsatisﬁes the QDYBE then ¯ARhas a dynamical repre- sentation realized in the space V, withπ0 V(λ) =R(λ). This representation is called the vector representation . However,

Question 357 multiple-choice

The Quantum Fourier Transform (QFT) is a fundamental operation in quantum computing, enabling efficient solutions to problems involving periodicity and underlying group structure. Its implementation and generalization to finite abelian groups are essential for key quantum algorithms and cryptographic applications. Which statement accurately describes the relationship between the QFT over a finite abelian group G and the decomposition of G into cyclic subgroups? 1) The QFT over G is defined as a single Hadamard transformation independent of subgroup structure. 2) The QFT over G is constructed as a tensor product of QFTs over its cyclic subgroups of prime power order. 3) The QFT over G requires an exponential number of gates for implementation due to subgroup interactions. 4) For abelian groups, the QFT circuit uses only controlled-NOT gates regardless of decomposition. 5) The QFT over G cannot be generalized beyond groups isomorphic to ZN for some N. 6) The QFT over G loses unitarity when applied to groups with more than one cyclic factor. 7) Decomposition into cyclic subgroups is irrelevant for the construction of the QFT over G.

✓ Correct Answer:

The correct answer is 2) The QFT over G is constructed as a tensor product of QFTs over its cyclic subgroups of prime power order..

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algorithms [ 8]. The QFT over ZN, the group of integers modulo Nunder addition, is a unitary operator FZNthat effects on a basis state as follows: |x/angbracketright/mapsto−→1√ N/summationdisplay y∈ZNωxy N|y/angbracketright,∀x∈ZN where ωN:=e2πi/Ndenotes a primitive Nth root of the unity. Its matrix representation is FZN=1√ N⎛ ⎜⎜⎜⎜⎜⎝11 1 ...1 1ω Nω2 N...ωN−1 N 1ω2 Nω4 N...ω2N−2 N............ 1ω N−1 Nω2N−2 N...ω(N−1)(N−1) N⎞ ⎟⎟⎟⎟⎟⎠ More succinctly, it is denoted by F ZN=1√ N/summationdisplay x,y∈ZNωxy N|y/angbracketright/angbracketleftx|. Further, we can derive QFT over any ﬁnite abelian group G. We know that any ﬁnite abelian group Gcan be expressed as a direct product of cyclic subgroups of prime power orders G∼=Zp1×···× Zpr. Thus, in this case the QFT over Gis the quantum operator FG=Fp1⊗···⊗ Fpr. 123 178 Page 4 of 26 J. Suo et al. |ϕ0 ··· • ··· • ··· • H |z0 |ϕ1 ··· • ··· • ··· H R2 |z1 ...........................|ϕ n−3 • ··· • |zn−3 |ϕn−2 • ··· H R2 ··· Rn−2 Rn−1 |zn−2 |ϕn−1 H R2 R3 ··· Rn−1 Rn |zn−1 Fig. 1 The circuit of quantum Fourier transform [ 10].|ϕ0/angbracketright,...,/vextendsingle/vextendsingleϕn−1/angbracketrightbig are input bits, and |z0/angbracketright,...,/vextendsingle/vextendsinglezn−1/angbracketrightbig are output bits. Rnis two-bit quantum controlled rotation Without loss of generality, assuming n=⌈ logN⌉, then the circuit of QFT over ZN, as depicted in Fig. 1, can be implemented exactly by usingn(n−1) 2of controlled rotation gates, plus with nHadamard gates, leading to the gate complexity O(n2). Recently, Su et al. [ 9] suggested that QFT over n-qubits can be approximate with O(nlogn)T-gates. 3 Quantum algorithms for ﬁnding periods A function over the domain Dis called periodic if there is a unique and smallest r>0 (called period) so that f(x)=f(x+r)holds for every x∈D. Say, the sine and cosine functions, respectively, have periods 2 πandπoverR. Although this deﬁnition does not require rto be integer and Dto be discrete, for the problems discussed in this survey, we only consider the settings of rbeing a positive integer and Dbeing a discrete ring, say ZorZn(for some n∈N). Intuitively, without any other heuristic information onf, say regarding fas a black box, any classical algorithm for determining whether fhas a period rneeds to evaluate fon, in the worse case, all elements in D, leading to the time complexity O(|D|)and space complexity O(1). However, quantum computers can work exponentially faster than any classical computers toward the period ﬁnding problem. The ﬁrst breakthrough on this issuecan be traced back to the landmark work due to Simon [ 11]. Simon’s algorithm is not only the ﬁrst algorithm that represents a substantial advance in relativized time complexity vs. classical computing, but also a turning point in the development ofquantum computation technology considering that it contains the key ingredients of the relevant algorithms that follow, including the notably Shor’s quantum algorithm for integer factoring problem [ 7]. V ery recently, Dong [ 12] proposed indistinguishable attack and key-recover attack toward one of the well-known cipher structure—the extended Feistel structures, including the typical block ciphers such as CAST256 andRC6. Simon’s algorithm is proposed to deal with

Question 358 multiple-choice

The Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) are fundamental tools in signal processing, often represented using matrix notation for clarity and computational efficiency. The structure and properties of the DFT matrix play a critical role in enabling fast algorithms for transforming data. Which property of the DFT matrix is essential for ensuring that the inverse DFT can be computed by simply taking the conjugate transpose and applying a normalization factor? 1) The DFT matrix is always symmetric. 2) The DFT matrix contains only real entries. 3) The DFT matrix is diagonalizable over the reals. 4) The DFT matrix has zero determinant. 5) The DFT matrix is sparse for all N. 6) The DFT matrix is unitary. 7) The DFT matrix is strictly upper triangular.

✓ Correct Answer:

The correct answer is 6) The DFT matrix is unitary..

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us eful. A similar approach is followed in [15, 18, 8, 20]. Van Loan [20] even states: “I am convinced that life as we know it would be considerably diﬀerent if, from the 1965 Cooley–Tukey paper onwards, the FFT community had made systematic and heavy use of matrix-vector notation!”. 1The scalar factors of the DFT and the inverse DFT need to multi ply to1 N. The choice1√ Nleads to unitary transformations. 3 Deﬁnition 7 (DFT matrix) .The DFT matrix is deﬁned as the unitary matrix FN:=1√ N ω0 Nω0 Nω0 N···ω0 N ω0 Nω1 Nω2 N···ωN−1 N ω0 Nω2 Nω4 N···ω2(N−1) N ............... ω0 NωN−1 Nω2(N−1) N ···ω(N−1)(N−1) N ∈CN×N, (6) whereωN:= exp(−2πı N). Using Deﬁnition 7, the DFT and IDFT become matrix-vector mul tiplications y=FNxandy=F∗ Nx, respec- tively. As an example, the ﬁrst four DFT matrices are given by : F1=/bracketleftbig 1/bracketrightbig , F2=1√ 2/bracketleftbigg1 1 1−1/bracketrightbigg , F3=1√ 3 1 1 1 1ω3ω2 3 1ω2 3ω3 , F4=1 2 1 1 1 1 1−ı−1ı 1−1 1 −1 1ı−1−ı , where we used that ωNsimpliﬁes for N= 1,2,and 4, and the equality ωm N=ω(mmodN) N . The 2×2 DFT matrix F2is also called the Hadamard matrix and denoted as H. In order to simplify notation, we will make use of the exponen t notation for the DFT matrix as introduced by Pease [15]: ˜FN= 0 0 0 ··· 0 0 1 2 ···N−1 0 2 4 ··· 2(N−1) ............... 0N−1 2(N−1)···(N−1)(N−1) , where only the exponents of ωNare listed in the matrix ˜FN. For the remainder of the paper, all exponent matrices will be indicated by a tilde. Note that by using exponent nota tion, an multiplication of ωNfactors becomes an addition of its exponents. 3.2 Radix- 2decomposition of the discrete Fourier transform matrix A straightforward implementation of the (inverse) DFT as a m atrix-vector product requires O(N2) operations. The radix-2 FFT algorithm, popularized by Cooley and Tukey[ 3], computes the DFT for vectors of length N= 2n in onlyO(Nlog2N) =O(2nn) operations. This speedup is achieved by a divide-and-conq uer approach which relates a permuted DFT matrix of dimension Nto a permuted DFT matrix of size N/2. Let us illustrate this matrix decomposition for the DFT matrix of dimension 8. Example 1. The exponent DFT matrix of dimension 8is given by: ˜F8= 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 0 2 4 6 8 10 12 14 0 3 6 9 12 15 18 21 0 4 8 12 16 20 24 28 0 5 10 15 20 25 30 35 0 6 12 18 24 30 36 42 0 7 14 21 28 35 42 49 mod 8≡ 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 0 2 4 6 0 2 4 6 0 3 6 1 4 7 2 5 0 4 0 4 0 4 0 4 0 5 2 7 4 1 6 3 0 6 4

Question 359 multiple-choice

In group theory, inductive techniques and commutator identities are crucial for analyzing the structure of nilpotent groups and the behavior of their subgroups under various operations. Understanding how subgroup inclusions propagate is essential for studying powers of subgroups and commutator subgroups. Which principle allows the inclusion $ [A^s, b]^p \subseteq A^{p^{k+1}} $ to be established for all $ s $ in a nilpotent group, given that $ [A^i, b]^p \subseteq A^{p^{k+1}} $ holds for all $ 1 \leq i \leq s-1 $? 1) The Frattini argument 2) The use of Sylow theorems 3) The extension of the lower central series 4) Application of the Zassenhaus lemma 5) The action of the automorphism group 6) The orbit-stabilizer theorem 7) Mathematical induction on $ n $ with commutator and p-th power identities

✓ Correct Answer:

The correct answer is 7) Mathematical induction on $ n $ with commutator and p-th power identities.

📚 Reference Text:

the proof of the theorem is to prove: Step III. U[A,bf ^ A pk for some k ^ 2, then [A, b] p ^ p k + l. Firstly note that (17) and the hypotheses of Step III give (18) [A p\ b] ^ A pk + l and [A, 2b] p ^ A pk+\ We establish Step III by showing that [An, b]p ^ Apk+] for all n. Our induction begins with the statement (19) [A 2, bf ^ A pk+\ To see this, observe that for a in A 2, (») [a,bf[a,2bf 2' ...[a,(p ~ \)b] p[a,pb] = 1. From (18) we can see that all the factors but the first and last are in A p From (7) we have [A2, (r, + r 2)b] ^ (A p2, Ap3, Apl) so that [A2, (p - 2)b] ^ {Ap2, Ai A pl). Thus [A2,pb] ^ ( [A 2, 2b]", [A 3, 2b] p, [A, 2bf). But (18) now gives that [A2, pb] ^ Apk + > and so [A2, bf =i A pk+\ Next we prove: If [A it b]p ^ ApkV' for 1 S i ^ s - 1 and s â 3, then (20) [A s, bf ^ A pk+\ Taking p-th roots in (19) and noting that k = 2, we get [A2, b] Si (A u Ap2, 4,.. .,4 +x, A pkk+2, Ad . . .>. When this is interpreted in AP and ps~2-th roots are taken, it gives (21) \A s,b]^ (A x,...,A s-X,Aps,Af+x,...). https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press 1042 J. BUCKLEY AND J. WIEGOLD From this we have immediately (22) [A p,b] ë (Ap2,...,Ap^],Apl). Next we show that (23) [A p, rjb] tk A"" + \ As usual we begin with an identity [a,rsb]p[a,(rs + l)^ 2' ...[a,(r s+p- 2)bf[a, (r s +p- 1)6] = 1, valid for a e A s. By (18), factors other than the first and last are in A 1' provided that r s ^ 1; if not there is nothing to prove. From (18) and (21), we see that [As,(rs + p - \)b) tk ([A u(rs + P ~ 2)b), . . ., [A s_h (rs +p- 2)b], . . . [A p, (rs +p- 2)b], A"k + \ Going back to (1), M „/-,/>]§ (A p2,Ap\ so from (18), [/t„(r, + r 2)b] =£ ([A 2,r2bf,AP k + '). Applying (19) we get that [Ax, (r, + r 2)b] S A"k + l and hence (24) [A ]9(p - 2)b] ^AP k+\ Now taking /7-th roots in the hypothesis in (20), we get for l^jfks- \ [Ar rsb] ^ (A,, A"2, Af,. . . ,AÙ^ 4+2, • • •) thus [Aj, (rs + p - 2)b] â ([AU(P - 2)b], [Ap2, (p - 2)b], [A p\p - 2)b]). We have just seen that [A„{p - 2)6] ^Apk + \ That [A{Ap ~ 2)b] =£p* + i https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press NILPOTENT EXTENSIONS 1043 follows from (19); and (18) then

Question 360 multiple-choice

Extensions of the Minimal Supersymmetric Standard Model (MSSM) often introduce hidden sector gauge symmetries to explain the nature and abundance of dark matter, with implications for direct detection and cosmological constraints. Models featuring either kinetically mixed U(1)x or hidden SUx gauge groups employ distinct mechanisms for dark matter interactions and mass spectra. In a hidden sector dark matter model with a kinetically mixed U(1)x and Dirac fermion or complex scalar dark matter candidates, which parameter is invoked to suppress late-time dark matter annihilation signals and must be unusually large to evade indirect detection constraints? 1) The gauge coupling constant of U(1)x 2) The mass splitting between dark matter and its superpartner 3) The kinetic mixing parameter ε 4) The DM-anti-DM asymmetry parameter 5) The relic abundance freeze-out temperature 6) The nucleon recoil energy threshold 7) The heavy connector matter mass scale

✓ Correct Answer:

The correct answer is 4) The DM-anti-DM asymmetry parameter.

📚 Reference Text:

Kinetic mixing of the hidden U(1)xwith hypercharge links the hidden vector to charged matter with an e ective coupling proportional to ecW. In the present case, the dark matter consists of Dirac fermions and complex scalars charged under U(1)x, and this allows a vectorial coupling of these states to the Xgauge boson. Together, these two features induce a vector-vector e ective operator (for mx&20 MeV) connecting the DM states to the proton that gives rise 23 to spin-independent (SI) scattering on nuclei. The -proton scattering cross section is p=2c2 We2g2 x2 n m4 x: (45) A similar expression applies to the scalar . This gives rise to an e ective SI cross section per nucleon (in terms of which experimental limits are typically quoted) of ~n= (Z2=A2)p (46) '21038cm22Z A2 1032gx 0:12n 1 GeV21 GeV mx4 : Comparing this result to the exclusions of low-mass DM from LUX [126], XENON10 S2 only analysis [127], CDMSLite [128] and CRESST-Si [129], we obtain the green exclusion regions shown in Figures 4 and 5. D. Summary This hidden U(1)xextension of the MSSM can account for the entire relic dark matter abundance in the aftermath of moduli reheating while being consistent with existing con- straints from direct and indirect detection. Even though the DM annihilation cross section is much larger than the standard thermal value, a strong DM-anti-DM asymmetry allows for a signi cant total density while suppressing DM annihilation signals at late times. Limits from direct detection searches can also be evaded for light DM masses below the sensitivity of current experiments. To achieve a strong DM asymmetry, a relatively large asymmetry parameter &103 is needed. We have not speci ed the dynamics that gives rise to the asymmetry in moduli reheating, but more complete theories of asymmetry generation suggest that values this large can be challenging to obtain [119{121, 125]. Furthermore, as in the symmetric hidden sector theory considered previously, the spectrum required for this mechanism to work requires scalar sequestering and scalar soft masses of the right size. VI. VARIATION #3: HIDDEN SU(N) The third extension of the MSSM that we consider consists of a pure supersymmetric SU(N)xgauge theory together with heavy connector matter multiplets charged under both SU(N)xand the MSSM gauge groups.12In contrast to the two previous extensions, we do not have to make any strong assumptions about the scalar soft mass parameters for the theory to produce an acceptable LSP relic density. In particular, this extension can work in the context of a mini-split spectrum where the scalar superpartners are much heavier than the gauginos [38{43]. 12See also Refs. [130, 131] for previous studies of this scenario in a slightly di erent context. 24 A.SU(N)xMass Spectrum and Con nement The hidden states below the TeV scale consist of the SU(N)xgluon and gluino. The hidden gluino soft mass is Mx=rxg2 x (4)2m3=2; (47) whererx= 3Nif it is generated mainly by AMSB e ects. In the discussion to follow, we will consider additional heavy matter charged under SU(N)xwith large supersymmetric mass

Question 361 multiple-choice

Quantum computing leverages unique phenomena such as superposition, interference, and entanglement to achieve computational advantages over classical systems. Understanding how these resources interact in algorithms like Shor’s algorithm is central to quantum algorithm design. In the context of Shor’s algorithm for factoring, which statement correctly describes the role of the Quantum Fourier Transform (QFT) regarding entanglement and computational speedup? 1) The QFT creates new entanglement among qubits, enabling access to all superposed states simultaneously. 2) The QFT collapses the superposition, reducing the computational advantage that entanglement provides. 3) The QFT manipulates interference patterns to reveal global properties like periodicity without increasing entanglement in periodic states. 4) The QFT destroys preexisting entanglement, leaving only product states for measurement. 5) The QFT enables direct measurement of all parallel outcomes via entanglement generation. 6) The QFT increases entanglement to amplify the correct answer during measurement. 7) The QFT substitutes superposition with classical parallelism, bypassing interference effects.

✓ Correct Answer:

The correct answer is 3) The QFT manipulates interference patterns to reveal global properties like periodicity without increasing entanglement in periodic states..

📚 Reference Text:

to how this speedup is made possible. Currently, the best ins ight we can oﬀer is that it is made possible by the combination of quantum superpositio n and quantum interference [30, 31]. Quantum superposition allows a sort of parallel co mputation, as all the states in the superposition go through the same unitary evolution in a n independent manner. Since the results of each of the states cannot be accessed directly , we cannot fully exploit quantum parallelism. Nevertheless, through quantum interference the diﬀerent parallel paths can 16 interact in a limited way, allowing us to access some global p roperties of the resulting states. Finding special situations where such a global property is t he solution to some computational problem (like the period in Shor’s algorithm is the solution to the factoring problem) is in fact the essence of quantum algorithm design. The role of ent anglement in this model of quantum speedup is to allow quantum parallelism to reach its full extent, since product states cover a very small range compared with all possible su perpositions [1]. To exemplify this model for speciﬁc algorithms, one must try to distinguish the role of quantum parallelism and quantum interference in each algor ithm. In the case of Shor’s algorithm, a superposition is built in the preprocessing st age, in a manner that does not make use of interference (modular exponentiation is perfor med on each computational basis state separately). This superposition in fact already cont ains the desired information (the period), and the QFT is merely needed to extract this informa tion (by canceling the shift). Therefore, it can be argued that the preprocessing stage is w here quantum parallelism is used, and the QFT introduces quantum interference. This cla im is supported by the fact that the QFT does not increase the entanglement of the regist er. VII. SUMMARY We have shown that periodic states play an important role in S hor’s factoring algorithm and pointed out that this is also true for other quantum algor ithms that rely on the QFT. Focusing on entanglement as a necessary resource for quantu m speedup, we set as our goal to explain the result presented in Ref. [7], according to whi ch the QFT hardly aﬀects the entanglement of periodic states. For this purpose we ana lyzed the entanglement of periodic states using the Groverian entanglement measure. We derived an approximated formula for the Groverian entanglement of periodic states a nd showed evidence to support it. Using this approximated formula, we presented a model th at explains the aforementioned result. Finally, we argued that this result and the model tha t explains it strengthen our understanding as to the source of quantum computational spe edup in Shor’s algorithm and in general. [1] A. Ekert and R. Jozsa, Phil. Trans. R. Soc. London A 356, 1769 (1998). 17 [2] R. Jozsa and N. Linden, Proc. R. Soc. London, Ser. A 459, 2011 (2003). [3] G. Vidal, Phys. Rev. Lett. 91, 147902

Question 362 multiple-choice

The classification of finite p-groups often relies on linear algebraic methods, including the use of matrices to encode group relations and the study of automorphism and isomorphism classes. Canonical forms and invariants help distinguish different group types and their explicit presentations. Which condition is necessary for two finite p-groups to be isomorphic over the field Fp using matrix-based classification techniques? 1) The existence of a diagonal matrix Y and a scalar μ ∈ Fp satisfying additive equations 2) The existence of an invertible matrix X and a scalar λ ∈ Fp* satisfying specific matrix equations 3) That both groups have abelian commutator subgroups 4) The groups share the same number of elements of order p 5) Their matrices can be reduced to upper triangular form only 6) The groups possess identical centralizer subgroups 7) Both groups are extensions of cyclic groups of order p

✓ Correct Answer:

The correct answer is 2) The existence of an invertible matrix X and a scalar λ ∈ Fp* satisfying specific matrix equations.

📚 Reference Text:

in the theorem. By Theorem 4.3 and 4.4,G∼=¯Gif and only if there exists X=/parenleftbiggx11x12 x21pn−mx22/parenrightbigg , an invertible matrix overFp, andλ∈F∗ psuch that /parenleftbigg¯w11 ¯w21/parenrightbigg =λ−1/parenleftbiggx110 x21x22/parenrightbigg/parenleftbiggw11 w21/parenrightbigg (4.8) and /parenleftbigg¯w12 ¯w22/parenrightbigg =λ−1|X|/parenleftbiggx11x12 0x22/parenrightbigg/parenleftbiggw12 w22/parenrightbigg . (4.9) By suitably choosing x21, that is, using an elementary row operation, we can simplify/parenleftbiggw11 w21/parenrightbigg to be one of the following three types: 14 (a1)/parenleftbiggw11 0/parenrightbigg wherew11/ne}ationslash= 0, (b1)/parenleftbigg0 w21/parenrightbigg wherew21/ne}ationslash= 0, (c1)/parenleftbigg0 0/parenrightbigg . In the following, we assume that both/parenleftbiggw11 w21/parenrightbigg and/parenleftbigg¯w11 ¯w21/parenrightbigg are such matrices. By Equation 4.8 and 4.9, it is easy to check that (i) diﬀerent type s give non-isomorphic groups, (ii) G∼=¯Gif and only if there exists X=/parenleftbiggx11x12 0x22/parenrightbigg , an invertible matrix overFp, andλ∈F∗ psuch that /parenleftbigg¯w11 ¯w21/parenrightbigg =λ−1/parenleftbiggx110 0x22/parenrightbigg/parenleftbiggw11 w21/parenrightbigg (4.10) and /parenleftbigg¯w12 ¯w22/parenrightbigg =λ−1|X|X/parenleftbiggw12 w22/parenrightbigg . (4.11) Similarly, we can simplify/parenleftbiggw12 w22/parenrightbigg to be one of the following two types (note that (w12,w22)/ne}ationslash= (0,0) byG3/ne}ationslash= 1): (a2)/parenleftbigg0 w22/parenrightbigg wherew22/ne}ationslash= 0, (b2)/parenleftbiggw12 0/parenrightbigg wherew21/ne}ationslash= 0. In the following, we assume that both/parenleftbiggw12 w22/parenrightbigg and/parenleftbigg¯w12 ¯w22/parenrightbigg are such matrices. By Equation 4.10 and 4.11, it is easy to check that (i) diﬀerent ty pes give non-isomorphic groups, (ii) G∼=¯Gif and only if there exists X= diag(x11,x22), an invertible matrix overFp, andλ∈F∗ psuch that /parenleftbigg¯w11 ¯w21/parenrightbigg =λ−1X/parenleftbiggw11 w21/parenrightbigg and/parenleftbigg¯w12 ¯w22/parenrightbigg =λ−1|X|X/parenleftbiggw12 w22/parenrightbigg . (4.12) By Equation 4.12, G∼=¯Gif and only if there exists X= diag(x11,x22), an invertible matrix over Fp, andλ∈F∗ psuch that w(¯G) =λ−1Xw(G)diag(1,|X|). By Table 4.1, /parenleftbigg w11 w21/parenrightbigg /parenleftbigg w12 w22/parenrightbigg w(G) λ X Remark w(¯G) Group (a1) (a2)/parenleftbiggw110 0w22/parenrightbigg 1 diag( w−1 11,d−1)w22w−1 11=νd2/parenleftbigg1 0 0ν/parenrightbigg (J1) (a1) (b2)/parenleftbigg w11w12 0 0/parenrightbigg 1 diag( w−1 11,w2 11w−1 12)/parenleftbigg 1 1 0 0/parenrightbigg (J2) (b1) (a2)/parenleftbigg 0 0 w21w22/parenrightbigg 1 diag( w2 21w−1 22,w−1 21)/parenleftbigg 0 0 1 1/parenrightbigg (J3) (b1) (b2)/parenleftbigg 0w12 w210/parenrightbigg 1 diag( d−1,w−1 21)w12w−1 21=νd2/parenleftbigg 0ν 1 0/parenrightbigg (J4) (c1) (a2)/parenleftbigg 0 0 0w22/parenrightbigg 1 diag( w−1 22,1)/parenleftbigg 0 0 0 1/parenrightbigg (J5) (c1) (b2)/parenleftbigg 0w12 0 0/parenrightbigg 1 diag(1 ,w−1 12)/parenleftbigg 0 1 0 0/parenrightbigg (J6) Table 4.1: Case 3 in Theorem 4.6 we get the groups of Type (J1)–(J6). /square By Theorem 4.2, we have the following theorem. 15 Theorem 4.7. LetGbe a ﬁnite p-group listed in Theorem 4.6. Then the following conclusions hold: (1)Imax= 1for the type (F); (2)Imax= 2for types (H1)–(H3), (I1)–(I3) ; (3)Imax=mfor types (G1)–(G3), (J2), (J4), (J6) ; (4)Imax=nfor types (J1), (J3), (J5) . 5 The case G3∼=CpandΦ(G′)G3∼=C2 p Suppose that Gis a ﬁnite p-group with G3∼=Cp, Φ(G′)G3∼=C2 pandG/Φ(G′)G3∼= Mp(n,m,1), where n >1 forp= 2 and n≥m. Let G/Φ(G′)G3=/an}bracketle{t¯a,¯b,¯c|¯apn=¯bpm= ¯cp= 1,[¯c,¯a] = [¯b,¯c] = 1/an}bracketri}ht. Then,withoutlossofgenerality, wemayassumethat G=/an}bracketle{ta,b,c/an}bracketri}htwhere[a,b] =c. Since Φ(G′)∼=Cp,cis of order p2. SinceG3∼=Cp,CG(G′) is maximal in G. If [c,b]/ne}ationslash= 1, then, without loss of generality, we may assume that [ c,a] = 1. In this case, the type ofCG(G′)/Φ(G′)G3is (pn,pm−1,p). If [c,b] = 1, then the type of CG(G′)/Φ(G′)G3is (pn,pm−1,p). 5.1 The type of CG(G′)/Φ(G′)G3is(pn,pm−1,p) Without loss of generality, we may assume that [ a,c] = 1 and [ b,c]/ne}ationslash= 1. Let

Question 363 multiple-choice

Quantum data compression leverages the ability of quantum computers to identify and exploit hidden symmetries in data, tasks that are often computationally prohibitive for classical algorithms. Photonic quantum processors, which encode information in properties of single photons, present a promising platform due to their unique degrees of freedom and low decoherence. Which feature specifically enables a photonic quantum processor to autonomously learn and compress classical databases with hidden subgroup symmetries that are hard for classical algorithms to detect? 1) Employing only polarization encoding of photons for data representation 2) Utilizing fixed quantum gates without tunable parameters 3) Implementing classical autoencoders for symmetry detection 4) Relying exclusively on time-bin encoding for quantum logic 5) Using non-variational quantum circuits with no training 6) Applying classical optimization methods outside the quantum processor 7) Training a variational quantum autoencoder on single photons encoded in path, polarization, and time-bin degrees of freedom with electronically controlled waveplate gates

✓ Correct Answer:

The correct answer is 7) Training a variational quantum autoencoder on single photons encoded in path, polarization, and time-bin degrees of freedom with electronically controlled waveplate gates.

📚 Reference Text:

Title: Photonic implementation of quantum hidden subgroup database compression Year: 2025 Paper ID: 0a02c524f8e74d047043584685f061150e753775 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0a02c524f8e74d047043584685f061150e753775 Abstract: We experimentally demonstrate quantum data compression exploiting hidden subgroup symmetries using a photonic quantum processor. Classical databases containing generalized periodicities-symmetries that are in the worst cases inefficient for known classical algorithms to be detect-can efficiently compressed by quantum hidden subgroup algorithms. We implement a variational quantum autoencoder that autonomously learns both the symmetry type (e.g., $\mathbb{Z}_2 \times \mathbb{Z}_2$ vs. $\mathbb{Z}_4$) and the generalized period from structured data. The system uses single photons encoded in path, polarization, and time-bin degrees of freedom, with electronically controlled waveplates enabling tunable quantum gates. Training via gradient descent successfully identifies the hidden symmetry structure, achieving compression by eliminating redundant database entries. We demonstrate two circuit ansatzes: a parametrized generalized Fourier transform and a less-restricted architecture for Simon's symmetry. Both converge successfully, with the cost function approaching zero as training proceeds. These results provide experimental proof-of-principle that photonic quantum computers can compress classical databases by learning symmetries inaccessible to known efficient classical methods, opening pathways for quantum-enhanced information processing.

Question 364 multiple-choice

Quantum compilers are essential for translating high-level quantum algorithms into instructions that can be executed on quantum hardware, enabling optimized and robust quantum software development. Their architecture and optimization techniques directly impact the efficiency and reliability of quantum computations. Which component of a quantum compiler is primarily responsible for transforming parsed high-level quantum code into an optimized intermediate representation before generating executable bytecode? 1) Error handling module 2) Intermediate representation layer 3) Quantum Virtual Machine 4) Quantum bytecode generator 5) Frontend parser 6) Qubit allocator 7) Algorithm simulator

✓ Correct Answer:

The correct answer is 2) Intermediate representation layer.

📚 Reference Text:

Title: A Quantum Computation Language-based Quantum Compiler Year: 2024 Paper ID: a831b556dc42a05d3ee6678fdf14c0a63eba7f8b Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a831b556dc42a05d3ee6678fdf14c0a63eba7f8b Abstract: Quantum computing is a revolution in computational theory that uses the principles of quantum mechanics to perform computations that are beyond the capacity of classical computers. It works with the special features of the quantum bits or qubits to encode and transform the information in manners conventional bits cannot imagine. A quantum compiler is a software tool designed to translate high-level quantum language code into executable quantum instructions. This paper aims to develop a quantum compiler for translating Quantum Computation Language source codes into quantum bytecodes which can later be executed on a Quantum Virtual Machine. This compiler will have a combinational structure that includes a frontend for interpreting the QCL code, an intermediate representation for optimizing quantum algorithms, and a backend for generating efficient quantum bytecode. Optimization techniques will be applied to improve the functioning of the quantum circuits. The compiler will also include an error handling module to handle the errors. The successful deployment of a quantum compiler for QCL has many implications and unlocks opportunities for advancements in quantum technologies.

Question 365 multiple-choice

Quantum algorithms have been instrumental in solving computational problems related to group theory and algebraic structures. The distinction between abelian and nonabelian groups plays a crucial role in the complexity of quantum solutions for hidden structure problems. Which problem has an efficient polynomial-time quantum algorithm under fixed degree and variable conditions, due to a more powerful oracle, even though its generalization, the Hidden Polynomial Problem, does not have such an algorithm? 1) Discrete Logarithm Problem 2) Graph Isomorphism Problem 3) Lattice Problems 4) Hidden Subgroup Problem for nonabelian groups 5) Hidden Radius Problem 6) Hidden Polynomial Graph Problem 7) Factoring Problem

✓ Correct Answer:

The correct answer is 6) Hidden Polynomial Graph Problem.

📚 Reference Text:

(OTKA,grants NK105645 and K77476). http://www.siam.org/journals/sicomp/42-5/86441.html †Centre for Quantum Technologies, National University of Singapore, Singapore 117543 (t.d3ck3r@gmail.com). ‡Institute for Computer Science and Control, Hungarian Academy of Sciences, Budapest, Hungary (Gabor.Ivanyos@sztaki.mta.hu). §LIAFA, University Paris 7, CNRS, 75205 Paris, France, and Centre for Quantum Technologies, National University of Singapore, Singapore 117543 (miklos.santha@liafa.univ-paris-diderot.fr). ¶Department of Electrical Engineering and Computer Science, University of Central Florida, Orlando, FL 32816-2362 (wocjan@eecs.ucf.edu). The work of this author was supported by NSFgrant CCF-0726771 and NSF CAREER award CCF-0746600. 1987 1988 T. DECKER, G. IVANYOS, M. SANTHA, AND P. WOCJAN query complexity, the problem is computationally solvable in quantum polynomial time for every abelian group [1, 2, 3]. Several attempts were made to extend the quantum solution of the abelian HSP. Most of the research focused on the HSP in nonabelian groups since these includeseveral algorithmically important problems. For example, it is known that eﬃcient solutions for the dihedral and the symmetric group would imply eﬃcient solutions for some lattice problems [4] and for graph isomorphism, respectively. While some progress has been made in this direction [5, 6, 7, 8, 9, 10, 11], the HSP for the dihe- dral and symmetric groups remains unsolved. It is already known that the methodsfor solving the abelian case fail for several nonabelian groups [12, 13]. The goal of obtaining eﬃcient quantum algorithms for larger classes of nonabelian groups turned out to be rather elusive. Another idea for generalizing the problem was proposed by Childs, Schulman, and Vazirani [14], who considered properties of algebraic sets hidden by black-box functions. One of these problems is the hidden polynomial problem ( HPP), where the hidden object is a polynomial. To recover it we have at our disposal an oracle whose level sets coincide with the level sets of the polynomial. Childs et al. [14]showed that the quantum query complexity of this problem is polynomial in the logarithm of the ﬁeld size, provided that the degree and the number of variables are held constant, leaving open the question of the time complexity. The authors alsoformulated computationally eﬃcient quantum procedures for some related problems, such as the hidden radius and the hidden ﬂat of centers. Nonetheless, to the best of our knowledge, no eﬃcient quantum polynomialtime algorithmhas been proposedfor the general HPP, not even for the simplest problem of hidden quadratic polynomials in one variable ( HQPP). In [15], Decker, Draisma, and Wocjan deﬁned a related problem that we refer to as the hidden polynomial graph problem ( HPGP) to distinguish it from the HPP. Here, similarly to the HPP, the hidden object is a polynomial, but the oracle is more powerful. They obtained a polynomial time quantum algorithm that correctly identiﬁes the hidden polynomial when the degree and the number of variables are considered to be constant. Their proof applies to all ﬁnite ﬁelds whose characteristic is not in a ﬁnite set of exceptional characteristics that depend on the degree of the polynomials. In this paper, we advocatea third possible approachfor ﬁnding hidden structures. We consider a

Question 366 multiple-choice

In topological quantum computing, braid group representations are used to realize quantum gates, with qubits embedded in tensor products of vector spaces associated with strands. The universality of such models depends on the density of braid group images in certain unitary groups and the ability to efficiently approximate quantum circuits. Which condition on the parameter ℓ ensures that the image of the 8-strand braid group representation is dense in the special unitary group SU, thereby enabling efficient approximation of any two-qubit unitary operation? 1) ℓ must be an even integer greater than 2 2) ℓ must be equal to 6 or less than 5 3) ℓ must be a prime number 4) ℓ must satisfy ℓ ≥ 5 and ℓ ≠ 6 5) ℓ must be odd and less than 10 6) ℓ must be a multiple of 4 7) ℓ must be less than or equal to 4

✓ Correct Answer:

The correct answer is 4) ℓ must satisfy ℓ ≥ 5 and ℓ ≠ 6.

📚 Reference Text:

subspace W(n)can be identiﬁed with a tensor product/circlemultiplytextn i=1Wi, where for each 1 ≤i≤n, we identify WiwithV, so that it is spanned by |t0/an}bracketri}htand|t1/an}bracketri}ht. In this subspace of V[2n,2n], each consecutive group of four strands is associated to a sin gle qubit in the following way. For each 1 ≤j < k ≤n−1, deﬁne the subset Σk j={σj,σj+1,...,σ k}of generators of Bn and writeBk j⊆Bnfor the subgroup generated by Σk−1 j. For each 1 ≤i≤n−2, the image of the 8 strand braid group B4(i+1) 4(i−1)+1under the representation π[2n,2n], when restricted to W(n), acts nontrivially only on the subsystem Wi⊗Wi+1. In fact, it rapidly approximates anyU∈SU(Wi⊗Wi+1). To see this, it suﬃces to look at the case i= 1. Note that W1⊗W2⊂V. The embedding V⊗V֒→Vlooks like |t0/an}bracketri}ht|t0/an}bracketri}ht /ma√sto→/vextendsingle/vextendsingle/vextendsingle1357 2468/angbracketrightBig |t0/an}bracketri}ht|t1/an}bracketri}ht /ma√sto→/vextendsingle/vextendsingle/vextendsingle1356 2478/angbracketrightBig |t1/an}bracketri}ht|t0/an}bracketri}ht /ma√sto→/vextendsingle/vextendsingle/vextendsingle1257 3468/angbracketrightBig |t1/an}bracketri}ht|t1/an}bracketri}ht /ma√sto→/vextendsingle/vextendsingle/vextendsingle1256 3478/angbracketrightBig . While this proves that dim V≥4, we mention that its dimension is 13 when ℓ= 5 and is 14 whenℓ >6. So it is clear that SU(W1⊗W2)⊂SU(V). Consider now the following proposition. Proposition 7.4 (Freedman, Larsen, Wang [13]) Letℓ≥5,ℓ/ne}ationslash= 6. The image of B8 underπ(2,ℓ)is dense in SU/parenleftBig V(2,ℓ)/parenrightBig . With this density result in hand, we are now equipped to prove Theorem 7.1. Proof of Theorem 7.1: Setǫ′=ǫ/m= Ω/parenleftbig 2−m/m). Together with the Solovay-Kitaev 26The Jones Polynomial: Quantum Algorithms and Applications in Quantum Complexity Theory Theorem, Proposition 7.4 implies that for each U′∈SU(W1⊗W2), there is a braid b∈B8 of lengthO/parenleftbig poly log(1/ǫ′)/parenrightbig =O/parenleftbig poly(m)/parenrightbig for whichπ(b)ǫ′-approximates U′, where we considerU′to be embedded in SU(V). Note that while some unitaries π(σi) will not be contained in SU(W1⊗W2), the construction above ensures that π(b) can be made arbitrarily close to any U′∈SU(W1⊗W2). Because of the four-periodicity of our encoding, the same ar guments apply for approxi- mating gates on any other adjacent pair of qubits. Now, let U=UmUm−1···U1as in the statement of the theorem. Without loss of generality, we may assume that each Uihas unit determinant. By (7.2), we may conclude that there is a braid b∈B4nof lengthO/parenleftbig poly(m)/parenrightbig for whichπ[2n,2n](b)ǫ-approximates the circuit U, and also that such a braid can be deter- mined on a classical computer in O/parenleftbig poly(m)/parenrightbig time. The rest of the theorem now follows by Lemma 7.2 and the expression (5.39) for the absolute value of the Jones polynomial of the plat closure; the normalization constant 1 /[2]2n−1 ℓis so because the braid has 4 nstrands./square 8 Complexity-theoretic applications of the Jones polynomi al Complexity theory characterizes the asymptotic consumpti on of resources, such as time and space, required to obtain the solutions of certain classes o f problems on a computer. Writing {0,1}∗for the set of all binary strings of ﬁnite length, a decision problem asks for the evaluation of some function f:{0,1}∗→ {0,1}on an arbitrary input. Such a function determines a languageL⊆ {0,1}∗deﬁned asL=f−1(1), so equivalently, the task is to determine, for each bit string xn, whetherxn∈L. Examples of languages include the set of all binary strings of even parity and the set of prime numbers, written

Question 367 multiple-choice

In quantum computing, the Quantum Fourier Transform (QFT) is a fundamental operation often extended to higher-dimensional systems such as qutrits, which possess three distinct quantum states. Implementing such transformations requires sophisticated mathematical and experimental techniques. Which method is commonly utilized to approximate the evolution operator when the effective Hamiltonian for a qutrit-based QFT gate is expressed as a sum of non-commuting components? 1) Grover’s search algorithm 2) Ramsey interferometry 3) Quantum error correction 4) Jordan-Wigner transformation 5) Quantum phase estimation 6) Trotter-Suzuki decomposition 7) Shor’s factoring algorithm

✓ Correct Answer:

The correct answer is 6) Trotter-Suzuki decomposition.

📚 Reference Text:

transform on a qutrit Before moving on to the numerical results for the QFT on system (11), we consider a simple example allow ing analytical solution and helping us to understand qualitatively the findings of Section IIA. Let us consider the QFT gate for a qutrit ( N = 3). Matrix (13) can be diagonalized by means of a sequence of selective rotations. Such sequences were explicitly found for N = 3, 5, and 7 in [21] and N = 4 and 8 in [22]. For the gate 3 GUF (13), in expression (7) we have 1 2 2 2 2 2 2 2 22 1 / 2 / 22/ 2 1 / 2g g g K g g g g g g    , (14) where 2 1sin ( / 2)g , 2cos 1/ 3g , and 2 arctg . The diagonalizing operator is 2 0 0 sin / 2 cos / 2 0 10 1 1 cos / 2 sin / 2 0 20 1 1 0 0 1P       . (15) The operators specifying the changes in the effective Hamiltonian are 1 2 2 † 1 1 2 1 1 2 1 12 2 1 1 1 1 11g g g m P P m g g g g g g       , 6 1 2 2 † 2 2 2 1 1 2 1 12(1 ) 2 2 2g g g m P P m g g g g g g  , (16) † 330 0 0 2 3 3 0 1 1 0 1 1m P P m  . Expressions (14) and (16) allow us to determine the effective Hamiltonian (9) required for the QFT implementation . Now , we need to find the method of its realization on system (11). The similar task for I = 1 was solved previously for a selective rotation operator [23]. We app ly the same approach to the QFT gate. We expre ss the effective Hamiltonian as eff mTH A B C   , (17) 12( ) , ( ) ,yy xxi I i I i I i I q q x z A e H t e B e H t e C I I      . (18) Substituting Hamiltonian (17) in the expression for the evolution operator and using the Trotter −Suzuki formula [24]  /2 /2 / /2 /2 ( ) 3(1/ )ri A r i B r i С r iB r iA r i A B Ce e e e e e O r        , (19) we arrive at the operator product, which can be presented as the pulse sequence 1 2 2 1/2 /2 /2 /2 { } { } { } { } { / } { } { } { } { }t r t r t r t r x x y y

Question 368 multiple-choice

Quantum algorithms often utilize labeled qubit states and probabilistic quantum operations to efficiently solve problems involving hidden structures, such as in the dihedral hidden subgroup problem. Advanced techniques like sieving and batch management are employed to refine quantum information and amplify desired results. Which quantum algorithmic technique utilizes iterative combination and measurement of qubit states labeled by integers, probabilistically mapping sums to differences, in order to efficiently solve the dihedral hidden subgroup problem? 1) Kuperberg sieve 2) Grover's search 3) Quantum Fourier transform 4) Shor's factoring algorithm 5) Phase estimation algorithm 6) Valiant's matchgate computation 7) Quantum walk algorithm

✓ Correct Answer:

The correct answer is 1) Kuperberg sieve.

📚 Reference Text:

any j2f1;::;mg:The input to the ﬁrst routine is the output of CSG , i.e., single-qubit states of the form (4.77), each with a corresponding label z chosen uniformly at random from f0;1;:::;2n1g. The output of routine jare qubits such that their labels have jmleast signiﬁcant bits equal to 0 and njmbits chosen uniformly at random. Each routine generates qubits whose labels have madditional bits set to 0. The process of combining the qubits is implemented as follows. Each routine jmaintains a batch of qubits, which is initially empty. When a new qubit comes in, the routine compares the m bits in positions (j1)m+1;:::;jmof its label to the bits in the same label positions in all of the qubits already in its batch. If no qubit already accumulated in the batch has matching label bits, the newly arrived qubit is added to the batch. If there is a match, the new qubit is combined with the matching one. 61 Proposition 4.5.5. There exists a procedure that with probability1 2implements the mapping (z1;jyz1i)+(z2;jyz2i)7!(z1z2;jyz1z2i) (4.86) Proof. Suppose CSG generates states jyz1i=1p 2 j0i+e2piz1b 2nj1i jyz2i=1p 2 j0i+e2piz2b 2nj1i (4.87) with labels z1andz2, respectively. The tensor product of these states is given by jyz1;yz2i=1 2 j00i+e2piz1b 2nj10i+e2piz2b 2nj01i+e2pi(z1+z2)b 2nj11i : (4.88) Application of a CNOT gate with the ﬁrst qubit playing the role of a control results in the stat jyz1;yz2i7!1 2 j00i+e2piz1b 2nj11i+e2piz2b 2nj01i+e2pi(z1+z2)b 2nj10i =1p 2 j0i+e2pi(z1+z2)b 2nj1i j0i +1p 2h e2piz2b 2nj0i+e2piz1b 2nj1i j1ii (4.89) =1p 2 jyz1+z2;0i+e2piz2b 2njyz1z2;1i : Measurement of the second qubit yields j1iwith probability1 2, leaving the ﬁrst qubit in the state jyz1z2i, up to an inconsequential global phase. The Kuperberg sieve discards the combined qubits of the form jyz1+z2i, and by iterating the combination process ensures that the last routine in the sequence outputs the desired state jy2n1i. Kuperberg showed [16] that given an input of 2O(pn)qubits, the sieve generates the desired state with high probability. 62 Quaternions’ potential in the world of quantum computation is just beginning to be ex- plored. Perhaps a quaternionic formulation of quantum mechanics applied to a quantum computa- tional framework could lead to an improvement of existing quantum algorithms or to the emergence of new ones. And if everything three-dimensional is ultimately quaternionic, quaternions could make signiﬁcant, even profound, contributions in subjects as diverse as cosmology and biology, shedding new light on the unfolding cosmos and the origins of life. As Hamilton put it [43], If some new truth, O Friend! thy toil discover, If thine eyes ﬁrst by some fair form be blest, Love it for what it is, and as a lover Gaze, or with joy receive thine honoured guest: The new-found thought set free, awhile may hover Gratefully near thee, but it cannot rest. 63 REFERENCES [1] Yu. I. Manin. The computable and incomputable. Sovetzkoe Radio, 1980. [2] R. Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 21:467-488, 1982. [3] R. Feynman. Quantum Mechanical Computers. Optics News, 11:11, 1985. [4] D. Deutsch. Quantum Theory, the

Question 369 multiple-choice

In abstract algebra, braces are structures that facilitate the study of solutions to the Yang-Baxter equation, which is fundamental in areas such as quantum groups and knot theory. The classification of simple left braces, especially those with abelian Sylow subgroups in their multiplicative group, is important for understanding the broader theory and constructing solutions. Which statement best reflects a surprising recent development regarding simple finite left braces with abelian Sylow subgroups in their multiplicative group? 1) These braces do not exist for any finite group order. 2) All such braces are isomorphic to cyclic groups. 3) Their multiplicative groups must be non-abelian simple groups. 4) They only occur when the additive group is trivial. 5) There is a unique brace of this type for each prime order. 6) There is a remarkable abundance of simple left braces with abelian Sylow subgroups. 7) Their construction is impossible using set-theoretic methods.

✓ Correct Answer:

The correct answer is 6) There is a remarkable abundance of simple left braces with abelian Sylow subgroups..

📚 Reference Text:

Title: An abundance of simple left braces with abelian multiplicative Sylow subgroups Year: 2018 Paper ID: 129679e39ca1b60a5a951da5b576bc053b2b1b1e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/129679e39ca1b60a5a951da5b576bc053b2b1b1e Abstract: Braces were introduced by Rump to study involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation. A constructive method for producing all such finite solutions from a description of all finite left braces has been recently discovered. It is thus a fundamental problem to construct and classify all simple left braces, as they can be considered as building blocks for the general theory. This program recently has been initiated by Bachiller and the authors. In this paper we study the simple finite left braces such that the Sylow subgroups of their multiplicative groups are abelian. We provide several new families of such simple left braces. In particular, they lead to the main, surprising result, that shows that there is an abundance of such simple left braces.

Question 370 multiple-choice

In quantum computing, the hidden subgroup problem (HSP) for nilpotent groups of bounded class has attracted interest due to its implications for efficient quantum algorithms. The problem often reduces to finding zero-sum subsequences in group-theoretic contexts, with efficiency depending on group structure and the properties of their prime divisors. Which of the following statements best explains why "smooth" nilpotent groups are considered particularly amenable to efficient polynomial-time quantum algorithms for the hidden subgroup problem? 1) Their group order is always a prime power, making all subgroups cyclic and easy to analyze. 2) They are always abelian, allowing direct application of Shor's algorithm without modification. 3) Every sequence over the group automatically contains a zero-sum subsequence of length equal to the group order. 4) All of their irreducible representations are one-dimensional, simplifying quantum Fourier sampling. 5) They have trivial center, ensuring unique factorization of elements and easy subgroup detection. 6) The prime divisors of their group order are polynomially bounded, enabling efficient zero-sum subsequence algorithms and thus efficient quantum solutions. 7) Their class is unbounded, allowing for arbitrary complexity in subgroup structure.

✓ Correct Answer:

The correct answer is 6) The prime divisors of their group order are polynomially bounded, enabling efficient zero-sum subsequence algorithms and thus efficient quantum solutions..

📚 Reference Text:

It is based on collecting relations organized in a d-dimensional hypercube rather than a square. (The method of doubling collisions is essentially identical with that described here in Proposition 5.) 123 Zero sum subsequences and hidden subgroups Page 21 of 23 14 6 Concluding remarks We have shown that the hidden subgroup problem in a nilpotent group Gof class bounded by a constant can be solved in polynomial time by an exact quantum algorithm provided that there is a polynomial time (that is, time poly (nlogp)) exact method that ﬁnds zero sum subsequences in sequences consisting of polynomially many elementsofZ n pfor prime divisors pof|G|. We have such a method for p=O(1). By Olson’s theorem [ 22], the shortest length for which a not necessarily polynomial time zero sum subsequence ﬁnding algorithm exists is around np. We propose the question of existence of a poly (np)-time algorithm for ﬁnding zero sum subsequences from sequences of length (np)dfor a sufﬁciently large constant das a problem for further research. A positive answer would imply existence of an exact polynomial time quantum algorithm for the case when |G|is smooth, that is, the prime factors of |G|are of size bounded by a polynomial in log |G|.E v e na non-exact method (e.g., a randomized algorithm) would be of great interest as, byProposition 3, it would give a new result in the non-exact setting: existence of an efﬁcient “probabilistic” quantum hidden subgroup algorithm for nilpotent groups of smooth order having O(1)-bounded nilpotency class. Even somewhat worse results would potentially lead to quantum hidden subgroup algorithms faster than the known ones. For the purposes of “probabilistic” quantum hidden subgroup algorithms even a method that ﬁnds a zero sum subsequence “on average,” that is for at least a 1 /poly(np) proportion of the possible sequences would be sufﬁcient. However, as the followingsimple worst-case to average-case reduction shows, at least in the randomized setting, the gain cannot be better than polynomial. Assume that the classical randomized algorithm Aﬁnds in time T=T(p,n)with probability at least δa subsequence of a random sequence of length S=S(p,n)of vectors from Z n p. Here, probability is taken for the uniform distribution of the array of the vectors together with the random bits of A. Then we can do the following. We start with an arbitrary sequence of1 δ·S2input vectors; we draw1 δ·S2uniformly random vectors, one for each input vector. Then we divide the input sequence into groups of length S, and to each input vector we add the corresponding random vector. Within each group, we apply procedure A.A st h es u m s are random vectors, in each group, procedure Asucceeds with probability at least δ and, with probability at least1 2,Awill succeed in at least Sgroups. If this is the case then we choose S“lucky” groups, in each group take the sum of the random vectors corresponding to the members of the zero sum subsequences. We apply algorithm A for these Ssums. It ﬁnds a non-trivial zero sum subsequence with probability

Question 371 multiple-choice

Quantum algorithms often leverage efficient implementations of classical signal transforms to achieve exponential speedups for tasks such as factoring and discrete logarithms. The structure and symmetries of these transforms, combined with quantum error correction, underpin the development of robust and powerful quantum computational techniques. Which statement accurately describes the quantum gate complexity required to implement the discrete Fourier transform for the Heisenberg group, and distinguishes it from other group-based and classical signal transforms? 1) It requires O gates, similar to the classical fast Fourier transform. 2) It can be implemented with O(log N) gates, matching the efficiency of simple abelian group transforms. 3) It requires O(log³N) gates, which is higher than the O(log²N) gate requirement for other listed transforms. 4) It uses O(N²) gates due to lack of exploitable group symmetries. 5) It can be efficiently realized using only O(1) gates for any sequence length N. 6) Its gate complexity is identical to that of the discrete cosine transform, at O(log²N). 7) It cannot be implemented on a quantum computer due to fundamental limitations.

✓ Correct Answer:

The correct answer is 3) It requires O(log³N) gates, which is higher than the O(log²N) gate requirement for other listed transforms..

📚 Reference Text:

the fact that the ability to compute a DFT in polylogarithmic time is the backbone of Shor’s algorithms for factoring and computing the discrete logarithm [Sho94]. A natural question is whether other signal transforms which have desirable feature ex- traction properties in classical signal processing can be used for quantum algorithms. In aseries of works [PRB99, RPB99, KR00, ABH +01, KR01, Röt02] it has been shown that many well-known signal transforms allow highly efﬁcient realizations on a quantum computer. In particular the following classes of unitary transformations have been shown to be efﬁciently implementable on a quantum computer: Discrete Fourier transforms for ﬁnite abelian groups [ABH+01, Röt02]. Generalized Fourier transforms for –2-groups with maximal cyclic normal subgroup [PRB99, Röt02]. –wreath products of the form G=A≀Z2,w h e r e Ais abelian [RB98, Röt02]. –Heisenberg groups over the ﬁnite ﬁelds F2n[Röt02]. Discrete cosine and sine transforms of types I, II, III, and IV [RB99, KR01, Röt02]. Discrete Hartley transforms [KR00, Röt02]. More precisely, we have shown that for discrete cosine transforms, discrete sine trans- forms, and discrete Hartley transforms O(log2N)elementary quantum gates are sufﬁcient to implement any of those transforms for input sequences of length N. The Fourier transforms for ﬁnite groups which have been mentioned above have the same computational complexity,except for the Heisenberg group where an implementation using O(log 3N)gates has been found. The underlying theory which allows to ﬁnd efﬁcient factorizations in the above mentioned cases relies on two techniques which have been developed at the Institut für Algorithmen und Kognitive Systeme. The ﬁrst technique is the so-called method of symmetry-based matrixfactorizations. Here a matrix M∈C n×nis said to have symmetry (G,φ,ψ )ifφandψare representations of a ﬁnite group Gand furthermore φ(g)·M=M·ψ(g),for all g∈G. The importance of symmetry in connection with generalized Fourier transforms was ﬁrst rec- ognized in the work of Th. Beth [Bet84]. An important feature of this approach is that classical 1.3 Quantum Error-correcting Codes 3 fast algorithms can be explained—and automatically derived—in terms of symmetry of ma- trices [Min93, Egn97, Püs98]. In the thesis of M. Rötteler [Röt02] symmetry-based matrix factorizations have been taken as a starting point for further optimizations. This has led to efﬁcient implementations of several generalized Fourier transforms, trigonometric transforms, and the Hartley transform. A second technique has been developed by A. Klappenecker and M. Rötteler and is de- scribed in [Röt02]and [KR03]. The basic idea is to reuse previously found factorizations for the construction of higher level operations. A prime example is given by the problem of im- plementing functions of unitary transformations, i. e., operations of the form f(U),w h e r e U is a unitary transformation of ﬁnite order and f(U)is also unitary. 1.3 Quantum Error-correcting Codes Owing to the high sensitivity of quantum mechanical systems to even small perturbations, means of error protection are essential for any computation or communication process basedon quantum mechanics. A general theory of quantum error-correcting codes (QECC) has been developed (see, e. g., [KL97]), but many algorithmic

Question 372 multiple-choice

Hopf algebras constructed from cyclic quivers incorporate roots of unity and exhibit rich algebraic structures, including specialized operations such as coproducts and antipodes. Understanding the interplay between the antipode and roots of unity is vital for analyzing their quantum properties. In a Hopf algebra built from the cyclic quiver Zn with deformation parameter q, an nth root of unity, which property does the antipode S satisfy for a generator α? 1) S^2(α) = qα 2) S^2(α) = α 3) S^2(α) = q^2α 4) S^2(α) = -α 5) S^2(α) = α + q 6) S^2(α) = q^{-1}α 7) S^2(α) = 0

✓ Correct Answer:

The correct answer is 1) S^2(α) = qα.

📚 Reference Text:

Π/^i,* = 1 we obtain that the common value of the beta's is a nih root of unity q. Finally we have Δ ι(fli) = Σ SJ ® ak + Σ ίfy ® skj+k=i j+k=i Actually the coassociativity property can be checked directly on this formula. The counit is extended by zero to the whole algebra. The antipode S has to be an anti-isomorphism of algebras, so S(ai)eS(Si) kZnS(si+1) = S-ikZns-i-ι. Let S(a?) = x&i-! with Xi in k. The condition (S(χ)l) (at) = 0 translates into xf = — qί + 1 and 462 C. Cibils this already gives the antipode for kZn. Notice that S2(αO = qah hence the antipode is of order Id, where a is the precise order of q. Remark 2.4. The above construction suggests that a tensor algebra TA(M) can be equipped with a Hopf algebra structure, provided that A is a Hopf algebra and M is an ,4-bimodule with some extra-structure. Indeed, if M is an ,4-bicomodule whose structure maps are ,4-bimodule maps, then TA(M) is a Hopf algebra. This fact has been established by Nichols [20], and Woronowicz in [29] has highlighted the role of these bicovariant bimodules (or Hopf bimodules) in relation to the braid equation. I am indebted to M. Rosso for pointing this out; in a forthcoming publication we are going to develop this aspect in order to obtain quantum groups based on the quiver of a group. Notice that the left normalisation we have made in the proof of the theorem corresponds to the fact that a Hopf bimodule is always free on the left with cotrivial left coaction, see [27, 29]. 3. Quotients and Covers In order to get two-sided ideals of kZn(q) preserved by A or to replace the root of unity q by an indeterminate, we first compute A of an arbitrary path yf of the quiver Zn. At the end of this section we consider the universal cover of the Hopf algebras that we have obtained. Recall that if x is a variable and m ^ u are positive integers, the Gauss polynomial I ) is -^—-, where mlx = mx(m — l)x. . . 2xlx, for mx =\ujx (m-u)lxu\x 1 + x + x2 + . . . + xm~1. We agree that 0!* = 1 and notice that II = ( 1. The Pascal relation holds: fm+ 1\ fm\ / m) = + χm-u+ί[ U Jx \U/χ \U — fm\This is used to prove that is actually a polynomial with coefficients in Z. \ujχ Lemma 3.1. Let kZn(q) be the Hopf algebra obtained from the quiver Zn and an nth root of unity q. Then , ^ ^ ίm^A(y7) = Σ Proof. For m = 1 the formula agrees with the comultiplication of an arrow obtained in the preceding section. We have A(yT+1) = Λ(ai+myf] = A(ai+m) Λ(ff). The proof follows by induction on m, using the Pascal relation. (d\ Lemma 3.2. Let q be a root of unity of order d.

Question 373 multiple-choice

Quantum learning algorithms are increasingly used to analyze the symmetry properties of quantum states and to solve problems fundamental to quantum computation and information. The identification of hidden subgroup structures in quantum systems has broad applicability, particularly for abelian groups. Which technique generalizes various quantum learning methods, such as Bell sampling and Bell difference sampling, by leveraging the quantum Fourier transform to efficiently identify hidden symmetry subgroups in abelian quantum systems? 1) Generalized Fourier sampling 2) Quantum amplitude amplification 3) Grover’s search algorithm 4) Density matrix renormalization 5) Quantum phase estimation 6) Classical hidden subgroup sampling 7) Variational quantum eigensolver

✓ Correct Answer:

The correct answer is 1) Generalized Fourier sampling.

📚 Reference Text:

Title: The abelian state hidden subgroup problem: Learning stabilizer groups and beyond Year: 2025 Paper ID: d2938cac459ca993c897d313f5ac10435f5b32b3 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d2938cac459ca993c897d313f5ac10435f5b32b3 Abstract: Identifying the symmetry properties of quantum states is a central theme in quantum information theory and quantum many-body physics. In this work, we investigate quantum learning problems in which the goal is to identify a hidden symmetry of an unknown quantum state. Building on the recent formulation of the state hidden subgroup problem (StateHSP), we focus on abelian groups and develop an efficient quantum algorithm that learns any hidden symmetry subgroup using a generalized form of Fourier sampling. We showcase the versatility of the approach in three concrete applications: These are learning (i) qubit and qudit stabilizer groups, (ii) cuts along which a state is unentangled, and (iii) hidden translation symmetries. Through these applications, we reveal that well-known quantum learning primitives, such as Bell sampling and Bell difference sampling, are in fact special cases of Fourier sampling. Our results highlight the broad potential of the StateHSP framework for symmetry-based quantum learning tasks.

Question 374 multiple-choice

Quantum algorithms for eigenvalue estimation rely on precise error management and efficient polynomial approximations to ensure accuracy and scalability. One prominent technique uses circuit transformations and expansions to encode and extract eigenvalue information in noisy quantum environments. Which approach allows a quantum circuit to efficiently approximate functions like cos(tx) on eigenvalues for energy estimation, providing explicit bounds on polynomial degree and error? 1) Fourier series expansion with direct amplitude encoding 2) Jacobi-Anger expansion combined with singular value transformation circuits 3) Lagrange interpolation using ancilla qubits 4) Grover’s algorithm with controlled phase rotations 5) Taylor series expansion in block-encoded unitaries 6) Classical Monte Carlo sampling of eigenstates 7) Quantum error correction with syndrome extraction

✓ Correct Answer:

The correct answer is 2) Jacobi-Anger expansion combined with singular value transformation circuits.

📚 Reference Text:

estimate in general results in an error of ±1. However, recall that when estimating future bits the value of ∆k/2nis subtracted oﬀ from λj. That meansthatwhenweerroneouslymeasurea 1, therestofthealgorithmproceedstomeasure λj−1/2ninstead. As a result, if the ﬁrst bit is wrong, then the algorithm will output ﬂoor(2nλj)−1instead. Since the algorithm is periodic in λjwith period 1, the output will be 2n−1if an error occurs when ﬂoor (2nλj) = 0. Notice that if we had allocated the error evenly between the nsteps, then we would have incurred an extra O(2nlogn)term in the above. Spreading the error via a geometric series avoids this, and we ﬁnd that we obtain better constant factors with this choice as well. This is because the k= 0term dominates, so we want to make δamp,kas large as possible. 2.2 Coherent Iterative Energy Estimation While for phase estimation we are given access to/summationtext je2πiλj|ψj/angbracketright/angbracketleftψj|, for energy estimation we have a block-encoding of/summationtext jλj|ψj/angbracketright/angbracketleftψj|. For phase estimation we could relatively easily synthesize a cosine in the eigenvalues, just by taking a linear combination with the identity. But for energy estimation we must build the cosine directly. To do so we leverage a tool employed by [GSLW18] to perform Hamiltonian simulation. The Jacobi-Anger expansion yields highly eﬃcient polynomial approximations to sin(tx) andcos(tx). To perform Hamiltonian simulation one then takes the linear combination cos(tx) +isin(tx). However, we only need the cosine component. Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 29 e2πiˆλ(k) =|∆k〉e2πi2−nn-k-1 e2πi2−n−1 e2πi22 −n−(k− 1) |ψj〉 U· e2πiφk U(k) signal=˜H • ˜HT |∆k〉 /k e2πiˆλ(k) |ψj〉 U(k) svt=H U(k) signaleiθ0Z U(k)† signaleiθ1ZH |∆k〉 /k |ψj〉 UPE= U(0) svt/1 U(1) svt/2 U(2) svt/3 U(3) svt| (2ﬂoor4λj)〉 |ψj〉 |ψj〉......(a)(b) (c) (d)Figure 4: Circuit diagram for the algorithm in Corollary 13. (a) and (b) are depictions of (108) and (113) respectively. (c) depicts the singular value transformation circuit obtained from Lemma 10 which interperses alternating applications of U(k) signalandU(k)† signalwith phase rotations eiθjZ, where the angles θj encode the polynomial approximating Aη→δamp(x2). (d) depicts the ﬁnal circuit assembled via Lemma 7, showing how each iterative estimator’s output becomes part of the |∆k/angbracketrightregister for the next, and how that register ﬁnally becomes the output. Lemma 14. Jacobi-Anger expansion. For anyt∈R+, and anyε∈(0,1/e), let: r(t/prime,ε/prime) :=the solution to ε/prime=/parenleftbiggt/prime r/parenrightbiggr such thatr∈(t/prime,∞), (150) R:=/floorleftbigg r/parenleftbigget 2,5 4ε/parenrightbigg /2/floorrightbigg (151) Jk(t) :=thek’th Bessel function of the ﬁrst kind (152) Tk(t) :=thek’th Chebyshev polynomial of the ﬁrst kind (153) pcos,t(x) :=J0(t) + 2R/summationdisplay k=1(−1)kJ2k(t)T2k(x) (154) Thenpcos,t(x)is an even polynomial of degree 2Rsuch that for all x∈[−1,1]: |cos(tx)−pcos,t(x)|≤ε. (155) Furthermore: r(t/prime,ε/prime)∈Θ/parenleftBigg t/prime+logε/prime−1 log(log(ε/prime−1))/parenrightBigg (156) Proof.These results are shown in Lemma 57 and Lemma 59 of [GSLW18], outlined in their section 5.1. Again, computation of the degree of the Jacobi-Anger expansion is a bit complicated, but the complexity is worth it due to the method’s high performance. Our approach is to Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 30 ﬁrst synthesize cos/parenleftBig πλ(k) j/parenrightBig to obtain a signal that oscillates to indicate bit k(λj),

Question 375 multiple-choice

Finite non-abelian simple groups are important mathematical structures with applications in modern cryptography, especially as quantum computing threatens traditional encryption methods. Their unique group-theoretic properties can inspire new cryptographic primitives and security assumptions. Which cryptographic construction leverages the ability to perform arbitrary computations on encrypted data without revealing the plaintext, and is being explored through the use of finite non-abelian simple groups for potential quantum-resistant security? 1) Symmetric key block cipher 2) Diffie-Hellman key exchange 3) Digital signature algorithm 4) Zero-knowledge proof system 5) Group-based hash function 6) Elliptic curve cryptography 7) Fully homomorphic encryption

✓ Correct Answer:

The correct answer is 7) Fully homomorphic encryption.

📚 Reference Text:

Title: Applications of Finite Non-Abelian Simple Groups to Cryptography in the Quantum Era Year: 2023 Paper ID: b0d9472b999d0c168217ea2e1c750f598e8b079c Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b0d9472b999d0c168217ea2e1c750f598e8b079c Abstract: The theory of finite simple groups is a (rather unexplored) area likely to provide interesting computational problems and modeling tools useful in a cryptographic context. In this note, we review some applications of finite non-abelian simple groups to cryptography and discuss different scenarios in which this theory is clearly central, providing the relevant definitions to make the material accessible to both cryptographers and group theorists, in the hope of stimulating further interaction between these two (non-disjoint) communities. In particular, we look at constructions based on various group-theoretic factorization problems, review group theoretical hash functions, and discuss fully homomorphic encryption using simple groups. The Hidden Subgroup Problem is also briefly discussed in this context.

Question 376 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) are foundational to breakthroughs in computational tasks such as factoring and graph isomorphism. The efficiency of these algorithms often depends on the structure of the underlying group and the measurement strategies employed. Which statement correctly characterizes the limitations of quantum Fourier sampling for solving non-Abelian Hidden Subgroup Problems efficiently? 1) Quantum Fourier sampling on a single register yields complete information for all non-Abelian HSPs. 2) Entangled measurements on only two registers are always sufficient for efficient non-Abelian HSP solutions. 3) Efficient solution of non-Abelian HSPs, such as for the symmetric group, requires measurements on O(log|G|) registers because single-register sampling provides exponentially little information. 4) Classical post-processing after quantum Fourier sampling is always efficient for non-Abelian groups. 5) The standard quantum Fourier transform always leads to efficient algorithms for graph isomorphism problems. 6) Multiple copies of the hidden subgroup state are unnecessary for efficient non-Abelian HSP algorithms. 7) Abelian and non-Abelian HSPs are solved with identical measurement strategies on quantum registers.

✓ Correct Answer:

The correct answer is 3) Efficient solution of non-Abelian HSPs, such as for the symmetric group, requires measurements on O(log|G|) registers because single-register sampling provides exponentially little information..

📚 Reference Text:

of the eﬃcient quantum algorithms for the HS P described above. Given that the query complexity of the HSP is polynomial, it is natural t o ask how tight this query complexity is. In particular it is natural to ask how many co pies of the hidden sub- group state must be supplied in order for there to exist a measurem ent on these copies which we can eﬃciently perform and which provides enough information to r econstruct the hidden subgroup. In such cases we say that the hidden subgroup can be in formation theoretically reconstructed. Note, however that being information theoretic ally reconstructible does not mean that there is an eﬃcient algorithm for the problem because the classical post processing required to reconstruct the hidden subgroup may not be tractab le. What is known about the number of copies needed to information theoretically construc t the hidden subgroup? On the one hand it is known that for certain groups, in particular for th e dihedral[40], aﬃne[12], and Heisenberg groups[41], measurements on a single register of th e hidden subgroup state is suﬃcient for information theoretic reconstruction of the hidden subgroup, and these quan- tum measurements can be eﬃciently enacted. However, it was show n by Moore, Russell and Schulman[17] that for the particular case of the symmetric group H SP, measurements on a single register of the hidden subgroup state reveal only an expone ntially small amount of information about the identity of the hidden subgroup. In particula r this means that if one 6How a Clebsch-Gordan Transform Helps to Solve ... makes measurements which act only on a single hidden subgroup prob lem state at a time, one cannot eﬃciently solve the HSP. This was subsequently extende d to two registers of the hidden subgroup state by Moore and Russell[18], and then, in the culm ination of this line of inquiry, Hallgren et al.[20] showed that this extends all the way up to the upper bound of Ettinger, Hoyer, and Knill. In other words, for the hidden subgrou p problem relevant to the graph isomorphism problem, if one attempts to eﬃciently solve this pr oblem on a quantum computer one is required to perform a measurement on O(log|G|) registers containing the hidden subgroup state in order to solve the problem. In particular if this measurement can be implemented, even adaptively, on less than kregisters, then this measurement will not be able to solve the HSP. The results of Hallgren et al.imply that in order to solve the HSP, one must perform measurements on multiple copies of the hidden subgroup state in ord er to eﬃciently solve the non-Abelian hidden subgroup problem. One important conseque nce of this is that the standard method combined with performing a quantum Fourier tran sform on the hidden subgroup state, a paradigm which works for many of the eﬃcient qu antum algorithms for the hidden subgroup problem, cannot be used to ﬁnd an eﬃcient qua ntum algorithm for the non-Abelian hidden subgroup problem.

Question 377 multiple-choice

Quantum information processing often relies on controlling spin systems, such as carbon-13 nuclei in molecules, with high precision. Accurately modeling decoherence and relaxation is crucial for understanding and improving quantum gate fidelity in these systems. In a multi-spin NMR quantum computing experiment, which statement best explains why the relaxation superoperator for carbon-13 spins can be effectively represented by an 8×8 diagonal Hadamard matrix in the product operator basis? 1) The spin relaxation processes induce strong cross-relaxation between all product operator components. 2) The RF field inhomogeneity leads to a non-diagonal relaxation matrix requiring off-diagonal corrections. 3) T1 and T2 relaxation times for carbon-13 and hydrogen spins are always identical in such experiments. 4) Mono-exponential decay of each carbon-13 product operator occurs independently with negligible cross-relaxation. 5) The Quantum Fourier Transform creates off-diagonal elements that cannot be described by a diagonal matrix. 6) The nuclear Overhauser effect dominates relaxation, making the matrix non-diagonal. 7) Quantum Process Tomography always reveals the detailed structure of the relaxation superoperator.

✓ Correct Answer:

The correct answer is 4) Mono-exponential decay of each carbon-13 product operator occurs independently with negligible cross-relaxation..

📚 Reference Text:

any desired unitary operation seeks to maximize the ﬁdelity between the desired unitary superoperator ¯Uth⊗Uthand the simulated superoperator Sop. The latter was obtained as an incoher- ent or Kraus operator over the 33 RF ﬁeld strengths in the experim entally measured RF inhomogeneity histogram, i.e. Sop(ρ) =33/summationdisplay ℓ=1pℓUℓρU† ℓ≡33/summationdisplay ℓ=1AℓρA† ℓ. (27) Thus, the gate ﬁdelity as deﬁned in Eqn. (7) can be calculated direct ly from the Kraus operators Amas Fe((¯Uth⊗Uth)†Sop)≡tr((¯Uth⊗Uth)†Sop) tr((¯U† th¯Uth)⊗(U† thUth))=1 6433/summationdisplay m=1tr((¯Uth⊗Uth)†(¯Am⊗Am)) (28) =1 6433/summationdisplay m=1tr((¯U† th¯Am)⊗(U† thAm)) =1 6433/summationdisplay m=1|tr(U† thAm)|2. In simulations including the four hydrogen spins, the sum on the right -hand side of this formula must be increased by another factor of 16 in taking the par tial trace over the hydrogen spins. 15 TABLE I: Measured carbon-13 decay rates for each type of product oper ator (s−1) in the alanine molecule used for the experiments. These incl ude the inverses of the relaxation times T1andT2for all three carbon-13 spins. The “x|y” subscripts indicate that, in every operator of the corresp onding product, all the labels to the left or to the right of the bar mu st be used together, thus indicating a total of two products with one ex pression. Type of Product Operator Decay Rate Type of Product Operator Decay Rate σ1 z 0.032±0.01σ2 z 0.345±0.01 σ3 z 0.583±0.01σ1 zσ2 z 0.282±0.01 σ1 zσ3 z 0.458±0.01σ2 zσ3 z 0.689±0.01 σ1 zσ2 zσ3 z 0.684±0.01σ1 x|y, σ1 x|yσ2 z, σ1 x|yσ3 z, σ1 x|yσ2 zσ3 z1.89±0.02 σ2 x|y, σ1 zσ2 x|y, σ2 x|yσ3 z, σ1 zσ2 x|yσ3 z3.19±0.01σ3 x|y, σ1 zσ3 x|y, σ2 zσ3 x|y, σ1 zσ2 zσ3 x|y1.68±0.01 σ1 x|yσ2 y|x, σ1 x|yσ2 y|xσ3 z, σ1 x|yσ2 x|y, σ1 x|yσ2 x|yσ3 z6.93±0.11σ1 x|yσ3 y|x, σ1 x|yσ2 zσ3 y|x, σ1 x|yσ3 x|y, σ1 x|yσ2 zσ3 x|y3.56±0.06 σ2 x|yσ3 y|x, σ1 zσ2 x|yσ3 y|x, σ2 x|yσ3 x|y, σ1 zσ2 x|yσ3 x|y6.81±0.10σ1 y|xσ2 x|yσ3 x|y, σ1 x|yσ2 y|xσ3 x|y, σ1 x|yσ2 x|yσ3 y|x13.48±0.81 σ1 x|yσ2 x|yσ3 x|y 14.58±0.20 The relaxation superoperator for the carbons in alanine was measu red in the absence of RF ﬁelds, and found to have 98 .5% of its norm along the diagonal in the product operator basis. This means that the various product operator components decay mono-exponentially, without cross-relaxation, so that it can be described by an 8 ×8 “Hadamard relaxation matrix” [23, 31]. This is shown pictorially in Fig. 5, while precise values for the various types of rates seen in the ﬁgure are given in Table I. To be complete, the relaxation superoperator should also be measu red as a function of the applied RF ﬁelds (and include memory eﬀects due to nuclear Ove rhauser eﬀects with the hydrogens [32]). The QFT, however, eﬃciently mixes the sta tes of the three carbonspins, so thatonaverage over itsimplementation thedecoh erence isindistinguishable from a uniform, isotropic attenuation of all the product operator components. That is to say, QPT on the QFT is not able to provide any details regarding the ph ysical relaxation 16 processes operative in

Question 378 multiple-choice

Quantum computing leverages principles such as superposition to achieve computational speedups that are not possible with classical computing architectures. Understanding the nature of quantum parallelism and its visualization tools is essential for designing and analyzing quantum algorithms. Which statement best describes the role of quantum dataflow diagrams in the study and optimization of quantum algorithms? 1) They measure the classical runtime of quantum algorithms using parallel processors. 2) They visualize how classical algorithms split tasks among multiple cores. 3) They depict only the physical hardware connections between qubits in a quantum computer. 4) They provide estimates of quantum algorithm scalability based on classical parallelism laws. 5) They graphically illustrate the flow of quantum operations and the exploitation of quantum parallelism in algorithm processes. 6) They represent the statistical likelihood of quantum error rates during computation. 7) They simulate the energy consumption patterns of quantum circuits.

✓ Correct Answer:

The correct answer is 5) They graphically illustrate the flow of quantum operations and the exploitation of quantum parallelism in algorithm processes..

📚 Reference Text:

Title: What is Quantum Parallelism, Anyhow? Year: 2024 Paper ID: 8902903b2f124062d42e18d0041f97cb9910221d Source: semantic-scholar URL: https://www.semanticscholar.org/paper/8902903b2f124062d42e18d0041f97cb9910221d Abstract: Central to the power of quantum computing is the concept of quantum parallelism: quantum systems can explore and process multiple computational paths simultaneously. In this paper, we discuss the elusive nature of quantum parallelism, drawing parallels with classical parallel computing models to elucidate its fundamental characteristics and implications for algorithmic performance. We begin by defining quantum parallelism as arising from the superposition of quantum states, allowing for the exploration of multiple computational paths in parallel. To quantify and visualize quantum parallelism, we introduce the concept of quantum dataflow diagrams, which provide a graphical representation of quantum algorithms and their parallel execution paths. We demonstrate how quantum parallelism can be measured and assessed by analyzing quantum algorithms such as the Quantum Fourier Transform (QFT) and Amplitude Amplification (AA) iterations using quantum dataflow diagrams. Furthermore, we examine the interplay between quantum parallelism and classical parallelism laws, including Amdahl's and Gustafson's laws. While these laws were originally formulated for classical parallel computing systems, we reconsider their applicability in the quantum computing domain. We argue that while classical parallelism laws offer valuable insights, their direct application to quantum computing is limited due to the unique characteristics of quantum parallelism, including the role of destructive interference and the inherent limitations of classical-quantum I/O. Our analysis highlights the need for an increased understanding of quantum parallelism and its implications for algorithm design and performance.

Question 379 multiple-choice

Quantum algorithms often rely on group representation theory to analyze and solve problems, with the quantum Fourier transform (QFT) playing a central role in transforming quantum states. Understanding the properties of regular representations and their decompositions is essential for tackling the hidden subgroup problem (HSP), particularly in non-abelian groups like the dihedral group. Which statement best explains why Schur's lemma is useful for analyzing hidden subgroup states in quantum algorithms involving group regular representations? 1) It allows the construction of efficient classical algorithms for all non-abelian hidden subgroup problems. 2) It guarantees that every group element corresponds to a unique quantum measurement outcome. 3) It proves that the quantum Fourier transform diagonalizes all operators on the Hilbert space. 4) It enables the decomposition of operators commuting with irreducible representations into scalar multiples of the identity, simplifying the structure of hidden subgroup states. 5) It provides an explicit basis for the computational representation in terms of group elements. 6) It ensures that the left and right regular representations act identically on the Hilbert space. 7) It shows that hidden subgroup states are always pure rather than mixed.

✓ Correct Answer:

The correct answer is 4) It enables the decomposition of operators commuting with irreducible representations into scalar multiples of the identity, simplifying the structure of hidden subgroup states..

📚 Reference Text:

, Cambridge University Press, Cambridge, 2000. [43] A. Peres, Quantum theory: concepts and methods , Fundamental theories in physics, vol. 72, Kluwer Academic Publishers, 1993. [44] O. Regev, A subexponential time algorithm for the dihedral hidden subgroup problem with polynomial space . arXiv:quant-ph/0406151. OPTIMAL MEASUREMENTS FOR THE DIHEDRAL HSP 23 [45] ,Quantum computation and lattice problems , SIAM Journal on Computing 33(2004), no. 3, 738–760. [46] ,New lattice based cryptographic constructions , Journal of the ACM 51(2004), no. 6, 899–942. [47] M. R¨ otteler and T. Beth, Polynomial-time solution to the hidden subgroup problem for a class of non-abelian groups . arXiv:quant-ph/9812070. [48] J. Serre, Linear representations of ﬁnite groups , Graduate Texts in Mathematics, vol. 42, Springer, New York, 1977. [49] P. W. Shor, Polynomial-time algorithms for prime factorization and discrete loga- rithms on a quantum computer , SIAM Journal on Computing 26(1997), no. 5, 1484– 1509. [50] D. R. Simon, On the power of quantum computation , SIAM Journal on Computing 26(1997), no. 5, 1474–1483. [51] N. J. A. Sloane, The on-line encyclopedia of integer sequences , 2004. http://www. research.att.com/ ~njas/sequences . [52] H. Yuen, R. Kennedy, and M. Lax, Optimum testing of multiple hypotheses in quan- tum detection theory , IEEE Transactions on Information Theory 21(1975), no. 2, 125–134. Appendix: Representation theory and the optimal measurement Many features of the hidden subgroup problem can be understood us- ing simple group representation-theoretic arguments. Here we present such arguments and demonstrate their application to the dhsp . Two important representations of a group Gfor the hspover that group are the left and right regular representations of G. These representations act on a Hilbert space spanned by vectors {|g/angbracketright}g∈Gas DL(g1)|g2/angbracketright=|g1g2/angbracketright DR(g1)|g2/angbracketright=|g2g−1 1/angbracketright. (71) Viewed as representations of the group algebra, these two representations are commutants of each other, i.e., DL(g1)DR(g2) =DR(g2)DL(g1). The hidden subgroup states ρHdeﬁned in (3) commute with the left regular rep- resentation: DL(g)ρH=ρHDL(g) for allg∈ G. Hence, via Schur’s lemma and the fact that the left and right regular representations are commutants, it is easy to show that a general hidden subgroup state can be expressed as (72) ρH=1 |G|/summationdisplay h∈HDR(h). The regular representations are reducible. Let ˆGbe a set of labels for a complete set of irreducible representations of G, and for any x∈ˆG, let Γ x(g) be thexth irreducible representation (irrep) matrix for the group element g∈ G. Letdxdenote the dimension of the xth irrep. Then there exists a basis of the Hilbert space {|g/angbracketright}g∈G, labeled by |x,/lscript,m /angbracketrightwithx∈ˆGand /lscript,m∈Zdx, such that DLandDRact as DL(g) =/circleplusdisplay x∈ˆGΓx(g)⊗IdxDR(g) =/circleplusdisplay x∈ˆGIdx⊗Γx(g) (73) 24 DAVE BACON, ANDREW M. CHILDS, AND WIM VAN DAM whereIdis thed-dimensional identity matrix. The unitary transforma- tion that transforms between the bases {|g/angbracketright}g∈Gand{|x,/lscript,m /angbracketright}x∈ˆG,/lscript,m∈Zdxis nothing but the Fourier transform over G, (74)QG:=1/radicalbig |G|/summationdisplay g∈G/summationdisplay x∈ˆG/summationdisplay /lscript,m∈Zdx/radicalbig dx[Γx(g)]/lscript,m|x,/lscript,m /angbracketright/angbracketleftg|. Here [Γ x(g)]/lscript,mis the matrix element in the /lscriptth row and mth column of the xth irrep at the group element g. If we perform the quantum Fourier transform over Gon the state ρH, we ﬁnd

Question 380 multiple-choice

Chiral multifold fermions in spin-orbit coupled crystals exhibit diverse spin textures that influence their topological and electronic properties. Understanding the factors that determine spin-momentum locking and winding numbers is crucial for engineering materials with novel quantum phenomena. Which of the following statements most accurately describes the possible range of integer winding numbers for spin textures around the Fermi surface in chiral multifold fermion systems? 1) The winding number is always limited to ±1. 2) The winding number only takes the values 0 and ±2. 3) The winding number cannot be negative and ranges from 0 to +3. 4) The winding number ranges from 0 to ±5. 5) The winding number is constrained to ±3 and ±6. 6) The winding number is strictly ±2 or ±4. 7) The winding number can take any integer value from 0 to ±7.

✓ Correct Answer:

The correct answer is 7) The winding number can take any integer value from 0 to ±7..

📚 Reference Text:

Title: Spin-momentum locking from topological quantum chemistry: Applications to multifold fermions Year: 2022 Paper ID: fcc277201b0eac62390f88be953c98656a1ac840 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/fcc277201b0eac62390f88be953c98656a1ac840 Abstract: In spin-orbit coupled crystals, symmetries can protect multifold degeneracies with large Chern numbers and Brillouin zone spanning topological surface states. In this work, we explore the extent to which the nontrivial topology of chiral multifold fermions impacts the spin texture of bulk states. To do so, we formulate a deﬁnition of spin-momentum locking in terms of reduced density matrices. Using tools from the theory of topological quantum chemistry, we show how the reduced density matrix can be determined from the knowledge of the basis orbitals and band representation forming the multifold fermion. We show how on-site spin orbit coupling, crystal ﬁeld splitting, and Wyckoﬀ position multiplicity compete to determine the spin texture of states near chiral fermions. We compute the spin texture of multifold fermions in several representative examples from space groups P 432 (207) and P 2 1 3 (198). We show that the winding number of the spin around the Fermi surface can take many diﬀerent integer values, from zero all the way to ± 7. Finally, we conclude by showing how to apply our theory to real materials using the example of PtGa in space group P 2 1 3.

Question 381 multiple-choice

Quantum machine learning models often rely on measurement operators that respect the underlying symmetries of quantum data, such as those induced by SU(2) transformations on qubits. The ability of these models to classify data depends crucially on the structure and reducibility of the representations in which these symmetries act. Which property of the SU(2) fundamental representation on single-qubit systems most directly explains why only measurement operators proportional to the identity are allowed when requiring SU(2)-invariance? 1) The fundamental representation is irreducible and admits only scalar multiples of the identity as operators commuting with all SU(2) unitaries. 2) The fundamental representation is reducible and allows block diagonal operators that are SU(2)-invariant. 3) The SU(2) group lacks any symmetry when acting on single-qubit systems, so any operator is allowed. 4) Only non-Hermitian operators commute with SU(2) transformations in the fundamental representation. 5) The fundamental representation supports non-trivial measurement operators due to entanglement in single qubits. 6) Schur-Weyl duality prohibits the use of the identity operator in SU(2)-invariant models. 7) Antisymmetric and symmetric subspaces of the fundamental representation enable arbitrary SU(2)-invariant measurements.

✓ Correct Answer:

The correct answer is 1) The fundamental representation is irreducible and admits only scalar multiples of the identity as operators commuting with all SU(2) unitaries..

📚 Reference Text:

(see Fig. 6(a)). Thus, we have a QML model of the form h(i) = Tr[iM]; (12) whereMis a single-qubit Hermitian measurement opera- tor. Then, the principles of GQML indicate that if his to respect the symmetries of the dataset, one needs to pick an invariant measurement operator (see Eq. (4)), i.e., an operatorMsuch that [M;Ug] = 0;8Ug2SU(2): (13) Since we know that we are working with the fundamen- tal representation of SU(2), we can use Example 3, and more speciﬁcally the fact that a unitary can be expressed asUg=c01+i(c1X+c2Y+c3Z)withj(c0;c1;c2;c3)j= 1, to see that the onlyoperators that commutes with any unitary in SU(2)are those proportional to the identity. Combining this result with Eqs. (12) and (13) shows that the only GQML model that is invariant under SU(2)is h(i) = Tr[i(c1)] =cTr[i] =cwhich is clearly incapable of classifying the data. The previous can be further under- stood from the fact that the fundamental representation ofSU(2)admits no (non-trivial) block diagonal represen- tation, i.e., the fundamental representation of SU(2)isir- reducible . As shown in Fig. 6(a), this absence of a block diagonal structure implies that the only possible measure- ment operator is the identity. This idea will return at a higher level when we learn Schur’s lemma (Theorem 6), which states that the only maps from an irrep to itself that commute with every unitary are scalar multiples of the identity c1. Next, let us consider a QML model as in Eq. (1) where we are allowed to act on two copies of the data states. These are known as quantum-enhanced experiments (one with quantum memories) [11, 25] (see Fig. 6(b)). That is, h(i) = Tr 2 iM ; (14) whereMis a Hermitian operator on two qubits. Now, the representation of the symmetry group is given by Ug Ug, 14 Figure 6. Fundamental and tensor representation of SU(2)and the task of classifying single-qubit states ac- cording to their purity. a) In a conventional experiment we want to classify the data by computing h(i) = Tr[iM]. Here, we are working with the fundamental representation of G=SU(2), which admits no non-trivial block diagonal struc- ture (is irreducible). Using Eq. (13) we ﬁnd that the only pos- sible measurement operator is M/1, in which case the model cannot classify the data. b) In a quantum-enhanced experiment we want to classify the data by computing h(i) = Tr 2 iM . Here, we are working with the tensor representation of G= SU(2), which admits a block diagonal structure (is reducible) in the symmetric-antisymmetric basis (see Example 5). Now the operatorsMsatisfying Eq. (15) are those that take the form M=c013Lc11. This will be revisited in greater generality when we reach the commutant structure theorem 7 and Schur- Weyl Duality VIID. Notably, by writing i=1 2(1+r), we canseethatinthesamesymmetric-antisymmetricbasis 2 ihas a component in the antisymmetric space given by1 4(1rr). and thus Eq. (4) indicates that we need to pick a measure- ment operator such that [M;U 2 g] = 0;8Ug2SU(2): (15) Take a look at the block

Question 382 multiple-choice

In group theory, varieties are classes of groups defined by group identities, and the distinction between finitely and non-finitely based varieties is fundamental to understanding their structure. Concepts such as the Specht property, Zorn's lemma, and verbal Noetherianity play significant roles in analyzing these varieties. Which of the following statements about just non-Specht varieties is correct? 1) They are generated exclusively by infinite groups. 2) Every Specht variety contains a just non-Specht subvariety. 3) All just non-Specht varieties are finitely based. 4) The intersection of any just non-Specht variety with AN2 is always non-Specht. 5) Only one just non-Specht variety exists in group theory. 6) Explicit examples of just non-Specht varieties have been fully constructed and classified. 7) There are infinitely many just non-Specht varieties, but no explicit examples have been found.

✓ Correct Answer:

The correct answer is 7) There are infinitely many just non-Specht varieties, but no explicit examples have been found..

📚 Reference Text:

Krasil'nikov [5] for arbitrary c) and each variety var (G) generated by a finite group G (Oates and Powell [10]). On the other hand, the variety N 2N2 of all (nilpotent of class at most 2)-by-(nilpotent of class at most 2) groups is known to be non-Specht (Vaughan-Lee [11]) as well as the variety ZAN 2 of all centre-by-Abelian-by-(nilpotent of class at most 2) groups (Gupta and Krasil'nikov [3]). A variety V is called just non-Specht or just non-finitely based if V is non-Specht but all proper subvarieties of V are Specht (equivalently, if V is non-finitely based but all proper subvarieties of V are finitely based). It follows easily from Zorn's lemma that each non-Specht variety contains a just non-Specht subvariety so just non-Specht varieties of groups 'form the border' between Specht and non-Specht varieties. It is known that there are infinitely many just non-Specht varieties of groups (Newman [7]) but no examples of such varieties are as yet known. The following theorem gives the first example of a non-finitely based subvariety V of the variety N 2N2 whose intersection with AN 2 is Specht. The variety V comes closest to being just non-Specht. We hope that it could give an approach to construct a just non-Specht variety of groups (a problem which remains open). Recall that a(x-y) = axyaTyx and a(x-y-z) = c?yza-yxzarv'yazyx. Let a"'-«»-'«« = (a1" "'-')"' for all k > 1. THEOREM 2. Let V be the variety of groups defined by the identities (3) [[x ux2,x3],[x4,x 5,x6],[x 1,xs]] = 1 and (4) T x x x •\<,xt,x i,X(,)(x 1,xt,x9)(xla,xn https://doi.org/10.1017/S1446788700002056 Published online by Cambridge University Press 332 C. K. Gupta and A. N. Krasil'nikov [4] Then the variety V is not Specht but the intersection variety V D AN 2 (which is defined by the identities [[x {, x2, *3], [JC 4, x5, x6]] = 1 and (4)) is Specht. REMARK. Theorem 2, in fact, remains valid if one replaces the identity (4) with the identity [x {, x2, Xi\{x*-Xi-xMxir5.-*6)(*7.-*8.-*9)-(*3*-2.Jr3*-l.*M) L*1,-*2>*3J Ut the variety of groups corresponding to the verbal subgroup U k so that U* is defined by the identities l\X\, X 2, X3\, L-*4> ^5-» ^aJJ = It lX\,X2,X3i =1. To prove that the variety V D AN 2 is Specht and the relatively free group F/ U defined in Theorem 1 is verbally Noetherian we need the following. PROPOSITION 1. For every positive integer k the relatively free group F/ U k is verbally Noetherian. Since V n AN 2 C U 5 and U k C U for all k > 2, Proposition 1 implies that the variety V n AN 2 is Specht and the relatively free group F/ U is

Question 383 multiple-choice

Quantum computing relies on efficient implementations of fundamental operations like the Quantum Fourier Transform (QFT), which is central to algorithms such as factoring and phase estimation. Optimizing these transformations is key for practical, error-corrected quantum devices. Which technique enables the reduction of T-count complexity in fault-tolerant QFT circuits from O(n log²(n)) to O(n log(n)) for a fixed approximation error? 1) Measurement and feedforward operations with phase gradient state reuse 2) Increasing the precision of small-angle rotations 3) Replacing all Clifford gates with Toffoli gates 4) Doubling the number of qubits used 5) Excluding the use of error correction codes 6) Utilizing only classical post-processing algorithms 7) Eliminating all measurement steps from the circuit

✓ Correct Answer:

The correct answer is 1) Measurement and feedforward operations with phase gradient state reuse.

📚 Reference Text:

Title: Approximate quantum Fourier transform with O(n log(n)) T gates Year: 2018 Paper ID: 561db83576eb62b572d725f71a4aa4df6fc5e128 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/561db83576eb62b572d725f71a4aa4df6fc5e128 Abstract: The ability to implement the Quantum Fourier Transform (QFT) efficiently on a quantum computer facilitates the advantages offered by a variety of fundamental quantum algorithms, such as those for integer factoring, computing discrete logarithm over Abelian groups, solving systems of linear equations, and phase estimation, to name a few. The standard fault-tolerant implementation of an n-qubit unitary QFT approximates the desired transformation by removing small-angle controlled rotations and synthesizing the remaining ones into Clifford+T gates, incurring the T-count complexity of O(nlog2(n))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n\,{\mathrm{log}}^{2}\,(n))$$\end{document}. In this paper, we show how to obtain approximate QFT with the T-count of O(nlog(n))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n\,{\mathrm{log}}\,(n))$$\end{document}. For brevity, the above figures omit the dependence on the approximation error ε, assuming the error is fixed. Our approach relies on quantum circuits with measurements and feedforward, and on reusing a special quantum state that induces the phase gradient transformation. We report asymptotic analysis as well as concrete circuits, demonstrating significant advantages in both theory and practice.

Question 384 multiple-choice

Quantum computing relies on efficient arithmetic operations to enable complex algorithms such as factoring and cryptography. Implementing integer multiplication with reduced circuit depth is crucial for practical, scalable quantum computations. Which technique allows quantum integer multiplication circuits to significantly reduce their circuit depth from exponential to polynomial order while maintaining scalability for any bit-width? 1) Employing Grover's search algorithm for multiplication 2) Utilizing the Quantum Fourier Transform with controlled rotations in the phase domain 3) Encoding multiplication using quantum annealing processes 4) Applying quantum error correction codes to each arithmetic operation 5) Relying solely on classical reversible logic gates in quantum circuits 6) Implementing measurement-based quantum computation for arithmetic 7) Using decoherence-free subspaces to perform multiplication

✓ Correct Answer:

The correct answer is 2) Utilizing the Quantum Fourier Transform with controlled rotations in the phase domain.

📚 Reference Text:

Title: Quantum Array Multiplier Year: 2023 Paper ID: a2669e3e99cba275b9c15bd659ee22c0dd41f299 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a2669e3e99cba275b9c15bd659ee22c0dd41f299 Abstract: This paper proposes a new and improved implementation of a quantum integer multiplier. Performing arithmetic computations is sometimes a necessary step in the implementation of quantum algorithms. In this work, Quantum Fourier Transform is used in order to perform scalable arithmetic in a generic bit-width quantum system. In the phase domain, addition can be implemented through accumulated controlled rotations on the qubits’ state. Leveraging this, and inspired by the classical implementation of an array multiplier, a new integer multiplier is fully designed and tested in a quantum environment.The depth of a quantum circuit is the number of computational steps necessary to completion, and it is a key parameter that reflects on the performance of the design. The new design reduces the quantum depth of the design from the exponential order of the previously proposed designs to polynomial order.

Question 385 multiple-choice

In algebraic topology, Morava K-theory is a generalized cohomology theory used to study the structure of classifying spaces of finite groups. Spectral sequences, module structures, and group extension techniques play a key role in analyzing when the odd-degree part of Morava K-theory vanishes for certain classes of groups. For a prime p > 2 and a p-group G with a normal subgroup H of index p such that the odd-degree Morava K-theory of BH vanishes, which of the following conditions is both necessary and sufficient for the odd-degree Morava K-theory of BG to vanish? 1) The Morava K-theory of BH is a permutation module for G/H ≅ Cp. 2) G is a minimal nonabelian p-group. 3) The action of G/H on K(n)(BH) is trivial. 4) G is a wreath product of two finite groups. 5) The Morava K-theory of BG is a permutation module for G/H. 6) H is an elementary abelian p-group. 7) The Atiyah–Hirzebruch spectral sequence for BG collapses at E2.

✓ Correct Answer:

The correct answer is 1) The Morava K-theory of BH is a permutation module for G/H ≅ Cp..

📚 Reference Text:

detail without placing an undue burden on the reader’s patience. 1 Preliminaries Our principal calculational tool shall be the Serre spectral sequence (1) E2DH.BQIK.BH//)K.BG/ associated to a group extension 1!Hi !G !Q!1 forKeither integral Morava K–theoryzK.n/or its mod pversion K.n/. Here H.BQIK.n/.BH//is the ordinary cohomology of Qwith coefﬁcients in the FpQ–module K.n/.BH/, the action of Qbeing induced by conjugation in Gas usual. This module structure can be quite messy, even in the simplest cases, since it involves the formal group law. Recall that as any complex oriented cohomology theory, Morava K–theory comes equipped with a formal group law induced from the tensor product of line bundles; we shall write the formal sum as “ CK.n/”. This spectral sequence is, via , a module over the Atiyah–Hirzebruch spectral sequence for BQ. In [4], Igor Kriz proved a beautiful theorem about the Serre spectral sequence associated to ﬁbrations over BCp. This theorem is one of the few practical tools for calculation; Kriz used it to great effect to supply the ﬁrst counterexample to the even degree conjecture. His result gives a useful criterion to decide whether a group Ghas even Morava K–theory. Algebraic & Geometric Topology , Volume 11 (2011) Morava K–theory of groups of order 32 505 Theorem 1.1 (Kriz [4]) LetGbe ap–group and Ha normal subgroup of index p withK.n/odd.BH/D0. Then K.n/odd.BG/D0if and only if the integral Morava K–theoryzK.n/.BH/is a permutation module for the action of G=HCp. For odd primes p, this condition is equivalent to saying that K.n/.BG/is a permuta- tion module, but for pD2this is trivially false (all F2C2–modules are permutation modules). At the other extreme, one could use extensions with a trivial action, such as central extensions. This has the advantage of not needing to know the Morava K–theory of the subgroup as a module for the quotient, which can be hard to determine, but the drawback is that the quotient is usually a large ( p–)group, whose mod pcohomology can be quite challenging. So one has to ﬁnd one’s way between two evils, and a combination of both strategies will often be our chosen line of attack. We conclude the section with a list of nonabelian p–groups whose Morava K–theory is known to be good. Theorem 1.2 (Hopkins–Kuhn–Ravenel [2]; Hunton [3]; Kriz [4]; Tezuka–Yagita [9]; Yagita [10; 11; 12; 13]) IfGbelongs to any of the following families of p–groups, thenK.n/odd.BG/D0: (a)wreath products of the form HoCpwithHgood [2; 3] , (b)metacyclic p–groups [10], (c)minimal nonabelian p–groups, ie, groups all of whose maximal subgroups are abelian [11], (d)groups of p–rank 2 [12], (e)elementary abelian by cyclic groups, ie, extensions V!G!Cwith V elementary abelian and Ccyclic [13; 4] . 2 Technical lemmas The ﬁrst result describes the Morava K–theory of a central product of a “good”group with an abelian group. Recall that a group Gis called a central product of two subgroups PandQifP\Qcontains a central subgroup ZofGsuch that Gis isomorphic to the quotient of PQby the inclusion of Zvia the diagonal. We denote the central product by PZQor simply

Question 386 multiple-choice

The evolution of chemical theory has been shaped by quantum mechanics, computational paradigms, and philosophical approaches to scientific concepts. Understanding how chemical properties emerge and are modeled is central to modern chemistry. Which approach emphasizes integrating abstractions and emergent properties across multiple scales to capture complex chemical behaviors, rather than merely linking phenomena at different levels? 1) Multiscale modeling 2) Hierarchical modeling 3) Valence bond theory 4) Density functional theory 5) Molecular orbital theory 6) Machine learning-based prediction 7) Quantum computing algorithms

✓ Correct Answer:

The correct answer is 2) Hierarchical modeling.

📚 Reference Text:

Title: The Evolving Quest for Chemical Understanding in the Quantum Age. Year: 2025 Paper ID: 027ee6d9bc757e720eba8b3964907ac3087b3af7 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/027ee6d9bc757e720eba8b3964907ac3087b3af7 Abstract: As quantum mechanics enters its second century, theoretical and computational chemistry stands at a pivotal transition. Traditional orbital-based approaches such as valence bond theory and molecular orbital theory and density-based frameworks like density functional theory have long provided the computational and conceptual foundations for the field. However, the advent of machine learning and quantum computers introduces completely new paradigms for representation, inference, and understanding. In this perspective, we examine how chemical understanding has been evolving in the past century through the lenses of ontology, epistemology, and emergence. We argue that chemical concepts, such as aromaticity, electronegativity, reactivity, and stereoselectivity, are not merely reducible to basic laws of physics but emerge as essential scaffolds linking chemical theories to chemical understanding. We propose a general scheme to obtain chemical understanding from the basic variables of chemical theories. Extending this scheme to deep learning and quantum computing, we suggest roadmaps to harvest chemical understanding from them and then advocate for hierarchical modeling as a new platform that moves beyond the constraints of multiscale modeling. Hierarchical modeling integrates abstraction across scales, captures emergent behaviors, and enables conceptual innovation for hierarchical systems. We conclude that the future of chemical understanding depends less on solving harder physical equations alone and more on epistemological shift characterized by conceptual pluralism, epistemic adaptability, and deeper appreciation of the multilayered ontological structure inherent to molecular systems.

Question 387 multiple-choice

Quantum algorithms have revolutionized the solving of certain group-theoretic and number-theoretic problems, particularly by exploiting quantum principles such as phase estimation and interference. These advancements have led to significant improvements in computational tasks relevant to cryptography and computational group theory. Which problem is a generalization that encompasses tasks such as order-finding, factoring, and discrete logarithms, and can be addressed using quantum eigenvalue estimation techniques? 1) Graph isomorphism problem 2) Quantum error correction 3) Non-Abelian group factorization 4) Abelian hidden subgroup problem 5) Hamiltonian simulation 6) Grover’s search problem 7) Quantum random walk

✓ Correct Answer:

The correct answer is 4) Abelian hidden subgroup problem.

📚 Reference Text:

Title: The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer Year: 1998 Paper ID: c4e61b8d75991fbcd774e462a8373effecb3216c Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c4e61b8d75991fbcd774e462a8373effecb3216c Abstract: A quantum computer can efficiently find the order of an element in a group, factors of composite integers, discrete logarithms, stabilisers in Abelian groups, and hidden or unknown subgroups of Abelian groups. It is already known how to phrase the first four problems as the estimation of eigenvalues of certain unitary operators. Here we show how the solution to the more general Abelian hidden subgroup problem can also be described and analysed as such. We then point out how certain instances of these problems can be solved with only one control qubit, or flying qubits, instead of entire registers of control qubits.

Question 388 multiple-choice

Quantum Fourier Transform (QFT) circuits are essential in quantum computing and can be constructed for systems with different radices, such as binary (qubits) or ternary (qutrits). Optimizing these circuits by simplifying their gate structure has direct implications on performance and efficiency. In which case does the magnitude of the multiplicative error introduced by omitting least significant controlled phase shift gates in the Quantum Fourier Transform circuit converge most rapidly to one, thus yielding better approximation properties? 1) Binary qubit systems with a small number of digits 2) Binary qubit systems with a large number of digits 3) Ternary qutrit systems with only Chrestenson gates omitted 4) Binary qubit systems without any phase error analysis 5) Multi-valued systems using only Hadamard gates 6) Binary qubit systems with all controlled phase shift gates included 7) Multi-valued quantum systems with higher radices such as qutrits and ququarts

✓ Correct Answer:

The correct answer is 7) Multi-valued quantum systems with higher radices such as qutrits and ququarts.

📚 Reference Text:

TRANSFORM BY HIGHER RADICES 205 The rest of the QFT circuit is constructed by implement- ing each term of this expanded exponential function. Pleasenote that, for each ternary digit (qutrit) x r, the superposition /C12/C120iþe/C02/C25ixr3/C0lj1iþe/C04/C25ixr3/C0lj2i is realized by the composition of a Chrestenson gate, followed by the phase shift, controlled by xkðk

Question 389 multiple-choice

Knot invariants such as the Jones polynomial play a pivotal role in quantum computational complexity, especially due to their connections with braid group representations and quantum circuits. Understanding how quantum processes can be encoded topologically is fundamental in both quantum information theory and mathematical physics. When representing a quantum circuit with m two-qubit gates acting on n qubits as a braid, which statement is correct regarding the number of strands and crossings required for this encoding? 1) The circuit can be encoded as a braid on 4n strands with a number of crossings that grows polynomially with m and n. 2) The circuit requires a braid on n strands with exponentially many crossings. 3) The circuit is represented by a braid on 2m strands with linear crossings in m. 4) The encoding uses a braid on n^2 strands with a constant number of crossings. 5) The circuit can only be simulated using a braid on 2n strands with exponential crossings. 6) The braid representation necessitates 8n strands and logarithmically many crossings. 7) The number of strands is independent of the number of qubits; only crossings depend on m.

✓ Correct Answer:

The correct answer is 1) The circuit can be encoded as a braid on 4n strands with a number of crossings that grows polynomially with m and n..

📚 Reference Text:

a suitable number of times, the absolute value of the average is reported. We om it the details as they are identical to the former case. Another procedure for this task makes direct use of the follo wing representation-theoretic formula for the absolute value of the Jones polynomial: 1 [2]p+r−1 ℓ/vextendsingle/vextendsingle/vextendsingleJ/parenleftbig ξp,r 1,1(b),e2πi/ℓ/parenrightbig/vextendsingle/vextendsingle/vextendsingle=[2]p−1 ℓ [2]p+r−1 ℓ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay λ∈Λ(2,ℓ) r[λ1−λ2+ 1] ℓTr/parenleftBig π(2,ℓ) [p,p]+λ(b)(t2p⊗1λ)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle =1 [2]r ℓ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay λ∈Λ(2,ℓ) r[λ1−λ2+ 1] ℓTr/parenleftBig π(2,ℓ) [p,p]+λ(b)(t2p⊗1λ)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(6.1) Recall that by (5.23), the numbers/braceleftBig Pλ=s(2,ℓ) λd(2,ℓ) λ:λ∈Λ(2,ℓ) r/bracerightBig sum to unity and thus deﬁne probabilities. The algorithm proceeds as in [2] by ﬁrst sele cting an irrep λ∈Λ(2,ℓ) rat random with probability Pλ≡s(2,ℓ) λd(2,ℓ) λ. Then, a path tin the Young graph Λ(2,ℓ) λis chosen uniformly at random. By the same arguments as in the HOMFLYPT case, one m ay then sample from random variables X,Y∈ {±1}with conditional expectations satisfying E[X+iY|λ,t] =/an}bracketle{tt2p|/an}bracketle{tt|π(2,ℓ) [p,p]+λ(b)|t2p/an}bracketri}ht|t/an}bracketri}ht. This procedure is then repeated, averaging the results unti l the desired precision is attained, after which the absolute value is reported. Using (6.1), it i s straightforward to show that /vextendsingle/vextendsingleE[X+iY]/vextendsingle/vextendsingle=1 [2]p+r−1 ℓ/vextendsingle/vextendsingle/vextendsingleJ/parenleftbig ξp,r 1,1(b),e2πi/ℓ/parenrightbig/vextendsingle/vextendsingle/vextendsingle. The remainder of the proof is identical to the HOMFLYPT situa tion. 7 Simulating quantum circuits with braids In this section, we show how to “compile” any quantum circuit onnqubits consisting of m two-qubit gates into a description of a braid on 4 nstrands with poly( m) crossings in such a way that the output of the circuit is encoded into an evaluati on of the Jones polynomial of the plat closure of the braid at any primitive ℓ’th root of unity with ℓ≥5,ℓ/ne}ationslash= 6. While this was ﬁrst proved in [12], our proof is simpler as it only uses the re presentation of B8corresponding to the rectangular diagram . There are uncountably many quantum circuits on nqubits, but countably many braids. Therefore, we will have to settle for approximations of quan tum circuits. Given unitaries U andU′and some 0 ≤ǫ≤1, we say that U ǫ-approximates U′if||U−U′||∞≤ǫ, where for a square matrix M,||M||∞denotes the operator norm , or largest singular value of M. Our main goal in this section will be to prove the following: 24The Jones Polynomial: Quantum Algorithms and Applications in Quantum Complexity Theory Theorem 7.1 LetU=UmUm−1···U1be a quantum circuit of length macting onnqubits and letǫ= Ω/parenleftbig 2−poly( n)/parenrightbig . Then there is a braid b∈B4nwithO/parenleftbig poly(m)/parenrightbig crossings such that Uisǫ-approximated by π(2,ℓ) [2n,2n](b). In this case, the following inequality holds: /vextendsingle/vextendsingle/vextendsingle/vextendsingle|/an}bracketle{t00···0|U|00···0/an}bracketri}ht|2−1 [2]2n−1 ℓ/vextendsingle/vextendsingle/vextendsingleJ(/tildewideb,e2πi/l)/vextendsingle/vextendsingle/vextendsingle2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ǫ. (7.1) Moreover, such a braid can be found from a description of the c ircuit inO/parenleftbig poly(m)/parenrightbig time on a classical computer. To prove Theorem 7.1 we must ﬁrst introduce some machinery. A ﬁnite set of gates in U(H) is said to be universal if it generates a dense subset of U(H). The following lemma implies that the measurement probabilities of an ǫ-approximation of a quantum circuit diﬀer by at most ǫfrom those of the desired circuit. The proof is in the appendi x. Lemma 7.2 LetUandU′bed×dunitaries for which U′ǫ-approximates U, whereǫ≤1.

Question 390 multiple-choice

In representation theory and quantum computation, the structure of group representations and their decomposition under subgroup actions is crucial for designing efficient algorithms. Techniques such as subgroup-adapted bases and combinatorial diagrams are commonly used to facilitate quantum Fourier transforms. Which of the following statements accurately describes the role of Gel'fand-Tsetlin bases in quantum algorithms for finite groups? 1) They guarantee that every group representation is automatically irreducible under all subgroups. 2) They ensure all matrix representations become scalar multiples of the identity for subgroup restrictions. 3) They provide a basis where every matrix representation becomes strictly upper triangular for any subgroup. 4) They are used to maximize the adapted diameter of the generating set in quantum algorithms. 5) They index coset representatives in the process of building subgroup towers. 6) They yield a basis in which group representations restrict to block diagonal matrices under any subgroup, facilitating efficient quantum computation. 7) They minimize the number of irreducible representations in all subgroup decompositions.

✓ Correct Answer:

The correct answer is 6) They yield a basis in which group representations restrict to block diagonal matrices under any subgroup, facilitating efficient quantum computation..

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embeddings of the cor respond- ing representation space into the representations of Gi+1and conversely, the edges entering a given representation ρ:Gi+1→U(Vρ)ofGi+1indexasetofmutuallyorthogonalsubspacesof Vρwhosedirectsumrepresentsthedecom- positionof Vρunderthe (restricted)actionof Gi. Thus,the pathsfromtherootnodeto a vertex ρ:Gi→U(Vρ)index a basis of Vρwith the following property: for any Gj0:∪j≤ℓ(S∩Gi)j⊇Ti}, and deﬁne the adapted diameter D =∑iDi. Then clearly any group element can be factored as a series of coset representatives,whichinturncanbefactoredasatotal ofa t mostDelementsof S. Of course, to perform the QFT efﬁciently we would like ρ(γ)to have a simple form for each γ∈S. Given a subgroup KK, andρis adapted to both HandK, thenρ(a)also possesses the block structure corresponding to ρ|H. This places an upper bound on Mof the maximum multiplicity with which representationsof Kappearin restrictionsof representationsof H. Thus we can minimize Mby choosinggenerators γinsidesubgroupsaslowonthetoweraspossible,whichcentr alizesubgroupsashighonthetoweraspossible. Forinstance,inthesymmetricgroup Snwetakethetowertobe Sn>Sn−1>···>{1},whereSiﬁxesallelements greater than i. LetSbe the set of pairwise adjacent transpositions (j,j+1); each of these is contained in Sj+1and centralizes Sj−1. The maximum multiplicity with which a representation of Sj−1appears in a representation of Sj+1 is 2, corresponding to the two orders in which we can remove tw o cells from a Young diagram. Since the adapted diameter is easily seen to be O(n2), this means that the ρ(α)can be carried out in O(n2)=polylog(|Sn|)elementary quantumoperations[3]. We will seethat asimilar situation obtainsforalargeclassofgroups. 4 Efﬁcient quantum Fourier transforms We describe our algorithm in this section. The algorithm per formsthe Fourier transform inductively on the tower of subgroups,usingthestructureoftheBrattelidiagramtoco nstructthetransformateachlevelfromthetransformatthe previouslevel. 5 Recallthatforeachlevelofourtowerofsubgroups G=Gm>Gm−1>···>G0={1}wehavechosenatransversal Tifortheleftcosetsof Gi−1inGi. Atthebeginningofthecomputation,werepresenteachgrou pelement gasaproduct α=αm···α1whereαi∈Ti. Thisstringbecomesshorterasweworkourwayupthetower,a ndafterhavingperformed theFouriertransformfor Githeremainingstring α=αm···αi+1indexesthecoset of

Question 391 multiple-choice

Quantum algorithms leverage mathematical symmetries in many-body systems to achieve computational advantages over classical methods, especially in the study of frustrated spin models and representation theory. The Schur-Weyl duality plays a foundational role in connecting unitary and permutation symmetries within quantum information science. Which statement correctly characterizes the mathematical relationship that enables efficient variational parameter updates on quantum hardware for systems with both SU(d) and Sn symmetries? 1) The SU(d) group is isomorphic to the Sn group, allowing direct translation of quantum operations. 2) Sn symmetry can only be exploited for classical neural networks, not quantum algorithms. 3) The Marshall-Lieb-Mattis theorem guarantees analytical solutions in all frustrated spin systems. 4) Young diagrams classify only fermionic states in quantum chromodynamics. 5) Schur-Weyl duality is irrelevant for tensor product spaces of qubits or qudits. 6) Representation theory of SU(d) is decoupled from permutation symmetries in quantum systems. 7) Schur-Weyl duality relates the representation theory of SU(d) and Sn in the tensor product space of n qudits, enabling efficient decomposition and computation using the Schur basis.

✓ Correct Answer:

The correct answer is 7) Schur-Weyl duality relates the representation theory of SU(d) and Sn in the tensor product space of n qudits, enabling efficient decomposition and computation using the Schur basis..

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that the measurements taken for variationally updating param- eters in Sn-CQA can be eﬃciently calculated on the Schur basis, while a similar conclusion is unlikely to be drawn classically. In the numerical par,t Sec. V, we illustrate the poten- tial of this framework by applying it to the problem of 020327-2 SPEEDING UP LEARNING QUANTUM... PRX QUANTUM 4,020327 (2023) ﬁnding the ground-state energy of J1-J2antiferromagnetic Heisenberg magnets, a gapless system with no known sign structure or analytical solution in quantum many-body the- ory. We compare our model with classical and quantum algorithms like Refs. [ 38,46,53]. We emphasize the conse- quence of the failure of the Marshall-Lieb-Mattis theorem [54–56] in the frustrated region with which classical neu- ral networks struggle to discover the sign structure to an admissible accuracy due to violation of global SU (2)sym- metry [ 57,58]. We include numerical simulation to show the eﬀectiveness of the Sn-CQA ansätze in ﬁnding the ground state with frustration using only O(pn2)param- eters for palternating layers. Noisy simulations are also provided to show the robustness of Sn-CQA. Our theoretical results about SU (d)symmetry can be reversed to exhibit Snpermutation symmetry. We deﬁne SU(d)-CQA on SU (d)-irrep blocks and leave it to future research work to explore its theoretical and experimen- tal potential. All statements and theorems discussed in the main text are proved in full detail in the Appendix. II. BACKGROUND ON REPRESENTATION THEORY OF THE SYMMETRIC GROUP In this section we deﬁne some of the mathematical con- cepts and notations used in the rest of the paper. Further details can be found in Refs. [ 49,50,59]. LetVbe a d-dimensional complex Hilbert space with orthonormal basis {e1,...,ed}. The tensor-product space V⊗nadmits two natural representations: the tensor-product representation πSU(d)of SU(d)acting as πSU(d)(g)(ei1⊗···⊗ ein):=g·ei1⊗···⊗ g·ein, where g·eikis the fundamental representation of SU (d), and the permutation representation πSnofSnacting as πSn(σ)(ei1⊗···⊗ ein):=eiσ−1(1)⊗···⊗ eiσ−1(n). We treat V⊗nas the Hilbert space of an n-qudit system. The so-called Schur-Weyl duality reveals how the above two representations are related. Schur-Weyl duality is widely used in quantum com- puting [ 60], quantum information theory [ 47], and high- energy physics [ 61]. In particular, in quantum chromody- namics it was used to decompose the n-fold tensor product of SU(3)representations. In that context, standard Young tableaux are referred to as Weyl tableaux and labeled by the three isospin numbers ( u,d,s). The underlying Young diagrams containing three rows λ=(λ1,λ2,λ3)are used to denote an SU (d)irreducible representation (irrep). There is another way using Young diagrams λ/prime=λ1−λ3,λ2−λ3) labeled by Dynkin integers via the highest weight vec- tors. In short, there are two conventions in the literature to denote SU (d)irreps. On the other hand, Snirreps can alsoFIG. 1. Young diagrams λ=(6, 3, 1)andλ/prime=(5, 2)represent the same SU (3)-irrep of highest weight (3, 2). However, in the context of Schur-Weyl duality, λcorresponds to an S10irrep whileλ/primegives an S7irrep. be denoted by Young diagrams [ 59]. Schur-Weyl duality says that irreps of SU (d)andSnare dual in the follow- ing sense and denoted by the

Question 392 multiple-choice

In the theory of algebraic groups, Serre’s notion of complete reducibility addresses how subgroups of reductive groups behave with respect to inclusion in parabolic and Levi subgroups. The study of these properties can be significantly affected by the nature of the base field, particularly when the field is non-perfect and separably closed. Which phenomenon demonstrates that the property of complete reducibility for a connected non-abelian subgroup of an exceptional group like F₄ can differ between the group G and the base field k? 1) The existence of an abelian subgroup that is G-completely reducible over both G and k 2) The presence of a finite subgroup that is not G-completely reducible over k 3) A non-connected subgroup being G-completely reducible over G but not over k 4) A connected non-abelian k-subgroup H that is G-completely reducible but not G-completely reducible over k 5) A Levi subgroup containing every subgroup of G 6) The universal complete reducibility of all subgroups over perfect fields 7) The equivalence of complete reducibility for all connected subgroups in F₄ regardless of the base field

✓ Correct Answer:

The correct answer is 4) A connected non-abelian k-subgroup H that is G-completely reducible but not G-completely reducible over k.

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Title: Complete reducibility of subgroups of reductive algebraic groups over non-perfect fields IV: An 𝐹4 example Year: 2020 Paper ID: 169ac0f2f593007c74ddd202e598dac711c23442 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/169ac0f2f593007c74ddd202e598dac711c23442 Abstract: Abstract Let 𝑘 be a non-perfect separably closed field. Let 𝐺 be a connected reductive algebraic group defined over 𝑘. We study rationality problems for Serre’s notion of complete reducibility of subgroups of 𝐺. In particular, we present the first example of a connected non-abelian 𝑘-subgroup 𝐻 of 𝐺 that is 𝐺-completely reducible but not 𝐺-completely reducible over 𝑘, and the first example of a connected non-abelian 𝑘-subgroup H ′ H^{\prime} of 𝐺 that is 𝐺-completely reducible over 𝑘 but not 𝐺-completely reducible. This is new: all previously known such examples are for finite (or non-connected) 𝐻 and H ′ H^{\prime} only.

Question 393 multiple-choice

In finite group theory, the analysis of centralizers, normalizers, and quotient groups is fundamental to understanding the structure and representation of large simple groups, especially those with connections to linear algebra and geometry. Groups like PSL(5,2) and SL3(2) often appear as symmetry groups acting on vector spaces and combinatorial objects. Which of the following is isomorphic to the quotient NG/FCG when F is an elementary abelian subgroup of order 8 and NG denotes its normalizer within a larger group? 1) S4 × (Z2)^3 2) SL4(2) ⋅ (Z3)^2 3) (Z2)^3 ⋅ SL3(2) 4) PSL(5,2) 5) (Z3)^3 × S3 6) GL3(2) × Z2 7) A5 × (Z2)^2

✓ Correct Answer:

The correct answer is 3) (Z2)^3 ⋅ SL3(2).

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Ca(Z0)0 = F, we have Cc;(A) ~ Nc;(F). Moreover NG(F)/FCG(F) = (Z2)3.SL3(2), and Cc(Z0)/ F lies in the normal subgroup (Z2)3 acting regularly on {11, .. ., ls}· It follows from (i) using a Frattini argument that Na(F) = F(NG(A) n Na(F)). ( *) Moreover, F n N0(A) induces S:i on the Klein group A/Z0. Thus there is a subgroup (r, s, t) ~ Na(A) n NG(F) inducing the regular normal subgroup (Z2)·' on {11, .. ., 18}. Now there exist i, j, k, l such that [r, a1 ... as]= (e;ei)'', [r, b1 ... b8] = (eke1)'2, where S1, s2 E {O, 1}. Replacing r by r(b;bi)''(aka1)9', we haver E Cc(A). Similarly we may takes, t E Ca(A), and so we have the subgroup (r, s, t) ~ Ca(A). Recall the set 11?f4-sets defined in Step 2, and that for R = {i, j, k, /} E 11, we set aR = a;aiaka,. Direct calculation gives . CF(A) =(A, e1> aR, bR I RE 11), and hence LOCAL MAXIMAL SUBGROUPS 41 Thus ICa(A)/Al = 210 and Ca(A)/A contains the elementary abelian subgroup CF(A)/A of order 27. We now show that X = Nc(A)f Cc(A) = L5(2). First note that for any a EA \Z0, there is a maximal torus I;, of F, and hence of G, such that (Z0, a)~ I;,. Now Na(T 0)/T0 = W(E8) = 2.0t(2). The non-trivial central ele ment here is Wo, the longest element of W(E8), and the corresponding element nw0 of Na(I;,) inverts every element of T,,. Moreover, A induces (nw0) on T0• Write Q=Q 1(02(T0))=(Z 2)8. Obviously (Z0,a)~Q, and Na(Ta) induces the group Ot(2) acting naturally on Q. Since (Z0, a) is z-pure, it is a totally singular 4-space in Q. Let Y be any 3-space in (Z0, a). Then there exists g E Na(T 0) such that Z~ = Y. Further, Cc(Y)0 = P. Also g normalizes ( T,,, nw0) ( = Ca(Q)), a group containing A, and hence A~ F8. Also A8 ~ F8, and so by the uniqueness of the conjugacy class of A8 in F8 given by (i), there exists f E FE such that A1 = A8. Then g1-I E Nc(A), zrr' = Y. As a was an arbitrary element of A\Z0, we deduce that X = Na(A)/C 0(A) is transitive on the set of 3-spaces in A. Moreover, the element a1a2 lies in N0(A) and induces a transvection in X. Consequently X = L5(2). By ( *) there is an element u E Nc(A) of order 3 which acts on {11, •.. , 18} as a product of two 3-cycles. Write Vi= CF(A)/A, Vi= Cc(A)/CF(A). Elementary calculation shows that dim CVi(u) = 3, dim Cv2(U) = 1, that [Vi, u] is a direct sum of two 2-spaces, and that dim [Vz, u] = 2. Now the non-trivial irreducible modules in characteristic 2 for X = L5(2) of dimension 10 or less are the natural module W of dimension 5, its dual W*, and the skew-squares A2W,

Question 394 multiple-choice

Quantum computing has the potential to revolutionize signal processing by enabling faster algorithms for fundamental operations such as the Discrete Fourier Transform (DFT) and convolution. Classical algorithms like the Fast Fourier Transform (FFT) are widely used, but quantum approaches promise substantial improvements in scalability and efficiency. Which of the following statements accurately describes the reported time complexity improvements of quantum algorithms for the 1D and 2D Discrete Fourier Transform (QDFT) compared to classical FFT algorithms? 1) 1D QDFT achieves O(NlogN) and 2D QDFT achieves O(N^2logN) 2) 1D QDFT achieves O(N^2) and 2D QDFT achieves O 3) 1D QDFT achieves O(sqrt) and 2D QDFT achieves O 4) 1D QDFT achieves O(logN) and 2D QDFT achieves O(NlogN) 5) 1D QDFT achieves O and 2D QDFT achieves O(N^2) 6) 1D QDFT achieves O(N^2logN) and 2D QDFT achieves O(N^3) 7) 1D QDFT achieves O(sqrt(logN)) and 2D QDFT achieves O(sqrt)

✓ Correct Answer:

The correct answer is 3) 1D QDFT achieves O(sqrt) and 2D QDFT achieves O.

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Title: Quantum Discrete Fourier Transform with Classical Output for Signal Processing Year: 2007 Paper ID: 8bcc3a85681b378f6bed8f21bcad18de9a5e04d8 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/8bcc3a85681b378f6bed8f21bcad18de9a5e04d8 Abstract: Discrete Fourier transform (DFT) is the base of modern signal or information processing. 1-Dimensional fast Fourier transform (1D FFT) and 2D FFT have time complexity O(NlogN) and O(N^2logN) respectively. Quantum 1D and 2D DFT algorithms with classical output (1D QDFT and 2D QDFT) are presented in this paper. And quantum algorithm for convolution estimation is also presented in this paper. Compared with FFT, QDFT has two advantages at least. One of advantages is that 1D and 2D QDFT has time complexity O(sqrt(N)) and O(N) respectively. The other advantage is that QDFT can process very long signal sequence at a time. QDFT and quantum convolution demonstrate that quantum signal processing with classical output is possible.

Question 395 multiple-choice

In quantum information theory, the classification of multipartite entangled states often involves advanced mathematical frameworks such as geometric invariant theory (GIT) and considerations of group actions on state spaces. The interplay between group symmetries, stabilizers, and quotient spaces is essential for understanding the structure and dimensionality of locally maximally entangled (LME) states. Which of the following correctly expresses how to compute the dimension of the quotient space of locally maximally entangled states under the action of the group G = SL(d1, C) ×.. × SL(dn, C), taking stabilizer subgroups into account? 1) Add the dimension of G to the dimension of the state space and subtract the dimension of the stabilizer. 2) Subtract the dimension of the state space from the dimension of G and also subtract the dimension of the stabilizer. 3) Use only the dimension of the state space, disregarding both the group and the stabilizer. 4) Add the dimensions of the state space, group, and stabilizer together. 5) Subtract the group dimension from the state space dimension, ignoring the stabilizer. 6) Subtract the group dimension from the state space dimension, then add the stabilizer dimension. 7) Compute the dimension by multiplying the dimensions of the state space and group, then subtracting the stabilizer dimension.

✓ Correct Answer:

The correct answer is 6) Subtract the group dimension from the state space dimension, then add the stabilizer dimension..

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function of kand check whether it is ever non-zero. Unfortunately, this turns out to be computationally hard for all but the smallest dimensions. Thus, we will instead make use of algebraic methods, to which which we turn in the next section. 5 Understanding SLMEusing geometric invariant the- ory In the previous section, we have reviewed how the space SLME/Kof LME states up to local unitary transformations is equivalent to the geometric invaria nt theory (GIT) quo- tientP(H)//Gof the full space of states by the group of local determinant-one invertible transformations G= SL(d1,C)×···×SL(dn,C). Starting from the latter description, it is possible to use the machine ry of geometric in- variant theory to characterize the space, providing explicit result s that tell us for which (d1,d2,...,dn) LME states exist, and give the dimension of the space in all nonempt y cases. Our rigorous results characterizing the GIT quotient SLME/Kappear in a companion math- ematics paper [BRV]. In this section, our goal is to present an overv iew of the results and their derivation. 31 Dimensionality of P(H)//G To understand the dimension of the quotient P(H)//G, we note that the original space P(H) =CPd1···dn−1has complex dimension d1d2···dn−1 while the group G= SL(d1,C)× ···×SL(dn,C) has complex dimension/summationtext i(d2 i−1). The latter is naively the dimension of a typicalG-orbit, so the dimension of P(H)//G, the space of these orbits, is naively the diﬀerence dim(P(H)//G)naive= dim(P(H))−dim(G) =/parenleftBiggn/productdisplay i=1di−1/parenrightBigg −n/summationdisplay i=1(d2 i−1)≡∆(d1,...dn). (5.1) This naive dimension can fail to be correct, however, if a typical poin t inHis invariant under some subgroup of Gwith positive dimension. For any point ψ∈ H, we deﬁne Sψ⊂Gto be the subgroup of Gthat leaves ψinvariant. This is called the stabilizer at position ψ. The dimension of Sψis equal to the dimension of the subspace of inﬁnitesimal transformations in the Lie algebra of Gthat leaves ψinvariant. As we review in [BRV, §3], there always exists some subgroup S⊂Gand a dense open subset ofU⊂ Hsuch that the stabilizer at any point in Uis conjugate to S. Moreover, dim(P(H)//G) = dim(P(H))−dim(G)+dim(S). (5.2) A key point here is that the group G= SL(d1,C)× ··· ×SL(dn,C) is semisimple.19If this dimension is −1, this means that a neighborhood of a generic point in the space His contained within the G-orbit of this point. In this case, it follows that the quotient is empty , since a point λxforλ<1 will then be in the G-orbit ofx, and by iterating the group action that gives this point, we will end up arbitrarily close to 0. Thus, the or bit of a generic point is unstable. For speciﬁc examples in which the dimensions are not too large, it is str aightforward to directly calculate dim( S) and thus the dimension of the quotient. For some ﬁxed point ψ, an element g=g1⊗1 1⊗···⊗1 1+···+1 1⊗···⊗1 1⊗gn (5.3) of the Lie algebra of G(wheregiis adi×ditraceless matrix) will be a generator of the stabilizer group if and only if g·ψ= 0. This gives dim Hlinear equations for the dim G variables (the elements

Question 396 multiple-choice

Quantum algorithms often leverage group-theoretic structures to solve problems like factoring and finding discrete logarithms. The Hidden Subgroup Problem (HSP) is central in this area, with the efficiency of algorithms measured relative to the group's size and representation. Which property enables large groups to be efficiently described and manipulated in the context of the Hidden Subgroup Problem? 1) Groups can always be decomposed into direct products of cyclic groups. 2) Any group can be specified by O(log|G|) generators, where |G| is the group's size. 3) Every group element can be uniquely encoded as a binary string. 4) Group operations are always efficiently computable without oracles. 5) All subgroups of a group are visible without the need for a function oracle. 6) The automorphism group of every group is always abelian. 7) Every group admits a semi-direct product decomposition.

✓ Correct Answer:

The correct answer is 2) Any group can be specified by O(log|G|) generators, where |G| is the group's size..

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•a groupGgiven as a set of generators, and •a function fgiven as an oracle, which is H-periodic for an unknown subgroup HofG, output a set of generators for H. Notice that any group Gcan be represented by a set of at most O(log|G|) generators, where |G|is the number of elements of G. We thus say that an algorithm solves the HSP over Gin polynomial time if it runs in time polynomial in log |G|. 2.2 Semi-direct product groups We now deﬁne the class of semi-direct product groups of cyclic grou ps. Deﬁnition 3. For any positive integers nandq, and any group homomorphism φfrom the group Zqinto the group of automorphisms of Zn, the semi-direct product group Zn⋊φZqis the set {(a,b)|a∈Zn,b∈Zq} with the group product (a1,b1)(a2,b2) := (a1+φ(b1)(a2),b1+b2). Because φhas to be a homomorphism and φ(a) must be an automorphism for every a∈Zq,φis completely deﬁned by setting φ(1)(1). The group Zn⋊φZqis generated by the two elements x= (1,0) andy= (0,1). Using the fact that φ(b)(a) =aφ(1)(1)b, we obtain the relation ybxa=xaφ(1)(1)byb, which will be used in almost all the group computations in this paper. 2.3 Black-box groups We will mainly consider the case where the group Gis input as a black-box group. A black-box group is a representation of a group where elements are represented by strings (of the same length). An oracle that performs the group product is available: given two strings rep resenting two elements aandb, the oracle outputs the string representing a·b. Moreover, we have another oracle that, given a string represen ting an elementa, computes a string representingthe inverse a−1. We will in Section 4 consider the most general case where the elements are not uniquely encoded. In this case an oracle is provided to check whether two strings represent the same element. We refer the reader to Babai and Sz emer´ edi [4] for the complete deﬁnition of black-box groups. In the quantum computation setting, the oracles have to be able to deal with quantum superpositions. These quantum black-box groups have been studied by Ivanyos, M agniez and Santha [13] and Watrous [27, 28]. The concept is the same as above but the oracles realizing gr oup operations are quantum. More precisely, we suppose that two oracles VGandV′ Gare available, such that VG(|g/an}bracketri}ht|h/an}bracketri}ht) =|g/an}bracketri}ht|gh/an}bracketri}ht V′ G(|g/an}bracketri}ht|h/an}bracketri}ht) =|g/an}bracketri}ht|g−1h/an}bracketri}ht for anygandhinG. In the case of a quantum back-box group with not necessarily uniq ue encoding, we suppose that the oracle checking whether two strings represent the same element is a quantum oracle too, although a classical oracle (i.e. an oracle not dealing with quantum sup erpositions) is actually suﬃcient for the algorithms in this paper. 3 Notice that any eﬃcient black-box algorithm gives rise to an eﬃcient a lgorithm whenever the oracle oper- ation can be replaced by eﬃcient procedures. Especially, when a mat hematical expression of the generators input to the algorithm is known, performing group operations can be done directly on the elements in poly- nomial time (in log |G|) for almost all natural groups, including

Question 397 multiple-choice

Quantum complexity theory explores the relationships between computational problems and quantum computation, often using mathematical structures such as braids and knot invariants. The complexity class QCMA consists of decision problems where a quantum verifier can efficiently check a classical proof. Which of the following statements correctly characterizes a QCMA-complete problem involving the Jones polynomial and plat closures of braids? 1) It asks whether the Jones polynomial of any braid's plat closure is exactly zero. 2) It requires finding a quantum circuit whose matrix element matches a given constant. 3) It asks, given a set of braids and a fixed integer, whether there exists a braid in the set whose normalized Jones polynomial of the plat closure is at least 3/4, or if it is at most 1/4 for all braids in the set. 4) It determines if a quantum witness can be efficiently verified by a classical circuit. 5) It involves checking whether every braid in a class has the same Jones polynomial value. 6) It asks whether the plat closure of a braid is topologically trivial. 7) It requires constructing a braid whose closure realizes the maximal possible Jones polynomial value.

✓ Correct Answer:

The correct answer is 3) It asks, given a set of braids and a fixed integer, whether there exists a braid in the set whose normalized Jones polynomial of the plat closure is at least 3/4, or if it is at most 1/4 for all braids in the set..

📚 Reference Text:

deﬁnition, Uis of length m=O/parenleftbig poly(n)/parenrightbig and acts on n′=O/parenleftbig poly(n)/parenrightbig qubits. For each xn, deﬁne the circuit Wxn=/circlemultiplytextn i=1Xxi, which acts as the identity on the ancillary qubits. Because /an}bracketle{t00···0|WxnUWxn|00···0/an}bracketri}ht=/an}bracketle{txn00···0|U|xn00···0/an}bracketri}ht, it follows by Theorem 7.1 that for any constant ǫ>0, there is a braid bof lengthO/parenleftbig poly(m)/parenrightbig = O/parenleftbig poly(n)/parenrightbig on 4n′strands which satisﬁes /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 [2]2n′−1 ℓ/vextendsingle/vextendsingle/vextendsingleJ(/tildewideb,e2πi/ℓ)/vextendsingle/vextendsingle/vextendsingle2 − |/an}bracketle{txn00···0|U|xn00···0/an}bracketri}ht|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ǫ. Moreover, there is an O/parenleftbig poly(n)/parenrightbig -time classical algorithm for determining such a braid from a description of WxnUWxn. By invoking the oracle A, we obtain a random variable Zsatisfying Pr/braceleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleZ−1 [2]2n′−1 ℓ/vextendsingle/vextendsingle/vextendsingleJ(/tildewideb,e2πi/ℓ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<1/8/bracerightBigg >3/4. Since |a2−b2| ≤2|a−b|for 0≤a,b≤1, the last two estimates can be combined via the triangle inequality to show that/vextendsingle/vextendsingleZ2− |/an}bracketle{txn00···0|U|xn00···0/an}bracketri}ht|2/vextendsingle/vextendsingle<1/4 +ǫwith probability at least 3/4. Choosing ǫ= 1/8 is suﬃcient to complete the proof. /square The complexity class QCMA is a certain quantum analog of NP, c onsisting of those lan- guagesLfor which a computationally unbounded oracle (generally pe rsoniﬁed as Merlin) can, with high probability, eﬃciently convince a computational ly bounded veriﬁer (Arthur) that eachxn∈Lis actually contained in L. Cheating is also discouraged, in that if xn/∈L, the probability that Merlin will convince Arthur otherwise should be small. Formally, we say that a language Lis in QCMA if there is a uniform family of quantum circuits {Un} of polynomial length such that for each length nstringxn∈L, there is a length mbinary 28The Jones Polynomial: Quantum Algorithms and Applications in Quantum Complexity Theory stringymfor whichP(xn,ym)≡ |/an}bracketle{txnym00···0|Un|xnym00···0/an}bracketri}ht|2≥3/4, while ifxn/∈L, we haveP(xn,ym)≤1/4 for every such ym. We remark that the usual deﬁnition of QCMA (see e.g. [3] for the original deﬁnition) is phrased in terms of measurement probabilities of a single output qubit; by the remarks following the deﬁnition above of BQP, the deﬁnition we give here is equivalent. For the same reasons as with BQP, the exact value 3 /4 of the success probability is not important; what is important is that the s quare of the matrix element is strictly bounded away from 1/2. Given a complexity class C, a problem is said to be C- hardif it is at least as hard as any other problem in the class C, in the sense that access to an oracle which immediately computes the solution of that problem allows the solution of any problem in the class C in polynomial-time. If that problem is also contained in the cl ass C, we say that it is C- complete . We will show that the following problem is QCMA-complete. Problem 8.1 (Increase Jones Plat) Given a braid b∈B2nwithO/parenleftbig poly(n)/parenrightbig crossings, a ﬁxed integer ℓ≥5,ℓ/ne}ationslash= 6, and a class of braids C2n⊂B2nof lengthO/parenleftbig poly(n)/parenrightbig for which membership can be decided in O/parenleftbig poly(n)/parenrightbig time on a classical computer, decide, with the promise that only these two cases can occur, whether ther e exists another braid c∈ C2n for which the absolute value of the normalized Jones polynom ial of the plat closure 1 [2]n−1 ℓ/vextendsingle/vextendsingle/vextendsingleJ(/tildewidercbc−1,e2πi/ℓ)/vextendsingle/vextendsingle/vextendsingle2 ≥3/4 or if it is ≤1/4for all braids c∈ C2n. Theorem 8.1 Increase Jones Plat

Question 398 multiple-choice

In module theory and the study of abelian groups, quasi-injective modules are those where every homomorphism from a submodule to the module extends to an endomorphism. The structure of divisible groups and their fully invariant subgroups plays a critical role in understanding quasi-injectivity. Which statement best describes the fully invariant subgroups of a torsion-free divisible abelian group? 1) They are precisely all subgroups generated by finite sets of elements. 2) They include all subgroups that contain the torsion subgroup. 3) They are exactly the subgroups isomorphic to direct sums of cyclic groups. 4) They consist of all subgroups invariant under automorphisms. 5) They are limited to the zero subgroup and the entire group itself. 6) They are characterized as subgroups that are also injective modules. 7) They include all subgroups whose elements are divisible by a fixed integer.

✓ Correct Answer:

The correct answer is 5) They are limited to the zero subgroup and the entire group itself..

📚 Reference Text:

Ga) = 0 for each a. Then G is quasi-infective if and only if each Ga is quasi-infective. 3. Divisible groups. Divisible groups are quasi-injective almost by accident, since their fully invariant subgroups are so easily identifiable. We first reduce to the torsion case with LEMMA 3.1. If G is divisible and H is fully invariant in G, then either H = G or H is a fully invariant subgroup of the torsion subgroup T of G. Proof. If H is contained in T then H is clearly fully invariant in T. Suppose then that H0 = H + T 9 e T. We show that H = G. Since every endomor- phism of G/T is induced by an endomorphism of G, and since H0 is a fully invariant subgroup of G properly containing T, H 0/T is a non-zero fully invariant subgroup of the torsion free divisible group G/T. Thus H0/T = G/T, or Ho = G, and the natural map v of G onto G/T induces an epimorphism of H onto G/T. Now choose an arbitrary summand Z(p°°) of T. Let ir be a projection of G/T onto a copy of Q and let p be a map of Q onto the chosen Z(pœ). Then pwv is an endomorphism of G which maps H onto Z(pœ). Since H is fully invariant in G, this yields Z(pœ) ^ H. Since Z(pœ) was an arbitrary summand of T, T ^ H. Then G = H+T = Has required. THEOREM 3.2. Every divisible group G is quasi-injective over its endomorphism ring. Proof. Let G be divisible with torsion subgroup T, fully invariant subgroup H, and let/ G Hom Ê((?) (77, G). If H = G, there is nothing to prove. Otherwise, by the lemma above, H is a fully invariant subgroup of T. Since every endo morphism of T is induced by an endomorphism of G we have, with a slight abuse of language, / G Hom^r) (H, T). It suffices now to show that T is quasi-injective over E(T) and by Corollary 2.4 we may assume T is a ^>-group. Since T is Z-injective, restriction of elements of E(T) to act in H is an epimor phism of E{T) onto E(H). Since the only fully invariant subgroups of a divisible ^-group Tare T itself, 0, and the /^-layers T[p n] = {x G T|^x = 0}, Hom z(i7, T) = E(H). Now clearly with these identifications, Hom E(T)(H, T) is contained in the center of E(H). But the center of E(H) is Z/pnZ if H is TT^J (and, otherwise, H = T) so each map in the center of E(H) is induced by multiplication by some p-adic integer, i.e., by some element of the center olE(T). For the next corollary, we use the term homogeneous p-group to denote a ^-group which is the direct sum of copies of Z/pnZ for some fixed n. COROLLARY 3.3. Homogeneous p-group s are quasi-injective. Proof. Such a group is fully invariant in its divisible

Question 399 multiple-choice

Quantum computing leverages superposition and interference to solve certain computational problems exponentially faster than classical algorithms. One foundational area involves the study of quantum algorithms for hidden structure problems across algebraic domains. Which of the following statements most accurately describes the generalization from the Hidden Subgroup Problem (HSP) to the Hidden Kernel Problem (HKP) in universal algebra? 1) HKP replaces group homomorphisms with lattice isomorphisms and seeks hidden automorphisms. 2) HKP restricts HSP to only solvable groups and studies their quotient rings. 3) HKP extends HSP by focusing on finding maximal ideals in commutative rings via quantum algorithms. 4) HKP transforms HSP by substituting group operations with Boolean functions and searching for fixed points. 5) HKP generalizes HSP by replacing subgroups with congruences, asking for the kernel of an oracle homomorphism between algebras. 6) HKP limits HSP to abelian groups and investigates the complexity of their coset representatives. 7) HKP modifies HSP by analyzing the spectral properties of group representations in non-abelian settings.

✓ Correct Answer:

The correct answer is 5) HKP generalizes HSP by replacing subgroups with congruences, asking for the kernel of an oracle homomorphism between algebras..

📚 Reference Text:

quantum gate which we will make use of is the Hadamard gate H, deﬁned by its action on the basis vectorsofBby H|0/a\}⌊ra⌋ketri}ht=1√ 2/parenleftbig|0/a\}⌊ra⌋ketri}ht+|1/a\}⌊ra⌋ketri}ht/parenrightbig,H|1/a\}⌊ra⌋ketri}ht=1√ 2/parenleftbig|0/a\}⌊ra⌋ketri}ht − |1/a\}⌊ra⌋ketri}ht/parenrightbig. Aconvenient formulafortheactionof H⊗nonanarbitrary n-qubitstate is H⊗n|x/a\}⌊ra⌋ketri}ht=1 2n/2/summationdisplay.1 a∈{0,1}n(−1)a·x|a/a\}⌊ra⌋ketri}ht, wherea·xisthedotproductof aandxmodulo2(regarded asZ2-vectors). 3 The HiddenKernelProblem Asdescribedin theintroduction,givenagroup G,thefunc- tionf:G→Xhidesa subgroupD≤Giff(/afii10069.ital)=f(h)if andonly if /afii10069.italD=hD. LICS’20,July8–11,2020,Saarbrücken,Germany Ma/t_thew Moore and TaylorWalenczyk /logicalandtext.1 /logicalortext.1 MPT∞ 0MPT∞ 1⊤ ⊥UA AP0 ≈Z2 APDMCloneGenerating Operations ⊥ ∅/logicalandtext.1x∧/y.alt, 0,1/logicalortext.1x∨/y.alt, 0,1 U ¬x,0 MPT∞ 0x∨(/y.alt∧z) MPT∞ 1x∧(/y.alt∨z) AP x+/y.alt+z AP0x+/y.alt Ax↔/y.alt, 0 DM maj(x,/y.alt,z) ⊤x∨/y.alt,¬x Figure1. DiagramofPost’sLattice,thelatticeofallclonesona2-ele mentdomain(i.e.booleanclones).Cloneswhichwillbe relevantlaterhavebeenlabelled,andoperationsgeneratingeac hofthemaregiveninthetabletotheright.Simon’sProblem correspondstothe HKPon the operationalclone ofthe group Z2,namelyAP0. HiddenSubgroupProblem(HSP) Input: groupG, oraclef:G→Xhiding some subgroup Task:determine the subgroup D≤Gthatfhides Aquantumpolynomial-timesolutiontotheHSPfor abelian Gisknown[ 11],andthiscasecontainsmanyfamousquan- tum algorithms exhibiting super-polynomial speedup over classicalsolutions.TheHSPfor non-abelianGhasbeensolved insomequitespeciﬁccasesandrestrictivesettings,butitre - mainsopenin general.Inspite offocusedresearch,no gen- eral polynomial-time quantum algorithm for the HSP has beenfound. Thatthe HSP hasbeeneﬃciently solved fora largeclass of groups (abelian ones) and an irregular constellation of groupsotherwise suggeststwo possibilities. 1. Thereisnoquantumpolynomial-timesolutiontothegen- eralHSP.Itissuspectedthatquantumalgorithmscannot solveNP-complete problems in polynomial time, and it maybe the casethat the generalHSP is NP-complete . 2. There is a quantum polynomial-time solution, but it re- quires a deeper insight into the problem than has been obtained over thepast 30years. Ineitherofthesecases,largeadditionalclassesofstructu res forwhichtheHSPhasaneﬃcientsolutionwouldbeincredi- blyvaluable.Tothisend,weproposeageneralizationoftheHSPtouniversalalgebraicstructures.Suchageneralization will extend the domain of the problem to classes of struc- tureswith additional gradationsof complexity,with the ex- pectationthatthesethesegradationswillbereﬂectedinthe complexityofthe generalized problem. 3.1 The HiddenKernelProblem TheHidden Subgroup Problem canbe framedas aproblem aboutthe quotientstructure inducedbythehiddensubgroup D,namelyG/D(thesetofcosetsof DinG).Inthisformula- tion, we are given Gand abijectivefunctionf:G/D→X as an oracle, and tasked with computing D≤G. The set G/Ddoesnothave group structure itself unless Dis a nor- mal subgroup. The progress which has been made on the general HSP begins with a deep analysis of what limited structureG/Ddoeshave forcertain classes ofgroups. Forabeliangroups,everysubgroupisnormal,sothestruc- ture ofG/Dis quite uniform. Restricting attention to only thenormalsubgroups yields the Hidden Normal Subgroup Problem. It is this problem which we generalize to obtain theHiddenKernel Problem . LetAbe an algebra and θa congruence ofA. Following thedeﬁnitionofahiddensubgroup,thefunction f:A→X hidesthecongruence θiff(a)=f(b)precisely when aθ b. The Hidden SubgroupProblem forUniversalAlgebras LICS’20,July8–11,2020,Saarbrücken,Germany TheHiddenKernelProblem(preliminaryversion) Input: algebraA, oraclef:A→Xhiding somecongruence Task:determine the congruence θofAthatfhides It is not hard to show that a function f:A→Xhides a congruence θif and only if fis a homomorphism with θ=ker(f).Thisobservation leadstothe generalstatement ofthe Hidden Kernel Problem. The HiddenKernel Problem(HKP) Input: similar algebrasAandB, homomorphism φ:A→B(asan oracle) Task:determine the congruenceker (φ) This generalization includes and extends the Hidden Nor- mal Subgroup Problem, and abstracts away the somewhat unnatural notion of“hiding”. 3.2 Simon’sProblemand Post’sLattice PerhapsthesimplestinstanceoftheHiddenSubgroupProb- lem is when the group is an elementary abelian 2-group, G=Zn 2. This instance of the HSP is known as Simon’s Problem[19], and historically is one of the ﬁrst examples of super-polynomial speedup when using a quantum algo- rithm. WewillconsiderauniversalalgebraicversionofSimon’s Problem — The Hidden Kernel Problem on A=Bn, where Bis a 2-element algebra. There are inﬁnitely many such B, andapriorithereseemstobeneitheranaturalstartingpoint nor a systematic way in which to proceed. One crucial ob- servation is the following. Observation 3.1. LetBandDbe universal algebras such thatB=Dand Clo(B) ⊆Clo(D). Each speciﬁc instance of HKP(Dn)is a speciﬁc instance of HKP (Bn). A solution to allinstancesofHKP (Bn)yieldsasolutiontoallinstancesof HKP(Dn). This observation

Question 400 multiple-choice

Finite p-groups are algebraic structures in group theory where the group's order is a power of a prime, and their classification reveals deep properties of group structure. The Frattini subgroup and the center play pivotal roles in understanding the constraints on possible groups generated by a limited number of elements. In the classification of three-generator finite p-groups where the Frattini subgroup Φ is contained in the center Z, what is the significance of this containment on the group's structure? 1) All non-generators commute with every element, restricting the possible group forms. 2) The group must be simple and non-abelian. 3) Every subgroup of the group is cyclic. 4) The group is necessarily abelian. 5) The group has a unique minimal normal subgroup of order p. 6) The commutator subgroup is trivial. 7) The group admits no non-trivial automorphisms.

✓ Correct Answer:

The correct answer is 1) All non-generators commute with every element, restricting the possible group forms..

📚 Reference Text:

Title: Finite p-groups with a minimal non-abelian subgroup of index p (III) Year: 2013 Paper ID: 3bb490382db9082487da240ab9c75eb21e98f0c5 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/3bb490382db9082487da240ab9c75eb21e98f0c5 Abstract: This paper finishes the classification of three-generator finite p-groups G such that Φ(G) ⩽ Z(G). This paper is a part of classification of finite p-groups with a minimal non-abelian subgroup of index p, and partly solves a problem proposed by Berkovich (2008).

Question 401 multiple-choice

Shor’s algorithm is a quantum algorithm for efficiently factoring large composite numbers, which relies on quantum principles to find the period of modular exponentiation functions. Understanding the role of quantum and classical components is key to grasping its effectiveness and implications for cryptography. Which step in Shor’s algorithm is specifically responsible for revealing the period information encoded in the quantum state, enabling classical extraction of factors of N? 1) Initial application of Hadamard gates to the top register 2) Selection of an integer a that is coprime to N 3) Quantum Fourier Transform applied to the top register 4) Measurement of the bottom register after modular exponentiation 5) Use of the Euclidean algorithm to process measurement outcomes 6) Initialization of the quantum computer with two registers 7) Repeating the algorithm if an unhelpful measurement result occurs

✓ Correct Answer:

The correct answer is 3) Quantum Fourier Transform applied to the top register.

📚 Reference Text:

to present the summary of the heart of Shor’s algorithm. The corresponding quantum circuit is displayed in Figure 3.2. Figure 3.2: Quantum Circuit for Period Finding j0 mi H m UfQFT NM j0 ni 29 1. Initialize a quantum computer in a state jy0i=j0 mij0 ni (3.24) where N22m<2N2;n=dlog2Ne 2. Prepare a superposition jy1i=1p 2m2m1 å x=0jxij0i (3.25) by applying mHadamard transforms to the top register. 3. Choose an integer asuch that 1

Question 402 multiple-choice

Quantum groups, hyperbolic geometry, and stochastic processes are deeply connected through mathematical constructs such as Uq(sl2), curvature parameters, and Markov transforms. The interplay between representation theory, random walks, and geometric limits reveals unifying principles in modern mathematics. In the unified framework involving the quantum group Uq^ħ(sl2), which explicit parameter relationship links the quantum deformation parameter q to the geometric curvature r? 1) q = r^2 2) q = e^{-r} 3) q = 1 / r 4) q = ħ * r 5) q = e^{r} 6) q = r / ħ 7) q = sqrt(r)

✓ Correct Answer:

The correct answer is 2) q = e^{-r}.

📚 Reference Text:

Title: Quantum SL2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ SL _2$$\end{document}, infinite curvature and Pitman’s 2M- Year: 2019 Paper ID: 168ff33fb03d18b7c837632ec9430aa0bb41eb4d Source: semantic-scholar URL: https://www.semanticscholar.org/paper/168ff33fb03d18b7c837632ec9430aa0bb41eb4d Abstract: The classical theorem by Pitman states that a Brownian motion minus twice its running infimum enjoys the Markov property. On the one hand, Biane understood that Pitman’s theorem is intimately related to the representation theory of the quantum group Uqsl2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {U}}}_q\left( {\mathfrak sl}_2 \right) $$\end{document}, in the so-called crystal regime q→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \rightarrow 0$$\end{document}. On the other hand, Bougerol and Jeulin showed the appearance of exactly the same Pitman transform in the infinite curvature limit r→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \rightarrow \infty $$\end{document} of a Brownian motion on the hyperbolic space H3=SL2(C)/SU2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {H}}}^3 = SL _2({{\mathbb {C}}})/ SU _2$$\end{document}. This paper aims at understanding this phenomenon by giving a unifying point of view. In order to do so, we exhibit a presentation Uqħsl2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {U}}}_q^\hbar \left( {\mathfrak sl}_2 \right) $$\end{document} of the Jimbo–Drinfeld quantum group which isolates the role of curvature r and that of the Planck constant ħ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar $$\end{document}. The simple relationship between parameters is q=e-r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=e^{-r}$$\end{document}. The semi-classical limits ħ→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar \rightarrow 0$$\end{document} are the Poisson–Lie groups dual to SL2(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ SL _2({{\mathbb {C}}})$$\end{document} with varying curvatures r∈R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \in {{\mathbb {R}}}_+$$\end{document}. We also construct classical and quantum random walks, drawing a full picture which includes Biane’s quantum walks and the construction of Bougerol–Jeulin. Taking the curvature parameter r to infinity leads indeed to the crystal regime at the level of representation theory (ħ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar >0$$\end{document}) and to the Bougerol–Jeulin construction in the classical world (ħ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar =0$$\end{document}). All these results are neatly in accordance with the philosophy of Kirillov’s orbit method.

Question 403 multiple-choice

In supersymmetric extensions of the Standard Model, hidden sectors with their own gauge groups can have significant cosmological implications, particularly when the lightest MSSM superpartner decays into hidden states. The presence of connector fields charged under both visible and hidden gauge groups enables interactions across these sectors. Which mechanism allows the lightest MSSM neutralino to decay into hidden sector particles, thereby addressing the moduli relic problem in supersymmetric models? 1) Kinetic mixing between the visible and hidden sector U(1) gauge bosons 2) Spontaneous breaking of the hidden SUₓ gauge symmetry 3) Integrating out heavy connector fields charged under both MSSM and hidden SUₓ gauge groups 4) Direct coupling of MSSM Higgs bosons to hidden sector scalars 5) Non-renormalizable gravitational interactions only 6) Anomaly-induced decays via Standard Model gauge bosons 7) Loop-induced flavor-changing neutral currents

✓ Correct Answer:

The correct answer is 3) Integrating out heavy connector fields charged under both MSSM and hidden SUₓ gauge groups.

📚 Reference Text:

F. ForF m3=2, the coecient rxwill be unchanged [26]. However, when F.m3=2, the value of rxcan be modi ed by an amount of order unity that depends on the soft masses of these states [30, 76]. We consider deviations in rxaway from the AMSB value but still of the same general size. Below the hidden gluino mass, the hidden sector is a pure SU(N)xgauge theory. It is therefore guaranteed to be asymptotically free, and the low-energy theory of hidden gluons should undergo a con ning transition at some energy scale xto a theory of massive glueball (and glueballino) bound states. The one-loop estimate of the con nement scale gives x=Mxexp 3rx 22Nm3=2 Mx : (48) Demanding that the SU(N)xgluino be lighter than the lightest MSSM neutralino typically forces xto be very small. For example, setting Mx<1000 GeV, rx= 3N, and requiring thatMx

Question 404 multiple-choice

In polynomial optimization problems involving symmetric polynomials, exploiting underlying algebraic structure and symmetry is essential for reducing computational complexity, especially in semidefinite programming (SDP) relaxations. Mathematical tools such as Gröbner bases, Veronese varieties, and symmetry constraints play a critical role in constructing efficient moment matrix representations. When optimizing symmetric polynomials for n = 3 variables using a second-level SDP relaxation, which mathematical property allows the reduction of the moment matrix dimension from 13 to 6, resulting in an exact relaxation at this level? 1) The identification of a reduced Veronese basis by systematically eliminating redundant monomials due to symmetry 2) The use of duality in linear programming to bound the objective function values 3) The introduction of additional slack variables to enforce convexity 4) Applying Lagrange multipliers to enforce equality constraints among variables 5) The computation of Hessian matrices to assess local optimality of solutions 6) The application of spectral decomposition to diagonalize the moment matrix 7) The use of combinatorial enumeration of all possible monomial terms of degree two

✓ Correct Answer:

The correct answer is 1) The identification of a reduced Veronese basis by systematically eliminating redundant monomials due to symmetry.

📚 Reference Text:

s2 12s3 12s2 12s13s2 12s23s4 12s3 12s13s3 12s23s2 12s13s12s2 12s2 13s2 12s13s23s2 12s23s12s2 12s23s13s2 12s2 23 s13s12s13s2 12s13s12s13s13s12s23s13s3 12s13s2 12s13s13s2 12s23s13s12s13s12s13s12s2 13s13s12s13s23s13s12s23s12s13s12s23s13s13s12s2 23 s23s12s23s2 12s23s12s13s23s12s23s23s3 12s23s2 12s13s23s2 12s23s23s12s13s12s23s12s2 13s23s12s13s23s23s12s23s12s23s12s23s13s23s12s2 23 s12s13s12s13s12s12s2 13s12s13s23s12s13s2 12s12s13s12s13s12s13s12s23s12s2 13s12s12s3 13s12s2 13s23s12s13s23s12s12s13s23s13s12s13s2 23 s2 13s2 13s12s3 13s2 13s23s2 13s2 12s2 13s12s13s2 13s12s23s3 13s12s4 13s3 13s23s2 13s23s12s2 13s23s13s2 13s2 23 s23s13s23s13s12s23s2 13s23s13s23s23s13s2 12s23s13s12s13s23s13s12s23s23s2 13s12s23s3 13s23s2 13s23s23s13s23s12s23s13s23s13s23s13s2 23 s12s23s12s23s12s12s23s13s12s2 23s12s23s2 12s12s23s12s13s12s23s12s23s12s23s13s12s12s23s2 13s12s23s13s23s12s2 23s12s12s2 23s13s12s3 23 s13s23s13s23s12s13s23s13s13s2 23s13s23s2 12s13s23s12s13s13s23s12s23s13s23s13s12s13s23s2 13s13s23s13s23s13s2 23s12s13s2 23s13s13s3 23 s2 23s2 23s12s2 23s13s3 23s2 23s2 12s2 23s12s13s2 23s12s23s2 23s13s12s2 23s2 13s2 23s13s23s3 23s12s3 23s13s4 23  leading to the second relaxation which would be a 13 ×13 SDP. However, the Veronese V2(3) has redundancies. For instance, s2 12= 1, etc. Eliminating redundancies (more precisely, picking a basis for the image of R⟨s⟩2inR⟨s⟩/IS3) leads to the reduced Veronese, which we shall by abuse of notation still call V2(3): V2(3) = (1, s12, s13, s23, s12s13, s12s23)∗. This yields a significantly smaller moment matrix pattern, M3 2= 1s12s13s23s12s13s12s23 s12s2 12s12s13s12s23s2 12s13s2 12s23 s13s13s12s2 13s13s23s13s12s13s13s12s23 s23s23s12s23s13s2 23s23s12s13s23s12s23 s13s12s13s2 12s13s12s13s13s12s23s13s2 12s13s13s2 12s23 s23s12s23s2 12s23s12s13s23s12s23s23s2 12s13s23s2 12s23.  Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 83 Applying the replacement rules of Example 4.8 simplifies this matrix further, M3 2= 1s12s13s23s12s13s12s23 s12 1s12s13s12s23s13s23 s13s12s23 1s12s13s23s12 s23s12s13s12s23 1s12s13 s12s23s13s23s12 1s12s13 s12s13s23s12s13s12s23 1  We thus obtain a smaller second level hierarchy SDP:ν2(h) = maxℓ12+ℓ13+ℓ23 s.t. 1ℓ12ℓ13ℓ23ℓ12,13ℓ12ℓ23 ℓ12 1ℓ12,13ℓ12,23ℓ13ℓ23 ℓ13ℓ12,23 1ℓ12,13ℓ23ℓ12 ℓ23ℓ12,13ℓ12,23 1ℓ12ℓ13 ℓ12,23ℓ13ℓ23ℓ12 1ℓ12ℓ13 ℓ12,13ℓ23ℓ12ℓ13ℓ12,23 1 ⪰0.(D.1) Since the matrix in Eq. (D.1) must be symmetric, we obtain ℓ12,23=ℓ12,13, reducing the number of unknowns by one,ν2(h) = maxℓ12+ℓ13+ℓ23 s.t. 1ℓ12ℓ13ℓ23ℓ12,13ℓ12ℓ23 ℓ12 1ℓ12,13ℓ12,13ℓ13ℓ23 ℓ13ℓ12,13 1ℓ12,13ℓ23ℓ12 ℓ23ℓ12,13ℓ12,13 1ℓ12ℓ13 ℓ12,13ℓ13ℓ23ℓ12 1ℓ12ℓ13 ℓ12,13ℓ23ℓ12ℓ13ℓ12,13 1 ⪰0 = 3.(D.2) Finally, since dimC[S3] = 6 is equal to the size of the reduced Veronese, the second relaxationν2(h) is automatically exact. Example D.1. Now consider n= 4, and I=IS4. In this case the replacement rules are9 s2 12→1, s13s12→s12s23, s2 13→1, s13s23→s12s13, s14s12→s12s24, s14s13→s13s34, s2 14→1, s14s24→s12s14, s14s34→s13s14, s23s12→s12s13, s23s13→s12s23, s23s14→s14s23, s2 23→1, s24s12→s12s14, s24s13→s13s24, s24s14→s12s24, s24s23→s23s34, s2 24→1, s24s34→s23s24, s34s12→s12s34, s34s13→s13s14, s34s14→s13s34, s34s23→s23s24, s34s24→s23s34, s2 34→1. (D.3) 9In order to obtain a Gr¨ obner basis (cf. Section 5) for the grlex order w.r.t. s12< s13< s14< s23< s24< s34, one needs to add three additional replacement rules to the ones given in Eq. (D.3), namely s13s14s23→s12s13s14, s14s23s24→s12s13s14, s14s23s34→s12s14s23. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 84 ThenV1(4) = (1,s12,s13,s14,s23,s24,s34)∗, and M4 1= 1s12s13s14s23s24 s12 1s12s13s12s14s12s23s12s24 s13s12s23 1s13s14s12s13s13s24 s14s12s24s13s34 1s14s23s12s14 s23s12s13s12s23s14s23 1s23s24 s24s12s14s13s24s12s24s23s34 1  leading to the SDP constraint  1ℓ12ℓ13ℓ14ℓ23ℓ24 ℓ12 1ℓ12,13ℓ12,14ℓ12,23ℓ12,24 ℓ13ℓ12,23 1ℓ13,14ℓ12,13ℓ13,24 ℓ14ℓ12,24ℓ13,34 1ℓ14,23ℓ12,14 ℓ23ℓ12,13ℓ12,23ℓ14,23 1ℓ23,24 ℓ24ℓ12,14ℓ13,24ℓ12,24ℓ23,34 1 ⪰0. As before, symmetry of this matrix yields a few additional linear constraints, namely ℓ12,13=ℓ12,23, ℓ12,14=ℓ12,24, ℓ13,14=ℓ13,34. The fullV2(4) has 43 entries, and its reduced form is V2(4) =/parenleftbig1, s12, s13, s14, s23, s24, s34, s12s13, s12s14, s12s23, s12s24, s12s34, s13s14, s13s24, s13s34, s14s23, s23s24, s23s34/parenrightbig∗, leading to the 18 ×18 moment matrix pattern  1s12s13s14s23s24s34s12s13s12s14s12s23s12s24s12s34s13s14s13s24s13s34s14s23s23s24s23s34 s12 1s12s13s12s14s12s23s12s24s12s34s13s14s23s24s34s12s13s14s12s13s24s12s13s34s12s14s23s12s23s24s12s23s34 s13s12s23 1s13s14s12s13s13s24s13s34s23s12s14s23s12s12s23s24s12s23s34s14s24s34s12s13s14s12s13s24s12s13s34 s14s12s24s13s34 1s14s23s12s14s13s14s12s13s24s24s12s23s34s12s12s23s24s34s12s13s34s13s23s12s13s14s12s14s23 s23s12s13s12s23s14s23 1s23s24s23s34s12s12s13s14s13s12s13s24s12s13s34s12s14s23s12s23s24s12s23s34s14s24s34 s24s12s14s13s24s12s24s23s34 1s23s24s12s13s34s12s12s14s23s14s12s13s14s12s23s24s13s12s13s24s12s23s34s34s23 s34s12s34s13s14s13s34s23s24s23s34 1s12s13s14s12s13s34s12s23s24s12s23s34s12s13s12s14s23s14s12s13s24s23s24 s12s23s13s23s12s14s23s12s12s23s24s12s23s34 1s13s14s12s13s13s24s13s34s14s23s23s24s23s34s12s14s12s24s12s34

Question 405 multiple-choice

Quantum fast Fourier transforms (QFFT) are foundational in many quantum algorithms, especially those relevant to cryptography and computational number theory. Recent advances have focused on making QFFT implementations exact for arbitrary orders, including large primes. Which technique enables the exact implementation of quantum fast Fourier transforms for arbitrary, particularly large prime, orders in quantum algorithms? 1) Amplitude amplification 2) Quantum error correction codes 3) Phase kickback 4) Quantum teleportation 5) Quantum annealing 6) Eigenvalue estimation 7) Entanglement purification

✓ Correct Answer:

The correct answer is 1) Amplitude amplification.

📚 Reference Text:

arXiv:quant-ph/0301093v1 17 Jan 2003Exact quantum Fourier transforms and discrete logarithm algorithms Michele Mosca and Christof Zalka Department of Combinatorics and Optimization University of Waterloo, Waterloo, Ontario Canada N2L 3G1 e-mail:mmosca@iqc.ca andzalka@iqc.ca August 27, 2018 Abstract We show how the quantum fast Fourier transform (QFFT) can be made exact for arbitrary orders (ﬁrst for large primes). For most quantum algorithms only the quantum Fourier transform of order 2nis needed, and this can be done exactly. Kitaev [9] showed how to approxi mate the Fourier transform foranyorder. Hereweshowhowhisconstru ctioncanbe made exact by using the technique known as “amplitude ampliﬁ cation”. Although unlikely to be of any practical use, this construct ion e.g. allows to make Shor’s discrete logarithm quantum algorithm exact. Thus we have the ﬁrst example of an exact non black box fast quantum al gorithm, thereby giving more evidence that “quantum” need not be prob abilistic. We also show that in a certain sense the family of circuits for the exact QFFT is uniform. Namely the parameters of the gates can be cal culated eﬃciently. 1 Introduction The “quantum fast Fourier transformation” (QFFT) plays an impor tant role in quantum algorithms. It is a unitary transformation that applies th e discrete Fouriertransformtotheamplitudesofaquantumregister. Thest andardversion has order 2nand is applied to a quantum register consisting of nqubits. It was found by Coppersmith [5] (see also Shor [11]). The construction is es sentially identical to the standard classical fast Fourier transform (FFT) . Like the FFT it generalises to orders which are a power of a small prime and more ge nerally to smooth numbers, thus integers who have only small prime factor s (see Cleve [4]). These constructions implement the desired unitary transform ation exactly. In contrast, so far no exact (and eﬃcient) constructions for ar bitrary or- ders have been known. For his “Abelian stabiliser problem” Kitaev [9] g ave an 1 approximate implementation based on “eigenvalue estimation”. Here we show how this eigenvalue estimation step can be made exact using “amplitud e am- pliﬁcation”. Amplitude ampliﬁcation [1] is a slight generalisation of Grove r’s algorithm, allowing to apply the square root speed up to any heuristic algo- rithm. Brassard and Høyer [2] used a variant of it to make Simon’s algo rithm exact. Finally we point out that an exact quantum Fourier transform for lar ge prime orders can be used to make Shor’s discrete logarithm algorithm exact. 2 The exact QFFT pfor large prime p The quantum Fourier transform of order (or “modulus”) Nacts on “computa- tional” basis states |x/an}⌊ra⌋k⌉tri}htas follows: QFFTN:|x/an}⌊ra⌋k⌉tri}ht → | Ψx/an}⌊ra⌋k⌉tri}ht=1 NN−1/summationdisplay y=0e2πixy N|y/an}⌊ra⌋k⌉tri}ht. For arbitrary, in particular non-smooth N, Kitaev [9] proposes to d o this in two steps (second part of section 5 in [9], see also the review by Jozsa [8]) : |x/an}⌊ra⌋k⌉tri}ht → | x,Ψx/an}⌊ra⌋k⌉tri}ht → | Ψx/an}⌊ra⌋k⌉tri}ht where, as usual, registers that “appear out of nowhere” are und erstood to have been initialised in the standard state |0/an}⌊ra⌋k⌉tri}ht. Similarly in the second step, one of the registers is reset

Question 406 multiple-choice

Quantum algorithms have significantly impacted computational group theory, enabling efficient solutions to problems that are classically difficult, such as those involving group orders, discrete logarithms, and constructive membership. Understanding how these algorithms interact with group structures and representations is key to advances in cryptography and algebraic computation. Which condition is essential for Shor’s algorithm to efficiently perform order finding and constructive membership testing in Abelian subgroups of finite groups? 1) The subgroup must be non-Abelian and contain elements of infinite order. 2) The group representation must allow for multiple encodings of each element. 3) The field over which the group is defined must be of characteristic zero. 4) Each group element must have a unique encoding in the computational representation. 5) The subgroup must be generated by elements of composite order only. 6) The group must be cyclic and of prime order. 7) The group operation must not be associative.

✓ Correct Answer:

The correct answer is 4) Each group element must have a unique encoding in the computational representation..

📚 Reference Text:

+ ν(G) or toZp wherepis a prime dividing |G|. Of course, if Gis solvable one can insist that the representations of all the cyclic factors be of the second kind. It turns out that for some of the tasks in the hypotheses of The orem 4 there are eﬃcient quantum algorithms. By Shor’s results [25], the oracle for c omputing discrete logarithms can be implemented by a polynomial time quantum algorithm. Also, a superset of the primes dividing |G|can be obtained in polynomial time by quantum algorithms in t he most natural cases. For example, if Gis a matrix group over a ﬁnite ﬁeld, say G≤GL(n,q) then such a superset can be obtained by factoring the number ( qn−1)(qn−q)···(qn−qn−1), the order of the group GL(n,q). The same method works even for factors of matrix groups over ﬁ nite ﬁelds. If Gis (a factor of) a ﬁnite matrix group of characteristic zero, then the situati on is even better because in that case the prime divisors of Gare of polynomial size. But in any case, one can note that the s uperset of the primes dividing the order of Gis only used in Theorem 4 to compute (and factorize) the order s of elements of Gas well as those of matrices over ﬁnite ﬁelds of size at most |G|. This latter task can also be achieved by a quantum algorithm in polynomial time. In addition, we remark that the algorithm for testing member ship can be understood in a stronger, constructive sense, (see Section 5.3 in [4]), which is the proper generali zation of the constructive membership test in the Abelian case. For this w e need the notion of a straight line program on a set of generators. This is a sequence of expressions e1,...,eswhere each eiis either of the form xi:=hwherehis a member of the generating set or of the form xi=xjx−1 kwhere 0< j,k < i . It turns out that for elements gofGone can also require that a straight line program expressing gin terms of the generators be returned. Therefore, one can im mediately derive from Theorem 4 the following result. Corollary 5. LetGbe a ﬁnite black-box group with not necessarily unique encod ing. Assume that the following are given: 5 (a) an oracle for computing the orders of elements of G, (b) an oracle for the constructive membership tests in eleme ntary Abelian subgroups of G. Then the following tasks can be solved by quantum algorithms of running time polynomial in the input size +ν(G): (i) constructive membership test in G, (ii)–(v) as in Theorem 4. 4 Quantum implementations Inthissection wewilldiscussseveralcaseswhentheremain ingtasksinthehypothesesofCorollary5 can also be eﬃciently implemented by quantum algorithms. 4.1 Unique encoding If we have auniqueencodingfor theelements of theblack-box groupGthen we can useShor’sorder ﬁnding method. As we will show, in that case there is also an eﬃ cient quantum algorithm for the constructive membership test in elementary (and non-eleme ntary) Abelian subgroups. Therefore we will get the following result. Theorem 6. Assume that Gis a

Question 407 multiple-choice

Optimizing quantum circuits for fault tolerance and resource efficiency is critical for practical quantum computation, especially when implementing algorithms like the Quantum Fourier Transform. Reducing the number of costly gates and leveraging advanced synthesis techniques can dramatically improve performance on error-corrected quantum hardware. Which technique directly enables the synthesis of quantum gates using cyclotomic integers by identifying integer relations among real numbers, leading to efficient gate decomposition? 1) PSLQ algorithm 2) Grover's algorithm 3) Shor's algorithm 4) Quantum phase estimation 5) Jordan-Wigner transformation 6) Quantum amplitude amplification 7) Variational quantum eigensolver

✓ Correct Answer:

The correct answer is 1) PSLQ algorithm.

📚 Reference Text:

can be replaced with the Zagate) by 4 T gates (8 ↦4), 9 CNOT gates (12 ↦3), 1 H gate (4 ↦3), and 1 Phase (P) gate (2 ↦1) at the cost of introducing 1 measurement and 1 classically-controlled con-trolled-Z operation. Note that the fault-tolerant cost of thoseoperations introduced is signi ﬁcantly lower than that of a single T gate, as the construction of the T gate itself requires both ameasurement and a classically controlled quantum correction 25. We now group the uncontrolled Zarotations into one layer (time slice), as shown in Fig. 3. This layer applies the transforma- tion that was coined the phase gradient operation in31, the induction of which by the addition circuit was ﬁrst reported in ref.32. Such a transformation can be implemented by a b-bit adder at the cost of 4 b+O(1) T gates31, so long as one has access to a special quantum state ψbþ1/C12/C12 i:¼1ﬃﬃﬃﬃﬃﬃ ﬃ 2bþ1pP2bþ1/C01 j¼0e/C02πij=2bþ1jji. The quantum state ψbþ1/C12/C12 ican be reused to induce phase gradient transformations in all n−1 sets of controlled-Zarotations. A schematic circuit diagram of our AQFT implementation is shownin Fig. 4. To construct the special ( b+1)-qubit state ψ bþ1/C12/C12 i,w eﬁrst apply H gates to the quantum register 00 :::0 ji and then exercise the gates Z, Z−1∕2,…,Z/C01=2b. The latter step is accomplished via approximating each Zaby RUS circuits33. Speci ﬁcally, we approximate complex number eiπabyz*∕z, where z2Z½ω/C138withFig. 1 AQFT with n=6 and b=3.Note that each of the n−1 sets of controlled- zagates are separated by the H gates. Fig. 2 Ancilla-aided, measurement/feed forward-based fault-tolerant controlled-Zagate. This construction improves on the known state- of-the-art in the quantum resource requirement (see main text fordetail), while enabling to decouple the control from the target,important for further optimization.Y. Nam et al. 2 npj Quantum Information (2020) 26 Published in partnership with The University of New South Wales1234567890():,; ω:=eiπ∕4being the cyclotomic integer obtained from the PSLQ Algorithm34. We choose r2Z½ﬃﬃﬃ 2p /C138randomly and search the solution y2Z½ω/C138of the norm equation ∣y∣2=2L−∣rz∣2with L¼dlogðjrzj2Þe35, such that V:¼1ﬃﬃ 2p Lðrz y /C0y/C3rz/C3Þis a unitary. We exactly synthesize the two-qubit gate ðV 0 0VyÞinto a Clifford +T circuit33,36. Upon measuring the second qubit and obtaining 0, the gate Zais successfully implemented. Otherwise, a Z error takes place and can be reversed at zero cost in the T gate count. Theexpected number of repetitions until success is 2L∕∣rz∣2.W e resorted to using this more complex algorithm as opposed to the simpler one given by refs35,36, as we already use quantum circuits with measurements and feedforward elsewhere in ourconstructions, and the RUS approach results in about 2.5-fold improvement 33in the number of the T gates required to obtain the desired Za. Local optimization Here we describe a local optimization of the AQFT circuit developed above, exploiting the fact that controlled-P andcontrolled-T gates have a special implementation, due to both P and T gates being a part of the Clifford +T library. We start by noting that the controlled-P gate may be implemented by two CNOT

Question 408 multiple-choice

In the mathematical theory of integrable systems and quantum groups, classical dynamical r-matrices are fundamental objects related to solutions of generalized Yang-Baxter equations. Their classification involves Lie algebraic structures, coupling constants, and symmetries. Which statement accurately describes the classification of classical dynamical r-matrices with a nonzero coupling constant ε in the context of simple Lie algebras? 1) They are all gauge-equivalent to the rational dynamical r-matrix associated with the Cartan subalgebra. 2) They correspond uniquely to automorphisms of the Weyl group acting on the Lie algebra. 3) They are classified by subsets X of the set of simple roots Π, with explicit formulas involving meromorphic functions ϕα(λ). 4) Their gauge equivalence classes are determined solely by the dimension of the Cartan subalgebra h. 5) They cannot be interpreted geometrically via Poisson-Lie groupoids. 6) They reduce to Belavin-Drinfeld classical r-matrices for all values of λ. 7) Their explicit form does not depend on the choice of reductive Lie subalgebra containing h.

✓ Correct Answer:

The correct answer is 3) They are classified by subsets X of the set of simple roots Π, with explicit formulas involving meromorphic functions ϕα(λ)..

📚 Reference Text:

rational dynamical r-matrix r(λ) and the basic trigonometric dynamical r-matrix rε(λ) have coupling constants 0 and εrespectively. 4 Classiﬁcation of classical dynamical r-matrices. In this section, we give the classiﬁcation of all dynamical r-matrices r:h∗→ g⊗gwhich have coupling constant ε∈C. 4.1. Gauge transformations. Consider the following operations on mero- morphic maps r:h∗→(g⊗g)h. 1.r(λ)/mapsto→r(λ)+/summationtext i0 eα∈leα⊗e−α−e−α⊗eα (λ,α). (4.1) It is clear that this is the image of the basic rational dynamical r-mat rix ofl under the embedding l⊂g. Theorem 4.1 ([EV1]). Any classical dynamical r-matrix r:h∗→(g⊗g)h with zero coupling constant is gauge-equivalent to rl(λ)for some l. 4.3. Classiﬁcation of dynamical r-matrices with coupling c onstant ε∈C∗.LetX⊂Π, and denote by /a\}bracketle{tX/a\}bracketri}ht ⊂∆ the set of all roots which are linear combinations of elements in X∪ −X. For any α∈∆ introduce a meromorphic function ϕα:h∗→Cby the following rule. Set ϕα(λ) =ε 2if α∈∆+\/a\}bracketle{tX/a\}bracketri}ht,ϕα(λ) =−ε 2ifα∈∆−\/a\}bracketle{tX/a\}bracketri}htand ϕα(λ) =ε 2cotanh/parenleftigε 2(λ,α)/parenrightig ifα∈ /a\}bracketle{tX/a\}bracketri}ht. Theorem 4.2 ([EV1]). LetX⊂Π. Set rX(λ) =ε 2Ω+/summationdisplay α∈∆ϕα(λ)eα⊗e−α. Thenrε X(λ)is a classical dynamical r-matrix with coupling constant ε. More- over, any classical dynamical r-matrix with coupling const antεis gauge-equivalent torε X(λ)for a suitable X⊂Π. Remarks. 1. The basic trigonometric dynamical r-matrix rε(λ) is obtained when we take X= Π. Moreover, the r-matrix rε X(λ) is equal to a limit of rε(λ−ν) whenνtends to inﬁnity in h∗in an appropriate direction. In other words, every classical dynamical r-matrix with nonzero coupling co nstantεis a limiting case of the basic trigonometric r-matrix. 2. LetWbe the Weyl group of g, and let w∈W. Letλ∈h∗tend to inﬁnity in a generic way in the Weyl chamber associated to w. Then limr1(λ) =1 2/summationdisplay ixi⊗xi+/summationdisplay α∈w(∆+)eα⊗e−α, which is the standard classical r-matrix corresponding to the polar ization of gassociated to w. Hence the basic trigonometric dynamical r-matrix r1(λ) interpolates all h-invariant classical (non-dynamical) r-matrices rsatisfying r+ r21= Ω, (up to the addition of a skew 2-form in Λ2h). 11 A classiﬁcationof all classicaldynamical r-matrices r:l∗→(g⊗g)lwheregis a simple Lie algebraand l⊂his given in [S]. This classiﬁcation generalizesboth the above classiﬁcation (when l=h) and the Belavin-Drinfeld classiﬁcation of classical r-matrices (when l= 0) (see Appendix A). 5 Classical dynamical r-matrices and Poisson- Lie groupoids In this section we give a geometric interpretation of the CDYBE. Let us ﬁrst brieﬂy recall the relationship between the classical Yang-Baxter e quation and the theory of Poisson-Lie groups, developed by Drinfeld. 5.1. Poisson-Lie groups. LetGbe a (complex or real) Lie group, let g be its Lie algebra and let O(G) be the algebra of regular functions on G. Let {,}:O(G)×O(G)→ O(G) be a Poisson structure on G. Let Π be the Poisson bivector ﬁeld, deﬁned by the relation

Question 409 multiple-choice

In quantum information theory, tensor network diagrams and permutation operators are essential tools for representing and manipulating multipartite quantum systems. Understanding the role of special unitary ensembles and their approximations is crucial for efficient quantum computation and benchmarking. Which property distinguishes a unitary k-design from a set of truly Haar-random unitaries, making k-designs more practical for quantum information tasks? 1) k-designs contain only diagonal unitary matrices 2) k-designs maximize entanglement entropy for all pure states 3) k-designs require exponential resources to implement for any k 4) k-designs replicate the full spectral distribution of Haar-random unitaries 5) k-designs always commute with permutation matrices 6) k-designs are limited to single-qubit operations 7) k-designs reproduce the first k moments of the Haar measure but are easier to implement

✓ Correct Answer:

The correct answer is 7) k-designs reproduce the first k moments of the Haar measure but are easier to implement.

📚 Reference Text:

as: ⟨ψ|AABAA=AAAA AT=AT=ABBA†A|A⟩⟩= , (140) where the equality is due to the transpose-trick|A⟩⟩=A⊗I|Ω⟩=I⊗AT|Ω⟩. In Tensor Network notation, the diagram below makes it clear that ⟨⟨A|B⟩⟩= Tr/parenleftbig A†B/parenrightbig : , (141) as well as the ABC-rule (104) : +++++ +++++ρℳ−1(O)BBBℳ−1(O)ℳ−1(O)ℳ−1(O)ρ AT=|ABC⟩⟩=BACBA=CT|B⟩⟩=ACT ,(142) whereA,B,C∈L/parenleftbig Cd/parenrightbig . In accordance with the Definition 23, a permutation matrix Vd(π), withπ∈Sk, maps the tensor product state |ψ1⟩⊗···⊗|ψk⟩to the tensor product state/vextendsingle/vextendsingleψπ−1(1)/angbracketrightbig ⊗···⊗/vextendsingle/vextendsingleψπ−1(k)/angbracketrightbig : Vd(π)|ψ1⟩|ψπ−1(1)⟩|ψπ−1(2)⟩|ψπ−1(k)⟩|ψ2⟩|ψk⟩= , (143) and it can be represented by a diagram with lines connecting the tensor space of the i-th element on the right with the the π(i)-th element on the left for each i∈[k]. For example, the identity operator Iand the Flip operator Fare represented as follows: /u1D540=/u1D53D= /u1D540= /u1D53D= , /u1D540=/u1D53D= /u1D540= /u1D53D= .(144) 19 The composition of permutations can be visualized by placing the corresponding diagrams next to each other. When considering a matrix A∈L/parenleftbig Cd/parenrightbig ⊗L/parenleftig Cd′/parenrightig , its vectorization is represented as: +++++ +++++ρℳ−1(O)BBBℳ−1(O)ℳ−1(O)ℳ−1(O)ρ AT=|ABC⟩⟩=BACBA=CT|B⟩⟩=ACTA . (145) For instance, the vectorization of the identity operator Iand the Flip operator Fare represented as: /u1D540=/u1D53D= /u1D540= /u1D53D= , /u1D540=/u1D53D= /u1D540= /u1D53D= . (146) Using this diagrammatic notation, we can express Eq. (123) E U∼µH[U⊗U∗] =1 d|I⟩⟩⟨⟨I|in Tensor Network notation as: EU∼µH  𝕀===|𝕀⟩⟩=UU*  =1 d 𝕀===|𝕀⟩⟩=UU* . (147) Similarly, Eq. (124) E U∼µH/bracketleftbig U⊗2⊗U∗⊗2/bracketrightbig =1 d2−1/bracketleftbig |I⟩⟩⟨⟨I|−1 d|I⟩⟩⟨⟨F|−1 d|F⟩⟩⟨⟨I|+|F⟩⟩⟨⟨F|/bracketrightbig can be rep- resented as: EU∼µH   =1 d2−1  −1 d −1 d +  . (148) Using the diagrammatic notation, we can easily see that: E U∼µH/bracketleftbig Ui1,j1U∗ i2,j2/bracketrightbig =1 dδi1,i2δj1,j2, (149) where we used E U∼µH/bracketleftbig Ui1,j1U∗ i2,j2/bracketrightbig =⟨i1,i2|E U∼µH[U⊗U∗]|j1,j2⟩andδa,brepresents the Kronecker delta. Similarly: E U∼µH/bracketleftbig Ui1,j1Ui2,j2U∗ i3,j3U∗ i4,j4/bracketrightbig =1 d2−1/bracketleftbigg δi1,i3δi2,i4δj1,j3δj2,j4−1 dδi1,i3δi2,i4δj1,j4δj2,j3/bracketrightbigg (150) +1 d2−1/bracketleftbigg −1 dδi1,i4δi2,i3δj1,j3δj2,j4+δi1,i4δi2,i3δj1,j4δj2,j3/bracketrightbigg ,(151) 20 where we used E U∼µH/bracketleftbig Ui1,j1Ui2,j2U∗ i3,j3U∗ i4,j4/bracketrightbig =⟨i1,i2,i3,i4|E U∼µH/bracketleftbig U⊗2⊗U∗⊗2/bracketrightbig |j1,j2,j3,j4⟩. An- other useful relation is the partial-swap-trick Tr2(A⊗BF) =AB, which can be visualized as: BA=A1 ABA2A3AkVd(πcyc)=Vd(πcyc)=AnA1BA=AB , (152) and thus we have the swap-trick formula Tr(A⊗BF) = Tr(AB). Similarly we also have Tr1(A⊗ BF) =BA. More generally, consider the cyclic permutation πcyc∈Sk, which corresponds to the unitary operator: Vd(πcyc) =/summationdisplay i1,...,ik∈[d]|i2,...,ik,i1⟩⟨i1,i2,...,ik|. (153) This unitary operator can be depicted diagrammatically as: BA=A1 ABA2A3AkVd(πcyc)=Vd(πcyc)=AkA1BA=AB . (154) Now, we want to calculate Tr2,...,n(A1⊗···⊗AnVd(πcyc)), and we will show that it simplifies toA1···An. We can verify this through direct calculation using resolutions of identities: Tr2,...,n(A1⊗···⊗AkVd(πcyc)) =/summationdisplay i1,...,ik∈[d]Tr2,···,k(A1⊗···⊗Ak|i2,...,ik,i1⟩⟨i1,i2,...,ik|) =/summationdisplay i1,...,ik∈[d]A1|i2⟩⟨i1|⟨i2|A2|i3⟩⟨i3|A3|i4⟩···⟨ik|Ak|i1⟩ =/summationdisplay i1,i2∈[d]A1|i2⟩⟨i1|⟨i2|A2A3···Ak|i1⟩=A1···Ak, (155) Alternatively, we can represent this calculation diagrammatically as: BA=A1 ABA2A3AkVd(πcyc)=Vd(πcyc)=AkA1BA=AB . (156) Hence, we have the generalized swap-trick , known as the cyclic-permutation-trick : Tr(A1⊗···⊗AkVd(πcyc)) = Tr(A1···Ak). (157) 21 7 Unitary designs Generating Haar random unitaries on a quantum computer can be a computationally expensive task, since most unitaries require an exponential number of elementary gates with respect to the number of qubits [3, 39, 63] to be implemented. However, for many applications in quantum information, only low-order moments of the Haar measure are needed [1, 10, 13–15, 21, 40]. This motivates the definition of unitary k-designs [64], which are distributions of unitaries that match the moments

Question 410 multiple-choice

Quantum circuit design for error-corrected hardware often focuses on minimizing resource-intensive gates, such as the T gate, especially in subroutines like the Approximate Quantum Fourier Transform (AQFT). Advanced techniques enable practical implementations of AQFT with dramatically reduced gate counts. Which combination of circuit optimizations results in the lowest T gate count for an AQFT implementation, achieving a resource scaling of 8n(log(n)/2) + 1.2 log²(n) T gates? 1) Discarding small rotations and using traditional controlled rotation circuits 2) Employing only Hadamard gates and omitting all controlled-Z rotations 3) Using textbook QFT circuits with O(n²) controlled rotations and no further optimizations 4) Implementing only integer adders without gradient states or RUS circuits 5) Applying RUS circuits without local circuit simplifications 6) Reducing controlled rotation cost from 8 to 4 T gates per rotation 7) Combining controlled-to-uncontrolled rotation mapping, integer adders, gradient states, RUS circuits, and local optimizations

✓ Correct Answer:

The correct answer is 7) Combining controlled-to-uncontrolled rotation mapping, integer adders, gradient states, RUS circuits, and local optimizations.

📚 Reference Text:

of algorithmic advantages and further low-level optimizations offered by the ﬁnal circuits to the following subsections. The entry point for our construction is the standard textbook implementation of the QFT circuit [ref. 26, page 219] using O(n2) parametrized controlled- Zarotations, where a∈{1∕2, 1∕4, . . ., 1 ∕2n−1}, and nHadamard gates. Recall that the AQFT may be obtained from the textbook circuit by simply discarding the rotations withparameter abelow a certain threshold, keeping only bcontrolled rotations per layer, with parameter bscaling logarithmically with n(see Fig. 1for an illustration). A standard fault-tolerant implementation of AQFT with /C24nlogðnÞ(removing lower order terms, and for simplicity furthermore assuming nstages of log ðnÞ gates) parametrized controlled rotations, choosing b¼logðnÞfor simplicity and to remove the dependence on the approximationerror, uses /C2424nlog 2ðnÞT gates since 8 T gates are employed to map controlled rotations into uncontrolled ones [ref.30, Fig. 10], and /C243 log ðnÞT gates are needed to approximate the uncontrolled rotations35. We optimize the above implementation by ﬁrst noting that mapping controlled rotations into uncontrolled ones may be doneusing only 4 T gates. This reduces the T gate count to/C2412nlog 2ðnÞ. We next notice that the uncontrolled rotations come in layers, and thus can be induced via adder, given access to a log ðnÞ-qubit gradient state32. Using an ef ﬁcient b-bit integer adder31with ~4 bT gates allows to reduce the T gate requirement from /C2412nlog2ðnÞto/C248nlogðnÞþ3 log2ðnÞ, where 8 =4+4T gates are employed to remove the control (4 T gates) and integer-add the target (4 T gates) per each controlled rotation, and3 log ðnÞT gates 35are used on each of log ðnÞqubits to synthesize the log ðnÞ-qubit gradient state, that is then reused. This is the most signi ﬁcant reduction giving improvements in both asymp- totic analysis and gate counts. We next apply RUS circuits toreduce the cost of state generation by a factor of about 2.533, leading to /C248nlogðnÞþ1:2 log2ðnÞT gates and ﬁnd local optimizations worth of ~8 nT gates further bringing down the T gate cost to the ﬁnalﬁgure of /C248nðlogðnÞ/C02Þþ1:2 log2ðnÞ, compared to the original /C2424nlog2ðnÞ. Details of the construction We start with an n-qubit AQFT whose construction relies on O(nb) controlled-Zagates with Za:¼10 0eiπa/C20/C21 ; where a∈{1∕2, 1∕4 ,... ,1 ∕2b}, for b:¼dlogne, and nHadamard (H) gates (see Fig. 1for an illustration with n=6 and b=3). Such a choice of bimplies a very speci ﬁc approximation error ε, whose analysis will be detailed in the next section. We unite theindividual controlled rotations into n−1 sets separated by the H gates, such as illustrated in Fig. 1. To implement a given controlled-Zarotation, we map its real- valued degree of freedom into that of the uncontrolled power ofPauli-Z, such as shown in Fig. 2. This implementation was developed by combining Kitaev ’s trick2with Toffoli- measurement construction of Jones29with our own choice of the relative the phase Toffoli gate, and custom circuit simpli ﬁca- tions. Our circuit improves over the one reported in [ref.30, Fig. 10] (note that the middle T gate in [ref.30, Fig. 10]

Question 411 multiple-choice

In quantum machine learning and quantum information theory, the use of symmetries and group representations plays a crucial role in structuring state spaces and understanding information processing limits. Block-diagonalization and duality concepts from representation theory are particularly important for designing algorithms and analyzing multipartite quantum systems. Which of the following best describes Schur-Weyl duality in the context of quantum systems with tensor product state spaces? 1) It establishes a one-to-one correspondence between irreducible decompositions under the unitary group U(d) and the symmetric group Sn, with their representations being mutual commutants. 2) It demonstrates that tensor product spaces are always fully symmetric under permutation. 3) It proves that quantum noise channels necessarily preserve all subspaces of the state space. 4) It asserts that block-diagonal structure only arises in classical systems, not quantum systems. 5) It provides a method for tracing out qubits to increase the dimension of irreducible representations. 6) It identifies the antisymmetric subspace as the only invariant subspace under group actions. 7) It shows that operator algebras and group symmetries are unrelated in quantum information theory.

✓ Correct Answer:

The correct answer is 1) It establishes a one-to-one correspondence between irreducible decompositions under the unitary group U(d) and the symmetric group Sn, with their representations being mutual commutants..

📚 Reference Text:

Crucially, if one wants to retrieve informa- tion outside of those sub-blocks (like we did in Fig. 6) one 30 Figure 10. QML and block-diagonal structure. Here we consider a GQML model of the form h(UgiUy g) = Tr W()UgiUy gWy()Mi , where the representation of the elements g2Ghave a block-diagonal structure as in Eqs. (31), while the equivariant neural network W()and the equivariant measurement operator Mhave a block-diagonal structure as in Eq. (32). When the states iare not invariant under the action of G, the model can only access the information of iencoded in the sub-blocks of the state associated to the irreps. On the other hand, there is information in the oﬀ-diagonal elements of ithat the QML model cannot access. needs to change the representation . This can be achieved, for instance, by allowing the QML model to act on mul- tiple copies of each i(i.e., going to a larger tensor rep- resentation of G) or in some cases by tracing out qubits (i.e., going to a smaller representation of G). We refer the reader to Ref. [62] for additional details on how GQML models can process information in the irreps. We mention in closing that this theme of block diagonalization impos- ing constraints on information processing appears in the study of quantum noise channels in the form of the “noise commutant” [78], and more generally that commutants are a central object of study across the theory of operator al- gebras. Taken together, the results in Fig. 10 perfectly showcase how a fundamental understanding on the representation of the symmetry group allows us to better understand how information is being processed in GQML models, and as such, how we can construct better models to solve a given task. D. Schur-Weyl duality Here, we give a concrete (and very useful) example of the interplay of a representation and its commutant described in the commutant structure of Theorem 7: Schur-Weyl du- ality. ThereareseveralformsofSchur-Weylduality, butwe will restrict our attention to the classic case between ten- sor representations of the unitary group U(d)21and index permuting representations of the symmetric group Sn. 21The representation theory over CofU(d)andSU(d)diﬀer only by a phase, which is irrelevant for complex representations.Consider the space H= (Cd) n. Ag2U(d)acts onH via the tensor representation U(see Example 5) Ug(v1 vn) :=Ugv1 Ugvn; and a2Snacts onHvia the index permutation repre- sentationP(see Example 1) P(v1 vn) :=v1(1) v1(n): The key to notice is that these representations are mutual commutants of each other. When this happens, Schur- Weyl duality states that the decomposition of HintoU(d) irreducibles Ukimmediately determines the decomposition ofHintoSnirreducibles, and vice versa. In other words, if there are Kdistinct irreducible invariant subspaces Uk ofH, then there is a one-to-one correspondence to Snir- reducible invariant subspaces Q:fUkg!fSnirrepsgand we can write H=Cd Cd=KX k=1Q(Uk) Uk:(33) To illustrate this, if we set n= 2, then (as we have seen similarly in Example 5), as representations of U(2), H=C2 C2=Sym2(C2)Alt2(C2): We have seen several times by now that in the

Question 412 multiple-choice

Commitment schemes are cryptographic protocols that enable a party to lock a value while keeping it hidden, with security based on group-theoretic assumptions and zero-knowledge proofs. Ensuring both hiding and binding properties is crucial for their security in applications like secure voting and digital cash. In a group-based commitment scheme where commitments are formed as c = g^x * h^r with randomly chosen r, which group property is specifically relied upon to ensure that a dishonest prover cannot efficiently open the same commitment to two different values? 1) The group has a small prime order. 2) The group supports fast inversion of exponentiation. 3) Randomness r is chosen from a small set. 4) The generator g is a quadratic residue. 5) The discrete logarithm problem is easy in the group. 6) The group order is rough, with few small divisors, making root extraction computationally hard. 7) All group elements are of order two.

✓ Correct Answer:

The correct answer is 6) The group order is rough, with few small divisors, making root extraction computationally hard..

📚 Reference Text:

honest prover PreceivesR←R(1k),p r o - duces(c,w)∈R, sendsctoVand ﬁnally conducts the protocol with V, who accepts with overwhelming probability in k. Knowledge Soundness. There exists a polynomial p()and an extractor M, such that for all provers P∗running in time at most t(k),Advκ,M,p(P∗,k)≤ ν(k). 4 TheCommitmentScheme Based on the above model, the goal is to make a commitment scheme with protocols to verify various claims on committed values. The basic scheme is thatthe veriﬁerV(the receiver of commitments) will run Gand send descr(G) (and more information to be described later) to the prover P(the committer). Set-Up.VrunsG(1 k) and chooses a random element h∈G, such that ord(h) isC(k)-rough (this can be done by raising a random element to power lG). NowVsetsg=hα, whereαis randomly chosen in [0 ..22B+k].Vsends descr(G),g,htoPand proves that g∈/angbracketlefth/angbracketright, by the standard zero-knowledge discrete log protocol with binary challenges: in one iteration of this, Vsends a=hRforarandom R∈[0..22B+2k].Pselectsarandombit b,andVreplies withz=R+bα.Pchecks that hz=agb. Repeating this ktimes results in a soundness error of 2−k, and the protocol is easily seen to be statistical zero-knowledge. This is not a very eﬃcient solution, but it only needs to bedone once and only in the set-up phase. Commit. Tocommittoaninteger x,Pchoosesratrandomin[0 ..2 B+k],sends c=gxhrtoV, and stores x,rfor later use. Open.To open a commitment, Pmust sendx,r,µsuch thatc=µgxhrand µlG= 1. An honest prover can always use µ= 1. Although this gives a dis- honest prover extra freedom, this in no way makes the commitment schemeweaker: the binding property still holds, as we argue below. Indeed, recallingourcommentsontherootassumption,onemaythinkoftheschemeastakingplace in the quotient group G/UwhereUis the subgroup in Gconsisting of all elements of C-smooth order. From this point of view, the opening condi- tion simply ensures that the prover opens something representing the sameelement ascinG/U(The quotient group is deﬁned canonically so that c≡¯c (modU) iﬀ there is a µ∈Usuch that ¯c=µc∈G). As for hiding, note that Pveriﬁes initially that g∈/angbracketlefth/angbracketright. Hence, since ris chosen with bit length at least k+log 2(ord(h)),cis statistically close to uniform in/angbracketlefth/angbracketright, for any value of x. As for binding, we consider any prover P∗who can create cand the corre- sponding valid distinct openings, ( µ,x,r) and (µ/prime,x/prime,r/prime). It follows that we get 134 IvanDamg˚ ardandEiichiroFujisaki µgxhr=c=µ/primegx/primehr/prime. Recall that Vcreatesgasg=hα. Plugging this in and raising both sides of the equation to lG, we get that hlG(α(x−x/prime)+(r−r/prime))=1 .W e can writeα=q·ord(h)+resfor integersq,reswith 0 ≤res< ord(h). Then fromP∗’s point of view, resis uniquely determined from g, whereas there is an exponentially small amount of information on q(the only source of information is the proof that g∈/angbracketlefth/angbracketrightwhich is statistical zero-knowledge). So P∗’s choice of x,x/prime,r,r/primeis (almost) independent of q. It followsl(α(x−x/prime)+(r−r/prime)) = 0 (as an integer) with probability exponentially small in k. Assuming this number is indeed non-zero, ord(h) must divide M:=α(x−x/prime)+(r−r/prime) (since the order isC(k)-rough). We can now use P∗to break the root assumption as follows: given input descr(G),h∈G, we choose gasVwould have done it, send descr(G),h,gto P∗and execute in the normal way the proof that g∈/angbracketlefth/angbracketright. With probability of at least 1 /l(k),hwill haveC(k)-rough order and everything has the same distribution

Question 413 multiple-choice

In group theory, word maps and their fibers provide insight into the structural and probabilistic properties of infinite and profinite groups. Concepts such as Hausdorff dimension and randomly free groups reveal the interplay between algebraic structure and randomness. Which statement best characterizes the "randomly free" property in profinite groups with infinitely many non-isomorphic non-abelian upper composition factors? 1) Every subgroup is free and has infinite rank. 2) Any finite set of elements generates a solvable subgroup with probability one. 3) For any integer d ≥ 1, randomly chosen d elements generate a free subgroup with probability one. 4) All word maps in the group are constant functions. 5) The group satisfies a nontrivial probabilistic identity for all words. 6) The set of elements of finite odd order always forms a normal subgroup. 7) Every nontrivial word has fibers of full Hausdorff dimension.

✓ Correct Answer:

The correct answer is 3) For any integer d ≥ 1, randomly chosen d elements generate a free subgroup with probability one..

📚 Reference Text:

Title: Words, Hausdorff dimension and randomly free groups Year: 2017 Paper ID: 4a297f65b87fcf7dd9d2063b2f9cdd0e8ce7a6fc Source: semantic-scholar URL: https://www.semanticscholar.org/paper/4a297f65b87fcf7dd9d2063b2f9cdd0e8ce7a6fc Abstract: We bound the size of fibers of word maps in finite and residually finite groups, and derive various applications. Our main result shows that, for any word $$1 \ne w \in F_d$$1≠w∈Fd there exists $$\epsilon > 0$$ϵ>0 such that if $$\Gamma $$Γ is a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors, then all fibers of the word map $$w:\Gamma ^d \rightarrow \Gamma $$w:Γd→Γ have Hausdorff dimension at most $$d -\epsilon $$d-ϵ. We conclude that profinite groups $$G := {\hat{\Gamma }}$$G:=Γ^, $$\Gamma $$Γ as above, satisfy no probabilistic identity, and therefore they are randomly free, namely, for any $$d \ge 1$$d≥1, the probability that randomly chosen elements $$g_1, \ldots , g_d \in G$$g1,…,gd∈G freely generate a free subgroup (isomorphic to $$F_d$$Fd) is 1. This solves an open problem from Dixon et al. (J Reine Angew Math (Crelle’s) 556:159–172, 2003). Additional applications and related results are also established. For example, combining our results with recent results of Bors, we conclude that a profinite group in which the set of elements of finite odd order has positive measure has an open prosolvable subgroup. This may be regarded as a probabilistic version of the Feit–Thompson theorem.

Question 414 multiple-choice

In integrable systems and quantum group theory, τ-functions play a central role in encoding solutions to nonlinear hierarchies, and their properties are often influenced by underlying symmetries and algebraic constraints. The interplay between classical and quantum symmetries, such as SL(2) and its q-deformed counterpart SLq(2), leads to rich mathematical structures. When the parameter λ equals 1/2 in a system involving Hirota-type equations and SL(2) symmetry, which of the following statements accurately describes the impact on the corresponding solution space for the τ-function? 1) The solution space becomes infinite-dimensional due to removal of all constraints. 2) The τ-function must be expressed solely in terms of Cartan subgroup elements. 3) The system reduces to the classic KP hierarchy with unrestricted solutions. 4) Arbitrary functions completely parametrize the general solution space without reduction. 5) The τ-function matrix elements always commute, regardless of parameter values. 6) The solution space is narrowed to a 3-parameter family of solutions, analogous to Virasoro or W-constraints in matrix models. 7) The q-deformation parameter is forced to be zero, returning the system to its classical limit.

✓ Correct Answer:

The correct answer is 6) The solution space is narrowed to a 3-parameter family of solutions, analogous to Virasoro or W-constraints in matrix models..

📚 Reference Text:

our equations to the Liouville one, which is also sometimes associated w ithSL(2) - but in a slightly diﬀerent sense. First of all, as all the Hirota-like equations, ( 34) can be rewritten as a (system of) ordinary diﬀerential equations, when expanded in po wers ofǫ=1 2(t−t′) and ¯ǫ=1 2(¯t−¯t′). For example, for λ=λ′we obtain from (34): coeﬃcient in front of ǫ: ∂τλ¯∂τλ−τλ∂¯∂τλ= 2λτ2 λ−1 2; coeﬃcient in front of ¯ ǫ: 2λτλ¯∂2τλ= (2λ−1)(¯∂τλ)2; ...(38) Ifλ=1 2, the ﬁrst one of these is just Liouville equation: ∂τ1 2¯∂τ1 2−τ1 2∂¯∂τ1 2=τ2 0= 1, (39) or ∂¯∂φ= 2eφ, τ1 2=e−φ/2, (40) while the second one, ¯∂2τ1 2= 0, (41) isveryrestrictiveconstraint. Itsroleistoreducethehugesetof solutionstoLiouvilleequation, τ1 2(t,¯t|g) = (1+A(t)B(¯t))/bracketleftigg∂A ∂t∂B ∂¯t/bracketrightigg−1 2 , (42) 12 parametrized by two arbitrary functionsA(t) andB(¯t), to the 3-parametric family (37). In this sense it plays the same role as the string equation (or Virasoro a ndW-constraints) in matrix models. In the language of inﬁnite-dimensional Grassmannian there are inﬁnitely many ways to embed SL(2) group into GL(∞) - and all the cases correspond to solutions (42) (with some A(t) andB(¯T)) to theSL(2) reduced Toda-lattice hierarchy (i.e. Liouville equation), - but the constraint (41) speciﬁes very concrete emb edding: that in the left upper corner ofGL(∞) matrix (associated with linearfunctionsA(t) andB(¯t)). 4τ-function for any g∈SLq(2) 4.1 General solution of SLq(2)bilinear equations In the previous section we derived Hirota-like equations for the SLq(2)τ-functions. Formally these equations were derived for any “group element” g, but so far we have explicitly checked only that they are indeed satisﬁed by a τ-fnction at g=I,τλ(t,¯t|g=I) = [1 +t¯t]2λ. It is of course easy to check this as well for any gfrom Cartan subgroup. However, restricting g to Cartan subgroup, we allow consideration of only (mutually) commu ting matrix elements /an}bracketle{tk|g|n/an}bracketri}htin (1). The purpose of this section is to show explicitly that there is no need in such restriction: the same equations are satisﬁed for arbitrary choice of matrix elements from the non-commutative coordinate ring A(G).9This result, when compared with the derivation of Hirota equations, implies that the set of the “group elements”, i.e. s olutions to ∆( g) =g⊗g, should be larger than Cartan subgroup. This is indeed true, if we con sidergas belonging to Uq(G|A(G)), which is deﬁned as the universal enveloping algebra (i.e. the set o f formal series in generators T±andq±T0) with coeﬃcients in A(G). Unfortunately we are not aware of any explicit construction of this type and present just an illustrative ex ample in section 4.2. 9To our knowledge, the idea to study non-commutative τ-functions was ﬁrst proposed in [10]. 13 Let us begin with the case of λ=1 2. Then τ1 2(t,¯t|g) =/an}bracketle{t+|g|+/an}bracketri}ht+¯t/an}bracketle{t+|g|−/an}bracketri}ht+t/an}bracketle{t−|g|+/an}bracketri}ht+t¯t/an}bracketle{t−|g|−/an}bracketri}ht= =a+b¯t+ct+dt¯t,(43) wherea,b,c,dare elements of the matrix T= a b c d  (44) with the commutation relations dictated by T TR=RT Tequation [11] ab=qba, ac=qca, bd=qdb, cd=qdc, bc=cb, ad−da= (q−q−1)bc.(45) Ifborcor both are non-vanishing, τ1 2(t,¯t|g) with diﬀerent values of time-variables t,¯tdo not commute. Still such τ1 2(t,¯t|g) does satisfy the same bilinear identity (29), moreover, for

Question 415 multiple-choice

In finite group theory, the study of non-abelian p-groups often involves analyzing subgroup structures and automorphism actions, particularly when groups contain quaternion subgroups. Understanding how centralizers and normal subgroups interact is key to classifying such groups. Which condition most strongly constrains the possible structure of a finite group containing a quaternion subgroup Q of order 8, when the group contains no abelian maximal subgroups? 1) The center Z is trivial 2) The group is cyclic and of order 16 3) All automorphisms of Q are inner 4) The centralizer C of Q is abelian 5) The existence of a minimal non-abelian centralizer C of Q 6) The group has order 24 7) Every element outside Q commutes with every element of Q

✓ Correct Answer:

The correct answer is 5) The existence of a minimal non-abelian centralizer C of Q.

📚 Reference Text:

writing again binstead of b0we may assume again ½v;b/C138¼1. In both cases we may assume ½v;b/C138¼1 so that hv;biGC4/C2C4.W e getG¼hv;bihai, where aacts by inversion on hv;biand so we have obtained a group of part (b6) of our theorem. Now suppose that jGj¼26. First assume that Ghas a normal subgroup Q¼ha;biGQ8and set ZðQÞ¼hzi. Since G0cFðGÞ¼W1ðGÞcZðGÞ; no element in G/C0Qinduces an outer automorphism on Q(ifxAG/C0Qinduces an outer automorphism on Q, then ðQhxiÞ0GC4). We get G¼Q/C3C, where C¼CGðQÞandQVC¼hzi. Also zis not a square in C. Indeed, if xACwith x2¼z,t h e nha;bxiGD8, a contradiction. We have exp ðCÞ¼4,jCj¼24and W1ðCÞ¼W1ðGÞGE4.I fCis abelian, then CGC4/C2C4, in which case zis a square inC, a contradiction. Hence Cis non-abelian. If Cis minimal non-abelian, then C¼hc;djc4¼d4¼1;cd¼c/C01iGH2: Since zis not a square in C, we must have z¼c2d2and this is the group of part (b7) of our theorem. If Cis not minimal non-abelian, then Chas a subgroup Q/C3GQ8 with Q/C3Vhzi¼f1g. We get G¼Q/C2Q/C3and this is a group of part (b5) of our the- orem. Now we treat the case where no quaternion subgroup is normal in G. Let Q¼ha;biGQ8be a quaternion subgroup of Gwith ZðQÞ¼hzi. Set W¼W1ðGÞ, so that FðGÞ¼WGE4,WcZðGÞand WVQ¼hzi. Since Qt=G, we have Q0¼hzi0G0which gives G0¼W. Since jGj¼26, exp ðGÞ¼4, and Ghas only three involutions, Gdoes not have an abelian maximal subgroup. Set H¼QW,s o thatHtG. Since Qt=G, there is cAG/C0Hsuch that ½Q;c/C138Ghzi. Interchanging aandbif necessary, we may assume ½a;c/C138Bhziand so ½a;c/C138¼uAW/C0hziand H¼Q/C2hui. Since ha;ciis minimal non-abelian (by Proposition 1.5) and a2¼z0½a;c/C138¼u, we have ha;ciGH2, so that zanduare the only involutions which are squares in ha;ci.I f c2¼z, then ðacÞ2¼a2c2½c;a/C138¼zzu¼uand ½a;ac/C138¼u, so that replacing cbyc0¼ac(if necessary), we may assume (writing again cinstead of c0) from the start that c2¼u. We consider the subgroup ha;bcinoting that ½a;bc/C138¼zuanda2¼z. It follows that ha;bciGH2so that ðbcÞ2Afzu;zg.I f zu¼ðbcÞ2¼b2c2½c;b/C138¼zu½c;b/C138, then ½c;b/C138¼1. If z¼ðbcÞ2¼b2c2½c;b/C138¼zu½c;b/C138, then ½c;b/C138¼uso that ½c;ab/C138¼uu¼1. Therefore, replacing bbyb0¼abif necessary and writing again binstead of b0, we may assume from the start that ½c;b/C138¼1. Hence aacts by inversion on the abelian subgroup hb;ciGC4/C2C4. Set M¼ha;b;ciand consider the abelian subgroup X¼ha;Wiof type ð4;2Þ. We have CMðXÞ¼X,300 Z. Janko Xt=G,A u t ðXÞGD8andG=XGE8. It follows that there is dAG/C0Mcentralizing X,s ot h a t ha;diWis abelian of order 24and therefore ha;diWGC4/C2C4. Replac- ingdwith a suitable element in ha;di/C0X, we may assume from the start that d2¼u.I f ½c;d/C138¼1, then cdwould be an involution in G/C0W, a contradiction. Hence ½c;d/C13801,½a;d/C138¼1 and G¼ha;b;c;di, and so it remains to determine the action of donhb;ciGC4/C2C4. We have hc;diGQ8orhc;diGH2and so ½c;d/C138Afu;z;uzg. Also ½b;d/C138¼1o rhb;diGH2and so ½b;d/C138Af1;z;ug.I f½b;d/C138¼z and ½c;d/C138¼u, then dacts by inversion on hb;ci; then adcentralizes hb;ciand so hb;cihadiis an abelian maximal subgroup of G, a contradiction. It follows that we have to consider the following eight possibilities for ½b;d/C138and ½c;d/C138. (1) If ½b;d/C138¼1 and ½c;d/C138¼u, thenhc;diGQ8andhc;ditG, a contradiction. (2) If ½b;d/C138¼1 and ½c;d/C138¼z, then ½b;cd/C138¼1 and ðcdÞ2¼c2d2½d;c/C138¼z¼b2so thatbcdis an involution in G/C0W, a contradiction. (3) If ½b;d/C138¼1a n d ½c;d/C138¼uz, then ½ad;c/C138¼zand ðc/C1adÞ2¼1 so that cadis an involution in G/C0W, a contradiction. (4) If ½b;d/C138¼zand

Question 416 multiple-choice

The Hidden Subgroup Problem (HSP) plays a central role in quantum algorithms and their efficiency depends crucially on the structure of the underlying group. Classification of group families based on how quantum methods can reconstruct subgroups is important for both computational feasibility and cryptographic security. Which classification accurately describes groups for which quantum algorithms can efficiently and practically identify hidden subgroups using polynomial resources? 1) Quantum information-theoretically reconstructible groups 2) Groups with irreducible representations only 3) Information-theoretically reconstructible groups 4) Nonabelian groups with matrix-valued homomorphisms 5) Fully reconstructible groups 6) Groups with exponentially many subgroups 7) Groups requiring brute-force quantum search

✓ Correct Answer:

The correct answer is 5) Fully reconstructible groups.

📚 Reference Text:

calls for a quantum measurement for each possible subgroup, and sincethere might be |G| Ω(log |G|)of these, it requires an exponential number of quantum operations. Our current understanding of the HSP, then, divides group families into three classes. I.Fully reconstructible . Subgroups of a family of groups {Gi}arefully recon- structible if the HSP can be solved with high probability by a quantum circuit of size polynomial in log |Gi|. II.Information-theoretically reconstructible . Subgroups of a family of groups {Gi} areinformation-theoretically reconstructible if the solution to the HSP for Giis de- Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE POWER OF STRONG FOURIER SAMPLING 941 termined information-theoretically by the fully measured result of a quantum circuit of size polynomial in log |Gi|. III.Quantum information-theoretically reconstructible . Subgroups of a family of groups {Gi}arequantum information-theoretically reconstructible if the solution to the HSP for Giis determined by the quantum state resulting from a quantum circuit of polynomial size in log |Gi|, in the sense that there exists a positive operator-valued measurement (POVM) that yields the subgroup Hwith constant probability but where it may or may not be possible to carry out this POVM with a quantum circuitof polynomial size. In each case, the quantum circuit has oracle access to a function f:G→S, for some set S, with the property that fis constant on each left coset of a subgroup H and distinct on distinct cosets. In this language, then, subgroups of abelian groups are fully reconstructible, while the result of [7] shows that subgroups of arbitrary groups are quantum information-theoretically reconstructible. The other work cited above has labored to place speciﬁcfamilies of nonabelian groups into the more algorithmically meaningful classes I and II. 1.2. Nonabelian Fourier transforms. In this section we give a brief review of nonabelian Fourier analysis but only to the extent needed to set down notation.We refer the reader to [9, 24] for a more complete exposition. Fourier analysis over a ﬁnite abelian group Aexpresses a function φ:A→Cas a linear combination of homomorphisms χ:A→C.I fA=Z p, for example, these are the familiar basis functions χt:z/mapsto→ωtz p, where ωpdenotes the pth root of unity e2πi/p. Any function φ:A→Ccan be uniquely expressed as a linear combination of these χt, and this change of basis is the Fourier transform. When Gis a nonabelian group, however, this same procedure cannot work: in particular, there are not enough homomorphisms of GintoCto span the space of allC-valued functions on G. To deﬁne a suﬃcient basis, the representation theory of ﬁnite groups considers more general functions, namely homomorphisms from Ginto groups of unitary matrices. Arepresentation of a ﬁnite group Gis a homomorphism ρ:G→U(d), where U(d) denotes the group of unitary d×dmatrices (with entries from C); the dimension d=dρis referred to as the dimension ofρ.I fρ:G→U(d) is a representation, a subspace WofCdis said to be invariant ifρ(g)(W)⊂Wfor all g. A representation is said to be irreducible if the only invariant subspaces are the trivial subspaces Cd and{/vector0}. For a function φ:G→Cand an irreducible representation

Question 417 multiple-choice

Quantum algorithms have shown notable efficiency in solving the Hidden Subgroup Problem (HSP) for groups with specific algebraic structures, particularly those involving semidirect products and finite fields. The nilpotency class, commutator subgroups, and group exponents play crucial roles in determining algorithmic complexity and tractability. In the context of quantum algorithms for HSP, which property of the semidirect product group constructed from Zm_p and Zp ensures that the subgroup generated by commutators and Fq is abelian and normal, facilitating efficient quantum solutions? 1) The group is cyclic and has prime order 2) The characteristic polynomial of the linear transformation A has distinct roots 3) The group exponent divides the order of Zp 4) The group has a center equal to Zm_p 5) The commutator subgroup is trivial 6) The commutator subgroup is abelian and normal 7) The matrix entries of Σ_{j=0}^{t-1} A^j are all constants

✓ Correct Answer:

The correct answer is 6) The commutator subgroup is abelian and normal.

📚 Reference Text:

the group Fg( F(d) q[x]) is of nilpotency class d+1. However, we can actually give a reduction to the HSPin a group of class d. To this end, observe that the hidden subgroup is a conjugate of the complement Fq. Therefore, it can be found in the subgroup generated by the commutator of FqwithF(d) q[x] (this is an abelian normal subgroup) and the complement Fq. This semidirect product group has nilpotency class d. Note that semidirect product groups Fg( F(d) q[x]) have commutative commutator subgroups and that their exponent is the characteristic of Fq. Therefore, for ﬁxed characteristic, we can apply Fact 2 (b) to obtain the following result. Corollary 5.6. Assume that qis a power of a ﬁxed prime p. Then we can solve HPGP(Fq,1,d)by a quantum algorithm in time polynomial in dandlogq. This corollary allows us to complete Fact 4 (b), because it can be applied to ﬁelds of characteristic in the set Edthat were left open. Since Edis ﬁnite, it follows that for ﬁxeddwe can solve HPGP(Fq,1,d) in quantum polynomial time for all ﬁnite ﬁelds. Together with Fact 4 (a) this improves the overall result of [15]: the HPGP(Fq,n,d) can be solved eﬃciently for all ﬁnite ﬁelds when nanddare constant. We further improve this result in the next section, where we present a more powerful reduction of the multivariate problem to the univariate case. We conclude this section by showing that the HSP for semidirect product groups of the form Zm p/multicloserightZpcan be reduced to a multidimensional analogue of the HPGP. This HSP is discussed in [5], where it is shown that the HSP for all possible subgroups 2002 T. DECKER, G. IVANYOS, M. SANTHA, AND P. WOCJAN can be reduced to the HSP in which the hidden subgroups are complements of Zm p. LetHbe such a subgroup. Following arguments of [5], we show that the cosets of Hcan be considered as level sets of a polynomial map from Zm+1 ptoZm pof the form y−Q(x), where y=(y1,...,y m)a n dQ(x)=(Q1(x),...,Q m(x)), eachQi(x)b e i n g a univariate polynomial over Zpof degree at most d.H e r ed≤min(m,p) depends on the structure of G(actually its nilpotency class). To this end, notice that the semidirect product structure is given by a linear transformation AonZm p. (This is the action of the generator 1 of ZponZm p.) We haveAp=I, whence B=A−IsatisﬁesBp=(A−I)p=Ap−Ip= 0. Therefore, there exists a smallest positive integer d≤min(m,p) such that Bd=0 . A subgroup Hvcomplementary to Zm pinZm p/multicloserightZpconsists of the powers of an element of the form ( v,1) for some v∈Zm p. With the map (5.1) Qv:/braceleftbiggZp→Zm p, t/mapsto→/summationtextt−1 j=0Ajv these powers are the pairs ( Qv(t),t)f o rt∈Zp, and the right cosets of Hvare the sets of the pairs ( Qv(t)+y,t)f o rt∈Zp,w h e r ey∈Zm p. It turns out that the entries of the matrix of/summationtextt−1 j=0Aj, as functions in t, are polynomials of degree at most dwith zero constant term (see [5]). Therefor e, the same holds for the coordinates Q(i)

Question 418 multiple-choice

In abstract algebra, the study of nilpotent groups often involves analyzing the behavior of commutators and their containment within subgroups, especially in the context of p-groups and extensions. Recursive and inductive arguments are commonly used to establish properties across all relevant elements and indices. Which concept plays a central role in demonstrating that higher commutators eventually become trivial in a nilpotent group? 1) Sylow theorem 2) Lower central series 3) Group homomorphism 4) Normalizer condition 5) Coset decomposition 6) Semidirect product structure 7) Orbit-stabilizer theorem

✓ Correct Answer:

The correct answer is 2) Lower central series.

📚 Reference Text:

. If we can show that the last term is, we will have that [a, (r 2 - \)b] p G A p\ and then we can repeat the argument if r 2 — 1 =2, until we have arrived at a proof that [a, b] p G Apl for all a <= A 2, which is Step I. To do this, note that [A2,r2b] ^ (A^A p2,AiAp2); so M2, (r2+p- 2)b] â < [^, (/> - 2)ft], M5, (/> - 2)b], [Ap„ (p - 2)6], ^ 2>. But since r, + r 2 ^ /? — 2, (5) gives that [^,0? - 2)b] ^A p2; similarly (6) and r 2 + r 3 = p ~ 2 give that [Ap2, (p - 2)b] ^ A p\ https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press 1038 J. BUCKLEY AND J. WIEGOLD Finally (8) with r 2 + r 3 ë p — 2 implies that [A% (p - 2)b] ^ A"\ We have now done enough to establish Step I. Step II. If [A h b]p S Apl for all / = 1, 2, . . . , s - 1, then [As, bf ^ A p\ Starting with the hypothesis that [As_„bf ^A p\ taking p-i\\ roots in A and using the fact that [As_]tb] ^Q s-M) we obtain [As_{,b]^ (A x,A^...M ps,A^+],...). Interpreting this statement in the sparse group A p gives do) [A>; s,b]^(A» 2,4,...,Aù^4 +2,---y, and taking p-th roots we have (11) [A s,b]S (A ],A2,Ap3,...,A" s+l,A'f+2,...). We show next that (12) [A s, rsb]p ^ A p\ As before we have for all a e A s, (p) [a, rJbY[a, (r s + \)b]^ 2' • • • [a, (r s + p - 2)b\ p X [a, (r, + p - \)b] = 1. When we commutate (10) with b r s times and use Step I, we find that all but the first and last factors of this identity are in A p . To show that the last factor is also in A p~, we commutate (11) (r s + p — 2) times with b obtaining (13) [A s, (rs + p - \)b] ^([A {,(rs+P- 2)6], [A 2, (rs +p- 2)6], [Ap, (rs + p - 2)b], .... K +1, (rs + p - 2)b], A" 1). Now we have four rather different situations to look at. From (5) we have [A^ip ~ 2)b] ta Ap\ so certainly [A^ (rs + p - 2)b] ^ A p\ https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press NILPOTENT EXTENSIONS 1039 For the second term in (13), we begin by taking p-th roots in the statement of Step I and obtain [A2,rJ,]^(A x,AP2,A^AP\ so that [A2,(rs +p- 2)6] ^ ([A {AP ~ 2)b], [A?2, (p ~ 2)6], [Ap3, (p - 2)6], A* 1). Now (5), (6) and (8) together with rx + r 2 = p — 2, r 2 + r 3 ^ p — 2 are sufficient to give [A2, (rs+p - 2)6]

Question 419 multiple-choice

Quantum algorithms have revolutionized search and attack strategies in cryptography, especially for problems like Learning With Errors (LWE). Understanding the impact of quantum oracles and complexity measures is crucial for assessing post-quantum security. In quantum attacks on LWE, which complexity measure is most relevant for expressing the algorithm's success probability, and why does it replace the classical measure? 1) Min-entropy, because it bounds worst-case unpredictability 2) Guessing complexity, because it aligns with the expected number of quantum guesses needed 3) Rényi entropy, due to its ability to generalize classical uncertainty 4) Statistical distance, as it quantifies distribution closeness 5) Collision entropy, reflecting likelihood of repeated guesses 6) Shannon entropy, as it measures average uncertainty 7) Mutual information, indicating shared randomness between variables

✓ Correct Answer:

The correct answer is 2) Guessing complexity, because it aligns with the expected number of quantum guesses needed.

📚 Reference Text:

to estim ate the output coeﬃcient. We note that the steps 12-14 could have be en in the oracle OWinstead, this would be completely equivalent (and save a small polynomial factor in the overall complexity). Step 16 corresp onds to Step 11 of Algorithm 2. The diﬀerence between the thresholds co mes from the way Theorem 3 works ( CvsC/D) and also because of the two-sided errors made by the oracle (factor η). See Theorem 6 for more details. –Steps 25-28 correspond to Step 5 of Algorithm 2. Speciﬁcally, the c lassical algorithm performs a sequential search by decreasing probability o f ˜senum. Our algorithm uses Lemma 3 to obtain a quantum speedup on this sear ch which requires to create an oracle. 5.2 Analysis of the Quantum Augmented Dual Attack We now analyse the quantum augmented dual attack given in Algorith m 3. In- formally, we ﬁrst establish that with constant probability the algorit hm outputs a guess for senumwhere the corresponding FFT scoring function’s score is above the chosen threshold C. Below, we will instantiate this theorem with appropri- ate choices of thresholds C(and number of samples D). Note that like [MAT22, Thm. 5.2], the following theorem is with respect to a ﬁxed tuple ( s,e). Theorem 6. Let(n,m,q,χ s,χe)be LWE parameters, let (β0,β1,kenum,kﬀt,klat,p,D,C, A,b,η) be the input of Algorithm 3. Let Lbe the list of vectors obtained at Line 3 of Algorithm 3. For any x>0, letSL x={˜senum:∃˜sﬀt,FL(˜senum,˜sﬀt)>x}. With probability at least 9/10,the algorithm returns a value in SL C∪{⊥}. Furthermore, ifSL (1+2η)C/n⌉}a⊔ionslash=∅then the algorithm returns a value in SL Cwith probability at least 9/10. 20 Proof.See Appendix C. Next, we establish that the parameter choices of C,Din Lemma 2 for Al- gorithm 2 allow us to also instantiate Algorithm 3 such that it succeeds with a comparable probability. Lemma 5. Let(n,m,q,χ s,χe)beLWEparameters, let (β0,β1,kenum,kﬀt,klat,p, C,D,η)be a tuple of parameters for Algorithm 3, and let 0< ν < 1. Fix (s,e)∈Zn q×Zm q. By choosing the parameters C,Daccording to Lemma 2 with µ=ν/2, andη/lessorequalslant√ 2πµ 8φfp(µ), Algorithm 3 returns senumwith probability at least 1−ν. Proof.See Appendix C. Next, we state the analogous result as [MAT22, Thm. 5.9] for our se tting, i.e. express the success probability with respect to χs,χerather than a ﬁxed tuple (s,e). The only diﬀerence compared to [MAT22, Thm. 5.9] is that we re- place the Shannon entropy H(X) by the guessing complexity G(X), as discussed in Section 3. Lemma 6. Let(n,m,q,χ s,χe)beLWEparameters, (β0,β1,kenum,kﬀt,klat,p, D,C,η)be parameters for Algorithm 3. Let 0< ν <1,µ=ν/2,η/lessorequalslant√ 2πµ 8φfp(µ). Denote byℓthe expected Euclidean length of the vectors returned by Alg orithm 1. LetG(X)as in Lemma 4. Let D≥Deq·Dround·Darg·Dfpfn(µ)andC= ˜φfp(µ)/radicalbig Darg·Dwith Deq= exp/parenleftigg 4/parenleftbiggπσsℓ q/parenrightbigg2/parenrightigg , D round=/productdisplay ¯s/negationslash=0 sin/parenleftig π¯s p/parenrightig π¯s p −2kfftχs(¯s) Darg=1 2exp/parenleftigg 2/parenleftbigg χe(0)+e−8π2α−2ℓ2 q2 (m+klat)/parenrightbiggm ·/parenleftbigg χs(0)+e−8π2ℓ2 q2(m+klat)/parenrightbiggklat/parenrightigg ≈1 2 Dfpfn(µ) = (φfp(µ)+φfn(µ))2·µ φfp(µ) =Φ−1/parenleftbigg 1−µ 2·G(χs)·pkfft/parenrightbigg and˜φfn(µ) =Φ−1/parenleftig 1−µ 2/parenrightig . Then, with probability at least 1−νthe algorithm returns senum. The proofisexactlythe sameasin [MAT22, Theorem5.9]except that wereplace the inequality E[Nenum(senum)]/lessorequalslant2kenumH(χs) by E[Nenum(senum)]

Question 420 multiple-choice

Quantum algorithms in knot theory often leverage representation theory and the efficient implementation of isometries to encode combinatorial and algebraic data. The computation of polynomial invariants, such as the Jones and HOMFLYPT polynomials, relies on specific algebraic structures and quantum circuit design. Which operation is essential for ensuring the invariance of polynomial evaluations when associating trace closures of braids to knot invariants in the context of quantum algorithms? 1) Applying random permutations to braid strands 2) Doubling all existing strands in the braid 3) Performing Markov moves, specifically conjugation and stabilization 4) Reversing the orientation of all strands 5) Swapping the first and last strands in the braid 6) Removing all crossings from the braid 7) Projecting onto the symmetric subspace of the Hilbert space

✓ Correct Answer:

The correct answer is 3) Performing Markov moves, specifically conjugation and stabilization.

📚 Reference Text:

following commutative diagram V(k,ℓ) nπ(k,n) n(b)/d47/d47 /d15/d15V(k,ℓ) n Rn ρ(k,ℓ) n(b)/d47/d47Rn./d79/d79 Now we show that the isometry Win (4.16) can be eﬃciently implemented. For this, we introducensystems with joint Hilbert space Cn, where each Ciis spanned by computational basis states {|0/an}bracketri}htCi,...,|n/an}bracketri}htCi}. The purpose of these systems is to store the column numbers of eachi. In addition, we introduce kregisters L1,...,Lkwith bases {|0/an}bracketri}ht,...,|n/an}bracketri}ht}which will store the shape of the tableau. Given a qubit encoding |r(t)/an}bracketri}htRnof a tableau t, we deﬁne the isometryW′:Rn→ RnCnLkas W′|r(t)/an}bracketri}htRn=|r(t)/an}bracketri}htRn|c(t)/an}bracketri}htCn|λ(t)/an}bracketri}htLk, 14The Jones Polynomial: Quantum Algorithms and Applications in Quantum Complexity Theory whereλ(t) is the vector of row lengths of t. This isometry can be eﬃciently implemented by ﬁrst preparing the additional systems in the state |0···0/an}bracketri}htCn|0···0/an}bracketri}htLk, after which the composition of local gates W′=W′ n···W′ 2W′ 1is applied, where each W′ iacts on RiCiLkas W′ i|r/an}bracketri}htRi|0/an}bracketri}htCi|λ/an}bracketri}htLk=|r/an}bracketri}htRi|λr+1/an}bracketri}htCi|λ′/an}bracketri}htLk whereλ′isλwith an additional box in row r. For each 1 ≤i < n , we then deﬁne local unitariesWiwhich act as RiRi+1CiCi+1Dias Wi|r/an}bracketri}htRi|r′/an}bracketri}htRi+1|c/an}bracketri}htCi|c′/an}bracketri}htCi+1|0/an}bracketri}htDi=|r/an}bracketri}htRi|r′/an}bracketri}htRi+1|c/an}bracketri}htCi|c′/an}bracketri}htCi+1|c−c′−(r−r′)/an}bracketri}htDi. We may thus write W=W′−1Wn−1···W2W1W′, where we uncompute the CnandLk registers. Because each WiandW′ ican be implemented with O/parenleftbig poly(n,k)/parenrightbig , we ﬁnd that W can be implemented with O/parenleftbig poly(n,k)/parenrightbig local gates. 5 Representation-theoretic formulae for the Jones and HOMF LYPT We will now illustrate how to use the unitary representation s of Section 3 to write expressions for the evaluations, at primitive roots of unity, of the Jone s and one-variable HOMFLYPT polynomials of the trace closure of a braid. We then review th e Temperley-Lieb algebra, which we use to derive a formula for the Jones polynomial of our gene ralized closure of a braid. Jones and HOMFLYPT polynomials of trace closures from the Ma rkov trace Here, we review the theory of Markov traces on Hecke algebras and show how they yield formulae for the one-variable HOMFLYPT polynomials of the t race closure of a braid. Our presentation is primarily based on Section 6 of [19] and Sect ion 3 of [38]. We urge the reader who seeks a deeper understanding to consult these ref erences. We also recommend [4]. Throughout this section, we ﬁx a primitive root of unity q=e2πi/ℓfor some integer ℓ≥3. The theory of Markov traces is best understood in light of the so-called Markov moves on braids. As mentioned earlier, the trace closures of two brai ds are isotopic iﬀ the braids are connected by a ﬁnite sequence of Markov moves. Given any b∈Bn, atype I Markov move consists of conjugating bby any braid x∈Bn, takingb/ma√sto→xbx−1. Atype II Markov move adds another strand to bwhich is twisted either clockwise or counterclockwise with then’th strand at the bottom of b, acting as b/ma√sto→σ± nι(b). It is clear that the trace closures of braids related by the ﬁrst Markov move are isotopic, since the top br aidxis untied by the bottom braidx−1. A picture more clearly illustrates the invariance of the tr ace closure under the second Markov move, as shown in Figure 5.4. These topological considerations are reﬂected algebraica lly as

Question 421 multiple-choice

In quantum computing, coherent control errors arise when the intended evolution of quantum systems is perturbed by unknown noise affecting the Hamiltonians that generate quantum gates. Understanding and mitigating the impact of such errors is crucial for building reliable and fault-tolerant quantum algorithms. Which design principle is most effective for enhancing the robustness of quantum circuits against coherent control errors caused by multiplicative noise in Hamiltonians? 1) Minimizing the total number of quantum gates in the circuit 2) Maximizing circuit depth to allow error averaging 3) Increasing the connectivity between qubits during computation 4) Using only Clifford gates in algorithm design 5) Focusing on shallow circuits with fewer entangling gates 6) Reducing the norms of the Hamiltonians that generate individual gates 7) Employing random gate sequences to scramble error accumulation

✓ Correct Answer:

The correct answer is 6) Reducing the norms of the Hamiltonians that generate individual gates.

📚 Reference Text:

Title: Robustness of quantum algorithms against coherent control errors Year: 2023 Paper ID: c33d6b31bbef715a5b397cd7222741392d67c1dc Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c33d6b31bbef715a5b397cd7222741392d67c1dc Abstract: Coherent control errors, for which ideal Hamiltonians are perturbed by unknown multiplicative noise terms, are a major obstacle for reliable quantum computing. In this paper, we present a framework for analyzing the robustness of quantum algorithms against coherent control errors using Lipschitz bounds. We derive worst-case fidelity bounds which show that the resilience against coherent control errors is mainly influenced by the norms of the Hamiltonians generating the individual gates. These bounds are explicitly computable even for large circuits, and they can be used to guarantee fault-tolerance via threshold theorems. Moreover, we apply our theoretical framework to derive a novel guideline for robust quantum algorithm design and transpilation, which amounts to reducing the norms of the Hamiltonians. Using the $3$-qubit Quantum Fourier Transform as an example application, we demonstrate that this guideline targets robustness more effectively than existing ones based on circuit depth or gate count. Furthermore, we apply our framework to study the effect of parameter regularization in variational quantum algorithms. The practicality of the theoretical results is demonstrated via implementations in simulation and on a quantum computer.

Question 422 multiple-choice

Permutation groups are mathematical structures used to study symmetries and reorderings of sets, with applications in areas such as economics and social choice theory. Certain subgroups exhibit unique properties related to invariance under permutation inversion, which can affect models of decision-making. Which subgroup of the permutation group on four elements is the smallest example of a "nudgable" group, characterized by the breakdown of cardinality invariance under inversion? 1) The cyclic group of order 4 2) The Klein four-group 3) The symmetric group S₄ 4) The alternating group A₄ 5) The dihedral group D₄ 6) The subgroup with 6 elements acting on 4 objects 7) The trivial group containing only the identity permutation

✓ Correct Answer:

The correct answer is 6) The subgroup with 6 elements acting on 4 objects.

📚 Reference Text:

Title: Non-nudgable subgroups of permutations Year: 2017 Paper ID: 73a457678a4ca6150ac4da1480bd4d03401dc8a5 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/73a457678a4ca6150ac4da1480bd4d03401dc8a5 Abstract: Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is called non-nudgable if the cardinality of this set always remains the same when replacing the initial permutation with its inverse. It is called nudgable otherwise. We show that all full permutation groups, standard dihedral groups, half of the alternating groups, and any abelian subgroup are non-nudgable. In the right probabilistic sense, it is thus quite likely that a randomly generated subgroup is non-nudgable. However, the other half of the alternating groups are nudgable. We also construct a smallest possible nudgable group, a 6-element subgroup of the permutation group on 4 elements.

Question 423 multiple-choice

In quantum computing, swap operations are essential for exchanging the states of qubits and play a significant role in the algebraic structure of quantum circuits. These operations can be combined with coefficients and signs to form complex identities that facilitate circuit design and optimization. Which of the following statements accurately describes the algebraic significance of linear combinations involving swap operators with integer coefficients and signs in quantum circuit theory? 1) They serve only as placeholders for measurement operators without affecting qubit states. 2) They represent classical logic gates implemented on classical bits, unrelated to quantum operations. 3) They are used exclusively to define error-correcting codes but not for basic quantum gates. 4) They define scalar multiples of single-qubit rotations without involving permutations. 5) They encode transformations within permutation groups, enabling the expression of complex multi-qubit interactions and symmetries in quantum systems. 6) They are mathematical constructs with no direct application to quantum hardware or algorithms. 7) They describe measurement outcomes rather than operators acting on the quantum state space.

✓ Correct Answer:

The correct answer is 5) They encode transformations within permutation groups, enabling the expression of complex multi-qubit interactions and symmetries in quantum systems..

📚 Reference Text:

2+Swap15 2−Swap15Swap24 2+Swap24 2−Swap24Swap25 2+Swap45 2 Swap12Swap15Swap12= Swap25 Swap12Swap15Swap13=−1 2−Swap12 2+Swap12Swap13 2+Swap12Swap15 2+Swap12Swap35 2−Swap13Swap15 2+ Swap13Swap25 2+Swap15 2−Swap15Swap23 2+Swap23 2−Swap23Swap25 2+Swap35 2 Swap12Swap15Swap14=−1 2−Swap12 2+Swap12Swap14 2+Swap12Swap15 2+Swap12Swap45 2−Swap14Swap15 2+ Swap14Swap25 2+Swap15 2−Swap15Swap24 2+Swap24 2−Swap24Swap25 2+Swap45 2 Swap12Swap2 15= Swap12 Swap12Swap15Swap23=−1 2+Swap12 2−Swap12Swap13 2+Swap12Swap15 2−Swap12Swap35 2+Swap13Swap15 2+ Accepted in Quantum2024-03-25, click title to verify. 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Published under CC-BY 4.0. 68 Swap14Swap25 2+Swap15 2−Swap15Swap24 2+Swap24 2−Swap24Swap25 2+Swap45 2 Swap12Swap45Swap23=−Swap12Swap13Swap45+ Swap12Swap45+ Swap13Swap45+ Swap23Swap45−Swap45 Swap12Swap45Swap24=−1 2+Swap12 2−Swap12Swap14 2−Swap12Swap15 2+Swap12Swap45 2+Swap14Swap15 2+ Swap14Swap25 2+Swap15 2−Swap15Swap24 2+Swap24 2+Swap24Swap25 2−Swap45 2 Swap12Swap45Swap25=−3 2+Swap12 2−Swap12Swap14 2−Swap12Swap15 2+Swap12Swap45 2−Swap14Swap15 2+ −Swap14Swap25 2+Swap15 2+Swap15Swap24 2+Swap24 2−Swap24Swap25 2+Swap45 2+ Swap14+ Swap25 Swap12Swap45Swap34= Swap12Swap34Swap35 Swap12Swap45Swap35=−Swap12+ Swap12Swap34−Swap12Swap34Swap35+ Swap12Swap35+ Swap12Swap45 Swap12Swap2 45= Swap12 Swap13Swap14Swap12=−1 2+Swap12 2−Swap12Swap13 2−Swap12Swap14 2+Swap12Swap34 2+Swap13Swap14 2+ Swap13Swap24 2+Swap14 2−Swap14Swap23 2+Swap23 2+Swap23Swap24 2−Swap34 2 Swap13Swap14Swap13= Swap34 Swap13Swap2 14= Swap13 Swap13Swap14Swap15=1 2−Swap13 2+Swap13Swap14 2+Swap13Swap15 2+Swap13Swap45 2+Swap14Swap15 2+ −Swap14Swap35 2−Swap15 2+Swap15Swap34 2−Swap34 2+Swap34Swap35 2−Swap45 2 Swap13Swap14Swap23=1 2−Swap12 2+Swap12Swap13 2+Swap12Swap14 2+Swap12Swap34 2+Swap13Swap14 2+ −Swap13Swap24 2−Swap14 2+Swap14Swap23 2−Swap23 2+Swap23Swap24 2−Swap34 2 Swap13Swap14Swap24=−1 2+Swap12 2−Swap12Swap13 2+Swap12Swap14 2−Swap12Swap34 2+Swap13Swap14 2+ Swap13Swap24 2−Swap14 2+Swap14Swap23 2+Swap23 2−Swap23Swap24 2+Swap34 2 Swap13Swap14Swap25= Swap13Swap14Swap25 Swap13Swap14Swap34= Swap14 Swap13Swap14Swap35=−1 2+Swap13 2+Swap13Swap14 2−Swap13Swap15 2−Swap13Swap45 2−Swap14Swap15 2+ Swap14Swap35 2+Swap15 2+Swap15Swap34 2−Swap34 2+Swap34Swap35 2+Swap45 2 Swap13Swap14Swap45=−1 2−Swap13 2+Swap13Swap14 2+Swap13Swap15 2+Swap13Swap45 2−Swap14Swap15 2+ Swap14Swap35 2+Swap15 2−Swap15Swap34 2+Swap34 2−Swap34Swap35 2+Swap45 2 Swap13Swap15Swap12=−1 2+Swap12 2−Swap12Swap13 2−Swap12Swap15 2+Swap12Swap35 2+Swap13Swap15 2+ Swap13Swap25 2+Swap15 2−Swap15Swap23 2+Swap23 2+Swap23Swap25 2−Swap35 2 Swap13Swap15Swap13= Swap35 Swap13Swap15Swap14=−1 2−Swap13 2+Swap13Swap14 2+Swap13Swap15 2+Swap13Swap45 2−Swap14Swap15 2+ Swap14Swap35 2+Swap15 2−Swap15Swap34 2+Swap34 2−Swap34Swap35 2+Swap45 2 Swap13Swap2 15= Swap13 Swap13Swap15Swap23=1 2−Swap12 2+Swap12Swap13 2+Swap12Swap15 2+Swap12Swap35 2+Swap13Swap15 2+ −Swap13Swap25 2−Swap15 2+Swap15Swap23 2−Swap23 2+Swap23Swap25 2−Swap35 2 Swap13Swap15Swap24= Swap13Swap15Swap24 Swap13Swap15Swap25=−1 2+Swap12 2−Swap12Swap13 2+Swap12Swap15 2−Swap12Swap35 2+Swap13Swap15 2+ Swap13Swap25 2−Swap15 2+Swap15Swap23

Question 424 multiple-choice

In quantum computing, the Hidden Subgroup Problem (HSP) provides a framework for many algorithms, including those that address graph isomorphism over non-abelian groups. The effectiveness of quantum algorithms often depends on the properties of coset states and the feasibility of entangled quantum measurements. Which statement best explains why quantum algorithms for non-abelian HSPs, such as those related to graph isomorphism, are challenging to implement efficiently? 1) Quantum algorithms cannot prepare coset states for non-abelian groups. 2) All non-abelian group problems can be solved with single-qubit measurements. 3) Efficient algorithms for non-abelian HSPs universally rely on classical post-processing. 4) Highly entangled quantum measurements on at least Ω(n log n) coset states are required, making practical implementation difficult. 5) Quantum speedups for non-abelian HSPs are due to the simplicity of their group structure. 6) Abelian and non-abelian HSPs are equally tractable for current quantum hardware. 7) Sampling uniform superpositions over isomorphic graphs is a well-understood and efficient quantum approach.

✓ Correct Answer:

The correct answer is 4) Highly entangled quantum measurements on at least Ω(n log n) coset states are required, making practical implementation difficult..

📚 Reference Text:

arXiv:quant-ph/0511148v1 15 Nov 2005Limitationsof QuantumCosetStatesforGraphIsomorphism Sean Hallgren,MartinR¨ otteler, and Pranab Sen NECLaboratories America, Inc. 4 IndependenceWay, Suite200 Princeton, NJ08540,U.S.A. {hallgren,mroetteler,pranab }@nec-labs.com Abstract It has been known for some time that graph isomorphism reduce s to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum co mputation are obtained by solving in- stances of the HSP. A common feature of the resulting algorit hms is the use of quantum coset states, whichencodethehiddensubgroup. Anopenquestionhasbeenh owharditistousethesestatestosolve graph isomorphism. It was recently shown by Moore, Russell, and Schulman [MRS05] that only an exponentially small amount of information is available fro m one, or a pair of coset states. A potential sourceof powerto exploitare entangledquantummeasuremen tsthat act jointly onmanystates at once. We show that entangled quantum measurementson at least Ω(nlogn)coset states are necessary to get useful information for the case of graph isomorphism, match ing an informationtheoretic upper bound. This may be viewed as a negative result because highly entang led measurements seem hard to imple- ment in general. Our main theorem is very general and also rul es out using joint measurementson few cosetstatesforsomeothergroups,suchas GL(n,Fpm)andGnwhereGisﬁniteandsatisﬁesasuitable property. 1 Introduction Almost all exponential speedups that have been achieved in q uantum computing are obtained by solving some instances of the Hidden Subgroup Problem (HSP). In part icular, the problems underlying Shor’s al- gorithms for factoring and discrete logarithm [Sho97], as w ell as Simon’s problem [Sim94], can be nat- urally generalized to the HSP: given a function f:G→Sfrom a group Gto a setSthat is constant on left cosets of some subgroup H≤Gand distinct on different cosets, ﬁnd a set of generators for H. Ideally, we would like to ﬁnd Hin time polynomial in the input size, i.e. log|G|. The abelian HSP [Kit95, BH97, ME98], i.e., when Gis an abelian group, lies at the heart of efﬁcient quantum al- gorithms for important number-theoretic problems like fac toring, discrete logarithm, Pell’s equation, unit group of anumber ﬁeldetc. [Sho97, Hal02,Hal05,SV05]. It has been known for some time that graph isomorphism reduce s to the HSP over the symmetric group [Bea97,EHK99a], anon-abelian group. Whilethenon-a belian HSPhasreceived muchattention asa result, efﬁcient algorithms are known only for some special classes of groups [IMS03, FIM+03, MRRS04, BCD05]. Ontheother hand, theHSPpresents asystematic wayt otryand approach thegraph isomorphism problem, and this approach is rooted in developing a deeper u nderstanding of how far techniques and tools that have worked in the abelian case can be applied. Tothe bes t of our knowledge, the only other approach to solve graph isomorphism on a quantum computer is by creati ng a uniform superposition of all graphs 1 isomorphic to a given graph. It has been proposed to create th is superposition via quantum sampling of Markov chains [AT03],however, very little isknown about th is. One of the key features of a quantum computer is that it can com pute functions in superposition. This fact alone does not lend itself to exponential speedups, for instance for unstructured search problems it merely leads to a

Question 425 multiple-choice

Quantum computing relies on foundational gates and principles to construct complex algorithms and circuits. Understanding how universal gate sets and error management enable scalable quantum operations is essential for advanced study in the field. Which statement correctly describes the Solovay-Kitaev Theorem’s significance in quantum computing? 1) It guarantees that any quantum state can be exactly replicated using only Toffoli gates. 2) It ensures any n-qubit unitary operation can be efficiently approximated to arbitrary accuracy using a finite universal gate set. 3) It states that phase shift gates alone are sufficient for universal quantum computation. 4) It limits the number of quantum gates required to simulate classical logic circuits. 5) It allows probabilistic outcomes to be converted into deterministic ones in quantum algorithms. 6) It demonstrates that only entangling gates are necessary for error correction. 7) It proves that measurement collapses do not affect computational accuracy in quantum circuits.

✓ Correct Answer:

The correct answer is 2) It ensures any n-qubit unitary operation can be efficiently approximated to arbitrary accuracy using a finite universal gate set..

📚 Reference Text:

0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 01 CCCCCCCCCCCCCCCCCCCCA: 18 The Toffoli gate is shown in Figure 2.4, with the AND operation ^deﬁned in Table 2.1. Figure 2.4: The Toffoli Gate controljxijxi controljyijyi targetjzi jz(x^y)i Table 2.1: The AND Operation xyx^y 11 1 10 0 01 0 00 0 The phase shift can be implemented as a controlled transformation CP=0 BBBBBBB@1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 eiq1 CCCCCCCA: (2.55) where the phase shift is applied to the target only when the control qubit is in the state j1i. An important ingredient in many quantum circuits is the controlled discrete phase gate CRk described by CRk:=j00ih00j+j01ih01j+j10ih10j+e2pi 2kj11ih11j =0 BBBBBBB@1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e2pi 2k1 CCCCCCCA(2.56) 19 2.3.3 Universal Quantum Gates In classical computation, complicated transformations are implemented as a sequence of simpler operations selected from a set of elementary gates. For example, any logical circuit can be simulated using only the AND and the NOT gate, and so these two gates form a set of universal logical gates. In quantum computation, a circuit designer can similarly rely on universal quantum gates. However, due to the probabilistic nature of quantum computation, a unitary transformation Vapproximates a desired unitary transformation Uto a speciﬁed level of accuracy [22], with the approximation error deﬁned by E(U;V)max jyik(UV)jyik (2.57) where kjcikp hcjci: (2.58) Theorem 2.3.1. The errors of approximating a series of unitary gates U 1;:::;Unby unitary trans- formations V 1;:::;Vnadd, so that E(UnUn1:::U1;VnVn1:::V1)E(Un;Vn)+E(Un1;Vn1)++E(U1;V1) (2.59) Proof. Suppose the desired unitary transformation U2U1is implemented by a unitary transforma- tionV2V1. Then E(U2U1;V2V1) =max jyik(U2U1V2V1)jyik =max jyik(U2U1V2U1+V2U1V2V1)jyik (2.60) =max jyik(U2V2)U1jyi+V2(U1V1)jyik: 20 Since U1andV2are unitary, E(U2U1;V2V1)E(U2;V2)+E(U1;V1): (2.61) and the result follows by induction. If a unitary transformation is to be implemented to a precision ewith Ngates, it is only necessary to implement each gate to a precision ofe N. It has been shown that all quantum circuits can be constructed by using only CNOT and single-qubit gates [27], which brings us to a deﬁnition of a universal quantum gate set. Deﬁnition 2.3.2. A set of gates Gis a universal quantum gate set if for every e>0,n2N, any nqubit unitary operator can be approximated to an accuracy eby a ﬁnite sequence of gates from setG. The universal gate sets commonly used in quantum computation are: 1. The set G=fCNOT ;H;Tg[23] 2. The set G=fCNOT ;H;S;Toffolig[24] The Solovay-Kitaev Theorem [25, 26] establishes that gates from a universal gate set approximate any unitary transformation to an arbitrary accuracy ewith the efﬁciency polylogarithmic in1 e. 2.4 Probabilistic Nature of Quantum Algorithms The nature of the quantum world dictates that quantum computers are

Question 426 multiple-choice

Quantum information science frequently leverages advanced mathematical frameworks to design algorithms and explore foundational phenomena. Category theory, group theory, and algebraic structures are central to developing unified models and practical quantum computing applications. Which mathematical concept enables a unified process-theoretic description of quantum algorithms, including constructions for the Fourier transform and Pontryagin duality, and forms the basis for generalizing unitary oracles and analyzing non-locality through phase groups? 1) Hilbert spaces 2) Lie algebras 3) Tensor networks 4) Boolean algebras 5) Topological groups 6) Dagger symmetric monoidal categories 7) Markov chains

✓ Correct Answer:

The correct answer is 6) Dagger symmetric monoidal categories.

📚 Reference Text:

Title: The Abstract Structure of Quantum Algorithms Year: 2015 Paper ID: 7c281027685fe2e650bdbd2d6c471f38b0a993f1 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/7c281027685fe2e650bdbd2d6c471f38b0a993f1 Abstract: Quantum information brings together theories of physics and computer science. This synthesis challenges the basic intuitions of both fields. In this thesis, we show that adopting a unified and general language for process theories advances foundations and practical applications of quantum information. Our first set of results analyze quantum algorithms with a process theoretic structure. We contribute new constructions of the Fourier transform and Pontryagin duality in dagger symmetric monoidal categories. We then use this setting to study generalized unitary oracles and give a new quantum blackbox algorithm for the identification of group homomorphisms, solving the GROUPHOMID problem. In the remaining section, we construct a novel model of quantum blackbox algorithms in non-deterministic classical computation. Our second set of results concerns quantum foundations. We complete work begun by Coecke et al., definitively connecting the Mermin non-locality of a process theory with a simple algebraic condition on that theory's phase groups. This result allows us to offer new experimental tests for Mermin non-locality and new protocols for quantum secret sharing. In our final chapter, we exploit the shared process theoretic structure of quantum information and distributional compositional linguistics. We propose a quantum algorithm adapted from Weibe et al. to classify sentences by meaning. The clarity of the process theoretic setting allows us to recover a speedup that is lost in the naive application of the algorithm. The main mathematical tools used in this thesis are group theory (esp. Fourier theory on finite groups), monoidal category theory, and categorical algebra.

Question 427 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) are foundational to many breakthroughs in quantum computing, particularly when dealing with non-abelian groups such as Weyl-Heisenberg groups. Techniques from representation theory and non-commutative Fourier analysis play a key role in making these algorithms efficient and practical. Which key feature distinguishes the improved quantum algorithm for the HSP over Weyl-Heisenberg groups from previous approaches, enabling lower quantum resource requirements? 1) Use of abelian group structures to simplify subgroup identification 2) Replacement of coset state analysis with classical post-processing 3) Application of tensor networks instead of Fourier analysis 4) Utilization of Grover's search to enhance speed 5) Reduction in the number of required qubits via state compression 6) Implementation of random sampling over group elements 7) Decrease in coset states per iteration from four to two

✓ Correct Answer:

The correct answer is 7) Decrease in coset states per iteration from four to two.

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Title: An Efficient Quantum Algorithm for the Hidden Subgroup Problem over Weyl-Heisenberg Groups Year: 2008 Paper ID: b9257e8a94113583ad31b87b3b0a6b09b36ba6fb Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b9257e8a94113583ad31b87b3b0a6b09b36ba6fb Abstract: Many exponential speedups that have been achieved in quantum computing are obtained via hidden subgroup problems (HSPs). We show that the HSP over Weyl-Heisenberg groups can be solved efficiently on a quantum computer. These groups are well-known in physics and play an important role in the theory of quantum error-correcting codes. Our algorithm is based on non-commutative Fourier analysis of coset states which are quantum states that arise from a given black-box function. We use Clebsch-Gordan decompositions to combine and reduce tensor products of irreducible representations. Furthermore, we use a new technique of changing labels of irreducible representations to obtain low-dimensional irreducible representations in the decomposition process. A feature of the presented algorithm is that in each iteration of the algorithm the quantum computer operates on two coset states simultaneously. This is an improvement over the previously best known quantum algorithm for these groups which required four coset states.

Question 428 multiple-choice

In the study of modular group representations on polynomial spaces, explicit matrix constructions are key to understanding the symmetries and invariants present in quantum algebra and related mathematical physics. The generators of SL(2,Z), typically represented by matrices S and T, play a central role in defining transformation properties of polynomial bases. When l is a prime number in the context of modular representations arising from quantum algebras, which property holds for the representation of SL(2,Z) on the polynomial basis V? 1) It decomposes into invariant subspaces of dimension two. 2) It is irreducible. 3) The T-matrix is always non-diagonalizable. 4) All eigenvalues of the S-matrix are equal to one. 5) The representation space is infinite-dimensional. 6) The matrices S and T commute. 7) The representation necessarily fails to satisfy the defining relations of SL(2,Z).

✓ Correct Answer:

The correct answer is 2) It is irreducible..

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from (3.91) by omitting one or two roots. If we multiply equation (3.120) and(3.121) with the projector πt(K) for suitable choices of aandbwe obtain these polynomials on the left hand side. The identities (3.117) - (3.119) follow using πt(K)Qj=πt(K)b(j+t). ✷ Lemma 18putsusnowinthepositiontodetermine theactionof Soonthepolynomials φk(X) and (X−b(k))φk(X). Insertion into (3.109) yields for k= 0,...,(m−1): So/parenleftBig πt(K)(X−b(k))φk(X)/parenrightBig =/summationdisplay b∈Z/lπb(K)∆tk b(X), (3.122) So/parenleftBig πt(K)φk(X)/parenrightBig =/summationdisplay b∈Z/lπb(K)Γtk b(X), (3.123) So/parenleftBig πt(K)φm(X)/parenrightBig =/summationdisplay b∈Z/lπb(K)∆tm b(X). (3.124) The polynomials ∆tk b(X) and Γtk b(X) are deﬁned by ∆tk b(X) := (3.125) 35 q√ ll−1/summationdisplay j=0(−1)j[j]! [l−1−j]!q(j+2b)(j+2t+1)l−1/productdisplay i=j+1(b(k)−b(i+t))l−1/productdisplay i=j+1(X−b(i+b)), and Γtk b(X) := (3.126) q√ ll−2/summationdisplay j=0l−2/summationdisplay s=j+1(−1)j[j]! [l−1−j]!q(j+2b)(j+2t+1)l−1/productdisplay i=j+1 i/ne}ationslash=s(b(k)−b(i+t))l−1/productdisplay i=j+1(X−b(i+b)) TheactionofSoonVisnow easily obtainedbysumming equations (3.122)to (3.124)over an appropriate range of tand combining them in (3.94). The result contains products of projections πb(X) with polynomials of the form/summationtext t∆kt b(X) and/summationtext tΓkt b(X) . The polynomials can be expanded for every weight binto the basisVdeﬁned in Lemma14; the coeﬃcients of PpandNpare obtained by the values and values of the derivatives of these polynomials at X=b(p) . By general construction the So- matrix has to map Vto itself. As in the remark following Lemma 15 this can be used to produce new families of partition identities. In order to ﬁnd the matrix coeﬃcients of the SL(2,Z) representation we need the following quantities: η(dA,dB) := (3.127) [dA]2(q−q−1)l l2min(dB,dA−1)/summationdisplay j=1dA/summationdisplay s=j+1[l−1−j]!(−1)j [j]q(j−dB)(j+1+dA−2s)× j/productdisplay i=1(q−q−1)[s−i][dA−s+i]j−1/productdisplay i=1/parenleftBig q(dB−i)−q−(dB−i)/parenrightBig , µ(dA,dB) := (3.128) [dA]2(q−q−1)l l2/summationdisplay s∈Z/ldB/summationdisplay j=1j/summationdisplay r=1[l−1−j]!(−1)j [j]q(j−dB)(j+1+dA−2s)× j/productdisplay i=1 i/ne}ationslash=r(q−q−1)[s−i][dA−s+i]j−1/productdisplay i=1/parenleftBig q(dB−i)−q−(dB−i)/parenrightBig and ρ(dB) :=(q−q−1)l−1 l2dB/summationdisplay j=1l/summationdisplay s=j+1[l−1−j]!q(j−dB)(j+1−2s) [j]× (3.129) 36 j/productdisplay i=1(q−q−1)[s−i]2j−1/productdisplay i=1/parenleftBig q(dB−i)−q−(dB−i)/parenrightBig The main result of the previous calculation - the SL(2,Z) representation on V- is described in the next theorem: Theorem 2 LetP0, N+ 0, N− 0, P1,...,N (m−1), Pmbe the ordered basis of Vas deﬁned in Lemma 14 . Then the following matrices deﬁne a projective SL(2,Z)representation. TheTomatrix is given by (3.112). TheSomatrix is given by the following formulae: 1. Fork= 0,...,m−1 So(N∓ k) =q√ lq−q−1 l[d± k]2m−1/summationdisplay p=0[d± kd∗ p] [d∗p]Pp +q√ l[d± k]2d± kPm+q√ lm−1/summationdisplay p=0/summationdisplay ǫ=±η(d± k,dǫ p)Nǫ p 2. Fork= 0,...,m−1 So(Pk) =q√ l[2d∗ k] [d∗ k]Pm +q√ lm−1/summationdisplay p=0/summationdisplay ǫ=±/parenleftbigg µ(d∗ k,dǫ p)+[2d∗ k] [d∗ k]3η(d+ k,dǫ p)+η(d− k,dǫ p) (q−q−1)/parenrightbigg Nǫ p 3. So(Pm) =q√ lPm+q√ lm−1/summationdisplay p=0/summationdisplay ǫ=±ρ(dǫ p)Nǫ p These matrices satisfy the relations /parenleftBig SoTo/parenrightBig3=γq−1q(1−m)1 I (So)2=q21 I Here the superscript ∗means that either + or −can be inserted yielding the same result. 37 5.) The Structure of the SL(2,Z)-Representation on V:For small values of lthe fol- lowing polynomial identities hold true: η(d+ k,d+ p) +η(d− k,d+ p) =η(d+ k,d− p) +η(d− k,d− p) and ρ(ρ+ p) =ρ(d− p) (3.130) In this case its easy to see that the representation contains an m+1-dimensional subrep- resentation spanned by the Nj’s andPm. Onthissubspacethe T-matrixisdiagonalandhaseigenvalues {q2j(j+1), j= 0,...,m}, i.e., one more than the ﬁnite m-dimensional representation obtained from the semisimpli- ﬁed representation category. For prime lit is not hard to see that the representation is irreducible. Also, for small lwe ﬁnd that it is ﬁnite. It is clear

Question 429 multiple-choice

In the theory of quantum groups, algebra generators obey modified commutation relations governed by a deformation parameter, and representations involve non-commutative structures. Coproducts play a central role in defining how group elements act on tensor products of representations. Which statement best characterizes the construction of group elements in quantum groups such as Uq(SL(2)) when considering their coproducts in representations? 1) Group elements are always linear combinations of the generators with real coefficients. 2) The coproduct of a group element is always the direct sum of the coproducts of individual generators. 3) Group elements in higher-spin representations are independent of the fundamental representation. 4) Universal formulas for group elements exist for all representations in quantum groups. 5) The commutation relations become trivial when the deformation parameter q equals 1. 6) The coproduct structure of group elements is always preserved under classical limits. 7) Group elements must be constructed so that their coproduct commutes with intertwining operators, and their explicit form depends on the chosen representation, with no universal formulas known.

✓ Correct Answer:

The correct answer is 7) Group elements must be constructed so that their coproduct commutes with intertwining operators, and their explicit form depends on the chosen representation, with no universal formulas known..

📚 Reference Text:

this to be true it is essential that commutation relations (45) are exact ly what they are. Indeed, the l.h.s. of the equation (29) (using (33) and (45)) is equal to −q1 2/radicalig M− ¯t(b+dt)/radicalig M+ ¯t′(a+ct′)+q−1 2/radicalig M− ¯t(a+ct)/radicalig M+ ¯t′(b+dt′) = = (q−1 2ab−q1 2ba)+(q−1 2cd−q1 2dc)tt′+(q−1 2cb−q1 2da)t+(q−1 2ad−q−1 2bc)t′= = (q−1 2t′−q1 2t)detqg,(46) which coincides with the r.h.s. of the equation (29). To perform the similar check for any half-integer-spin representa tion, let us note that the 14 corresponding τ-function can be easily written in terms of τ1 2. Indeed, |n/an}bracketri}htλ=/parenleftig qT0⊗T−+T−⊗q−T0/parenrightign|0/an}bracketri}htλ−1 2⊗|0/an}bracketri}ht1 2= =q−n/2/parenleftig |n/an}bracketri}htλ−1 2⊗|0/an}bracketri}ht1 2+[n]qλ|n−1/an}bracketri}htλ−1 2⊗|1/an}bracketri}ht1 2/parenrightig ; λ/an}bracketle{tn|=λ−1 2/an}bracketle{t0|⊗1 2/an}bracketle{t0|=/parenleftig qT0⊗T++T+⊗q−T0/parenrightign= =q−n/2/parenleftig λ−1 2/an}bracketle{tn|⊗1 2/an}bracketle{t0|+[n]qλ λ−1 2/an}bracketle{tn−1|⊗1 2/an}bracketle{t1|/parenrightig .(47) Thus λ/an}bracketle{tk|g|n/an}bracketri}htλ=q−k+n 2/bracketleftig λ−1 2/an}bracketle{tk|g|n/an}bracketri}htλ−1 2/an}bracketle{t+|g|+/an}bracketri}ht+qλ[n]λ−1 2/an}bracketle{tk|g|n−1/an}bracketri}htλ−1 2/an}bracketle{t+|g|−/an}bracketri}ht+ +qλ[k]λ−1 2/an}bracketle{tk−1|g|n/an}bracketri}htλ−1 2/an}bracketle{t−|g|+/an}bracketri}ht+q2λ[k][n]λ−1 2/an}bracketle{tk−1|g|n−1/an}bracketri}htλ−1 2/an}bracketle{t−|g|−/an}bracketri}ht/bracketrightig(48) or, in terms of generating ( τ-)functions: τλ(t,¯t|g) =/radicalig M− tM− ¯t/parenleftig τλ−1 2(t,¯t|g)/parenleftig a+qλ¯tb+qλtc+q2λt¯td/parenrightig/parenrightig . (49) Applying this procedure recursively we get: τλ(t,¯t|g) =τλ−1 2(q−1 2t,q−1 2¯t|g)τ1 2(qλ−1 2t,qλ−1 2¯t|g) = ifλ∈Z/2=τ1 2(q1 2−λt,q1 2−λ¯t|g)τ1 2(q3 2−λt,q3 2−λ¯t|g)...τ1 2(qλ−1 2t,qλ−1 2¯t|g),(50) i.e. for half-integer λ τλis a polynomial of degree 2 λina,b,c,d. For example, τ1(t,¯t|g) =τ1 2(q−1 2t,q−1 2¯t|g)τ1 2(q1 2t,q1 2¯t|g) = = (a+q−1 2¯tb+q−1 2tc+q−1t¯td)(a+q1 2¯tb+q1 2¯tc+qt¯td) = =a2+(q1 2ab+q−1 2ba)¯t+(q1 2ac+q−1 2ca)t+b2¯t2+ +(qad+bc+cb+q−1da)t¯t+c2t2+(q1 2bd+q−1 2db)t¯t2+ +(q1 2cd+q−1 2dc)t2¯t+d2t2¯t2(51) Using the relations like q1 2ab+q−1 2ba= [2]q1 2ba= [2]q−1 2abetc. (52) 15 one gets for this case τ1(t,¯t|g) =a2+[2]q−1 2ab¯t+[2]q−1 2act+b2¯t2+([2]qbc+[2]da)t¯t+c2t2+ +[2]q−1 2dbt¯t2+[2]q−1 2dct2¯t+d2t2¯t2.(53) With this explicit expression, one can trivially make the calculations simila r to (refcheck) in order to check manifestly equation (29) for λ= 1,λ′=1 2,1 and equation (30) for λ=λ′=1 2. Thus, we showed explicitly (for the case of SLq(2)) that the quantum bilinear identities have as many solutions as the classical ones, provided the τ-function is allowed to take values in non-commutative ring A(G). 4.2 Comment on the notion of “group element” To avoidpossible confusionwe present here anillustrative example of how theelements gfrom the universal enveloping algebra could look like, in order to guarante e that the “group co- product” ∆( g) =g⊗gcommutes with any intertwining operator. In its turn the intertwinin g operators are deﬁned as commuting with the “algebra coproduct” ∆(T)/ne}ationslash=T⊗I+I⊗T. Construction of such elements gbecomes possible if gis allowed to take values in Uq(G|A(G)): the universal enveloping algebra with coeﬃcients in the non-commut ative “coordinate ring” A(G) (i.e. made out of a,b,c,d-like objects). Unfortunately we know such gonly in a form which strongly depends on particular representation: more univer sal formulas still remain to be found. As everywhere in this paper, we restrict our example to the simplest case: this time to a product of two spin-1 2representations of Uq(SL(2)), still this will be enough to illustrate the most obscure points of the story. So the claim is that the “group element” in a spin-1 2representation is equal to g=αqT0+βq−T0+bT++cT−, (54) with α=aq1 2−dq−1 2 q−q−1, β=−aq−1 2+dq1 2 q−q−1. (55) 16 Then ∆(g) =g⊗g=α2qT0⊗qT0+β2q−T0⊗q−T0+ +αβqT0⊗q−T0+βαq−T0⊗qT0+bc(T+⊗T−+T−⊗T+)+ +(αqT0+βq−T0)⊗bT++bT+⊗(αqT0+βq−T0)+ +(αqT0+βq−T0)⊗cT−+cT−⊗(αqT0+βq−T0).(56) The r.h.s. does not look like any combination of ∆( T±)

Question 430 multiple-choice

In the representation theory of finite-dimensional algebras, understanding the relationship between projective indecomposable modules and simple modules is fundamental, especially for degenerate versions of classical algebras such as the $0$-Schur algebra. Functorial connections between module categories of related algebras can reveal deeper structural insights and facilitate categorification. In the context of the $0$-Schur algebra $\mathbf{S}_0(n,r)$, what is the role of the "top" of a projective indecomposable module? 1) It is the largest simple quotient, providing a classification of simple modules. 2) It is the submodule generated by all vectors of highest weight. 3) It is the kernel of the module homomorphism to the regular representation. 4) It coincides with the radical of the projective module. 5) It determines the Loewy length of the module. 6) It is the largest projective summand contained in the module. 7) It coincides with the socle of the module.

✓ Correct Answer:

The correct answer is 1) It is the largest simple quotient, providing a classification of simple modules..

📚 Reference Text:

Title: Representation Theory of $0$-Schur Algebras and Related Categories Year: 2025 Paper ID: 45b749c28a244d1c8547999403e1fa6ba5cd70a1 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/45b749c28a244d1c8547999403e1fa6ba5cd70a1 Abstract: Jensen, Su, and Yang described the projective indecomposable modules of the $0$-Schur algebra $\mathbf{S}_0(n,r)$ using its geometric realization. In this paper, the simple modules of $\mathbf{S}_0(n,r)$ are identified by computing the tops of the projective indecomposable modules. Furthermore, functorial relations among the module categories $\mathbf{H}_r(0)$\textsf{-mod}, $\mathbf{S}_0(n,r)$\textsf{-mod}, and $U_0(\mathfrak{gl}_n)$\textsf{-mod} are examined, where $\mathbf{H}_r(0)$ denotes the $0$-Hecke algebra and $U_0(\mathfrak{gl}_n)$ denotes the degenerate quantum group.

Question 431 multiple-choice

Quantum computing leverages mathematical group structures to solve complex computational problems, with the Hidden Subgroup Problem (HSP) serving as a foundational framework for many quantum algorithms. The relationship between the decision and search versions of HSP varies across different families of groups, impacting algorithmic design and efficiency. For which type of group does the polynomial-time equivalence between the decision and search versions of the Hidden Subgroup Problem specifically depend on the group's order possessing the property of smoothness? 1) Cyclic groups 2) Abelian groups 3) Permutation groups 4) Dihedral groups 5) Symmetric groups 6) Alternating groups 7) General linear groups

✓ Correct Answer:

The correct answer is 4) Dihedral groups.

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Title: The central nature of the Hidden Subgroup problem Year: 2006 Paper ID: d81f25ad9c0ede684d5179933442501df7298b07 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d81f25ad9c0ede684d5179933442501df7298b07 Abstract: We show that several problems that figure prominently in quantum computing, including Hidden Coset, Hidden Shift, and Orbit Coset, are equivalent or reducible to Hidden Subgroup for a large variety of groups. We also show that, over permutation groups, the decision version and search version of Hidden Subgroup are polynomial-time equivalent. For Hidden Subgroup over dihedral groups, such an equivalence can be obtained if the order of the group is smooth. Finally, we give nonadaptive program checkers for Hidden Subgroup and its decision version.

Question 432 multiple-choice

In mathematical physics, quantum Chern-Simons theory explores deep connections between topology, gauge theory, and quantum groups. The study of quantum representations of mapping class groups, especially for genus one surfaces, is central to understanding symmetries in quantum field theories. Which mathematical tool is generalized to provide explicit formulas for the quantum representation of the mapping class group associated with genus one surfaces and complex gauge groups? 1) Fourier-Mukai transform 2) Radon transform 3) Laplace transform 4) Mellin transform 5) Heisenberg group representation 6) Weil-Gel'fand-Zak transform 7) Haar measure decomposition

✓ Correct Answer:

The correct answer is 6) Weil-Gel'fand-Zak transform.

📚 Reference Text:

Title: The Genus One Complex Quantum Chern-Simons Representation of the Mapping Class Group Year: 2016 Paper ID: 06a9097ffedf42f9fbda7664d78e2722049b8b98 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/06a9097ffedf42f9fbda7664d78e2722049b8b98 Abstract: In this paper we compute explicitly, following Witten's prescription, the quantum representation of the mapping class group in genus one for complex quantum Chern-Simons theory associated to any simple and simply connected complex gauge group $G_{\mathbb C}$. We use a generalization of the Weil-Gel'fand-Zak transform to exhibit an explicit expression for the representation.

Question 433 multiple-choice

In group theory, varieties classify groups by shared identities, and wreath products are a central construction used to understand group extensions and automorphism groups. The interplay between nilpotent groups and abelian groups of finite exponent is crucial in determining when varieties generated by composite groups coincide with products of varieties. Which structural property must an abelian group $B$ of exponent $n$ possess to ensure that the variety generated by the wreath product $A\,Wr\,B$, where $A$ is nilpotent of finite exponent $m$, equals the product of varieties generated by $A$ and $B$? 1) $B$ must contain a subgroup isomorphic to $C_{d}^c \times C_{n/d}^\infty$, where $c$ is the nilpotency class of $A$, and $d$ is the largest divisor of $n$ coprime to $m$. 2) $B$ must be a finite cyclic group of order $m$. 3) $B$ must be isomorphic to $C_{m}^\infty$ for some $m$ dividing $n$. 4) $B$ must be a direct product of $c$ copies of $C_{n}$, where $c$ is the nilpotency class of $A$. 5) $B$ must be a finite abelian group of exponent dividing $m$. 6) $B$ must contain a subgroup isomorphic to $C_{n}^c \times C_{d}^\infty$, where $d$ divides $n$ and is coprime to $c$. 7) $B$ must have exponent equal to the nilpotency class of $A$.

✓ Correct Answer:

The correct answer is 1) $B$ must contain a subgroup isomorphic to $C_{d}^c \times C_{n/d}^\infty$, where $c$ is the nilpotency class of $A$, and $d$ is the largest divisor of $n$ coprime to $m$..

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Title: A Classification Theorem for Varieties Generated by Wreath Products of Groups Year: 2016 Paper ID: 644eafc07879ee48bc4940165d090844a3f9614d Source: semantic-scholar URL: https://www.semanticscholar.org/paper/644eafc07879ee48bc4940165d090844a3f9614d Abstract: We suggest a criterion under which for a nilpotent group of finite exponent $A$ and for an abelian group $B$ the variety $var(A \,Wr\, B)$ generated by their wreath product $A \,Wr\, B$ is equal to the product of varieties $var(A)$ and $var(B)$ generated by $A$ and $B$. Namely the equality holds if and only if either the group $B$ is not of some non-zero exponent; or if $B$ is of a non-zero exponent $n$, and $B$ contains a subgroup isomorphic to $C_{d}^c \times C_{n/d}^\infty$, where $c$ is the nilpotency class of $A$, $d$ is the largest divisor of $n$ coprime with $m$, $C_{d}^c$ is the direct power of $c$ copies of the cycle $C_d$ of order $d$, $C_{n/d}^\infty$ is the direct power of countably many copies of the cycle $C_{n/d}$ of order $n/d$. This criterion continues our previous work on cases when the similar criterions were given for wreath products of abelian groups or of finite groups. Also, this is a generalization of known results in literature, which solve the same problem for much more restricted cases. Some applications of the criterion are considered at the end of paper.

Question 434 multiple-choice

Quantum algorithms often exploit algebraic properties of groups to efficiently solve problems such as the hidden subgroup problem. Understanding the structure of abelian groups is especially important for reducing the query complexity of these algorithms. Which approach is crucial for minimizing the number of queries required to identify a hidden subgroup in an abelian group using structured algorithms? 1) Using random sampling of group elements without regard to group structure 2) Searching for a set of independent generators and employing a carefully chosen sequence of generating pairs 3) Applying brute-force search across all possible subgroup elements 4) Ignoring subgroup structure and focusing solely on output collisions 5) Utilizing only the identity element in the query process 6) Partitioning the group elements arbitrarily without reference to their algebraic properties 7) Relying exclusively on classical bitwise XOR operations

✓ Correct Answer:

The correct answer is 2) Searching for a set of independent generators and employing a carefully chosen sequence of generating pairs.

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L i for abelian groups may be a factor of/radicalbig logmsmaller than that of Algorithm 2. They achieve the stronger b ound by searching for a structured set of independent generators for the hidde n subgroup, and by using a partially nested sequence of generating pairs with ﬁnely tuned size. Th is entails a detailed analysis of the structure of the hidden subgroup. It is not clear whether we can achieve the same query complexity without resorting to these ideas. Acknowledgements. A.N. is grateful for the opportunity to teach quantum comput ation at the undergraduate level, which prompted this work. He is also gr ateful to Kanstantsin Pashkovich for several helpful discussions, especially for a course-c orrection early in this work. He thanks William Slofstra for a pointer to relevant literature, and Z ekun Ye for bringing Ref. [13] to his attention. This research is supported in part by a Discovery Grant from NSERC Canada. 2 The Simon Problem We start with a simple, deterministic algorithm for the Simo n problem that matches the query complexity of the asymptotically optimal classical algori thm. Theorem 2.1. There is a deterministic algorithm that makes 2⌊k/2⌋+2⌈k/2⌉queries and solves the Simon Problem with parameter k. Moreover, if the input function hid es the non-zero element s, the algorithm ﬁnds s. Finally, any deterministic algorithm for the problem requires at least q queries, where q is the smallest positive integer such that (q 2)≥2k−1. Proof. Letl:=⌊k/2⌋. Viewing elements of Zk 2ask-bit strings, we query the function fat all 2l elements of the form u0k−l, where u∈Zl 2, and at all 2k−lelements of the form 0lv, where v∈Zk−l 2. If the function is distinct at all these points, we say fis injective. Otherwise, we say that fis not injective, and output x+y, where x/\⌉}atio\slash=yand x,yare a colliding pair (i.e., are such that f(x) = f(y)). If the function is injective, the above algorithm outputs th e correct answer. Suppose fhides a non-zero element s∈Zk 2. Let abe the projection of sonto the ﬁrst lcoordinates, and bbe the 3 projection of sonto the last k−lcoordinates. Then a0k−l+0lb=s. As f(a0k−l) = f(0lb), the above algorithm detects a collision and computes scorrectly. Now consider any deterministic algorithm for the problem th at makes tqueries such that /parenleftbiggt 2/parenrightbigg ≤ 2k−2 . Then we argue that there is an injective function f0, and a function f1that hides a non-zero el- ement s, such that f0and f1agree on all the telements queried. Suppose the tqueries that the algorithm makes are x1,x2, . . . , xt. We take f0to be any injective function. Consider the set of ele- ments S:=/braceleftbig xi+xj:i,j∈[t],i/\⌉}atio\slash=j/bracerightbig . We have|S|≤(t 2)≤2k−2, so there is at least one non-zero element in Zk 2\S. Let sbe one such element. By deﬁnition of s, we have xi+s/\⌉}atio\slash=xjfor any i,j∈[t]. Thus there is a function f1that equals f0at all the points xi,i∈[t], and also hides s. Note that for k≥2, if(q 2)≥2k−1, we have q2≥2k+1−2, and therefore q≥ 2(k+1)/2(1−1/2k)1/2≥ 2(k+1)/2−2−(k−1)/2. Since

Question 435 multiple-choice

In quantum computing, the quantum Fourier transform (QFT) over the symmetric group Sn is crucial for algorithms that exploit permutation symmetries. Group-theoretic techniques such as coset factorization and centralizer decomposition are key to efficient implementation of these algorithms. For a permutation σ in the symmetric group Sn, which of the following accurately describes the structure of its centralizer Z(σ)? 1) It is a direct product over cycle types, with each factor being a wreath product S_ck ≀ Z_k corresponding to cycles of length k in σ. 2) It is always isomorphic to the cyclic group Zn, regardless of σ's cycle structure. 3) It is a simple group with no nontrivial subgroups. 4) It is the full symmetric group Sn itself for any σ. 5) It consists only of the identity permutation. 6) It forms an alternating group An for every σ. 7) It decomposes as a direct sum of abelian groups corresponding to each cycle in σ.

✓ Correct Answer:

The correct answer is 1) It is a direct product over cycle types, with each factor being a wreath product S_ck ≀ Z_k corresponding to cycles of length k in σ..

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for Sn, the symmetric group. With such algorithms, the QFT itself is straightforward: 1. Convert the basis |g,h∗/a\}bracketri}htto|t,z,h∗/a\}bracketri}ht, wheret∈Thandz∈Z(h). 2. Rewriting it as |h∗,t,z/a\}bracketri}ht, apply the QFT over Z(h) on the third register conditioned on h∗in the ﬁrst register. We obtain the basis |h∗,t,ρ,i,j/a\}bracketri}ht, whereρ∈/hatwideZ(h). Reordering again yields the basis of ( 3.8) above. This gives us the QFT over D(G). 3.3.1 An eﬃcient QFT over D(Sn) In light of the discussion above, we show that one can uniform ly compute the quantum Fourier transform over centralizers in Snand, additionally, uniformly factor along a canonical tran sversal of each centralizer. It follows that we can compute the QFT over D(Sn). First, let us determine the centralizer of an element σofSnconsisting of c1one cycles, c2two cycles, etc. For an element τ∈Z(σ), we haveτσ=στand henceτστ−1=σ:τﬁxesσunder conjugation. Recall that the conjugation action by τis given by the permutation action of τon the labels of the cycle representation 10 ofσ; thus, for each k, thek-cycles ofσare ﬁxed under the action ( a1... a k)/ma√st≀→(τ(a1)...τ(ak)). At a coarser level, the permutation τﬁxes, as a set, the elements of Wk={t∈{1,...,n}|tlies in ak-cycle ofσ}, and it follows that Z(σ) is the direct product Z(σ1)×···×Z(σn), whereσkis the product of the ckcycles of lengthkappearing in σandZ(σk) is the centralizer of σk(in the subgroup of permutations on Wk). Observe that when ( τ(a1)... τ(at)) = (a1... a t), the element τ(restricted to{a1,...,a t}) lies in the cyclic subgroup generated by ( a1,...,a t). It follows that on Wkthe element τcan be written as the product of two permutations, one which cyclicly permutes each k-cycle, and one which permutes the k-cycles. This subgroup of permutations on Wkis isomorphic to the wreath product Sck≀Zk=Sck⋉Zck kand we conclude that Z(σ)∼=n/circleplusdisplay k=1Sck⋉Zck k. Coset factorization LetHbe a subgroup of GandTa transversal of HinG(that is, a set containing one representative from each left coset of HinG). (H,G)-coset factorization is the problem of expressing an arbitrary element gofGas a product th, whereh∈Handt∈T. Observe that if K

Question 436 multiple-choice

The Haar measure is a foundational tool in the analysis of topological groups, particularly in the study of compact and matrix Lie groups, and plays a crucial role in quantum information theory and representation theory. Its properties enable averaging and integration over groups in a way that preserves group symmetries. Which property of the Haar measure on a compact Lie group ensures that integrating a function over the group remains unchanged when the function is translated by any group element? 1) Absolute continuity with respect to Lebesgue measure 2) Invariance under automorphisms 3) Left (or right) invariance under group translation 4) Finiteness of the group order 5) Symmetry under complex conjugation 6) Multiplicativity over disjoint subsets 7) Commutativity of group multiplication

✓ Correct Answer:

The correct answer is 3) Left (or right) invariance under group translation.

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compact Lie group deﬁnes, after proper rescaling, a probability distribution on a compact Lie group. This means we can describe sizes of “nice” (i.e., Borel) subsets HGby integrat- ing1 jGjR HGd, wherejGj=R Gd(this is eﬀectively a partition function in probability theory), and it means we can now integrate functions f:G!R, like the trace norm Trjj, overtheseregions. Forinstance, ifwehadaniceopen regionHsurrounding the identity 1, perhaps describing a ball of"perturbations of 1, we could now meaningfully integrate over that. ButwhatmakestheHaarmeasurespecialistheso-called left-invariance (orright-invariance , depending on conven- tion).17This implies the following relationship for any in- tegrable function f:G!R: Z Gf(gx)d(x) =Z Gf(x)d(x);for anyg2G: One fruitful interpretation of this deﬁnition is that this gives achange of variables formula for integration over ar- bitrary Lie groups. To illustrate this, let us just think about calculus. The real numbers with the addition G= (R;+)form a matrix Lie group, and using the standard Lebesgue measure on R18we can integrate real functions. Then, writing gx=g+xandd(x) =dx, we have Z G=Rf(g+x)dx=Z G=Rf(x)dx: The Haar measure is essentially the unique probability measure with this change of variable property, and given how useful change of variables is in any application of cal- culus, it is a short stretch of the imagination to see why this should be critical to any analysis on Lie groups. Find- ing closed form expressions for Haar measures of matrix Lie groups in terms of the more familiar Lebesgue measure from calculus boils down to combining standard methods for integration on manifolds using Jacobians with the “left- invariant vector ﬁeld” interpretation of the Lie algebra [51]. We mention brieﬂy that for any ﬁnite group G, the Haar measure is just the counting measure: Z Gf(x)d(x) =X x2Gf(x): 17For a wide class of groups, called “unimodular”, left invariance and right invariance are the same. This includes compact and semisim- ple Lie groups. 18...which is not a probability measure since (R) =1, but that is not the point here.The good news going forward is that we generally will be integrating over the whole group, and we only really need these formal properties, not the particular form of the measure. In the following two sections, we describe two fundamental ideas which have close ties to Haar inte- grals: twirling and commutants. B. Twirling Twirling is a Haar-averaging technique that appears in a variety of disparate contexts, including entanglement- theory and quantum error correction [65], randomized benchmarking [66, 67], quantum process tomography [68, 69] classical shadows [70, 71], and barren plateau analy- sis [13–15, 18, 20, 64], to say nothing of the numerous ap- plications within representation theory. The following dis- cussion is a rephrasing of the standard material presented in [53]. Everyformoftwirlinghasthefollowingstructureincom- mon. Take a representation Rof a group Gover a vector spaceV. Suppose that we want to ﬁnd the subspace of all trivial invariant subspaces VG: VG:=fvjRgv=vg: Then the twirl operator TG:V!Vis the linear map TG(v) =1 jGjZ GRgvd(g): The twirl operator is in fact a projection onto VG, soT2 G= TGandIm(TG) = VG. At a more abstract level, the

Question 437 multiple-choice

In quantum information theory, projection operators and representation theory concepts such as Schur-Weyl duality and Kostka numbers are crucial for analyzing the structure of tensor product spaces and for performing hypothesis testing between quantum states. These mathematical tools underpin results on distinguishability and error exponents in quantum statistical experiments. Which statement accurately describes a key property of Kostka numbers in the context of quantum hypothesis testing and representation theory? 1) The Kostka number K_{f,λ} is positive if and only if the vector f is majorized by λ. 2) The Kostka number K_{f,λ} is always equal to the dimension of the Hilbert space. 3) Kostka numbers cannot be used to construct permutation-invariant quantum states. 4) If the Kostka number K_{f,λ} is zero, then f and λ must be equal. 5) Kostka numbers are unrelated to the decomposition of tensor powers in quantum information theory. 6) Majorization between f and λ is never relevant for the positivity of Kostka numbers. 7) Kostka numbers provide upper bounds on quantum relative entropy in hypothesis testing.

✓ Correct Answer:

The correct answer is 1) The Kostka number K_{f,λ} is positive if and only if the vector f is majorized by λ..

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valid for all Pf,λsatisfying tr {Pf,λPn,s}= 0. It generally holds that Pf,λ≤/summationtext λPf,λ=:PfandPf,λ≤/summationtext fPf,λ=:Pλand thus by [44] and the basic estimate Lemma 2.6 in [20] followd by Pinsker’s inequality we get the uppe r bound tr{ρ⊗n sPf,λ} ≤min{tr{ρ⊗n sPf},tr{ρ⊗n sPλ}} (49) ≤(2n)d2min{2−nD(¯λ/bardblrs),2−nD(¯f/bardbl˜rs)} (50) ≤(2n)d22−nαε2. (51) SetXs:={(f,λ) :/bardbl¯f−˜rs/bardbl> ε/logicalandtext/bardbl¯λ−rs/bardbl> ε}. From deﬁnition of our basic object it can be seen that the inequality/summationtext f/summationtext λPf,λ= /BD⊗nholds, so for an arbitrary s∈Swe have 1 = tr{ρ⊗n s/summationdisplay f/summationdisplay λPf,λ} (52) = tr{ρ⊗n sPn,s}+tr{ρ⊗n s( /BD⊗n Cd−Pn,s)} (53) ≤tr{ρ⊗n sPn,s}+(2n)d2max (f,λ)∈Xstr{ρ⊗n sPf,λ} (54) ≤tr{ρ⊗n sPn,s}+(2n)2d22−nαε2(55) ≤tr{ρ⊗n sPn}+(2n)2d22−nαε2. (56) It follows inf s∈Str{ρ⊗n sPn} ≥1−2−n(αε2−2d2 nlog(2n)). (57) Therefore, lim n→∞inf s∈Str{Pnρ⊗n s}= 1, (58) as required. By linearity of the trace, the results immediately imply th at we can also (asymptotically) distinguish the sequence (conv( {ρ⊗n s}s∈S)n∈Nof sets from the sequence ( σ⊗n)n∈N. Proof of the lower bound on the error exponent in Theorem 2. Let an operator 0 ≤A≤ /BD⊗nbe given that satisﬁes inf s∈Str{Aρ⊗n s} ≥1−ν (59) for some ν∈(0,1). Note that we can without loss of generality assume that Ais permutation-invariant, since permuting an Athat is not invariant does change neither tr {Aρ⊗n}nor tr{Aσ⊗n}. But then by Schur-Weyl duality we can write A=/summationdisplay λAλ, (60) where 0≤Aλ≤cλPλfor some set of real numbers cλ. This leads us to tr{σ⊗nAλ}=/summationdisplay λ,f,j2n/summationtext i¯f(i)logtitr{Pf,λ,jAλPf,λ,j}, (61) 10 wherePf,λ,jare any set of mutually orthogonal andirreducible projections such that Pf,λ=/summationtext jPf,λ,j for all (f,λ). On the other hand by Lemma 5 in [27] (which is identical to Lemma 9 in [28]) we get /summationdisplay f,λ,jPf,λ,jAPf,λ,j≥(2n)−d2A, (62) and it follows 1−ν≤(2n)d2/summationdisplay f,λ,jtr{Pf,λ,jAPf,λ,jρ⊗n}. (63) It is clear that Pf,λ,jAPf,λ,j=cf,λ,jPf,λ,jfor every triple ( f,λ,j) and a suitable choice of coeﬃcients 0≤cf,λ,j≤1 by permutation-invariance of A. Therefore there is at least one ( f,λ,j) such that /bardbl¯f−˜r/bardbl ≤ n−1/4,/bardbl¯λ−r/bardbl ≤n−1/4and cf,λ,j≥(2n)−2d2(1−ν−2−α√n). (64) But this implies that for this triple ( f,λ,j) we have (and at this point one sees again that it is crucial that we deﬁned the Vfusing the eigenbasis of σ, since this ensures that σ⊗nand every one of the Pf,λ,j commute) tr{σ⊗nA} ≥tr{σ⊗nAλ} (65) =/summationdisplay f′,itr{σ⊗nPf′,iAλPf′,i} (66) ≥2n/summationtext i¯f(i)logtitr{Pf,λ,jAλPf,λ,j} (67) = 2n/summationtext i¯f(i)logtitr{Pf,λ,j}cf,λ,j (68) ≥2n/summationtext i¯f(i)logtidim(Fλ)1−ν−2−α√n (2n)2d2 (69) ≥2n(/summationtext i¯f(i)log(ti)−H(¯λ)−2d6 nlog(2n))1−ν−2−α√n (2n)2d2 (70) ≥2−n(D(ρ/bardblσ)−Θ(n,n−1 4,d)−2d6 nlog(2n)+1 nlog(1−ν−2−α√n (2n)2d2)), (71) So the function Θ′(n,ν,d) := Θ(n,n−1 4,d)−2d6 nlog(2n)+1 nlog(1−ν−2−α√n (2n)2d2) (72) does the job. Proof of Theorem 3. Let us ﬁrst state some known facts about Kostka numbers. It ho ldsKf,λ>0⇔ f≺λ. ButKf,λis just the number of ways to ﬁll the tableau λwith numbers from [ d] such that the numbers in the ﬁlling are weakly increasing in each row, read from left t o right, and strictly increasing along the columns (read from top to bottom). Thus Kf,λ>0⇔tr{Pf,λ}>0. There is actually another important equivalence that we shall use: f≺λ⇔f=Dλfor some doubly stochastic matrix D. Now assume Kf,λ>0 holds. Thus f=Dλfor some doubly stochastic matrix D. In this case we may deﬁne a state ρby the following procedure: Pick any basis {ei}d i=1. Choose a unitary matrix U=/summationtext i,juij|ei/an}bracketri}ht/an}bracketle{tej|such that

Question 438 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) over non-abelian groups are a focus of research due to their complexity and relevance to computational tasks such as factoring and graph isomorphism. Representation theory plays a key role in analyzing the quantum resources required for solving HSP efficiently. For which group family do efficient quantum algorithms for the Hidden Subgroup Problem provably require entangled measurements across at least Ω(n) quantum registers, due to the properties of their irreducible representations and subgroup structure? 1) Dihedral groups D_n for all n ≥ 2 2) Finite abelian groups with non-trivial centers 3) Symmetric groups S_m for all m ≥ 5 4) Cyclic groups of prime order 5) Quaternion groups Q_8 6) Alternating groups A_n for n < 5 7) Nilpotent groups of class 2

✓ Correct Answer:

The correct answer is 3) Symmetric groups S_m for all m ≥ 5.

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assumption that |G|>λ(1+κ)/parenleftBig/summationtext ρ∈/hatwideGdρ/parenrightBig holds. This implies that there exists a constant γ1>0such thatδ1≤γn 1. Forthe parameter δ2inTheorem 2weobtain δ2= 2k(1+2kε)δ1/2 1+3kε+3k |G|n/summationdisplay σ∈Sεd2 σ ≤2k/parenleftBigg 1+2k/parenleftbigg 1−2|C(h)| |G|/parenrightbiggn/c/parenrightBigg γn/2 1+3k/parenleftbigg 1−2|C(h)| |G|/parenrightbiggn/c +3k/parenleftbiggλ(1+κ) |G|/parenrightbiggn . Now, since our assumption implies that |G|>λ(1+κ), we obtain that there exists a constant γ2>0such thatδ2≤γn 2. Hence, wehave proved the following corollary toTheorem 2. 19 Corollary 19. LetGbe a ﬁnite group and let h∈Gbe an involution. Let /hatwideGdenote the set of irreps of Gand letΛ :={σ∈/hatwideG:|χσ(h)|=dσ}. Suppose that |G|>/parenleftbig/summationtext σ∈Λd2 σ/parenrightbig/parenleftBig/summationtext ρ∈/hatwideGdρ/parenrightBig holds. Then any efﬁcient algorithm operating on coset states that distingu ishes between the case when the hidden subgroup is a conjugate of the subgroup /a\}bracketle{t(h,...,h)/a\}bracketri}ht ≤Gn, and the case when the hidden subgroup is the identity subgroup in Gn,needs to makemeasurements entangled across Ω(n)registers. Recently, Alagic, Moore and Russell [AMR05] showed that any measurement on a single coset state givesexponentiallylittleinformationaboutahiddensubg roupinthegroup Gn,whereGisﬁxedandsatisﬁes asuitable condition. Theircondition on Gisweaker than ourcondition inCorollary 19,but theyonly pr ove lowerboundsforalgorithmsmeasuringonecosetstateatati me. Theyalsogiveseveralexamplesoffamilies of groups satisfying their condition, including all non-ab elian ﬁnite simple groups. In fact, the condition of Corollary 19 holds for all families of groups Gconsidered in their paper, showing that efﬁcient coset stat e based algorithms solving the HSP for their families of group sGnneed to make measurements entangled acrossΩ(n)registers. From Corollary 19,it iseasy to prove Corollary 20 viathe Cau chy-Schwartz inequality. Corollary 20. LetGbe a ﬁnite group and let h∈Gbe an involution. Let /hatwideGdenote the set of irreps of Gand letΛ :={σ∈/hatwideG:|χσ(h)|=dσ}. Suppose that |G|1/2>|/hatwideG|1/2/parenleftbig/summationtext σ∈Λd2 σ/parenrightbig holds. Then any efﬁcient algorithm operating on coset states that distingu ishes between the case when the hidden subgroup is a conjugate of the subgroup /a\}bracketle{t(h,...,h)/a\}bracketri}ht ≤Gn, and the case when the hidden subgroup is the identity subgroup in Gn,needs to makemeasurements entangled across Ω(n)registers. Using Corollary 20, weprove the following result. Corollary 21. Any efﬁcient algorithm operating on coset states that disti nguishes between the case when the hidden subgroup is a conjugate of the subgroup /a\}bracketle{t(h,...,h)/a\}bracketri}ht ≤(Sm)nwhereh∈Smis any involution andm≥5isﬁxed,andthecasewhenthehiddensubgroupistheidentity subgroupin (Sm)n,needstomake measurements entangled across Ω(n)registers. Thesameholds also when m= 4andh= (1,2)∈S4. Proof.LetG=Sm, wherem≥5, and lethbeany involution in G. Recall that for m≥5all irreps ofSm of degree greater than 1are faithful [JK81, Theorem 2.1.13], and that the center of Smis trivial. Since for faithfulσ∈/hatwiderSmwe have that |χσ(h)|=dσimplies that his in the center, we obtain that |χσ(h)|1. HenceΛ ={11,alt}consists of the trivial and the alternating character only a nd weobtainthat/summationtext σ∈Λd2 σ= 2. Sinceform≥5wehavethat |G|1/2=√ m!>2/radicalbig p(m) =|/hatwideG|1/2/summationtext σ∈Λd2 σ, wherep(m)denotes the partition number of m,the statement for m≥5follows from Corollary 20. Form= 4andh= (1,2)we observe that the set Λis again given by Λ ={11,alt}. We verify that the condition |S4|1/2=√ 24>2√ 5 =|/hatwiderS4|1/2/summationtext σ∈Λd2 σholds. Hence the statement for this case also follows from Corollary 20. Acknowledgments Wethank Andrew Childs, Fr´ ed´ eric Magniez and UmeshVazira ni for helpful discussions and comments. References [AMR05] G. Alagic, C. Moore, and A. Russell. Strong Fourier s ampling fails over Gn. ArXiv preprint quant-ph/0511054,

Question 439 multiple-choice

In quantum computing, simulating the time evolution of quantum systems often involves representing Hamiltonians using block-encoding techniques, which facilitate efficient algorithmic implementation. Error quantification, especially via norms such as the diamond norm, is vital for ensuring the accuracy of quantum channel constructions in these simulations. When constructing a quantum channel to simulate the evolution operator eiHt using block-encoding and amplitude amplification, what is the tightest upper bound on the diamond norm error between the constructed channel and the exact evolution if the underlying block-encoding has ε accuracy? 1) ε 2) 2ε 3) 3ε 4) 4ε 5) ε² 6) √ε 7) 2√ε

✓ Correct Answer:

The correct answer is 4) 4ε.

📚 Reference Text:

postselection succeeds, and the probability that the postselection fails. The Hamiltonian simulation method is extremely accurate, letting us obtain block- encodingswithanerrorthatdecaysexponentiallyinthequerycomplexityto UH. Thus, the error contribution of this channel is almost entirely negligible in the performance analysis. The purpose of the argument below is really to provide a Lemma analogous to Lemma 4 that lets us convert error bounds in spectral norms on block-encodings to diamond norms. Thatway,ourentireerroranalysisiscompletelyformal,asitshouldbeforafaircomparison of all algorithms involved. Notably, an ε-accurate block-encoding of a unitary is not the same thing as an ε-accurate implementation of that unitary: the latter implies a channel with diamond norm error 2εwhile the prior yields an error 4εdue to the ancilla registers. Lemma 17. Hamiltonian simulation. SayUHis a block-encoding of a Hamiltonian H. Then, for any t>0andε>0there exists a quantum channel that is ε-close in diamond norm to the channel induced by the unitary eiHt. This channel can be implemented using 3·r/parenleftbigget 2,ε 24/parenrightbigg + 3 (183) queries to controlled- UHor controlled- U† H. Proof.This is an extension of Theorem 58 of [GSLW18], which states that there exists a block-encoding UAof a matrix Asuch that|A−eiHt|≤εwith query complexity 3r/parenleftbiget 2,ε 6/parenrightbig. This result leverages the Jacobi-Anger expansion (Lemma 14) to construct approximate block-encodings of sin(tH)andcos(tH)and uses linear combinations of unitaries to ap- proximateeiHt/2. Then it uses oblivious amplitude ampliﬁcation to get rid of the factor of1/2, obtaining UA. IfUHis a block-encoding with aancillae, then UAhasa+2ancillae. All that is left to do is to turn this block-encoding into a quantum channel that approximately implements eiHt. To do so, we just initialize the ancillae to |0a+2/angbracketright, apply UA, and trace out the ancillae. We can write this channel Λas: Λ(ρ) :=/summationdisplay i(/angbracketlefti|⊗I)UA(|0a+2/angbracketright/angbracketleft0a+2|⊗ρ)U† A(|i/angbracketright⊗I) (184) To ﬁnish the theorem we must select an εso that the error in diamond norm of Λto the channel implemented by eiHtis bounded by δ. To do so, we write Λas a sum of postselective channels Λi(ρ): Λi(ρ) := (/angbracketlefti|⊗I)UA(|0/angbracketright/angbracketleft0|⊗ρ)U† A(|i/angbracketright⊗I) (185) That way Λ =/summationtext iΛi. If we let ΓeiHt:=eiHtρe−iHt, then we can bound the error in diamond norm using the triangle inequality: |Λ−ΓeiHt|/diamondmath≤|Λ0−ΓeiHt|/diamondmath+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay i>0Λi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle /diamondmath(186) Now we proceed to bound the two terms individually. Observe that: Λ0(ρ) := (/angbracketleft0|⊗I)UA(|0/angbracketright/angbracketleft0|⊗ρ)U† A(|0/angbracketright⊗I) =AρA†(187) Since|A−eiHt|≤ε, we can invoke Lemma 4 and see that |Λ0−ΓeiHt|/diamondmath≤2ε, and we are done with the ﬁrst term. Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 36 To bound|/summationtext i>0Λi|/diamondmath, we ﬁrst observe that/summationtext i>0Λi= Λ−Λ0. Second, we observe that theΛiare all positive semi-deﬁnite, so |Λi(ρ)|1=Tr(Λi(ρ)). Plugging in the deﬁnition of the diamond norm, we can compute: /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay i>0Λi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle /diamondmath= sup ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay i(Λi⊗I)(ρ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle 1(188) ≤sup ρTr/parenleftBigg/summationdisplay i(Λi⊗I)(ρ)/parenrightBigg (189) = sup ρTr((Λ⊗I)(ρ)−(Λ0⊗I)(ρ)) (190) = 1−infρTr(AρA†) (191) If we letE:=eiHt−Aso that|E|≤ε, and plug into the above expression, we get: Tr(AρA†) =Tr(eiHtρe−iHt−Eρe−iHt−eiHtρE†+EρE†)≥1−2ε+ε2(192) Putting everything together we obtain |Λ−ΓeiHt|/diamondmath≤2ε+ 2ε−ε2≤4ε. So if we select ε:=δ/4we obtain the desired bound. The above proof uses the trick for the proof of the block-measurement theorem that we mentioned in the introduction. We will need it again in the proof Theorem 19.

Question 440 multiple-choice

In quantum computing, precise implementation of quantum gates is crucial for achieving high-fidelity operations, especially in systems using qudits realized by quadrupole nuclei. The global phase of a quantum gate and the arrangement of energy levels in the underlying Hamiltonian can influence control strategies and gate performance under time constraints. When synthesizing a quantum Fourier transform (QFT) gate for qudit systems based on quadrupole nuclei, which statement best describes the relationship between global phase, Hamiltonian energy levels, and minimum implementation time? 1) The minimum implementation time is always independent of the global phase chosen for the gate. 2) The global phase of the QFT gate determines the number of energy levels in the quadrupole nucleus required for implementation. 3) Changing the global phase has no impact on the optimal pulse shapes found by control algorithms for gate synthesis. 4) For different choices of global phase, the effective Hamiltonian and minimum gate implementation time can vary, affecting optimal control solutions. 5) The implementation time is solely determined by the qudit dimension and not by the energy level arrangement. 6) The global phase only affects measurement probabilities and not gate synthesis or control strategies. 7) Quadrupole nuclei with spins greater than 1/2 cannot realize qudit gates with varying global phases.

✓ Correct Answer:

The correct answer is 4) For different choices of global phase, the effective Hamiltonian and minimum gate implementation time can vary, affecting optimal control solutions..

📚 Reference Text:

Title: Global phase and minimum time of quantum Fourier transform for qudits represented by quadrupole nuclei Year: 2013 Paper ID: 73ba658c7b02d215191b20516ba14ea79c93b8b3 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/73ba658c7b02d215191b20516ba14ea79c93b8b3 Abstract: We demonstrate the relation between a global phase of the quantum gate and the layout of energy levels of its effective Hamiltonian required for implementing the gate for minimum time. By an example of the quantum Fourier transform gate for a qudit represented by a quadrupole nucleus with the spin I = 1, the effective Hamiltonians and minimum implementation times for different global phases are found. Using numerical optimal control methods, the problem of the global phase in searching for the optimal pulse shape is considered in detail for the quantum Fourier transform gate at I = 1, 3/2, 2, and 5/2. It is shown that at the constrained control time the gradient algorithms can converge to the solutions corresponding to different global phases or the same global phase with different minimum times of the gate implementation.

Question 441 multiple-choice

Quantum Phase Estimation (QPE) is a foundational algorithm in quantum computing used to estimate eigenvalues of unitary operators, with applications in quantum simulation, chemistry, and group theory. Its complexity and accuracy depend on parameters such as the number of bits used for phase estimation and the structure of the input state. Which statement best describes how the Quantum Phase Estimation algorithm enables efficient identification of projectors in the center of the group algebra C(Sn) for large symmetric groups? 1) It applies QPE to arbitrary non-Hermitian operators without requiring knowledge of the group structure. 2) It relies on measuring random basis states and ignores eigenvalue information from characters. 3) It uses QPE on superpositions, but the probability of significant error grows as the group size increases. 4) It employs a fixed number of bits for phase estimation irrespective of the group's complexity. 5) It constructs unitary operators using random group elements, without reference to cycle central elements. 6) It achieves exponential complexity in the group size due to the need for high-precision measurements. 7) It applies QPE to Hermitian operators built from cycle central elements, choosing the number of bits for phase estimation to guarantee exact eigenvalue identification, resulting in polynomial complexity in the group size.

✓ Correct Answer:

The correct answer is 7) It applies QPE to Hermitian operators built from cycle central elements, choosing the number of bits for phase estimation to guarantee exact eigenvalue identification, resulting in polynomial complexity in the group size..

📚 Reference Text:

proved (see section 5.2 in [7]) that the probability of getting bafter a measurement mobeys the probability boundP(|m−b|>e)e)<|βk|2/epsilon1that remains small, for constant βk. Thus, the procedures which follow here will adapt in this case and will remain accurate. Complexity. The runtime complexities of QPE has been also discussed in [ 7]. The query complexity isO(t), as one requires tqueries of the CU-operators. On the other hand, the gate complexity, that enumerates the number of gates used to deliver an output, is O(t2) (tHadamard gates, tCU-operators, the inverse QFT requires t+ (t−1) +···+ 2 + 1H and C-Rkgates, and therefore asymptotically leads to O(t2)gates). Here and after, we summarize these statements as: Query complexity ∈O(t); Gate complexity ∈O(t2). In the projector detector tasks for algebras related to Snwhich we consider in the following sections, where we will use unitary operators for QPE, which are exponentials of the cycle central operators discussed in section 2.1, we will be choosing the number of bits tto allow exact determination of the relevant eigenvalues which have known integrality properties. As a result this will translate into n-dependent complexities which are bounded by polynomials inn, subject to the assumption that k∗(n)∈O(nα)withα<1/2which we described in section 2.1.1. 3 Quantum Phase Estimation for projectors in the centre of C(Sn) In this section, we focus on a quantum algorithm for detecting projectors in the centre Z(C(Sn))of the group algebra C(Sn)of the symmetric group Sn. Bob prepares a projector – 10 – JHEP05(2023)191PRas a quantum state in the Hilbert space Z(C(Sn)). Alice is tasked with identifying the projector, i.e identifying the Young diagram R. Her procedure uses QPE to obtain the eigenvalues of appropriate Hermitian operators (the cycle central elements described in section 2.1) acting on the Hilbert space. These eigenvalues are expressible in terms of normalized characters of irreducible representations of Sn. We estimate the query and gate complexities of the algorithm using their standard deﬁnitions. We are interested in particular in the cycle central elements Tk,k∈[ [1,n] ], corresponding to partitions µ= [k,1n−k]with one cycle of length kand remaining cycles of length 1, described in more detail in section 2.1. ConsiderUk=e2πi χmax kTkandUkPR=e2πi χmax k/hatwideχR(Tk)PR, using the deﬁnition ( 2.8)

Question 442 multiple-choice

Quantum algorithms such as Shor's are pivotal in computational number theory, especially for tasks like factoring large composite numbers. The implementation of these algorithms often involves quantum circuits that simulate modular exponentiation and phase estimation using specialized libraries and hardware. In a quantum circuit designed for order-finding as part of Shor's algorithm, which operation is typically the most resource-intensive and requires careful implementation to ensure reversibility? 1) Quantum Fourier transform 2) Measurement of qubits 3) Application of Hadamard gates 4) Initialization with PauliZ gates 5) Modular exponentiation 6) Qubit reset operations 7) Classical post-processing

✓ Correct Answer:

The correct answer is 5) Modular exponentiation.

📚 Reference Text:

�soSt€�tzr 6t€no€trzon �{qtzn Ðt�s strs |~{lkltwt�Þ �sowok€� |{€t�t�o tz�oro~r€�ms �sk�yry{nNB4q{~6/lt� tz�oro~€yB6kznNB91\so €tþo{q �so~ort€�o~ ðt€5n-5B5�6-5B<1 St€�tzr 61Kðky|wo {q�€kro {q�so�oy|wk�o Order_Finding1 import pennylane asqml import QuantumOperations asq #define initial parameters N=5 y=3 bits_for_register_with_a_number =3 bits_for_x_register =2�bits_for_register_with_a_number +2 #define wires with all registers wires=[i for iinrange(bits_for_x_register+bits_for_register_with_ a_number�5+2)] #device Ltr;1Jom{y|{ €t�t{z {qMODULAR_EXPONENTIATION mt~m�t� tz�{ w{Ðo~/wo�o w{|o~k�t{z€1 \so nom{y |{€t�t{z ~okwtþo€ y{n�wk~ oð|{zoz �tk�t{zyx y{nNrt�oz 6/lt� tz�oro~€x.y.kznN1 s��|�>22n{t1{ ~r243146;42u {�~zkw1|{zo 135;47:51r33; PLOS ONEKqqtmtoz� ~okwt�k�t{z {q}�kz��y |~tyt�t�o� q{~[s{~)� kwr{~t�sy ��tzr Wozz�S kzowtl~k~� WSV[ VUK �s��|�>22n{t1{~ r243146; 42u{�~zkw1|{ zo135;47 :5 Q�w� 47.5355 4424; dev =qml.device(‘default.qubit’, wires =wires, shots =10000, analytic =None) #circuit def func(N, y,bits_for_x_register,input_): #insert input for iinrange(len(wires)): ifinput_[i] ==1: qml.PauliX(wires =wires[i]) #circuit q.Order_Finding(N, y,bits_for_x_register,wires =wires) return qml.probs(wires=[0, 1,2,3,4,5,6,7]) #QNode circuit =qml.QNode(func,dev) #Run calculations for given parameters with the #register wires_N initialized asbinary Nand #register wires_z −asbinary 1 measurements_probabilities =circuit (5, 3,bits_for_x_register, [0, 0,0,0,0,0,0,0]+[1, 0,0]+[0, 0,0]+[0, 0,0,0]+[0, 0,0] +[1, 0,1]+[0]) Zo€�w�€ {qyok€�~oyoz�€ tz�som{z€�~�m�on mt~m�t� k~o|��tz�{ �so�k~tklwo �yok€�~oyoz�€i |~{lkltwt�to€� k€kzk~~kÞ1 Fq�o~ |{€�/|~{mo€€tzr. Ðomkzro��so|~{lkltwt�Þ nt€�~tl��t{z {qyok/ €�~oyoz�€ k€no|tm�on tzLtr<1Kkms lk~m{~~o€|{zn€ �{k|k~�tm�wk~ yok€�~oyoz� {��m{yo �sk� mkzlotz�o~|~o�on k€kzo€�tyk�o {qs2r.Ðso~ort€�so{~no~ {qyy{n�w{Nkznst€€{yo tz�oro~1 Oz|k~�tm�wk~. tqyok€�~oyoz� sk€�soq{~y �x4x5444w.�soz �soo€�tyk�o {qs2rt€�soz�ylo~ 31x4x5444tz�soltzk~Þ ~o|~o€oz�k�t{z. kzn |{€€tlwo �kw�o€ {qrmkzlo~om{z€�~�m�on q~{y �st€ o€�tyk�o1 Fmm{~ntzr �{�sokwr{~t�sy. yok€�~oyoz�€ Ðt�s strs |~{lkltwt�Þ m{~~o€|{zn �{o€�t/ yk�o€ �sk� k~omw{€o �{�so�~�o �kw�o {qs2r1 Oz{�~mk€o. �soq{�~ yok€�~oyoz�€ Ðt�s �sor~ok�o€� |~{lkltwt�to€ k~o�333333w. �433333w. �444444w kzn �343333w1 \so€o yok€�~oyoz�€ m{~~o€|{zn �{~o|~o€oz�k�t{z€ {qs2rtz�soq{~y 31333333 B3.31433333 B425. 31444444 B:62:7 kzn 31343333 B427. ~o€|om�t�owÞ1 O�mkzlo€ooz �sk� �soq{�~�s ~o€�w� rt�o€ k|~{|o~ �kw�o {qr.€tzmoyry{nNB67y{n 9B41 9\~kz€|twk�t{z kzn ~o€{�~mo/o€�tyk�t {z L{~~okw/wtqo ty|woyoz�k�t{z€ {q�sort�oz kwr{~t�sy t�t€ty|{~�kz� �{�~kz€wk�o �sonom{y|{€t/ �t{z tz�{ k€o�{qrk�o€ �sk� k~ozk�t�o q{~k|wk�q{~y {q�sotz�o~o€�1 \st€ |~{mo€€ {q�~kz€wk�t{z t€mkwwon �~kz€|twk�t{z. kznq{~ykzÞ q~kyoÐ{~v€ |k~�tm�wk~ no�ktw€ {q�st€|~{mo€€ k~oz{� oð|wktzon �{k�€o~1 Fw�s{�rs ~o€�w�€ {q�~kz€|twk�t{z {q�oz�tyo€ mkzlokmmo€€on. t�t€z{�mwok~ �{Ðstms oð�oz� �s{€o ~o€�w�€ mkzlo~owtklwo q{~o€�tyk�t{z {q~o€{�~mo€ q{~kzkwr{~t�sy. €�ms k€ z{z/Iwtqq{~n rk�o m{�z� kznoðom��t{z �tyo oð|~o€€on tz�sono|�s {qk�~kz€|twon kwr{~t�sy1 \so kn�kz�kro {q�€tzr WozzÞSkzo |kmvkro q{~�so~okwtþk�t{z {q�sokwr{~t�sy t€�sokltwt�Þ �{~�zkwr{~t�sy€ {zntqqo~oz� |wk�q{~y€1 O�kww{Ð€ q{~nt~om� m{y|k~t€{z {qkwr{~t�sy€) |o~q{~/ ykzmo q{~ntqqo~oz� |wk�q{~y€. Ðstms t�€owq mkzlok€�luom� {q~o€ok~ms *€oo. q{~tz€�kzmo. d4;f+1 _t�s tzm~ok€o tzsk~nÐk~o)€ m{y|��tzr mk|kltwt�to€. t�Ðtwwlosk~no~ �{m{y|two kwr{~t�sy€ q{~ ntqqo~oz� |wk�q{~y€. €{�ztqton q~kyoÐ{~v €�ms k€WozzÞSkzo wtl~k~Þ ytrs� |~{�tno l{�s �{{w€ q{~~okwtþtzr kwr{~t�sy€ kzn �~kz€|k~ozmÞ tz~�wo€ {qnom{y|{€tzr �so€o kwr{~t�sy€ �{�sowo�ow {qzk�t�o rk�o€1 \{tww�€�~k�o �st€~ok€{ztzr. Ðo|~o€oz� k€ty|wtqton |~{�{m{w q{~�~kz€|twk�t{z Ðstms t€ no~t�on q~{y d422n{t1{~ r243146; 42u{�~zkw1|{ zo135;47 :5 Q�w� 47.5355 4524; Fnnt�t{zkwwÞ. �so{~trtzkw Ð{~v d4 oqqtmtozmÞ {q~okwtþk�t{z tz�o~y€ {qrk�o m{�z�€ kzn €ty|wtmt�Þ {qoð|{/ €t�t{z1 Zokwtþk�t{z {q�~kz€|twk�t{z �omszt}�o€ m{�wn loy{~o €{|st€�tmk�on k€Ðoww. kzn t� ~oyktz€ �zmwok~ Ðstms |k~�tm�wk~ kwr{~t�sy€ Ðtwwlo{qr~ok�o~ tz�o~o€� tz�soq��~o1 914Uk�t�o rk�o€ q{~�~k||on/t{z }�lt�€1 Uk�t�o €tzrwo/}�lt� rk�o Ðt�s

Question 443 multiple-choice

In quantum computing and representation theory, semistandard Young tableaux (SSYTs) and Gelfand-Tsetlin (GT) patterns are essential for describing basis vectors of group representations used in algorithms such as the Schur transform. Operators acting on these basis elements are critical for implementing efficient quantum algorithms that exploit symmetry. When the operator J(l)₀ acts on the basis vectors labeled by SSYTs or GT patterns in the Schur transform, how is its action mathematically characterized in terms of the label multiplicities? 1) By summing all possible replacements of l with l+1 regardless of tableau validity 2) By multiplying the total number of boxes in the tableau by l 3) By taking the product of μ_l and μ_{l+1} 4) By adding the counts of l and l+1 and dividing by two 5) By subtracting μ_{l+1} from μ_l 6) By computing (μ_l - μ_{l+1}) divided by two 7) By assigning zero unless all l+1 labels are changed to l

✓ Correct Answer:

The correct answer is 6) By computing (μ_l - μ_{l+1}) divided by two.

📚 Reference Text:

jand applying Hand an element of J. This means that from the SSYT ˜j, we get a sum that involves SSYTs obtained by replacing some box labeled l+ 1 bylas long as it gives a valid SSYT (since if two boxes labeled lare in the same column, then by lemma 2, that SSYT is taken to zero). Now we show that the action of J(l) 0is as described in 2.5. To see this note that El,lacting on the computational basis counts the number of ls in the basis vector, which means that it counts the number of boxes labeled lin the SSYT basis. Therefore since J(l) 0= (1/2)(El,l−El+1,l+1), the action of J(l) 0on our basis is (µl−µl+1)/2. Using the translation from SSYT to GT patterns described in section 2.5, for a GT patternMdeﬁned as M= m1,d m2,d ... m d,d m1,d−1 ... m d−1,d−1 ...... m1,2m2,2 m1,1 , (4.8) 18 we have µl=l/summationdisplay k=1(mk,l−mk,l−1). (4.9) Thus the action of J(l) 0can be seen to be J(l) 0|M/angbracketright=/bracketleftBiggl/summationdisplay k=1mk,l−1 2(l+1/summationdisplay k=1mk,l+1+l−1/summationdisplay k=1mk,l−1)/bracketrightBigg |M/angbracketright. (4.10) 5 Conclusions We have presented an ecient algorithm for a high dimensional Schur transform that runs in time O(poly( n, logd,log 1//epsilon1)). This improves exponentially in the dimension over the prior work of Bacon, Chuang and Harrow [5]. As mentioned above, Harrow's thesis [18] contains a way to make the unitary group approach of [5] polynomial in logd. Our algorithm is novel in that it uses the representation theory of the symmetric group rather than that of the unitary group. Another interesting feature is that it uses only the quantum Fourier transform and generalized phase estimation (which is also based on the QFT) and essentially no new tools. A potentially useful feature of this algorithm that could be a primitive for other problems is the circuit for a Fourier transform over induced representations. Several permutation modules, which are induced representations encode important problems that include element distinctness and collision nding. The subroutines to block diagonalize permutation modules could provide Fourier analytic algorithms to these problems and generalize to solve other problems which have permutational symmetry. 6 Acknowledgments We would like to thank Aram Harrow for useful discussions and for explaining the algorithms in his thesis. We would like to thank John Wright for useful discussions and for comments on an earlier draft. We would also like to thank the anonymous referees of TQC for useful feedback. We are also grateful to the referees of Quantum for several suggestions that helped improve the paper. References [1] Arne Alex, Matthias Kalus, Alan Huckleberry, and Jan von Delft. A numerical algorithm for the explicit calculation of su (n) and sl (n, c) clebsch{gordan coecients. Journal of Mathematical Physics , 52(2): 023507, 2011. DOI: 10.1063/1.3521562. 2 [2] Robert Alicki, Slawomir Rudnicki, and Slawomir Sadowski. Symmetry properties of product states for the system of n n-level atoms. Journal of Mathematical Physics , 29:1158{1162, 1988. DOI: 10.1063/1.527958. 1 [3] D Bacon. Decoherence, control, and symmetry in quantum computers . PhD thesis, University of

Question 444 multiple-choice

In condensed matter physics, quantum transport in disordered systems is deeply influenced by symmetries present in the Hamiltonian and how they are affected by disorder. Particle-hole symmetry and its breaking can lead to nontrivial collective excitations and impact observable transport properties. Which statement most accurately describes the implications of spontaneously broken particle-hole symmetry in a disordered quantum system with respect to quantum transport behavior? 1) It invariably leads to insulating behavior due to localization of all electronic states. 2) It introduces a finite energy gap that suppresses low-energy excitations. 3) It results in a massless mode whose presence supports robust, diffusive quantum transport up to two-loop order in perturbation theory. 4) It causes the disappearance of all transport channels, completely halting conductivity. 5) It ensures the diffusion coefficient becomes entirely independent of microscopic Hamiltonian details. 6) It induces strong nonlinear corrections to transport at the lowest perturbative order. 7) It enforces perfect symmetry in the electronic density of states at all energies.

✓ Correct Answer:

The correct answer is 3) It results in a massless mode whose presence supports robust, diffusive quantum transport up to two-loop order in perturbation theory..

📚 Reference Text:

Title: Robust quantum transport at particle-hole symmetry Year: 2021 Paper ID: 06c00d190f3defbfa0b56e5631d259232ba6483b Source: semantic-scholar URL: https://www.semanticscholar.org/paper/06c00d190f3defbfa0b56e5631d259232ba6483b Abstract: We study quantum transport in disordered systems with particle-hole symmetric Hamiltonians. The particle-hole symmetry is spontaneously broken after averaging with respect to disorder, and the resulting massless mode is treated in a random-phase representation of the invariant measure of the symmetry group. We compute the resulting fermionic functional integral of the average two-particle Green's function in a perturbation theory around the diffusive limit. The results up to two-loop order show that the corrections vanish, indicating that the diffusive quantum transport is robust. On the other hand, the diffusion coefficient depends strongly on the particle-hole symmetric Hamiltonian we choose to study. This reveals a connection between the underlying microscopic theory and the classical long-scale metallic behaviour of these systems.

Question 445 multiple-choice

Quantum Hamiltonian simulation is central to quantum computing and control, where the complexity of implementing desired dynamics depends on both the interaction structure and control capabilities. Coupling matrices and their spectral properties offer a tractable framework for analyzing the limits of simulation efficiency. When simulating a time-reversed Hamiltonian (−H) in a system of n coupled qubits (d=2), which of the following statements best describes the fundamental lower bound on time overhead using explicit coupling matrix constructions? 1) The time overhead is independent of the system size n. 2) The time overhead scales exponentially with n. 3) The time overhead is bounded below by n−1. 4) The time overhead is determined solely by the rank of the Hamiltonian. 5) The time overhead equals the number of available control pulses. 6) The time overhead is always less than n/2. 7) The time overhead depends only on the interaction graph's connectivity, not on n.

✓ Correct Answer:

The correct answer is 3) The time overhead is bounded below by n−1..

📚 Reference Text:

a convex optimization problem. To derive lower bounds on the time overhead and the number of time steps it is possible to work with the coupling matrices instead of the Hamiltonians themselves. This is because thecondition in Eq. (1.3) translates into ˜J=N/summationdisplay j=1τjOjJOT j, (1.4) where Ojare orthogonal transformations corresponding to the local operations Ujforj= 1,...,N [WRJB02a]. The advantage is that the size of the coupling matrices grows only linearly with n, whereas the size of the Hamiltonians grows exponentially. LetHand˜Hbe arbitrary Hamiltonians with coupling matrices Jand˜J, respectively. A necessary condition that Hcan simulate ˜Hwith time overhead τfollows from Eq. (1.4) and Uhlmann’s theorem [WJB02, WRJB02a]: If ˜H≤τHthen we necessarily have k/summationdisplay i=1λi(˜J)≤τk/summationdisplay i=1λi(J), fork=1,...,nm ,w h e r e λ1(˜J)≥λ2(˜J)≥ ··· ≥ λnm(˜J)andλ1(J)≥λ2(J)≥ ··· ≥ λnm(J)denote the eigenvalues of ˜JandJ, respectively. This so-called spectral majorization criterion has still to be satisﬁed after rescaling the strengths Jkl;αβand˜Jkl;αβof correspond- ing interactions in Hand˜Hby the same real factor skl∈R. The reason is that the same sequence of local operations may be used for the rescaled problem. One of the nicest applications of spectral majorization theory is to derive lower bounds on time-reversal schemes [JWB02], i. e., to simulate −HwhenHis the quantum computer’s Hamiltonian. This is the usual problem of efﬁcient refocusing techniques in NMR. For the case of coupled qubits, i. e., d=2, we consider the Hamiltonian H=/summationtext k

Question 446 multiple-choice

Quantum algorithms for group-theoretic problems have the potential to revolutionize cryptography and computational complexity, especially when dealing with non-abelian groups such as the dihedral group. The dihedral hidden subgroup problem (DHSP) is a pivotal example where quantum speedups are explored. Which of the following best characterizes the quantum algorithm's approach for solving the dihedral hidden subgroup problem in terms of extracting the hidden subgroup information? 1) It uses Grover's search to directly identify elements fixed by the hidden subgroup. 2) It applies a quantum phase estimation procedure after random sampling of group elements. 3) It performs a quantum character transform to obtain a group representation, then iteratively pairs unfavorable qubits to reach a target representation that reveals the hidden subgroup through measurement. 4) It relies exclusively on amplitude amplification to distinguish between rotations and reflections in the group. 5) It computes the classical Fourier transform of the oracle outputs and applies sieve techniques to the result. 6) It measures all qubits immediately after the initial transform to obtain subgroup information without further manipulation. 7) It reduces the problem to factoring integers using Shor's algorithm.

✓ Correct Answer:

The correct answer is 3) It performs a quantum character transform to obtain a group representation, then iteratively pairs unfavorable qubits to reach a target representation that reveals the hidden subgroup through measurement..

📚 Reference Text:

Title: A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem Year: 2003 Paper ID: 43809e75c048ff3973b333d0070464c20508a6cb Source: semantic-scholar URL: https://www.semanticscholar.org/paper/43809e75c048ff3973b333d0070464c20508a6cb Abstract: We present a quantum algorithm for the dihedral hidden subgroup problem (DHSP) with time and query complexity $2^{O(\sqrt{\log\ N})}$. In this problem an oracle computes a function $f$ on the dihedral group $D_N$ which is invariant under a hidden reflection in $D_N$. By contrast, the classical query complexity of DHSP is $O(\sqrt{N})$. The algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group. The algorithm begins as usual with a quantum character transform, which in the case of $D_N$ is essentially the abelian quantum Fourier transform. This yields the name of a group representation of $D_N$, which is not by itself useful, and a state in the representation, which is a valuable but indecipherable qubit. The algorithm proceeds by repeatedly pairing two unfavorable qubits to make a new qubit in a more favorable representation of $D_N$. Once the algorithm obtains certain target representations, direct measurements reveal the hidden subgroup.

Question 447 multiple-choice

In algebraic graph theory, Cayley digraphs and their generalizations are studied to understand the interplay between group actions and graph symmetries. The structure of certain groups determines the possible automorphism properties of associated digraphs. Which of the following statements correctly describes the classification of finite non-abelian BCI-groups in relation to bi-Cayley digraph isomorphisms? 1) All finite non-abelian BCI-groups are simple groups. 2) Finite non-abelian BCI-groups must have cyclic Sylow subgroups of every prime order. 3) Only abelian groups can be finite BCI-groups. 4) Finite non-abelian BCI-groups always have non-solvable composition factors. 5) Finite non-abelian BCI-groups must be nilpotent. 6) Finite non-abelian BCI-groups are characterized by having trivial automorphism groups. 7) Finite non-abelian BCI-groups are completely classified using kernel isomorphism theory and explicit subgroup structure results.

✓ Correct Answer:

The correct answer is 7) Finite non-abelian BCI-groups are completely classified using kernel isomorphism theory and explicit subgroup structure results..

📚 Reference Text:

Title: On kernel isomorphisms of $m$-Cayley digraphs and finite $2$PCI-groups Year: 2025 Paper ID: 48761c1453621af191dc048e7a7b09c13136d531 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/48761c1453621af191dc048e7a7b09c13136d531 Abstract: The isomorphism problem for digraphs is a fundamental problem in graph theory. In this paper, we consider this problem for $m$-Cayley digraphs which are generalization of Cayley digraphs. Let $m$ be a positive integer. A digraph admitting a group $G$ of automorphisms acting semiregularly on the vertices with exactly $m$ orbits is called an $m$-Cayley digraph of $G$. In our previous paper, we developed a theory for $m$-Cayley isomorphisms of $m$-Cayley digraphs, and classified finite $m$CI-groups for each $m\geq 2$, and finite $m$PCI-groups for each $m\geq 4$. The next natural step is to classify finite $m$PCI-groups for $m=2$ or $3$. Note that BCI-groups form an important subclass of the $2$PCI-groups, which were introduced in 2008 by Xu et al. Despite much effort having been made on the study of BCI-groups, the problem of classifying finite BCI-groups is still widely open. In this paper, we prove that every finite $2$PCI-group is solvable, and its Sylow $3$-subgroup is isomorphic to $Z_3, Z_3\times Z_3$ or $Z_9$, and Sylow $p$-subgroup with $p\not=3$ is either elementary abelian, or isomorphic to $Z_4$ or $Q_8$. We also introduce the kernel isomorphisms of $m$-Cayley digraphs, and establish some useful theory for studying this kind of isomorphisms. Using the results of kernel isomorphisms of $m$-Cayley digraphs together with the results on $2$PCI-groups, we give a proper description of finite BCI-groups, and in particular, we obtain a complete classification of finite non-abelian BCI-groups.

Question 448 multiple-choice

In quantum algorithms dealing with finite groups, Fourier sampling and Schur sampling are techniques used to extract information about hidden subgroups by measuring properties of irreducible representations (irreps) in quantum states. Understanding the effect of permutation invariance and irrep multiplicities is crucial for analyzing the effectiveness of such algorithms. When k weak Fourier samplings are performed on registers in a quantum algorithm for a finite group, under what condition does the resulting probability distribution for observed irrep types become independent of the hidden subgroup, thereby preventing extraction of information about the subgroup? 1) When all observed irreps are identical across all registers 2) When the dimension of the largest irrep is maximized 3) When only the order of the irrep sequence is measured 4) When the registers are not permuted after measurement 5) When each observed irrep is different, resulting in a multiplicity-free outcome 6) When the hidden subgroup has index one in the group 7) When the number of samples k equals the order of the group

✓ Correct Answer:

The correct answer is 5) When each observed irrep is different, resulting in a multiplicity-free outcome.

📚 Reference Text:

Fourier sampling in the usual way, measuring the irrep label for each register, then we will typically obtain a state that is no longer permutation invariant. Ho wever, since the irrep labels are identically distributed for each register, the order in which the irreps appear carries no information. Only the typeof the irreps, i.e., the number of times each irrep appears, is important. Thus, it is suﬃcient to perform what we might call weak Fourier type sampling , in which we only measure the irrep type. Equivalently, we could perform complete weak Fourier sampling and then randomly permute the kregisters, thereby discarding the (irrelevant) in- formation about the order of the irreps, and producing a permuta tion-invariant state to use for weak Schur sampling. (In fact, since the distribution under wea k Schur sampling has the permutation invariance (11), it is not actually necessary to explicitly symmetrize the state.) We begin by performing weak Fourier sampling. The hidden subgroup s tateρHdeﬁned in (3) has the following block structure in the Fourier basis: ρH∼=1 |G|/circleplusdisplay σ∈ˆGIdimVσ⊗/summationdisplay h∈Hσ(h)∗(13) =:/summationdisplay σ∈ˆGPr(σ)IdimVσ dimVσ⊗ρH,σ. (14) Here the probability of observing the irrep σis Pr(σ) =dimVσ |G|/summationdisplay h∈Hχσ(h)∗, (15) and the state conditioned on this observation is ρH,σ. Weak Fourier sampling consists of mea- suring the label of the irrep. Assume we have observed σ. Then the resulting density operator is ρH,σ=1/summationtext h∈Hχσ(h)/summationdisplay h∈H|σ/an}bracketri}ht/an}bracketle{tσ|⊗σ(h)∗. (16) We explicitly include the label |σ/an}bracketri}htto reﬂect the fact that σwas observed. Repeating weak Fourier sampling ktimes, we obtain the state ρH,σ=ρH,σ1⊗···⊗ρH,σk, (17) 6 whereσ:= (σ1,σ2,...,σ k)∈ˆGkmay be viewed either as the actual outcome of the kinstances of weak Fourier sampling, or merely as a representative of the irrep type, as discussed above. Given this state, the conditional probability of observing the partit ionλis Pr(λ|σ) = tr(Π λρH,σ) (18) =dimPλ k!/summationdisplay π∈Skχλ(π)tr[P(π)ρH,σ]. (19) Note that tr[ P(π)ρH,σ] = 0 ifπ(σ)/ne}ati≀nslash=σ, whereπ(σ) = (σπ−1(1),...,σ π−1(k)). Now we are ready to prove Theorem 1: Proof of Theorem 1. Let us assume that σis multiplicity-free, i.e., that all the σi’s are diﬀerent. In this case the traces are zero for all π/ne}ati≀nslash= 1, where 1 is the identity element of Sk. Then Pr(λ|σ) =dimPλ k!χλ(1) trρH,σ=(dimPλ)2 k!, (20) which is nothing but the Plancherel distribution over ˆSk, and which in particular is independent ofthe hiddensubgroup H. Thisshowsthat wecannotextractanyinformationabout Hprovided that we have obtained a multiplicity-free σ. It remains to show that the probability of obtaining a type σthat is not multiplicity-free is vanishingly small. This can be seen as follows. From (15), we have Pr(σ)≤(dimVσ)2 |G||H| (21) since the absolute value of the character χσat any group element can be at most dim Vσ. If we perform weak Fourier sampling on two copies of ρH, then the probability of obtaining the same irrep twice is Pr(2 irreps same) =/summationdisplay σ∈ˆGPr(σ)2≤max σ∈ˆGPr(σ)≤d2 max |G||H|, (22) wheredmax:= maxσ∈ˆGdimVσdenotes the maximal dimension of any irrep of G. Forkcopies, we can upper bound the probability that any two irreps out of the kobserved irreps are the same by

Question 449 multiple-choice

In quantum algorithms for group-theoretic problems, representation theory and symmetries of group actions play a pivotal role in designing efficient measurement strategies. Quantum states associated with hidden subgroups can be analyzed via decompositions into irreducible representations and exploitation of invariance properties. Which property of the multi-copy hidden subgroup state is essential for leveraging representation-theoretic tools such as Schur’s lemma and character projection operators to solve the Hidden Subgroup Conjugacy Problem? 1) The state is always separable across all copies. 2) The state is invariant under the left regular representation of the group. 3) Each copy is independently sampled from different subgroups. 4) The state is expressed solely in the computational basis. 5) The state is invariant only under abelian group actions. 6) The state forms a simple tensor product without symmetrization. 7) The state is invariant under the m-fold tensor product of the right regular representation, enabling decomposition into irreducible representations.

✓ Correct Answer:

The correct answer is 7) The state is invariant under the m-fold tensor product of the right regular representation, enabling decomposition into irreducible representations..

📚 Reference Text:

h∈HRR(g)RR(h)RR(g−1), (37) so that ρ[H]=1 |G|/summationdisplay g∈GRR(g)ρHRR(g−1). (38) In other words, ρ[H]isρHsymmetrized over the right regular representation. This implies thatρ[H]is invariant under the right regular representation: RR(g)ρ[H]RR(g−1) =ρ[H]. (39) Thus we can again apply Schur’s lemma. Since ρHalready only has support over the right regular representation, we can deduce that ρ[H]in the|µ,rµ,lµ/an}bracketri}htbasis is given by ρ[H]=/circleplusdisplay µcµ(H)Idµ⊗Idµ. (40) We can use the character projection operator, which projects o nto a given irrep space, Cµ=dµ |G|/summationdisplay g∈Gχµ(g)∗RR(g), (41) D. Bacon 15 whereχµ(g) is the character of the µth irrep evaluated at group element g, along with our simple representation of the hidden subgroup states to determine cµ(H), cµ(H) =1 d2µTr[Cµρ[H]] =1 |G|dµ/summationdisplay h∈Hχµ(h)∗. (42) What does our expression for the hidden subgroup conjugacy sta te in Eq. 40 tell us? It tells us that, without loss of generality, if we are trying to solve the H SCP using a single copy of the hidden subgroup state, then we can without loss of generalit y perform a QFT over Gand then measuring the irrep label µ. Further all of the information, if there is enough information to reconstruct which set of conjugate subgroups th e hidden subgroup belongs to, can be derived from such a measurement. 5.2 Multiple Copy Case Above we have argued that ρ[H]is the appropriate state to consider when attempting to describe algorithms for the HSCP when we have been given a single cop y of the hidden subgroup state. What state should we consider for the multiple cop y case? At ﬁrst glance oneistempted toanswer ρ⊗m [H]. However,this isthestatewhich isrelevantifweareattempting to identify the which set of conjugate subgroups a hidden subgrou p state belongs to, and we are given diﬀerent hidden subgroup states from this set every time we produce a hidden subgroup state. This however, is not our case, as we are still quer ying a function which hides a ﬁxed subgroup H. Thus the actual state which is relevant for the HSCP is instead ρ[H],m=1 |G|/summationdisplay g∈Gρ⊗m gHg−1. (43) The nice thing about Eq.(43) isthat it representsastate which has been averagedoverthe m-fold direct product of the right regular representation. The m-fold direct product of right regular representations is the representation of Ggiven by RR(g)⊗m, i.e. the tensor product of the right regular representation acting in the same manner on ev ery tensor product space. Indeed we can express Eq. (43) as ρ[H],m=1 |G|/summationdisplay g∈GRR(g)⊗mρ⊗m HRR(g−1)⊗m. (44) This implies, as before, that ρ[H],mis invariant under the m-fold direct product of the right regular representation, RR(g)⊗mρ[H],mRR(g−1)⊗m=ρ[H],m. (45) This is the symmetry which we are going to exploit to help us solve the mu lticopy HSCP. In order to exploit the symmetry in Eq. (45) we begin by ﬁrst rewritin gρ[H]. In particular we rewrite ρ[H]in a basis in which every hidden subgroup state copy has been subjec t to a QFT over G: ρ[H],m=1 |G|k+1/summationdisplay g∈Gm/circlemultiplydisplay i=1/bracketleftBigg/circleplusdisplay µi/summationdisplay hi∈HDµi(ghig−1)⊗Idµi/bracketrightBigg . (46) 16How a Clebsch-Gordan Transform Helps to Solve ... Exchanging the product of sums with a sum of

Question 450 multiple-choice

In representation theory and mathematical physics, modular group actions on vector spaces play a crucial role in constructing and analyzing Topological Quantum Field Theories (TQFTs). The decomposition of representations and the use of tensor products are essential tools for understanding symmetry and modularity in this context. For l = 5, which statement correctly describes the relationship between the 4-dimensional representation VS and the structure of SL(2,Z) representations in the context of TQFT? 1) VS is isomorphic to the direct sum of two standard SL(2,Z) representations. 2) VS is spanned by the same basis as the 3-dimensional representation VN. 3) VS decomposes into three irreducible representations, none of which are tensor products. 4) VS can be written as a tensor product of the standard and semisimple representations of SL(2,Z). 5) VS is invariant under all bilinear forms derived from integrals on genus one surfaces. 6) VS is characterized by degeneracies that cannot be resolved using doubles. 7) VS has an explicit matrix representation identical to that of VN.

✓ Correct Answer:

The correct answer is 4) VS can be written as a tensor product of the standard and semisimple representations of SL(2,Z)..

📚 Reference Text:

o-dimensional sub- representations V=VN+VS. Here the subspace VNis spanned by N0=N+ 0+N− 0andP1withSandTacting as: TN= 1 0 0q SN=q√ 3 −1−2 3(q−q−1) (q−q−1) 1  (3.134) This subrepresentation VShas basis vectors/tildewiderPo=Po+1 (q−q−1)Noand/tildewidestNo:=N+ o−2N− o for which theSandTmatrix have the form of the standard representation: TS= 1 0 1 1 SS=qγq 0−1 1 0  (3.135) 40 Forl= 5 the matrices are given for the basis P0, N+ 0, N− 0, P1N+ 1, N− 1, P2: To= 1 0 0... 0 0 0... 0 ...... 1 1 0... 0 0 0... 0 ...... −4 0 1... 0 0 0... 0 ······ ······ ······...······ ······ ······...······ 0 0 0... q−10 0... 0 ...... 0 0 0...−q−13 [2]q−10... 0 ...... 0 0 0...q−12 [2]0q−1... 0 ······ ······ ······...······ ······ ······...······ 0 0 0... 0 0 0...q2 (3.136) So=q√ 5 0−ξ ξ... 0 ξ(−1+2[2]) ξ(1−2[2])... 0 ...... ξ(2+4[2]) [2] 0...ξ(2−6[2])1 5(−2+[2])1 5(2−6[2])...ξ(−4+2[2]) ...... ξ(4[2]−23) [2] 0...ξ(−23+19[2])1 5(−2+[2])1 5(2−6[2])...ξ(−4+2[2]) ······ ······ ······...······ ······ ······...······ 0−ξ ξ... 0 −ξ[2] ξ[2]... 0 ...... −ξ(12+14[2])−3−2[2] 0...ξ(18+16[2])1 5(−2+[2])1 5(2+4[2])...−ξ(6+2[2]) ...... ξ(13+11[2])−3−2[2] 0...−ξ(7+9[2])1 5(−2+[2])1 5(2+4[2])...−ξ(6+2[2]) ······ ······ ······...······ ······ ······...······ [2] 4 ξ ξ...−1−[2]ξ(2−2[2])ξ(3−3[2])... 1  (3.137) 41 where ξ=q−q−1 5. This representation decomposes into two irreducible representat ionsVNandVS, with dim(VN) = 3 and dim(VS) = 4. The three dimensional representation is spanned by the vectors N0=1 q−q−1(N+ 0+N− 0)N1=1 q−q−1(N+ 1+N− 1) and P2. The representation matrices are TN= 1 0 0 0q−10 0 0 q2 SN=q√ 5 [2]−[2] 2 −(3+2[2]) [2] (4+2[2]) 1 (1−[2]) 1 (3.138) The four dimensional representation is spanned by /tildewiderP0:=P0−[2]N0/tildewiderP1:=1 [2](P1+(3+2[2]) N1) and /tildewiderN0:= (N+ 0−4N− 0)/tildewiderN1:=1 [2]2(−3N+ 1+2N− 1). With ordering/tildewiderP0,/tildewiderN0,/tildewiderP1,/tildewiderN1we ﬁnd the matrices: TS= 1 0... 0 0 ... 1 1... 0 0 ··· ···...··· ··· 0 0...q−10 ... 0 0...q−1q−1 SS=q(q−q−1)√ 5 0−1... 0−[2] ... 1 0... [2] 0 ··· ···...··· ··· 0−[2]... 0 1 ... [2] 0...−1 0 (3.139) 42 Now it is easy to see that this can be written as a tensorproduct of SL(2,Z) represen- tations: VS∼=Vstan⊗Vsemis, In order to denote the isomorphism /tildewiderPi→vP⊗wi/tildewiderNi→vN⊗wi we introduce bases {vP,vN}and{w0,w1}ofVstanandVsemisrespectively. For these bases we can write TN=Tstan⊗Tsemis SN=qSstan⊗Ssemis with Tstan= 1 0 1 1 Sstan= 0−1 1 0  (3.140) and Tsemis= 1 0 0q−1 Ssemis=(q−q−1)√ 5 1 [2] [2]−1  (3.141) Conclusion The results of Chapter 2 show that a the construction of a univers al TQFT should include two features. One is to avoid degeneracies by considering only doub les. The fact that the projective phases and the proofs of modular relations are mos t conveniently given in terms the bilinear forms and moduli deﬁned from the integrals is an in dication that this is the correct language also for constructions at higher genus . In view of the glueing operations described in the introduction the genus one case can in f act be thought of as a basicbuilding bloc. It shouldbepossibletounderstand moreconcept ually theappearance of the ﬁnite representation we know from the semisimple theory as a tensorproduct with the standard representation rather than a sub representation .

Question 451 multiple-choice

Quantum simulation of lattice models often requires efficient methods to switch between real-space and momentum-space representations, especially in systems with translation symmetry and multiple energy bands. The Quantum Quasi-Fourier Transform (QQFT) is a key protocol for enabling such transformations on scalable quantum hardware. In simulating a d-dimensional k-band model on a simple cubic lattice using QQFT, which statement correctly describes both the mathematical structure and practical implementation of the d-dimensional QQFT operator? 1) The d-dimensional QQFT is implemented by a single global unitary that entangles all lattice directions simultaneously. 2) The d-dimensional QQFT operator is a direct sum of one-dimensional QQFT matrices, each corresponding to a lattice direction. 3) The QQFT in d dimensions requires exponentially increasing operation depth with respect to d. 4) Each 1D QQFT must be applied to every site individually, followed by a global phase evolution. 5) The d-dimensional QQFT operator is a Kronecker product of d identical one-dimensional QQFT matrices, realized by performing d successive 1D QQFTs along each direction. 6) The QQFT for higher dimensions requires non-local interactions across all lattice sites in every direction. 7) The operation depth for the d-dimensional QQFT is independent of the number of dimensions.

✓ Correct Answer:

The correct answer is 5) The d-dimensional QQFT operator is a Kronecker product of d identical one-dimensional QQFT matrices, realized by performing d successive 1D QQFTs along each direction..

📚 Reference Text:

identical to the QQFT. D. General framework of Hamiltonian engineering with QQFT In this section, we show how to simulate a generic k-band model in a d-dimensional space for arbitrary kandd. The model Hamiltonian has a translation symmetry and is gener- ally expressed as ˆH=X n ;n0 J ; nn0ˆ y ;rnˆ ;rn0; (22) where J ; (nn0)is the hopping amplitude, and ˆ ;rnis the ﬁeld operator at the position rn=Pd i=1niai, with the orbital index =1;2:::k. Here a1,a2;,addenote the primitive vectors of a Bravais lattice in a d-dimensional space. Since our scheme is to emulate the momentum-space of the target- ing model with the real-space lattice of a physical system, it is natural to assume a simple cubic geometry for the physi- cal lattice, which corresponds to setting a cubic grid for the momentum-space Brillouin zone. This does not compromise the generality of the Hamiltonian engineering protocol. We assume the lattice size is Lin the direction of each ai, and then the total number of sites is Ld. Just as in the one-dimensional single-band case, we employ the Fourier transform and factorize the evolution operator ( ˆU) 5 into ˆVeiˆHDTˆVy, with ˆHD=X m; ; ˆ y ;rmH (m)ˆ ;rm; (23) ˆV=exp0BBBBBB@2i LX ;n;mAn;mˆ y ;rnˆ ;rm1CCCCCCA; (24) where Ais a Hermitian matrix deﬁned by !A n;m= !nm=p Ldwith!=ei2=L. The matrixH(m) is the Fourier transform of the tunneling matrix, i.e., H (m)=X nJ ; (n)ei2nm. The phase evolution ( eiˆHDT) is local, and then can be simu- lated easily. More importantly, the QQFT in a d-dimensional space can be factorized into a sequence of one-dimensional QQFTs, which are along the directions of aiwith i= 1;2;;d, respectively. To see it, we notice !nm=dY i=1!nimi, therefore, the matrix !Ais the Kronecker product of dsame matrices, which reads !A=!¯A !¯A !¯A = !¯A 1 1 1 !¯A 1 1 1 !¯A ;(25) where !¯A n;m=!nm=p Lis the one-dimensional QQFT ma-trix and 1 is the identity matrix. The second line of Eq. (25) tells us that !Ais a product of dmatrices, i.e., !A=dY i=1!Ai, where!Aiis the Kronecker product of !¯Aand(d1)iden- tity matrices. According to Eq. (3), we immediately know that the factorization of !Aindicates a factorization of ˆV, i.e., ˆV=dY i=1ˆV(i)with ˆV(i)=!ˆ yAiˆ . Therefore, the d-dimensional QQFT ( ˆV) can be realized by successively performing ˆV(1), ˆV(2),,ˆV(d). According to the deﬁnition of !Ai, we see that the corre- sponding operator ˆV(i)involves only the hopping in the di- rection of aibut not in the other (d1)directions. There- fore, ˆV(i)on a d-dimensional cubic lattice is equivalent to the QQFT on a one-dimensional lattice in the ai-direction, which can be e ectively constructed by using the numerical decom- position scheme for generic unitary operations38, or by us- ing the analytic sequence introduced in Sec. II B. Since the Fourier transform across di erent spatial dimensions is sep- arable, the Hamiltonian sequence depth in a d-dimensional QQFT is exactly dtimes of that in a one-dimensional QQFT. Ad-dimensional QQFT involves d-cyles of

Question 452 multiple-choice

In the study of finite p-groups, particularly those of small order, structural properties such as abelianness, metacyclicity, and the nature of maximal subgroups are key to their classification. The behavior of subgroup indices, exponents, and the presence of elementary abelian normal subgroups influences which group configurations can exist. Which of the following statements accurately describes the structure of a finite non-metacyclic p-group of order greater than 24, where every maximal subgroup is minimal non-abelian? 1) Its center is always of order p³. 2) It contains no abelian subgroups of rank greater than 2. 3) Its quotient by any normal subgroup is always cyclic. 4) It is necessarily isomorphic to a direct product of cyclic groups. 5) It has only abelian maximal subgroups. 6) It is isomorphic to a group enumerated in classification propositions, and all its maximal subgroups are minimal non-abelian. 7) It must have exponent p³.

✓ Correct Answer:

The correct answer is 6) It is isomorphic to a group enumerated in classification propositions, and all its maximal subgroups are minimal non-abelian..

📚 Reference Text:

non-abelian. On the other hand, MVHZi0>f1g, and so G=Ais elementary abelian, a contradiction. Case (ii). We suppose that p¼2 and we may assume that Gis non-metacyclic. Let E be a minimal non-metacyclic subgroup of G. We use Proposition 1.9 and see that in cases (b), (c) and (d) of that proposition, Eis non-abelian and has an abelian maxi- mal subgroup. It follows that in these cases Emust be minimal non-abelian, which is not the case. Hence EGE8, which is case (a) of Proposition 1.9. We claim that when- ever Ais a maximal abelian subgroup of rank at least 3 in GandK>Ais a sub- group of Gsuch that jK :Aj¼4, then jKj¼26andKis isomorphic to the group of Proposition 1.7(a), so that K0¼ZðKÞ¼FðKÞ¼W1ðKÞ¼W1ðAÞGE8 and Ais abelian of type ð4;2;2Þ. Indeed, if X>Ais a maximal subgroup of Kcontaining A,t h e n jX :Aj¼2 and therefore Xis minimal non-abelian with E8GW1ðAÞ¼W1ðXÞ. Let Ybe a maximal subgroup of Kdistinct from X. Since K is not minimal non-abelian, Yis non-abelian. But YVXð24all of whose maximal subgroups are minimal non-abelian. It follows that Kis isomorphic to the group of Proposition 1.7(a) and our claim is proved. Suppose that K0Gand let L>Kbe a subgroup of Gwith jL:Kj¼2. Letting L=Kact on the seven subgroups Ai=W1ðAÞof order 2 in K=W1ðAÞGE8(where all Ai are abelian of type ð4;2;2Þand they are also maximal abelian subgroups in G), we see that one of them, Ai0say, is certainly normal in L. We note that L=Ai0is a non- cyclic group of order 8 since K=Ai0GE4. Let K/C3=Ai0be a maximal subgroup of L=Ai0distinct from K=Ai0. By the preceding paragraph, K/C3is isomorphic to

Question 453 multiple-choice

Representation theory and algebraic topology intersect in the study of exceptional Lie groups, where tools like $K$-theory and computational algebra programs reveal deep structural relationships. Operations such as Adams operations and exterior powers play a central role in understanding these connections at specific primes. Which computational tool is commonly used to explicitly calculate operations like the exterior power $\lambda^2$ in the representation ring $R(E_8)$? 1) GAP 2) MAGMA 3) LiE 4) SageMath 5) Mathematica 6) Maple 7) Pari/GP

✓ Correct Answer:

The correct answer is 3) LiE.

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Title: From Representation Theory to Homotopy Groups Year: 1998 Paper ID: 3f5c778596800186262fa58797fd4173bb53e364 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/3f5c778596800186262fa58797fd4173bb53e364 Abstract: Introduction Representation theory and $\psi^2$ in $K$-theory Nice form for $\psi^2$ in $PK^1(E_8)_{(5)}$ and $PK^1(X)$ Determination of $v_1^{-1}\pi_{2m}(E_8 5)$ Determination of $v_1^{-1}\pi_{2m-1}(E_8 5)$ Calculation of $v_1^{-1}\pi_\ast(E_8 3)$ LiE program for computing $\lambda^2$ in $R(E_8)$ Analysis of $F_4$ and $E_7$ at the prime $3$ References.

Question 454 multiple-choice

Quantum walk algorithms leverage efficient circuit designs to simulate complex quantum phenomena, often utilizing advanced gate decompositions and statistical methods to validate performance on quantum hardware. Optimizing these implementations is key for scalable quantum computing, especially on current noisy devices. Which feature most critically enables the scalability of multi-qubit diagonal unitary approximations in quantum walk circuits for large system sizes? 1) The total circuit depth increases linearly with the number of qubits. 2) The gate count depends exponentially on the number of qubits. 3) Circuit optimization relies primarily on hardware-specific gate sets. 4) The implementation requires additional ancilla qubits for each new qubit added. 5) The circuit size for diagonal unitary approximation depends on the error threshold, but is independent of the number of qubits for sufficiently large systems. 6) The circuit must include a separate Quantum Fourier Transform for each qubit. 7) The accuracy of results is solely determined by the connectivity of the quantum device.

✓ Correct Answer:

The correct answer is 5) The circuit size for diagonal unitary approximation depends on the error threshold, but is independent of the number of qubits for sufficiently large systems..

📚 Reference Text:

unitary diagonal matrices Λ,Λ′of eq. (15) for the multi-qubit system. In [13], a method is described to eﬃciently implement multi-qubit diagonal unitary matrices, using CNOT gates and single qubit rotations of the form Rθ=e−iθ 2Z=/bracketleftBigg e−iθ 20 0eiθ 2/bracketrightBigg , (17) where Z=/bracketleftbigg 1 0 0−1/bracketrightbigg . The approximation in [13] uses partial Walsh-Fourier series. The circ uit size depends on the approximation error ǫasO(1/ǫ), and is independent of nfor large enough n. That implementation can be utilized with a simple modiﬁcation to implement Cv(Λ) eﬃciently. Given an eﬃcient circuit prescribed in [13] for the diagonal unitary ma trix Λ, suppose that some qubit in the circuit has a rotationof the form Rθfrom eq. (17) applied to it. We replace this rotation with Cv(Rθ), the|v/an}⌊ra⌋ketri}ht-controlled Rθapplied to that qubit, Cv(Rθ) =/bracketleftbigg I20 0Rθ/bracketrightbigg , Substituting |v/an}⌊ra⌋ketri}ht-controlled versions for all such rotations on all qubits would give us Cv(Λ). In a similar manner, we can obtain C¯v(Λ) by substituting C¯v(Rθ), which can be constructed as in eq. (16) if necessary, for the rotations Rθthat occur in the circuit for Λ. With C¯v(Λ) andCv(Λ′) thus constructed, we obtain Cv(Λ,Λ′) =C¯v(Λ)Cv(Λ′) =Cv(Λ′)C¯v(Λ). We use this eﬃcient implementation of Cv(Λ,Λ′) and the 2ndimensional QFT implementa- tion to obtain an eﬃcient implementation of σfrom eq. (15). An eﬃcient QW algorithm for the given Λ ,Λ′follows easily. 4.Simulating QW with QFT based shift on quantum computers We test the QW algorithm just described with the form of propagatio n from Section 2.1 and QFTbased shift on three IBM quantum computers [16]: IBMQX2, IBMQX London and IBM16Melbourne, which are 5, 5 and 14 qubit machines respectively. The re ader may ﬁnd the machine gate and coupling maps in Appendix A Section A.1. IBM Qiskit [17] provides the API to access the computers and develop the python based co de, transpile it, and run it. The lattice sizes, walk step pairs in our simulations are (4 ,1), (4,2), and (8,1) correspond- ing ton= 2,3 qubits. For all the simulations, we use the same scattering matrix S=1√ 2/bracketleftbigg 1i i1/bracketrightbigg . 12 EFFICIENT AND SCALABLE QUANTUM WALK ALGORITHMS After initializing the walk joint velocity-position state, we run the algo rithm for the given number of evolutionary steps. Then we measure the ﬁnal state. For each experiment, i.e., a set of lattice size, initial state, and step s of evolution, we execute the algorithm 1024 times (called “shots”) to generate a dis tribution on measured states.4The circuit is transpiled (compiled to quantum gates) using optimizatio n level 3 of the transpiler prior to executing each experimental run. At this lev el, the transpiler takes in to account the noise properties and the connectivity of the device . We repeat these runs, for each experiment and each machine, num erous times. We show the result from the run with the smallest ℓ1distance from the ideal distribution. Note that theℓ1distance between two probability distributions PandQover a ﬁnite, discrete variable i∈ Iis ℓ1(P,Q) =1 2/summationdisplay i∈I|P(i)−Q(i)|. It takes values in [0 ,1] range.

Question 455 multiple-choice

In group theory, the study of finite $ p $-groups involves analyzing their subgroup structures, often using characteristic matrices to encode relationships among generators. The indices of certain special subgroups provide key insights into the group's overall structure. Which of the following conditions must be met for a finite $ p $-group $ G $ to have $ I_{\max} = 2 $, where $ I_{\max} $ is the maximal index of an $ A_1 $-subgroup, and the characteristic matrix $ w $ satisfies specific algebraic requirements? 1) $ n = m = 3 $ and all entries of $ w $ are quadratic residues modulo $ p $ 2) $ I_{\min} = 1 $ and $ I_{\max} = p $ 3) The group has an $ A_1 $-subgroup of index $ p $ and $ n \geq 2 $ 4) The characteristic matrix is the zero matrix and $ p $ is odd 5) $ n = m = 2 $ and certain entries of $ w $ are quadratic non-residues modulo $ p $ 6) The group is abelian and all indices are equal to 1 7) $ I_{\max} \geq n $ for all choices of $ n $ and $ m $

✓ Correct Answer:

The correct answer is 5) $ n = m = 2 $ and certain entries of $ w $ are quadratic non-residues modulo $ p $.

📚 Reference Text:

[ b,c] =x. ThenG3=/an}bracketle{tx/an}bracketri}ht. Sinceapn∈Φ(G′)G3, we may assume that apn=cw11pxw12. By a similar reason, we may assume that bpm=cw21pxw22. Letw(G) = (wij). Then we get a 2×2 matrix over Fp.w(G) is called a characteristic matrix of G. Notice that w(G) will be changed if we change the generators a,b. We also call a,ba set of characteristic generators of w(G). Theorem 5.1. Suppose that pis an odd prime, Gis a ﬁnite p-group such that G3∼= Cp,Φ(G′)G3=C2 p,G/Φ(G′)G3∼=Mp(n,m,1)and the type of CG(G′)/Φ(G′)G3is (pn,pm−1,p)wheren≥m≥2. Letw(G) = (wij)be a characteristic matrix of G. Then the following conclusions hold: (1)Imin≥2. That is, Gdoes not have an A1-subgroup of index p; (2)Imax≥n; (3)Imax= 2if and only if n=m= 2and one of the following holds: (i)w11= w12= 0andw22/ne}ationslash= 0;(ii)w2 11−4w12is a quadratic non-residue modular p. ProofWe getImaxandIminby investigating all maximal subgroups of G. LetN= /an}bracketle{ta,bp,c,x/an}bracketri}htandMi=/an}bracketle{taib,ap,c,x/an}bracketri}ht. Then all maximal subgroups of GareNandMi where 0≤i≤p−1. 16 (1) Ifw12/ne}ationslash= 0 orw22/ne}ationslash= 0, then N=/an}bracketle{ta,bp/an}bracketri}ht∗/an}bracketle{tc/an}bracketri}ht ∈ A2by Lemma 2.3. If w12=w22= 0, thenN=/an}bracketle{ta,bp/an}bracketri}ht ∗ /an}bracketle{tc/an}bracketri}ht × /an}bracketle{tx/an}bracketri}ht ∈ A3by Lemma 2.3. By calculation, M′ i=/an}bracketle{tcp,x/an}bracketri}ht= Φ(G′)G3and hence Mi/ne}ationslash∈ A1. ThusGdoes not have an A1-subgroup of index p. (2) LetD=/an}bracketle{tc,b/an}bracketri}ht. ThenD∈ A1and|G:D|=pn. HenceImax≥n. (3) IfImax= 2, then w12/ne}ationslash= 0 orw22/ne}ationslash= 0 by the proof of (1), and n=m= 2 by (2). LetA=/an}bracketle{tap,bp,c,x/an}bracketri}ht,Br=/an}bracketle{taibcr,ap,cp,x/an}bracketri}htandCst=/an}bracketle{taibasp,catp,cp,x/an}bracketri}htwhere 0≤r,s,t≤p−1. Then A,BrandCstare all maximal subgroups of Mi. Since G∈ A3,Mi∈ A2. HenceA,BrandCstare abelian or minimal non-abelian. First of all, Ais the unique abelian maximal subgroup of Mi. Since w12/ne}ationslash= 0 orw22/ne}ationslash= 0,Br=/an}bracketle{taibcr,ap/an}bracketri}ht ∈ A 1. Since [ catp,aibasp] =ctpx−1and (aibasp)p2= c(iw11+w21)pxiw12+w22and (catp)p=c(tw11+1)pxtw12,Cst∈ A1if and only if the following equation set about iandthas no solution. /braceleftBigg iw11+w21=−t(iw12+w22) (5.1.1) tw11+1 =−t2w12 (5.1.2)(5.1) By (5.1), i=tw21+t2w22. Hence Cst∈ A1if and only if Equation (5.1.2) has no solution. If w12= 0, then w11= 0. Hence (i) holds. If w12/ne}ationslash= 0, then w2 11−4w12is a quadratic non-residue modular p. Hence (ii) holds. Conversely, if (i) or (ii) holds and n=m= 2, then, by using the above argument, it is easy to see Imax= 2. /square Theorem 5.2. LetGbe a ﬁnite 2-group suchthat G3∼=C2,Φ(G′)G3=C2 2,G/Φ(G′)G3∼= M2(n,m,1)and the type of CG(G′)/Φ(G′)G3is(2n,2m−1,2)wheren≥m≥2. Let w(G) = (wij)be a characteristic matrix of G. Then the following conclusions hold: (1)Imin≥2. That is, Gdoes not have an A1-subgroup of index 2; (2)Imax≥n; (3)Imax= 2if and only if n=m= 2and one of the following holds: (i)w22= 1 andw21= 0;(ii)w22= 0andw21=w12= 1. ProofWe getImaxandIminby investigating maximal subgroups of G. LetN= /an}bracketle{ta,b2,c,x/an}bracketri}htandMi=/an}bracketle{taib,a2,c,x/an}bracketri}ht. Then all maximal subgroups of GareN,M0and M1. (1) Ifa2n=c2, thenN=/an}bracketle{ta,b2/an}bracketri}ht × /an}bracketle{tca2n−1/an}bracketri}ht ∈ A 2by Lemma 2.3. If a2n=x, thenN=/an}bracketle{ta,b2/an}bracketri}ht ∗ /an}bracketle{tca2n−1/an}bracketri}ht ∈ A 2. Ifb2m=c2, thenN=/an}bracketle{ta,b2/an}bracketri}ht × /an}bracketle{tcb2m−1/an}bracketri}ht ∈ A 2. Ifb2m=x, thenN=/an}bracketle{ta,b2/an}bracketri}ht ∗ /an}bracketle{tcb2m−1/an}bracketri}ht ∈ A2. Ifa2n=c2ixiandb2m=c2jxj, then N=/an}bracketle{ta2,b/an}bracketri}ht × /an}bracketle{tc/an}bracketri}ht ∈ A3by Lemma 2.3. By calculation, M′ i=/an}bracketle{tc2,y/an}bracketri}ht= Φ(G′)G3and henceMi/ne}ationslash∈ A1. To sum up, Gdoes not have an A1-subgroup of index 2. (2) LetD=/an}bracketle{tc,b/an}bracketri}ht. ThenD∈ A1and|G:D|= 2n. HenceImax≥n. (3) IfImax= 2, then n=m= 2 by (2). LetA=/an}bracketle{ta2,b2,c,x/an}bracketri}ht,Br=/an}bracketle{taibcr,a2,c2,x/an}bracketri}htandCst=/an}bracketle{taiba2s,ca2t,c2,x/an}bracketri}htwhere r,s,t= 0,1. Then A,BrandCstare all maximal subgroups of Mi. SinceG∈ A3, Mi∈ A2. HenceA,BrandCstare abelian or minimal non-abelian. 17 First of all,

Question 456 multiple-choice

Quantum Fourier transform (QFT) is a foundational operation in quantum computing, essential for algorithms that achieve exponential speedup. Implementing QFT in systems composed of qudits (quantum elements with more than two levels) and hybrid architectures presents unique challenges and opportunities. Which of the following statements most accurately describes a technical advantage of using hybrid qudit systems in quantum computing? 1) They allow quantum error correction codes to be applied without modification. 2) They enable more compact and potentially robust quantum registers by combining quantum elements of different dimensions. 3) They ensure that all universal gate sets from qubit systems can be directly transferred to qudit systems. 4) They eliminate the need for controlled-phase gates in quantum algorithms. 5) They restrict the implementation of QFT to only two-level quantum systems. 6) They prevent higher-dimensional entanglement from being utilized in quantum communication. 7) They guarantee that gate decompositions for qubits are optimal for qudits.

✓ Correct Answer:

The correct answer is 2) They enable more compact and potentially robust quantum registers by combining quantum elements of different dimensions..

📚 Reference Text:

arXiv:1503.06624v1 [quant-ph] 23 Mar 2015Implementation of the quantum Fourier transform on a hybrid qubit-qutrit NMR quantum emulator Shruti Dogra,∗Arvind,†and Kavita Dorai‡ Department of Physical Sciences, Indian Institute of Scien ce Education & Research Mohali, Sector 81 Mohali, Manauli PO 140306 Punjab India. The quantum Fourier transform (QFT) is a key ingredient of se veral quantum algorithms and a qudit-speciﬁcimplementation of the QFT is hence an importa nt step toward the realization of qudit- based quantumcomputers. This work develops a circuit decom position of the QFT for hybridqudits based on generalized Hadamard and generalized controlled- phase gates, which can be implemented using selective rotations in NMR. We experimentally implem ent the hybrid qudit QFT on an NMR quantum emulator, which uses four qubits to emulate a single qutrit coupled to two qubits. I. INTRODUCTION The size of a quantum register can be increased ei- ther by increasing the number of qubits (in the standard model ofa quantum computer) orby increasingthe num- ber of accessible logical states in each quantum element i.ebyusingqudits(quantumdigits)[1–5]. Forquditcom- puting to become a reality, it is essential to concretize the theoretical framework for computation in terms of a set of quantum gates that are universal and can imple- ment any algorithm eﬃciently. It is however not always straightforward to generalize a qubit gate to the qudit scenario and in some cases, more than one kind of gen- eralization is possible, each retaining some features of the original qubit gate [6]. A gate library for qudits has been proposed, containing one-qudit and two-qudit gates such as the CINC gate, generalized SWAP and general controlled-X (GCX) gates [7–11]. Several qudit comput- ing proposals using diﬀerent physical systems have been designed to incorporate “hybrid” qudits of diﬀerent di- mensions [12–14], wherein multi-qudit gates are general- ized to transform two or more qudits of diﬀerent dimen- sions. A set of hybrid quantum gates have been designed to act on qudits ofdiﬀerent dimensions, including the hy- brid SUM, SWAP, Toﬀoli and Fredkin gates[15, 16]. Qu- dits havebeen used as a resourcein securequantum com- munication [17], quantum key distribution [18], quantum computing [19] and quantum cloning [20]. NMR has been a fruitful testbed for implementing quantuminformationprocessingideasandisnowincreas- inglybeingusedtomanipulatequdits. NMRquditimple- mentations include using quadrupolar nuclei oriented in a liquid crystal [21] for information processing, studying the dynamics of nonclassical correlations [22] and ﬁnding the parity of a permutation using a single qutrit [23] and a single ququart [24]. Logic gates and pseudopure states for a ququart have been implemented using23Na and7Li nuclei oriented in a liquid-crystalline matrix [25, 26]. ∗shrutidogra@iisermohali.ac.in †arvind@iisermohali.ac.in ‡kavita@iisermohali.ac.inThe quantum Fourier transform (QFT) needs to be speciﬁcally tailored for qudit computers as it plays a key role in several quantum algorithms such as factor- ization [27], quantum phase estimation [28] and the hid- den subgroup problem [29]. An eﬃcient decomposition of the QFT for qubits achieves an exponential speedup over the classical fast Fourier transform [30], and has been experimentally implemented by several groups on diﬀerent physical systems [31–35]. Decompositions of the QFT

Question 457 multiple-choice

In Topological Quantum Field Theory (TQFT), algebraic objects are assigned to surfaces in a way that encodes their topology and symmetries, and specialized constructions are used to capture features such as punctures, genus, and mapping class group actions. The assignment of these objects often requires additional data and structural choices to reflect the interplay between geometry and algebra. Which of the following is a mechanism by which the TQFT functor acquires a projective phase when composing cobordisms, and quantifies the non-additivity of the signature in the context of four-manifolds? 1) The Frobenius number 2) The Jones polynomial 3) The Verlinde formula 4) The Euler characteristic 5) The character space isomorphism 6) The Maslov index 7) The universal coefficient theorem

✓ Correct Answer:

The correct answer is 6) The Maslov index.

📚 Reference Text:

(1.5). In Cothe answer is XP2∼=/summationtext j∈Ij⊙j∨and inR(A) the module XP2 is given byAwithA⊙2-action aiven by a⊙b.(x) =axS(b). Moreover,FP3=⊗implies thatXP3∼=/summationtext ij∈Ii⊙j⊙(i⊗j)∨orXP3=A⊗AwithA⊙3-action ( a⊙b⊙c).(x⊗y) = (ax⊗by)∆(S(c)). This allows us to identify the object associated to the puncture d torusT′with∂T′=S1by contracting the objects XP3andXP2along the category of theS1∐S1-boundary pieces. In Cothe resulting object is XT′=/summationtext j∈Ij⊗j∨and in R(A) by the module Awith adjoint action. The objects of all other surfaces are now found easily by sewing along circles. For example the surface Σ g,1of genus gwith one puncture is assigned to X⊗g T′. The object of the ( n+1)-punctured sphere Pn+1has object XPn+1=/summationtext iki1⊙...⊙in⊙(i1⊗...⊗in)∨inCoand inR(A) the module XPn+1=A⊗n where theA⊙(n+1)-action is given by the obvious generalization of the cases n= 2,3. The object for a general compact, orientable surface is found by sew ingXPn+1andXΣg,1. For R(A) this gives for example the module HomA(A⊗g,A⊗n) of intertwiners for one of the A-actions. Let us discuss the case g= 1, n= 1 in some more detail. The mapping class group D=πo/parenleftBig Diff(T′,∂T′)/parenrightBig maps by (1.2) into End(T′) so that we obtain in R(A) an action 5 ofDonAintertwining the adjoint action. Following (1.9) we obtain a represent ation of the modular group πo(Diff(T)) onV(T) =Hom(1,T), which for R(A) is just the restriction of the Daction to the center Z(A) =Hom(1,A). In order to interpret the rest of (1.9) recall that for R(A) the natural transformations of the identity functor are given by the action of central elements of A. In particular the generator θofπ1(Diff(S)) acts onAasad(v) wherev=θ(A) is the central “ribbon element”, see [RT]. The Dehn twist along the boundary can also be given by S4whereSis the standard generator of D. The restriction of S4=ad(v) to the center is clearly trivial. The second generator of D, the Dehn twist at a handle, Tis given by the action of θon the constituent XP2, i.e., by multiplication of vonA. The deﬁnition of a TQFT we presented so far is not quite complete. Cle arly, there are many ways of sewing up a surface Σ so we have many ways to constru ct the object XΣ. For example instead of using the center of Aas the vectorspace for the closed torus V(T) we can also choose the space Hom(A,1) - which is isomorphic to the space of characters onA- or we could have chosen the endomorphism set End(XP2) =EndA⊙2(A). These spaces are isomorphic to each other but there is no one canonical is omorphism identifying two of them. Instead the sewing procedure used to ﬁnd the objec t deﬁnes a surface with a cut diagram or decoration. Thus we should take as objects of the cobordism categories surfaces Σ together with a Lagrangian subspace of H1(Σ,R) which nust be compatible with the cobording 3-manifolds. The functor of the TQFT is now allowed to have projective phases. This means for two cobordisms M1andM2with a common, decorated boundary component that Φ(M1M2) =cµΦ(M1)Φ(M2) (1.10) whereµistheMaslovindexofatripleofLagrangiansubspaces deﬁnedbythe cobordisms. It also measures the non-additivity of the signature of the 4-manif olds cobording the Mi

Question 458 multiple-choice

Exceptional Lie groups, such as E7 and E8, possess rich subgroup structures that are deeply intertwined with their root systems and automorphism properties. Understanding involutions and their centralizers is pivotal in the classification of these subgroups and their symmetries. In the group E7, which of the following statements accurately describes a subgroup E consisting of two involutions and its centralizer? 1) E is isomorphic to (Z₂)^3, and its centralizer is D4 × A1. 2) E is an extraspecial group of order 8, with centralizer E6 × A1. 3) E is a cyclic group of order 4, and its centralizer is A7. 4) E is elementary abelian of order 8, with centralizer A1 × D6. 5) E is isomorphic to S₃, and its centralizer is E6 × D4. 6) E is an elementary abelian group of order 2, with centralizer A7 × D4. 7) E is isomorphic to (Z₂)^2, and its centralizer is E6 × D4 (adjoint type).

✓ Correct Answer:

The correct answer is 7) E is isomorphic to (Z₂)^2, and its centralizer is E6 × D4 (adjoint type)..

📚 Reference Text:

maximal commuting product of fundamental subgroups SL2 in G. Let Z(J;) = (e;) for 1 ::;;;; i::;;;; 8. Then with suitable ordering, (i) e1 ... e8 = l, (ii) (ev ... , e8) = ((e1) x ... X (e8) )/R, where R = ( e1 e2e3e8, e1 e4e5e8, e2e4e6e8, eJe4e1es), Proof. Choose a subsystem (A1)8 in the £8 root system. The elements e; are the elements hrx(-1) as described in [9, Chapter 6] for the roots a in the subsystem. A direct check using these elements gives the assertions. LEMMA 2.15. Let r = 2 and G = E7. Then E = (Z2)2, C0(E) =Ex D4 (D4 of adjoint type), and No(E)/Cc(E) = S3. Moreover, Land N0.,(E) are as in Table 1, and Eis determined up to G0-conjugacy. Proof. First note that G has precisely three classes of involutions, with centralizers A1 D6, A7( w2) and 7JE6 ( w2). Of these, the first lifts to a class of involutions in E7, while the other two lift to elements of order 4. Let (z)=Z(G). By Lemma 2.9(i), Eis either extraspecial or elementary abelian. Suppose that Eis extraspecial. Then e is conjugate to ez for every e EE, and hence C0(e)0 < C0(e) for each e EE#. Hence Lemmas 2.6 and 2.9(ii) imply that each element of E# has centralizer A7( w2). Consequently, every element of E\(z) has order 4, and so E = Q8 and E = (e, /)with [e,f] = z. Let y be an involution of G with centralizer T1E6( w2) and let d be an involution in C0(y )\C0(y )0• (To see that d exists, let T be a maximal torus of 7!E6 and let T ~11 .•• 17, a maximal commuting product of fundamental subgroups SL2• Let S; be a fundamental reflection in l; and set d = s1 •.• s7• Then cxd = -a for each root a and so d induces a graph automorphism on the factor £6. Finally, d is an involution.) Write C = ( C0(y )0)' = £6• The proof of Proposition 2. 7 shows that there is an involution h in a fundamental subgroup SL2 of K = C n C0(d) such that d, dh are representatives of the two classes of involutions in Cd. Suppose that d, dh E y0. Then C0(d, dh)0 = (C0(d)0 n C0(h))0 = T1A1A5• Bence d8 = dh implies that g normalizes T1• Then g normalizes Cc(T 1) == CG(y) 0. lut d, dh are not conjugate in Nc(C0(y)0) = C0(y), a contradiction. Thus d, dh ·e not both conjugates of y. Now d E C(y)\C(y) 0, so [d, y] -=F 1 and therefore :;(d)0 < Cc(d). Similarly C0(dh )0 < C0(dh). It follows that at least one of d, dh conjugate to an element of E#. Consequently, we may choose e E En such that y commute but e, y do not commute. LOCAL MAXIMAL SUBGROUPS 35 We now have the two non-conjugate involutions f, y E C(e)\C(e) 0• Moreover, since e E C(y)\C(y)0,

Question 459 multiple-choice

Quantum state compression algorithms are being developed to efficiently store and process quantum information, with significant implications for quantum thermodynamics and energy extraction. These advances hinge on leveraging quantum properties such as translational symmetry and correlations in spin systems. Which modification is specifically implemented when adapting a variational quantum compression algorithm for quantum states with translational symmetry, and why is this significant for reconstructing the original state? 1) Replacing all measurement gates with swap gates to preserve entanglement 2) Introducing an additional layer of classical processing to minimize total entropy 3) Utilizing a cost function based on trace distance rather than infidelity 4) Adding extra Hadamard gates to increase basis state mixing 5) Maximizing degeneracy by applying repeated Uf gates throughout the circuit 6) Omitting Hadamard and Uf gates and employing a cost function based on infidelity between input and output states 7) Encoding spin sequences using only Pauli-Z measurements to exploit pseudo-randomness

✓ Correct Answer:

The correct answer is 6) Omitting Hadamard and Uf gates and employing a cost function based on infidelity between input and output states.

📚 Reference Text:

to the com- pression of pictures (regarded as seq uential data), audio, and stock price data, where approximate reconstructio ni sa l l o w e d ,a sl o n ga st h ea p p r o x - imation meets the respective requirements. We are presently investigatingsuch applications. Application to quantum state compression Our algorithm can be adapted to comp ress quantum states satisfying translational symmetry. Speci ﬁcally, consider a set of quantum states S¼ f∣φ/C11 jTh0∣φ/C11 ¼∣φ/C11 gthat is translationally invariant under the action of the operator Th0,d eﬁned as Th0∣xi¼∣x/C3h0/C11 ; ð19Þ for all computational basis ∣xi,w h e r e/C3represents the group operation of G assigned to the bit strings x’s. Assume that there is a source that randomly generates ∣φ/C11/C10naccording to a probability density μ∣φi, i.e.,ncopies of a state ∣φ/C11 2Ssatisfying the translational sym metry. Then our variational HSP algorithm can be applied to compress the quantum states generated by thesource, with two modi ﬁcations. First, we do not need Hadamard gates and U f, and the variational QFT is applied directly to the quantum state ∣φ/C11 ,a s shown by the following circuit, ð20Þ Secondly, the cost function Eq. ( 17) is replaced by the in ﬁdelity between the input state and the reconstructed state. See Supplementary Note E for amore detailed description of this compr ession algorithm for quantum states. Quantum thermodynamical advantage implied by the algorithm Now we discuss the thermodynamic implications of our results. Quantum advantages for thermodynamical tasks are being sought in coherent accu-rate information processing 33, in charging time of batteries34,a n di nt h e efﬁciencies of thermal machines35.H e r ew ea r g u et h a tt h eq u a n t u mc o m - putational advantage of HSP algorithms translates into a thermodynamicadvantage in energy harvesting, enabling more work extracted from a sourcewithin a ﬁxed amount of time. Information can fuel energy harves ting together with heat energy; having the knowledge of the system will enable work extraction that isotherwise impossible, e.g. using Szilard engine 36–38. The free energy of n degenerate 2-level systems is F¼U/C0kBTSln 2¼kBTP ipilnpi,w h e r e Tis the temperature of the heat bath, kBis the Boltzmann ’s constant, U=0i s the internal energy, S¼/C0P ipilog2piis the Shannon entropy and piis the probability for the system being in energy level i. Transforming the system into the thermal state gives a free energy difference ðn/C0SÞkBTln 2, which is t h ea m o u n to fw o r kt h a tc a nb ee x t r a c t e d37. As depicted in Fig. 4, we consider a source that generates pseudo- random spin sequences, where spin up encodes bit 0 and spin down encodesbit 1. Correlations among spins can be exploited to extract work, even if themarginal distributions of each spin are uniformly random 37. For example, if two spins are correlated in such a way that they are pointing

Question 460 multiple-choice

Quantum algorithms for the hidden subgroup problem often leverage properties of group representations and measurement statistics to reconstruct hidden subgroups efficiently. In particular, subgroup conjugacy and distinguishability play crucial roles in determining sample complexity and algorithm feasibility. In the context of reconstructing hidden conjugate subgroups in q-hedral groups using quantum measurements, which property ensures that the order of a non-normal subgroup is sufficient to determine its conjugacy class? 1) Each order corresponds to a unique conjugacy class except for normal subgroups 2) All subgroups of the same order are conjugate regardless of normality 3) The group is abelian, so all subgroups are normal 4) Measurement outcomes are always orthogonal for different subgroups 5) The total variation distance between coset states is zero 6) Every subgroup has multiple conjugacy classes for a given order 7) The representation labels are non-random and fixed for each subgroup

✓ Correct Answer:

The correct answer is 1) Each order corresponds to a unique conjugacy class except for normal subgroups.

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has rank ( p−1)/q, and the total probability that we observe ρis (p−1)/p=1−1/pas before. Then these four partial measurements determine k, remove the eﬀect of the coset, and determine that the row has one of two values, uoru+ 1. Up to an overall phase we can write this as a two-dimensional vector: 1√ 2/parenleftbiggωkaub p ωkau+1b p/parenrightbigg . Our only goal in doing this is to create a one-qubit state where the relative phase between the two basis vectors depends on the conjugate b. Moreover, the relative phase is multiplied by the irrep label k, which is uniformly random. As a result, the typical angle between the states corresponding to any two distinct cosets b, b/primewill be Ω(1), and a simple measurement yields a constant variation distance between them. To make this precise, apply the Hadamard transform 1√ 2/parenleftbigg11 1−1/parenrightbigg and measure according to the resulting basis. The probability that we observe the ﬁrst or second basis vector is then cos2θand sin2θ, respectively, where θ=(kau(a− 1)bπ)/p. Now when we observe a q-dimensional representation ρkofNk, the observed label kis uniformly distributed over Z∗ p/Zq. Moreover, when we perform the POVM in step (iv) above, the uwe observe is uniformly distributed over Zq. It follows that the coeﬃcient m=kau(u−1) is uniformly distributed over Z∗ p. For any two distinct b,b/prime, the total variation distance is then 1 2(p−1)/summationdisplay m∈Z∗p/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos 2πmb p−cos2πmb/prime p/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsinglesin 2πmb p−sin2πmb/prime p/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg . Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 950 MOORE, ROCKMORE, RUSSELL, AND SCHULMAN This we rewrite 1 p−1/summationdisplay m∈Z∗ p/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos 2πmb p−cos2πmb/prime p/vextendsingle/vextendsingle/vextendsingle/vextendsingle =1 2(p−1)/summationdisplay m∈Zp/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos2πmb p−cos2πmb/prime p/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≥1 4(p−1)/summationdisplay m∈Zp/parenleftbigg cos2πmb p−cos2πmb/prime p/parenrightbigg2 =p 4(p−1)>1 4. (Adding the m= 0 term contributes zero to the sum in the second line. In the third line we use the facts that |x|≤x2/2 for all |x|≤2, the average of cos2xis 1/2, and the two cosines have zero inner product.) Since the total variation distance between any two distinct conjugates is bounded below by a constant, we can distinguish between the pdiﬀerent conjugates with only O(logp) = poly( n) samples. Thus, hidden conjugates in Apare information- theoretically reconstructible, completing the proof. By embedding the q-hedral groups in Apas in Theorem 2, we can generalize Theorem 3 to the q-hedral groups ( qnot necessarily prime) as follows. Let pbe prime, letqbe a divisor of p−1, and let q/primebe a divisor of q. The q-hedral group Zq/multicloseleftZ p has a unique normal subgroup K∼=Zq/prime/multicloseleftZ p, in which there is a maximal subgroup Hq/primeof order q/prime. Moreover, all nonnormal subgroups of order q/primeare conjugates of Hq/prime. Theorem 4. Ifp, q, q/prime, and Hq/primeare as above, then the hidden conjugates of Hq/prime inZq/multicloseleftZ pare information-theoretically reconstructible. We now wish to information-theoretically reconstruct all subgroups of the aﬃne andq-hedral groups. We can do this by using the fact that, except for normal sub- groups, there is a unique conjugacy class of each order q/prime, namely the conjugates of Hq/prime. Thus if we can determine the order of H, this determines

Question 461 multiple-choice

Efficient computation of the discrete Fourier transform (DFT) often leverages tensor decompositions and Kronecker products, but their application poses significant challenges for classical computers due to tensor rank growth and redundancy. Quantum computers, by contrast, can exploit these structures for potentially exponential speedups in certain linear algebra problems. Which statement best explains why quantum computers can outperform classical computers in tensor-based DFT algorithms involving Kronecker product structures? 1) Quantum computers avoid increasing tensor rank by using classical parallelism. 2) Classical computers can always reduce tensor rank to one after each operation. 3) Quantum computers rely solely on matrix multiplication for efficient DFT computation. 4) Classical computers can use redundancy in product formulations to minimize computation time. 5) Quantum computers are limited by O(n²) complexity when computing tensor-based DFTs. 6) Quantum computers exploit quantum parallelism and entanglement, allowing efficient operations on tensor products and leveraging cancellation through interference. 7) Classical computers natively support entanglement, making tensor operations faster than on quantum hardware.

✓ Correct Answer:

The correct answer is 6) Quantum computers exploit quantum parallelism and entanglement, allowing efficient operations on tensor products and leveraging cancellation through interference..

📚 Reference Text:

transf orm of a rank-1 tensor even on a classical computer. In this section, we show that this is not the case. However, in the next section, we will discuss a key feature of a quantum computer that allows to take full advantage of the K ronecker product structure of the decomposition given by Theorem 3, and yields an algorithm that only require sO(n2) operations. Recall from Theorem 1 that F′ ncan be decomposed as F′ n= (I2⊗F′ n−1)(In−1⊕Ωn−1)(H⊗In−1), where the factorIn−1⊕Ωn−1can be further decomposed according to (16). The multiplica tion of the rightmost factor of F′ nwith (20) yields (H⊗In−1)xn= (Hx1)⊗x2⊗···⊗xn. Note that this multiplication only requires a constant numb er of operations to produce Hx1, which is independent of the problem size. Furthermore, the result remains a rank- 1 tensor. As long as we keep the result in this rank-1 tensor product form, no additional computation is required . 8 Next, we consider multiplying one factor of In−1⊕Ωn−1derived in Theorem 3 with a rank-1 tensor of the form (20). This amounts to a product of the matrix (18), where k=n−1, with the vector xn, yielding yn=/bracketleftBig E1⊗Ii−1⊗I2⊗In−i−1+E2⊗Ii−1⊗Ri+1⊗In−i−1/bracketrightBig xn = (E1x1)⊗x2⊗···⊗xn+(E2x1)⊗x2⊗···⊗(Ri+1xi+1)⊗···⊗xn. (21) Thus, this multiplication also requires a constant number o f operations. However, the rank of the product is generally increased to 2, unless one of the components of x1is zero. Ifynwere toremain arank-1tensor after successive multiplicat ions ofall factors of In−1⊕Ωn−1, thecomplexity of multiplying the last two factors of F′ nwithxnwould be O(n). Consequently, by applying this estimate recursively to F′ k, fork=n−1,n−2,...,1, the overall complexity of F′ nxcalculation would be O(1+2+···+n) = O(n2), even on a classical computer. Unfortunately, as we can already see from (21), that success ive applications of factors of the form E1⊗Ii−1⊗ I2⊗In−i+E2⊗Ii−1⊗Ri+1⊗In−ifori= 1,2,...,ntend to increase the rank of the tensor. Nonetheless, as we saw in the proof of Theorem 3, the product formulation (16) is a redundant representation where most terms cancel out. If instead we decompose the diagonal matrix as In−1⊕Ωn−1=E1⊗I2⊗···⊗I2+E2⊗R2⊗···⊗Rn, (22) based on (4) and Lemma 4, then we get that the product of the dia gonal matrix with a rank-1 tensor is given by (In−1⊕Ωn−1)xn= (E1x1)⊗x2⊗···⊗xn+(E2x1)⊗(R2x2)⊗···⊗(Rnxn). (23) So we see that in general the rank increases to 2 after multipl ication with the diagonal matrix, just like after multiplication with a single term of the product representa tion (21). The redundant product representation (16) is more expensiv e to evaluate classically than (23), but will allow an eﬃcient implementation on a quantum computer. Although, in a classical algorithm, we can attempt to reduce the rank of each product before the multiplication with the next diagonal matrix is initiated, which is the appr oach taken in [7, 16] for a diﬀerent tensor represen- tation, it is, in general, not clear how much reduction can be achieved. Even if a rank reduction can be achieved, it will not come free. Consequently, the complexity of compu ting the discrete Fourier transform on a classical computer through a sequence of tensor products

Question 462 multiple-choice

In quantum many-body physics, tensor networks are used to efficiently represent complex entangled states, especially when symmetries from groups like SU are present. Group representation theory underpins how these networks are constructed and analyzed. Which procedure ensures that a trivalent vertex in an SU-symmetric tensor network can correspond to an invariant quantum state under group action? 1) Selecting edges labeled only by fundamental representations 2) Decomposing tensor products without considering the trivial representation 3) Choosing group representations for the edges so that their tensor product contains the identity (trivial) representation 4) Assigning the same representation to all edges regardless of their combination 5) Using only the dual representation for every edge in the network 6) Ignoring Clebsch-Gordon coefficients in vertex construction 7) Restricting tensor networks to bivalent (two-legged) vertices only

✓ Correct Answer:

The correct answer is 3) Choosing group representations for the edges so that their tensor product contains the identity (trivial) representation.

📚 Reference Text:

×SU(N)×SU(N). Example 6: states from representation theory tensor networ ks For general n, we will now argue that LMEstates obtained using our representation theory construction can always be represented by tensor networks with trivalent vertices, where the edges at a vertex are labelled by representations of our group who se tensor product contains the identity and tensors at each vertex correspond to the Clebsc h-Gordon type coeﬃcients describing how the trivial representation is constructed from the tensor product. To see this, we note that for a given set of representations R1,···,Rnassociated to the elementarysubsystems, wecandeterminethetensorproductof representationsrecursivelyby ﬁrst decomposing the tensor product of any pair of representat ions into a sum of irreducible representations, and then repeating this procedure (now with n−1 total representations) for each element in the sum. For each possible representation in the full tensor product, we can associate a tree graph with trivalent vertices, with the graph stru cture showing our choice for how to pair up representation and the edges labeled by the repr esentations obtained in the intermediate steps. For our application, the ﬁnal representa tion should be the trivial representation. To construct the corresponding LME state exp licitly, we can interpret the graph as a tensor network, with vertices corresponding to Clebsc h-Gordon coeﬃcients that tell us how the representation for each outgoing edge is obtained f rom the tensor product of the two representations associated with incoming edges at that vertex. This is shown in ﬁgure 3a. We can represent the state using an equivalent tensor network wh ere all the trivalent vertices 17 R R R R RRRR1 1 2 3 4 5678 R R R R RRRR1 1 2 3 4 5678 R6dR7R8 dda) b) Figure 3: Tensor networks describing LME states constructed fr om irreducible representa- tions of a group. On the left, trivalent vertices correspond to Cleb sch-Gordon type tensors describing how the representation corresponding to the outgoing leg arises from the tensor product of representations corresponding to the incoming legs. O n the right, the trivalent vertices correspond to tensors constructing the trivial repres entation from the product of the three incoming representations. Small bivalent vertices represen t tensors giving the trivial representation from the product of a representation and its dua l. have incoming legs (i.e. a PEPS network) making use of a simple represe ntation theory fact: for any representations R1andR2whose tensor product contains R3, the tensor product ofR1,R2, and the dual representation R∗ 3contains the trivial representation. Thus, we can replace the Clebsch-Gordon coeﬃcient for a vertex with two inc oming and one outgoing leg with the related tensor constructing the trivial representatio n from the R1,R2,R∗ 3and a tensor constructing the trivial representation from R3andR∗ 3. For the case of SU(2), this means that we are using 3 j-symbols rather than Clebsch-Gordon coeﬃcients. This is indicated in ﬁgure 3b. For a given set of representations ( R1,···,Rn), we will often have a number of copies of the trivial representation in the tensor product. These

Question 463 multiple-choice

Modern theoretical physics explores extensions to gauge symmetry that may explain dark matter and dark energy through unconventional field and charge models. Some proposals involve imaginary charges and novel particles that interact via generalized electromagnetic fields. Which concept involves particles with imaginary charge that generate large-scale electromagnetic fields with negative energy density and are proposed as dark matter candidates? 1) Allotons 2) Axions 3) Sterile neutrinos 4) WIMPs (Weakly Interacting Massive Particles) 5) Primordial black holes 6) Chameleons 7) MACHOs (Massive Compact Halo Objects)

✓ Correct Answer:

The correct answer is 1) Allotons.

📚 Reference Text:

Title: Generalization of Abelian Gauge Symmetry and the Dark Matter and Energy Problem Year: 2010 Paper ID: e898cd458c4b6cf26f6de7d44a7501c91af0f565 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/e898cd458c4b6cf26f6de7d44a7501c91af0f565 Abstract: A commutative generalization of the gauge symmetry group is proposed. The two-parametric family of two-connected abelian Lie groups is obtained. The necessity of existence of so-called imaginary charges and electromagnetic fields with negative energy density (dark photons) is derived. The possibilities when the overall Lagrangian represents a sum or difference of two identical Lagrangians for the visible and hidden sectors (i.e. copies of unbroken) are ruled out by the extended symmetry. The distinction between the two types of fields resides in the fact that for one of them current and electromagnetic kinetic terms in Lagrangians are identical in sign, whereas for another type these terms are opposite in sign. As a consequence, and in contrast to the common case, like imaginary charges attract and unlike charges repel. Some cosmological issues of the proposed hypothesis are discussed. Particles carrying imaginary charges ("allotons") are proposed as dark matter candidates. Such a matter would be imaginary charged on a large scale for the reason that dark atoms would carry non-compensated charges. Consequently, there exist (dark) electromagnetic fields with negative energy density on cosmological scales (the reviving of the idea of Faradayan cosmology). This leads to the hypothesis that the modern state of the Universe is radiation-dominated by dark photons with negative energy density that is the source of the observed late-time cosmological acceleration. This provides an explanation for the small value of the cosmological constant as a renormalized vacuum energy.

Question 464 multiple-choice

Blind Quantum Computation (BQC) is a cryptographic protocol in quantum computing that enables clients with limited quantum capabilities to securely outsource quantum computations to powerful servers. Achieving privacy in delegated quantum algorithms, such as the Quantum Fourier Transform (QFT) on entangled states, is central to the development of secure quantum cloud services. Which protocol design specifically increases privacy assurance by performing the Quantum Fourier Transform (QFT) across two Bell pairs involving qubits 1 and 3? 1) Delegated QFT protocol on a single qubit using classical encryption 2) Blind QFT protocol for qubits 2 and 4 using separable states 3) Primary BQC protocol for QFT on qubits 1 and 2 with one Bell state 4) Enhanced BQC protocol for QFT on qubits 1 and 3 involving two Bell states 5) Generalized BQC protocol for arbitrary combinations of qubits and Bell states 6) QFT protocol using non-entangled quantum states for outsourcing 7) Basic BQC protocol employing QFT only on qubit 1 without entanglement

✓ Correct Answer:

The correct answer is 4) Enhanced BQC protocol for QFT on qubits 1 and 3 involving two Bell states.

📚 Reference Text:

Title: Blind quantum computation for quantum Fourier transform on Bell states Year: 2017 Paper ID: 0c182785510cabacb6bd16888ef4c2cd8a0361ef Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0c182785510cabacb6bd16888ef4c2cd8a0361ef Abstract: It is called blind quantum computation(BQC) that a client who has limited quantum technologies can delegate her quantum computing to a server who has fully-advanced quantum computers. But the privacy of the client's quantum inputs, algorithms and outputs is still a challenge. To realize a secure BQC, we mainly study how to hide quantum fourier transform (QFT) performed on Bell states. In this paper, three cases are considered as follows. For the first case, we design primary BQC protocols of QFT performed on qubits 12 of belonging to $\{|\phi^\pm\rangle_{12},$ $|\psi^\pm\rangle_{12}\}$ with relevant circuits. To strengthen security, we construct enhanced BQC protocols of QFT performed on qubits 13 of any two Bell states $|\xi\rangle_{12}\otimes|\theta\rangle_{34}$ with relevant quantum circuits. Featured the property of stronger security, we give generalized BQC protocols of QFT performed on qubits 13 and 24 of any two Bell states with relevant quantum circuits respectively. At last, we analyze and prove the blindness and correctness.

Question 465 multiple-choice

In the study of Hopf algebras and quantum groups, the Drinfeld double D is a central construction with important implications for representation theory and categorical invariants. Understanding the properties of integrals, cointegrals, and the antipode is crucial for exploring its algebraic and categorical structure. Which of the following statements correctly describes the modulus of the Drinfeld double D for a finite-dimensional Hopf algebra A? 1) The modulus of D is always equal to 1, regardless of the choice of A. 2) The modulus of D is given by the square of the modulus of A. 3) The modulus of D is equal to the inverse of the modulus of A, that is, a_D = g⁻¹. 4) The modulus of D coincides with the modular function of the dual algebra A*. 5) The modulus of D is determined by the dimension of A and not by its group-like elements. 6) The modulus of D vanishes unless A is semisimple. 7) The modulus of D is equal to the modulus of A for any Hopf algebra A.

✓ Correct Answer:

The correct answer is 3) The modulus of D is equal to the inverse of the modulus of A, that is, a_D = g⁻¹..

📚 Reference Text:

.−1:D(A)/tildewidest−→A⊗A∗=End(A) is the projector onto the space of left cointegrals and is giv en by: AR⊗1−−→A⊗Ao⊗A1⊗τ−−→A⊗A⊗Ao·⊗1−−→A⊗Ao∆⊗1−−→ A⊗A⊗Ao 1⊗τ−−→A⊗Ao⊗A1⊗S2−−→A⊗Ao⊗A1⊗<,>−−−→A Proof :SinceS(p) =x.S(µ) andS−1(x) andS(µ) are right integrals it is clear that 1.) implies that pis an invariant with respect to right and left multiplication of AandAo. This shows 2.). Assuming that 1.)is true we show 3.): ⊲⊳−1(µ⊗S−1(x)) =/angbracketleftBig S(µ′),S−1(x)′/angbracketrightBig/angbracketleftBig µ′′,S−1(x)′′′/angbracketrightBig S−1(x)′′⊗µ′′(2.39) by(2.28)=ω−1/angbracketleftBig S(µ′),S(x′′′)/angbracketrightBig/angbracketleftBig µ′′′,S(x′)/angbracketrightBig S(x′′)⊗µ′′ =ω−1/angbracketleftBig µ′,x′′/angbracketrightBig/angbracketleftBig µ′′′,S(x′)′′/angbracketrightBig S(x′)′⊗µ′′ =ω−1/summationdisplay ij/angbracketleftBig µ′,x′′/angbracketrightBig/angbracketleftBig µ′′′,e′′ j/angbracketrightBig/angbracketleftBig µ′′,ei/angbracketrightBig/angbracketleftBig fj,S(x′)/angbracketrightBig e′ j⊗fi =ω−1/summationdisplay ij/angbracketleftBig µ′′,e′′ jei/angbracketrightBig/angbracketleftBig fj,Sβl(µ′)/angbracketrightBig e′ j⊗fi =ω−1/summationdisplay ij/angbracketleftBig fj,Sβlβr(e′′ jei)/angbracketrightBig e′ j⊗fi by(2.30)=ω−1/summationdisplay ij/angbracketleftBig fj,S2(e′′ jei)/angbracketrightBig e′ j⊗fiby(1.))=x⊗S(m) (2.40) 15 If we apply 1⊗S2to both sides of the last equation and use (2.28) we ﬁnd P=/summationdisplay ij/angbracketleftBig fj,S2(e′′ j)ei/angbracketrightBig e′ j⊗fi=/summationdisplay j/angbracketleftBig f′′ j,S2(e′′ j)/angbracketrightBig e′ j⊗f′ jby(2.12)=/summationdisplay ij/angbracketleftBig fj,S2(e′′ je′′ i)/angbracketrightBig e′ je′ i⊗fi, which is precisely the equation given in 3.). The same formula has been proven in [Dr1] using the theory of Hopf modules directly. From (2.28) we ﬁnd S(µ)(x) =µ(ax) = 1 so thatPas in (2.40) is a rank one projection. It remains to show the ﬁrst par t of the proposition. For this purpose we need two identities for the integra ls, namely S2/parenleftBig (S(µ)′′⊳a−1)α/parenrightBig ⊗S(µ)′=S2/parenleftBig (S(µ)⊳a−1)′′α/parenrightBig ⊗(S(µ)⊳a−1)′ by(2.28)=S2(µ′′α)⊗µ′ by(2.35)=µ′⊗µ′′= ∆(µ) (2.41) and /parenleftBig (S2(x′′)a−1)↽ α/parenrightBig ⊗x′by(2.37)= ((x′)↽ α)⊗x′′ = (x ↽ α)′⊗(x ↽ α)′′ by(2.28)=S−1(x)′⊗S−1(x)′′= ∆(S(x)−1) (2.42) Inserting (2.41) and (2.42) into the expression for the bicrossed p roduct of pin (2.39) we ﬁnd: ⊲⊳−1(µ⊗S−1(x)) =/angbracketleftBig S(µ′),S−1(x)′/angbracketrightBig /angbracketleftBig µ′′′,S−1(x)′′′/angbracketrightBig S−1(x)′′⊗µ′′ =/angbracketleftBig S3/parenleftBig (S(µ))′′′⊳a−1)α/parenrightBig , S2((x′′a−1)↽ α)/angbracketrightBig /angbracketleftBig S(µ)′′,x′′/angbracketrightBig x′⊗S(µ)′ =/angbracketleftBig α−1/parenleftBig a⊲S(S(µ)′′′)/parenrightBig ,(x′′′a−1)↽ α/angbracketrightBig /angbracketleftBig S(µ)′′,x′′/angbracketrightBig x′⊗S(µ)′ =/angbracketleftBig S(S(µ)′′′), x′′′/angbracketrightBig /angbracketleftBig S(µ)′′,x′′/angbracketrightBig x′⊗S(µ)′ =/angbracketleftBig S(µ)′′S(S(µ)′′′), x′′/angbracketrightBig x′⊗S(µ)′=x⊗S(µ).(2.43) Henceµ.S−1(x) =S(µ.S−1(x)) =x.S(µ) and we have shown 1.)of the proposition. ✷ Thanks to the simple comultiplicative structure of D(A) it is much easier to ﬁnd the integral. Since S−1(µ) is a right integral of Aand since we have opposite comultiplication onAoarightintegral of D(A) is given by µD(λ.y) :=λ(x)µ(S(y)) for all λ∈Ao,andy∈A. (2.44) Its properties are described next: Proposition 6 Letxandµbe the left integrals of Aas in (2.22). Then 16 1.µDis a right integral of D(A)withµD(p) = 1. 2. The modulus (as deﬁned in 2.26) of D(A)isaD=g−1. 3.µD∈C(D(A)), in particular µD(λ.y) =ωµD(y.λ)∀y∈A, λ∈Ao. Proof :Part1.)is clear. Also, it follows directly from the deﬁnitions of the moduli of Athat 1⊗µD∆(h) =α.a−1µD(h).2.)follows if we apply the antipode and use that S(µD) is a left integral of D(A). In order to show 3.)we observe that the equation for right integrals analogous to (2.35) is µD(S2(k)h) =µD(h(k ↽ α−1 D)). Together with Proposition 5 2.) this shows µD∈C(D(A)). In the case where kandhare in the special subalgebras we use (2.28) to show µD(S2(λ)y) =ω−1µD(λy) which yields the last equation in part 3.). ✷ The fact that the right integral of a double algebra is invariant unde r the coadjoint ac- tionallowsustoidentifyasanobject intherepresentation categor yofD(A), namelywith the integral of the“braided algebra” [Ly] of the category. Befor e we explain this aspect in more detail in the next section let us discuss a few more consequen

Question 466 multiple-choice

Distributed quantum computing leverages multiple quantum processors to solve computational problems more efficiently, particularly in scenarios where the structure of the problem allows for quantum speedup. Simon's problem and its generalizations are key benchmarks for demonstrating the advantages of quantum algorithms in both centralized and distributed settings. Which property distinguishes an exact distributed quantum algorithm for generalized Simon’s problem from probabilistic or approximate variants, while also enabling exponential speedup and reducing active qubits per oracle for NISQ devices? 1) Utilization of error-correcting codes to mitigate decoherence in all circuit layers 2) Guarantee of the correct solution with certainty via quantum amplitude amplification 3) Implementation of randomized classical post-processing to filter measurement outcomes 4) Introduction of adaptive measurement schedules to minimize overall gate count 5) Application of variational optimization for oracle parameter selection 6) Use of entanglement swapping to distribute quantum information between nodes 7) Encoding solution candidates with non-orthogonal quantum states for compression

✓ Correct Answer:

The correct answer is 2) Guarantee of the correct solution with certainty via quantum amplitude amplification.

📚 Reference Text:

for Simon’s problem [ 12]. Not only can Algorithm 2and Algorithm 5solve Simon’s problem, but they can also handle the generalized Simon’s problem. Notably, Algorithm 5is exact. However, the algorithm [12] cannot solve the generalized Simon’s problem and lacks exactness. As a result, Algorithm 2and Algorithm 5have better generalisability. Algorithm 5not only has better generalisability, but also has the advantage of being ex act. 15 Table 4 Comparison of Algorithm 2and Algorithm 5with algorithm in [ 12]. Algorithms Precision Generalisability Algorithm 2 InaccuracyThe generalized Simon’s problem Algorithm 5 ExactThe generalized Simon’s problem The algorithm in [ 12]Inaccuracy Simon’s problem 8 Conclusion In this paper, we have characterized the structure of the generaliz ed Simon’s problem in distributed scenario. Based on the structure, we have designed a corresponding distributed quantum algorithm. Then we have further utilized qu antum amplitude ampliﬁcationtechniquetomakeouralgorithmexact.Thenumberofactualf unctioning qubits for each oracle in our algorithm is reduced, which reduces the c ircuit depth and helps reduce circuit noise, making our algorithm easier to be impl emented in the current NISQ era. Our algorithm oﬀers exponential acceleration compared to the distribut ed classical algorithm. When compared to the centralized quantum algorithm for the gen eral- ized Simon’s problem, our algorithm has the advantage of a simpler impleme ntation for the oracle, which is crucial for implementing quantum query algori thms. Addi- tionally, compared to the best previously proposed distributed quan tum algorithm for Simon’s problem, our exact distributed quantum algorithm designed for the generalized Simon’s problem provides better generalizability and e xactness. We have discovered that understanding the fundamental structur e of a problem is crucial for designing an exact distributed quantum algorithm that accur ately solves it. In future research, our algorithm can guide the development of exact distributed quantum algorithms to tackle the hidden subgroup problem. Moreover, t he ideas and methods employed in our algorithm’s design hold the potential for desi gning exact distributed quantum algorithms for solving various other problems. Acknowledgments. ThisworkissupportedinpartbytheNationalNaturalScience Foundation of China (No. 61876195) and Funda¸ c˜ ao para a Ciˆ encia e Tecnologia, Insti - tuto de Telecomunica¸ c˜ oes Research Unit, ref. UIDB/50008/2020, UIDP/50008/2020. Appendix A Proof of Proposition 1 Proposition 1 Letφ= 2arctan/parenleftBig/radicalBig 2n−t−r−kl 3·2n−t−r−kl−4/parenrightBig ,ϕ= arccos/parenleftBig 2n−t−r−kl−1−1 2n−t−r−kl−1/parenrightBig , where r=|Y|−1. Then Q/vextendsingle/vextendsingle/vextendsingleS⊥ l,02tm,S(T)/angbracketrightBig =|ΨX⟩. ProofFrom Eq. ( 24), we can write R0(φ) as follows. R0(φ) =I⊗n−t+2t+1m−/parenleftBig 1−eiφ/parenrightBig/vextendsingle/vextendsingle/vextendsingle0n−t,02t+1m/angbracketrightBig/angbracketleftBig 0n−t,02t+1m/vextendsingle/vextendsingle/vextendsingle. (A1) 16 From the deﬁnitions of RA(ϕ,Y),|ΨX⟩and|ΨY⟩, we have/parenleftBig RA(ϕ,Y)⊗I⊗2t+1m/parenrightBig |ΨX⟩=eiϕ|ΨX⟩. (A2) /parenleftBig RA(ϕ,Y)⊗I⊗2t+1m/parenrightBig |ΨY⟩=|ΨY⟩. (A3) LetU(A,φ) =−AR0(φ)A†, then based on Eq. ( 26),Qcan be written as Q=U(A,φ)/parenleftBig RA(ϕ,Y)⊗I⊗2t+1m/parenrightBig . (A4) ForU(A,φ), we have U(A,φ) =−AR0(φ)A† =−A/parenleftBig I⊗n−t+2t+1m−/parenleftBig 1−eiφ/parenrightBig/vextendsingle/vextendsingle/vextendsingle0n−t,02t+1m/angbracketrightBig/angbracketleftBig 0n−t,02t+1m/vextendsingle/vextendsingle/vextendsingle/parenrightBig A† =/parenleftBig 1−eiφ/parenrightBig/parenleftBig A/vextendsingle/vextendsingle/vextendsingle0n−t,02t+1m/angbracketrightBig/angbracketleftBig 0n−t,02t+1m/vextendsingle/vextendsingle/vextendsingleA†/parenrightBig −I⊗n−t+2t+1m =/parenleftBig 1−eiφ/parenrightBig/vextendsingle/vextendsingle/vextendsingleK⊥,02tm,S(T)/angbracketrightBig/angbracketleftBig K⊥,02tm,S(T)/vextendsingle/vextendsingle/vextendsingle−I⊗n−t+2t+1m =/parenleftBig 1−eiφ/parenrightBig/parenleftbig |ΨX⟩+|ΨY⟩/parenrightbig/parenleftbig ⟨ΨX|+⟨ΨY|/parenrightbig −I⊗n−t+2t+1m.(A5) Since|⟨Y⟩|= 2rand|S⊥ l|= 2n−t−kl, according to the deﬁnition of |ΨX⟩and|ΨY⟩, we have ⟨ΨX|ΨX⟩= 1−2r+kl+t−n. ⟨ΨY|ΨY⟩= 2r+kl+t−n. ⟨ΨX|ΨY⟩= 0.(A6) Thus, we can obtain the following equations. Q|ΨX⟩=U(A,φ)/parenleftBig RA(ϕ,Y)⊗I⊗2t+1m/parenrightBig |ΨX⟩ =eiϕU(A,φ)|ΨX⟩ =eiϕ/parenleftBig/parenleftBig 1−eiφ/parenrightBig/parenleftbig |ΨX⟩+|ΨY⟩/parenrightbig/parenleftbig ⟨ΨX|+⟨ΨY|/parenrightbig −I⊗n−t+2t+1m/parenrightBig |ΨX⟩ =eiϕ/parenleftBig 1−eiφ/parenrightBig/parenleftbig |ΨX⟩+|ΨY⟩/parenrightbig/parenleftbig ⟨ΨX|+⟨ΨY|/parenrightbig |ΨX⟩−eiϕ|ΨX⟩ =eiϕ/parenleftBig 1−eiφ/parenrightBig ⟨ΨX|ΨX⟩/parenleftbig |ΨX⟩+|ΨY⟩/parenrightbig −eiϕ|ΨX⟩ =eiϕ/parenleftBig 1−eiφ/parenrightBig/parenleftBig 1−2r+kl+t−n/parenrightBig/parenleftbig |ΨX⟩+|ΨY⟩/parenrightbig −eiϕ|ΨX⟩ =eiϕ/parenleftBig (1−eiφ)(1−2r+kl+t−n)−1/parenrightBig |ΨX⟩+eiϕ(1−eiφ)(1−2r+kl+t−n)|ΨY⟩. (A7) Q|ΨY⟩=U(A,φ)/parenleftBig

Question 467 multiple-choice

In quantum computing, the Hadamard gate plays a central role in transforming qubit states and enabling quantum algorithms to exploit superposition and interference. Understanding how quantum oracles and basis changes interact is key to appreciating early quantum algorithmic speedups. Which of the following statements correctly describes how the Hadamard gate is utilized in solving Deutsch’s XOR problem with a quantum oracle? 1) The Hadamard gate is applied after measurement to amplify the probability of the correct answer. 2) The Hadamard gate is used to convert the dual basis back into the standard basis before applying the quantum oracle. 3) The Hadamard gate prepares an entangled state that distinguishes the function’s output without further measurement. 4) The Hadamard gate creates a superposition input, enabling the quantum oracle to encode global properties of the function via interference. 5) The Hadamard gate is used exclusively to measure the parity of the function outputs after two oracle calls. 6) The Hadamard gate acts as the quantum oracle, computing f: Bⁿ → B on the input qubit. 7) The Hadamard gate applies a phase flip only when the function output is balanced.

✓ Correct Answer:

The correct answer is 4) The Hadamard gate creates a superposition input, enabling the quantum oracle to encode global properties of the function via interference..

📚 Reference Text:

The dual basis of Bdenoted by {|0′/an}bracketri}ht,|1′/an}bracketri}ht}is deﬁned by |0′/an}bracketri}ht=1√ 2(|0/an}bracketri}ht+|1/an}bracketri}ht) |1′/an}bracketri}ht=1√ 2(|0/an}bracketri}ht − |1/an}bracketri}ht) (1) Hwill denote the fundamental unitary matrix H=1√ 2/parenleftBigg 1 1 1−1/parenrightBigg (2) ThusH2=IandHinterchanges the standard and dual bases. In terms of real geometry the dual basis lies on the 45◦lines between the orthogonal directions |0/an}bracketri}ht and|1/an}bracketri}htandHis the transformation given by reﬂection in a line at angle π/8 to the |0/an}bracketri}htdirection. Thus the eigenvectors of H(parallel and perpendicular to the mirror line) are cosπ 8|0/an}bracketri}ht ±sinπ 8|1/an}bracketri}htbelonging to λ=±1 respectively. We will see later that His also the Fourier transform on the group B. The elements of Bnarenbit strings. If x= (x1,...,x n) andy= (y1,...,y n) are inBnthen we write x⊕y= (x1⊕y1,...,x n⊕yn)∈Bn x·y= (x1y1⊕ · · · ⊕xnyn)∈B (the operations on the RHS’s being addition and multiplicat ion mod 2 in B.) Note thatx·yis the parity of the number of places where xandyboth have a bit value of 1. Early Days The earliest quantum algorithms [1, 2] were concerned with a situation in which we are given a “black box” or oracle that computes a function f:Bn→Band we are required to decide whether a certain “global” property ( i.e. a joint property of 2 all the function values) holds of f. For quantum computation the black box is given as a unitary transformation Ufonn+ 1 qubits given in the standard basis by Uf:|x1/an}bracketri}ht |x2/an}bracketri}ht...|xn/an}bracketri}ht/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright input|y/an}bracketri}ht −→ |x1/an}bracketri}ht|x2/an}bracketri}ht...|xn/an}bracketri}ht|y⊕f(x1,...,x n)/an}bracketri}ht (3) (We will often abbreviate |x1/an}bracketri}ht|x2/an}bracketri}ht...|xn/an}bracketri}htas|x/an}bracketri}htforx∈Bn.) Thus ifyis initially set to 0 the value of fmay be read from the last qubit. For our ﬁrst problem, referred to as Deutsch’s XOR problem [1 ], we haven= 1 so thatfis one of the four possible functions f:B→B. We are to decide whether f(0)⊕f(1) is 0 or 1. Equivalently we wish to decide whether fis a constant function or a “balanced” function (where balanced means that ftakes one value 0 and one value 1). Clearly any classical computer requires evaluati ngftwice to decide this. According to Deutsch’s original method [1], the problem may be solved on a quantum computer after running Ufonlyoncebut the algorithm succeeds only with probability 1 2(and we know when it has been successful). The method is simpl y to run Ufon the input superposition1√ 2(|0/an}bracketri}ht+|1/an}bracketri}ht) yielding the state1√ 2(|0/an}bracketri}ht |f(0)/an}bracketri}ht+|1/an}bracketri}ht|f(1)/an}bracketri}ht). Writing this state in the dual basis we have the four possibilities gi ven by the two constant functions: 1√ 2(|0/an}bracketri}ht|f(0)/an}bracketri}ht+|1/an}bracketri}ht|f(1)/an}bracketri}ht) =/braceleftBigg1√ 2(|0/an}bracketri}ht|0/an}bracketri}ht+|1/an}bracketri}ht|0/an}bracketri}ht) =1√ 2(|0′/an}bracketri}ht|0′/an}bracketri}ht+|0′/an}bracketri}ht|1′/an}bracketri}ht) 1√ 2(|0/an}bracketri}ht|1/an}bracketri}ht+|1/an}bracketri}ht|1/an}bracketri}ht) =1√ 2(|0′/an}bracketri}ht|0′/an}bracketri}ht − |0′/an}bracketri}ht |1′/an}bracketri}ht) and the two balanced functions: 1√ 2(|0/an}bracketri}ht|f(0)/an}bracketri}ht+|1/an}bracketri}ht|f(1)/an}bracketri}ht) =/braceleftBigg1√ 2(|0/an}bracketri}ht|0/an}bracketri}ht+|1/an}bracketri}ht|1/an}bracketri}ht) =1√ 2(|0′/an}bracketri}ht|0′/an}bracketri}ht+|1′/an}bracketri}ht|1′/an}bracketri}ht) 1√ 2(|0/an}bracketri}ht|1/an}bracketri}ht+|1/an}bracketri}ht|0/an}bracketri}ht) =1√ 2(|0′/an}bracketri}ht|0′/an}bracketri}ht − |1′/an}bracketri}ht |1′/an}bracketri}ht) Now measure the second qubit in the dual basis. If the result i s 0′(which occurs with probability1 2in every case) then we have lost all the information about the function f. If the result is 1′then measurement of the ﬁrst qubit will reliably distinguis h between constant and balanced functions. In our second algorithm [2], referred to as Deutsch’s algori thm, we are given nand a functionf:Bn→B. It is promised that fis either

Question 468 multiple-choice

Quantum algorithms offer significant speedups for certain computational problems, such as finding collisions in functions. The r-collision problem seeks to identify r distinct inputs that map to the same output, with quantum query complexity being a central measure of algorithmic efficiency. What is the quantum lower bound for the number of queries required to solve the r-collision problem for a function with domain size d, and which algorithmic technique contributes to achieving a nearly optimal query complexity? 1) Ω(√d/r), with Grover's algorithm facilitating the nearly optimal query complexity 2) Ω(log d/r), with the quantum Fourier transform enabling optimal complexity 3) Ω(d/r), with the Hadamard test providing the speedup 4) Ω(√d log r), using amplitude amplification for optimality 5) Ω(1), with the swap test as the main technique 6) Ω(d^2/r^2), with phase estimation as the key subroutine 7) Ω(√r/d), with quantum walks supplying the efficiency

✓ Correct Answer:

The correct answer is 1) Ω(√d/r), with Grover's algorithm facilitating the nearly optimal query complexity.

📚 Reference Text:

transformation from the standard basis toB. We also assume we can perform its inverse R†. By considering the case where the basis B(or equivalently U) is known, it is clear that the quantum lower bound for the usual collision problem [1] implies an Ω(3/radicalbig d/r) lower bound on the number of queries for the quantum collision problem as well. We pre sent an algorithm for this problem that requires only O(3/radicalbig d/rlogd/r) queries, nearly matching the lower bound. The basic tool used in this algorithm is the following ampliﬁed version of t he swap test: Lemma 10 (Ampliﬁed swap test with small disturbance). Suppose we are given a quan- tum stateρsatisfying 1 2/bardblρ−|ψ/an}bracketri}ht/an}bracketle{tψ|/bardbl1≤ǫ (78) where |ψ/an}bracketri}ht=/summationdisplay iai|αi/an}bracketri}ht⊗m|βi/an}bracketri}ht⊗m|γi/an}bracketri}ht (79) with/an}bracketle{tγi|γj/an}bracketri}ht=δi,jand/summationtext i|ai|2= 1. We further assume that for each i,θi:=/an}bracketle{tαi|βi/an}bracketri}htis equal to either 0 or 1. Then there exists a unitary operation U, implementable with O(m)gates, such that1 2/vextenddouble/vextenddoubleUρU†−|ψ′/an}bracketri}ht/an}bracketle{tψ′|/vextenddouble/vextenddouble 1≤ǫ+21−m/2(80) where |ψ′/an}bracketri}ht:=/summationdisplay iai(−1)θi|αi/an}bracketri}ht⊗m|βi/an}bracketri}ht⊗m|γi/an}bracketri}ht. (81) Proof.The procedure for implementing Uis as follows. First perform the m-fold swap test without the ﬁnal measurements. Conditioned on any swap test sho wing that the states are orthogonal, record this answer in an ancilla qubit; otherwise, recor d that they are the same. Apply a phase of −1 in the “same” case. Finally, uncompute the ancilla qubit along with all the swap tests. To analyze U, we initially assume we are acting on |ψ/an}bracketri}ht/an}bracketle{tψ|. Let Π denote the projector that results from each swap test returning the symmetric outcome, so thatU= 1−2Π. Since the swap test will always project |αi/an}bracketri}ht|αi/an}bracketri}htonto the symmetric space, we have Π|αi/an}bracketri}ht⊗m|βi/an}bracketri}ht⊗m|γi/an}bracketri}ht=|αi/an}bracketri}ht⊗m|βi/an}bracketri}ht⊗m|γi/an}bracketri}ht (82) wheneverθi= 1. On the other hand, if θi= 0 then Π|αi/an}bracketri}ht⊗m|βi/an}bracketri}ht⊗m|γi/an}bracketri}ht=1√ 2m|ηi/an}bracketri}ht⊗m|γi/an}bracketri}ht (83) where|ηi/an}bracketri}ht:=1√ 2(|αi/an}bracketri}ht|βi/an}bracketri}ht+|βi/an}bracketri}ht|αi/an}bracketri}ht). Combining these, we obtain U|ψ/an}bracketri}ht=/summationdisplay iai/parenleftbig (−1)θi|αi/an}bracketri}ht⊗m|βi/an}bracketri}ht⊗m−δθi,021−m 2|ηi/an}bracketri}ht⊗m/parenrightbig |γi/an}bracketri}ht (84) =|ψ′/an}bracketri}ht−/summationdisplay iaiδθi,021−m 2|ηi/an}bracketri}ht⊗m|γi/an}bracketri}ht. (85) 15 Now the ﬁdelity between U|ψ/an}bracketri}htand|ψ′/an}bracketri}htis F/parenleftbig |ψ′/an}bracketri}ht/an}bracketle{tψ′|,U|ψ/an}bracketri}ht/an}bracketle{tψ|U†/parenrightbig =|/an}bracketle{tψ′|U|ψ/an}bracketri}ht|2=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−/summationdisplay iaiδθi,021−m 2/an}bracketle{tψ′|ηi/an}bracketri}ht⊗m|γi/an}bracketri}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (86) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−/summationdisplay i|ai|2δθi,021−m/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ≥/vextendsingle/vextendsingle1−21−m/vextendsingle/vextendsingle2≥1−22−m.(87) Finally, 1 2/vextenddouble/vextenddoubleUρU†−|ψ′/an}bracketri}ht/an}bracketle{tψ′|/vextenddouble/vextenddouble 1≤1 2/vextenddouble/vextenddoubleUρU†−U|ψ/an}bracketri}ht/an}bracketle{tψ|U†/vextenddouble/vextenddouble 1+1 2/vextenddouble/vextenddoubleU|ψ/an}bracketri}ht/an}bracketle{tψ|U†−|ψ′/an}bracketri}ht/an}bracketle{tψ′|/vextenddouble/vextenddouble 1(88) ≤1 2/bardblρ−|ψ/an}bracketri}ht/an}bracketle{tψ|/bardbl1+21−m 2≤ǫ+21−m 2, (89) where (88) uses the triangle inequality and (89) uses ﬁrst the unita ry invariance of the trace norm and then (87) and (30). We will use this ampliﬁed swap test to prove Theorem 11. The quantum r-collision problem has query complexity O(3/radicalbig d/rlogd/r). Proof.We give an explicit algorithm based on the quantum algorithm for the us ual collision problem [11]. We ﬁrst outline that original algorithm and then show how to extend it to the quantum problem. The algorithm builds a table of the ﬁrst3/radicalbig d/ritems and uses Grover’s algorithm to search the remaining items for an entry of the table. Set K:={1,...,3/radicalbig d/r}and¯K:={1,...,d}−K. Then the algorithm consists of two steps: 1. Build the table L:={(i,f(i)) :i∈K}. 2. Use Grover search to ﬁnd j∈¯Ksuch thatF(j) = 1, where the function F:¯K→ {0,1} is deﬁned by F(j) = 1 if there is i∈Kwithf(i) =f(j) andF(j) = 0 otherwise. The ﬁrst step uses O(3/radicalbig d/r) queries. In the second step, the function Fmust be computed reversibly, so two queries ( RandR†) are used for each call to F. Note that it is important that Fcan

Question 469 multiple-choice

In heterotic M-theory, constructing hidden sector bundles is essential for exploring new physics beyond the Standard Model, often leveraging advanced geometric techniques such as bundle extensions on Calabi-Yau manifolds. Slope-stability and supersymmetry preservation are critical for ensuring phenomenological viability in these constructions. When two line bundles are embedded into the SU(3) factor of the E6×SU(3) subgroup within the hidden sector E8 gauge group on the Schoen threefold and extended to a non-Abelian SU(3) bundle, what is the number of inequivalent extension branches that generalizes the solution space compared to using a single line bundle? 1) Three 2) Four 3) Six 4) One 5) Five 6) Seven 7) Eight

✓ Correct Answer:

The correct answer is 3) Six.

📚 Reference Text:

Title: Hidden Sectors from Multiple Line Bundles for the B−L$B-L$ MSSM Year: 2021 Paper ID: 7890f22542f9cc607af1b38532387d1d135f8c71 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/7890f22542f9cc607af1b38532387d1d135f8c71 Abstract: We give a formalism for constructing hidden sector bundles as extensions of sums of line bundles in heterotic M‐theory. Although this construction is generic, we present it within the context of the specific Schoen threefold that leads to the physically realistic B−L$B-L$ MSSM model. We discuss the embedding of the line bundles, the existence of the extension bundle, and a number of necessary conditions for the resulting bundle to be slope‐stable and thus N=1$N=1$ supersymmetric. An explicit example is presented, where two line bundles are embedded into the SU(3)$SU(3)$ factor of the E6×SU(3)$E_{6} \times SU(3)$ maximal subgroup of the hidden sector E8 gauge group, and then enhanced to a non‐Abelian SU(3)$SU(3)$ bundle by extension. For this example, there are in fact six inequivalent extension branches, significantly generalizing that space of solutions compared with hidden sectors constructed from a single line bundle.

Question 470 multiple-choice

Solving equations in modular arithmetic is crucial in number theory and cryptography, especially when working with composite moduli and elliptic curves. Powerful results like the Chinese Remainder Theorem and specialized case analysis enable efficient problem-solving and cryptanalysis. When using the Chinese Remainder Theorem (CRT) to solve equations modulo a composite number N in cryptographic applications, which of the following best describes the main advantage provided by CRT? 1) It transforms all equations into linear equations over the integers. 2) It allows decomposition of the problem into separate equations modulo each prime power dividing N. 3) It guarantees that every equation has a unique solution modulo N. 4) It enables direct computation of elliptic curve isogenies without considering modular constraints. 5) It eliminates the need to check quadratic residuosity conditions on parameters. 6) It automatically ensures the invertibility of all modular parameters. 7) It restricts the solution space to trivial solutions when all coefficients are zero.

✓ Correct Answer:

The correct answer is 2) It allows decomposition of the problem into separate equations modulo each prime power dividing N..

📚 Reference Text:

. ⊓ ⊔ C Full proof of Lemma 5.8 We provide a full proof for Lemma 5.8. Proof.Let us write x1ast1+s1i,x2ast2+s2i,xasu1+v1iandyasu2+v2i fort1,s1,t2,st,u1,v1,u2,v2∈Z. Then Equation (7) is equivalent to /braceleftBigg (t1t2+qs1s2)+(−t1s2+s1t2)i=u1+v1imodNR, (t1t2−qs1s2)+(t1s2+s1t2)i=u2+v2imodNR.(10) Letζbe an integer such that 2 ζ≡1 modN, and letM1=ζ(u1+u2),M2= ζq′(u1−u2),M3=ζ(v1+v2) andM4=ζ(v2−v1). Equation (10) is equivalent to the following:   t1t2=M1modN, s1s2=M2modN, s1t2=M3modN, t1s2=M4modN.(11) SinceN=/producttexti=k i=1ℓei i,byChineseReminderTheorem,Equation(11)hasasolution (t1,s1,t2,s2) inZ4if and only if the following equation has a solution in Z4for eachi.   t1t2=M1modℓei i, s1s2=M2modℓei i, s1t2=M3modℓei i, t1s2=M4modℓei i.(12) We claim that Equation (12) has a solution in Z4if and only if M1M4−M2M3≡ 0 modℓei i. We prove this claim by considering the following three cases: 1. At least one of Mj’s is invertible in Z/ℓei i, and we assume its M1without loss of generality. Then ( t1,s1,t2,s2) = (1,M3 M1,M1,M4) is a solution. 2. None of Mj’s is invertible but at least one of Mj’s is nonzero in Z/ℓei i. Let e= minj=1,2,3,4{valℓi(Mj)}, reducing Equation (12) to Z/ℓei−e iwill give us a solution ( t1,s1,t2,s2) inZ/ℓei−e isince we are back to the previous case. Thinking ( t1,s1,t2,s2) as inZand (pet1,pes1,t2,s2) is a solution of Equation (12). 34 M. Chen, M. Imran, G.Ivanyos, P.Kutas, A.Leroux, C. Petit 3.M1,M2,M3,M4are all zero in Z/ℓei iZ, then (0,0,0,0) is a solution. The condition M1M4−M2M3≡0 modℓei iholds for each iif and only if M1M4− M2M3≡0 modN. Plugging in the values of Mj’s shows that this is equivalent ton(x)≡n(y) modN. Therefore, in this case, lifting the solutions modulo each ℓei iwill give rise to a solution of Equation (7). ⊓ ⊔ D IsERP for general N In the main body of the paper we restricted ourselves to Nwith not too many prime factors due to the fact that if Nhas too many prime factors, then a randomnumberisunlikelytobeasquaremodulo N.Suchaquadraticresiduosity condition appears in our strategy to solve the PQLP, more precisely for the discriminant 9. Here we present several approaches that could remove the rest riction onN. D.1 Torsion-point attacks The worst case for our algorithm is when Nis smooth and has many diﬀerent prime factors. This is however exactly the case where classical tor sion-point attacksareuseful. Forsimplicity we assumethat the starting curv eisy2=x3+x but the general approach should follow in a similar fashion. Let Mbe a divisor ofNof sizepoly(logp). Let the secret isogeny be φ:E1→E2. First we try to ﬁnd an endomorphism θ∈End(E1) and an integer dsuch that the degree of ψ=φ◦θ◦ˆφ+dis a powersmooth number B. Since we know how φacts on theB-torsion we can compute ψ. Now the usual step here would be to compute Ker(ψ−d)∩E2[N] but theN-torsion is not deﬁned over a small extension ﬁeld. Instead we decompose φ=φ1◦φ2where the degree of φ1isM. Now instead we compute Ker(ψ−d)∩E2[M] which can be accomplished in polynomial time asMis powersmooth. This intersection will be the kernel of ˆφ1(modulo some technical detailed already handled in [ Pet17]). Thus we can compute ˆφ1and translate the isogeny representation and we now have a smaller Nto work with. We can do this iteratively to handle smooth (and

Question 471 multiple-choice

In quantum thermodynamics and quantum information theory, the extraction of thermodynamic work from information-bearing quantum systems can be enhanced by exploiting quantum algorithms and operations. One area of research explores the advantage of quantum extractors compared to classical methods when applied to pseudo-random spin sources with hidden subgroup structures. Which of the following best explains why quantum extractors can outperform classical extractors in maximizing work extraction from spin sequences exhibiting hidden subgroup structures? 1) Quantum extractors can measure spin states non-destructively, preserving information for repeated extraction cycles. 2) Quantum extractors employ efficient algorithms for the hidden subgroup problem, enabling superior pattern discovery and compression, resulting in greater extractable work. 3) Quantum extractors use probabilistic gates, which inherently increase the entropy of the system and thus the work extracted. 4) Quantum extractors rely on reversible logic gates, ensuring no energy loss during computation. 5) Quantum extractors generate more heat during operation, which can be converted into usable work. 6) Quantum extractors can access hidden classical correlations inaccessible to classical algorithms. 7) Quantum extractors utilize error correction codes that directly convert computational errors into thermodynamic work.

✓ Correct Answer:

The correct answer is 2) Quantum extractors employ efficient algorithms for the hidden subgroup problem, enabling superior pattern discovery and compression, resulting in greater extractable work..

📚 Reference Text:

to either both upor both down, e.g. 1 2ð∣00i00h∣þ∣11i11h∣Þ,t h e nw ec a na p p l yaC N O T operation and transform it to1 2ð∣0i0h∣þ∣1i1h∣Þ/C10∣0i0h∣, and can therefore deterministically extract kBTln 2 work from the second bit. More generally, we can use a unitary transformation to convert non-uniformly distributed long spin sequences into shorter, uniformly-random spin sequences, with a ﬁxed number of spins appended at the end, as shown in Fig. 4.To demonstrate quantum energy harvesting advantage, consider a competition between classical and quan tum intelligent extractors working on a pseudo-random spin source. The classi cal extractor uses classical neural nets or algorithms, while the quantum counterpart leverages our algorithmon a quantum computer. Speci ﬁcally, let P/C26f0;1g nbe the set of all the binary strings with length nthat have the same hidden subgroup structure H

Question 472 multiple-choice

Quantum algorithms for group-theoretic problems often exploit properties of finite Abelian groups and utilize techniques such as quantum Fourier sampling and probabilistic analysis. Efficiency and error reduction are crucial in practical implementations, particularly when solving the Hidden Translation problem in vector spaces over finite fields. In the context of a quantum algorithm for the Hidden Translation problem over Zn_p, which probabilistic tool is specifically used to bound deviations in the number of useful samples for ensuring the algorithm's success probability? 1) Hoeffding's inequality 2) Chernoff’s bound 3) Bayes’ theorem 4) Azuma’s inequality 5) Stirling's approximation 6) Chebyshev's inequality 7) Jensen’s inequality

✓ Correct Answer:

The correct answer is 2) Chernoff’s bound.

📚 Reference Text:

/angbracketlefty/angbracketright. Lemma 3.2 implies that every coset of/angbracketlefty/angbracketrightcontains an element outside of R. This proves that |Rk|/|Vk|≤(p−1)/p fork=0,...,p−1. This completes the proof of the lemma. We now specify the algorithm TranslationFinding and prove that, with high probability, it ﬁnds the hidden translation in quantum polynomial time when pis constant. TranslationFindingf(Zn p) 0. Iff0(0) =f1(0)then output 0. 1.N←13p/parenleftbign+p−2 p−1/parenrightbig . 2. Fori=1,...,N do (zi,bi)←FourierSamplingf(Zn p×Z2). 3.{y1,...,y M}←{zi:bi=1}. 4. Fori=1,...,M doYi←y(p−1) i . 5. Solve the system of linear equations Y1·U=1,...,Y M·U=1. 6. If there are no solutions or more than one solution, then abort. 7. Let 1≤j≤nbe such that the coefficient of xp−1 j is 1 in U. 8. Letv=(v1,...,v n)∈Zn pbe such that vj=1 andvkis the coordinate of xkxp−2 j inUfork/negationslash=j. 9. Find 04p/parenleftbign+p−2 p−1/parenrightbig ,b yM a r k o v ’ s inequality, the solution Uis unique with probability at least 3 /4. Thus, the total probability of aborting is less than 1 /2. Corollary 3.6. Letpbe a prime. Then the problem of Hidden Trans- lation(Zn p)c a nb es o l v e di nq u a n t u mt i m e (n+p)O(p)log(1/ε)with error εusing t=Θ (p(n+p)p−1log(1/ε))accesses to the oracles for f0,f1. Proof. We perform two modiﬁcations in the algorithm TranslationFinding . First, togeterror ε, theinteger Nismultiplied byO(log(1 /ε)). Moreover,weassumed in the algorithm that there is an oracle for f=(f0,f1), which was used to choose fb knowing b. This is not possible in general when f0andf1are given by two distinct oracles. Therefore we replace the oracle access |x/angbracketright|b/angbracketright|0/angbracketrightS/mapsto→|x/angbracketright|b/angbracketright|fb(x)/angbracketrightSby |x/angbracketright|b/angbracketright|0/angbracketrightS|0/angbracketrightS/mapsto→|x/angbracketright|b/angbracketright|fb(x)/angbracketrightS|f1−b(−x)/angbracketrightS. This type of quantum oracle corresponds to the function f/prime=(f/prime 0,f/prime 1), where f/prime 0(x)= (f0(x),f1(x)) andf/prime 1(x)=(f1(x),f0(−x)). Obviously, f/prime 0is injective and f/prime 0(x)= f/prime 1(x+u). We can

Question 473 multiple-choice

Computational group theory leverages algorithmic techniques to efficiently solve problems related to subgroups of the symmetric group, especially via strong generating sets and stabilizer chains. These methods are foundational for modern computational algebra systems and complexity analysis. Which key property ensures that coset representatives selected during the construction of a strong generator set for a subgroup of the symmetric group can be uniquely identified and supports unambiguous computation in algorithms? 1) Each coset representative has maximal order in its coset 2) Each coset representative fixes all previously stabilized points 3) Each coset representative is an involution 4) Each coset representative commutes with all elements of the subgroup 5) Each coset representative belongs to the center of the group 6) Each coset representative is a random element of its coset 7) Each coset representative is the lexicographically least element in its coset

✓ Correct Answer:

The correct answer is 7) Each coset representative is the lexicographically least element in its coset.

📚 Reference Text:

goal is to compute a strong generator set for Gx≤Snusing MEMB as subroutine. As we have ﬁxed the input, we will sometimes drop the subscript and write Ginstead of the group Gx. Our goal is to design an FPSPPalgorithm for ﬁnding the coset represen- tatives ofG(i)inG(i−1)for eachiin the tower of subgroups 1=G(n−1)≤ G(n−2)≤...≤G(1)≤G(0)=G. Starting with G(n−1), which is trivial, the algorithm will build a strong generator set for G(i)in decreasing order of iuntil ﬁnally it computes a strong generator set for G(0)=G. Thus, it suﬃces to describe how the algorithm will compute the coset representatives of G(i)in G(i−1)assuming that a strong generator set for G(i)is already computed. To this end we ﬁrst deﬁne a language: 7 L={/angbracketleftx,0n,S,i,j,π /angbracketright |πis a partial permutation that pointwise ﬁxes 1 ,...,i −1 and mapsitoj,S⊆Gx,and/angbracketleftS/angbracketrightpointwise ﬁxes [ i], and ∃g∈G(i−1) x such that ig=j,lex−least( /angbracketleftS/angbracketrightg) and extends π}. Here, we use lex −least(Hg) for a group Hto denote the lexicographically least permutation in the coset Hg. Partial permutation πis part of instance /angbracketleftx,0n,S,i,j,π /angbracketright, as we will be using Las oracle to do a preﬁx search for the lexicographically least g∈G(i−1)such thatig=j. We now describe an NP machine Nthat accepts L. Description of Machine N; Input :/angbracketleftx,0n,S,i,j,π /angbracketright Verify using MEMB that S⊆G(i); Guessg∈Sn; ifg∈G(i−1)andig=jandgextendsπandg= lex−least( /angbracketleftS/angbracketrightg)then ACCEPT ; end else REJECT ; end Clearly,Nis an NP machine that accepts L. The crucial point is that if ig=jthen for every element h∈ /angbracketleftS/angbracketrightg,ih=j. Also, using the algorithm in Theorem 3.1 the lexicographically least element of /angbracketleftS/angbracketrightgcan be computed in polynomial time. Claim 4.2 If/angbracketleftS/angbracketright=G(i)then the number of accepting paths of Non input /angbracketleftx,0n,S,i,j,π /angbracketrightis either 0or1. In general, on input /angbracketleftx,0n,S,i,j,π /angbracketright,Nhas either 0or|G(i)| |/angbracketleftS/angbracketright|. Proof of Claim . Suppose /angbracketleftx,0n,S,i,j,π /angbracketrightis inLand/angbracketleftS/angbracketright=G(i). Notice that if for someg∈G(i−1)we haveig=j(forj >i ), then /angbracketleftS/angbracketrightgconsists of all elements inG(i−1)that mapitoj. Thus the only guessed element g∈Snby the machine Nthat leads to acceptance corresponds to the unique lexicographically least element of /angbracketleftS/angbracketrightg. On the other hand, if /angbracketleftS/angbracketrightis a proper subgroup of G(i)then we can easily see thatG(i)gcan be written as a disjoint union of |G(i)|/|/angbracketleftS/angbracketright|many right cosets of/angbracketleftS/angbracketright. Thus, in general Nwould have |G(i)|/|/angbracketleftS/angbracketright|many accepting paths if /angbracketleftx,0n,S,i,j,π /angbracketrightis inL. We are now ready to describe an FPLalgorithm for FIND-GROUP . The algorithm is designed in a way that it will query Lfor some /angbracketleftx,0n,S,i,j,π /angbracketright only if /angbracketleftS/angbracketright=G(i), thereby ensuring that it makes only UP-like queries to L. Finally, by Lemma 2.3 we can convert this algorithm to an FPSPPalgorithm. 8 Ci:=∅for every 0 ≤i≤n−2; {Ciwill ﬁnally be a complete set of coset representatives of G(i+1)in G(i).} Di:=∅for every 0 ≤i≤n−2; Dn−1= 1; {Diwill ﬁnally be a strong generator set for G(i)for eachi.} fori:=n−1down to 1do {Diis already computed at the beginning of the ithiteration and at the end of the ithiteration we have Di−1} Letπ: [i−1]→[n] be the partial permutation that ﬁxes all elements from 1 toi−1 {in casei= 1 this is the everywhere undeﬁned partial permutation } forj:=i+ 1tondo π/prime:=π[i:=j]; if/angbracketleftx,0n,Di,i,j,π/prime/angbracketright ∈Lthen {There is

Question 474 multiple-choice

In modern representation theory, diagrammatic frameworks such as spiders provide combinatorial and categorical approaches to understanding tensor invariants and morphisms in quantum groups and Lie algebras. These tools are especially valuable for the study of symmetries, quantum link invariants, and bases in higher algebraic structures. Which property of spiders makes them particularly suited for constructing bases of invariant spaces in rank 2 simple Lie algebras such as A2, B2, and G2? 1) Their ability to encode central elements of the universal enveloping algebra 2) Their use in classifying all possible outer automorphisms of Lie algebras 3) Their correspondence with highest weight vectors in modules 4) Their formulation as braided monoidal categories with ribbon structures 5) Their combinatorial generators and relations yielding explicit bases related to Lusztig's canonical bases 6) Their capacity to define universal R-matrices for quantum groups 7) Their incorporation of symmetric group actions on tensor products

✓ Correct Answer:

The correct answer is 5) Their combinatorial generators and relations yielding explicit bases related to Lusztig's canonical bases.

📚 Reference Text:

Title: Spiders for rank 2 Lie algebras Year: 1996 Paper ID: fec45cac5a7a8cb771414a86863b806dab01e5db Source: semantic-scholar URL: https://www.semanticscholar.org/paper/fec45cac5a7a8cb771414a86863b806dab01e5db Abstract: A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namelyA2,B2, andG2. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger.

Question 475 multiple-choice

In quantum information theory and geometric invariant theory, polytopes are used to analyze the set of possible reduced states and entanglement properties of multipartite quantum systems. The Borel polytope associated with a tensor encodes constraints on marginals that arise from group actions and scaling procedures. Which statement about the Borel polytope defined for a tensor in a product of vector spaces is correct? 1) It is a convex polytope whose vertices are all rational points. 2) It is always a non-convex set with irrational vertices. 3) Its vertices correspond only to pure product states in the Hilbert space. 4) It cannot be described using linear inequalities or convex geometry. 5) The Borel polytope never arises in the context of quantum marginal problems. 6) Its structure depends exclusively on the unitary group without reference to classical tensors. 7) The Borel polytope has vertices that are necessarily transcendental numbers.

✓ Correct Answer:

The correct answer is 1) It is a convex polytope whose vertices are all rational points..

📚 Reference Text:

1988. [Stu08] Bernd Sturmfels. Algorithms in invariant theory . Springer, 2008. [VDDM03] FrankVerstraete,JeroenDehaene,andBartDeMoor. Normalformsandentanglement measures for multipartite quantum states. Physical Review A , 68(1):012103, 2003. arXiv:quant-ph/0105090 ,doi:10.1103/PhysRevA.68.012103 . 53 [VMC08] Frank Verstraete, Valentin Murg, and J Ignacio Cirac. Matrix product states, projected entangledpairstates,andvariationalrenormalizationgroupmethodsforquantum spin systems. Advances in Physics , 57(2):143–224, 2008. arXiv:0907.2796 ,doi:10. 1080/14789940801912366 . [VW17] Michele Vergne and Michael Walter. Inequalities for moment cones of ﬁnite- dimensional representations. Journal of Symplectic Geometry , 15(4):1209–1250, 2017. arXiv:1410.8144 ,doi:10.4310/JSG.2017.v15.n4.a8 . [Wal14] Michael Walter. Multipartite Quantum States and their Marginals . PhD thesis, ETH Zurich, 2014. arXiv:1410.6820 . [WDGC13] MichaelWalter,BrentDoran,DavidGross,andMatthiasChristandl. Entanglement polytopes: multiparticle entanglement from single-particle information. Science, 340(6137):1205–1208, 2013. arXiv:1208.0365 ,doi:10.1126/science.1232957 . [Wil13]Mark M Wilde. Quantum information theory . Cambridge University Press, 2013. arXiv:1106.1445 ,doi:10.1017/CBO9781139525343 . A Appendix A.1 Borel polytope HereweincludeanelementarydescriptionoftheBorelpolytope BpXqforXPTenpn0;n1;:::;ndq. and prove that it is indeed a polytope with rational vertices. By the remarks after the proof of Proposition 2.11 we have pPBpYqif and only if 0 pcapacityppYqq2inf bPBtrpbYq| ppbq|2: It’seasytoseethat log capacitypqisconcavein! Thisimmediatelyimplies Bpqisconvex, but we can say more: Theorem A.1. BpXqis a polytope with rational vertices. This follows from a more detailed description of BpXq, which requires some technical deﬁnitions. The ideas are very similar to the elementary derivation of the tensor case of the Hilbert-Mumford criterion [ BGO17]. In the uniform case, a density matrix can be scaled to uniformmarginalsifin everyorthonormalbasis,thediagonalof (regardedasaclassicaltensor) can be scaled by diagonal matrices to uniform marginals. We ﬁnd a similar criterion, but the reduced density matrix will be blown up (much as in Section 4, and the diagonal must be scaled to certain nonuniform marginals. Let n:pn1;:::;ndq. Deﬁnition A.2 (unitary family) .Apd;nq-unitary family is a tuple UpUpiq j;iPrds;jPrnisq whereUpiq jis ajjunitary matrix. 54 LetSbe the set of pairs pj;lqsuch thatj pjp1q;:::;jpdqq P rn1srndsandl plp1q;:::;lpdqqPrjp1qsrjpdqs. We useSas an index set for Tenpn11 2q;:::;pn11 dqpR¥0q. Deﬁnition A.3 (expanded classical tensor) .Given apd;nq-unitary family U, denote byCp;Uqthe element of Tenpn11 2q;:::;pn11 dqpR¥0qgiven by Cp;Uqj;l:pUjjql;l; whereuj;ldenotesthetuple pUp1q jp1q;:::;Updq jpdqqandjdenotespjp1q;:::;jpdqq. ForanyCPTenpn11 2q;:::;pn11 dqpR¥0q, we deﬁne SupppCqtpj;lqPS:Cj;l¡0u: We may now state a description of BpYq. We use the shorthand ppiq jppiq jppiq j1, where pni10. Proposition A.4. pPBpYqif and only if for every pd;nq-basis-family U, there is a tensor DPTenpn11 2q;:::;pn11 dqpR¥0q satisfying SuppDSuppCpY;Uq (31) and ¸ pj;lqPS:jpiqj;lpiqlDj;lppiq j for alliPrds;jPrnis;lPrjs: (32) That is,Dhas asithclassical margin the vector pppiq j:jPrnis;lPrjsqPRpni1 2q ¥0. Before we prove Proposition A.4, we use it to prove Theorem A.1. Proof of Theorem A.1: FixSuppCpY;Uq. By Farkas’ lemma, the existence of Dsatisfying Eq. (31) and Eq. (32) is equivalent to the following statement: Every sequence apapiq j;l;iPrds;jPrnis;lPrjsqof numbers satisfying d¸ i1apiq jpiq;lpiq¥0for allpj;lqPSuppCpY;Uq (33) also satisﬁes d¸ i1ni¸ j1ppiq jl¸ i1ajpiq;lpiqpiq¥0: (34) Sincethesetof asatisfying33isaconvexconewithﬁnitelymanyconstraints,itisgeneratedbya ﬁnite set of rational vectors; it is enough to check that 33 implies 34 on that ﬁnite set of rational vectors; this implies the Borel polytope is indeed a polytope with rational vertices (there are only a ﬁnite number of possibilities for SuppCpY;Uq!) 55 Before we prove Proposition A.4, we must prove a lemma. We wish to characterize

Question 476 multiple-choice

Parity encoding is an alternative quantum computation paradigm where logical qubits are represented by the parity of physical qubit subsets. This approach offers potential advantages in scalability and circuit efficiency for quantum algorithms. Which feature of parity encoding directly enables more natural implementation of higher-order quantum interactions, especially in hardware with limited qubit connectivity? 1) Encoding individual logical qubits in single physical qubits 2) Reliance on global entangling gates for state preparation 3) Mapping logical qubits to parity qubits representing the parity of subsets 4) Use of decoherence-free subspaces for error mitigation 5) Direct embedding of classical bits into quantum registers 6) Application of SWAP gates to reroute quantum information 7) Restriction to nearest-neighbor coupling architectures

✓ Correct Answer:

The correct answer is 3) Mapping logical qubits to parity qubits representing the parity of subsets.

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PHYSICAL REVIEW A 106, 042442 (2022) Applications of universal parity quantum computation Michael Fellner ,1,2,*Anette Messinger ,2Kilian Ender ,1,2and Wolfgang Lechner1,2,† 1Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria 2Parity Quantum Computing GmbH, 6020 Innsbruck, Austria (Received 19 May 2022; accepted 16 September 2022; published 27 October 2022) We demonstrate the applicability of a universal gate set in the parity encoding, which is a dual to the standard gate model, by exploring several quantum gate algorithms such as the quantum Fourier transform and quantumaddition. Embedding these algorithms in the parity encoding reduces the circuit depth compared to conventionalgate-based implementations while keeping the multiqubit gate counts comparable. We further propose simpleimplementations of multiqubit gates in tailored encodings and an efﬁcient strategy to prepare graph states. DOI: 10.1103/PhysRevA.106.042442 I. INTRODUCTION In recent decades, there has been an enormous effort to develop novel strategies for quantum computation [ 1–7], including measurement-based [ 8,9] or adiabatic [ 10] quan- tum computation, complementing the gate-based paradigmof quantum computation. In Ref. [ 11] we proposed univer- sal quantum computation in the Lechner-Hauke-Zoller (LHZ)scheme [ 12] as a dual to the conventional gate model. Recent achievements in quantum hardware development on variousqubit platforms [ 13–20] might soon allow for experimental realizations of well-known quantum algorithms for reason-able system sizes. Nevertheless, a fundamental challengeof state-of-the-art quantum devices remains the interqubitconnectivity on quantum chips [ 21,22]. This is especially pressing because a long-range and dense (ideally all-to-all)connectivity is a crucial ingredient for many key quantumalgorithms, unless algorithm-speciﬁc preprocessing steps areperformed [ 23]. Unless efﬁcient native implementations for long-range interactions as in, for example, the surface code[24,25] exist, the connectivity problem is often dealt with by utilizing resource-intensive SWAP gates which, apart from requiring error-prone two-qubit gates, render a paralleliza-tion of gates difﬁcult. Although there exist several quantumrouting techniques [ 26,27], this is in particular problematic for scalability of devices beyond the noisy intermediate-scalequantum (NISQ) era [ 28]. An alternative way to address the connectivity issue and, as a side effect, allow for the native implementation of higher-order interactions, was introduced with the parity encoding[12,29]. The parity encoding maps nlogical qubits (with operators ˜ σ)t o K>nphysical qubits (parity qubits, with *m.fellner@parityqc.com †wolfgang.lechner@uibk.ac.at Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.operators σ), encoding the relative alignment (parity) along thezaxis of two or more logical qubits such that for any state |ψ/angbracketrightin the code space ˜σ(i) z˜σ(j) z|ψ/angbracketright=σ(ij) z|ψ/angbracketright. (1) Here the superscripts correspond to qubit labels. In order to deal with the additional degrees of freedom in the physicalHilbert space, parity constraints of the form C l|ψ/angbracketright:=σ(l1) zσ(l2) zσ(l3) z/bracketleftbig σ(l4) z/bracketrightbig |ψ/angbracketright=|ψ/angbracketright (2) are introduced as stabilizers of the code space, which is also referred to as the constraint-fulﬁlling subspace HCF.T h ei n - dices lirepresent pairs of logical qubits. In every constraint, each logical index must occur

Question 477 multiple-choice

In supersymmetric model building, resolving naturalness problems and generating appropriate mass spectra for superpartners are critical challenges. Gauge group structure and symmetry considerations play central roles in determining allowed terms and particle properties. Which solution is proposed to avoid a problematic Fayet-Iliopoulos term associated with hypercharge in dynamical supersymmetry breaking models? 1) Embedding hypercharge into a non-Abelian gauge group 2) Introducing explicit soft SUSY-breaking terms for gauginos 3) Adding extra U(1) gauge symmetries with tuned D-terms 4) Utilizing anomaly cancellation through new fermion generations 5) Relying on radiative corrections to generate the required mass terms 6) Breaking R symmetry spontaneously in the visible sector 7) Implementing gravity mediation with large Planck-scale masses

✓ Correct Answer:

The correct answer is 1) Embedding hypercharge into a non-Abelian gauge group.

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Title: Supersymmetry, naturalness, and dynamical supersymmetry breaking. Year: 1992 Paper ID: 0152f78e88c1e8a84485a10029732a0a3c3c10fc Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0152f78e88c1e8a84485a10029732a0a3c3c10fc Abstract: Models with dynamical supersymmetry (SUSY) breaking have the potential to solve many of the naturalness problems of hidden sector supergravity models. We review the argument that in a generic supergravity theory in which supersymmetry is {ital dynamically} broken in the hidden sector, only tiny Majorana masses for gauginos are generated. This situation is similar to that of theories with continuous {ital R} symmetries, for which Hall and Randall have suggested that gluino masses could arise through mixings with an octet of chiral fields. We note that, in hidden sector models, such mixing can only occur if the auxiliary {ital D} field of a U(1) gauge field has an expectation value. This in turn gives rise to a catastrophically large Fayet-Iliopoulos term for ordinary hypercharge. To solve this problem it is necessary to unify hypercharge at least partially in a non-Abelian group. We consider, also, some general issues in models with continuous or discrete {ital R} symmetries, noting that it may be necessary to include SU(2)-triplet fields, and that these are subject to various constraints. In the course of these discussions, we consider a number of naturalness problems. We suggest that the so-called {mu} problem'' is not a problem, and pointmore » out that in models in which the axion decay constant is directly related to the SUSY-breaking scale, squarks, sleptons, and Higgs particles generically acquire huge masses.« less

Question 478 multiple-choice

Efficient optimization of quantum circuits is vital due to the exponential growth of the search space as circuit length and gate options increase. Advanced algorithms are often required to find optimal circuits with minimal resources, especially given the limitations of current quantum hardware. Which optimization algorithm consistently finds the shortest quantum circuits with significantly fewer samples than traditional sampling and evolutionary methods, making it particularly effective for variable-length quantum circuit design tasks? 1) Random sampling 2) Genetic algorithm 3) Particle filter 4) Exhaustive search 5) Simulated annealing 6) Monte Carlo Graph Search (MCGS) 7) Greedy search

✓ Correct Answer:

The correct answer is 6) Monte Carlo Graph Search (MCGS).

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In the first experiment we compared the opti- mization of quantum circuits of length 1, 2, . . ., 7, for a given predefined set of elementary quan- tum gates. Note, for a database of elementary gates of size 32 and a circuit length of 7 there are 327∼3.5×1010different quantum circuits possible; the search space grows exponentially and is already infeasibly large (for exhaustive searches) for larger circuits. In Fig 4 we present the comparison of four different baselines, based on a naive random sampling, a genetic algo- rithm (Sec 3.2), a particle filter (Sec. 3.3), and simulated annealing (Sec. 3.4), respectively, along with our proposed Monte Carlo Graph Search. The x-axis records the circuit length required for a predefined quantum circuit. The complexity of the optimization problem increas- ing exponentially with circuit length. Note, that the yaxis is scaled with the log function, so that a linear slope of indicates an exponential growth in complexity. The error bars depict the standard deviation. For this experiment, the amount of experiments/observations was set to 10. In comparison to the four baselines, our pro- posed Monte Carlo Graph Search requires far fewer samples to reach a solution. One expla- nation for this is that unnecessary samples (e.g., leading to cycles, etc.) are efficiently avoided. In the next experiment we analyze the ef- ficiency of the generated quantum circuits in terms of the amount of required gates. As the computation graph contains cycles, a valid ques- FIG. 4. Amount of required samples for quantum circuit optimization for differing circuit lengths. Compared are random sampling, a genetic algo- rithm, a particle filter, simulated annealing and the proposed MCGS model. The bars depict the stan- dard deviation. The diagram is illustrated with re- spect to a logarithmic yaxis scale. One observes in this way the the MCGS already requires an order of magnitude fewer samples than the four baselines. FIG. 5. Optimal circuit length versus circuit length of different architecture search algorithms. tion is, if standard sampling based approaches for QAS can lead to solutions which require more gates than necessary, resulting in ineffi- cient code. The approaches of Gibbs-Sampling, simulated annealing, and the proposed MCGS model allow one to optimize quantum circuits where the resulting code length is not fixed. Due to the diffusion and crossover steps, it is not clear how one can do this for the particle 9 filter and the genetic algorithms, so these ap- proaches were omitted for this experiment. In Figure 5 the mean and standard deviation of the random sampling, simulated annealing, and MCGS for varying optimal circuit length tasks is shown. The x-axis is labelled by the optimal circuit length and the y-axis by the required cir- cuit lengths (including standard deviation) of the different optimizers. The optimal graph is a straight line, which was achieved by our pro- posed MCGS. Thus, the MCGS always finds the optimal length, whereas the sampling and annealing schemes more often tend to find in- efficient solutions. Thus, the proposed MCGS

Question 479 multiple-choice

In algebraic approaches to quantum optimization problems, swap algebras and polynomial ideals interact with representation theory and graph structures. Understanding how degree bounds and support graphs influence semidefinite programming hierarchies is crucial for solving problems like Quantum Max Cut. Which statement correctly identifies the upper bound on the degree of swap variable polynomials relevant for the semidefinite program hierarchies addressing the Quantum Max Cut problem? 1) The maximum degree is always n−1 for any swap variable polynomial. 2) The maximum degree equals the number of swap variables, n(n−1)/2. 3) The degree bound is determined by the dimension of the swap algebra. 4) There is no fixed upper bound; degrees can be arbitrarily high. 5) The maximum degree is restricted to n/3 for computational efficiency. 6) The maximum degree of swap variable polynomials is ⌈n/2⌉. 7) The degree depends solely on the number of irreducible representations in the decomposition.

✓ Correct Answer:

The correct answer is 6) The maximum degree of swap variable polynomials is ⌈n/2⌉..

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of the symmetric group Sn. Lemma 3.9. The function of kin Eq. (2.35) is strictly increasing for 0≤k≤n 2. Proof. Simply observe that the function’s derivative w.r.t. kis 2(n+ 1)−4k≥2(n+ 1)−2n= 2>0. Proposition 3.10. LethKn∈Aswap n denote the Hamiltonian polynomial for the n-clique. Then the ideal In−k,kof polynomials which annihilate ρ[n−k,k], is generated by Iswaptogether withhKn−ηn−k,k. Proof. LetAλ ndenote the simple algebra of the irrep ρλofSnforλ⊢n. It is isomorphic toMχλ(e)(C). By Corollary 3.8, Aswap n∼=⌊n 2⌋/circleplusdisplay k=0A[n−k,k] n. (3.18) With this notation, Aswap n/In−k,k∼=A[n−k,k] n. By Lemma 2.11 and Lemma 2.12, hKn−ηn−k,k∈In−k,k. Since by Lemma 3.9, the map k∝⇕⊣√∫⊔≀→ηn−k,kis injective for 0≤k≤n/2, the image under the isomorphism of Eq. (3.18) of hKn−ηn−k,k∈In−k,kis zero in the k-th entry and a nonzero number in each of the others. In particular, Aswap n/I(hKn−ηn−k,k)∼=A[n−k,k] n∼=Aswap n/In−k,k. SinceI(hKn−ηn−k,k)⊆In−k,k, this concludes the proof. 3.4 Structure of Swap Variable Polynomials Now we describe some further properties of the Swap Algebra. The main result of this subsection is a sharp upper bound of ⌈n/2⌉on the maximum degree of elements in the swap algebra, along with simplified form for elements in this algebra. Later, we will observe that this upper bound gives a corresponding upper bound on the level at which the hierarchy of semidefinite programs constructed in Section 4 gives an exact solution to the Quantum Max Cut problem. We head in that direction via a lemma. Define the support graph Gpof a polynomial pin the variables sijto have vertices corresponding to integers which appear as indices iorjfor somesijinp, with (i,j)an edge iffsijappears in p. A polynomial in Iswapis called degree reducing provided it has exactly one highest degree term. 3.4.1 Degree reducing polynomials in Iswap Lemma 3.11. Forn≥4and1≤i,j,k,ℓ≤nwith no two indices being equal, there is a degree reducing polynomial in Iswapwhose highest degree term has the form (1)sjksiksℓk. a term with support graph a three edge star; (2)sijsjkskℓ,sjksijskℓ, orsjkskℓsij, a term with support graph a three edge line. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 26 Observe that any of the three edge line monomials equals a star graph monomial modulo a single triangle pair substitution. Here a triangle pair relation is a polynomial of the form sijsjk−sjkski. It lies in Iswap and relates behavior of edge pairs in the triangle which is the support of the polynomial. Proof. Straightforward matrix multiplication can be used to verify the following matrix identity: 2 SwapjkSwapikSwapℓk= 1−Swapij−Swapiℓ−Swapjk−Swapkℓ+ SwapijSwapik + SwapijSwapiℓ+ SwapijSwapkℓ+ SwapikSwapiℓ −SwapikSwapjℓ+ SwapiℓSwapjk+ SwapjkSwapjℓ, which implies the same identity for the sijby Theorem 3.6. The proof of the three-edge-line case and the final paragraph of the lemma amount to the identities sijsjkskℓ=sijsjℓsjk, sjksijskℓ=siksjksℓk, sjkskℓsij=sjℓsjksij so we can always reduce to the first case. Lemma 3.12. Forn≥6and1≤i,j,k,a,b,c≤nwith no two indices being equal, there is a degree reducing polynomial in Iswapwhose highest degree term has the form sijsjksabsbc, a term whose support graph is two disjoint two edge lines. Proof. It is straightforward to verify the following identity for 26×26matrices from which the lemma follows using Theorem

Question 480 multiple-choice

Quantum circuit uniformity concerns how quantum circuits are constructed for varying input sizes, impacting both theoretical complexity and practical implementation. Different models of uniformity define what information and computational resources are available during circuit construction and execution. Which model of quantum circuit uniformity allows the computation of gate parameters dynamically from the actual input during the computation, thereby enabling the deterministic (derandomized) execution of algorithms like Shor's factoring algorithm? 1) Finitely based uniform circuit families 2) Infinitely based uniform circuit families 3) Input-computed uniform circuit families 4) Non-uniform circuit families with post-selected measurements 5) Oracle-accessed circuit families with purely classical gates 6) Circuit families restricted to smooth modulus only 7) Uniform circuit families limited to fixed finite gate sets

✓ Correct Answer:

The correct answer is 3) Input-computed uniform circuit families.

📚 Reference Text:

poly(log m)) built knowing mrather than the input size which is essentially log m. This model is apparently stronger than the uniform circuit family in the sense of Nishimura and Ozawa. However, in some applications when mis known to be a member of a set that can be computed in time polynomial from the sizeof the input for the speci c application, the Mosca-Zalka implementation of QFTmcan be used as an ingredient of a uniform circuit family in the strict sense. Besides smooth numbers, these cases include numbers msuch as the order of the multiplicative group of a nite eld of small characteristic or, more generally, the order of the general linear group over such elds. We remark that Mosca [16] proposed an even more powerful model in which it is allowed to apply a gate U with parameter when (a description of) emerges during the computation. (To be more speci c, Mosca proposed allowing gates mapping j ijxitoj iU jxi.) Using that model, he successfully "derandomized" Shor's factoring algorithm. The square-free decom- position algorithm of Li et al. [17] also appears to require application of certain gates with M. Imran and G. Ivanyos 773 parameters that depend on the divisors of the number Nto factor and hence are not known to be classically computable from the input in polynomial time. (To be more speci c, they useQFTd1to create the uniform superposition of integers between 1 and d1 wheredis a "random" divisor of Npopping up during the computation. Approaches using amplitude ampli cation directly could also do the job, but all we are aware of need one or more gates with parameters depending on d.) For the problems considered in this paper, issues with emerging divisors can be circumvented, therefore we only consider the three models discussed above. In increasing order of power, these are (i) nitely based uniform circuit families; (ii) in nitely based uniform circuit families; and (iii) circuit families uniform in the sense that the parameters of the gates are computed from the input. In (i) and (ii), uniformity is understood in the strict sense, that is, parameters of the gates are computed from the input size. By a polynomial time quantum algorithm with oracles we mean a family of polynomial size circuits with oracles which, apart from the oracles, is uniform in the strictest sense like model (i), that is, the input size parametrizes the circuits, and the gates are from a nite set. When we do not specify the number of invocations of the oracles in a statement, the number of calls is included in the bound for the circuit size. We state our results in a form when the quantum Fourier transform is accessed via an oracle. This makes it easier to derive the di erent implications in the three models discussed above. Our algorithms will also be purely unitary, that is, no measurements will be used. The output will be a state in a separate register and will not be entangled with the

Question 481 multiple-choice

The three-body problem is a central challenge in dynamical systems theory, with profound implications in physics and mathematics. Modern approaches explore its behavior in various geometrical frameworks and seek to uncover hidden structural features and efficient computational methods. Which development most directly enables a reduction in the mathematical order of the general three-body problem, simplifying both analytical and numerical treatment? 1) Discovery of hidden symmetries in the internal motion within a curved configuration space 2) Introduction of external time as an additional dynamical parameter 3) Reformulation of the system in a strictly flat Euclidean space without local coordinates 4) Application of standard conservation laws for momentum and energy alone 5) Use of purely stochastic methods without consideration of underlying symmetries 6) Construction of geodesic flow equations without analyzing system invariants 7) Numerical integration using generic algorithms for stiff differential equations

✓ Correct Answer:

The correct answer is 1) Discovery of hidden symmetries in the internal motion within a curved configuration space.

📚 Reference Text:

Title: The General Three-Body Problem in Conformal-Euclidean Space: Hidden Symmetries and New Properties of a Low-Dimensional Syste Year: 2024 Paper ID: 10aa3c254739cdf100bd2a3c68414ed4a77a9b4c Source: semantic-scholar URL: https://www.semanticscholar.org/paper/10aa3c254739cdf100bd2a3c68414ed4a77a9b4c Abstract: Despite the huge number of research into the three-body problem in physics and mathematics, the study of this problem still remains relevant both from the point of view of its broad application and taking into account its fundamental significance for the theory of dynamical systems. In addition, to solve the problem of quantum-to-classical transition, it is important to answer the question: is irreversibility fundamental to the description of the classical world? To answer this question, we considered a reference classical dynamical system, the general three-body problem, formulating it in conformal Euclidean space and rigorously proving its equivalence to the Newtonian three-body problem. It is shown that a curved configuration space with a local coordinate system reveals new hidden symmetries of the internal motion of a dynamical system, which makes it possible to reduce the problem to a 6th order system instead of the known 8th order. The most important consequence of this consideration is that the chronologizing parameter of the motion of a system of particles, which we call internal time, is in the general case irreversible, which is characteristic of the general three-body problem. An equation is derived that describes the evolution of the flow of geodesic trajectories, with the help of which the entropy of the system is constructed. New criteria for assessing the complexity of a low-dimensional dynamic system and the dimension of stochastic fractal structures arising in three-dimensional space are obtained. An effective mathematical algorithm has been developed for the numerical simulation of the general three-body problem, which is traditionally a difficult-to-solve system of stiff ordinary differential equations.

Question 482 multiple-choice

In quantum information theory, operator frames and their duals play a crucial role in quantum state reconstruction and measurement. Concepts such as completeness relations, the frame operator, and informationally complete POVMs are fundamental for ensuring robust and accurate quantum measurements. Which statement most accurately characterizes the relationship between bi-orthogonality and the uniqueness of the canonical dual frame in operator frame theory on a Hilbert space? 1) Bi-orthogonality guarantees that every dual frame is orthogonal to the original frame. 2) Bi-orthogonality indicates that the frame operator is not invertible. 3) Uniqueness of the canonical dual frame occurs only if the frame is overcomplete. 4) Bi-orthogonality and the uniqueness of the canonical dual frame are unrelated concepts. 5) The canonical dual frame is always unique, regardless of bi-orthogonality. 6) Bi-orthogonality and uniqueness of the canonical dual frame are equivalent; if the frame is bi-orthogonal, the canonical dual is unique. 7) Bi-orthogonality requires the use of continuous frames for uniqueness.

✓ Correct Answer:

The correct answer is 6) Bi-orthogonality and uniqueness of the canonical dual frame are equivalent; if the frame is bi-orthogonal, the canonical dual is unique..

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the norm will be the Frobenius norm ||A||2 2=/radicalbig Tr[A†A]. For{Ξi}an operator frame there exists another frame {Θi}—called dual frame — giving the operator expansion in the form A=/summationdisplay iTr[Θ† iA]Ξi. (5) The completeness relation of the frame and its dual reads/summationdisplay i/an}bracketle{tψ|Ξi|φ/an}bracketri}ht/an}bracketle{tϕ|Θ† i|η/an}bracketri}ht=/an}bracketle{tψ|η/an}bracketri}ht/an}bracketle{tϕ|φ/an}bracketri}ht, (6) for anyφ,ϕ,ψ,η ∈ H. For continuous sets, the sums in Eqs. (5) and (6) are replaced by integrals. Given a frame {Ξi}, generally the dual set is not unique. However, all duals {Θi}of a given frame can be obtained via the linear relation [21] |Θi/an}bracketri}ht/an}bracketri}ht=F−1|Ξi/an}bracketri}ht/an}bracketri}ht+|Yi/an}bracketri}ht/an}bracketri}ht−/summationdisplay j/an}bracketle{t/an}bracketle{tΞj|F−1|Ξi/an}bracketri}ht/an}bracketri}ht|Yj/an}bracketri}ht/an}bracketri}ht, (7) whereYiare arbitrary, and the positive and invertible operator Fwrites F=/summationdisplay i|Ξi/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tΞi|. (8) The operator Fis called ”frame operator” in frame theory, whereas the set of ope rators corresponding to the vectors F−1|Ξi/an}bracketri}ht/an}bracketri}htthrough the above isomorphism is the so-called Informationally complete measurements and groups represe ntation 4 ”canonical dual” frame. On the other hand, given an arbitrary set of operators {Ξi}, the invertibility of Fin Eq. (8) implies that the set is a frame. Notice that if the frame is bi-orthogonal, namely /an}bracketle{t/an}bracketle{tΞi|F−1|Ξj/an}bracketri}ht/an}bracketri}ht=δij, then the canonical one is the unique dual frame. One can also prove the converse statement [17], when ce bi-orthogonality is equivalent to uniqueness of the canonical dual frame. From the above considerations it follows that a POVM {Πi}is info-complete if and only if the corresponding operator F=/summationtext i|Πi/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tΠi|is invertible. From linearity, by comparing Eq. (4) and (5), one can see that a dual frame of an info -complete POVM provides a data processing function as fi(O) = Tr[Θ† iO], whence Eq. (7) allows a useful ﬂexibility in the data-processing, with the possibility of minimizing the st atistical error of the estimation by varying the free operators Yi. Since the number of elements of an operator frame for Hcannot be smaller than d2, an info-complete POVM is necessarily not orthogonal, whence it is ov ercomplete. Viceversa, it is simple to prove that an arbitrary frame for operato rs inHmade of positive operators {Ki}allows to construct an info-complete POVM. In fact, since the operatorS≡/summationtext iKiisinvertible, theset {˜Ki=S−1/2KiS−1/2}satisﬁesthecompleteness relation/summationtext i˜Ki=I. 3. Group-theoretic techniques The representation theory of groups provides the easiest way to construct frames made of unitary operators. Consider for example a unitary irreducible re presentation (UIR) {Ug, g∈G}of a compact group Gon the Hilbert space H. From the Shur’s lemma, one has/integraldisplay Gdµ(g)UgOU† g= Tr[O]I , (9) wheredµ(g) denotes the left-invariant measure normalized as/integraltext Gdµ(g) =d. As one can see, Eq. (9) is equivalent to Eq. (6) with {Ug}self-dual operator frame. On the other hand, the direct construction of info-complete POVM’s is not as simp le, since it involves the searching of frames of positiveoperators. A way to construct info-complete POVM’s is suggested by Eq. (9). For any density matrix νthe set of positive operators Πg=UgνU† g (10) provides a resolution of the identity, whence {Πg}is a POVM. Moreover, the POVM is info-complete iﬀ the operator F=/integraldisplay Gdµ(g)|Πg/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tΠg|=/integraldisplay Gdµ(g)Ug⊗U∗ g|ν/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tν|U† g⊗(U∗ g)†,(11) is invertible, where we used Eq. (2). Representation theory allows t o evaluate the

Question 483 multiple-choice

In quantum information theory, unitary designs and group structures like the Pauli and Clifford groups are crucial for simulating randomness and implementing error correction protocols. Understanding the properties and limits of such groups is foundational for efficient quantum computation and benchmarking. Which statement correctly characterizes the Clifford group in the context of unitary designs? 1) The Clifford group forms a unitary 3-design but does not form a 4-design. 2) The Clifford group forms a unitary 2-design and a 4-design. 3) The Clifford group forms only a unitary 1-design. 4) The Clifford group cannot be efficiently implemented in quantum circuits. 5) The Clifford group acts as an orthogonal projector onto the symmetric subspace. 6) The Clifford group generates the entire set of Haar-random unitaries. 7) The Clifford group includes only Pauli matrices and the identity operator.

✓ Correct Answer:

The correct answer is 1) The Clifford group forms a unitary 3-design but does not form a 4-design..

📚 Reference Text:

µH/parenrightig2/bracketrightbigg −2 Tr/bracketleftig A(k) ηA(k) µH/bracketrightig (181) =E |ψ⟩,|ϕ⟩∼η/bracketleftig |⟨ψ|ϕ⟩|2k/bracketrightig + Tr/bracketleftbigg/parenleftig A(k) µH/parenrightig2/bracketrightbigg −2 Tr/bracketleftig A(k) ηA(k) µH/bracketrightig , (182) where in the second equality we used cyclicity of the trace. Additionally, we have: Tr/bracketleftbigg/parenleftig A(k) µH/parenrightig2/bracketrightbigg = Tr /parenleftigg P(d,k) sym dim (Symk(Cd))/parenrightigg2  (183) = Tr/bracketleftigg P(d,k) sym (dim (Symk(Cd)))2/bracketrightigg (184) =/parenleftbig dim/parenleftbig Symk(Cd)/parenrightbig/parenrightbig−1, (185) where in the first equality we used Eq.(95), in the second equality the fact that P(d,k) sym squares to itself (Theorem 16) and in the last equality the fact that P(d,k) sym is the orthogonal projector on the symmetric subspace (Theorem 16). Similarly, we can show that Tr/bracketleftig A(k) ηA(k) µH/bracketrightig is equal to /parenleftbig dim/parenleftbig Symk(Cd)/parenrightbig/parenrightbig−1. Here, dim/parenleftbig Symk(Cd)/parenrightbig can be calculated using Proposition 17, which shows that it is equal to/parenleftbigd+k−1 k/parenrightbig . Therefore, we can conclude. 25 It is worth noting that any uniform distribution νdefined over a set of unitaries S={Ui}d2 i=1 that forms a basis for L(Cd)and satisfies Tr/parenleftig U† iUj/parenrightig =dδi,jconstitutes a 1-design. This can be easily proven by computing the frame potential as follows: F(k=1) ν =1 |S|2d2/summationdisplay i,j=1/vextendsingle/vextendsingle/vextendsingleTr/parenleftig UiU† j/parenrightig/vextendsingle/vextendsingle/vextendsingle2 =1 d4d2/summationdisplay i,j=1d2δi,j= 1 = dim(span( Vd(π) :π∈S1)), (186) and applying Proposition 34. Therefore, the uniform distribution defined over the Pauli basis ˜P:={I,X,Y,Z}⊗nis a1-design, where nis the number of qubits and d= 2n. The Flip operator can be elegantly represented in terms of the Pauli basis using the following expression: F=/summationdisplay P,Q∈˜P1 d2Tr((P⊗Q)F)P⊗Q=/summationdisplay P,Q∈˜P1 d2Tr(PQ)P⊗Q=/summationdisplay P∈˜P1 dP⊗P, (187) where we wrote the Flip operator in the Pauli basis and used the swap-trick . Using this, it is also evident that the Pauli group forms a 1-design (according to Definition 27). In fact, for any O∈L/parenleftbig Cd/parenrightbig , we have: 1 |˜P|/summationdisplay P∈˜PPOP†=1 d2/summationdisplay P∈˜PTr2((I⊗O)(P⊗P)F) =1 dTr2((I⊗O)FF) =Tr(O) d=E U∼µH/bracketleftbig UOU†/bracketrightbig , where in the first step we used the partial-swap-trick (Eq.(152) ) and in the last step (Eq. (50)). An important set of unitaries is the Clifford group Cl(n)[68] i.e. the set of unitaries which sends the Pauli group Pnin itself under the adjoint operation: Cl(n):={U∈U(2n):UPU†∈Pnfor allP∈Pn}, (188) wherePn:={ik}3 k=0×{I,X,Y,Z}⊗n. It can be proven that the uniform distribution over the Clifford group, forms a 3-design for all d= 2n[69, 70], but it fails to be a 4-design [71]. Moreover, it can be shown that any Clifford circuit can be implemented with O(n2/log(n)) gates from the set {H,CNOT,S}where H,CNOT andSare the Hadamard, Controlled-NOT and Phase gate, respectively [63, 72]. Furthermore, there are efficient algorithms to sample uniformly from the Clifford group [73, 74]. This is of fundamental importance for applications in quantum computing, since if we are interested in reproducing moments of the Haar measure up to the third moment, it is sufficient to sample efficient quantum circuits that correspond to Clifford unitaries (instead of sampling from the Haar measure, which would require implementing quantum circuits with an exponential number of gates in the number of qubits [63]). 8 Approximate unitary designs In many cases, having an exact unitary design may not be necessary, and

Question 484 multiple-choice

In quantum circuit simulation using tensor networks, maintaining isometric tensors is essential for numerical stability and efficient contraction, especially in the context of simulating quantum Fourier transform (QFT) circuits. Understanding the conditions that guarantee isometry at different sites and the behavior of the Schmidt coefficients is crucial for scalable simulations. Which property of a tensor at a specific site in a QFT circuit ensures isometry when combined with a Hadamard gate and a copy tensor, and is formally defined such that fixing its vertical indices yields always-diagonal matrices horizontally? 1) Vertical block-diagonality 2) Unitary tensor structure 3) Full rank across all indices 4) Symmetry under permutation of indices 5) Horizontal diagonality 6) Maximal Schmidt rank 7) Invariance under phase gate application

✓ Correct Answer:

The correct answer is 5) Horizontal diagonality.

📚 Reference Text:

completely stabilize the zip-up algorithm, one wants to show that combining two sites besides the orthogonality centers results in the isometric tensors(H2) This is already true for sites j0+2t o n−1, i.e., sites with phase tensors, since one can apply Eq. (F6) to reduce the following tensor diagram to identity: (H3) For sites 1 to j0this is trivial since they are untouched and one already has isometric tensors from previous iterations. Therefore, the only problematic site is site j0+1. That is, we cannot guarantee the following diagram is equivalent to two identity tensors: (H4) However, we show that if the tensor U[j0+1] is hori- zontally diagonal , then site j0+1 with an Hgate and a copy tensor also forms an isometric tensor. By “horizon- tally diagonal,” we mean that by our setting vertical indices of the tensor U[k] to some values, the resulting matrix in the horizontal direction is always diagonal. Equivalently, one can describe it as the diagram (H5) where the xtensor is a projector onto |x/angbracketrightwith x∈{0, 1}. This property allows us to express the contraction of U[k] and its conjugate transpose as a sum (H6) which also equals identity since U[k] is guaranteed to be isometric. One can then evaluate a diagram like that in 040318-26 QUANTUM FOURIER TRANSFORM... PRX QUANTUM 4,040318 (2023) Eq.(H4) as a sum over two diagrams: (H7) where whether xequals 0 or 1, the right part of the dia- gram is evaluated to identity up to a factor of 1 /2 by the introduction of an extra sum (H8) which is due to the fact that the square of any element of Hequals 1 /2. Therefore, one can simplify Eq. (H7) to (H9) As already discussed, this entirely relies on U[j0+1] being horizontally diagonal. We conjecture this is true, because for the ﬁrst j0phase MPOs the bottom part k>j0are all formed by horizontally diagonal phase tensors. In addition, we have done numerical simulations for nup to 64 qubits and veriﬁed this is the case. Since the QFT has site tensors approaching a ﬁxed form as nbecomes large, we expect this behavior to hold for larger n. Even if this does not hold, the sites k/negationslash=j0+1 will all form isometric tensors; thus, the zip-up algorithm will not be very unstable in any case. APPENDIX I: SCHMIDT COEFFICIENTS OF INTERMEDIATE MPOS Assume we have exactly contracted the ﬁrst j0MPOs. Using the same techniques as introduced in Sec. II B,w e can obtain the Schmidt coeﬃcients of the current MPO by reducing them to singular values of the operator /Omega1n,j,j0=/summationdisplay x,x/primeexp⎛ ⎝imin(j,j0)/summationdisplay l=1n−j/summationdisplay m=1xlx/prime mπ 2l+m−1⎞ ⎠ ×|xj···x1/angbracketright/angbracketleftxj···x1|⊗| x/prime 1···x/prime n−j/angbracketright/angbracketleftx/prime 1···x/prime n−j|, (I1)and the eigenvalue value equation is 1 2n2min(j,j0)−1/summationdisplay y=0sin(π(x−y)/2j) sin(π(x−y)/2n)vk,y n,j,j0=λk n,j,j0vk,x n,j,j0. (I2) By setting 2 N/primeW=2min(j,j0)−jfor results in Appendix C, we conclude that for max (2,(eπ/4)2min(j,j0)−j)≤k≤ 2min(j,j0)−1, /radicalBig λk n,j,j0≤1√ kexp/parenleftbigg −2k+1 2log/parenleftbigg4k+4 2min(j,j0)−jeπ/parenrightbigg/parenrightbigg . (I3) Forj>j0, the Schmidt coeﬃcients decay even faster than those of Qn.F o r j≤j0, the Schmidt coeﬃcients are iden- tical to those of Qn(which makes

Question 485 multiple-choice

Quantum computing with qudits—quantum systems with d possible states—relies on efficient circuit design due to hardware constraints. Advanced mathematical approaches, including Riemannian geometry and generalized Gell-Mann matrices, are crucial for evaluating and optimizing quantum circuit complexity. Which statement accurately describes the role of Riemannian geometry in optimizing quantum circuits for qudit systems? 1) It provides a probabilistic model for error correction in quantum circuits. 2) It establishes a commutative algebra structure for multi-qudit operations. 3) It enables the formulation of circuit optimization as a minimal geodesic problem, where the shortest path corresponds to the most efficient sequence of quantum gates. 4) It restricts quantum circuit design to use only single-qudit gates. 5) It defines the energy spectrum of qudits through eigenvalue analysis. 6) It eliminates the need for a cost function by ensuring all unitary operations have equivalent complexity. 7) It prescribes a unique Hamiltonian for every unitary transformation without consideration of resource constraints.

✓ Correct Answer:

The correct answer is 3) It enables the formulation of circuit optimization as a minimal geodesic problem, where the shortest path corresponds to the most efficient sequence of quantum gates..

📚 Reference Text:

a qudit state limits the number of quantum gates in thecircuit. Therefore, we need to design more ef ﬁcient algorithms and circuits. A method exists to do a general systematic evaluation of the circuit ef ﬁciency with the mathematical techniques of Riemannian geometry [ 126]. By reforming the quantum circuits designing problems as a geometric problem,we are able to develop new quantum algorithms or to exploringand evaluating the full potential of the quantum computers. Thisevaluation is able to generalized to qutrit systems, where the leastamount of the gates required to synthesize any unitary operationis given [ 100]. To begin with, we assume that the operations done by the quantum circuit can be described by a unitary evolution U derived from the time-dependent Schrödinger equation d U/dt/equals −iHUwith the boundary condition t f,U(tf)/equalsU. The complexity of realizing Ucan be characterized by a cost function F[H(t)]on the Hamiltonian control H(t). This allow us to de ﬁne a Riemannian geometry on the space of unitary operations[122]. Finding the minimal geodesics of this Riemannian geometry is equivalent to ﬁnding the optimal control function H(t)of synthesizing the desired U. Now we transform the problem of calculating a lower bound to the gate number to ﬁnding the minimal geodesic distance between the identity operation Iand U. Instead of Pauli matrices for the qubit representation of the Hamiltonian, the qutrit version of Hamiltonian is expanded in terms of the Gell-Mann matrices. Here we give an explicit form of the Gell-Mannmatrices representation in d-dimension [ 109] which is used for qutrit (where d/equals3) as well as other qudit systems in the later partof the section. Let e jkdenote the d×dmatrix with a one in the (j,k) elements and 0 s elsewhere, a basis can be described as ud jk/equalsejk+ekj,1≤j

Question 486 multiple-choice

Knot theory employs algebraic and combinatorial methods to study knots and links, using polynomials and group theory to distinguish and classify these topological objects. One important tool is the HOMFLYPT polynomial, which generalizes several classical knot invariants. Which invariant is obtained by specializing the HOMFLYPT polynomial to the case k = 2 and is central to both knot theory and quantum computation due to its connection with braid group representations? 1) Alexander polynomial 2) Conway polynomial 3) Kauffman polynomial 4) Whitehead polynomial 5) Bracket polynomial 6) HOMFLYPT polynomial 7) Jones polynomial

✓ Correct Answer:

The correct answer is 7) Jones polynomial.

📚 Reference Text:

crossing, it k eeps track of which string goes above or below the other. Usually, this information is conve yed by leaving a gap in the string of the undercrossing. A classic result in knot theory states that two unoriented links are equivalent if and only if the link projection of one can be tra nsformed into that of the other by a ﬁnite sequence of Reidemeister moves (see e.g. any of the texts listed above). The HOMFLYPT and Jones polynomials Another way of (partially) distinguishing links involves a ssigning an invariant to each link so that links which are equivalent up to isotopy have the same invariant. The HOMFLYPT polynomial [16, 32]H− →L(t,x)∈Z[t±1,x±1] is one such invariant. It is inductively deﬁned by theskein relation tHL+(t,x)−t−1HL−(t,x) =xHL0(t,x) (2.1) (see Figure 2.1) together with the fact that the HOMFLYPT of t he trivial knot (that is, a single unknotted circle) is unity. In this paper, we are conc erned with various one-variable specializations of this invariant. For each integer k≥2 there is a corresponding one-variable 4The Jones Polynomial: Quantum Algorithms and Applications in Quantum Complexity Theory /d63/d63/d63/d63 /d31/d31/d63/d63/d63/d63 /d127/d127/d127/d127/d127/d127/d127/d127/d127/d127/d127/d127 L+/d127/d127/d127/d127 /d127/d127/d127/d127/d127/d127 /d31/d31/d63/d63/d63/d63/d63/d63/d63/d63/d63/d63 L−/d2/d2 /d28/d28 L0 Fig. 2.1. L+,L−,L0denote three oriented links that diﬀer in a small region, as s ymbolized by the above diagrams, but are otherwise the same. We say that th e crossing corresponding to L+is a positive crossing, and the one corresponding to L−is a negative crossing. HOMFLYPT polynomial in Z[q±1/2]: H(k) − →L(q) =H− →L(qk/2,q1/2−q−1/2) (2.2) This is also known as the slkinvariant. Restricting to k= 2 yields the famous Jones polyno- mial: J− →L(q)≡H(2) − →L(q) =H− →L(q,q1/2−q−1/2). If− →Lconsists ofmunlinked copies of the trivial knot, setting t=qk/2andx=q1/2−q−1/2 in (2.1) implies that H(k) − →L(q) =/parenleftbiggqk/2−q−k/2 q1/2−q−1/2/parenrightbiggm−1 . The exponentiated quantity on the right is known as a quantum integer and is easily shown to satisfy qk/2−q−k/2 q1/2−q−1/2=qk−1 2+qk−1 2−1+···+q−k−1 2 (2.3) ifq/ne}ationslash= 1. By continuity, we consider this expression to hold as q→1, recovering the ordinary integerk. In this paper, we will only be interested in evaluations of t hese invariants at primitive roots of unity q=e2πi/ℓ, whereℓis a nonnegative integer. To this end, we deﬁne for each integer kand each nonnegative integer ℓ, the constantc [k]ℓ=qk/2−q−k/2 q1/2−q−1/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle q=e2πi/ℓ=sin(πk/ℓ) sin(π/ℓ). A value which will occur frequently, especially when analyz ing the Jones polynomial, is [2]ℓ=q1/2+q−1/2/vextendsingle/vextendsingle/vextendsingle q=e2πi/ℓ= 2 cos(π/ℓ). (2.4) The Jones polynomial will be the central object of our study, although we will also provide algorithms for approximating the one-variable HOMFLYPT po lynomials. Where convenient, we sometimes write J/parenleftbig− →L,q/parenrightbig ≡J− →L(q). cOur deﬁnition here departs from the usual convention [38, 12 ] which deﬁnes [ k]q=qk/2−q−k/2 q1/2−q−1/2for arbitrary q. This expression evaluates to ours when qis a primitive ℓ’th root of unity, which is the case of interest to us. We warn that other authors (e.g. [19]) use a diﬀerent conv ention for the quantum integer and q. Pawel Wocjan and Jon Yard 5 The braid group The braid group Bnonnstrands is generated by

Question 487 multiple-choice

The Hidden Subgroup Problem (HSP) is a central challenge in both quantum computing and computational group theory, with algorithms designed to identify hidden subgroups within finite groups using oracle functions. The efficiency of these algorithms often depends on the structure of the group and the properties of the hidden subgroup. Which of the following statements correctly describes the asymptotic query complexity of a quantum algorithm for solving the Hidden Subgroup Problem over a finite abelian group G of order n with hidden subgroup H of order m? 1) It is O(n log n) 2) It is O(m^2) 3) It is O(√(n log n)) 4) It is O(log n / log m) 5) It is O(√(n/m)) 6) It is O(n/m) 7) It is O(m/n)

✓ Correct Answer:

The correct answer is 5) It is O(√(n/m)).

📚 Reference Text:

x∈S1and y∈S2, we will ﬁnd f(g−a) = f(b), and can compute h. This is the key idea underlying the algorithm. For sets S1,S2⊆G, deﬁne S−1 1as the set S−1 1:=/braceleftbig x−1:x∈S1/bracerightbig . Following our notation for the product of sets of group elements, S−1 1S2=/braceleftbig x−1y:x∈S1,y∈S2/bracerightbig . Algorithm 1 (Find-Collision) implements the above idea with a geometrically decreasing s equence of guesses for the order of H. We show in Theorem 5.2 that the algorithm is correct and has th e query complexity we seek for a large class of groups. Theorem 5.2. Algorithm 1 (Find-Collision) solves the Hidden Subgroup Pr oblem over any ﬁnite group G. If the order of the group is n and that of the hidden subgroup H i s m, the algorithm has query complexity as stated below. 1. If G is abelian, the algorithm has query complexity O(√ n/m). 2. If G is not abelian, the algorithm has query complexity O(√ nlnn). Further, if G has a sub- group of order n 1such that n /m≤n1≤κn/m for some κ≥1, then the query complexity isO(/radicalbig (κn/m)ln(κn/m)). Proof. Since the algorithm outputs a collision only when it ﬁnds one, it gives the correct answer when the oracle function fis injective. Suppose the function fhides a non-trivial subgroup H, so that m≥2. If the group is abelian, it has a subgroup of order n/m[8, Corollary 2.4, page 77], and n/m≤ n/2. If it does not ﬁnd a collision in an earlier iteration, Algo rithm 1 ﬁnds a collision in an iteration with k=ℓ≥0, where ℓis such that n/m∈/bracketleftbig n/2ℓ+1,n/(2ℓ+1)/bracketrightbig . This is due to the reasoning given after Lemma 5.1. It thus outputs the correct answer. We haveℓ=⌈log2m⌉−1, and initially, k=⌈log2n⌉−1≥0. By Theorem 3.5, we have |S1|,|S2|≤2/radicalbig |G1|in every iteration with queries. 8 Algorithm 1: Find-Collision( G,f) Input : group Gof order n, with n>1; oracle for f:G→Sthat hides a subgroup Output : injective , orcollision a,b∈G 1Letkbe the integer lsuch that n∈(2l, 2l+1]; 2ifG is abelian then k0←0 ; 3else k0←− 1 ; 4while k≥k0do 5 Find, if there is one, a subgroup G1≤Gwith the largest order in/bracketleftbig n/2k+1,n/(⌊2k⌋+1)/bracketrightbig ;/* the expression ⌊2k⌋is required to correctly handle the case k=k0=−1*/ 6 ifsuch a subgroup G 1exists then 7 Find a generating pair S1,S2forG1which minimizes max {|S1|,|S2|}; 8 ifG1=Gthen g←e, the identity element of G; 9 else g←any element in G\G1; 10 LetR←S−1 1∪S2∪/parenleftbig S−1 1{g}/parenrightbig ; 11 Query fat all the elements in R; 12 iff(z) = f(y)for some z ,y∈R such that z/\⌉}atio\slash=ythen return collision z,y; 13 k←k−1 ; 14returninjective So the query complexity of the algorithm is at most ⌈log2n⌉−1 ∑ k=ℓ3·2/radicalbig n/2k≤ 6/radicalbigg n 2ℓ∑ i≥01√ 2i ≤ 12(1+√ 2)√ n/m, asℓ≥log2m−1. The bound on the query complexity when m=1 is the same as that for m=2, as the algorithm executes all the iterations until k=0. Part 1 of the theorem thus follows. Suppose Gis non-abelian. If the algorithm does not ﬁnd a collision in ea rlier iterations, when k=−1, we have G1=G,

Question 488 multiple-choice

In quantum computing, multivalued logic circuits extend binary qubit systems to higher-dimensional qudit systems, such as ternary (q=3) logic. This enables the construction of generalized quantum gates and transforms, including the Chrestenson Transform and the Quantum Fourier Transform, which leverage the mathematical properties of q-th roots of unity. Which of the following statements accurately describes the Chrestenson Transform when applied to a ternary (q=3) quantum system? 1) It uses binary phase shifts and produces only two possible superposition states. 2) It constructs its transformation matrix using third roots of unity, and the n-qudit transform is built via the n-fold Kronecker product of univariate transforms. 3) It reduces to a classical discrete Fourier transform when q=3. 4) It cannot be implemented in parallel and requires sequential application of all gates. 5) It depends solely on real-valued entries in its transformation matrix for q=3. 6) It is identical to the Walsh-Hadamard Transform regardless of the value of q. 7) It requires binary fractional notation for encoding both input and output states.

✓ Correct Answer:

The correct answer is 2) It constructs its transformation matrix using third roots of unity, and the n-qudit transform is built via the n-fold Kronecker product of univariate transforms..

📚 Reference Text:

1a/C3a 1aa/C32 43 5: The gate is unitary, which can be verified by multi- plying it by its Hermitian conjugate. By performing sucha multiplication and applying identities a 3¼1and 1þaþa2¼0, we obtain the identity matrix. Multivalued quantum gates used in 3-valued implemen- tation include a controlled phase shift gate, which applies tothe incoming signal the multiplication by a factor e ði/C11xÞ, where xis the input controlling the amount of shift. Unlike in the binary case, there are three possible amounts by which the signal is phase-shifted, depending on the value of x. 4.2 Quantum Chrestenson Transform The Chrestenson Transform in the q-valued case is given by means of orthogonal expansion: jxi!1ﬃﬃﬃﬃﬃqnpX y2f0;1;/C1/C1/C1;q/C01gne/C02/C25ixy qjyi: The multipliers are equal to the qthcomplex roots of the unity. Please note that, by setting q¼2, we obtain the quantum Walsh-Hadamard Transform as a special case. For practical calculation, the input and output vectors to the Chrestenson Transform are expressed in terms of theirq-ary expansions: x¼X n i¼1xiqi;y¼Xn i¼1yiqi and the decomposable states are jxi¼jx0ijx1ijx2i/C1/C1/C1jxn/C01i andjyi¼jy0ijy1i/C1/C1/C1jyn/C01i. The transform matrix is then the n-fold Kronecker product of univariate transforms: CH n¼CH/C10CH/C10/C1/C1/C1/C10 CH: Further, the quantum Chrestenson Transform can be written in the recursive form. In the case when q¼3and byrecalling that the third root of unity is a¼e/C02/C25i 3, we obtain the following matrix for the n-variable Chrestenson Transform: CH n¼CH n/C01CH n/C01 CH n/C01 CH n/C01aCH n/C01a2CH n/C01 CH n/C01a2CH n/C01aCH n/C012 43 5: Similarly to the Walsh-Hadamard case, the transform is performed in parallel by nChrestenson gates. We note that the implementation primitives have been elaborated uponin [13] in terms of ion-trap quantum devices, where it was shown that the implementation using multiple-valued gates naturally fits the technology. 4.3 QFT Using Ternary Gates We now present the Quantum Fourier Transform over3 nelements implemented using ternary quantum circuits. Analogously to the binary case in Section 3.2.1, the QFTfactorization is conducted as follows [14]: QFT njxi¼1ﬃﬃﬃﬃﬃ 3npX y2f0;1;2gne/C02/C25ixy 3njyi ¼1ﬃﬃﬃﬃﬃ 3npX y2f0;1;2gne/C02/C25i 3nxPn l¼1yl3ljy0y1/C1/C1/C1yn/C01i ¼1ﬃﬃﬃﬃﬃ 3np/C10n/C01 l¼0/C12/C120iþe/C02/C25ix3/C0lj1iþe/C04/C25ix3/C0lj2ihi ¼1ﬃﬃﬃﬃﬃ 3np j0iþe/C02/C25i0:x0j1iþe/C04/C25i0:x0j2i/C0/C1 /C1/C1/C1 /C1/C1/C1 j 0iþe/C02/C25i0:xn/C01/C1/C1/C1x1x0j1iþe/C04/C25i0:xn/C01/C1/C1/C1x1x0j2i/C0/C1 :ð2Þ In the final form, we employ in this case the ternary fractional number notation: 0:xl/C01xl/C02/C1/C1/C1x0¼3/C01xl/C01þ3/C02xl/C02þ/C1/C1/C1þ 3/C0lx0: The actual circuit for computing the transform is derived by realizing each bracket in (2) with ternary quantum gates.Starting with the leftmost bracket, the whole expression Tðx 0Þ¼1ﬃﬃﬃ 3pj0iþe/C02/C25i0:x0j1iþe/C04/C25i0:x0j2i/C0/C1 can be realized by a single linear and unitary gate. By considering all three possible values for jx0>, we obtain exactly the action of the Chrestenson gate and its matrix representation is the same as matrix CH given in Section 4.1. In the final step of deriving the transform, we expand x in terms of its ternary encoding. Considering the term with3 lin the denominator, we express expð/C02/C25ix= 3lÞfrom (2): exp/C02/C25i3n/C01xn/C01þ3n/C01xn/C01þ/C1/C1/C1 3lxlþ/C1/C1/C1þ 3x1þx0/C0/C1 3l/C18/C19 ¼expð/C02/C25iKÞexp/C02/C25ixl/C01 3/C18/C19 /C1/C1/C1exp/C02/C25ix 1 3l/C01/C18/C19 exp/C02/C25ix 0 3l/C18/C19 : Since Kin this expression is an integer, the first term is always equal to 1. Hence, xland higher terms in the ternary expansion of xare omitted.ZILIC AND RADECKA: SCALING AND BETTER APPROXIMATING QUANTUM FOURIER

Question 489 multiple-choice

Quantum circuit compilation is a critical process in adapting algorithms to the practical limitations of current quantum hardware, particularly in the NISQ era. Variational quantum circuits (VQCs) and quantum state tomography play important roles in both error mitigation and state verification. Which of the following best describes the primary advantage of substituting a segment of a quantum circuit with a trained variational quantum circuit (VQC) in the context of NISQ hardware? 1) It eliminates the need for classical simulation entirely. 2) It guarantees perfect fidelity regardless of hardware errors. 3) It increases the number of quantum gates required for computation. 4) It prevents the need for quantum state tomography during verification. 5) It automatically adapts to any quantum hardware without further configuration. 6) It enables robust execution by approximating complex operations with hardware-friendly, trainable circuits. 7) It ensures that all entanglement structures in the original circuit are preserved without modification.

✓ Correct Answer:

The correct answer is 6) It enables robust execution by approximating complex operations with hardware-friendly, trainable circuits..

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with further calculations. With this in mind, we envision our project as a compiler. The compiler’s role would involve initially simulating the probability distribution, training the VQC, and subsequently substituting the original circuit with the trained VQC before the circuit is sent for execution. It’s important to note that this might require additional steps, such as comprehensive quantum state tomography. From the insights gained in this project, we have concluded that the linear entangler is a favored choice for NISQ hardware in our particular case. However, this conclusion could be subject to revision when the problem’s complexity increases and demands greater expressibility than linear entanglers can provide. In other words, if we intend to replace a different part of the circuit, as opposed to the QPE, we may need a different VQC setting to achieve optimal performance. The systematic analysis of the target circuit and the selection of an appropriate circuit represent future research directions in the context of this “compiler” approach, aimed at reducing errors in quantum computation. CODE AVAILABILITY We provide our source codes for project LearnQPE on GitHub: https://github.com/Abeeekoala/LearnQPE. ACKNOWLEDGMENT C.Y .L. and C. H. A. L. would like to express their gratitude to Xanadu for hosting QHack2023. This work is an exten- sion of their QHack2023 winning project, LearnQPE. This research received support from the IBM Quantum Researchers Program. K.C. is grateful for the financial support from the Turing Scheme for the Imperial Global Fellows Fund. REFERENCES [1] A. Steane, “Quantum computing,” Reports on Progress in Physics , vol. 61, no. 2, pp. 117, Feb. 1998. [2] J. Preskill, “Quantum Computing in the NISQ Era and Beyond,” Quantum , vol. 2, pp. 79, Aug. 2018. [3] F. Leymann and J. Barzen, “The bitter truth about gate-based quantum algorithms in the NISQ era,” Quantum Science and Technology , vol. 5, no. 4, pp. 044007, Sep. 2020. [4] M. Prest and K.-C Chen, “Quantum-Error-Mitigated Detectable Byzan- tine Agreement with Dynamical Decoupling for Distributed Quantum Computing,” arXiv , arXiv:2311.03097, Nov. 2023. [5] M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information,” Cambridge University Press , Dec. 2000. [6] B. Duan, J. Yuan, C.H. Yu, J. Huang, C.Y . Hsieh, “A survey on HHL algorithm: From theory to application in quantum machine learning,” Physics Letters A , vol. 384, no. 24, p. 126595, 2020 [7] J. Biamonte et al., “Quantum machine learning,” Nature , vol. 549, no. 7671, pp. 195–202, 2017. [8] M. Schuld, I. Sinayskiy, and F. Petruccione, “An introduction to quantum machine learning,” Contemporary Physics , vol. 56, no. 2, pp. 172–185, 2015. [9] C. Ciliberto et al., “Quantum machine learning: a classical perspective,” Proc. R. Soc. A , vol. 474, no. 2209, pp. 20170551, 2018 [10] P.-L. Dallaire-Demers and N. Killoran, “Quantum generative adversarial networks,” Phys. Rev. A , vol. 98, no. 1, pp. 012324, Jul. 2018. [11] S. Lloyd and C. Weedbrook, “Quantum Generative Adversarial Learn- ing,” Phys. Rev. Lett. , vol. 121, no. 4, pp. 040502, Jul. 2018. [12] J. Tian et

Question 490 multiple-choice

In quantum computational approaches to knot theory, representations of algebras such as the Hecke and Temperley-Lieb algebras are mapped to quantum circuits using combinatorial data from Young tableaux and diagrams. These representations are crucial for efficiently approximating knot invariants and understanding the structure of quantum algorithms for topological problems. In the Jones-Wenzl representation π(k,ℓ)_n, which of the following statements correctly describes how the generator π(k,ℓ)_n(e_i) acts on a basis vector |t⟩ labeled by a tableau t in T(k,ℓ)_n? 1) It always acts as the zero operator regardless of the positions of i and i+1 in the tableau. 2) It acts as multiplication by q if i and i+1 are in the same row. 3) It swaps |t⟩ and |s_i(t)⟩ for any tableau t in T(k,ℓ)_n. 4) It acts as the identity operator if i and i+1 are in the same row, and as zero otherwise. 5) It acts as the identity operator if i and i+1 are in the same column, and as zero otherwise. 6) It projects onto the sum |t⟩ + |s_{i+1}(t)⟩ for all tableaux. 7) It acts nontrivially only when i and i+1 occupy the corner boxes of the tableau.

✓ Correct Answer:

The correct answer is 5) It acts as the identity operator if i and i+1 are in the same column, and as zero otherwise..

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For eachi,π(k,ℓ) n(ei) is a projection which is block diagonal in the T(k,ℓ) nbasis with blocks of size either one or two. The size two blocks correspond to dist inct pairs of tableaux tandsi(t) which are contained in T(k,ℓ) n. LettingVi,t= span/braceleftbig |t/an}bracketri}ht,|si(t)/an}bracketri}ht}, the restriction of π(k,ℓ) n(ei) to Vi,tis the rank one projection onto/radicalBig aℓ/parenleftbig di(t)/parenrightbig |t/an}bracketri}ht+/radicalBig aℓ/parenleftbig −di(t)/parenrightbig |si(t)/an}bracketri}ht, written π(k,ℓ) n(ei)/vextendsingle/vextendsingle Vi,t= aℓ/parenleftbig di(t)/parenrightbig/radicalBig aℓ/parenleftbig di(t)/parenrightbig aℓ/parenleftbig −di(t)/parenrightbig /radicalBig aℓ/parenleftbig di(t)/parenrightbig aℓ/parenleftbig −di(t)/parenrightbig aℓ/parenleftbig −di(t)/parenrightbig . (3.13) For the remaining t∈T(k,ℓ) n, setπ(k,ℓ) n(ei)|t/an}bracketri}ht=|t/an}bracketri}htifiandi+ 1 are in the same column of t; otherwise set π(k,ℓ) n(ei)|t/an}bracketri}ht= 0. This deﬁnes the action of π(k,ℓ) n(ei) on all ofV(k,ℓ) n. Using (3.7), this representation maps each gito the unitary matrix π(k,ℓ) n(gi) =q1V(k,ℓ) n−(1 +q)π(k,ℓ) n(ei). (3.14) Abbreviating d=di(t), we may use (2.3) to express the restriction to each Vi,tas π(k,ℓ) n(gi)/vextendsingle/vextendsingle Vi,t=−q1/2 qd/2 [d]ℓ/radicalBig 1−1 [d]2 ℓ /radicalBig 1−1 [d]2 ℓ−qd/2 [d]ℓ . (3.15) Ifiandi+ 1 are in the same column of t, this gives π(k,ℓ) n(gi)|t/an}bracketri}ht=−|t/an}bracketri}ht, while if they are in the same row we have π(k,ℓ) n(gi)|t/an}bracketri}ht=q|t/an}bracketri}ht. Because permutations cannot change the shapes of tableaux, it follows that this representation is reducib le. The irreducible components are labeled by the ( k,ℓ)-diagrams, so that π(k,ℓ) n=/circleplusdisplay λ∈Λ(k,ℓ) nπ(k,ℓ) λ, where each π(k,ℓ) λacts nondegenerately only on the subspace V(k,ℓ) λ=/braceleftbig |t/an}bracketri}ht:t∈T(k,ℓ) λ}, and V(k,ℓ) n=/circleplusdisplay λ∈Λ(k,ℓ) nV(k,ℓ) λ. 12The Jones Polynomial: Quantum Algorithms and Applications in Quantum Complexity Theory 4 Eﬃcient quantum circuits for unitary Jones-Wenzl represe ntations We will now show, for any integers 1 < k < ℓ , how to implement the corresponding Jones- Wenzl representations π(k,ℓ) ndeﬁned in (3.14) on a quantum computer. The basic idea is to embed these representations into a tensor product space (c. f. Section 5.2 of [33]). Our circuits resemble those of [2], where the k= 2 case is treated, although we require the combinatorial tools of Young diagrams and tableaux introduced in the previ ous section for higher values of k. These circuits allow us to give a new quantum algorithm in Se ction 6 for approximating the family of one-variable HOMFLYPT polynomials H(k)of the trace closures of braids. We begin with some background on quantum circuits. The state space of a quantum computer has the structure of a Hi lbert space H. Denoting byU(H) the group of unitary matrices acting on H, aquantum gate is anyU∈U(H). By thestandard quantum circuit model , we shall mean quantum computers whose state space decomposes into a ﬁnite number of localized two-dimensiona l subsystems, or qubits, so that H= (C2)⊗n. We ﬁx a preferred orthonormal basis for the Hilbert space of each qubit, which we call the computational basis : |0/an}bracketri}ht ≡/parenleftbigg1 0/parenrightbigg |1/an}bracketri}ht ≡/parenleftbigg0 1/parenrightbigg . The state space of His thus spanned by tensor products of these basis vectors, wh ich we write as/braceleftbig |xn/an}bracketri}ht:xn∈ {0,1}n/bracerightbig , where |xn/an}bracketri}ht ≡ |x1/an}bracketri}ht|x2/an}bracketri}ht···|xn/an}bracketri}ht ≡ |x1/an}bracketri}ht ⊗ |x2/an}bracketri}ht ⊗ ··· ⊗ |xn/an}bracketri}ht. We will abbreviate the set of all gates

Question 491 multiple-choice

In quantum algorithms for the Hidden Subgroup Problem (HSP), the design of optimal measurements plays a crucial role in distinguishing hidden subgroups from quantum states. The symmetry properties of the group and its representations can strongly influence the effectiveness of these measurement strategies. Which property of a group and its subgroups ensures that the Pretty Good Measurement (PGM) is optimal for the Hidden Subgroup Problem in multi-copy cases? 1) The group is non-abelian and has a trivial center 2) The subgroups are all normal in the parent group 3) The group is cyclic of prime order 4) The hidden subgroups form Gel’fand pairs with the parent group 5) The group admits only one irreducible representation 6) The subgroups are all maximal subgroups 7) The group is simple and non-solvable

✓ Correct Answer:

The correct answer is 4) The hidden subgroups form Gel’fand pairs with the parent group.

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probability the PGM is optimal. Further Moore and Russell showed that the PGM is optimal f or the multi-copy case when all of the hidden subgroups form Gel’fand pairs with the parent group. It is easy to see that for the abelian HSP, the optimal measurement is not always the PGM. For instance, suppose that the group is G=Z2and the hidden subgroup is either the entire group, {e,r}or the trivial subgroup, {e}with equal probability. Then the hidden subgroup states are simply ρ0=1 2(|e/an}bracketri}ht+|r/an}bracketri}ht)(/an}bracketle{te|+/an}bracketle{tr|), ρ1=1 2(|e/an}bracketri}ht/an}bracketle{te|+|r/an}bracketri}ht/an}bracketle{tr|) (10) Deﬁning |±/an}bracketri}ht:=1√ 2(|e/an}bracketri}ht±|r/an}bracketri}ht), we can express these as ρ0=|+/an}bracketri}ht/an}bracketle{t+|andρ1=1 2(|+/an}bracketri}ht/an}bracketle{t+|+|−/an}bracketri}ht/an}bracketle{t−|). It is then easy to calculate that the PGM has measurement operators E0=2 3|+/an}bracketri}ht/an}bracketle{t+|andE1=1 3|+/an}bracketri}ht/an}bracketle{t+|+|−/an}bracketri}ht/an}bracketle{t−|. (11) The probability of successfully identifying the subgroup is then Psucc=1 2Tr[E0ρ0+E1ρ1] =2 3. However, if we had chosen the optimal measurements F0=|+/an}bracketri}ht/an}bracketle{t+|andF1=|−/an}bracketri}ht/an}bracketle{t−|, then this succeeds with probability3 4. We therefore see that the PGM is not optimal. We can thus conclude that while the PG M is optimal for a plethora of HSPs, it is not always so. Therefore, an important unsolved question in the o ptimal measurement approach to the HSP is to determine the optimal measurement for the HSP beyond the cases where the PGM is optimal or Ip’s measurement is optimal (see [23] and [24] for the near optimality of the PGM). In this paper we derive the optimal single copy measurement for the HSP when the subgroups are chosen from all possible subgroups with equal probability. We will show that this meas urement is a hybrid between the PGM and the measurement procedure elucidated by Ip. III. OPTIMAL SINGLE COPY HIDDEN SUBGROUP PROBLEM MEASUREME NT We begin by reviewing properties of hidden subgroup states and the n turn to identifying the optimal measurement. We ﬁnd the optimal measurement by choosing a measurement ansat z that is a hybrid between the PGM and the measurement described by Ip [15]. A. Properties of the Hidden Subgroup State The hidden subgroup state, given by Eq. (2) above, has many symm etries and properties that we can use for constructing the optimal measurement. Here we review these pro perties, and the reader is referred to older papers like [25] where these properties are described in more detail. 4 First, deﬁne the left and right regular representations of G. The left regular representation is the representation ofGgiven by DL(g′)|g/an}bracketri}ht=|g′g/an}bracketri}ht. Similarly the right regular representation is deﬁned by DR(g′)|g/an}bracketri}ht=|g(g′)−1/an}bracketri}ht. The hidden subgroup state can, because it is symmetric under the left r egular representation, DL(g)ρHDL(g−1) =ρH, be written as a sum over right regular representation elements. In fa ct it can be written in the particularly simple form ρH=1 |G|/summationdisplay h∈HDR(h). (12) If we deﬁne the subgroup projection operator PH:=1 |H|/summationdisplay h∈HDR(h), (13) then we can express the hidden subgroup state as ρH=|H| |G|PH. The reason for doing this is to notice that PHis a projector, P2 H=PHand to further note the role of this projector in the construction of the optimal measurements like those constructed by Ip [15]. While

Question 492 multiple-choice

In computational group theory and additive combinatorics, the Davenport constant and algorithmic solutions to zero-sum problems in finite abelian groups have important implications for computational complexity and cryptography. Efficient algorithms for zero-sum subsequence detection are especially significant in the context of elementary abelian p-groups. What is the exact value of the Davenport constant for the elementary abelian p-group Zn_p, where n is a positive integer and p is prime? 1) n(p + 1) 2) 1 + n(p − 1) 3) n^2p 4) n(p − 1) 5) 2n(p − 1) 6) p^n − n + 1 7) np − 1

✓ Correct Answer:

The correct answer is 2) 1 + n(p − 1).

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subgroup: One can also apply twists with certain nice automorphisms of the group that do not change the hidden subgroup too much. Unfortunately, such automorphisms do not exist in general nilpotent groups of class greater than two. The methods of [ 20,21] offer efﬁcient solution to the HSP in certain nilpotent groups of higher class, again with potentially large prime factors in their orders. Thesegroups have a normal subgroup with an abelian factor group of restricted kind (e.g., cyclic). These methods as well as that of [ 17] are of highly non-exact nature. Probably, the technique of [ 20] can be made exact with some efforts. The Davenport constant S (A)of a ﬁnite abelian group Ais the smallest number s such that any sequence of selements of Acontains a non-empty subsequence adding up to the zero element of A. The name comes from that H. Davenport proposed 123 Zero sum subsequences and hidden subgroups Page 5 of 23 14 determining S(A)in the case when Ais the ideal class group of a number ﬁeld as a measure for non-uniqueness of factorization of the integers of the ﬁeld. The general problem has become a famous question of additive combinatorics. Olson[22] determined the exact value of the Davenport constant of p-groups; in particular for Z n pit is 1 +n(p−1). What we are looking for is an “effective” Davenport constant: What is the smallest number S/prime=SB(A)such that from any sequence ofS/primeelements of A, algorithm Bﬁnds a non-empty zero sum subsequence in time polynomial in S/primelog|A|(roughly this is the bit size of the input sequence). In this paper we give a deterministic algorithm Brunning in time poly (n), that, for p= O(1), given a sequence of SB(Zn p)=poly(n)vectors from Zn p, returns a zero sum subsequence. Note that when we consider Zpas a ﬁeld and Zn pas an n-dimensional vector space then ﬁnding a zero sum subsequence is equivalent to represent zeroas a non-trivial linear combination of the given vectors with coefﬁcients 0 and 1 only. Alternatively, it is equivalent to ﬁnding a non-trivial solution of a system of nhomogeneous linear congruences modulo pinS B(Zn p)variables. Actually, our algorithm is based on elementary linear algebra: We ﬁnd many linear dependencies in subsets of the given vectors and combine these relations in an iteration so that the large coefﬁcients eventually cancel out. The structure of the rest of the paper is the following. In Sect. 2,w eg i v es o m e background material on exact quantum procedures, on nilpotent black-box groups andon computations with them, on (hidden) subgroups states and their puriﬁcations and present methods to convert subgroup states in the entire group to those in subgroups and—in certain very easy cases—in factor groups. In Proposition 1, the existence of an exact polynomial time reduction from the HSP in general nilpotent groups to the problem HSMC in semi-elementary groups is proved in Sect. 3. Section 4is devoted to converting several copies of a subgroup state in a semi-elementary group to acopy of a

Question 493 multiple-choice

In finite group theory and quantum algorithms, the Hidden Subgroup Problem (HSP) involves finding a hidden subgroup of a group using character functions and sampling techniques. Algebraic methods are often employed to convert subgroup conditions into polynomial equations over finite fields. When addressing the nonzero inner product condition y·u ≠ 0 in Zn_p, which transformation enables the conversion of this condition into a set of linear equations over Zp, making the problem solvable in polynomial time for fixed prime p? 1) Mapping y·u into the kernel of the group homomorphism 2) Expressing y·u as a sum of cyclic permutations 3) Raising y·u ≠ 0 to the (p−1)-th power and linearizing in the (p−1)-th symmetric power of Zn_p 4) Encoding y·u as a system of quadratic equations in Zn_p 5) Applying the discrete Fourier transform to y·u 6) Projecting y·u onto the subgroup generated by u 7) Representing y·u using the exterior algebra of Zn_p

✓ Correct Answer:

The correct answer is 3) Raising y·u ≠ 0 to the (p−1)-th power and linearizing in the (p−1)-th symmetric power of Zn_p.

📚 Reference Text:

whence the statement follows as χy(u) = 1 if and only if y∈u⊥and/summationtext y∈G|1−χy(u)|2= 2|G|−2/summationtext y∈Gχy(u)=2|G|. WhenG=Zn p, the value χy(u)=e2πi py·udepends only on the inner product y·u overZp,a n dy∈u⊥exactly when y·u= 0. Therefore every ( y,1) generated satisﬁes y·u/negationslash= 0. Thus the output distribution is diﬀerent from the usual one obtained for the abelian Hidden Subgroup where only vectors orthogonal to the hidden subgroup are generated. We overcome the main obstacle, which is that we do not know the actual value of the inner product y·u, by raising these inequations to the power (p−1). They become a system of polynomial equations of degree at most ( p−1) sinceap−1= 1 for every nonzero a∈Zp. In general, solving systems of polynomial equations over any ﬁnite ﬁeld is NP-complete. But using the other special feature of our distribution, which is that the probability of sampling ( y,1) depends only on the inner product y·u, we are able to show that—for ﬁxed prime p—after a polynomial number of samplings, our system of equations has a unique solution with constant probability, and the solution can be found in deterministic polynomial time. To solve our system of polynomial equations, we linearize it in the ( p−1)th sym- metric power of Zn p. We think of Zn pas ann-dimensional vector space over Zp.F o r 10 FRIEDL, IVANYOS, MAGNIEZ, SANTHA, AND SEN ap r i m en u m b e r pa n da ni n t e g e r k≥0, letZ(k) p[x1,...,x n]b et h e kth symmetric power of Zn pwhich will be thought of as the vector space, over the ﬁnite ﬁeld Zp, of homogeneous polynomials of degree kin variables x1,...,x n. The monomials of degree(p−1)forma basis of Z(p−1) p[x1,...,x n], whose dimension is therefore/parenleftbign+p−2 p−1/parenrightbig , which is polynomial in nwhenpis constant. Z(1) p[x1,...,x n] is isomorphic to Zn pas a vector space. For two vectors Y1,Y2∈Z(p−1) p[x1,...,x n], we denote their standard inner product over the monomial basis by Y1·Y2. For every y=(a1,...,a n)∈Zn pand positive integer k, we deﬁne y(k)∈ Z(k) p[x1,...,x n] as the polynomial (/summationtextn j=1ajxj)k.F o r y=(a1,...,a n),z= (b1,...,b n)i nZn p, and positive integers k,l, we deﬁne the product y(k)z(l)∈ Z(k+l) p[x1,...,x n] as the polynomial (/summationtextn i=1aixi)k(/summationtextn j=1bjxj)l.N o wo b s e r v et h a ti f u=(u1,...,u n) is the hidden translationvector, then the vector u∗∈Z(k) p[x1,...,x n] which for every monomial xe1 1···xennhas coordinate ue1 1···uennsatisﬁesy(p−1)·u∗= (y·u)p−1. Therefore each linear inequation y·u/negationslash=0o v e r Zn pwill be trans- formed into the linear equation y(p−1)·U=1o v e r Z(p−1) p[x1,...,x n], where Uis ad i mZ(p−1) p[x1,...,x n]-sized vector of unknowns. We will see below that the vectors y(p−1)span the space Z(p−1) p[x1,...,x n]w h e n yranges over Zn p. Moreover, in what is the main part of our proof, we show in Lemma3.4thatwheneverthespanof y(p−1)forthesamples yisnotZ(p−1) p[x1,...,x n], our sampling process furnishes with probability at least 1 /pa vector z∈Zn psuch thatz(p−1)is linearly independent from

Question 494 multiple-choice

In computational algebraic geometry and representation theory, algebraic structures such as ideals, Gröbner bases, and braid group representations are essential for understanding the symmetries and computational properties of quantum systems and polynomial rings. Tools like monomial orderings and ideal quotients play a significant role in constructing and analyzing these structures. Which construction method allows general polynomial rings to be formed from commutative rings by using ideal quotients and Gröbner basis techniques, and also implies the existence of toric deformations of projective varieties over finite fields? 1) Clifford algebra representations and group cohomology 2) Direct sums of primitive spectra and braid group modules 3) Construction of order domains with Gröbner bases and ideal quotients 4) Tensor products of maximal ideals and monomial lattices 5) Localization at regular cones and lexicographic orderings 6) Intersection of anti-particle modules and unfaithful representations 7) Homological algebra with fundamental group actions

✓ Correct Answer:

The correct answer is 3) Construction of order domains with Gröbner bases and ideal quotients.

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between the ideal Iand (x1) is given by I ∩(x1) =tI+(1−t)(x1)∩C[x1,...,xn] (32) wheret∈C[11]. Therefore, although we cannot generally ﬁnd the constructio n of the anti-particle representation of this most general many-body qua ntum system at ﬁnite tem- perature, without considerable eﬀort, we can easily identify that t he basis functions which 11 have notin them will generate I ∩(x1), and in fact these elements form an exact Gr¨ obner basis, as will now show. The key point is that although usually we want t he representa- tion of the braid group to be formed via a known algebra X(such as the Cliﬀord algebra) and its twist Y(for a faithful representation of the fundamental group of the quantum sys- tem [8]), in (25), what we can see from the above is that it is more natu ral to braid the intersection between the primitive and maximal spectra of X, andX, as deﬁned via the quotient ideal ( I:J) in (31), which in this case we would deﬁne via the speciﬁc ideal in (11). This generalises the notion of the anyon in (3) to an object wh ich can be used to form an exact representation of the Braid group of a general man y-body quantum system at ﬁnite-temperatures, even when the unitary representation of t he fundamental group of the system deﬁned in (3) is unfaithful. V. GR ¨OBNER BASES The simple properties of ideal quotients, which closely follow of those of complete regular fans we have discussed previously, are that for Iand{Ik}1≤k≤rbeing ideals in C[x1,...,xn], andfa polynomial in C[x1,...xn], it follows that /parenleftBiggr/intersectiondisplay k=1Ik/parenrightBigg :I=r/intersectiondisplay k=1(Ik:I),I:/parenleftBiggr/summationdisplay k=1Ik/parenrightBigg =r/intersectiondisplay k=1(I:Ik) (33) Hence the simple prescription given in [11] for constructing a basis fo r an ideal quotient is; calculate /an}bracketle{tg1,...gp/an}bracketri}ht=I ∩/an}bracketle{tf/an}bracketri}ht, followed by /an}bracketle{tg1/f,...g p/f/an}bracketri}ht=I:f, then ﬁnally/intersectiontextr k=1(I:gp). The further subtlety which makes this construction a Gr¨ obner ba sis is the ordering of the monomials xα, speciﬁcally, identifying the largest monomial of fwith respect to theordering which is denoted LM(f,≺). The deﬁnition of a Gr¨ obner basis that follows [11] is a set of polynomials Gwherethereexists some g∈GsuchthatLM(g,≺) dividesLM(f,≺). Wecan then make the connection between the weight associated with each cone within a complete regular fan (the aiin (21)), and the support associated with each monomial order, fo llowing [22], by deﬁning order domains which are a class of commutative ring. F or this deﬁnition, the monomial ordering is chosen as follows; xα≻M,τxβifMα≻Mβ, orxα≻τxβif Mα=Mβ, whereMis anr×nmatrix with entries in Z≥0with linearly independent rows. It follows the order domain, deﬁned as Fq[x1,...,xs]/I, can be constructed from a general projective variety deﬁned over a ﬁnite ﬁeld (with poles at only one sm oothFqrational 12 point), and moreover, that the Gr¨ obner bases imply the existenc e of the toric deformations of this general variety [24]. However, the important question for us, as separate from the construction of order domains, is if we can construct a general po lynomial ring from these commutative rings, and from (33) the answer quite clearly is

Question 495 multiple-choice

In algebraic combinatorics and group theory, the study of CI-groups and Schur rings involves examining how group automorphisms interact with combinatorial structures such as Cayley graphs and subset constructions. Understanding the criteria for CI-groups is essential for classifying symmetries and isomorphisms in finite groups. Which of the following most accurately describes the obstruction that prevents W×V from being a CI-group when an AH-subset S is constructed such that (w, H(w)) lies in the Schur ring generated by S for all w in E? 1) The existence of a nontrivial center in W×V that fixes all Cayley graphs. 2) The failure of S to generate the full group ring over W×V. 3) The non-existence of a subgroup of W×V containing every connection set as a coset. 4) The inability to express every automorphism of Cayley graphs as an inner automorphism. 5) The presence of elements in S that are invariant under all automorphisms of W×V. 6) The presence of repeated coefficients in the sums of elements in S, violating Schur-Wielandt principle conditions. 7) The fact that S is an AH-subset for which (w, H(w)) lies in the generated Schur ring for all w in E, making S not a CI-subset and thus W×V not a CI-group.

✓ Correct Answer:

The correct answer is 7) The fact that S is an AH-subset for which (w, H(w)) lies in the generated Schur ring for all w in E, making S not a CI-subset and thus W×V not a CI-group..

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we proved that there exists no linear function /Lambda1such that (f−/Lambda1)(w) ∈H(w) for everyw∈E. Proposition 1 IfSis anAH-subset such that (w,H(w)) ∈/angbracketleft /angbracketleftS/angbracketright/angbracketright for every w∈E, (1) thenSis not a CI-subset of W×V.In particular W×Vis not a CI-group (/angbracketleft/angbracketleftS/angbracketright/angbracketright denotes the Schur ring generated by S). Proof The following argument mimics Proposition 4 in [ 6]. Assume Sis a CI- subset. Since f∈Hom H(W,V ) , it is easy to check that Cay (W×V,T)f= Cay(W×V,Tf)for any simple quantity TinAH(see Proposition 6 in [ 6]). So, since Sis anAH-subset, we have Cay (W×V,Sf)=Cay(W×V,S)f∼=Cay(W×V,S) . Therefore, since Sis a CI-subset, we get Sf=Sgfor some gin Aut (W×V). Then, Cay(W×V,S)f=Cay(W×V,Sf)=Cay(W×V,Sg)=Cay(W×V,S)g. There- forefg−1is an automorphism of Cay (W×V,S) . Now, using Equation (1), the reader can verify (see Theorem 2 .4i n[ 6]) that fg−1is an automorphism of Cay (W×V,(w,H(w))) for every w∈E. Thus (w,H(w))f=(w,H(w))gfor every w∈E. Hence (w,H(w)) is a CI-subset for everyw∈Eand [(w,H(w)) ]w∈E∼=Cay[(w,H(w))f]w∈E. Now, Proposition 7 in [ 6] yields that there exists /Lambda1∈Hom(W,V ) such that (f− /Lambda1)(w) ∈H(w) for every w∈E, a contradiction. This proves that Sis not a CI-subset of W×V. Therefore W×Vis not a CI- group. /square Proof of Theorem 2Like in Theorem 1, it is enough to prove that W×Vis not a CI-group (see the ﬁrst paragraph of Sect. 5 .1i n[ 5]). Let Sbe theAH-subset /parenleftbigg/uniondisplay i∈X(0,εi)/parenrightbigg/uniondisplay/parenleftbigg/uniondisplay i,j∈X,i/negationslash=j(ei+ej,H(e i+ej))/parenrightbigg/uniondisplay (en,H(e n)) /uniondisplay (e−en,H(e−en))/uniondisplay/parenleftbigg/uniondisplay 1≤j≤p−1(je2j,H(e 2j))/parenrightbigg/uniondisplay (e,H(e)) /uniondisplay/parenleftbigg/uniondisplay 1≤j≤p−1(je2j−1,H(e 2j−1)+{ ±ε})/parenrightbigg . J Algebr Comb (2007) 26: 343–355 353 By Proposition 1, it remains to prove that Equation (1)holds for the set S. Let us denote by Athe Schur ring /angbracketleft/angbracketleftS/angbracketright/angbracketright. Case p ≥5.The element (S+S)◦Slies inAand it can be written as /summationdisplay (a,b)∈Sλ(a,b)(a,b) , where λ(a,b) is the number of solutions of the equation (‡)( u , x ) +(v,y) =(a,b) with(u,x),(v,y) ∈S. It can be easily shown that (S+S)◦S=0T1/unionmulti2pn−2T2/unionmulti2(n−2)T3/unionmulti(2(n−2)/unionmulti2pn−2)T4 /unionmulti(2(n−2)/unionmulti4pn−2)T5/unionmulti2(n−1)T6/unionmulti(2(n−1)/unionmulti6pn−2)T7 where T1=/unionmultidisplay i∈X(0,εi)/unionmulti(e−en,H(e−en)), T2=(e,H(e)) , T3=/unionmultidisplay i

Question 496 multiple-choice

Quantum computing leverages linear algebraic structures such as Kronecker products, diagonal matrices, and projection operators to design efficient implementations of quantum algorithms, including the quantum Fourier transform (QFT). Understanding how these constructs enable computational advantages is central to the development of scalable quantum circuits. Which mathematical property primarily enables the efficient cancellation of mixed terms in the Kronecker product decomposition used for implementing the quantum Fourier transform in tensor product spaces? 1) Non-singularity of diagonal matrices 2) Commutativity of matrix multiplication 3) Orthogonality of tensor products 4) Nilpotency of projection matrices 5) Idempotency of diagonal operators 6) Associativity of the Kronecker product 7) Orthogonality of projection matrices (e.g., $E_1E_2 = 0$)

✓ Correct Answer:

The correct answer is 7) Orthogonality of projection matrices (e.g., $E_1E_2 = 0$).

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=ωj 2n+1=ωj12n−1 2n+1ωj22n−2 2n+1···ωjn−121 2n+1ωjn20 2n+1, (13) forj∈N: 0≤j≤2n−1. We can rewrite (13) as Ωn(j,j) =R2n−1 n+1(j1,j1)R2n−2 n+1(j2,j2)···R21 n+1(jn−1,jn−1)R20 n+1(jn,jn). (14) It follow from Lemma 3 that (14) simpliﬁes to Ωn(j,j) =R2(j1,j1)R3(j2,j2)···Rn(jn−1,jn−1)Rn+1(jn,jn), (15) such that (12) follows from Lemma 2, which extends trivially to diagonal matrices. Our objective is to decompose Ik⊕Ωkinto a product of kmatrices, each of which can be written as the sum of Kronecker products of 2 ×2 matrices. The follow theorem and its proof shows how this ca n be done. Theorem 3. The diagonal operators Ik⊕Ωk∈C2k+1×2k+1from Theorem 2 admit the decomposition Ik⊕Ωk=k/productdisplay i=1/bracketleftBig E1⊗Ii−1⊗I2⊗Ik−i+E2⊗Ii−1⊗Ri+1⊗Ik−i/bracketrightBig , (16) whereRiis deﬁned in Deﬁnition 9. Proof.We start by splitting the diagonal matrix Ik⊕Ωkinto 2 terms using (4), which yields Ik⊕Ωk=E1⊗Ik+E2⊗Ωk. (17) Next, we rewrite these terms in a redundant form based on the m ixed product identity of Kronecker products (2). Using the property E1=E2 1, we get for the ﬁrst term E1⊗Ik=E1⊗I2⊗···⊗I2, = (E1⊗I2⊗I2⊗···⊗I2)(E1⊗I2⊗I2⊗···⊗I2)···(E1⊗I2⊗I2⊗···⊗I2), =k/productdisplay i=1E1⊗Ii−1⊗I2⊗Ik−i, and using Lemma 4 and the property E2=E2 2, the second term results in E2⊗Ωk=E2⊗R2⊗···⊗Rk+1, = (E2⊗R2⊗I2⊗···⊗I2)(E2⊗I2⊗R3⊗···⊗I2)···(E2⊗I2⊗I2⊗···⊗Rk+1), =k/productdisplay i=1E2⊗Ii−1⊗Ri+1⊗Ik−i. Hence, the sum of Kronecker products (17) is equal to the prod uct of sums, i.e., Ik⊕Ωk=k/productdisplay i=1E1⊗Ii−1⊗I2⊗Ik−i+k/productdisplay i=1E2⊗Ii−1⊗Ri+1⊗Ik−i, =k/productdisplay i=1/bracketleftBig E1⊗Ii−1⊗I2⊗Ik−i+E2⊗Ii−1⊗Ri+1⊗Ik−i/bracketrightBig , because the mixed terms in the latter expression cancel out d ue to the property E1E2= 0. The order of the terms in the Kronecker product expression (1 6) can be changed because of the speciﬁc form of theRi+1matrices. 7 Lemma 5. The Kronecker product matrices in (16)satisfy E1⊗Ii−1⊗I2⊗Ik−i+E2⊗Ii−1⊗Ri+1⊗Ik−i (18) =I2⊗Ii−1⊗E1⊗Ik−i+Ri+1⊗Ii−1⊗E2⊗Ik−i, (19) fori= 1,...,k. Proof.SplittingI2andRi+1in theirE1andE2components, i.e., I2=E1+E2, Ri+1=E1+ω2i+1E2, yields E1⊗Ii−1⊗I2⊗Ik−i+E2⊗Ii−1⊗Ri+1⊗Ik−i=E1⊗Ii−1⊗E1⊗Ik−i+E1⊗Ii−1⊗E2⊗Ik−i+ E2⊗Ii−1⊗E1⊗Ik−i+E2⊗Ii−1⊗ω2i+1E2⊗Ik−i, =E1⊗Ii−1⊗E1⊗Ik−i+E1⊗Ii−1⊗E2⊗Ik−i+ E2⊗Ii−1⊗E1⊗Ik−i+ω2i+1E2⊗Ii−1⊗E2⊗Ik−i, = (E1+E2)⊗Ii−1⊗E1⊗Ik−i+ (E1+ω2i+1E2)⊗Ii−1⊗E2⊗Ik−i, =I2⊗Ii−1⊗E1⊗Ik−i+Ri+1⊗Ii−1⊗E2⊗Ik−i, where we subsequently used the scalar shift and distributiv ity properties (1), and combined the terms with E1, respectively E2, in the (i+1)st position. Observe that (16) splits Ik⊕Ωkinto a product of kterms where each term has the form (18). Lemma 5 allows us to modify each term in the product (16) to an alternative re presentation (19). In what follows (section 5.3), we will see that every single term in the product corresponds to a speciﬁc instance of an elementary operation on a quantum computer and that the combination of Theorem 3 and l emma 5 leads to a new class of algorithms for implementing Ik⊕Ωk. 5 Quantum Fourier transform on a quantum computer We now discuss how the decomposition shown in section 4 enabl es the discrete Fourier transform to be computed eﬃciently on a quantum computer. 5.1 Quantum Fourier transform of rank-1 tensors on a classic al computer The Kronecker product form that appears in (16) allows for a v ery eﬃcient application to a rank-1 tensor of the form xn=x1⊗x2⊗···⊗xn, (20) wherexi∈C2are vectors of length 2. The multiplication of each factor in (16) with (20) takes O(n) operations, whereas multiplying the diagonal matrix In⊕Ωnwith a vector of length 2ntakesO(2n) operations. Therefore, it is conceivable that the decomposition of the DFT matrix de rived in the previous section may yield a more eﬃcient algorithm for performing a discrete Fourier

Question 497 multiple-choice

Quantum algorithms are increasingly being developed for advanced data processing tasks such as interpolation, image analysis, and signal reconstruction. Leveraging quantum transforms and specialized circuits can offer computational advantages over classical approaches, especially for large or complex datasets. Which quantum interpolation method is specifically designed to efficiently handle natural data such as images and audio by adapting the principles of the classical Discrete Cosine Transform? 1) Quantum Wavelet Transform-based interpolation 2) Quantum Hadamard Transform-based interpolation 3) Quantum Phase Estimation-based interpolation 4) Quantum Cosine Transform (QCT)-based interpolation 5) Quantum Singular Value Decomposition-based interpolation 6) Quantum Fourier Transform (QFT)-based interpolation 7) Quantum Grover Search-based interpolation

✓ Correct Answer:

The correct answer is 4) Quantum Cosine Transform (QCT)-based interpolation.

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Title: Efficient quantum interpolation of natural data Year: 2022 Paper ID: 13dce77618ca2798bc112b50ad100fa067382ee4 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/13dce77618ca2798bc112b50ad100fa067382ee4 Abstract: We present eﬃcient methods to interpolate data with a quantum computer that complement uploading techniques and quantum post-processing. The quantum algorithms are supported by the eﬃcient Quantum Fourier Transform (QFT) and classical signal and imaging processing techniques, and open the door of quantum advantage to relevant families of data. We showcase a QFT interpolation method, a Quantum Cosine Transform (QCT) interpolation geared towards natural data, and we improve upon them by utilizing a quantum circuit’s capabilities of processing data in superposition. A novel circuit for the QCT is presented. We demonstrate the methods on probability distributions and quantum encoded images, and discuss the precision of the resulting interpolations.

Question 498 multiple-choice

Quantum algorithms often exploit group structures to solve problems more efficiently, with techniques like Quantum Fourier Sampling (QFS) underpinning breakthroughs such as Shor’s algorithm. However, certain group-theoretic problems remain resistant to these methods, especially in the context of nonabelian groups and post-quantum cryptography. Which statement correctly characterizes the limitations of quantum algorithms based on Fourier sampling for the Hidden Subgroup Problem (HSP) in nonabelian groups like the symmetric group? 1) Weak Fourier sampling reliably distinguishes all subgroups in nonabelian groups, enabling efficient solutions to Graph Isomorphism. 2) Entangled measurements on multiple coset states always improve the subgroup distinguishability for the symmetric group. 3) Quantum Fourier Sampling provides exponential speedups for nonabelian HSPs, including those underlying Graph Isomorphism. 4) The success of Fourier sampling in nonabelian groups matches its effectiveness in abelian groups for cryptographic applications. 5) Naive quantum Fourier sampling approaches are sufficient to solve Graph Isomorphism efficiently using quantum computers. 6) Even strong quantum Fourier sampling fails to distinguish certain subgroups in the symmetric group, limiting the applicability of quantum algorithms to problems like Graph Isomorphism. 7) Quantum algorithms based on QFS universally solve all instances of the Hidden Subgroup Problem, regardless of group structure.

✓ Correct Answer:

The correct answer is 6) Even strong quantum Fourier sampling fails to distinguish certain subgroups in the symmetric group, limiting the applicability of quantum algorithms to problems like Graph Isomorphism..

📚 Reference Text:

they are rarely used in practice because of their comparatively larg e public key (see remark 8.33 in [14]). The discovery of successful quantum attacks on R SA and El Gamal, however, has changed the landscape: as suggested by Ryan [20] and Bernstein et al. [1], if “post-quantum” security guarantees can be made for the McEliece cryptosystem, this may compensate for its comparatively expensive computational demands. Quantum Fourier sampling. Quantum Fourier Sampling (QFS) is the key ingredi- ent in nearly all known efﬁcient quantum algorithms for algebraic problems, includ- ing Shor’s algorithms for factorization and discrete logarithm [22] and Simon’s algo- rithm [24]. Shor’s algorithm relies on quantum Fourier sampling over the cyclic group ZN, while Simon’s algorithm uses quantum Fourier sampling over Zn 2. In general, these algorithms solve instances of the Hidden Subgroup Problem (HSP) over a ﬁnite group G. Given a function fonGwhose level sets are left cosets of some unknown subgroup H

Question 499 multiple-choice

In the study of Lie groups and Lie algebras, the exponential map serves as a bridge connecting infinitesimal symmetries to global group elements. The surjectivity and decomposition properties of this map have important implications for both mathematics and physics. Which statement about the exponential map in matrix Lie groups is correct? 1) For any matrix Lie group, the exponential map is always surjective onto the group. 2) In matrix Lie groups with infinitely many disconnected components, every element is a single exponential of a Lie algebra element. 3) For non-compact matrix Lie groups, every element can be written as a single exponential of a Lie algebra element. 4) Any element of a matrix Lie group can be decomposed into a product of at most three exponentials of Lie algebra elements. 5) The exponential map is never surjective for compact, connected matrix Lie groups. 6) Matrix Lie group elements corresponding to reflections cannot be written as products of exponentials. 7) Only simply connected matrix Lie groups allow any element to be written as a product of exponentials.

✓ Correct Answer:

The correct answer is 4) Any element of a matrix Lie group can be decomposed into a product of at most three exponentials of Lie algebra elements..

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preimages of 1and1must be disconnected). But even when Gis connected, we are not guaranteed that the exponential map is surjective. So— when is it surjective? In general, that is a nuanced ques- tion, but the proposition below gives one useful instance where this is so.Proposition 2. IfGis a compact connected Lie group, then the exponential map is surjective. In practice however, this is not too terrible a constraint: whenGis connected, we can still write Lie group elements as products of exponentials of Lie algebra elements, thanks to the next proposition. And if Gconsists of ﬁnitely many disconnected components, one can easily work on the con- nected component and then move to other components by actingviaaﬁnitegroup. Theprototypehereis O(d), which consists ofSO(d)andO(d)SO(d). We can use Lie theory toworkonSO(d),thenextendouranalysisto O(d)SO(d) by multiplying by any map of determinant 1. Let us now consider the following proposition: Proposition 3. LetGbe a connected Lie group and let g2G. Then we can ﬁnd X1;X2;:::;Xm2gsuch that g=eX1eX2:::eXm: Note that while in Hall [51] the result is only stated for matrix Lie groups, a large result in Phillips [61] (which Phillips attributes to Djoković) in fact ensures this holds for any connected Lie group, and mcan be chosen to be m63. An explicit example of this appears when one uses Euler angles to decompose any single qubit unitary into three rotations U=ei 1Xei 2Yei 3Z. Now, for the quantum practioner, we need to know how the correspondence furnished by the exponential map passes through to representations—remember, physical symmetries come from representations of groups, not the groups themselves. First, a deﬁnition. Notice the similar- ity to Lie group representation: Lie group representations are homomorphisms into GL(V), the invertible matrices fromV!V, while Lie algebra representations are homo- morphisms into gl(V), the matrices from V!Vwith the commutator [;]as the Lie bracket. Deﬁnition 7. Let gbe a Lie algebra and Vbe a ﬁnite di- mensional vector space. A representation rofgacting on Vis a mapr:g!gl(V)that is a Lie algebra homomor- phism, a linear map satisfying r([X;Y ]) = [r(X);r(Y)];for allX;Y2g: The dimension of the representation ris deﬁned by dim(r) = dim(V). Every example in this section consists of Lie algebra rep- resentations. All three of these arise from their correspond- ing Lie group representations: the key, perhaps unsurpris- ingly, is the exponential map combined with the tangent space deﬁnition of a Lie algebra. 17 We now present a series of theorems which capture the relationship between Lie group and Lie algebra represen- tations. First, every matrix Lie group representation gives rise to a Lie algebra representation. Theorem 2 (Lie group reps induce Lie algebra reps) .Let Gbe a matrix Lie group with Lie algebra g. IfRis a representation of GonV, then there exists a unique rep- resentation rofgonVgiven by r(X) =d dt R(etX) t=0;for allX2g: We callrthe representation of ginduced by R. This next pair of theorems form a partial converse to the previous theorem.8 Theorem3 (LiealgebrarepslifttosimpleLiegrouprepre- sentations) .LetGbe a simply connected matrix Lie group, and letrbe a representation of the corresponding Lie al-

Question 500 multiple-choice

In computational algebraic number theory, the efficiency of algorithms for computing the unit group and class group of number fields depends critically on growth bounds for class numbers. Quantum algorithms increasingly leverage these structures through reductions to problems like the Hidden Subgroup Problem (HSP). Which statement accurately describes a key conjecture that enables efficient quantum algorithms for computing the unit or class group of real cubic fields? 1) The class number of real cubic fields always equals one for all sufficiently large discriminants. 2) Every real cubic field possesses a cyclic unit group for arbitrary values of m. 3) The automorphism group of real cubic fields is always simple and known for all values of m. 4) The plus part of the class number h+(m) of real cubic fields is bounded by a polynomial in m. 5) Kummer extensions cannot be used for representing periods and elements in totally real fields. 6) Efficient computation of class groups requires the existence of a small dual basis in all Galois fields. 7) Quantum algorithms for the HSP are only efficient for imaginary quadratic fields.

✓ Correct Answer:

The correct answer is 4) The plus part of the class number h+(m) of real cubic fields is bounded by a polynomial in m..

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to compute. A recent work compiled existing conjec- tures and pushed further the numerical computations, so one can give a precise form to the folklore conjecture. Conjecture 1 (Assumption 2 in [18]). For all integers m, h+(m)≤poly(m) for a ﬁxed polynomial poly. 18 Razvan Barbulescu and Adrien Poulalion Consequence of the conjecture : class number in real cubic ﬁe lds without quan- tum computers Marie-Nicole Gras [26] proposed an algorithm to compute cla ss numbers of cyclic cubic ﬁelds in polynomial time with respec t to a bound on h. Ifmis the conductor of a cyclic cubic ﬁeld, then Conjecture 1 imp lies thathis polynomial in m. Note also that Schoof [47] proposed an algorithm for Q(ζm)+ with prime mwhich is faster when a small bound on h+(m)is known. Reduction of unit group computation to HSP Theorem 45 Under Conjecture 1 the unit group of K=Q(ζm)+and its class group can be computed by an HSP algorithm in polynomial time an d space. Proof. LetN= poly(m)!and note that Conjecture 1 implies that O∗ K⊂C1/N. Letm′=ϕ(m)/2−1and letε1,...,ε m′be a basis of C//an}⌊ra⌋k⌉tl⌉{tζm,−1/an}⌊ra⌋k⌉tri}ht. Note that it can be computed using the HNF algorithm from the vectors vjof Lemma 41. Since K is totally real, its only roots of unity are ±1. For anyc∈C, we call vectorization ofcand denote vect(c)the unique (m′+1)-tuple in (e0,e1,...,e m′)∈Z/2Z× (Z/NZ)m′such thatc= (−1)e0/producttextm′ j=1εej j. We set f:Z/2Z×(Z/NZ)m′→/braceleftbigg canonical representations of Kummer extensions/bracerightbigg (e0,e1,...,e m′)/ma√sto→K(ζN,N/radicalBig (−1)e0/producttextm′ j=1εej j))(6) We claim that the period of fis period(f) ={vect(u)|u∈(OK)∗}. Indeed, let (ej)and(zj)be such that f(e0,...,e m′) =f(e0+z0,...,e m+zm′) and setε= (−1)e0/producttext jεej jandζ= (−1)z0/producttext jεzj j. We haveK(ζN,N√ε) =K(ζN,N√εζ). Letd:= [K(ζN,N√ε) :K(ζN)]. Then the criterion of isomorphism of Kummer extensions [34, page 58] states thatN√ζ belongs toK(ζN). ButN√ζ∈Rand the maximal real subﬁeld of K(ζN)isK, soN√ζ∈ O∗ K. Conversely, ifN√ζ∈ O∗ Kit is direct that (z0,...,z m′)is a period off. Finally, to compute the complexity of the algorithm, we writ ea=P(b)for a=bO(1)anda= polylog(b)fora= (logb)O(1). By [33], the cost of the HSP is polylog(N). Since, for any n,n!≤nn, we have time= polylog(N) =P(poly(m)log(poly(m))) =P(P(m)) =P(m). 5 Conclusion and open questions 1. The unit group algorithms follow a parallel path to those o f other problems like the class group e.g Buchmann’s algorithm computes the t wo groups together and the quantum algorithm for class groups [9] is a g eneralization of the one of Hallgren et al [22]. It is interesting to have a pr ecise estimation of the number of qubits for class groups depending on the boun d on the class number, on the computation of discrete logarithms in the cla ss group etc. The special case of cyclotomic ﬁelds in quantum algorithms f or unit groups 19 2. The idea of a small dual basis can be generalized: a) to all a belian ﬁelds; b) to Galois ﬁelds with simple group of automorphisms and kno wn units e.g. [31]. 3. The possibility of exploiting automorphisms of arbitrar y order depends on the possibility of a Buchmann-Pohst algorithm for OK-lattices. One

Question 501 multiple-choice

The hidden subgroup problem (HSP) is central to quantum computing, with wide-ranging implications for quantum algorithms and cryptography. The dihedral group's HSP in particular is crucial for understanding the security of post-quantum cryptosystems and the limits of quantum algorithmic techniques. Which statement best explains why the quantum hardness of the hidden subgroup problem for the dihedral group is significant for post-quantum cryptography? 1) It enables efficient quantum algorithms for factoring large integers, threatening RSA cryptosystems. 2) It underpins the security of cryptosystems based on hard lattice problems, since efficient quantum solutions could break these schemes. 3) It provides a polynomial-time quantum solution to the graph isomorphism problem. 4) It demonstrates that Fourier sampling works equally well for all finite groups. 5) It shows that non-abelian group structure has no impact on quantum algorithm efficiency. 6) It removes the need for coset sampling in quantum algorithms for non-abelian groups. 7) It proves that quantum state cloning is always possible for cryptographic applications.

✓ Correct Answer:

The correct answer is 2) It underpins the security of cryptosystems based on hard lattice problems, since efficient quantum solutions could break these schemes..

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Research Article Imin Chen* and David Sun The dihedral hidden subgroup problem https://doi.org/10.1515/jmc-2022-0029 received October 06, 2022; accepted August 23, 2023 Abstract: The hidden subgroup problem (HSP) is a cornerstone problem in quantum computing, which captures many problems of interest and provides a standard framework algorithm for their study based on Fourier sampling, one class of techniques known to provide quantum advantage, and which succeeds for some groups but not others. The quantum hardness of the HSP problem for the dihedral group is a critical questionfor post-quantum cryptosystems based on learning with errors and also appears in subexponential algorithms for constructing isogenies between elliptic curves over a ﬁniteﬁeld. In this article, we give an updated overview of the dihedral hidden subgroup problem as approached by the “standard ”quantum algorithm for HSP on ﬁnite groups, detailing the obstructions for strong Fourier sampling to succeed and summarizing other known approaches and results. In our treatment, we “contrast and compare ”as much as possible the cyclic and dihedral cases, with a view to determining bounds for the success probability of a quantumalgorithm that uses mcoset samples to solve the HSP on these groups. In the last sections, we prove a number of no-go results for the dihedral coset problem (DCP), motivated by a connection between DCP and cloning of quantum states. The proofs of these no-go results are then adapted to give nontrivial upper bounds on thesuccess probability of a quantum algorithm that uses mcoset samples to solve DCP. Keywords: quantum computation, hidden subgroup problem MSC 2020: 81P94, 68Q12, 20C05, 14H52 1 Introduction LetGbe aﬁnite group and Ha hidden subgroup of G. A function →fGS : , whereSis a known ﬁnite set, which is constant on left H-cosets and takes distinct values on distinct left H-cosets, is called a separating function for the subgroup H. The hidden subgroup problem (HSP) is the problem of ﬁnding generators for the hidden subgroup H, given access to evaluations of a separating function fforH. The HSP can be solved in polynomial time using a quantum computer when Gis an abelian group [1 ,2] and has been extensively studied for many classes of ﬁnite groups [3,4]. Many problems can be cast in terms of the HSP and there is a “natural ”standard quantum algorithm based on the quantum Fourier transform, which is typically used to study the HSP. For instance, Shor ’s integer factorization algorithm [5 ] can be described in terms of the HSP for cyclic groups [6], and in this case, the standard algorithm succeeds in yielding an e ﬃcient quantum algorithm. Another example is the HSP on the symmetric group, which can be used to solve the graph isomorphism problem [7 –10], but here the standard algorithm fails to provide an e ﬃcient quantum algorithm [11]. A polynomial time quantum algorithm for solving the HSP on dihedral groups would imply a polynomial time quantum algorithm to solve certain hard lattice problems that are considered intractable using classical  * Corresponding author: Imin Chen, Department of

Question 502 multiple-choice

In quantum algorithms for group-theoretic problems, superpositions over cosets and subgroup state conversions play a central role in efficiently extracting hidden subgroup information. The mathematical structure of groups, their subgroups, and associated operators are fundamental to designing such procedures. Which process enables the conversion of a purification of an operator associated with a subgroup H of a group G into a purification for a subgroup L of G by transferring parts of the state to the purifying subsystem, specifically focusing on the intersection H∩L? 1) Measurement in the Fourier basis 2) Decomposition via tensor products 3) Projection onto the invariant subspace 4) Restriction 5) Normalization of coset representatives 6) Pushing onto quotient groups 7) Amplification of eigenvalue multiplicity

✓ Correct Answer:

The correct answer is 4) Restriction.

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the vectors vi corresponding to nonzero eigenvalues λiis an orthonormal system of vectors in V.T h e vectors videpend on the choice of the basis |ui/angbracketright(i=1,..., m). Notice that the only nonzero eigenvalue of /Xi1G,His1 kwith multiplicity k, where k=|G:H|. The coset superpositions give an orthonormal basis of the corresponding eigenspace. Thus, if |ψ/angbracketright is a puriﬁcation of /Xi1G,Hthen a Schmidt decomposition of |ψ/angbracketrightwhich is a puriﬁcation of/Xi1G,His of the form |ψ/angbracketright=1√ k/summationtextk i=1|ui/angbracketright|vi/angbracketrightwhere |u1/angbracketright,...,|uk/angbracketrightis an arbitrary orthonormal basis of the1 k-eigenspace of /Xi1G,Hand|v1/angbracketright,...,|vk/angbracketrightis an orthonormal 123 Zero sum subsequences and hidden subgroups Page 9 of 23 14 system of V. In particular, if X={a1,..., ak}then by taking |ui/angbracketright=| aiH/angbracketrightand by deﬁning |v(x)/angbracketright=|vi/angbracketrightforx∈aiH, we obtain |ψ/angbracketright=1√ k/summationtext a∈X|aH/angbracketright|v(a)/angbracketright= 1√|G|/summationtext x∈G|x/angbracketright|v(x)/angbracketright. /intersectionsq/unionsq 2.4 Basic subgroup state conversions Given a subgroup LofG, a copy of (a puriﬁcation of) the subgroup state /Xi1G,H can be “converted” to a copy of (a puriﬁcation of) /Xi1L,H∩Lby replacing |x/angbracketrightwith the decomposition |βL(x)/angbracketright|αL(x)/angbracketrightobtained by the method outlined in Sect. 2.2forx∈G and “ignoring” |αL(x)/angbracketright(passing this part to the purifying subsystem). To see this, let 1√|G|/summationtext x∈G|x/angbracketright|ψ(x)/angbracketrightbe a puriﬁcation of /Xi1G,H, with |ψ(x)/angbracketrightand|ψ(y)/angbracketrightare equal if and only if y−1x∈H, and orthogonal otherwise. Then, the substitution gives the state1√|L|/summationtext x∈L|x/angbracketright1√|G:L|/summationtext y∈Y|y/angbracketright|ψ(yx)/angbracketrightwhere Y={αL(z):z∈G}.N o wi f x1,x2∈Lare from the same left coset of H∩Lthen|ψ(yx1)/angbracketright=|ψ(yx2)/angbracketrightfor every y∈Yand hence the states1√|G:L|/summationtext y∈Y|y/angbracketright|ψ(yxi)/angbracketrightare equal ( i=1,2), while otherwise they do not overlap as for y1,y2∈Yeither |y1/angbracketrightand|y2/angbracketrightare orthogonal or (for y1=y2)|ψ(y1x1)/angbracketrightand|ψ(y1x2)/angbracketrightare orthogonal. We shall refer to this procedure asrestriction . The term is justiﬁed by that in the standard version of the HSP , one could obtain an instance of the HSP in the subgroup Lby restricting the “hiding function” toL. Similarly, assume that Lis a normal subgroup of Gcontained in H. Then a copy of (a puriﬁcation of) the subgroup sate /Xi1G,Hcan be converted to a copy of (a puriﬁcation of)/Xi1G/L,H/Lby replacing xwith|αL(x)/angbracketright|βL(x)/angbracketrightand passing |βL(x)/angbracketrightto the purifying subsystem. This corresponds to the technique called “pushing” in [ 6,31]. 3 A group-theoretic reduction In this section we prove the following. Proposition 1 Let G be a nilpotent black-box group of class at most c and assume that the prime factors of |G|are given as part of the input and that for each such prime p the quantum Fourier transform modulo a multiple of p and its inverse can be implemented by an efﬁcient exact quantum procedure. Then, the HSP in G can bereduced by an exact procedure in time poly(log/lscript)topoly(log|G|)instances of the problem HSMC in semi-elementary quotient groups of subgroups of G. (The elements of G are assumed to be uniquely encoded by strings of length /lscript.) Proof A ﬁnite nilpotent group Gis the direct product of its Sylow subgroups. Therefore any subgroup His the product of its Sylow subgroups. The p-Sylow subgroup of H isP∩Hwhere Pis the p-Sylow subgroup of G. The Sylow subgroups of Gcan be computed using the method outlined in Sect. 2.2. One can convert subgroup states in Gto subgroup states in Pusing restriction, see Sect. 2.4. In the rest of

Question 503 multiple-choice

Supersymmetric extensions of the Standard Model often introduce hidden sectors with new gauge symmetries and multiplets to provide viable dark matter candidates. Communication between the hidden and visible sectors can occur through mechanisms such as gauge kinetic mixing. In a supersymmetric model featuring a hidden sector with U(1)x gauge symmetry and chiral multiplets H and H0, what is the primary low-energy interaction that allows MSSM neutralinos to decay into the hidden sector? 1) Scalar portal coupling via Higgs mixing 2) Neutrino oscillation between sectors 3) Gravitational interactions 4) Gauge kinetic mixing between U(1) gauge groups 5) Strong force mediation via gluons 6) Yukawa interactions with hidden sector fermions 7) Thermal contact through shared reheating phase

✓ Correct Answer:

The correct answer is 4) Gauge kinetic mixing between U(1) gauge groups.

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multiplets HandH0with charges xH;H0=1. Motivated by the scaling relation of Eq. (16), we take the characteristic gauge coupling and mass scale of the hidden sector to be signi cantly less than electroweak, along the lines of Refs. [82{84]. The LSP of the extended theory will therefore be the lightest hidden neutralino. We also assume that the only low-energy interaction between the hidden and visible sectors is gauge kinetic mixing. Among other things, this allows the lightest MSSM superpartner to decay to the hidden sector. In this section we the investigate the contribution of the hidden LSP to the dark matter density following moduli reheating as well as the corresponding bounds from indirect and direct detection. A. Setup and Spectrum The hidden superpotential is taken to be WHS=WMSSM0HH0; (20) and the soft supersymmetry breaking terms are Lsoftm2 HjHj2+m2 H0jH0j2+ b0HH0+1 2MxeXeX+ h:c: : (21) The only interaction with the MSSM comes from supersymmetric gauge kinetic mixing in the form LZ d2 2X B ; (22) whereXandBare theU(1)xandU(1)Y eld strength super elds, respectively. We assume that the gaugino mass is given by its AMSB value [83] Mx=bxg2 x (4)2m3=2; (23) wherebx= 2 andgxis theU(1)xgauge coupling. Since pure AMSB does not provide a viable scalar spectrum in the MSSM, we do not impose AMSB values on the scalar soft terms in the hidden sector. However, we do assume that they (and 0) are of similar magnitude to their AMSB values, on the order of ( g2 x=162)m3=2. This could arise if the dynamics that leads to a viable MSSM spectrum also operates in the hidden sector and that its e ects are proportional to the corresponding gauge coupling. 12 For a range of values of 0and the soft terms, the scalar components of HandH0will develop vacuum expectation values, hHi=sin;hH0i=cos : (24) Correspondingly, the hidden vector boson Xreceives a mass mx=p 2gx : (25) The scalar mass eigenstates after U(1)xbreaking consist of two CP-even states hx 1;2(withhx 1 the lighter of the two) and the CP-odd state Ax. The fermionic mass eigenstates are mixtures of the hidden Higgsinos and the U(1)xgaugino, and we label them in order of increasing mass as x 1;2;3. Full mass matrices for all these states can be found in Refs. [85, 86]. B. Decays to and from the Hidden Sector Kinetic mixing allows the lightest MSSM neutralino to decay to the hidden sector. It can also induce some of the hidden states to decay back to the SM. We discuss the relevant decay modes here. The MSSM neutralinos connect to the hidden sector through the bino. For AMSB gaugino masses, the bino soft mass is signi cantly heavier than that of the wino, and the lightest neutralino 0 1tends to be nearly pure wino. Even so, it will have a small bino admixture given by the mass mixing matrix element N11. In the wino limit, it can be approximated by [49] jN11j=cWsWm2 Z(M2+ sin 2 ) (M1M2)(2M2 2): (26) With this mixing, the lightest MSSM neutralino will decay to

Question 504 multiple-choice

Quantum error correction utilizes mathematical structures such as stabilizer codes and graph codes to protect quantum information from noise. These codes are intimately connected to symplectic geometry and physical implementations in quantum computing. Which property ensures that a symplectic code used for quantum error correction is robust against certain types of errors due to its mathematical structure? 1) Having a generator matrix with full rank 2) Possessing an adjacency matrix with exclusively nonzero entries 3) Commutativity of its stabilizer generators under matrix multiplication 4) Encoding using only Pauli Z operators 5) Being represented by a unique graph structure 6) Being self-orthogonal under the symplectic inner product 7) Utilizing non-symmetric quadratic forms

✓ Correct Answer:

The correct answer is 6) Being self-orthogonal under the symplectic inner product.

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by SQ={exp(2πi/p)q(a)XaZaM y|a∈Fn psuch that Ba=0}. The matrices My∈Fn×n p andB∈Fk×n pare submatrices of the adjacency matrix Γdeﬁning the quadratic form qwhich can be written as Γ=/parenleftbiggMx B Bt My/parenrightbigg . Additionally, the orthogonal projection onto the joint eigenspace for the eigenvalue 1of the operators in SQcoincides with Q, showing that Qindeed is a stabilizer code. In order to show that any stabilizer code deﬁned over the ﬁnite ﬁeld Fpmcan be realized as a graph code, we ﬁrst note that those stabilizer codes can equivalently be considered as stabi-lizer codes over F p(see [AK01]). To any stabilizer code corresponds a (classical) symplectic codeCover Fp2. The generator matrix of Ccan be written as (X|Z)where X,Z∈F(n−k)×n p andXZt−ZXt=0. As the code Cis self-orthogonal, i. e., contained in the orthogonal code C⊥with respect to the symplectic inner product on Fn p2, there exists a self-dual code Dwith C⊆D =D⊥⊆C⊥. We can choose a generator matrix for Dof the form G/prime:= (X/prime|Z/prime)=/parenleftBigg X Z /tildewideX /tildewideZ/parenrightBigg . Similar to Gauß’ algorithm, the matrix G/primecan be transformed into standard form (I|C)where Cis symmetric [Gra02a, Gra02b]. Quantum mechanically, the transformations correspond to 1.4 Efﬁcient Decomposition of Quantum Operations into Given One-parameter Groups 5 local unitary operations which do not change the error-correcting properties of the QECC. Applying the same transformations to the subcode CofD, we obtain an equivalent code whose generator matrix can be written as (D|DC)where D∈F(n−k)×n. Finally, we obtain the symmetric matrix Γ: =/parenleftbigg0 B Bt C/parenrightbigg , where DBt=0. Using this matrix Γas adjacency matrix in (1.1), we obtain a graph code which is equivalent to the stabilizer code corresponding to C. Hence, for any stabilizer code over Fpmthere exists an equivalent graph code. Figure 1.1: Non-isomorphic graphs which all yield gra phical quantum codes that are equivalent to the CSS code [[ 7,1,3] ]. In general, the corresponding graph is not uniquely determined, as illustrated in Fig. 1.1. All four graphs yield equivalent QECC, but the graphs are non-isomorphic. Therefore it is not straightforward to relate basic properties of the graph and the error-correcting propertiesof the resulting graph code. But the possibility to choose among different graphs results indifferent interaction Hamiltonians which may yield more efﬁcient algorithms for encoding (cf. Section 1.5). 1.4 Efﬁcient Decomposition of Quantum Operations into Given One-parameter Groups The time evolution of all quantum mechanical systems is governed by the Schrödinger equa- tion. To control the time evolution of a concrete physical implementation, we apply different Hamilton operators which are supposed to be time independent. This gives us the ability to implement computational operators on the state space. Equipped with mone-parameter groups {t/mapsto→exp(−˙ıH1t),t/mapsto→exp(−˙ıH2t),...,t /mapsto→exp(−˙ıHmt)} generated by the different Hamilton operators Hj, the goal is to build up each unitary Uas a product of elements of the given one-parameter groups: U=l/productdisplay j=1exp(−˙ıHpjtpj),where Hpj∈{H1,H2,...,Hm}andtpj∈R,tpj>0. 6 1 Algorithms for Quantum Systems — Quantum Algorithms As an efﬁciency measure we consider the minimal number of terms needed to express each unitary as a product of the given one-parameter groups. This efﬁciency measure

Question 505 multiple-choice

In quantum physics and group theory, the interplay between symmetries, operators, and the structure of eigenspaces is fundamental for understanding conserved quantities and degeneracies. The representation theory of groups provides powerful techniques for analyzing how symmetries manifest in quantum systems. Which statement best describes the structure of a nondegenerate (one-dimensional) eigenspace of a Hermitian operator H under the action of a group representation U that commutes with H? 1) The eigenspace can always be decomposed into multiple irreducible representations. 2) The eigenvector in the eigenspace may be mapped to vectors in different eigenspaces by U. 3) The eigenspace necessarily contains eigenvectors with different eigenvalues. 4) The symmetry group action can permute vectors within the eigenspace but not preserve eigenvalues. 5) The sole eigenvector must be invariant under all symmetries that commute with the operator. 6) The eigenspace forms a reducible representation under the group action. 7) The group representation always breaks the symmetry of the eigenspace.

✓ Correct Answer:

The correct answer is 5) The sole eigenvector must be invariant under all symmetries that commute with the operator..

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manifold G). Once this is accomplished, we apply the following proposition: Proposition 5. LetU:G!GL(H)be a representation of a groupG, and letHbe a Hermitian operator such that [Ug;H] = 0for allg2G. Then, for any eigenvector j i ofHwith eigenvalue ,Ugj iis also an eigenvector of H of eigenvalue . Proof.Observe that HUgj i=UgHj i=Ugj i: Observation 1. Warning: As discussed earlier, this does not mean thatj i=Ugj i! An illustrative counterexample is to takeH=1onC2and to consider the representation U:Z2!GL(C2)given byU1=1andU=X. Then certainlyj0i;j1iare eigenvectors of H, butUj0i=j1i. In general, this means that representations can (but do not have to) permute within eigenspaces of H, but they cannot map between eigenspaces. In other words, since [Ug;H] = 0,His simultaneously block diagonalized with Ug, and distinct eigenspaces necessarily inhabit diﬀerent blocks. So, what does this do for us? This means that the eigenspaces of H(often interpreted as energy eigenspaces) are invariant subspaces of the representation U, and we can use the powerful methods of representation theory to furtheranalyzethem. Takealookattwoexamples,onedis- crete and one continuous, of this approach in Box 11 and 24This shows up quite prominently in the analysis of the Hydrogen atom: for a treatment which is more Lie-theoretic in ﬂavor, see Hall’s other book [96].Box 12. In the former, we see that translation invariant Hamiltonians have energy eigenspaces which are invariant undertranslations. Heedingthewarning, thisdoesnotnec- essarily mean that every eigenstate is translation invariant: for instance, in a process known as dimerization [97], the Majumdar-Ghosh model [98] has a translation-invariant Hamiltonian, but its 2 ground states are only 2-periodic. In this case, translation by 1 site maps between the two ground states (and so of course preserves the energy). To translate to the language of representation theory, we would say the ground state space forms an irreducible rep- resentation for this lattice translation symmetry. While a simple case, the theme is quite deep: when an eigenspace forms an irreducible representation, we need only ﬁnd a single eigenvector—the rest can be generated by acting on this vector with the representation (in the above example, acting via the translation operator). Al- ternatively, when the eigenspace forms a reducible repre- sentation, restricting to an invariant subspace allows us to continue extracting structure via further decomposition. As a special case of this lemma, note that if one possesses a nondegenerate eigenstate, i.e. one whose eigenspace is one- dimensional, that state automatically inherits the symme- tries of its Hamiltonian. B. Deadfalls: geometric invariants Representations have close ties to geometry—indeed, many of the physical symmetries we described a moment ago are highly geometric in nature. An obvious applica- tion to QML is when the data itself is invariant under the symmetry group [25, 30, 60]. But often, geometric invari- ants have a way of hiding in plain sight: in fact, even the deﬁnitions of the classical Lie groups can be phrased as “the transformations which leave a tensor invariant”. Just as in the case of the inner product, other tensors (like bi- linear forms and the determinant) carry rich geometric in-

Question 506 multiple-choice

Quantum algorithms can solve certain group-theoretic problems far more efficiently than classical algorithms, especially within abelian and solvable groups. Encoding group elements and handling error robustness are essential for reliable quantum computation in these scenarios. Which property is guaranteed by the ArbitraryAbelianTCS procedure when solving the Translating Coset problem in black-box abelian groups? 1) It always finds the exact generators of the coset with zero error. 2) It outputs a classical list of group elements rather than a quantum state. 3) It runs in strictly polynomial time in the group size. 4) It rejects the input when no solution is found. 5) It produces an output quantum state within trace distance ε of a desired ideal state. 6) It cannot be applied to factor groups of solvable groups. 7) It relies on classical algorithms for discrete logarithm and order finding.

✓ Correct Answer:

The correct answer is 5) It produces an output quantum state within trace distance ε of a desired ideal state..

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Xdenotes a set of generators for G|φ1/angbracketright andua group element such that |u·φ1/angbracketright=|φ0/angbracketright. When no solution exists, we simply 2The sum is over all elements u∈Gand all lists Xconsisting of at most log |G|elements of G.I f there were no errors, αu,Xwould be zero for pairs ( u,X) which do not describe the coset translating φ0toφ1. Due to errors of the algorithm, some such coeﬃcients can be nonzero, although very small. 16 FRIEDL, IVANYOS, MAGNIEZ, SANTHA, AND SEN request the algorithm of Corollary 3.8 to set X=∅, and let ube any group element, instead of rejecting. Letvbe a ﬁxed group element such that |v·φ1/angbracketright=|φ0/angbracketright. Because of the choice of the parameters and by Corollary 3.8, the states |u/angbracketleftX/angbracketright/angbracketrightand|vG|φ1/angbracketright/angbracketrightare of trace distanceatmost ε. Thisimpliesthattheﬁnalstateofthealgorithmisoftracedistance at mostεfrom the following state without garbage: |φ0/angbracketright⊗t|φ1/angbracketright⊗t|vG|φ1/angbracketright/angbracketright. For an arbitrary abelian group G, we can modify procedure Elementary - AbelianTCS (G,α,t) by replacing the algorithm of Corollary 3.8 with an adapted version of Kuperberg’s subexponential method (see Theorem 7.1 of [31]) to solve Translating Coset . (We only need modiﬁcations to Kuperberg’s algorithm like the ones to TranslationFinding described in the proof of Corollary 3.8: We use the “conditionally swapped pairs of functions” trick presented in the proof of Corol-lary 3.6 and simulate the oracle with input quantum states.) Let us call the resulting procedure ArbitraryAbelianTCS (G,α,t). We obtain the following. Corollary 4.6. LetGbe a black-box abelian group with unique encoding of length/lscript.L e tαbe a group action of G,a n dl e t t=2 Ω(√ log|G|)be a positive inte- ger. Then ArbitraryAbelianTCS (G,α,t)implements Translating Coset Super- position (G,α,t)with error εin quantum time /lscriptO(1)2O(√ log|G|)log(1/ε). 4.1.2. Compatible encodings. We will apply recursion into factor groups of solvable groups. Therefore we need an eﬃcient procedure to design a unique encoding for these factor groups. Moreover, for the purpose of our algorithm we will require this encoding to be compatible with the originalencodingof the groupin the followingsense. Definition 4.7. LetGbe a black-box group with unique encoding encof length /lscript. LetNbe a normal subgroup of G. A unique encoding enc NforG/Niscompatible with encif 1.for every x∈G,t h e r ei s y∈xNsuch that encN(xN)=enc(y); 2.the partial unitary |enc(x)/angbracketright|0/angbracketright/mapsto→|enc(x)/angbracketright|encN(xN)/angbracketright,w h e r ex∈G,c a nb e implemented in quantum time poly(/lscript). Note that if Ghas encoding length /lscript, then a compatible encoding for G/Nalso has encoding length /lscript. From now on, we assume for simplicity that we have at our disposal a multiple r of|G|such that r=O (/lscript). This multiple is given or computed once for a group, and we keep the same value for all its subgroups. This assumption is reasonable since for solvable groups the cardinality of Gcan be computed in time poly( /lscript)[ 4 6 ] . In the following theorem, we assume for simplicity that we have at our disposal a zero-error quantum algorithm for computing the generalized discrete logarithm and for order ﬁnding. Theorem 4.8. LetGbe a black-box solvable group with unique encoding encof length/lscript.L e tNbe a normal subgroup of Gsuch that G/Nis abelian.

Question 507 multiple-choice

Quantum computation often leverages multi-level quantum systems (qudits) and hybrid gates to optimize circuit complexity and resource requirements. Techniques involving generalized gates and graph-theoretical structures are used to enhance the efficiency of implementing multi-qubit-controlled operations. In designing a scalable n-qubit Toffoli gate using qudits and controlled-Z gates arranged by a tree topology, what is the minimum required local dimension di of a qudit node with ki connections to guarantee circuit feasibility? 1) di must be equal to the number of qubits in the circuit 2) di must be twice the number of connections ki 3) di must be less than the number of connections ki 4) di must be at least the number of connections ki plus one 5) di must be exactly three for all nodes 6) di must be odd and greater than ki 7) di must be one less than the number of connections ki

✓ Correct Answer:

The correct answer is 4) di must be at least the number of connections ki plus one.

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two lines represent two control qubits and the third line represents a target qutrit that has threeaccessible levels. The initial and ﬁnal quantum states of the quantum information carrier are encoded in the |0〉and |1〉. The His the generalized Hadamard gate such that H|0〉/equals1/⎥radicaltpext⎥radicaltpext 2√ [|0〉+|1〉], H|1〉/equals1/⎥radicaltpext⎥radicaltpext 2√ [|0〉-|1〉]andH|2〉/equals|2〉.X agate is a qutrit gate such that Xa|0〉/equals|2〉andXa|2〉/equals|0〉withXa|1〉/equals|1〉. With the control being qubit, the target being qudit, the two qudit gate in this case is a hybrid gate. Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504 7Wang et al. Qudits and High-Dimensional Quantum Computing two-qubit gates [ 125]; this method offers a signi ﬁcant resource reduction. Furthermore, these schemes can be extended to more general quantum circuits such as the multi-qudit-controlled-unitary gate C nU. The previous method turns the target qubit into a qudit; another method simpli ﬁes the Toffoli gate by using only qudits and treating the ﬁrst two levels of the qudit as qubit levels and other levels as auxiliary levels. The reduction in thecomplexity of Toffoli gate is accomplished by utilizing thetopological relations between the dimensionality of the qudits,where higher qudit levels serve as the ancillas [ 89]. Suppose we have a system of nqudits denoted as Q i,i∈{1,... ,n}and each qudit has dimension di≥2. Qudits are initialized into pure or mix states on the ﬁrst two levels, i.e., the qubit states, and zero population for the other levels,i.e., the auxiliary states. This scheme assumes the ability toperform single-qubit operations. We can apply the desirableunitary operation on the qubit states and leave the auxiliarystates unchanged. We also assume that we have the ability tomanipulate the auxiliary levels by a generalized inverting gate X m Xm|0〉/equals|m〉,Xm|m〉/equals|0〉,Xm|y〉/equals|y〉,fory≠m,0.(58) At the same time, the two-qubit CZgates are applied according to certain topological connections between qudits. We introduce asetEof ordered pairs (i,j), such that i,j∈{1,... ,n},i

Question 508 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) rely on group-theoretic properties and measurement strategies to identify hidden subgroups efficiently. In the context of graph isomorphism, specialized group constructions and coset states play a crucial role in the formulation and solution of the problem. Which of the following statements correctly characterizes the role of the involutive swap element (π, π^{-1}, 1) in the wreath product Sn ≀ S_2 when applied to the quantum reduction of graph isomorphism for rigid graphs? 1) It always generates a nontrivial subgroup regardless of whether the graphs are isomorphic. 2) It encodes the automorphism group of each individual graph. 3) It distinguishes non-isomorphic graphs by being present in all hidden subgroups. 4) It acts as the trivial element in the hidden subgroup for isomorphic graphs. 5) It generates the hidden subgroup in the case when the graphs are isomorphic and rigid. 6) It represents the identity element in Sn ≀ S_2 for any pair of graphs. 7) It prevents the use of coset state measurements in quantum algorithms for graph isomorphism.

✓ Correct Answer:

The correct answer is 5) It generates the hidden subgroup in the case when the graphs are isomorphic and rigid..

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solve the HS P using coset states in groups PSL(2,Fpm), GL(n,Fpm), and groups of the form Gn, whereGa constant-sized group satisfying a suitable property, including all groups (Sm)nwherem≥4is aconstant. Thecase of (S4)nisinteresting, because there isan efﬁcient algorithm for the HSPmaking joint measurements on nO(1)states using the orbit coset techniques of [FIM+03]. However, the orbit coset approach creates coset states not just for the hidden subgroup H, but also for various subgroups of the form HN, whereN/triangleleftequal(S4)n. This example suggests that one way to design efﬁcient algorithms for the HSP making highly entang led measurements may be to use coset states for subgroups of Gother than just the hidden subgroup H. Recently, Childs and Wocjan [CW05] proposed a hidden shift approach to graph isomorphism. They established a lower bound for the total number of hidden shif t states required and also showed that a single hidden shift state contains exponentially little informat ion about the isomorphism. Our results generalize boththeirboundsandimplythat o(nlogn)hiddenshiftstatescontainexponentially littleinformat ion about the isomorphism. The chief technical innovation required to prove our main th eorem is an improved upper bound for the second moment of theprobability of observing aparticular m easurement outcome aswevaryover different candidate hidden subgroups. In particular, we give a new and improved analysis of the projection lengths of vectors of the form b⊗bonto homogeneous spaces of irreducible representations of a group. The earlier works [MRS05, MR05] tried to bound these projection lengths using simple geometric methods. As a result, their methods failed beyond k= 2for the symmetric group. Instead, we make crucial use of the representation-theoretic structure of the projection ope rators as well as the structure of the vectors, in order 3 toproveupper bounds ontheprojection lengths better than t hose obtainable bymeregeometry. Thisallows us toprove ageneral theorem that applies with large kfor many groups. Finally,wealsoproveasimplelowerboundonthetotalnumbe rofcosetstatesrequiredbyanyalgorithm tosolvetheHSPinagroup G. Thislowerboundgivesasimpleproofofthefactthatdistin guishing ahidden reﬂection from theidentity subgroup inthe dihedral group DnrequiresΩ(logn)coset states. 2 Preliminaries 2.1 Graphisomorphism andHSP The usual reduction of deciding isomorphism of two n-vertex graphs to HSP in S2nactually embeds the problem into a proper subgroup of S2n, namely,Sn≀S2[EHK99a]. The elements of Sn≀S2are tuples of the form (π,σ,b)whereπ,σ∈Snandb∈Z2with the multiplication rule (π1,σ1,0)·(π2,σ2,b) := (π1π2,σ1σ2,b)and(π1,σ1,1)·(π2,σ2,b) := (π1σ2,σ1π2,1⊕b). The embedding of Sn≀S2inS2n treats{1,...,2n}asaunion of {1,...,n}∪{n+1,...,2n}withπ,σpermuting theﬁrst and second sets respectively when b= 0, andπpermuting the ﬁrst set onto the second and and σpermuting the second set onto the ﬁrst when b= 1. There is an element of the form (π,π−1,1), called an involutive swap , in the hidden subgroup iff thetwo graphs areisomorphic. Additionally, if the two graphs are rigid, i.e., have no non- trivial automorphisms, then the hidden sub- group istrivial iftheyarenon-isomorphic, andisgenerate d by(π,π−1,1)iftheyareisomorphic where πis the unique isomorphism from the ﬁrst graph onto the second. T his element (π,π−1,1)is of order two, and is a conjugate in Sn≀S2ofh:= (e,e,1)wheree∈Snis the identity permutation. Viewed as an element ofS2n,h= (1,n+1)(2,n+2)···(n,2n). The set of conjugates of hinSn≀S2is the set of all involu- tive swaps (π,π−1,1),π∈Sn, and corresponds exactly to all the isomorphisms possible b etween the two

Question 509 multiple-choice

In quantum information theory, locally maximally entangled (LME) states are central for understanding multipartite entanglement, especially when considering equivalence under local unitary transformations. The dimension and existence of the space of such states for given subsystem sizes have deep connections to combinatorial mathematics and quantum marginal compatibility. For a tripartite quantum system with subsystems of dimensions A, B, and C, what is the formula for the expected dimension of the quotient space of locally maximally entangled states under local unitary transformations? 1) A + B + C − ABC 2) ABC − A² − B² − C² + 2 3) A² + B² + C² − 2ABC 4) AB + AC + BC − (A + B + C) 5) (A − 1)(B − 1)(C − 1) 6) A²B²C² − 2 7) AB + BC + CA − A² − B² − C²

✓ Correct Answer:

The correct answer is 2) ABC − A² − B² − C² + 2.

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max(g max(/vectord)−3,0). If∆(/vectord)<−2, thenR≤0andSLME/Kis a single point for R= 0and empty for R<0. Both of these results follow from a recursive algorithm for the dimen sion ofSLME/Kgiven as Theorem 5.1 below. Using this algorithm, we can also provide some mo re explicit results for the dimension of SLME/K. Generally, for ﬁxed ( d1,...,dn−1), there is a range of values dn−1≤dn< d∗for which theSLME/Kis not empty and its dimension ∆( /vectord) is the naive dimension of the quotient P(H)//G. Ford∗≤dn≤d1···dn, this naive dimension is negative,7but there are sporadic values ofdnin this range for which the space SLME/Kis nevertheless non-empty; it is given by a single point unless ∆ = −2. Forn= 3, we are able to characterize the sporadic cases more explicitly. We ﬁnd that they are all of the form ( d1,d2,d3) = (A,fi,fi+1) where (fi,fi+1) are successive terms in a Fibonnaci-like sequence deﬁned by fi+1=Afi−fi−1 (1.9) with (f0,f1) = (k,kA) for some positive integer kor (f0,f1)∈SA, whereSAis a ﬁnite set of pairs deﬁned in Proposition 5.4. The results for dimensions (2 ,B,C) and (3,B,C) are displayed in ﬁgure 2. Relation to the quantum marginal problem and its solution Deciding whether LME states exist for given subspace dimensions is a particular case of the quantum marginal problem : given a collection of density operators associated to subsystems a of a multipart system, can these arise as the reduced density ope rators from a quantum state of the full system? This problem has received considerable at tention in the past and has applications in quantum chemistry (see, for example [TEVS16, TV S17]). In the case where the subsystems do not overlap and the full syst em is assumed to be in a pure state, a general answer has been provided by Klyachko [Kly04 ].8Klyachko’s criterion is a set of inequalities on the spectra for the density operators, wh ich may be expressed 7Speciﬁcally, d∗is deﬁned by setting ∆ = −2. 8See [Walt14] for a review. For a discussion of the quantum marginal p roblem using a diﬀerent approach from Klyachko’s, see [MT17, VW17] and references therein. 7 B-AC-B0 0 A=2 0 0 B-AC-B A = 3 Figure 2: Properties of the space SLME/Kof locally maximally entangled states up to local unitary transfomations for tripartite systems with dimension s (A= 2,B,C) and (A= 3,B,C). In each plot, circles correspond to values of ( A,B,C) (plotted with axes ( B−A) and (C−B) for ﬁxedA) for which the necessary conditions B≤C≤ABfor the existence of LME states are satisﬁed. Empty circles indicate cases where LME states do not exist. Blue ﬁlled circles (on the left in the A= 3 case) indicate cases where the dimension of SLME/Kis equal to the expected dimension ABC−A2−B2−C2+ 2 of the quotient P(H)/G. Black circles are non-empty cases with negative expected dimensio n but for which SLME/Kis a single point. Red circles are non-empty cases with expected dimen sion -2 for whichSLME/Kis equivalent to a K¨ ahler manifold describing unit vectors in R3adding to zero with equivalence under SO(3) rotations.

Question 510 multiple-choice

Quantum algorithms often rely on precise manipulation of quantum states using unitary operators and phase parameters to achieve desired computational outcomes. Orthogonality and phase estimation play crucial roles in ensuring accurate measurement and isolation of quantum states. In designing a quantum algorithm that requires the superposition Q(|ΨX⟩ + |ΨY⟩) to be orthogonal to |ΨY⟩, which mathematical technique is most directly employed to express the necessary phase parameters in terms of geometric relationships between quantum states? 1) Linear interpolation of amplitudes 2) Fourier series decomposition 3) Eigenvalue decomposition 4) Gram-Schmidt orthonormalization 5) Probability amplitude normalization 6) Complex conjugation of inner products 7) Trigonometric functions such as arctan and arccos relating inner products and amplitudes

✓ Correct Answer:

The correct answer is 7) Trigonometric functions such as arctan and arccos relating inner products and amplitudes.

📚 Reference Text:

RA(ϕ,Y)⊗I⊗2t+1m/parenrightBig |ΨY⟩ =U(A,φ)|ΨY⟩ =/parenleftBig/parenleftBig 1−eiφ/parenrightBig/parenleftbig |ΨX⟩+|ΨY⟩/parenrightbig/parenleftbig ⟨ΨX|+⟨ΨY|/parenrightbig −I⊗n−t+2t+1m/parenrightBig |ΨY⟩ =/parenleftBig 1−eiφ/parenrightBig/parenleftbig |ΨX⟩+|ΨY⟩/parenrightbig/parenleftbig ⟨ΨX|+⟨ΨY|/parenrightbig |ΨY⟩−|ΨY⟩ =/parenleftBig 1−eiφ/parenrightBig ⟨ΨY|ΨY⟩/parenleftbig |ΨX⟩+|ΨY⟩/parenrightbig −|ΨY⟩ =/parenleftBig 1−eiφ/parenrightBig 2r+kl+t−n/parenleftbig |ΨX⟩+|ΨY⟩/parenrightbig −|ΨY⟩ =(1−eiφ)2r+kl+t−n|ΨX⟩−/parenleftBig (1−eiφ)(1−2r+kl+t−n)+eiφ/parenrightBig |ΨY⟩.(A8) 17 By making sure the resulting superposition Q(|ΨX⟩+|ΨY⟩) has inner product zero with |ΨY⟩, then based on Eq. ( A7) and Eq. ( A8), we can obtain the following equation. eiϕ(1−eiφ)(1−2r+kl+t−n) = (1−eiφ)(1−2r+kl+t−n)+eiφ. (A9) Denote by b= 1−2r+kl+t−n. (A10) Then according to Eq. ( A9), we have b=e−iϕ/parenleftbigg b+1 e−iφ−1/parenrightbigg =e−iϕ/parenleftbigg b+1 cosφ−1−isinφ/parenrightbigg =e−iϕ/parenleftBigg b+cosφ−1 (cosφ−1)2+sin2φ+isinφ (cosφ−1)2+sin2φ/parenrightBigg =e−iϕ/parenleftbigg b−1 2+isinφ 2−2cosφ/parenrightbigg .(A11) Taking the square of b, we can further have b2=/parenleftbigg b−1 2/parenrightbigg2 +sin2φ 4(1−cosφ)2. (A12) Arrange Eq. ( A12) to get 4b−1 =sin2φ (1−cosφ)2 = cot2φ 2.(A13) From Eq. ( A13), we can obtain φ= 2arctan/parenleftBigg/radicalbigg 1 4b−1/parenrightBigg = 2arctan /radicalBigg 2n−t−r−kl 3·2n−t−r−kl−4 .(A14) Sincebis a real number, Eq. ( A11) also needs to be a real number, and we can further obtain ϕ= arccos/parenleftBigg b−1 2 b/parenrightBigg = arccos/parenleftBigg 2n−t−r−kl−1−1 2n−t−r−kl−1/parenrightBigg .(A15) □ References [1] Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Informati on. Cambridge university press, Cambridge (2010) 18 [2] Shor, P.W.: Polynomial-time algorithms for prime factorization and dis crete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997) [3] Grover, L.K.: A fast quantum mechanical algorithm for database search. I n: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pp. 212–219. ACM, New York, USA (1996) [4] Harrow, A.W., Hassidim, A., Lloyd, S.: Quantum algorithm for linear syst ems of equations. Phys. Rev. Lett. 103, 150502 (2009) [5] Preskill, J.: Quantum Computing in the NISQ era and beyond. Quant um2, 79 (2018) [6] Buhrman, H., R¨ ohrig, H.: Distributed quantum computing. In: Rovan , B., Vojt´ aˇ s, P. (eds.) Mathematical Foundations of Computer Science 2003, p p. 1–20. Springer, Berlin, Heidelberg (2003) [7] Beals, R., Brierley, S., Gray, O., Harrow, A., Kutin, S., Linden , N., Shepherd, D., Stather, M.: Eﬃcient distributed quantum computing. Proc. Math. Phys. Eng. Sci.469(2153) (2013) [8] Li, K., Qiu, D., Li, L., Zheng, S., Rong, Z.: Application of distribut ed semi- quantum computing model in phase estimation. Inf. Process. Lett. 120, 23–29 (2017) [9] Caleﬃ, M., Cacciapuoti, A.S., Bianchi, G.: Quantum internet: From communi- cation to distributed computing! In: Proceedings of the 5th ACM Inte rnational Conference on Nanoscale Computing and Communication, pp. 1–4. ACM, New York, USA (2018) [10] Avron, J., Casper, O., Rozen, I.: Quantum advantage and noise reduction in distributed quantum computing. Phys. Rev. A 104, 052404 (2021) [11] Qiu, D., Luo, L., Xiao, L.: Distributed Grover’s algorithm (2022). https://arxiv. org/abs/2204.10487. [12] Tan, J., Xiao, L., Qiu, D., Luo, L., Mateus, P.: Distributed quantum algor ithm for simon’s problem. Phys. Rev. A 106, 032417 (2022) [13] Xiao, L., Qiu, D., Luo, L., Mateus, P.: Distributed shor’s algorithm. Qu antum Inf. Comput. 23, 27–44 (2022) [14] Li, H., Qiu, D., Luo, L.: Distributed exact quantum algorithms for Deut sch-Jozsa problem (2023). https://arxiv.org/abs/2303.10663. [15] Xiao, L., Qiu, D., Luo, L., Mateus, P.: Distributed Quantum-classical Hybrid

Question 511 multiple-choice

Quantum computing systems are highly sensitive to environmental noise, which can cause decoherence and impact the performance of algorithms. The Approximate Quantum Fourier Transform (AQFT) and Quantum Fourier Transform (QFT) are two circuit architectures whose effectiveness depends on the interplay between gate count, approximation error, and noise strength. In the presence of strong environmental decoherence (high δ), which strategy most effectively maximizes the quality factor Q in AQFT networks, and why? 1) Increasing the number of single-qubit gates while keeping two-qubit gate count constant 2) Using the full QFT regardless of noise strength 3) Distributing decoherence evenly across all gates and wires 4) Minimizing the degree of approximation (using maximum m) 5) Applying error correction only to two-qubit gates 6) Reducing the number of gates by selecting m less than L, thus using AQFT instead of QFT 7) Maximizing the entanglement between qubits during transformation

✓ Correct Answer:

The correct answer is 6) Reducing the number of gates by selecting m less than L, thus using AQFT instead of QFT.

📚 Reference Text:

j/an}bracketri}ht, where the average /an}bracketle{t/an}bracketri}htis taken over diﬀerent realisations of the phase ﬂuctuations within a given period of time (cf. [12] ). The diagonal elements do not depend on φ, whereas the oﬀ–diagonal term /an}bracketle{tc0c1ei2φ/an}bracketri}htaverages to zero for a suﬃciently long period of time. The latter approach to decoherence is very convenient for nu merical simulations and was chosen for the purpose of this paper. It is similar to t he Monte Carlo wave function method used in quantum optics [14]. 6 AQFT and Decoherence We have analysed decoherence in the AQFT networks assuming t hat the environment introduces a random phase ﬂuctuation in a qubit probability amplitudes each time the qubit is aﬀected by gate B. In our model we have not attached any decoherence eﬀects to gate A. In most of the suggested physical realisations the single q ubit operations are quite fast, whereas the conditional logic needed in two– qubit operations is often much harder to produce, which makes these operations slower than the single qubit ones, and often much more susceptible to decoherence. For in stance, in the ion trap model proposed by Cirac and Zoller [15] single qubit operati ons require only one laser pulse interacting with one atom, whereas in a two–qubit oper ation two subsequent laser pulses are needed, and the atoms involved must form an e ntangled state with a trap phonon mode between the pulses. In any case, introducin g decoherence to gate Aoperations and ‘wires’ in the network would not aﬀect our res ults much, only the time scale for decoherence would change. We have quantiﬁed the performance of the AQFT by introducing the quality factor Q. It is simply the probability of obtaining an integer which i s closest to any integer multiple of 2L/r, when the state of the register is measured after the transformation. In the decoherence free environment analy sed in Sec. 4 we obtain Q= 1 for integer values of 2L/rand for a randomly selected rthe quality factor Q is of the order 4 /π2for the QFT and of the order of8 π2sin2/parenleftig 4 πm L/parenrightig for the AQFT of degree m. In Fig. 6 we show how the quality factor Qbehaves as a function of mandδ (which characterises the strength of the coupling to the env ironment). For δ >0 7 the maximum of Qis obtained for m < L . Thus in the presence of decoherence one should use the AQFT rather than the QFT. This ‘less is more’ result can be easily understood. The AQFT means less gates in the network and because each Bgate introduces phase ﬂuctuations the approximate network generates less decoherence as compare d to the regular QFT network. By decreasing mwe eﬀectively decrease the impact of decoherence. On the other hand decreasing mimplies approximations which reduces the quality factor. This trade-oﬀ between the two phenomena results in the maxim um value Qfor m∈[1,L]. It is worth pointing out that for δ= 0 (no decoherence) Qremains almost constant for

Question 512 multiple-choice

Post-quantum digital signature algorithms are being developed to withstand attacks from quantum computers and to maintain efficient public-key sizes for practical deployment. Multivariate cryptography and algebraic structures such as hidden groups and non-commutative algebras offer promising directions in this field. Which property most directly enables certain post-quantum signature schemes to achieve smaller public-key sizes compared to traditional multivariate cryptography approaches? 1) Dependence on classical discrete logarithm problems 2) Use of exponentiation in hidden groups that produces random-looking systems of equations over large finite fields 3) Reliance on lattice-based problem hardness 4) Implementation using only abelian group structures 5) Requirement for very large key sizes to ensure security 6) Dependence on symmetric-key primitives 7) Utilization of hash-based signature frameworks

✓ Correct Answer:

The correct answer is 2) Use of exponentiation in hidden groups that produces random-looking systems of equations over large finite fields.

📚 Reference Text:

Title: A new type of digital signature algorithms with a hidden group Year: 2023 Paper ID: d5e96812dd083fd8e5f22a30010203747f93e063 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d5e96812dd083fd8e5f22a30010203747f93e063 Abstract: The known designs of digital signature schemes with a hidden group, which use finite non-commutative algebras as algebraic support, are based on the computational complexity of the so-called hidden discrete logarithm problem. A similar design, used to develop a signature algorithm based on the difficulty of solving a system of many quadratic equations in many variables, is introduced. The significant advantage of the proposed method compared with multivariate-cryptography signature algorithms is that the said system of equations, which occurs as the result of performing the exponentiation operations in the hidden group, has a random look and is specified in a finite field of a higher order. This provides the ability to develop post-quantum signature schemes with significantly smaller public-key sizes at a given level of security.

Question 513 multiple-choice

Quantum algorithms often rely on group-theoretic problems, with the hidden subgroup problem (HSP) serving as a foundation for breakthroughs in areas like cryptography and computational complexity. Approaches to solving HSP in nonabelian groups use advanced techniques from representation theory and quantum Fourier analysis. In which scenario does the strong standard method for quantum Fourier sampling become strictly necessary for reconstructing hidden subgroups, compared to weaker methods such as measuring only representation names or using abelian approaches? 1) When working with abelian groups where all subgroup information is accessible via representation names 2) When solving the HSP for certain nonabelian groups, such as $q$-hedral and affine groups, where full representation data is required 3) When the subgroup is trivial and does not affect the structure of the group 4) When all subgroups are normal and easily distinguishable using classical algorithms 5) When random basis measurement provides complete information about the hidden subgroup 6) When the group has only one irreducible representation 7) When weak standard methods always outperform strong methods for all group types

✓ Correct Answer:

The correct answer is 2) When solving the HSP for certain nonabelian groups, such as $q$-hedral and affine groups, where full representation data is required.

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Title: The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts Year: 2005 Paper ID: 0ffce04825147d96afbdbbb84869f4fb9c0fd1a3 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0ffce04825147d96afbdbbb84869f4fb9c0fd1a3 Abstract: Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup $H$ of a group $G$ must be determined from a quantum state $\psi$ over $G$ that is uniformly supported on a left coset of $H$. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of $\psi$ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups $H$ of the $q$-hedral groups, i.e., semidirect products ${\mathbb Z}_q \ltimes {\mathbb Z}_p$, where $q \mid (p-1)$, and in particular the affine groups $A_p$, can be information-theoretically reconstructed using the strong standard method. Moreover, if $|H| = p/ {\rm polylog}(p)$, these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the “forgetful” abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems, generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently.

Question 514 multiple-choice

In the framework of the AdS/CFT correspondence, the encoding of bulk information into the boundary theory is closely related to quantum error correction and the structure of symmetry operators. The relationship between bulk and boundary symmetries has significant implications for the realization of global symmetries in holographic quantum gravity. Which property must bulk global symmetry operators satisfy when mapped to the boundary theory to ensure compatibility with quantum error correction in holography? 1) They must factorize as tensor products of local symmetry operators acting on each boundary subregion. 2) They must act trivially on all boundary subregions except one. 3) They must be non-splittable and supported on the entire boundary. 4) They must correspond only to gauge symmetries in the boundary theory. 5) They must commute with all local boundary operators. 6) They must be reconstructible only from the union of all boundary subregions. 7) They must act as non-local operators that cannot be decomposed on the boundary.

✓ Correct Answer:

The correct answer is 1) They must factorize as tensor products of local symmetry operators acting on each boundary subregion..

📚 Reference Text:

operator in the bulk, as its position asymptotically approaches the boundary, becomes dual to a local operator on the boundary. It follows that a bulk localsymmetry implies a corresponding symmetry of the dual boundary theory. Furthermore, a global symmetry operator of the boundary CFT is splittable [16]; that is, it can be expressed as a tensor product of many operators, eachCONTINUOUS SYMMETRIES AND APPROXIMATE QUANTUM … PHYS. REV . X 10,041018 (2020) 041018-23 supported on a small region. In coding theory language, the encoding isometry Vwhich maps bulk to boundary has the property VUL¼UCFTV; ð81Þ where ULis the bulk symmetry operator and UCFTis the corresponding CFT symmetry operator. Because the CFTsymmetry is splittable, we may consider decomposing theCFT into small spatial subregions fA kgand infer that UCFT¼⊗ kWk; ð82Þ where Wkis a CFT operator supported on Ak. Next, we would like to see that the boundary subregions can be chosen so that erasure of any Akis correctable. This point is most naturally discussed using the language ofoperator algebra quantum error correction [6]. We consider the subalgebra Aof logical operators which are supported on a subregion of the bulk. Each logical operator O L∈A can be “reconstructed ”as a physical operator OCFTacting on the boundary using the encoding isometry V: VOL¼OCFTV: ð83Þ What we wish to show is that, for any OLinA, and for each boundary subregion Ak, the reconstructed boundary oper- atorOCFTcan be chosen to have support on the comple- mentary boundary subregion Ac k. This property ensures that, for the bulk subalgebra A, erasure of boundary region Akis correctable. The argument showing that erasure of boundary sub- region Akis correctable is illustrated in Fig. 6. Associated with each boundary subregion Akis a bulk subregion ak which is called the entanglement wedge of Ak. The AdS/ CFT code has these important properties [6]: (i) A bulk operator supported in bulk subregion akcan be recon- structed as a boundary operator supported in boundarysubregion A k. This property is called subregion duality . (ii) Furthermore, a bulk operator supported in the bulkcomplement a c kof bulk subregion akcan be reconstructed as a boundary operator supported in the boundary comple- ment Ac kof boundary subregion Ak. This property is called complementary recovery . It follows from complementary recovery that if the bulk subalgebra Ais supported in ac k, then erasure of boundary subregion Akis correctable for the subalgebra A. This result is the key fact that we need. As in Fig. 6, for any fixed subregion a0of the bulk, we can choose the decomposition of the boundary into subregions fAkgsuch that a0lies outside the entanglement wedge of each Ak. Therefore, the algebra Aof bulk operators supported on a0has the feature that erasure of each Akis correctable for the algebra A. This result completes the first step of the Harlow-Ooguri argu-ment, showing that a bulk global symmetry operator U Lmust be transversal in the sense we assume in this paper —it factorizes as a tensor product of boundary operators, each of which is supported

Question 515 multiple-choice

In quantum information theory, the structure of operators invariant under unitary transformations is characterized using group representation theory and concepts such as Schur-Weyl duality. The commutant of the unitary group action is central to understanding symmetries and decompositions in multipartite quantum systems. For a two-copy quantum system ($k=2$), which set of operators precisely spans the commutant of the unitary group action on $\mathbb{C}^d \otimes \mathbb{C}^d$? 1) Only the flip operator $F$ 2) All diagonal projectors $\sum |i,i\rangle\langle i,i|$ 3) All permutation matrices of $S_d$ 4) The identity $I$ and all diagonal projectors 5) The identity operator $I$ and the flip operator $F$ 6) All rank-one projectors 7) The set of all Hermitian operators

✓ Correct Answer:

The correct answer is 5) The identity operator $I$ and the flip operator $F$.

📚 Reference Text:

the trace, we obtain the linear system of equation: /braceleftigg Tr(O) =cI,Od2+cF,Od Tr(FO) =cI,Od+cF,Od2. (56) Solving this system, we obtain cI,O=Tr (O)−d−1Tr (OF) d2−1andcF,O=Tr (FO)−d−1Tr (O) d2−1. We have shown how to obtain the formulas for the first and second moment operator by utilizing the property that the moment operator lies in the commutant (Lemma 6.2) and that the commutant is the span of the permutation matrices Comm(U(d),k) = span/parenleftig Vd(π) :π∈Sk/parenrightig for k= 1,2, which are particular instances of Schur-Weyl duality. Now, we will provide explicit proofs for these cases. Example 14. Comm(U(d),k= 1) = span/parenleftig I/parenrightig , (57) Comm(U(d),k= 2) = span/parenleftig I,F/parenrightig . (58) Proof. Consider an element A∈Comm(U(d),k= 1). In the canonical basis decomposition, Acan be written as: A=d/summationdisplay i,j=1Ai,j|i⟩⟨j|=d/summationdisplay i=1Ai,i|i⟩⟨i|+d/summationdisplay i,k=1 j̸=iAi,j|i⟩⟨j|, (59) whereAi,j:=⟨i|A|j⟩. SinceAis in the commutant, then UAU†=Aholds for all unitary matrices U. By choosing a unitary Uthat sends vector |k⟩to its negative|k⟩→−|k⟩while leaving the other basis vectors unchanged, it follows that, for all k̸=l∈[d], the term Ak,l|k⟩⟨l|is equal to its negative, and hence, Ak,l= 0. Therefore, we have that if Ais in the commutant, then A=/summationtextd i=1Ai,i|i⟩⟨i|. Now, for all k̸=l∈[d], by choosing a unitary matrix Uthat exchanges the vectors|k⟩and|l⟩while leaving the other basis states unchanged, we derive that all diagonal elementsAk,kshould be equal each other for all k∈[d]. Thus, we have A=c/summationtextd i=1|i⟩⟨i|=cI, where c:=A1,1. Therefore an operator can only be in the commutant if and only if it is proportional to the Identity I. Thus, we have shown the first formula we aimed to prove. 8 Consider now Q∈Comm(U(d),k= 2).Qcan be decomposed as: Q=d/summationdisplay i,j,k,l =1Qi,j;k,l|i,j⟩⟨k,l|, (60) whereQi,j;k,l:=⟨i,j|Q|k,l⟩. SinceQis in the commutant, then U⊗2QU†⊗2=Qholds for all unitary matrices U. For all sets of indices {i,j}̸={k,l}, letmbe a index such that m∈{i,j}but m /∈{k,l}or vice versa (there is at least one). By choosing unitaries that send a basis vector to its negative, while leaving unchanged the other basis vectors, it follows that, for all {i,j}̸={k,l}∈[d], the termsQi,j;k,l|i,j⟩⟨k,l|are equal to their negatives, and hence, Qi,j;k,l= 0. Therefore, if Qis in the commutant, it must have the following form: Q=d/summationdisplay i=1Qi,i;i,i|i,i⟩⟨i,i|+d/summationdisplay i,j=1 i̸=jQi,j;i,j|i,j⟩⟨i,j|+d/summationdisplay i,j=1 i̸=jQi,j;j,i|i,j⟩⟨j,i|. (61) Again, by choosing a unitary matrix Uthat exchanges the vectors |i,j⟩with a different vector |k,l⟩while leaving the other basis states unchanged, and using U⊗2QU†⊗2=Q, we derive that: Q=ad/summationdisplay i=1|i,i⟩⟨i,i|+bd/summationdisplay i,j=1 i̸=j|i,j⟩⟨i,j|+cd/summationdisplay i,j=1 i̸=j|i,j⟩⟨j,i| (62) wherea:=Q1,1;1,1,b:=Q1,2;1,2andc:=Q1,2;2,1. Note that we can also write Qas: Q= (a−b−c)d/summationdisplay i=1|i,i⟩⟨i,i|+bI+cF. (63) Since IandFcommute with U⊗2for allU∈U(d), thenQis in the commutant if and only if (a−b−c)/summationtextd i=1|i,i⟩⟨i,i|is also in the commutant. We now show that/summationtextd i=1|i,i⟩⟨i,i|is not in the commutant, which implies that Qis in the commutant if and only if ( a−b−c) = 0, i.e., if and only if Qis a linear combination of only I andF. To see this, suppose that/summationtextd i=1|i,i⟩⟨i,i|is in the commutant. Then, it should commute withU⊗2for all unitaries U. We can choose the (Discrete Fourier Transform) unitary Usuch that U|k⟩=1√ d/summationtextd j=1ω(j−1)(k−1)|j⟩, whereω:= exp/parenleftbig i2π d/parenrightbig , and obtain the following relation: d/summationdisplay i=1|i,i⟩⟨i,i|=d/summationdisplay i=1U⊗2|i,i⟩⟨i,i|U⊗2†. (64) However,

Question 516 multiple-choice

The Hidden Subgroup Problem (HSP) is a foundational challenge in quantum computing, especially relevant for group structures such as semi-direct products of cyclic groups. Efficient quantum algorithms for solving the HSP depend crucially on the algebraic properties of the underlying group. Which property of the semi-direct product group $ G = \mathbb{Z}_N \rtimes \mathbb{Z}_{q^s} $, where N is a product of prime powers and q is an odd prime not equal to any $ p_i $, ensures that the HSP can be efficiently solved using quantum algorithms? 1) The presence of a normal abelian subgroup in $ G $ 2) The requirement that q is a divisor of N 3) The group being simple and non-cyclic 4) The existence of a nontrivial group action of $ \mathbb{Z}_{q^s} $ on $ \mathbb{Z}_N $ 5) Specific arithmetic conditions on the primes and exponents that allow reduction to direct product cases with known quantum solutions 6) The group having a unique maximal subgroup 7) The restriction to groups where all elements have prime order

✓ Correct Answer:

The correct answer is 5) Specific arithmetic conditions on the primes and exponents that allow reduction to direct product cases with known quantum solutions.

📚 Reference Text:

the restriction of ftoB, which hides H2inB. The solution of the HSP in the groups AandBwith functions f1andf2determines generators for the subgroups H1 andH2, respectively. The group Ais abelian and the group Bwas addressed in [5,6] and recently generalized by [4]. Therefore, we obtain the following result: Theorem 3.1. LetNbe a positive integer with prime factorization pr1 1:::prnn,qan odd prime such that q6=piandsa positive integer. De ne the semi-direct product group G=ZNo Zqsfor an 2Z N. Lett2f1;:::;sgbe the smallest positive integer such that qt= 1. Let us assume that there exist a 1knsuch thatqtjpk1andq-pi1 for alli6=k. Then there exists an ecient quantum algorithm that solves the HSP in the semi-direct product groups ZNo Zqs. 4 Conclusion We have addressed the HSP on the semi-direct product groups G=ZNoZqswhereN is factorized as N=pr1 1:::prnnand given a 1ts, there exists a 1knsuch thatqt dividespk1q-pi1 for alli6=k. By employing an isomorphism between ZNoZqsandProceeding Series of the Brazilian Society of Applied and Computational Mathematics, Vol. 3, N. 2, 2015. DOI: 10.5540/03.2015.003.02.0045 020045-4 © 2015 SBMAC 5 the direct product of Zpro Zqswith cyclic groups we have shown that the HSP can be reduced to similar HSPs the solutions of which are already known. This provides a new ecient solution for the HSP on G. Acknowledgements The authors would like to tank the CNPq and Faperj for nancial support and the anonymous reviewers for their valuable comments and suggestions to improve the manuscript. References [1] R. Beals, Quantum computation of Fourier transforms over symmetric groups, Proc. 29th ACM Symp. on Theory of Computing, pages 48{53, New York, (1997). [2] D. P. Chi, J. S. Kim and S. Lee, Quantum algorithms for the hidden subgroup problem on some semi-direct product groups by reduction to Abelian cases, Physics Letters A, pages 114{116, (2006). [3] A. M. Childs and W. van Dam, Quantum algorithms for algebraic problems, Rev. Mod. Phys., 82(1): 1{52, (2010). [4] W. van Dam and S. Dey, Hidden Subgroup Quantum Algorithms for a Class of Semi-Direct Product Groups, Proc. of 9th Conference on the Theory of Quantum Computation, Communication and Cryptography, TQC14, pages 110{117, (2014). [5] D. N. Gon calves, R. Portugal, and C. M. M. Cosme, Solutions to the hidden sub- group problem on some metacylic groups, Proc. 4th Worshop on Theory of Quantum Computation, Communication and Cryptography, LNCS, Springer-Verlag, (2009). [6] D. N. Goncalves and R. Portugal, Solution to the Hidden Subgroup Problem for a Class of Noncommutative Groups, Quantum Physics, Abstract quant-ph/1104.1361, (2011). [7] C. Lomont, The Hidden Subgroup Problem - Review and Open Problems, Quantum Physics, Abstract quant-ph/0411037, (2004). [8] M. Mosca, Quantum algorithms, Encyclopedia of Complexity and Systems Science, pages 7088{7118, (2009). [9] O. Regev, Quantum Computation and Lattice Problems, SIAM Journal on Comput- ing, 33(3):738{760, (2004). [10] D. R. Simon, On the Power of Quantum Computation, Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pages 116{123, (1994).Proceeding Series of the Brazilian Society of Applied and Computational Mathematics, Vol. 3, N. 2, 2015. DOI: 10.5540/03.2015.003.02.0045 020045-5 © 2015

Question 517 multiple-choice

Quantum algorithms often rely on solving hidden subgroup problems (HSPs), particularly in groups with complex structures relevant to cryptography and computational mathematics. Distinguishing between various quantum Fourier sampling techniques is crucial for understanding the limitations and powers of these algorithms. In the context of nonabelian hidden subgroup problems, which statement best describes a key advantage of strong Fourier sampling over weak Fourier sampling for certain group families? 1) Strong Fourier sampling always yields polynomial-time algorithms for all nonabelian groups. 2) Weak Fourier sampling reconstructs hidden subgroups for all q-hedral and affine groups efficiently. 3) Random basis measurement and the abelian reduction are strictly more powerful than strong Fourier sampling for cryptographically motivated groups. 4) Strong Fourier sampling can efficiently reconstruct hidden subgroups in q-hedral and affine groups where weak sampling fails. 5) Strong and weak Fourier sampling are equivalent for all groups, regardless of their structure. 6) Weak Fourier sampling provides complete information about representation matrices in high-dimensional cases. 7) Hidden shift problems cannot be addressed by any Fourier sampling technique.

✓ Correct Answer:

The correct answer is 4) Strong Fourier sampling can efficiently reconstruct hidden subgroups in q-hedral and affine groups where weak sampling fails..

📚 Reference Text:

Title: The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts Year: 2005 Paper ID: 0ffce04825147d96afbdbbb84869f4fb9c0fd1a3 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0ffce04825147d96afbdbbb84869f4fb9c0fd1a3 Abstract: Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup $H$ of a group $G$ must be determined from a quantum state $\psi$ over $G$ that is uniformly supported on a left coset of $H$. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of $\psi$ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups $H$ of the $q$-hedral groups, i.e., semidirect products ${\mathbb Z}_q \ltimes {\mathbb Z}_p$, where $q \mid (p-1)$, and in particular the affine groups $A_p$, can be information-theoretically reconstructed using the strong standard method. Moreover, if $|H| = p/ {\rm polylog}(p)$, these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the “forgetful” abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems, generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently.

Question 518 multiple-choice

Kohn-Sham Density Functional Theory (KSDFT) is a foundational computational method used to determine electronic properties of molecules and solids by solving quantum mechanical equations. Its efficiency and accuracy depend on the nature of the orbitals, the computational algorithms, and the handling of electron interactions. Which statement about the occupied Kohn-Sham orbitals in KSDFT is correct regarding their role in determining physical properties? 1) Only the specific functional form of each occupied orbital affects physical observables. 2) Changing the occupied orbitals via any arbitrary nonlinear transformation alters the physical properties. 3) Physical properties are determined solely by the highest occupied orbital and not the full set of occupied orbitals. 4) Linear degenerate transformations of occupied orbitals change observable quantities. 5) Only the unoccupied orbitals contribute to the electron density relevant for observables. 6) The electron density and thus physical properties depend on the ordering of occupied orbitals. 7) Any non-degenerate linear transformation of the set of occupied Kohn-Sham orbitals yields the same physical properties.

✓ Correct Answer:

The correct answer is 7) Any non-degenerate linear transformation of the set of occupied Kohn-Sham orbitals yields the same physical properties..

📚 Reference Text:

Kohn-Sham density functional theory. For a given atomic con gura- tion withMatoms at locations fRIgM I=1, KSDFT solves the nonlinear eigenvalue problem ^H[^;fRIg]^ i="i^ i; ^(r) =neX i=1 ^ i(r) 2 ;Z ^ i(r)^ j(r) dr=ij:(2.1) For simplicity we omit the spin degeneracy. The number of electrons is neand the eigenvaluesf"igare ordered non-decreasingly. The lowest neeigenvaluesf"igne i=1are called the occupied state energies, and f"igj>n eare called the unoccupied state en- ergies. We assume "g:="ne+1"ne>0. Here"neis often called the highest occupied molecular orbital (HOMO), "ne+1the lowest unoccupied molecular orbital (LUMO), and hence "gthe HOMO-LUMO gap. For extended systems, if "gis uni- formly bounded away from zero as the system size increases, the quantum system is an insulating system [27]. The eigenfunctions f^ igne i=1de ne the electron density ^ (r), which in turn de nes the Kohn-Sham Hamiltonian (2.2) ^H[^;fRIg] =1 2 + ^Vc[^] +^Vxc[^] +Vion[fRIg]: 3 Here is the Laplacian operator for the kinetic energy of electrons, ^Vc[^](r)Z^(r0) jrr0jdr0 is the Coulomb potential, and ^Vcdepends linearly with respect to the electron density ^.^Vxc[^] depends nonlinearly with respect to ^ , and characterizes the many body exchange and correlation e ect. Vion[fRIg] is an external potential depending explic- itly on the ionic positions, and describes the electron-ion interaction potential and is independent of ^ . Because the eigenvalue problem (2.1) is nonlinear, it is often solved iteratively by a class of algorithms called self-consistent eld iterations (SCF) [27], until (2.1) reaches self-consistency. In a nite dimensional discretization of Eq. (2.2), let Nbe the number of degrees of freedom. Using a large basis set such as the plane-wave basis set, we have N=cneand cis a large constant that is often 102104. Due to this large constant, we explicitly distinguish Nandnein the complexity analysis below. For large scale systems, the cost for storing the Kohn-Sham orbitals is O(Nne), and the cost for computing them is generallyO(Nn2 e) and scales cubically with respect to ne. In modern KSDFT calculations the Hartree-Fock exact exchange term is also often taken into account in the form of hybrid functionals [35, 3]. The computational cost for this step not only scales asO(Nn2 e) but also has a large pre-constant. When the self-consistent solution of the Kohn-Sham equation is obtained, the existence of nite HOMO-LUMO gap has important implications on the collective behavior of the occupied Kohn-Sham orbitals f^ igne i=1. Since any non-degenerate linear transformation of the set of Kohn-Sham orbitals yields exactly the same physical properties of a system, the physically relevant quantity is the subspace spanned by the Kohn-Sham orbitals f^ igne i=1. Various e orts [15, 30, 29, 18, 14, 34] have been made to utilize this fact and to nd a set of localized orbitals that form a compressed representation of a Kohn-Sham subspace. In other words, we nd a set of functions f^'igne i=1whose span is the same as the span of f^ igne i=1. Compared to each Kohn-Sham orbital ^ iwhich is delocalized in the real space, each compressed orbital

Question 519 multiple-choice

In quantum mechanics, the geometric (Berry) phase is a topological phenomenon arising when a system's parameters are adiabatically cycled, and its properties can be described using advanced mathematical tools from geometry and representation theory. These include concepts such as line bundles, holonomies, flag manifolds, and topological charges. Which mathematical theorem links the classification of complex line bundles over flag manifolds to representations of compact connected semisimple Lie groups, thereby enabling the computation of topological charges associated with geometric phases? 1) Atiyah–Singer Index Theorem 2) Noether’s Theorem 3) Stokes’ Theorem 4) Gauss–Bonnet Theorem 5) Borel–Weil–Bott Theorem 6) Hodge Decomposition Theorem 7) Poincaré Duality

✓ Correct Answer:

The correct answer is 5) Borel–Weil–Bott Theorem.

📚 Reference Text:

Title: Geometric phase, bundle classification, and group representation Year: 1993 Paper ID: bcd2f45d753b49a2ed1365ef9d0cc3f77ca92bd1 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/bcd2f45d753b49a2ed1365ef9d0cc3f77ca92bd1 Abstract: The line bundles that arise in the holonomy interpretations of the geometric phase display curious similarities to those encountered in the statement of the Borel–Weil–Bott theorem of the representation theory. The remarkable relationship between the mathematical structure of the geometric phase and the classification theorem for complex line bundles provides the necessary tools for establishing the relevance of the Borel–Weil–Bott theorem to Berry’s adiabatic phase. This enables one to define a set of topological charges for arbitrary compact connected semisimple dynamical Lie groups. These charges signify the topological content of the phase. They can be explicitly computed. In this paper, the problem of the determination of the parameter space of the Hamiltonian is also addressed. It is shown that, in general, the parameter space is either a flag manifold or one of its submanifolds. A simple topological argument is presented to indicate the relation between the Riemannian structure on the parameter spa...

Question 520 multiple-choice

Graduate computing education often emphasizes technical proficiency, but there is growing interest in pedagogical methods that foster engagement and inclusivity. Strategies such as storytelling and providing historical context are being explored alongside other active learning approaches. Which of the following strategies is identified as both simplifying complex ideas and enhancing student engagement, while also being noted as underutilized in highlighting the contributions of diverse individuals in computing? 1) Gamification 2) Project-based learning 3) Peer instruction 4) Flipped classroom 5) Collaborative coding exercises 6) Case studies 7) Storytelling

✓ Correct Answer:

The correct answer is 7) Storytelling.

📚 Reference Text:

Title: Exploring Graduate Computing Students' Perceptions of Storytelling and Historical Context in STEM Education Year: 2025 Paper ID: bf8fc0f774bbd7f8cc22832469c589d7f10147c0 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/bf8fc0f774bbd7f8cc22832469c589d7f10147c0 Abstract: Computing education often emphasizes technical mastery while overlooking strategies that foster engagement and inclusivity. This study explores how storytelling and historical context can bridge that gap, particularly at the graduate level where such approaches are underexplored. A mixed-methods survey of 26 graduate students included Likert-scale and open-ended questions. 85% of participants believed storytelling could have a long-term impact on student interest, while historical context received a mean score of 2.77/5, indicating mixed views. Thematic analysis revealed three key themes: storytelling enhanced engagement by simplifying complex ideas; historical context added relevance by connecting theory to real-world applications; and representation was limited, with storytelling underutilized to highlight diverse contributions. While no significant group differences emerged quantitatively, qualitative responses suggest storytelling boosts engagement but has an unclear impact on inclusivity. These findings support integrating narrative and historical elements into computing education to increase student connection and diversity awareness. Future work should involve larger samples and compare storytelling to other engagement strategies like gamification or project-based learning.

Question 521 multiple-choice

Quantum phase estimation algorithms are foundational in quantum computing, enabling the determination of eigenphases of unitary operators with high precision. Modifications to these algorithms can enhance their robustness and adapt them to specialized computational requirements. Which of the following statements accurately describes a key modification made to the standard quantum phase estimation algorithm to estimate the floor of (2nλj) with improved reliability? 1) The algorithm repeats the estimation process M times and selects the median value using a quantum sorting network to guarantee that the probability of correct estimation exceeds 1/2 + η when α > 1/2. 2) The algorithm replaces the inverse quantum Fourier transform with a classical post-processing step to minimize query complexity. 3) The algorithm only prepares a superposition over time indices without applying controlled-U operations, relying on classical amplification for accuracy. 4) The phase estimation procedure is altered to estimate the ceiling of (2nλj) instead of the floor, boosting success probability for all values of α. 5) The modification involves applying a Hadamard transformation after each controlled-U operation to eliminate garbage qubits. 6) The algorithm reduces the number of required qubits by omitting the median amplification step and relying on a single estimate. 7) The algorithm uses majority voting among multiple runs instead of the median to select the final estimate when α ≤ 1/2.

✓ Correct Answer:

The correct answer is 1) The algorithm repeats the estimation process M times and selects the median value using a quantum sorting network to guarantee that the probability of correct estimation exceeds 1/2 + η when α > 1/2..

📚 Reference Text:

Then the phase estimator has query complexity: (2n−1)·/ceilingleftBigg ln(δ−1 med) 2η2/ceilingrightBigg (38) and hasn/ceilingleftbigg ln(δ−1 med) 2η2/ceilingrightbigg qubits of garbage. Proof.We begin with the α>1/2case which is simpler, and then extend to the α≤1/2 case. The following is a review of phase estimation, which has been modiﬁed to estimate ﬂoor(2nλj)rather than round (2nλj): 1. Prepare a uniform superposition over times: |+n/angbracketright|ψj/angbracketright=1√ 2n/summationtext2n−1 t=0|t/angbracketright|ψj/angbracketright. 2. ApplyUto the|ψj/angbracketrightregisterttimes, along with an additional phase shift of−2πt(1−α) 2n+1 to account for ﬂooring: →1√ 2n2n−1/summationdisplay t=0|t/angbracketright⊗Ute−2πit(1−α) 2n+1|ψj/angbracketright=1√ 2n2n−1/summationdisplay t=0e2πit/parenleftbig λj−1−α 2n+1/parenrightbig |t/angbracketright|ψj/angbracketright(39) Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 11 3. Apply an inverse quantum Fourier transform to the |t/angbracketrightregister: →1 2n2n−1/summationdisplay t=02n−1/summationdisplay x=0e2πit/parenleftbig λj−1−α 2n+1/parenrightbig e−2πi 2ntx|x/angbracketright|ψj/angbracketright (40) =2n−1/summationdisplay x=0/bracketleftBigg 1 2n2n−1/summationdisplay t=0e2iπt/parenleftbig λj−x 2n−1−α 2n+1/parenrightbig/bracketrightBigg |x/angbracketright|ψj/angbracketright (41) =2n−1/summationdisplay x=0β/parenleftBig λ(x) j/parenrightBig |x/angbracketright|ψj/angbracketright (42) where in the ﬁnal line we have deﬁned: λ(x) j:= 2nλj−x−1−α 2(43) β(λ(x) j) :=1 2n2n−1/summationdisplay t=0e2πitλ(x) j/2n. (44) 4. Letηandδmedbe as in the theorem statement, and let: M:=/ceilingleftBigg log(δ−1 med) 2η2/ceilingrightBigg (45) Repeat the above process for a total of Mtimes. If we sum /vector xover{0,...,2n−1}M, then we obtain: →/summationdisplay /vector xM/circlemultiplydisplay l=1β/parenleftBig λ(xl) j/parenrightBig |kl/angbracketright⊗|ψj/angbracketright (46) 5. Run a sorting network, e.g. [Beals&12], on the Mestimates, and copy the median into the output register. The query complexity analysis is straightforward: each estimate requires 2n−1ap- plications of controlled- U, and there are Mestimates. So all we must do is demonstrate accuracy. It is clear that the process above implements a map of the form: |0n/angbracketright|0...0/angbracketright|ψj/angbracketright→2n−1/summationdisplay x=0eiϕj,x√pj,x|x/angbracketright|garj,k/angbracketright|ψj/angbracketright (47) for some probabilities pj,xand phases ϕj,x, since this is just a general description of a unitary map that leaves the |ψj/angbracketrightregister intact. This way of writing the map lets us bound the error in diamond norm to the ideal map |0n/angbracketright|0...0/angbracketright|ψj/angbracketright→eiϕj|ﬂoor(λj2n)/angbracketright|garj/angbracketright|ψj/angbracketright (48) (selectingϕj:=ϕj,ﬂoor(λj2n)and|garj/angbracketright:=|garj,ﬂoor(λj2n)/angbracketright) without uncomputing while still employing the standard analysis of phase estimation which just reasons about the probabilities pj,x. These can be bounded from a median ampliﬁcation analysis of the probabilities/vextendsingle/vextendsingle/vextendsingleβ/parenleftBig λ(x) j/parenrightBig/vextendsingle/vextendsingle/vextendsingle2. Consider a vector of Mestimates/vector x. Then: pj,x:=/summationdisplay /vector xwhere median (/vector x)=xM/productdisplay l=1/vextendsingle/vextendsingle/vextendsingleβ(λ(xl) j)/vextendsingle/vextendsingle/vextendsingle2(49) Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 12 First we show that probabilities/vextendsingle/vextendsingle/vextendsingleβ/parenleftBig λ(x) j/parenrightBig/vextendsingle/vextendsingle/vextendsingle2associated with the individual estimates are bounded away from1 2. Using some identities we can rewrite: |β(λ(x) j)|2=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 2n2n−1/summationdisplay t=0e2iπtλ(x) j/2n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (50) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 2ne2iπλ(x) j−1 e2iπλ(x) j/2n−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (51) =1 4nsin2(2πλ(x) j) + (cos(2πλ(x) j)−1)2 sin2(2πλ(x) j/2n) + (cos(2πλ(x) j/2n)−1)2(52) =1 4n2−2 cos(2πλ(x) j) 2−2 cos(2πλ(x) j/2n)(53) =1 4nsin2(πλ(x) j) sin2(πλ(x) j/2n)(54) Using the small angle approximation sin(θ)≤θfor0≤θ≤π, we can derive: |β(λ(x) j)|2=1 4nsin2(πλ(x) j) sin2(πλ(x) j/2n)≥1 4nsin2(πλ(x) j) π2(λ(x) j)2/4n=sin2(πλ(x) j) π2(λ(x) j)2=:γ(λ(x) j)(55) γ(λ(x) j)is a very tight lower bound to |β(λ(x) j)|2, and is plotted in Figure 1. We would like this probability to be larger than1 2+ηfor someηwhenλjsatisﬁes a rounding promise. This is guaranteed if we select: η:=γ/parenleftbigg1−α 2/parenrightbigg −1 2(56) as in the theorem statement. From the ﬁgure, we see that when α > 1/2thenη > η 0. This fact actually is not necessary for this

Question 522 multiple-choice

Probabilistic algorithms play a critical role in state estimation and optimization within fields such as robotics, computer vision, and artificial intelligence. Understanding the mechanics and interrelations of particle filters, simulated annealing, and Monte Carlo Tree Search is essential for effective application in complex and uncertain environments. Which of the following best describes the role of the measurement model $ p(z_t|x_t) $ in Bayesian filtering algorithms for dynamical systems? 1) It predicts the next state given the previous state and control input. 2) It encodes the initial uncertainty about the system's starting condition. 3) It defines the acceptance probability of moves in simulated annealing. 4) It guides the exploration-exploitation balance in Monte Carlo Tree Search. 5) It relates the hidden state of the system to the observed data, allowing corrections to predictions. 6) It determines the mutation rate in genetic algorithms used for state estimation. 7) It builds the search tree by simulating random playouts in decision-making algorithms.

✓ Correct Answer:

The correct answer is 5) It relates the hidden state of the system to the observed data, allowing corrections to predictions..

📚 Reference Text:

distri- bution of the state xtof a dynamical system at time tconditioned on the data, p(xt|zt, ut). This posterior is estimated via the following re- cursive formula p(xt|zt, ut) = ηtp(zt|xt)Z p(xt|ut, xt−1)p(xt−1|zt−1, ut−1)dxt−1, (2) where ηtis a normalization constant. Three probability distributions are required for such a particle filter: (1) a so-called measure- ment model ,p(zt|xt), which gives the probabil- ity of measuring ztwhen the system is in state xt; (2) a control model ,p(xt|ut, xt−1), which models the effect of a control uton the system state. It provides the probability that the sys- tem is in state xtafter executing control utat state xt−1; and (3) an initial state distribution p(x0) is required, to specify the user’s knowl-edge about the initial system state, see also [37]. In computer vision, the so-called condensation algorithm is a well-known example of how to perform a conditional density propagation for visual tracking [38]. The implementation of a particle filter can be very similar to that of a genetic algorithm. It can be based on Mindependent random vari- ables ξi 0, (i= 1, . . . , M ) with a probability den- sityp(x0). Based on the underlying distribu- tion, e.g., representing a fitness score, Mof 4 these variables are selected ξi k→ˆξi kand dif- fused using a mutation-like operation, yielding a new set ξi k+1. C. Simulated annealing Simulated Annealing (SA) is another prob- abilistic technique to approximating the opti- mum of a given function [39]. The name de- rives from annealing in metallurgy where the process involves heating and a controlled cool- ing of a material to change and control its phys- ical properties. As an optimization scheme the algorithm works iteratively with respect to time tgiven a state xt. At each step, the simulated annealing heuristic samples a neighboring state ˆxtof the current state xt. Then a probabilistic decision is made to decide whether to move to the new state xt+1= ˆxtor to remain in the for- mer state xt+1=xt. The probability of making the transition from the current state xtto the new state ˆ xtis defined by an acceptance proba- bility function P(e(xt), e(ˆxt), T). The function e(x) evaluates the energy of this state, which is in our case the fitness score given by the opti- mization task (e.g. the ℓ2-loss). The parameter Tis a time-dependent variable dictating the be- havior of the stochastic process according to a cooling scheme or annealing schedule. The P function is typically chosen in such a way that the probability of accepting an uphill move de- creases with time and it decreases as the dif- ference e(ˆxt)−e(xt) increases. Thus, a small increase in error is likely to be accepted so that local minima can be avoided, whereas a larger error increase is not likely to be accepted. A typical function for Ptakes the form P(e(xt), e(ˆxt), T)∝exp −e(ˆxt)−e(xt) kT , (3) with kadamping factor k >0.D. Monte Carlo tree search Monte-Carlo tree search (MCTS) is a heuris- tic search algorithm for decision processes [29].

Question 523 multiple-choice

In the study of compact, connected Lie groups and their representations, understanding the relationship between the group, its Lie algebra, and associated geometric structures is crucial for applications in geometry and theoretical physics. The moment polytope and its membership problem are central topics in this area, linking algebraic and geometric perspectives. Which statement correctly describes the relationship between a compact, connected Lie group G and its simply-connected covering group ~G in the context of representation theory and moment polytopes? 1) G and ~G share the same Lie algebra and moment polytope, and representations of G can be extended to ~G without changing irreducibility. 2) G and ~G always have different Lie algebras, making their moment polytopes non-comparable. 3) Only G, not ~G, possesses a well-defined weight lattice for its representations. 4) The moment polytope of ~G is strictly larger than that of G due to the covering group structure. 5) Irreducibility of representations is lost when passing from G to its covering group ~G. 6) The Lie algebra of ~G determines the entire structure of G, including its global topology. 7) ~G and G generally have different Weyl groups, leading to distinct root systems.

✓ Correct Answer:

The correct answer is 1) G and ~G share the same Lie algebra and moment polytope, and representations of G can be extended to ~G without changing irreducibility..

📚 Reference Text:

k(~ A;~ B;~ C)(diagA;diagB;diagC)=kkF: In view of (6), our algorithm will therefore never accept if (A;B;C)=k62Kron (m). We conclude that the problem KronPolytope is in NP. Subsections 2.2 and 2.3 together establish Theorem 1. 3. The general membership problem. We now turn to the membership problem for the moment polytope associated with an arbitrary ﬁnite-dimensional unitary representation Mof a compact, connected Lie group G. We start with some classical facts from the theory of Lie groups and Lie algebras (see, e.g., [ 24,20]). Recall that the Lie algebra of Gis of the form g=g1 gn, 10 P. BÜRGISSER, M. CHRISTANDL. K.D. MULMULEY, M. WALTER where each gjis either the compact real form of a complex simple Lie algebra or one- dimensional abelian. The Lie algebra gdoes not determine Gcompletely. However, any compact, connected Ghas a ﬁnite covering group ~G, which is a product G1Gn where each factor is either the unique (up to isomorphism) connected, simply-connected Lie groupGjwith simple Lie algebra gjor a one-dimensional torus U(1)with Lie algebraiR. BothGand ~Ghave the same Lie algebra and we can reconstruct ~Gfrom g. Furthermore, any representation MofGcan be extended to a representation of ~G, and it is irreducible for Gif and only if it is irreducible for ~G. In particular, both groupsGand ~Glead to the same moment polytope: G(M) = ~G(M). On the one hand, this explains why it suﬃces for our purposes to specify the Lie algebra only, as we will do in section 3.1 below. On the other hand, this also allows us to assume without loss of generality that G=~G=G1Gnin our analysis, and we shall do so throughout the remainder of this section. In particular, this allows us to make the following choices in a coherent way: For each Dynkin diagram, we ﬁx a corresponding compact, connected, simply- connected Lie group Gand simple Lie algebra g, and choose once and for all a maximal torusTGand positive roots P(G). Let gC=g Cdenote the complexiﬁcation. For each root , we choose a basis vector e 2igin the corresponding root space of gCas well as the corresponding coroot h 2itsuch that the commutation relations of sl2(C), i.e., [e ;e ] =h etc., are satisﬁed. The coroots h span a lattice Git of maximal rank, with basis given by the simple coroots (i.e., those corresponding to simple roots). The elements of the dual basis of the simple coroots are known as the fundamental weights, they form a basis of the weight lattice Git. Together, the e for arbitrary roots and the simple coroots h determine a basis of ig. Finally, we equipigwith the Killing form, which is positive deﬁnite on ig, normalized such that the long roots have norm one. For a one-dimensional torus G=U(1), we haveG=Tandig=it=R. We identify both the weight lattice Gand its dual lattice GwithZ, choose 12Zas a basis vector of both lattices, and use the standard inner product on R. Given an arbitrary group G=G1Gn, the above choices determine a maximal torus T, Lie algebra t, Weyl

Question 524 multiple-choice

Quantum information processing often utilizes qudits—quantum systems with more than two levels—to enable richer computational possibilities. The quantum Fourier transform (QFT) must be adapted for systems where qudits of different dimensions are present, such as those implemented via nuclear magnetic resonance (NMR). Which of the following statements correctly describes the total number of quantum states in a hybrid system composed of two qubits and one qutrit? 1) There are 6 quantum states because each particle can occupy a unique state. 2) There are 9 quantum states, given by 3 × 3 since the qutrit dominates the state count. 3) There are 8 quantum states, as only the qubits contribute to the total state count. 4) There are 16 quantum states, calculated as 2 × 2 × 4 for two qubits and a four-level system. 5) There are 12 quantum states, found by multiplying 2 × 2 × 3 for two qubits and one qutrit. 6) There are 18 quantum states, based on the sum of all possible individual states. 7) There are 24 quantum states, derived from a permutation of the three particles.

✓ Correct Answer:

The correct answer is 5) There are 12 quantum states, found by multiplying 2 × 2 × 3 for two qubits and one qutrit..

📚 Reference Text:

superposition of all basis states of the qudit, with equal amplitudes, withitsactiononthe pthquditinan N-qudit system given by [15, 36]: Fp|xj/an}bracketri}ht=1√ dd−1/summationdisplay yk=0exp/parenleftbigg2πιxjyk d/parenrightbigg |yk/an}bracketri}ht xj,yk∈[0,1,2,...(d−1)] (2) Ford= 2 (qubit) and d= 3 (qutrit) the Fourier gate reduces to the Hadamard gate ( H) and the Chrestenson gate (C) respectively: H=1√ 2/parenleftbigg 1 1 1−1/parenrightbigg C=1√ 3 1 1 1 1e2πι 3e4πι 3 1e4πι 3e2πι 3  (3) For a system of N“hybrid” qudits, each of diﬀerent di- mensions d1,d2,···dN, the action of a two-qudit hybridcontrolled-rotation gate RH j,k(withjbeing the control qudit and kthe target qudit) is given by [15, 16]: RH j,k|x0···xj···xk···xN−1/an}bracketri}ht= exp/parenleftBigg −2πιxjxk/producttextk p=jdp/parenrightBigg |x0···xj···xk···xN−1/an}bracketri}ht (4) wheredpis the dimension of the pthqudit;k > jhas been assumed here, in order to explicitly deﬁne the ac- tionofthe gateonthe N-qudit basisstate, howeverinter- changing jandkdoes not alter the gate operation. For dp=d= 2, thehybridtwo-quditcontrolled-rotationgate reduces to the standard two-qubit controlled-rotation gateRj,k. B. Hybrid Qudit QFT The action of the QFT on the basis vectors of a hybrid qudit system is given by [15, 36]: QFT|x/an}bracketri}ht=1√ DN−1/summationdisplay k=0exp/parenleftbigg2πιxy D/parenrightbigg |y/an}bracketri}ht(5) where the states |y/an}bracketri}hthave the same form as |x/an}bracketri}ht. The circuit for implementing the QFT on Nhy- brid qudits can be decomposed as a set of single-qudit Fourier gates interspersed with two-qudit controlled- rotation gates: F1RH 1,2RH 1,3···RH 1,NF2RH 2,3···FN−1RH N−1,NFN. ForN qudits each of a diﬀerent dimension dj, the total dimen- sion of the Hilbert space is D=/producttextN j=1dj, and it has to be kept in mind while implementing hybrid controlled- rotation gates that the dimensions of the control and target qudits may be diﬀerent. After the implementa- tion of the QFT using the above decomposition, the bit values of the resultant state appear in the reverse order. A sequence reversal can be achieved either by applying a series of multi-valued permutation gates [6] or by reading oﬀ the result in the reverse order. III. NMR IMPLEMENTATION A. The hybrid QQT system We consider a QQT system with a Hamiltonian gov- erned by [45, 46] HQQT=ω1I1z+ω2I2z+Ω3I3z+DQ(3I2 3z−I2) +J12I1zI2z+J′ 23I2zI3z+J′ 13I1zI3z(6) where 1, 2 and 3 label the qubits and the qutrit respec- tively,Iizis the z-component of the magnetization vector of theithqubit(qutrit), I2=/summationtext iI2 3i, i=x,y,zis the total magnetization of the qutrit, and ω1,ω2,Ω3denote 3 (a) (b)00000001 0010 0100 10000011 0101 0110 1001 1010 11000111 1011 1101 11101111 000′001′010′100′002′011′101′110′012′102′111′112′ 123 456 789 10111212 34 56 78 910 1112 1314 1516 FIG. 1. Energy level diagrams: (a) For a QQQQ system showing 16 energy levels corresponding to four coupled spin - 1/2 particles. Spin states are given below each energy level, 0 and 1 correspond to spin states +1 2and−1 2respectively. There are 32 transitions originating from single spin ﬂips. Transitions belonging to spins 1 and 2 are shown in blue and brown respectively while transitions belonging to spins 3 a nd 4 are shown in purple. (b) For a hybrid QQT system showing 12 energy levels and 20 transitions with energy levels label ed in the computational basis; 0′, 1′and 2′correspond to the qutrit

Question 525 multiple-choice

Quantum algorithms often rely on the Quantum Fourier Transform (QFFT) and carefully engineered measurement strategies to ensure efficient and predictable outcomes. Achieving a desired success probability is critical for optimizing amplitude amplification techniques. When adjusting a quantum algorithm so that its success probability is exactly 1/4 prior to amplitude amplification, which method enables this precise probability setting? 1) Increasing the number of QFFT repetitions per run 2) Discarding all measurement results except the largest values of $ y $ 3) Replacing modular arithmetic with classical averaging 4) Introducing an auxiliary qubit in a superposed state using a specific rotation angle 5) Using a higher-order polynomial expansion for the QFFT 6) Doubling the group order $ p $ in all calculations 7) Restricting input values $ x $ to only even numbers

✓ Correct Answer:

The correct answer is 4) Introducing an auxiliary qubit in a superposed state using a specific rotation angle.

📚 Reference Text:

clear that if we measure y=⌊x·N/p⌋we will calculate the correct x. For a given x, smaller ymay also lead to the correct x, but here we would like to eliminate this contribution to the success pro bability, as it will lead to a simpler expression. Given a yit is possible to eliminate these cases by also checking y >⌈y·p/N⌉·N/p−1 and “throwing away” y’s which do not satisfy this. (Note that in order to obtain an algorithm Awith a certain success probability, we can think as if this were a non-reve rsible algo- rithm including measurements and classical computations. Such an a lgorithm can then easily be turned into a unitary Awhich, besides the intended answer, also produces some “garbage”.) So now the success probability pxfor correctly gettingxfrom|Ψx/an}⌊ra⌋k⌉tri}htis: px=f2(⌊xN/p⌋−xN/p) =f2(xNmodp p). To get the instance independent success probability of the uniform ised algo- rithm, we average this over all x= 0...p−1: ¯p=1 pp−1/summationdisplay x=0f2(xNmodp p) =1 pp−1/summationdisplay k=0f2(k/p) where we have used that Nandpare coprime and so for each xthere is exactly onek. 4.1 Eﬃciently calculating the success probability Forlarge pthissumofcourseiswellapproximatedbythecorrespondingintegr al, which (for large N) is approximately 0.4514. Here we show that for each pand N, the success probability can be approximated eﬃciently in the sense that the computation time is polynomial in the number of (e.g. decimal) digits we w ant 6 to compute. The following method achieves this in a simple way, althoug h it is probably not the best one could do. Note that f2(z) can be expanded in a (fast converging) power series in xand 1/N. (To compute ¯ ptoddigits we will only use polynomially many terms in d.) Now each power zmofzcan be summed separately, giving: 1 pp−1/summationdisplay k=0(k/p)m=1 pm+1Sm(p) =1 pm+1m+1/summationdisplay i=0Am,ipi where for each power mthe coeﬃcients Am,ican be calculated (in various ways) intimepolynomialin m. (Astraightforwardwayistosimplysolvetheequations resulting from Sm(p+1)−Sm(p) =pmfor theAm,i. E.g. for m= 1 we get the familiar formula/summationtextp−1 k=0k=p(p−1)/2.) 4.2 Adjusting the success probability Once we have calculated the success probability p(to arbitrary precision) for a givenp, we can use this to modify the algorithm Aso that it will succeed exactly with probability 1/4, so that just one iteration of amplitude ampliﬁcat ion leads to an exact algorithm. One way to do this is to add a qubit prepared in s tate cos(α)|0/an}⌊ra⌋k⌉tri}ht+sin(α)|1/an}⌊ra⌋k⌉tri}htwithpsin2(α) =1 4and additionally require for success that this qubit be in state |1/an}⌊ra⌋k⌉tri}ht. The preparation of this qubit will now require the one “strange” gate in our algorithm, although its rotation angle αcan be computed eﬃciently in the above sense. 5 Further remarks and observations 5.1 Generalisation to arbitrary orders The construction of the exact QFFT qeasily generalises to arbitrary orders q. Above we only needed the primality of the order for (eﬃciently) comp uting the success probability. And there we only needed that N= 2nandqshould be coprime. Things can easily be adjusted for the case when qis even. Either we can modify (a bit) the calculation of the success probability, or we ca n consider the QFFT

Question 526 multiple-choice

Indirect detection of dark matter relies on searching for astrophysical signals, such as gamma rays and cosmic microwave background (CMB) distortions, produced by dark matter annihilation or decay. The expected strength of these signals depends sensitively on the assumed spatial distribution of dark matter in galactic halos. Which of the following dark matter halo profiles is characterized by a constant-density core in its central region, potentially resulting from baryonic feedback or non-standard dark matter interactions? 1) Navarro-Frenk-White (NFW) profile 2) Einasto profile 3) Contracted NFW profile 4) Isothermal sphere profile 5) Burkert profile 6) Cored NFW profile 7) Plummer profile

✓ Correct Answer:

The correct answer is 6) Cored NFW profile.

📚 Reference Text:

from observations of the di use photon ux by the INTEGRAL [92], COMPTEL [93], EGRET [94], and Fermi [95] experiments. For the GeV-scale dark matter masses we are considering, these observations are expected to give the strongest constraints [96, 97].10We also study bounds from the e ects of DM annihilation during recombination on the cosmic microwave background (CMB) [98, 99]. The continuum photon ux from x 1pair annihilation into hidden vectors is given by d dE =hvi !XX 8m2 dNtot dE Z dl2(l); (30) wherehvi !XXis the thermally averaged annihilation rate at present, (l) is the dark matter density along the line of sight l, anddNtot =dE is the total di erential photon yield per annihilation, de ned as dNtot dE X fBRfdNf dE (31) where BR fis the branching fraction of the XX state into the nal state f. In our calculations, we use the results of Refs. [100, 101] to estimate the partial yields dNf =dE by interpolating between the results for the values of m andm =mxlisted in these studies. For the dark matter density pro le, we consider four distributions that span the range of reasonable possibilities: Navarro-Frenk-White (NFW) [102], Einasto [103, 104], contracted [96], and cored NFW [96]. These take the forms (r)/8 >>>>>>>>>< >>>>>>>>>: r Rs 1 +r Rs21 (NFW) e2= [(r Rs) 1](Einasto) r Rs 1 +r Rs3 1 (contracted) rc+(rrc) (rrc) Rs 1 +rc+(rrc) (rrc) Rs21 (cored): (32) Here,ris the radial distance from the GC and is a step function. Following Refs. [51, 96], we x the scale radius to be Rs= 20 kpc and the Einasto parameter = 0:17. For the contracted pro le we set = 1:4 and for the cored pro le we set the core radius to be rc= 1 kpc, as in Ref. [96]. In all four cases, we x the overall normalization such that (r= 8:5 kpc) = 0:3 GeV=cm3. 10We have also examined constraints from monochromatic photon line searches and found the continuum constraints signi cantly more stringent for the small values of allowed by xed target experiments. 16 Using these halo pro les, we are able to compute the gamma-ray uxes from hidden dark matter created in moduli decays and compare them to limits derived from observations of the GC and the di use gamma-ray background. For the GC signal, we use the limits on hvi=m2 REmax EmindE dNtot =dE computed in Ref. [96] in several energy bins [ Emin;i;Emax;i] and each of the four DM pro les described above. For the di use gamma ray background, we use the ux limits compiled and computed in Ref. [97]. In addition to measurements of cosmic gamma rays, observations of the CMB also pro- vide a signi cant limit on DM annihilation [98, 99]. The energy released by dark matter annihilation around the time of recombination will distort the last scattering surface, and hence a ect the CMB anisotropies. The limit derived from this e ect is [105{107] f 2 2 cdmhviCMB(2:421027cm3=s)m GeV

Question 527 multiple-choice

Quaternion algebras and their orders play a central role in computational number theory and isogeny-based cryptography, with algorithms often relying on linear algebra over finite fields, ideal arithmetic, and decomposition techniques. Ensuring certain algebraic properties, such as norms and coprimality, is crucial for the correctness and efficiency of these algorithms. In advanced algorithms for manipulating elements in quaternion algebras, which of the following most directly ensures that a constructed element Fⁿ(μ₀) is a square modulo a composite integer N, thereby enabling the solution of specific norm equations as required for the algorithm’s progression? 1) Using ideals with norm equal to a large prime greater than p³ 2) Applying the Chinese Remainder Theorem to combine local solutions 3) Selecting elements with coprime trace and norm 4) Computing greatest common divisors of N with auxiliary parameters 5) Ensuring the discriminant of the order is squarefree modulo N 6) Choosing elements whose norm is smooth with respect to small primes 7) Constructing and solving a linear system over the finite field F₂

✓ Correct Answer:

The correct answer is 7) Constructing and solving a linear system over the finite field F₂.

📚 Reference Text:

same dimension, therefore, repeating Step 3 con stant number oftimes will giveriseto a linearsystem over F2that is solvable.We haveensured thatFgenerated in Step 8 is such thatF n(µ0)is a square modulo N, hence Step 9 makes sense. Similar to what is discussed in the proof of Lemma 5.3, St ep 12 succeeds with probability at least 1 /O(log2p). To make sure this algorithm gets passed to Step 14, we need O(log2p) many solutions from Step 10. According to Lemma 5.5, this happens if gcd( N,t)gcd(N,s) p3. This algorithm ﬁrst computes a connecting ideal I′ fromO0toO, then computes an equivalent left O0-idealIof normMwhose right order is α−1Oαfor some nonzero α∈ Bp,∞usingKLPTM(I′). ThisKLPT′ algorithm outputs Iandα. 5.2 The main algorithm In this section, we ﬁrst present a strategy that solves Problem 4.1 0 for special ordersO0. Then, we expand this strategy to address more general orders O. Let σ0∈ O0be as in Problem 4.10, the method proceeds as follows. 1. Findσ′ 0∈R+Rjsuch thatσ′ 0=σ0modNO0. Since [O0:R+Rj] =D, Dσ0∈R+Rj. LetD′∈Zbe such that D′D≡1 modN, suchD′exists since (D,N) = 1, then σ′ 0=D′Dσ0∈R+Rjandσ′ 0=σ0modNO0. By an abuse of notation, we will use σ0to denoteσ′ 0in what follows. 2. Writeσ0=A+BjwithA,B∈R,letγbetheoutputof RepresentInteger′ R+Rj(N,A,B). Intuitively, γis anelement in R+Rjthat haspowermoothnormand satisﬁes extra properties to ensure the next step has a solution. 3. Findα1,α2,α3∈Rjsuch thatσ0=α1γα2γα3modNO0. This is the main technical point in this method, we introduce it as Problem 5.7 and discu ss it in detail in Section 5.3. 4. Findγi∈ O0such thatγi=λiαimodNO0,n(γi) is powersmooth and (λi,N) = 1fori= 1,2,3.Theseareexactlytheoutputsof StrongApproximationps(N,αi). 5. The element γ1γγ2γγ3∈ O0satisﬁes that σ0=λγ1γγ2γγ3modNO0with n(γ1γγ2γγ3) powersmooth and λcoprime to N. In general, let Obe a maximal order in Bp,∞, and letnI>p3be a random integer that is coprime to N. Letn′ I∈Zsuch thatn′ InI≡1 modN. LetI,α be the outputs of KLPTnI(O0,O) such that Iis a connecting ideal from O0 toO′:=α−1Oαandn(I) =nI. We then have inclusions nIO′⊆ O0and nIO0⊆ O′, therefore nIα−1σ0α∈ O0. Letσ∈PQLPO0(nIα−1σ0α), thenσ= nIα−1σ0αmodNO0. Multiplying nIwith both sides of the equation yields that nIσ=n2 Iα−1σ0αmodNO′. Multiplying n′2 Ion both sides gives that n′ Iσ= Title Suppressed Due to Excessive Length 25 α−1σ0αmodNO′. Sincen′ Iis coprime to N,σ∈PQLPO′(α−1σ0α), therefore ασα−1∈PQLPO(σ0). We summarize the discussions above and present Algorithm 3. Algorithm 3: PQLPO(N,σ0) Input:An odd integer Nthat is of size polynomial in pand has O(loglogp) distinct prime factors, a maximal order O, and an element σ0∈Osuch that (n(σ0),N) = 1. Output: σ∈PQLP O(σ0). 1Compute D′such that D′D≡1 modN. 2σ0←D′Dσ0 3Writeσ0asA+BjwithA,B∈R 4γ←RepresentInteger′ R+Rj(N,A,B) 5α1,α2,α3←QuaternionDecomposition (σ0,γ,N)(Algorithm 4) 6λi,γi←StrongApproximationps(N,αi) fori= 1,2,3 7σ←γ1γγ2γγ3 8Randomly generate nI> p3that is coprime to N. 9I,α←KLPT nI(O0,O) 10Return αγ1γγ2γγ3α−1. 5.3 Quaternion decomposition In this section, we discuss how to perform Step 3 from the strateg y outline. We start with introducing a new problem. Problem 5.7 (Quaternion Decomposition). LetNbe an odd integer, and O0,R be as deﬁned in Section 5.1. Let σ0=A+Bj,γ=C+Dj∈R+Rjbe such that: i) (n(γ),N)

Question 528 multiple-choice

Localization algorithms in quantum chemistry are crucial for efficiently constructing molecular orbitals, enabling accurate modeling and interpretation of electronic structure in various molecular systems. Advances that improve computational speed while maintaining numerical stability and orbital sparsity are highly sought after for large-scale simulations. Which statement best characterizes the two-stage (randomized plus refinement) orbital localization algorithm when applied to systems such as the borane-ammonia complex and alkane chains? 1) It consistently produces higher condition numbers than traditional methods, resulting in unreliable orbital localization. 2) It fails to adapt to changes in molecular configuration, grouping orbitals incorrectly during dissociation events. 3) It yields localized orbitals with significantly lower sparsity than the original SCDM algorithm, increasing computational cost. 4) It groups all orbitals into separate disconnected fragments even in systems with a single connected group. 5) It requires substantially more computational time than the original selected columns method, making it impractical for large molecules. 6) It sacrifices localization quality for speed, resulting in visually and quantitatively inferior orbitals. 7) It achieves comparable localization quality and sparsity to the original method, maintains low condition numbers for numerical stability, and is significantly faster.

✓ Correct Answer:

The correct answer is 7) It achieves comparable localization quality and sparsity to the original method, maintains low condition numbers for numerical stability, and is significantly faster..

📚 Reference Text:

alkane and water examples. 4.1. BH 3NH 3.First, we demonstrate the performance of the approximate col- umn selection method for the dissociation process of a BH 3NH3molecule. The main purpose of this example is to demonstrate that our approximate localization algo- rithm is equally e ective when localized orbitals form disconnected groups or a single group. Figure 4.1 shows the localized orbitals for three atomic con gurations, where the distance between B and N atoms is 1.18, 3.09, and 4.96 Bohr, respectively and also shows the locality for orbitals computed by Algorithms 1, 3, and 4 for each of the three atomic con gurations. Here we plot an isosurface of the orbitals at a value of 2:5102: We nd that Algorithm 4 automatically identi es that the localized orbitals should be treated as two disconnected groups for the dissociated con guration, and as one single group for the bonded con guration. We see that in all cases, the random- ized method works quite well on its own. Furthermore, after applying Algorithm 4 to the output of the randomized method, the sparsity of the orbitals is nearly in- distinguishable from that of the original SCDM algorithm. In these three scenarios the condition number of P:;C;equivalently ( C;:);is never larger than three. Here we letCeCdenote the nal set of columns we have selected via the last QRCP in Algorithm 4 as it corresponds to the columns of the density matrix from which we ultimately build the localized basis. 13 1234567 basis vector index, sorted10152025303540percent non-zeroRandomized Refined Original SCDM 1234567 basis vector index, sorted10152025303540percent non-zeroRandomized Refined Original SCDM 1234567 basis vector index, sorted10152025303540percent non-zeroRandomized Refined Original SCDM Fig. 4.1: Sparsity (left) of localized orbitals computed by Algorithms 1, 3, and 4 based on fraction of non-zero entries after truncation and orbital isosurfaces (right) at 2:5102generated by Algorithm 4 when using the output of Algorithm 3 as input. Three di erent con gurations moving from the bonded con guration (top) to the dissociated con guration (bottom). 4.2. Alkane chain. Our second example is the alkane chain (atomic con gura- tion shown in Figure 4.2). Similar to the BH 3NH3, this example is not computation- ally expensive, but con rms that the approximate localization algorithm is e ective even when all localized orbitals form one large connected group. We demonstrate that 14 Table 4.1: Runtime for localization algorithms as applied to an alkane chain. Operation time (s) Matrix-matrix multiplication Q 0.2713 Randomized version, Algorithm 3 0.0261 Re nement step, Algorithm 4 0.7176 Total cost of our two stage algorithm 1.0149 Original algorithm, Algorithm 1 6.9412 the re nement process still achieves the desired goal. In this example N= 820;125 andne= 100. Fig. 4.2: Atomic con guration of the alkane chain Figure 4.3 shows histograms of the fraction of non-zero entries after truncation for the randomized method, the re nement procedure applied to the output of the randomized method, and our original algorithm. We observe that the randomized method actually serves to localize the orbitals rather

Question 529 multiple-choice

Quantum computing leverages the properties of quantum mechanics to solve certain problems much more efficiently than classical computers. Abelian hidden subgroup problems, such as Simon's problem, are important benchmarks for demonstrating quantum algorithmic speedup and practical quantum advantage. Which technique, when applied to superconducting quantum processors, directly improves both the speedup exponent and the range of Hamming weights solvable in quantum algorithms for restricted Abelian hidden subgroup problems? 1) Increasing the number of qubits on the device 2) Using quantum error correction codes 3) Optimizing gate fidelity through hardware calibration 4) Implementing dynamical decoupling sequences 5) Running algorithms at lower temperatures 6) Employing classical post-processing methods 7) Enhancing qubit connectivity through advanced chip design

✓ Correct Answer:

The correct answer is 4) Implementing dynamical decoupling sequences.

📚 Reference Text:

Title: Demonstration of Algorithmic Quantum Speedup for an Abelian Hidden Subgroup Problem Year: 2024 Paper ID: a6c9d3d70d188cb2e2cd6db208da3842ea9e4b16 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a6c9d3d70d188cb2e2cd6db208da3842ea9e4b16 Abstract: Simon’s problem is to find a hidden period (a bitstring) encoded into an unknown 2-to-1 function. It is one of the earliest problems for which an exponential quantum speedup was proven for ideal, noiseless quantum computers, albeit in the oracle model. Here, using two different 127-qubit IBM Quantum superconducting processors, we demonstrate an algorithmic quantum speedup for a variant of Simon’s problem where the hidden period has a restricted Hamming weight w. For sufficiently small values of w and for circuits involving up to 58 qubits, we demonstrate an exponential speedup, albeit of a lower quality than the speedup predicted for the noiseless algorithm. The speedup exponent and the range of w values for which an exponential speedup exists are significantly enhanced when the computation is protected by dynamical decoupling. Further enhancement is achieved with measurement error mitigation. This case constitutes a demonstration of a bona fide quantum advantage for an Abelian hidden subgroup problem. Published by the American Physical Society 2025

Question 530 multiple-choice

In quantum computing, solving the Hidden Subgroup Problem (HSP) for complex non-abelian groups is fundamental to the development of advanced algorithms and cryptographic protocols. The efficiency of these algorithms often depends on the quantum resources required, especially the degree of entanglement and the number of coset states measured jointly. What is the minimum entanglement scaling required, in terms of the number of registers, for any efficient quantum algorithm solving the HSP in matrix groups such as SL(2, Fq) or GL(2, Fq)? 1) Ω(log q) entangled registers 2) Ω(√q) entangled registers 3) Ω(q^2) entangled registers 4) O(1) entangled registers 5) Ω(q log q) entangled registers 6) Ω(q) entangled registers 7) O(log log q) entangled registers

✓ Correct Answer:

The correct answer is 1) Ω(log q) entangled registers.

📚 Reference Text:

coset state s that solves the hidden subgroup problem in G= PSL(2,Fq)in polynomial time has to make joint measurements on k= Ω(log|G|) = Ω(q)coset states. 4.3 Special andgeneral lineargroups Corollary 16. Anyalgorithm solving theHSPin SL(2,Fq)orGL(2,Fq)efﬁciently usingcoset statesneeds tomake measurements entangled across k= Ω(logq)registers. Proof.ByCorollary 15any algorithm solving theHSPin PSL(2,Fq)efﬁciently using coset states needs to makemeasurements entangled across k= Ω(logq)registers. Thestatement nowfollowsfromLemma1by using the facts that PSL(2,Fq)∼=SL(2,Fq)/ζ(SL(2,Fq))and thatSL(2,Fq)≤GL(2,Fq). Corollary 17. Any algorithm solving the HSP in GL(n,Fpm)efﬁciently using coset states needs to make measurements entangled across k= Ω(n(mlogp+logn))registers. Proof.SinceGL(n,Fpm)contains all n×npermutation matrices, a lower bound of k= Ω(nlogn)fol- lows from Corollary 14 and Lemma 1. Also, we can use the embedd ing ofGL(2,Fpnm)≤GL(2n,Fpm) via/parenleftbigga b c d/parenrightbigg /ma√st≀→/parenleftbiggMaMb McMd/parenrightbigg ,where for each x∈Fpnmthe matrixMx∈GL(n,Fpm)realizes mul- tiplication by xwith respect to a ﬁxed basis of FpnmoverFpm. Hence by Lemma 1 we obtain that for the HSP inGL(2n,Fpm)at least as much entanglement is necessary as in case of GL(2,Fpnm). The latter has been bounded by Ω(nmlogp)inCorollary 16. 4.4 Direct products ofthe form Gn In this section we show that for a large class of ﬁnite groups G, efﬁcient algorithms for HSP for direct products of the form Gn, wheren≥1, require entangled measurements on at least k= Ω(n)coset states. LetGbe a ﬁnite group and let /hatwideG={σ1,...,σ m}denote the irreducible representations of G. Recall that the centralizer C(g)of an element g∈Gis the subgroup C(g) :={c∈G:cg=gc}. Lethbe an involution in G, andletσ∈/hatwideG. Theneither |χσ(h)|=dσor|χσ(h)| dσ<1−2|C(h)| |G|holds [Gal94]. Wedeﬁne ε:= (1−2|C(h)| |G|)t,wheret=t(n)isafunction of ntobe determined later. Theirreps of Gn,wheren≥1, aregiven by σ:=σ1⊗...⊗σn,whereσi∈/hatwideG. WeletΛ :={σ∈/hatwideG: |χσ(h)|=dσ},λ:=/summationtext σ∈Λd2 σ, andµ:=/summationtext σ∈/hatwideG\Λd2 σ=|G|−λ. Thefollowing property of the set Sε:=/braceleftbigg σ∈/hatwideGn:|χσ(h,...,h)| dσ≥ε/bracerightbigg holdsforourchoiceoftheparameter ε: ifσ∈Sεthennecessarily atleast n−tpositionsσihavetobefrom Λ, i.e., have to satisfy |χσi(h)|=dσi. Indeed, otherwise we would have more than tpositionsσjin each of which|χσj(h)| dσj≤1−2|C(h)| |G|, making the product less than ε. We next give an estimate for the quantity /summationtext σ∈Sεd2 σappearing in Theorem 2. For that we require the following lem ma for estimating the tail of the binomial distribution. 18 Lemma18. Letα,β >0, letn≥1, and lett=n/c, wherec>α+β β. Then n/summationdisplay ℓ=n−t/parenleftbiggn ℓ/parenrightbigg αℓβn−ℓ≤/parenleftBigg α/parenleftbiggce(α+β) α/parenrightbigg1/c/parenrightBiggn . Proof.Wehave that n/summationdisplay ℓ=n−t/parenleftbiggn ℓ/parenrightbigg αℓβn−ℓ= (α+β)nn/summationdisplay ℓ=n−t/parenleftbiggn ℓ/parenrightbigg/parenleftbiggα α+β/parenrightbiggℓ/parenleftbiggβ α+β/parenrightbiggn−ℓ ≤(α+β)n/parenleftbiggn n−t/parenrightbigg/parenleftbiggα α+β/parenrightbiggn−t =αn/parenleftbiggn t/parenrightbigg/parenleftbiggα+β α/parenrightbiggt ≤αn/parenleftbiggne(α+β) tα/parenrightbiggt =/parenleftBigg α/parenleftbiggce(α+β) α/parenrightbigg1/c/parenrightBiggn , where the ﬁrst inequality follows from the union bound on pro babilities and the second one from/parenleftbign t/parenrightbig ≤/parenleftbigne t/parenrightbigt. Supposeweﬁx ℓ≥n−tlocations forputtinginirrepsfrom Λ. Thecontribution ofthisconﬁguration to/summationtext σ∈Sεd2 σisthe sum of products of squares of dimensions of ℓirreps from Λandn−ℓirreps from/hatwideG\Λ, which simpliﬁes to λℓµn−ℓ. Lettingα:=λ,β:=µ, andt=n/c, with some constant cto be determined later, weobtain the following bound from Lemma18: /summationdisplay σ∈Sεd2 σ≤n/summationdisplay ℓ=n−t/parenleftbiggn ℓ/parenrightbigg λℓµn−ℓ≤λn/parenleftBigg/parenleftbiggce|G| λ/parenrightbigg1/c/parenrightBiggn . Hence, for any given κ >0we can ﬁnd a constant c >0such that/summationtext σ∈Sεd2 σ≤λn(1+κ)nholds for alln≥c. Note that the same upper bound applies to/summationtext σ∈Sεdσ|χσ(h,...,h)|. Also, observe that /summationtext ρ∈/hatwideGndρ=/parenleftBig/summationtext ρ∈/hatwideGdρ/parenrightBign . Now,wecan bound the parameter δ1used inTheorem 2: δ1≤ε+1 |G|n/parenleftBigg/summationdisplay σ∈Sεdσ|χσ(h,...,h)|/parenrightBigg /summationdisplay ρ∈/hatwideGndρ  ≤/parenleftBigg/parenleftbigg 1−2|C(h)| |G|/parenrightbigg1/c/parenrightBiggn +λn(1+κ)n |G|n /summationdisplay ρ∈/hatwideGdρ n . For the following we make the

Question 531 multiple-choice

Quantum signal processing leverages specialized transforms to manipulate and analyze quantum states, with strict requirements for mathematical properties such as unitarity. Generalizations of classical transforms, like the Fractional Fourier Transform, present unique challenges and opportunities for quantum implementation. Which property did the reformulation of the Weighted Fractional Fourier Transform (WFRFT) specifically ensure to enable its adoption in quantum circuit designs for reversible and probability-preserving operations? 1) Linearity 2) Commutativity 3) Idempotence 4) Unitarity 5) Orthogonality 6) Hermiticity 7) Time-invariance

✓ Correct Answer:

The correct answer is 4) Unitarity.

📚 Reference Text:

Title: Quantum Weighted Fractional Fourier Transform Year: 2022 Paper ID: 86508e833b2f1d17406551a41f68f5348f514422 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/86508e833b2f1d17406551a41f68f5348f514422 Abstract: Quantum Fourier transform (QFT) is an important part of many quantum algorithms. However, there are few reports on quantum fractional Fourier transform (QFRFT). The main reason is that the definitions of fractional Fourier transform (FRFT) are diverse, while some definitions do not include unitarity, which leads to some studies pointing out that there is no QFRFT. In this paper, we first present a reformulation of the weighted fractional Fourier transform (WFRFT) and prove its unitarity, thereby proposing a quantum weighted fractional Fourier transform (QWFRFT). The proposal of QWFRFT provides the possibility for many quantum implementations of signal processing.

Question 532 multiple-choice

Modern cryptographic systems increasingly utilize algebraic number theory and lattice structures to secure communications, especially in the era of quantum computing. Understanding the relationships between number field invariants and computational hardness is crucial for developing quantum-resistant encryption schemes. Why is the discriminant of a number field particularly important in the context of lattice-based cryptography and the security of related cryptographic algorithms? 1) It quantifies the complexity and density of the lattice, affecting both computational efficiency and security bounds. 2) It determines the characteristic polynomial of the associated ideal lattices. 3) It guarantees that the principal ideal problem remains hard for all number fields. 4) It establishes the existence of a unit group with cryptographic significance. 5) It provides a direct measure for the dimension of the embedding space. 6) It controls the trace function of the number field elements. 7) It serves as the generator of the ring of integers for cryptographic constructions.

✓ Correct Answer:

The correct answer is 1) It quantifies the complexity and density of the lattice, affecting both computational efficiency and security bounds..

📚 Reference Text:

phic encryption and the ensuing eﬀorts to make the systems more eﬃcient and more secure, assumptions related to num- ber ﬁelds have been used. These systems are set up based on high degree number ﬁelds. In [GH11], a version of the prin- cipal ideal problem where a special generator is the secret was used as the hardness assumption. The Ring-LWE prob- lem which forms the basis in [LPR10, BV11] assumes that ﬁnding short vectors in ideal lattices of high degree number ﬁelds is hard. To summarize, the constant degree assumptions are bro- ken by quantum algorithms. The relatively recent high de- gree number ﬁeld assumptions about computing short vec- tors are still open in terms of security against quantum com- puters. However, in this paper we show that it is now pos- sible to eﬃciently compute the unit group in these number ﬁelds, which could move towards understanding whether the new homomorphic cryptosystems really are secure against quantum computers. 2. NUMBER-THEORETIC BACKGROUND In the following Kwill denote a number ﬁeld of degree n overQ, and Owill denote its ring of integers. When we want to consider Oas a lattice in E=Rs×Ctwiths+2t=n(see below), we will write O. We use bold letters to designate elements of Eand vectors in general. If{b1,...,bn}is a basis for a lattice Λ ⊆Rn, letBbe the matrix B= (b1,...,bn) which is composed of column vectors bk. Thend(Λ) := |det(B )|is the unit cell volume of the lattice Λ generated by the basis.Elements of Kcan be conveniently represented by using the embeddings of Kinto the ﬁeld of complex numbers. In general, there are nsuch embeddings, which break into s real ones and tcomplex-conjugate pairs: τ1,...,τ s:K→R, τs+1,...,τ s+t,τs+1,...,τs+t:K→C (s+ 2t =n). Each element z∈Kis mapped to the corresponding conju- gate vector τ(z) =“ z(1),...,z(s),z(s+1),...,z(s+t),z(s+1), ...,z(s+t)”T ∈Rs×C2t, where the last tcoordinates are re- dundant. Thus, Kis embedded into E=Rs×Ct. Conjugate vectors are added and multiplied coordinate-wise. Many useful functions on Kextend naturally to E. For exam- ple, the algebraic trace and norm are deﬁned for arbitrary conjugate vectors: tr(z) =sX j=1z(j)+s+tX j=s+1` z(j)+z(j)´ , N(z) =sY j=1z(j)s+tY j=s+1|z(j)|2. Both these functions take real values. As far as the additive structure is concerned, the ring Eis simply an n-dimensional real space. We can deﬁne a Euclidean inner product on Eby letting /angbracketleftx,y/angbracketright= tr(x y) =sX j=1x(j)y(j)+ 2s+tX j=s+1“` Rex(j)´` Rey(j)´ +` Imx(j)´` Imy(j)´” . The length of a vector with respect to this inner product is denoted by /bardblz/bardbl. Now let {ω1,...,ω n}be some basis (over Z) for the ring Oof integral elements in K. One way to characterize O is by its “multiplication table”, i.e., the decomposition of ωjωkintoωlwith integer coeﬃcients. In the conjugate vec- tor representation, Obecomes a lattice O⊆Ewith basis {z1,...,zn}. From the computational perspective, it is im- portant to have some upper bound on the length of the basis vectors or, equivalently, on the coeﬃcients in the multipli- cation table. To this end, we use the notion of discriminant, which is deﬁned as the determinant of the

Question 533 multiple-choice

Digital inclusion strategies for marginalized groups often involve innovative approaches to overcoming linguistic and technological barriers. Understanding how low-literate adults participate in digital society can inform more equitable design and policy interventions. Which tactic specifically enables low-literate individuals to convert written text into more accessible formats using advances in AI-driven accessibility technologies? 1) Non-written communication 2) Optimal character recognition 3) Informal support structures 4) Formal support structures 5) Translation software 6) Social network integration 7) Multimodal interface usage

✓ Correct Answer:

The correct answer is 2) Optimal character recognition.

📚 Reference Text:

Title: Bypassing digital literacy: Marginalized citizens’ tactics for participation and inclusion in digital societies Year: 2024 Paper ID: 97ea408893a7fd956749a99e13555bb3c0fbfc9f Source: semantic-scholar URL: https://www.semanticscholar.org/paper/97ea408893a7fd956749a99e13555bb3c0fbfc9f Abstract: This article asks what digital literacy tactics low-literate Dutch adults employ to bypass their low-literacy to be able to participate in digital society, and what the consequences are for their socio-digital exclusion and inclusion. It contributes to a better understanding of the impact of digitalization for low-literate citizens, and the linguistic and digital barriers encountered in everyday life. Drawing upon participant observations and semi-structured interviews with low-literate adult citizens in four libraries, a community center, and a school for adult education (N = 73), this article develops a taxonomy of five tactics which enables low-literate citizens to digitally participate despite their linguistic and digital barriers: (1) informal support structures, (2) formal support structures, (3) non-written communication, (4) translation software, and (5) optimal character recognition. We show how these tactics of appropriating the affordances of information and communications technologies (ICTs), and making use of social networks enable low-literate Dutch citizens to participate in socially situated manners, making use of social support structures and digital literacies developed in relation to “foreign” languages. Consequently, this study counters the stigma on such marginalized groups, who are often assumed to be unable or unwilling to participate, and presents them as not adhering to the dominant discourse of participatory culture. Hence, the added value of this study is threefold: (1) it centers the capabilities of low-literate citizens stemming from social capital and obfuscated linguistic potential, (2) it gives visibility toward more hidden everyday (digital) practices of marginalized subgroups with a larger distance toward the digital society, and (3) it foregrounds the lived experiences of the user and their (limited) use of ICTs, and how tactics are developed and practiced to bypass linguistic and/or digital barriers showing situated agency and problem-solving capacities. We argue that digital literacies should not be considered as a prerequisite for digital participation and inclusion, as our findings show that low-literate Dutch citizens are a highly diverse group that are capable of participating, despite their low (digital)-literacy. However, they do so in socially situated and non-written manners, in line with their digital and linguistic capabilities and barriers.

Question 534 multiple-choice

In quantum computing using nuclear magnetic resonance (NMR), operations on qubits are achieved by applying precisely controlled radio frequency (rf) pulses, with the efficiency of quantum algorithms often determined by pulse design and spin-spin interactions. The quantum Fourier transform (QFT) is a key algorithm whose practical implementation relies on optimizing both gate timing and pulse selectivity. Which of the following statements best explains why the experimental bottleneck in NMR-based quantum Fourier transform circuits is most closely associated with the time-cost of quantum gates, rather than the total number of gates? 1) The duration of controlled-phase shift gates depends on the phase angle and the qubit interaction strength, making gate times the dominant factor regardless of gate count. 2) Increasing the number of gates always results in exponential scaling of total computation time, even with fast gate operations. 3) Each gate in an NMR QFT circuit is executed instantaneously, so only the number of gates matters. 4) Gate time-costs in NMR are negligible compared to classical computation delays. 5) Pulse selectivity allows for simultaneous execution of all gates, eliminating time constraints. 6) Only single-qubit gates contribute to the total time in NMR implementations. 7) The number of gates directly determines the duration of rf pulses, making gate count the main bottleneck.

✓ Correct Answer:

The correct answer is 1) The duration of controlled-phase shift gates depends on the phase angle and the qubit interaction strength, making gate times the dominant factor regardless of gate count..

📚 Reference Text:

shown in the circuits). is the total non-selective Hadamard operator on all qubits, i.e. a single π/2 radio frequency pulse. TheUgates in Eqn. (5) are U1=H2B1,2H−1 2 = (UCNOT)1/2 1,2 Un−2=Hn−1(B1,n−1B2,n−1...Bn−2,n−1)H−1 n−1 = (UCNOT)1/2n−2 1,n−1(UCNOT)1/2n−3 2,n−1..(UCNOT)1/2 n−2,n−1 Un−1=Hn(B1,nB2,n...Bn−1,n)H−1 n = (UCNOT)1/2n−1 1,n(UCNOT)1/2n−2 2,n..(UCNOT)1/2 n−1,n (6) They correspond to single spin rotations conditioned on the status of all the other spins involved in the opera- tion. Since they are single spin rotations, they can be implemented as single radio frequency pulses. The con- ditioning on the state of the other spins is achieved by making them selective on speciﬁc transitions. As an ex- ample, for a system of three qubits, the operation Un−1 in this case is given by a fourth-root of a controlled-NOT gateon qubits one andthree, followedby a square-rootof a controlled-NOT gate on qubits two and three, with the third qubit being the target qubit in both cases. These gates thus involve three transitions of the third qubit: 100→101,110→111, and 010 →011. Since these are unconnected transitions, rotations in the subspace of 3 these transitions can be achieved simultaneously. Many pulsed irradiation schemes for such precise selective ex- citation exist in NMR, mostly involving shaping the ex- citation proﬁle of the rf waveform [22]. The entire QFT operation in this decomposition therefore reduces to a sequence of nradio frequency pulses. It scales linearly with the number of qubits, and we will denote it as the “parallel” implementation. III. TIME-COST OF THE QFT The main issue in the experimental implementation of quantum algorithms is not the number of logical opera- tions per se, but the actual time-cost of each logicaloper- ation/quantum gate. The Utransformations in Eqn. (5) are no longer two-qubit phase-shift gates but correspond tosquare-andhigher-rootsofcontrolled-NOToperations on speciﬁc qubits. They can be implemented experimen- tally using multiqubit gates that perform manipulations on qubits simultaneously. Since the NMR Hamiltonian has terms connecting multiple pairs of qubits, such mul- tiqubit gates can be directly implemented. The quantum circuits for both serial and parallel implementations of the QFT are shown in Figure 1. The most expensive operation in the serial implemen- tation of the QFT is the controlled-phase shift gate Bjk. The ideal time-cost is computed assuming all gates take the same amount of time. However experimentally, the controlled-phase shift gate requires a time τjkpropor- tional to the desired phase rotation angle θjk(related to the “distance” ( k−j) of the qubits), and inversely proportional to the interaction Jjkbetween the qubits. The magnitude of the interaction Jjkand hence the time cost depends on the speciﬁc experimental quantum com- puting technology under consideration. For liquid-state NMR,Jjkis the electron-mediated scalar coupling be- tween the qubits. For our calculations, we assume the Jjk’s to be of the same order of magnitude for all qubits, represented by a universal constant coupling J. The ac- tual time cost of the serial decomposition of the QFT, involving only one-qubit Hadamards and the two-qubit phase-controlled gate is Tser=nδ+n−1/summationdisplay j=0n/summationdisplay k=j+1τjk=nδ+κn−1/summationdisplay j=0n/summationdisplay k=j+12j−k =nδ+κ(n−1+2−n) ≈O(n) (7) whereδis the time-cost of each single-qubit

Question 535 multiple-choice

Quantum algorithms can exploit hidden symmetries in data to achieve efficient compression, especially for datasets with complex group-theoretic structures. Quantum autoencoders combine these algorithms with variational techniques, enabling practical compression and recovery of sequence data on quantum devices. Which aspect distinguishes quantum autoencoders from classical autoencoders when compressing data with unknown group-theoretic symmetries? 1) They require prior statistical analysis of data distributions before compression. 2) They only compress data efficiently if all symmetries are explicitly known in advance. 3) They use purely classical optimization methods for identifying data structures. 4) They rely on redundancy detection rather than symmetry exploitation. 5) They outperform classical methods only in cases of low-dimensional data. 6) They variationally identify hidden subgroup structures and utilize quantum algorithms to compress data, achieving exponential speed-up over classical algorithms. 7) They cannot produce a compressed representation that allows for lossless data recovery.

✓ Correct Answer:

The correct answer is 6) They variationally identify hidden subgroup structures and utilize quantum algorithms to compress data, achieving exponential speed-up over classical algorithms..

📚 Reference Text:

Title: Information compression via hidden subgroup quantum autoencoders Year: 2023 Paper ID: 1049c07d275df61cb160423713d502f9602241b2 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/1049c07d275df61cb160423713d502f9602241b2 Abstract: We design a quantum method for classical information compression that exploits the hidden subgroup quantum algorithm. We consider sequence data in a database with a priori unknown symmetries of the hidden subgroup type. We prove that data with a given group structure can be compressed with the same query complexity as the hidden subgroup problem, which is exponentially faster than the best-known classical algorithms. We moreover design a quantum algorithm that variationally finds the group structure and uses it to compress the data. There is an encoder and a decoder, along the paradigm of quantum autoencoders. After the training, the encoder outputs a compressed data string and a description of the hidden subgroup symmetry, from which the input data can be recovered by the decoder. In illustrative examples, our algorithm outperforms the classical autoencoder on the mean squared value of test data. This classical-quantum separation in information compression capability has thermodynamical significance: the free energy assigned by a quantum agent to a system can be much higher than that of a classical agent. Taken together, our results show that a possible application of quantum computers is to efficiently compress certain types of data that cannot be efficiently compressed by current methods using classical computers.

Question 536 multiple-choice

Quantum integrable systems are deeply connected to the theory of Lie algebras and their quantum deformations, which provide the algebraic structure underlying many solvable models in mathematical physics. The representation theory of these algebras, including infinite-dimensional cases, is crucial for understanding the symmetries and solutions in such systems. Which mathematical structure directly relates the representation theory of Lie algebras and quantum groups to the Quantum Inverse Scattering Method in the study of integrable systems? 1) Virasoro algebra 2) Yangian 3) Heisenberg algebra 4) Clifford algebra 5) Affine Lie algebra 6) Cartan subalgebra 7) Weyl group

✓ Correct Answer:

The correct answer is 2) Yangian.

📚 Reference Text:

Title: Representation Theory and Quantum Integrability Year: 2004 Paper ID: 2feab6b131a769626c94d01d58a3e36eca49dd66 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/2feab6b131a769626c94d01d58a3e36eca49dd66 Abstract: We describe new constructions of the infinite-dimensional representations of U( $\mathfrak{g}$ ) and U q ( $\mathfrak{g}$ ) for $\mathfrak{g}$ being $\mathfrak{g}\mathfrak{l}$ (N) and $\mathfrak{s}\mathfrak{l}$ (N). The application of these constructions to the quantum integrable theories of Toda type is discussed. With the help of these infinite-dimensional representations we manage to establish a direct connection between the group theoretical approach and the Quantum Inverse Scattering Method based on the representation theory of the Yangian and its generalizations.

Question 537 multiple-choice

In categorical quantum mechanics, the concept of strong complementarity plays a pivotal role in unifying algebraic structures and quantum phenomena. Extensions to infinite-dimensional Hilbert spaces have broadened the scope of this framework in quantum theory and information science. Which of the following is identified as the fundamental resource responsible for quantum advantage in algorithms such as those solving the Hidden Subgroup Problem? 1) Quantum error correction codes 2) Entangled measurement bases 3) Strong complementarity 4) Bell inequalities 5) Decoherence-free subspaces 6) Quantum teleportation protocols 7) No-cloning theorem

✓ Correct Answer:

The correct answer is 3) Strong complementarity.

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Title: Categorical quantum dynamics Year: 2017 Paper ID: ddb40dfeae627a3d76ab23f1f75842fa4eb4a9ef Source: semantic-scholar URL: https://www.semanticscholar.org/paper/ddb40dfeae627a3d76ab23f1f75842fa4eb4a9ef Abstract: We use strong complementarity to introduce dynamics and symmetries within the framework of CQM, which we also extend to infinite-dimensional separable Hilbert spaces: these were long-missing features, which open the way to a wealth of new applications. The coherent treatment presented in this work also provides a variety of novel insights into the dynamics and symmetries of quantum systems: examples include the extremely simple characterisation of symmetry-observable duality, the connection of strong complementarity with the Weyl Canonical Commutation Relations, the generalisations of Feynman's clock construction, the existence of time observables and the emergence of quantum clocks. Furthermore, we show that strong complementarity is a key resource for quantum algorithms and protocols. We provide the first fully diagrammatic, theory-independent proof of correctness for the quantum algorithm solving the Hidden Subgroup Problem, and show that strong complementarity is the feature providing the quantum advantage. In quantum foundations, we use strong complementarity to derive the exact conditions relating non-locality to the structure of phase groups, within the context of Mermin-type non-locality arguments. Our non-locality results find further application to quantum cryptography, where we use them to define a quantum-classical secret sharing scheme with provable device-independent security guarantees. All in all, we argue that strong complementarity is a truly powerful and versatile building block for quantum theory and its applications, and one that should draw a lot more attention in the future.

Question 538 multiple-choice

Quantum algorithms often rely on the efficient implementation of the Quantum Fourier Transform (QFT) over various abelian groups, which is a critical component for solving problems like the hidden subgroup problem. The practical construction of quantum circuits for QFT and amplitude amplification depends on mathematical properties of the underlying group and the complexity of Boolean function design. Which of the following statements correctly describes a major contribution by Mosca and Zalka to quantum algorithm implementation over abelian groups? 1) They provided a method for approximate QFT over only smooth values of $ m $. 2) They proved that QFT modulo $ m $ can always be implemented by a single Turing machine for any $ m $. 3) They developed an exact QFT implementation method for general $ m $ using circuits of size polynomial in $ \log m $. 4) They constructed efficient Boolean functions for amplitude amplification exclusively for $ m = 2 $. 5) They showed that uniform quantum circuit families and quantum Turing machines are not equivalent for polynomial time computations. 6) They established that QFT on non-abelian groups is inherently inefficient for all values of $ m $. 7) They demonstrated that amplitude amplification cannot be used for hidden subgroup identification over $ \mathbb{Z}_n^m $.

✓ Correct Answer:

The correct answer is 3) They developed an exact QFT implementation method for general $ m $ using circuits of size polynomial in $ \log m $..

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to other abelian groups G, e.g., toG=Zn m, provided that two problems become eciently solved. One of them is just the exact computation of the quantum Fourier transform modulo m. At the time of publishing [6], this was known for smooth m, see the manuscripts of Cleve [11] and Coppersmith [12]. The exact quantum Fourier transform proposed by Mosca and Zalka [10] solves this problem for 772 An exact quantum hidden subgroup algorithm and applications to solvable groups generalm. The other problem is constructing a Boolean function on Gwith certain properties that support amplitude ampli cation to obtain new information about the hidden subgroup with certainty. To our knowledge, apart from the case m= 2, no such construction has been published although the authors of [6] were planning to investigate the issue for groups of smooth order. In this paper, using ideas from the discrete logarithm algorithm of Mosca and Zalka, we actually solve a modi ed version of the problem for general m: we construct a function with the desired properties, though on Gf0;1g, rather than on G. See the remarks at the end of Subsection 3.1 for details. Models for exact quantum computations. Bernstein and Vazirani de ne complexity classes related to exact quantum computing (such the class EQP) based on quantum Turing machines in [2]. Unfortunately, in contrast to bounded error quantum computing, the power of this model turns to be quite restricted for the purposes of exact versions of Fourier sampling based approaches. Indeed, as it is pointed out by Nishimura and Ozawa in [13], the quantum Fourier transform QFTmmodulomcan be implemented by a single Turing machine only for a nite set of integers m. Therefore, we consider more powerful models based on quantum circuits . We refer the reader to the paper [14] by Nishimura and Ozawa for the details of the def- inition of uniform quantum circuit families. Here we provide a brief summary. A quantum circuit family is a sequence of circuits built from a given set of elementary gates, called the basis. Uniformity means that the circuit Cs, used for input length s, is required to be con- structible by a deterministic Turing machine in time polynomial in the size of Csincluding also that the numerical parameters of the gates should be computable by a classical deter- ministic algorithm in time polynomial in the circuit size and a prescribed precision. Uniform quantum circuit families with nite bases are equivalent to quantum Turing machines at least for polynomial time computations, see [15]. Cleve [11] and Coppersmith [12] noticed that the quantum Fourier transform QFTmofZmfor smooth m(that is, when all the prime factors ofmare polynomially small) can be exactly computed by an in nitely based uniform circuit family of polynomial size. Later, Mosca and Zalka [10] proposed a method for implement- ingQFTmexactly by a circuit family of size polynomial in log m. That family is uniform in the sense that the parameters of some gates in the circuit for QFTmcan be eciently (in time

Question 539 multiple-choice

Lattice-based cryptography, and in particular the Learning With Errors (LWE) problem, relies on the hardness of finding a secret vector given noisy linear equations, making it a leading candidate for post-quantum security. Advances in attack methods have shifted the practical cost of breaking these schemes, especially as quantum algorithms become relevant. Which approach enables a quantum speedup for efficiently detecting whether any Fourier coefficient in a high-dimensional array exceeds a threshold, thus improving the effectiveness of dual attacks against LWE? 1) Grover's search applied to modulus switching 2) Classical brute-force enumeration of secret vectors 3) Gradient descent optimization on the Fourier spectrum 4) Parallel random sampling of secret candidates 5) Amplitude estimation applied to the FFT threshold problem 6) Integer linear programming for coefficient selection 7) Lattice reduction using BKZ algorithm

✓ Correct Answer:

The correct answer is 5) Amplitude estimation applied to the FFT threshold problem.

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In [GJ21] the search over part of the secret vector is realised usin g a Fast Fourier Transform style algorithm and the search space is signiﬁcan tly reduced by roughly considering only the most signiﬁcant bits of this part of th e secret. In [MAT22] this last step is replaced by “modulus switching” [BV11, AFF P14, KF15, GJS15] which further provides signiﬁcant performance gain s. Overall, these neweriterationsofthedual attackrelatethe searchspac eto the underlying secret in such a way that large dimensions can now be covered even w hen the norm of the secret vector is not very small (previous versions of t he dual attack relied on, say, coeﬃcients si∈ {−1,0,1}). Thus, with this new generation of dual attacks, unstructured se arch starts again to play a bigger role in costing attacks on LWE. It is therefore n atural to ask what performance gains can be obtained by tackling this unst ructured search using a Grover-like quantum algorithm. More precisely, [MAT2 2] relies on two diﬀerent kinds of unstructured search: –Secret guessing: part of the secret is exhaustively searched unt il a match is found. Since the secret is generated according to a discrete Gaus sian of small width, a signiﬁcant speedup can be obtained by starting the search with the most likely values of the secret ﬁrst. The expected complexity of th is step is known as the guessing complexity. –FFT threshold: given a list of values in a n-dimensional array, and a thresh- old, the problem is to decide whether one of the coeﬃcients of the Fo urier transform of the array is above the threshold. This problem arises when trying to determine whether the secret guess was correct by dist inguishing between a uniform distribution and a Gaussian one. Contributions. After some preliminaries in Section 2, we provide a quantum versionofthedualattackof[MAT22].Speciﬁcally,ourimprovement saretwofold. In Section 3 we give a more rigorous analysis of the (classical and qua n- tum) expected complexity of guessing a vector (whose coeﬃcients are) drawn from a modular discrete Gaussian. In [MAT22], the authors estimate d this com- plexity as the exponential of the entropy which is known not to be co rrect in general [Mas94]. We show that this complexity is indeed related to the entropy 2 in the case of a (modular) discrete Gaussian, albeit up to an exponen tial factor in the dimension. In Section 4 we show how to obtain a quantum speedup on the search for Fast Fourier Transform (FFT) coeﬃcients above a given threshold . This was left as an open problem in [MAT22]. Here, we leverage the relative sparse ness of the FFT and use amplitude estimation to estimate the Fourier coeﬃcients . In Section 5 we provide and analyse a quantum augmented dual atta ck util- ising our guessing and mean estimation results. In Section 6, we then estimate the impact of our algorithm on the cos t of solving instances of lattice-based schemes with parameters taken from the liter- ature. We will refer to such

Question 540 multiple-choice

In computational group theory and quantum algorithms, the task of detecting projectors associated with Young diagrams in symmetric groups is notable for exhibiting significant differences in classical and quantum query complexities. The representation theory of symmetric groups, particularly regarding irreducible representations and their dimensions, plays a crucial role in determining algorithmic efficiency for such problems. For symmetric groups S_n with n ≥ 2, which statement accurately characterizes the one-dimensional irreducible representations over the complex numbers? 1) There are exactly n one-dimensional irreducible representations, corresponding to each cycle type. 2) Only the trivial representation is one-dimensional for all n ≥ 2. 3) All irreducible representations are one-dimensional for n ≥ 2. 4) Every Young diagram gives rise to a one-dimensional irreducible representation. 5) The only one-dimensional irreducible representations are the trivial and sign representations. 6) The number of one-dimensional irreducible representations equals the number of elements of order two. 7) One-dimensional irreducible representations are associated with two-row Young diagrams.

✓ Correct Answer:

The correct answer is 5) The only one-dimensional irreducible representations are the trivial and sign representations..

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∆/parenleftBigg χR(Tk) dR/parenrightBigg (5.23) We want this uncertainty to be less than 1: then, by the same routine that ensures us that we can use the classical algorithm to detect the normalized character, we set ∆/parenleftBigg χR(Tk) dR/parenrightBigg =/epsilon1||X||||Y||≤1 (5.24) – 22 – JHEP05(2023)191and obtain, for k/lessmuchn /epsilon1≤1 ||X||||Y||=/radicalBigg n! d2 R./radicalBigg k n(n−1)...(n−k+ 1)∼/radicalBigg n! d2 R/radicalBigg k nk(5.25) Making use of the upper bound /epsilon1∗on/epsilon1, we write a lower bound on the query complexity which we call Q∗ k Q∗ k(R) =1 (/epsilon1∗)2=nk k.d2 R (n!)(5.26) which holds for k= 2,3,...,k∗(n). Taking the worst case, this complexity is maximised for the largest dR. Keeping in mind (dmax R)2≤(n!−2)(5.15), we are led to Q∗ k(Rmax)≤nk k.(n!−2) (n!)∼nk k(5.27) This is the behavior of the largest query complexity for each k. The total query complexity is obtained by summing them: Q∗(Rmax) =k∗(n)/summationdisplay k=2Q∗ k(Rmax) =k∗(n)/summationdisplay k=2nk k≥nk∗(n) k∗(n)=nnα−α(5.28) where we assume k∗(n) =O(nα)at largen(for some 0< α < 1/2, withα= 1/4being a conjecture as discussed in section 2.1). This shows that the query complexity of this algorithm is much larger than that of the quantum algorithm with n2αlogn, see (3.10). 5.2 Discussion and future directions 5.2.1 Interpretation We are ﬁnding, for the Young diagram projector detection task, that the quantum phase estimation methods perform exponentially better than the classical analog we have deﬁned using randomised algorithms (compare for example (3.3)with(5.17)). This may be due to the intrinsically quantum nature of our task which may bring it closer to the exponential quantum improvements envisioned in [ 58]. There is no computational cost of converting classical vector data to quantum states in our discussion of section 3. In practical recommendation tasks it has been found that the randomised algorithms only diﬀer polynomially in complexity from the quantum tasks [ 29, 59]. 5.2.2 One dimensional representations We have asserted, in arriving at (5.16), that the one-dimensional representations of the symmetric groups are the trivial and the sign representations. That there are no further one-dimensional irreps over CforSn(n≥2) is discussed in [ 60–62]. – 23 – JHEP05(2023)1915.2.3 More general choices of V We have formulated the randomised classical algorithm in terms of the regular representation. We could also use a more general reducible representation which contains all irreps Rin its decomposition into irreducibles. If Vcontains every irrep R, then its dimension is at least /summationdisplay RdR (5.29) It is known that this sum is equal to the number of permutations σwhich square to 1. This means that σonly has one and two-cycles. Let kbe the number of 2-cycles. /summationdisplay RdR=⌊n 2⌋/summationdisplay k=0n! 2kk!(n−2k)!(5.30) The summand is maximum when k= (n/2). The value is then n! (n/2)!2n/2(5.31) A lower bound is therefore D−=n! (n/2)!2n/2(5.32) An upper bound is D+= (n/2)D− (5.33) 6 Detection of the Kronecker projector in K(n) Permutation centralizer algebras K(n)related to C(Sn), which arise in the enumeration of tensor model observables and the computation of their correlators [ 31,63] have been used to give constructive integer matrix algorithms for computing Kronecker coeﬃcients

Question 541 multiple-choice

In finite group theory and its representation theory, understanding the relationship between centralizers, subgroups, and the structure of induced representations is fundamental. These concepts play a key role in the decomposition of tensor products and the analysis of group symmetries, especially in applications such as quantum computing. In the decomposition of the tensor product of two induced representations ρ↑G and σ↑G over a finite group G, which of the following determines the multiplicity of a particular irreducible representation appearing in the Clebsch-Gordan decomposition? 1) The order of the group G 2) The number of elements commuting with both generators 3) The index of Z(g)∩Z(hd) in Z(ghd) 4) The dimension of the trivial representation 5) The number of double coset representatives in G 6) The rank of the group algebra 7) The number of irreducible representations of G

✓ Correct Answer:

The correct answer is 3) The index of Z(g)∩Z(hd) in Z(ghd).

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of times (for any other dthe procedure is the same). In order to count this, consider t he two groups Z(gh) and Z(g)∩Z(h). It is easy to see that the latter is a subgroup of the former. If it is a strict subgroup, then there exist elements in Z(gh) which do not commute with either gorhor both. It turns out that in fact, any element of Z(gh) either commutes with both gandhor does not commute with both. To see this, notice that if sis an element not in Z(g)∩Z(h) and it commutes with gbut noth, then we have that gh= (gh)s=gshs=ghs. But this implies that hs=hand soscommutes with h. This means that any element inZ(gh) that does not commute with g, also does not commute with hand vice versa. Now for any non-trivial coset representative sofZ(g)∩Z(h) inZ(gh), we have that gh=sghs−1=gshs. Sincesdoes not commute with gandh, we have produced a pair ( gs,hs) distinct from ( g,h) such that their product is the same. For distinct coset representatives sandt, the pairs ( gs,hs) and (gt,ht) are distinct since if not, then gs=gtandhs=ht. This means that st−1∈Z(g)∩Z(h) which is not possible since s andtare in distinct cosets of Z(g)∩Z(h). Therefore, for each distinct coset representative sofZ(g)∩Z(h) inZ(gh), we get a distinct pair ( gs,hs) such that gshs=gh. The same argument holds for any non-trivial d. To show that these are all the possible pairs, we make use of t he above theorem. When ρandσare both trivial, equation ( 7.1) takes the form ([g],tr)⊗([h],tr)∼=/circleplusdisplay d/parenleftig tr↑Z(ghd) Z(g)∩Z(hd)/parenrightig ↑G. This shows that the number of times [ ghd] appears in [ g]·[hd] is the index of Z(g)∩Z(hd) inZ(ghd). With all this in hand, we can determine the Clebsch-Gordan de composition. We can re-write equa- tion ( 7.1) to get ρ↑G⊗σ↑G∼=/circleplusdisplay d/parenleftig/parenleftbig ρ↓Z(g)∩Z(h)d⊗σ↓Z(g)∩Z(h)d/parenrightbig ↑Z(ghd)/parenrightig ↑G, by taking the induction in two stages. Now suppose that/parenleftbig ρ↓Z(g)∩Z(h)d⊗σ↓Z(g)∩Z(h)d/parenrightbig ↑Z(ghd)breaks up into irreps of Z(ghd) as /parenleftbig ρ↓Z(g)∩Z(h)d⊗σ↓Z(g)∩Z(h)d/parenrightbig ↑Z(ghd)∼=/circleplusdisplay µ∈/hatwiderZ(ghd)mρ,σ,µµ, 24 wheremρ,σ,µ is the multiplicity of the irrep µ. We now obtain the Clebsch-Gordan decomposition as ρ↑G⊗σ↑G∼=/circleplusdisplay d/circleplusdisplay µ∈/hatwiderZ(ghd)mρ,σ,µ(µ↑G). (7.2) Since this decomposition is obtained by considering the act ion ofGalone, we need to check if the action of h∗is consistent with it. We do this in the next section as we deve lop the transform. 7.1 Clebsch-Gordan transform 7.1.1 Fluxon irreps We ﬁrst describe the Clebsch-Gordan transform for irreps of D(G) of the type ([ g],tr) (ﬂuxon irreps). For this case, we give an eﬃcient transform for any ﬁnite group. F or two ﬂuxon irreps, the Clebsch-Gordan decomposition states that ([h],tr)⊗([g],tr)∼=/circleplusdisplay d/parenleftig tr↑Z(ghd) Z(g)∩Z(hd)/parenrightig ↑G∼=/circleplusdisplay d/circleplusdisplay µ∈/hatwiderZ(ghd)mtr,tr,µ(µ↑G). We give the transform in two steps, one for each of the above tw o isomorphisms. In the ﬁrst step, we need to convert from the basis |ha,gb/a\}bracketri}htto|(ghd)k,t/a\}bracketri}ht, wheretis an element of the transversal TdofZ(g)∩Z(hd) in Z(ghd). Once we ﬁx the double coset representatives dand the transversal t(for eachd), this transformation is straightforward. Given |h′,g′/a\}bracketri}ht, we determine their product and the double coset representa tivedthat this product belongs to.

Question 542 multiple-choice

Topological quantum computing relies on mathematical structures to model and manipulate quantum information through the exchange of anyons. Ribbon fusion categories (RFCs) are central in describing the braiding operations that correspond to quantum logic gates in these systems. Which property of ribbon fusion categories is particularly essential for ensuring that the physical braiding of anyons can be systematically mapped to robust quantum logic gate operations? 1) The existence of a universal object in the category 2) The ability to define a tensor product on objects without associativity 3) A strict commutativity law for all morphisms 4) The absence of fusion rules among objects 5) The presence of a braiding structure compatible with fusion and ribbon twists 6) The lack of representation-theoretic connections to quantum groups 7) The requirement that all morphisms be invertible

✓ Correct Answer:

The correct answer is 5) The presence of a braiding structure compatible with fusion and ribbon twists.

📚 Reference Text:

Title: The ribbon category framework for topological quantum computing Year: 2022 Paper ID: 3a09c13b178c5218167e551716665fe43e41b979 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/3a09c13b178c5218167e551716665fe43e41b979 Abstract: . This expository article supplies the mathematical background underpinning the braid representation calculator introduced in [ Abo22]; those representations describe the sets of logic gates available to a topological quantum computer for processing encoded qubits. Assuming little background in category theory, we ﬁrst recall the notion of a ribbon fusion category (RFC), collecting most of the necessary deﬁnitions. Then we discuss how certain RFCs arise from the representation theory of quantum groups. We explore the braiding in these categories in detail, since it is essential for the quantum computing application.

Question 543 multiple-choice

Lattice-based cryptography relies on the computational hardness of certain problems in high-dimensional lattices, making it a promising candidate for post-quantum security. Understanding reduction algorithms and attack techniques is vital for evaluating and choosing secure cryptographic parameters. Which statement correctly describes the impact of quantum sieving algorithms compared to classical sieving in lattice reduction attacks? 1) Quantum sieving algorithms have polynomial cost in the block size parameter. 2) Quantum sieving algorithms require fewer lattice basis vectors than classical sieving. 3) Quantum sieving algorithms achieve an exponential speedup, with the cost exponent reduced from 0.292 to 0.257 times the dimension parameter. 4) Quantum sieving algorithms are only applicable to lattices with orthogonal bases. 5) Quantum sieving algorithms increase the number of short vectors found compared to classical sieving. 6) Quantum sieving algorithms are less efficient than BKZ for all dimensions. 7) Quantum sieving algorithms eliminate the need for lattice basis reduction.

✓ Correct Answer:

The correct answer is 3) Quantum sieving algorithms achieve an exponential speedup, with the cost exponent reduced from 0.292 to 0.257 times the dimension parameter..

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For anyx∈Zq, denote by ˜ x∈x+qZthe unique integer such that |˜x|/lessorequalslantq−1 2. We extend this notion to vectors in x∈Zn qcomponentwise. In other words, ˜xis the lift from ZqtoZcentered on 0. We deﬁne /⌊ard⌊lx/⌊ard⌊lforx∈Zn qas/⌊ard⌊l˜x/⌊ard⌊l. 2.1 Lattices A lattice Lis a discrete subgroup of Rd. We can represent it as {/summationtextxi·bi|xi∈Z} wherebiarethe columns ofamatrix B, wemaywrite L(B). IfBhasfull column rank, we call Ba basis. While the central object of this work, the dual attack, critically re lies on lattice reduction, such as the BKZ algorithm, we mostly make blackbo x use of these algorithms here. Thus, we refer the reader to e.g. [GJ21, MAT22] for details. In particular, the blackbox use we make of lattice reduction algorithms and, critically, lattice sieving algorithms is captured in Algorithm 1. Algorithm 1: Short Vectors Sampling Procedure [GJ21] Input:A basisB=/bracketleftbig b0...bd−1/bracketrightbig for a lattice and 2 ≤β0,β1∈Z/lessorequalslantdandD. Output: A list ofDvectors from the lattice. 1L={}. 2fori∈[0,⌈D/Nsieve(β1)⌉−1]do 3Randomise the basis B. 4Run BKZ-β0to obtain a reduced basis b′ 0,...,b′ d−1. 5Run a sieve in dimension β1on the sublattice spanned by b′ 0,...,b′ β1−1to obtain a list of Nsieve(β1) vectors and add them to L. 6returnL In Algorithm 1 the BKZ- β0call performs lattice reduction with parameter β0where the cost of the algorithm scales at least exponentially with β0. The BKZ algorithm proceeds by making polynomially many calls to an SVP orac le. In this work, this oracle is instantiated using a lattice sieving algorithm which is also called explicitly in Algorithm 1 with parameter β1. Such a sieving algorithm outputsNsieve(β1) short vectors in the lattice L(B) and has a cost exponential 4 inβ1. The magnitude Nsieve(β1) also grows exponentially with β1but slower than the cost of sieving. We will write TBKZ(d,β0) for the cost of running BKZ- β0in dimension dandTsieve(β1) for the cost of sieving in dimension β1. We may instantiate the lattice sieve with a classical algorithm [BDGL16] w hich has a cost of 20.292β1+o(β1). We may also instantiate the lattice sieve with a quantum augmented variant of sieving [LMv13, Laa15, AGPS20, CL2 1] which has a cost of 20.257β1+o(β1). Thus, according to the best known algorithms we haveTBKZ(d,β0)∈poly(d)·2Θ(β0)andTsieve(β1)∈2Θ(β1). 2.2 Learning with Errors The Learning with Errors problem (LWE) is deﬁned as follows. Deﬁnition 1 (LWE). Letn,m,q∈N, and letχs,χebe distributions over Zq. Denote by LWEn,m,χ s,χethe probability distribution on Zm×n q×Zm qobtained by sampling the coordinates of the matrix A∈Zm×n qindependently and uniformly overZq, sampling the coordinates of s∈Zn q,e∈Zm qindependently from χsand χerespectively, setting b:=A·s+emodqand outputting (A,b). We deﬁne two problems: –Decision-LWE. Distinguish the uniform distribution over Zm×n q×Zm qfrom LWEn,m,χ s,χe. –Search-LWE. Given a sample from LWE n,m,χ s,χe, recover s. Dual Attack. Dual-lattice attacks, or simply “dual attacks”, on LWE and related problems were introduced in [MR09]. In its simplest form it proc eeds as follows. Given either ( A,A·s+e) or (A,u) where ( A,u) are uniform and w.l.o.g s,eare short [ACPS09], the attack ﬁnds short xjs.t.xT j·A≡0. Then, we either obtainxT j·A·s+/an}⌊rack⌉⊔l⌉{⊔xj,e/an}⌊rack⌉⊔ri}h⊔=/an}⌊rack⌉⊔l⌉{⊔xj,e/an}⌊rack⌉⊔ri}h⊔or/an}⌊rack⌉⊔l⌉{⊔xj,u/an}⌊rack⌉⊔ri}h⊔. The former follows a distribution with small entries,

Question 544 multiple-choice

Quantum error correction is crucial for preserving quantum information in the presence of noise, particularly for large-scale quantum systems. Approximate error correction criteria enable code designers to ensure resilience without requiring complete characterization of the noise. Which condition is a sufficient basis-dependent criterion for approximate quantum error correction in large-dimensional systems, particularly when dealing with erasures at known locations? 1) All Kraus operators must commute with the logical operators of the code. 2) The fidelity between any two encoded logical basis states must be exactly zero. 3) The code must be covariant under all possible symmetry groups of the system. 4) The environment must have full knowledge of the logical information after erasure. 5) The local reduced diagonal states of encoded logical basis states should be close to a fixed state, and off-diagonal elements should have small norm. 6) The trace norm of all diagonal elements must vanish for correctability. 7) Only exact Knill-Laflamme conditions can guarantee approximate error correction.

✓ Correct Answer:

The correct answer is 5) The local reduced diagonal states of encoded logical basis states should be close to a fixed state, and off-diagonal elements should have small norm..

📚 Reference Text:

with explicit recovery procedures. Intuitively, if we consider erasures at known locations, we can expect that if all local reduced states of code words look alike independently of the logical information, then the code performs well. That is, if for each individualsubsystem each code word has the same reduced state, then,because the environment gets access only to those indi- vidual reduced states, it obtains no information about the code word and the erasure is, thus, correctable. Thisintuition is correct in the exact case, but in the approximate case the fact that the entanglement fidelity is defined with a “stabilization ”over a reference system poses an additional challenge [43]. Our solution is to consider how logical operators of the form jxihx 0jare encoded, where fjxigis any fixed basis of the logical system. In the case of a singleerasure at a known location, we define ρ x;x0 i¼trAnAi½Eðjxihx0jÞ/C138; ð31Þ noting that ρx;x0 iis a quantum state if x¼x0but is not even necessarily Hermitian for x≠x0. Our criterion then states the following: If the states ρx;x iare approximately indepen- dent of x, and if each ρx;x0 iforx≠x0has a very small norm, then the code is a good approximate error-correcting code against erasure of subsystem i.CONTINUOUS SYMMETRIES AND APPROXIMATE QUANTUM … PHYS. REV . X 10,041018 (2020) 041018-11 Our criterion is a variant of the approximate Knill- Laflamme conditions of B´ eny and Oreshkov [30] in a particular basis. The nontrivial part of our condition is thesimple and practical conditions on how to bound the errorparameter ϵ worst of the code. Proposition 4. For any completely positive, trace- preserving maps EandN(the encoding and noise channels, respectively), letdN∘Ebe a complementary channel of N∘E. Fixing a basis of logical states fjxig,d e f i n e ρx;x0¼dN∘Eðjxihx0jÞ: ð32Þ Assume that there exists a state ζ,a sw e l la sc o n s t a n t s ϵ, ν≥0, such that Fðρx;x;ζÞ≥ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1−ϵ2p ; ð33aÞ kρx;x0k1≤νforx≠x0: ð33bÞ Then, the code Eis an approximate error-correcting code with an approximation parameter satisfying ϵworstðN∘EÞ≤ϵþdLﬃﬃﬃνp; ð34Þ where dLis the logical system dimension. If one of several noise channels is applied at random and it is known which one occurs, then Eq. (34) holds for the overall noise channel if the assumptions above are satisfiedfor each individual noise channel. Note that the criterion holds for any arbitrary noise channel, not only for erasures at known locations. The proof of Proposition 4 is given in Supplemental Material, Sec. C [37]. Our criterion is a sufficient condition for a code to be approximately error correcting, but the condition is notnecessary. When the criterion does not apply, we cannotdraw any conclusion about the code ’s performance. We note that our criterion does not make reference to individual Kraus operators of the noise channel, as theKnill-Laflamme conditions or their approximate versionsdo[26,30] . This property eases its application to large- dimensional physical quantum systems. VI. EXAMPLES OF COVARIANT CODES Here, we study three classes of covariant codes that illustrate the behavior of our bound in

Question 545 multiple-choice

In the study of finite non-abelian 2-groups, subgroup structure and the behavior of involutions play a crucial role in classifying possible group types. The interplay between the center, Frattini subgroup, and maximal normal abelian subgroups is especially significant for understanding these groups' architecture. Which statement about finite non-abelian 2-groups with a unique maximal normal abelian subgroup A of exponent 4 is TRUE? 1) The center Z is always equal to the maximal normal abelian subgroup A. 2) Every element of G must have order 2. 3) The group necessarily has more than three involutions. 4) The subgroup W1 is always outside the Frattini subgroup F. 5) The exponent of G is equal to the exponent of A, both being 4. 6) The group is always metacyclic. 7) The quotient G/F is always abelian.

✓ Correct Answer:

The correct answer is 5) The exponent of G is equal to the exponent of A, both being 4..

📚 Reference Text:

the group of Proposition 1.7(a). It follows that W1ðAÞ¼ZðK/C3Þso that W1ðAÞ¼ZðLÞ. SetAi0¼ha;W1ðAÞi, where oðaÞ¼4 and L=Ai0acts faithfully on Ai0stabilizing the chain Ai0>W1ðAÞ>f1g. In particular, there is lALsuch that al¼aa2¼a/C01. Henceha;liis a metacyclic minimal non-abelian subgroup, so that W1ðAÞGha;li. This is a contradiction since W1ðAÞcentralizes ha;li. Hence K¼Gand we have ob- tained the group of part (c) of Theorem 1.2.298 Z. Janko 4 Proof of Theorem 1.3 LetGbe a ﬁnite non-abelian 2-group all of whose minimal non-abelian subgroups are isomorphic to Q8orH2¼ha;bja4¼b4¼1;ab¼a/C01i. Let Abe a maximal normal abelian subgroup of G, so that W1ðGÞ¼W1ðAÞcZðGÞ(Lemma 1.15) and W1ðAÞ4. Then by Proposition 1.10, jG :Aj¼2 and if xAG/C0A, then oðxÞ¼4 and xinverts each element in A=W1ðAÞ. Let aAAwith oðaÞ¼8. Then ax¼a/C01zwith zAW1ðAÞso that ½a;x/C138¼a/C02zis of order 4 and so jG0j>2. IfGhas more than one abelian maximal subgroup, then jG :ZðGÞj ¼4 and Proposition 1.14 gives jGj¼2jZðGÞj jG0jand jG0j¼2, a contradiction. Hence Ais a unique abelian maximal subgroup of G. In what follows we assume that exp ðAÞ¼expðGÞ¼4. In view of Propositions 1.11 and 1.12 we may also assume that both Q8andH2are subgroups of G.W e haveFðGÞcW1ðAÞcZðGÞand jFðGÞjd4. If x,yare elements of order 4 in G with ½x;y/C13801, then ½x;y/C138is an involution in ZðGÞand so (by Proposition 1.5) hx;yiGH2orQ8, which implies that in case x20y2, we have hx;yiGH2and yxAfy/C01;yx2g. Considering G=FðGÞwe get G¼K/C2V, where W1ðAÞ¼FðGÞ/C2V, exp ðVÞc2, W1ðKÞ¼FðKÞ¼FðGÞcZðKÞandAVKis a maximal normal abelian subgroup ofK. We have to determine the structure of Kand therefore in the sequel we may assume that K¼Gand so W1ðAÞ¼W1ðGÞ¼FðGÞcZðGÞ. First assume that jFðGÞj ¼4, so that our group Ghas exactly three involutions. By Proposition 1.13, Ghas a metacyclic normal subgroup Msuch that G=Mis elemen- tary abelian of order at most 4. However exp ðMÞc4 and so jMjc24and jGjc26. Since both H2andQ8are subgroups of G, we have jGjd25. Suppose that jGj¼25. In this case Gis non-metacyclic since exp ðGÞ¼4. IfGwere minimal non-metacyclic, then each minimal non-abelian subgroup of Gwould be iso- morphic to H2(see Propositions 1.9 and 1.11), contrary to our assumptions. But G has only three involutions and so Proposition 1.9 implies that Ghas a minimal non- metacyclic subgroup H¼Q/C2huiwith Q¼ha;biGQ8,ZðQÞ¼hzianduan in- volution. Since FðGÞ¼hz;uiGE4, we may assume (interchanging uanduzif nec- essary) that there is an element vAG/C0Hsuch that v2¼u.I fG0¼Q0¼hzi, then QtG,G=QGC4and vinduces an inner automorphism on Q. In that case G¼QC GðQÞwith QVCGðQÞ¼hziandCGðQÞ=hziGC4. But exp ðGÞ¼4 and so CGðQÞsplits over hziand we get GGQ8/C2C4which is a group of part (b4) of our theorem. We assume that G0¼hz;ui¼W1ðGÞ¼FðGÞ.I f ½a;v/C138Ahziand ½b;v/C138Ahzi, then G¼ha;b;viimplies that G=hziis abelian and so G0¼hzi, con- trary to our assumption. It follows that we may assume (interchanging aandbif nec- essary) that ½a;v/C138Ahz;ui/C0hzi. Since a2¼z0½a;v/C138, we have ha;viGH2and so ½a;v/C138¼u¼v2andva¼v/C01. We have ½a;bv/C138¼½a;b/C138½a;v/C138¼zu0a2¼zand therefore ha;bviGH2. It follows that ðbvÞ2¼zuorz.I f zu¼ðbvÞ2¼b2v2½v;b/C138¼zu½v;b/C138;On minimal non-abelian subgroups in ﬁnite p-groups 299 then ½v;b/C138¼1. If z¼ðbvÞ2¼b2v2½v;b/C138¼zu½v;b/C138; then ½v;b/C138¼u, which implies ½v;ab/C138¼½v;a/C138½v;b/C138¼uu¼1 and so in this case replac- ingbbyb0¼abwe get ½v;b0/C138¼1. Hence

Question 546 multiple-choice

In group theory and computational complexity, the commutator length of an element in a group reflects the minimal number of commutators needed to express that element. The computational difficulty of determining commutator length has significant implications for algorithms in algebra and cryptography. Which statement best explains why there is no known polynomial-time algorithm for deciding whether the commutator length of an element in a non-abelian free group is at most a given number, assuming P ≠ NP? 1) The commutator length function is undefined for non-abelian free groups. 2) Commutator length in non-abelian free groups can always be computed in constant time. 3) The problem reduces to solving systems of linear equations, which are always tractable. 4) Determining if the commutator length is at most a given value is NP-complete, implying computational intractability under standard complexity assumptions. 5) Non-abelian free groups have trivial commutator subgroups, making computation unnecessary. 6) All elements in non-abelian free groups are products of a single commutator. 7) Polynomial-time algorithms exist for all group-theoretic decision problems.

✓ Correct Answer:

The correct answer is 4) Determining if the commutator length is at most a given value is NP-complete, implying computational intractability under standard complexity assumptions..

📚 Reference Text:

Title: Computing commutator length is hard Year: 2020 Paper ID: aa63b42585a7ef22ed626313460b9de5c82d2e55 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/aa63b42585a7ef22ed626313460b9de5c82d2e55 Abstract: The commutator length $cl_G(g)$ of an element $g \in [G,G]$ in the commutator subgroup of a group $G$ is the least number of commutators needed to express $g$ as their product. If $G$ is a non-abelian free groups, then given an integer $n \in \mathbb{N}$ and an element $g \in [G,G]$ the decision problem which determines if $cl_G(g) \leq n$ is NP-complete. Thus, unless P=NP, there is no algorithm that computes $cl_G(g)$ in polynomial time in terms of $|g|$, the wordlength of $g$. This statement remains true for groups which have a retract to a non-abelian free group, such as non-abelian right-angled Artin groups. We will show these statements by relating commutator length to the \emph{cyclic block interchange distance} of words, which we also show to be NP-complete.

Question 547 multiple-choice

Knowledge graph embedding leverages mathematical structures to represent entities and relations for tasks like link prediction. Advanced models increasingly use concepts from group theory to capture complex relational patterns. Which approach allows knowledge graph embedding models to represent relations as elements of continuous non-Abelian groups, thereby capturing cycles, inverses, and richer symmetries in relational data? 1) Embedding relations as group elements within group action spaces, such as using SO(3) or SU(2) 2) Mapping relations to scalar values through vector addition 3) Representing relations as diagonal matrices in commutative (Abelian) groups 4) Encoding relations using random initialization without algebraic constraints 5) Assigning relations as fixed binary values in a discrete space 6) Modeling relations with simple dot product operations between entity vectors 7) Utilizing Boolean logic gates for relation representation in the graph

✓ Correct Answer:

The correct answer is 1) Embedding relations as group elements within group action spaces, such as using SO(3) or SU(2).

📚 Reference Text:

Title: A Group-Theoretic Framework for Knowledge Graph Embedding Year: 2019 Paper ID: 0ba58e5040b9e3f330f032ba9ea0f5aebe21f213 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0ba58e5040b9e3f330f032ba9ea0f5aebe21f213 Abstract: We demonstrated the existence of a group algebraic structure hidden in relational knowledge embedding problems, which suggests that a group-based embedding framework is essential for designing embedding models. Our theoretical analysis explores merely the intrinsic property of the embedding problem itself hence is model-independent. Motivated by the theoretical analysis, we have proposed a group theory-based knowledge graph embedding framework, in which relations are embedded as group elements, and entities are represented by vectors in group action spaces. We provide a generic recipe to construct embedding models associated with two instantiating examples: SO3E and SU2E, both of which apply a continuous non-Abelian group as the relation embedding. Empirical experiments using these two exampling models have shown state-of-the-art results on benchmark datasets.

Question 548 multiple-choice

In algebraic number theory and algebraic geometry, the structure of quadratic forms over discrete valuation rings (DVRs) depends crucially on properties of the base ring, including the invertibility of specific elements. The diagonalizability of modules equipped with symmetric bilinear forms is an essential result in this area. Which property of the ring R ensures that every module with a non-degenerate symmetric bilinear form over a discrete valuation ring is diagonalizable? 1) R is a local Artinian ring 2) 2 is a unit in R 3) R is a principal ideal domain 4) The value group of the valuation is infinite 5) All ideals of R are maximal 6) The module is decomposable 7) The bilinear form is alternating

✓ Correct Answer:

The correct answer is 2) 2 is a unit in R.

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have: COROLLARY 1. // ord. ^tjeZ and generate the unit ideal, then ψ is bijective (for the matrix (stfi3)). COROLLARY 2. // each ord. sίu is in Q, or less generally, if the value group of the valuation is contained in Q,then ψ is bijective. Appendix Quadratic forms over Z& Let R be a discrete valuation ring, M a finite free β-module and (,): M X M —> R a symmetric bilinear map. The following lemma is easy linear algebra. LEMMA 1. Let n19 ->'9nseM and N be the R-submodule generated by the n^s. If άet.((nί9nj)) is a unit in R, then the n/s are R-linearly independent and M = N 0 2V1. We say that M is decomposable if M — N Θ N' with N and N' non- zero submodules of M and orthogonal M is diagonalizable if it is the orthogonal sum of 1-dimensional submodules and M is primitive if there exist m,mf eM with (m,mθ a unit in R. THEOREM 1. If 2 is a unit in R, then M is diagonalizable. Proof. We may assume M primitive. Let m,m' eM with (m,m') a https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press ABELIAN VARIETIES 95 unit. Then (m + m', m + mf) = (m, m) + (m\ m') + unit. So there exists n € M with (n, n) a unit. By Lemma 1, M = Rn 0 (Rri)L and we use in- duction on the dimension. COROLLARY 1. Let ^ be a symmetric matrix over Z6 (t, Φ 2). Then there exists an invertible matrix S over Z£ such that Ss/S1 is diagonal. Suppose now that 2 is not a unit in R. LEMMA 2. // M is primitive and indecomposable, then dim. M < 2. Proof. If there exists meM with (m,m) a unit then by Lemma 1, M — Rm 0 (Rm)1. So M — i?m and dim. M = 1. Suppose that (m, m) is in the maximal ideal of R for all m. Choose mx and m2 with (mx, m2) a unit. By Lemma 1 and indecomposibility, M = Rmλ + Rm2. THEOREM 2. For any M there exists a diagonalizable R-module N such that the orthogonal direct sum of M and N is diagonalizable. Proof. We may assume M primitive and indecomposable. By Lem- ma 2 we may assume M generated by ex and e2 with (e19 ej, (e2, e2) in the maximal ideal and (el9 e2) a unit. Replacing e2 by a multiple we may assume (eu e2) = — 1. Let N = Re2 with (β3, e3) = 1. Then {ex + e3, β2 + e3) = 0. Since (^ + e3, eγ + e3) and (e2 + e3, e2 + e3) are units we conclude from Lemma 1 that M 0 N admits an orthogonal basis consisting of βi + &3> &2 + &3 ^nd one other vector. Remark. The proof of Theorem 2 shows the following: if π is a generator of the maximal ideal of

Question 549 multiple-choice

Quantum Phase Estimation (QPE) is an essential algorithm in quantum computing, often used to find eigenvalues of unitary operators for applications such as factoring and quantum chemistry. Variational quantum circuits (VQCs) are hybrid algorithms designed to optimize quantum operations, especially useful on noisy intermediate-scale quantum devices. Which approach allows quantum phase estimation to be performed with shallower quantum circuits and improved resilience to hardware noise, making it more suitable for current quantum hardware? 1) Increasing the number of qubits in the QPE register 2) Implementing error correction codes in standard QPE 3) Using deeper circuits with additional quantum gates 4) Employing classical post-processing to refine QPE results 5) Running QPE exclusively on photonic quantum computers 6) Relying solely on the inverse Quantum Fourier Transform 7) Approximating QPE with variational quantum circuits (VQCs)

✓ Correct Answer:

The correct answer is 7) Approximating QPE with variational quantum circuits (VQCs).

📚 Reference Text:

Title: Learning Quantum Phase Estimation by Variational Quantum Circuits Year: 2023 Paper ID: 7271aa7bb9bfa6e36f40a047dbe4ba0f1f2b0514 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/7271aa7bb9bfa6e36f40a047dbe4ba0f1f2b0514 Abstract: Quantum Phase Estimation (QPE) stands as a pivotal quantum computing subroutine that necessitates an inverse Quantum Fourier Transform (QFT). However, it is imperative to recognize that enhancing the precision of the estimation inevitably results in a significantly deeper circuit. We developed a variational quantum circuit (VQC) approximation to reduce the depth of the QPE circuit, yielding enhanced performance in noisy simulations and real hardware. Our experiments demonstrated that the VQC outperformed both Noisy QPE simulation and standard QPE on real hardware by reducing circuit noise. This VQC integration into quantum compilers as an intermediate step between input and transpiled circuits holds significant promise for quantum algorithms with deep circuits. Future research will explore its potential applicability across various quantum computing hardware architectures.

Question 550 multiple-choice

In the study of finite group cohomology and representation theory, particular attention is given to the structure and properties of groups of order 32, including their centers, abelian subgroups, and extensions. Advanced methods such as spectral sequences and Morava K-theory are used to analyze these groups' classifying spaces. Which group of order 32 has its center isomorphic to C2 × C2, possesses a unique index 2 abelian subgroup, and requires the use of the Serre spectral sequence for computing its Morava K-theory due to the limitations of central and abelian extension methods? 1) The dihedral group D32 2) The semidihedral group SD32 3) The extraspecial group of order 32 4) Group 38 with center generated by ⟨a², c²⟩ 5) The generalized quaternion group Q32 6) The elementary abelian group C2 × C2 × C2 × C2 × C2 7) The direct product D8 × C2 × C2

✓ Correct Answer:

The correct answer is 4) Group 38 with center generated by ⟨a², c²⟩.

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a;bD1;c1acDa3;c1bcDa2b3i G41Dha;b;cja4Db4Dc2Da;bD1;cacDa3b2;cbcDa2bi Algebraic & Geometric Topology , Volume 11 (2011) 514 Björn Schuster All these groups have centre C2C2with quotient C3 2, a unique index 2 abelian subgroup, and 14 conjugacy classes of elements. This sufﬁces to conclude that they all have the same Euler characteristic n;2D1 216nC8n1 24n: The calculation of their Morava K–theories appears to require new arguments and is thus deferred. 4.6 Groups 42 and 43 G42andG43are extraspecial and were dealt with in [8], using a variant of Lemma 2.3 . 4.7 Groups 44–48 Presentations of G44–G48are given by: G44Dha;b;cja8Db2Dc2Db;cD1;babDa1;cacDa5i G45Dha;b;cja8Dc2D1;b2Da4;b;cD1;b1abDa1;cacDa5i G46Dha;b;cja4Db2Dc2Da;c2D1;a;a;cDb;cD1;babDaci G47Dha;b;cja8Db2Dc2Db;cD1;babDac;cacDa5i G48Dha;b;cja8Dc2D1;b2Da4; b;cD1;b1abDac;cacDa5i For each group, the centre ZisC2with quotient either D8C2(forG44andG45) or162c, the group we called H1in Section 3, in the other cases. All ﬁve groups have Euler characteristic n;2D7 48n3 44n: G44,G46andG47have a normal rank 3 elementary abelian subgroup with cyclic quotient, so they are covered by Theorem 1.2 (e). Furthermore, G44can be written as ha;ciÌhbiQD16ÌC2, and G45is a nonsplit version of that group, ie, ﬁts into an extension 1!QD16!G45!C2!1with the same action as for G44. Similarly, G48is a nonsplit version of G47Dha;ciÌhbiQD16ÌC2. Thus Theorem 1.1 implies that G45andG48are good, too. 4.8 Groups 49–51 G49is dihedral, G50semidihedral, and G51a generalised quaternion group, hence already covered in the literature, eg by [12]. Algebraic & Geometric Topology , Volume 11 (2011) Morava K–theory of groups of order 32 515 We have seen that the theory described in the previous sections covers 47 groups, leaving four to be calculated. Closer inspection also shows that the Morava K–theory of those 47 groups is equidistributed (by which we mean rankK.n/K.n/2i.BG/D rankK.n/K.n/0.BG/1for any i60modjvnj); we omit the details. 5 The remaining cases 5.1 Group 38 The centre of G38isZDha2;c2iC2C2. The representation ring has three generators ; ; of dimension one, inﬂated from G=Z, with .a/D1, .b/D .c/D1, .a/D .c/D1, .b/D1, and .a/D .b/D1, .c/D1. Let ADha;b;c2iC4C2C2and;2RAbe deﬁned by .a/Di,.b/D.c2/D1 and.a/D.b/D1,.c2/D1. The irreducible representations of G38are r s t, r;s;t2f0;1g, andDIndG A./, , DIndG A./, , DIndG A./ , . The group Ais a maximal abelian subgroup, but we found neither the central extension Z!G!G=ZnorA!G!G=Asuitable for calculation: in the spectral sequence for the central extension, the ﬁrst differential d3produces zero divisors, and for the second, we found no easy way to see that all invariants for the action of the quotient onK.n/.BA/are permanent cycles, as the rank of K.n/.BG38/would suggest. Thus we consider the normal subgroup EDhbiZwith quotient VC2C2and consider the Serre spectral sequence of the extension (5) 1!E!G38!V!1: Deﬁne nontrivial characters ;;2RE by the quotients E=ha2;bi,E=ha2;c2i, and E=hb;c2i, respectively. Then we obtain the following restrictions: G38 E 1 1.1C/ 2.1C/ Letx;y;z2K.n/.BE/be the Euler classes of ;; , respectively. Therefore K.n/.BE/K.n/x;y;z=.x2n;y2n;z2n/. Conjugation by ais trivial on RE, andcD,cD, butcD. This gives the following action of VonK.n/.BE/: axDx;ayDy;azDz;cxDx;cyDy;czDyCK.n/z: From the above table we furthermore see x2DResG Ec2./. The Serre spectral sequence has E2–page E2DH.BVIK.n/.BE//: Algebraic & Geometric Topology , Volume 11 (2011) 516 Björn Schuster Write MWDK.n/.BE/, then MDM0x=.x2nC1/, where we know M0from the wreath product calculation in Lemma 2.5.

Question 551 multiple-choice

In representation theory and quantum information, the Kronecker polytope is a geometric object that encodes constraints on the possible spectra of quantum systems and combinatorial structures arising from tensor products. Understanding its facets and membership is central to problems in computational complexity and mathematical physics. Which of the following pairs, used to define non-trivial facets of the Kronecker polytope, must satisfy three conditions—admissibility, trace, and determinant—for its associated inequalities to characterize Kron(m) within the positive Weyl chamber? 1) Young tableaux 2) Cartan subalgebra elements 3) Highest weight vectors 4) Schur elements 5) Standard basis vectors 6) Killing vectors 7) Ressayre elements

✓ Correct Answer:

The correct answer is 7) Ressayre elements.

📚 Reference Text:

operatorsAacting on Hilbert space, we denote by kAkthe operator norm, by kAk1 the trace norm, and by kAkFthe Frobenius norm. Finally, in section 3 we introduce normskkigandkkigfrom the Killing form of a Lie algebra g. 2. The Kronecker polytopes. In this section, we will prove our complexity result for the Kronecker polytopes (Theorem 1). While this result can also be obtained as a consequence of our general result (Theorem 2), which we prove in section 3 below, the exposition in this section contains all essential ideas while not requiring expert knowledge in representation theory. Our notation and terminology will match precisely the one used in section 3 below. 2.1. Inequalities for the Kronecker polytopes. Letm > 0be a positive integer. We deﬁne P(m) :=f(xA;xB;xC)2R3m:Pm i=1xA;i=Pm j=1xB;j=Pm k=1xC;k= 1g. We note that P(m)is an aﬃne space of dimension 3(m1). We will call (m) :=f(ei;ej;ek) :i;j;k = 1;:::;mgP(m)the set of weights, where eidenotes the i-th standard basis vector of Rm, andN(m) :=f(eiej;0;0) :i > jg[f (0;eiej;0) :i>jg[f (0;0;eiej) :i>jgthe set of negative roots . For any MEMBERSHIP IN MOMENT POLYTOPES IS IN NP AND CONP 5 H= (HA;HB;HC)2(Zm 0)3andz2Z, we deﬁne the following three subsets: (H=z) =f'2(m) :'H=zg; (H

Question 552 multiple-choice

Discrete quantum walks are a computational framework for simulating quantum systems on grids, often employed to study phenomena like angular momentum and Landau quantization. The choice of coordinate systems and basis transformations can significantly affect the conservation laws and symmetry properties in such simulations. Which mathematical operation is essential for aligning the spinor basis with the physical rotational symmetry when rewriting the Dirac Equation from Cartesian to polar coordinates in discrete quantum walk simulations? 1) Applying a Fourier transform to the spinor wavefunction 2) Diagonalizing the Hamiltonian in the Cartesian basis 3) Symmetrizing the Laplacian operator on the grid 4) Introducing a gauge field that depends only on radial distance 5) Scaling all grid points by a constant factor 6) Replacing Pauli matrices with identity operators 7) Performing a basis transformation in spin space to a polar basis

✓ Correct Answer:

The correct answer is 7) Performing a basis transformation in spin space to a polar basis.

📚 Reference Text:

article is to construct angular momen- tum for DQWs. In standard theories, angular momen- tum is conserved under rotations if the system has ro- tational symmetry. We want to obtain such a theorem in the simplest manner possible, so we de ne the DQW on a polar space grid, which by de nition has a natu- ral rotational symmetry. We then simulate the so-called cylindrical Landau levels. The material is organized as follows. We rst transcribe the usual, at-spacetime (1 + 2)D Cartesian Dirac Equation (CDE) into a Po- lar Dirac Equation (PDE), then construct a DQW which simulates the PDE in the presence of an arbitrary elec- tromagnetic eld. We exhibit the angular momentum of this DQW, establish its conservation in electromagnetic elds with axial symmetry and nally construct the dis- crete Landau levels and show by numerical simulation that these converge to the usual continuous-spacetime Landau levels as the step if the spacetime grid tends to zero. These results are summarized and discussed in the nal section. Appendix A o ers an alternative derivation of the PDE while Appendix B presents the literal com- putations behind the construction of the Landau levels. THE POLAR DIRAC EQUATION In (1 + 2)D at spacetime, the CDE can be written DA B B= 0; (1) with the operator Dde ned by D= i( 0@t+ 1@x+ 2@y)m; (2) where (t;x;y ) are Minkowski coordinates in the at (1 + 2)D spacetime and mis the mass of the particle. The indices ( A;B)2fL;Rg2refer to components on a cartesian, point-independent spin basis which we denote by (bL;bR). In this basis, the operators are represented by the Pauli matrices: [( 0)A B] =1;[( 1)A B] = i2;[( 2)A B] = i3;(3)arXiv:2007.06211v2 [quant-ph] 19 Jul 2020 2 where the notation [( i)A B] represents the matrix formed by the components of the operator iin the basis ( bL;bR). We also introduce, for further use, a metric in spin space, de ned by AB=1 ifA=B 0 otherwise: (4) To obtain the PDE from the CDE, one must proceed in two steps. The rst one consists in introducing the polar coordinates ( r;) in the plane and use the relations x=rcos,y=rsin, to express the partial derivatives @xand@yappearing inDin terms of @rand@. This delivers D= i 0@t+ i~ 1()@ri r~ 2()@m; (5) with ~ 1() = 1cos+ 2sin; (6) and ~ 2() = 1sin 2cos: (7) The second step consists in changing basis in spin space. The new, so-called polar basis ( b;b+), is de ned by b= cos 2bLi sin 2bR; (8) b+=i sin 2bL+ cos 2bR; (9) and we denote by MB aand (M1)b Athe change of basis matrices: B=M()B a a; b= (M1())b A A; (10) where h M()B ai = ei 21;h (M1())b Ai = ei 21:(11) In the new basis, the components of the operators 0, ~ 1 and ~ 2, read [( 0)a b] =1; [(~ 1)a b] = i2; [(~ 2)a b] = i3;(12) and components of the operator Dare Da b=

Question 553 multiple-choice

Quantum computing leverages group-theoretic problems to achieve exponential speedups for certain tasks, with the Hidden Subgroup Problem (HSP) playing a key role in algorithmic breakthroughs. The complexity of solving the HSP varies significantly between abelian and non-abelian group structures. Which class of non-abelian groups has recently been shown to admit efficient quantum algorithms for the Hidden Subgroup Problem by exploiting homomorphic reductions to abelian cases? 1) Symmetric groups of arbitrary order 2) General dihedral groups 3) Matrix groups over finite fields 4) Semi-direct product groups of the form Z_N ⋉ Z_{q^s} with specific factorization constraints 5) Alternating groups 6) Free groups 7) Quaternion groups

✓ Correct Answer:

The correct answer is 4) Semi-direct product groups of the form Z_N ⋉ Z_{q^s} with specific factorization constraints.

📚 Reference Text:

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics An Ecient Quantum Algorithm for the Hidden Subgroup Problem over some Non-Abelian Groups Demerson Nunes Gon calves1 Coordena c~ ao de Licenciatura em F sica, CEFET, Petr opolis, RJ Renato Portugal2 Coordena c~ ao de Ci^ encia da Computa c~ ao, LNCC, Petr opolis, RJ Tharso D. Fernandes3 Departamento de Matem atica Pura e Aplicada, UFES, Alegre, ES Programa de P os-Gradua c~ ao em Modelagem Computacional, LNCC, Petr opolis, RJ Abstract . The hidden subgroup problem (HSP) plays an important role in quantum com- putation, because many quantum algorithms that are exponentially faster than classical algorithms are special cases of the HSP. In this paper we show that there exist a new ef- cient quantum algorithm for the HSP on groups ZNo ZqswhereNis an integer with a special prime factorization. Keywords . Quantum Algorithms, Hidden Subgroup Problem, Quantum Computational Group Theory 1 Introduction The most important problem in group theory in terms of quantum algorithms is called hidden subgroup problem (HSP) [7]. The HSP can be described as follows: given a group Gand a function f:G!Xon some set Xsuch thatf(x) =f(y) i xH=yHfor some subgroupH, the problem consists in determining a generating set for Hby querying the functionf. We say that the function fhides the subgroup HinGor thatfseparates the cosets ofHinG. A quantum algorithm for the HSP is said to be ecient when the running time isO(poly(logjGj)). There are many examples of ecient quantum algorithms for the HSP in particular groups [10,11]. It is known that for nite abelian groups, the HSP can be solved eciently on a quantum computer [7]. On the other hand, an ecient solution for a generic non-abelian group is not known. Two important groups in this context are the symmetric and the dihedral groups. An ecient algorithm for solving the HSP for the former would imply in an ecient solution for the graph isomorphism problem [1] and for 1demerson.goncalves@gmail.com 2portugal@lncc.br 3tharso.fernandes@ufes.brProceeding Series of the Brazilian Society of Applied and Computational Mathematics, Vol. 3, N. 2, 2015. Trabalho apresentado no III CMAC - SE, Vitória-ES, 2015. DOI: 10.5540/03.2015.003.02.0045 020045-1 © 2015 SBMAC 2 the latter would solve instances of the problem of nding the shortest vector in a lattice, which has applications in cryptography [9]. In this article, we describe a new ecient quantum algorithm to solve the HSP in the speci c class of non-abelian groups, i.e., the semi-direct product groups of the form G=ZNoZqs, where,Nis factorized as pr1 1:::prnnand there exists a 1 knsuch that qt(qodd prime) divides pk1 andqdoes not divide pi1 for alli6=k. Te parameter t2f0;1;:::;sgcharacterizes the group as can be checked in Sec. 2. Our algorithm was inspired by the work of Chi et. al. [2] that solved the HSP in ZNo Zp, wherepis a prime that does not divide pi1 for any of the prime factors piofN. As in [2], we use the homomorphic properties of G=ZNo Zqsto reduce the problem to similar ones with known ecient solutions.

Question 554 multiple-choice

Quantum algorithms for phase, energy, and amplitude estimation are foundational to many quantum computing applications, but their implementation depends critically on the ability to maintain or discard quantum coherence during computation. The distinction between incoherent and coherent algorithms affects feasibility, accuracy, and resource requirements in quantum information processing. Which scenario fundamentally prevents the use of uncomputation to clean up ancilla qubits in quantum estimation algorithms? 1) When the quantum output is a superposition of multiple estimates 2) When the output is a classical probability distribution 3) When the input state is an eigenstate of the Hamiltonian 4) When iterative phase estimation is used without quantum Fourier transform 5) When amplitude estimation is performed additively 6) When adaptivity allows quantum states to pause for classical computation 7) When the state can be prepared and measured inexpensively

✓ Correct Answer:

The correct answer is 1) When the quantum output is a superposition of multiple estimates.

📚 Reference Text:

allowed to measure the quantum state, either because we are given many copies of the input state, or because we have the ability to prepare it inexpensively. Alternatively, we can assume a type of adaptivity where quantum states wait patiently without decohering while a classical computer performs a computation on the side. This assumption sometimes lets us repair the input state after using it [MW05, Temme&09, Poulin&17]. But most importantly, incoherent phase estimation algorithms usually require that the input state is an eigenstate of the unitary or Hamiltonian in question. If enough of these assumptions hold, then ‘iterative phase estimation’ [Kit95] removes an enormous amount of conceptual and computational overhead. The estimate can be extracted one bit at a time, thereby removing the QFT. The accuracy of each bit can be ampliﬁed individually via many classical samples, removing the need for the quantum sorting network. The awkward notion of an ‘ n-bit estimate’ can be removed and replaced with a traditional additive-error estimate [AA16]. Iterative phase estimation has seen many reﬁnements and has been applied to gate set tomography and ground state energy estimation [Higgins07, KLY15, LT21]. An incoherent iterative approach also permits direct simpliﬁcation of amplitude estimation [AR19, GGZW19]. However, for some applications of phase estimation maintaining coherence remains crucial. The original strategy for quantum matrix inversion [HHL08], quantum Metropolis sampling [Temme&09, YA11, Lemi&19], and a protocol for partition function estimation [Mon15], thermal state preparation, and Bayesian inference [HW19, AHNTW20] all violate the assumptions above. The eigenvalues must be estimated while preserving the superpo- sition, and there is no guarantee that the input state is an eigenstate. These applications of ‘coherent’ phase estimation motivate the main question of this paper: does there exist a conceptually and computationally simpler algorithm for performing the transformation (1) while remaining coherent? That is: the input state is not necessarily an eigenstate, we are given exactly one copy, and there is no adaptive interaction with a classical computer. In this paper we answer this question in the aﬃrmative. We present simpliﬁed coher- ent algorithms for phase estimation, energy estimation, and amplitude estimation. None of these algorithms require a QFT, and they can be made arbitrarily close to the ideal transformation without a quantum sorting network to compute a median. These algo- rithms are about 14x to 20x faster than traditional phase estimation in terms of their query complexity, a performance metric that neglects the fact that they also require fewer ancilla qubits. However, we also observe that there are unavoidable barriers to estimation in super- position. Consider an estimation algorithm that outputs a superposition of two estimates ˆλ(1) jandˆλ(2) j, both of which could be pretty close to the true value λj: /summationdisplay jαj|0n/angbracketright|ψj/angbracketright→/summationdisplay jαj/parenleftBig ξj|λ(1) j/angbracketright+ζj|λ(2) j/angbracketright/parenrightBig |ψj/angbracketright (2) However, it is a well known fact [BBBV97] that uncomputation cannot work in such a Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 2 situation (unless one of ξjorζjis≈0). Thus, any algorithm that actually makes use of the estimates must necessarily

Question 555 multiple-choice

Quantum circuit optimization is essential for enabling practical and scalable quantum computing, especially when fault tolerance and efficient state preparation are required. Genetic algorithms have emerged as a promising tool for discovering resource-efficient circuit designs under hardware constraints. Which approach specifically targets the reduction of error-prone T gates in quantum state preparation circuits, thereby enhancing fault tolerance in architectures such as the surface code? 1) Utilizing genetic algorithms to optimize circuits with Clifford + T gate sets 2) Increasing the number of Clifford gates while keeping T gates constant 3) Employing Grover's algorithm for circuit synthesis 4) Replacing all T gates with Hadamard gates 5) Using brute-force search to generate all possible gate sequences 6) Implementing only Pauli gates in circuit construction 7) Focusing exclusively on minimizing circuit depth without regard to gate types

✓ Correct Answer:

The correct answer is 1) Utilizing genetic algorithms to optimize circuits with Clifford + T gate sets.

📚 Reference Text:

Title: T-Count Optimizing Genetic Algorithm for Quantum State Preparation Year: 2024 Paper ID: cec0c9059476fdf828c3a7807b10ff9d627e5a0e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/cec0c9059476fdf828c3a7807b10ff9d627e5a0e Abstract: Quantum state preparation is a crucial process within numerous quantum algorithms, and the need for efficient initialization of quantum registers is ever increasing as demand for useful quantum computing grows. The problem arises as the number of qubits to be initialized grows, the circuits required to implement the desired state also exponentially increase in size leading to loss of fidelity to noise. This is mainly due to the susceptibility to environmental effects of the non-Clifford T gate, whose use should thus be reduced as much as possible. In this paper, we present and utilize a genetic algorithm for state preparation circuits consisting of gates from the Clifford + T gate set and optimize them in T-Count as to reduce the impact of noise. Whilst the method presented here does not always produce the most accurate circuits in terms of fidelity, it can generate high-fidelity, non-trivial quantum states such as quantum Fourier transform states. In addition, our algorithm does automatically generate fault tolerantly implementable solutions where the number of the most error prone components is reduced. We present an evaluation of the algorithm when trialed against preparing random, Poisson probability distribution, W, GHZ, and quantum Fourier transform states. We also experimentally demonstrate the scalability issues as qubit count increases, which highlights the need for further optimization of the search process.

Question 556 multiple-choice

Quantum simulators enable the study of complex lattice gauge theories and non-perturbative phenomena using controllable experimental systems, such as ultracold atoms in optical lattices. Choices made in discretizing the Dirac Hamiltonian, including the selection between Wilson and staggered fermions, significantly affect experimental resource requirements and feasibility. In the context of simulating two-dimensional lattice quantum electrodynamics with ultracold atoms, which advantage do Wilson fermions offer compared to staggered fermions? 1) They require a larger number of atoms per lattice site. 2) They introduce additional unwanted particle species on the lattice. 3) They simplify the experimental setup by reducing the degrees of freedom needed to simulate dynamical gauge fields. 4) They make it impossible to observe non-perturbative phenomena like the Schwinger mechanism. 5) They necessitate the use of three-dimensional optical lattices. 6) They complicate the measurement of real-time dynamics in lattice gauge theories. 7) They increase the computational resources required for quantum simulation.

✓ Correct Answer:

The correct answer is 3) They simplify the experimental setup by reducing the degrees of freedom needed to simulate dynamical gauge fields..

📚 Reference Text:

Title: Quantum simulation of lattice gauge theories using Wilson fermions Year: 2018 Paper ID: cd9e2f0bc4ef2f3203909ce6755647e34235c8c5 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/cd9e2f0bc4ef2f3203909ce6755647e34235c8c5 Abstract: Quantum simulators have the exciting prospect of giving access to real-time dynamics of lattice gauge theories, in particular in regimes that are difficult to compute on classical computers. Future progress towards scalable quantum simulation of lattice gauge theories, however, hinges crucially on the efficient use of experimental resources. As we argue in this work, due to the fundamental non-uniqueness of discretizing the relativistic Dirac Hamiltonian, the lattice representation of gauge theories allows for an optimization that up to now has been left unexplored. We exemplify our discussion with lattice quantum electrodynamics in two-dimensional space-time, where we show that the formulation through Wilson fermions provides several advantages over the previously considered staggered fermions. Notably, it enables a strongly simplified optical-lattice setup and it reduces the number of degrees of freedom required to simulate dynamical gauge fields. Exploiting the optimal representation, we propose an experiment based on a mixture of ultracold atoms trapped in a tilted optical lattice. Using numerical benchmark simulations, we demonstrate that a state-of-the-art quantum simulator may access the Schwinger mechanism and map out its non-perturbative onset.

Question 557 multiple-choice

Quantum field theory (QFT) in curved spacetime presents unique challenges, particularly in analyzing phenomena near strong gravitational fields such as black holes. Nonperturbative methods like the functional renormalization group (FRG) are often required to study complex quantum effects in these regimes. Which equation serves as the central tool in the functional renormalization group formalism, governing the evolution of the effective action for both bosonic and fermionic fields? 1) Klein-Gordon equation 2) Schrödinger equation 3) Dirac equation 4) Lippmann-Schwinger equation 5) Callan-Symanzik equation 6) Wetterich equation 7) Kadanoff-Baym equation

✓ Correct Answer:

The correct answer is 6) Wetterich equation.

📚 Reference Text:

Title: Nonperturbative Aspects of Quantum Field Theory in Curved Spacetime Year: 2023 Paper ID: d3b41af9ce34afa520411e24f1d3217a56b03661 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d3b41af9ce34afa520411e24f1d3217a56b03661 Abstract: Quantum field theory in curved spacetime is perhaps the most reliable framework in which one can investigate quantum effects in the presence of strong gravitational fields. Nevertheless, it is often studied by means of perturbative treatments. In this thesis, we aim at using the functional renormalization group -- a nonperturbative realization of the renormalization group -- as a technique to describe nonperturbative quantum phenomena in curved spacetimes. The chosen system is an Unruh--DeWitt particle detector coupled to a scalar quantum field. We discuss how to formulate such a system in terms of an action and how one can compute its renormalization group flow in the case of an inertial detector in flat spacetime, for simplicity. We learn, however, that the results are divergent in the limit in which the detector's energy gap vanishes. Possible workarounds are discussed at the end. This thesis also presents a review of quantum field theory in curved spacetimes by means of the algebraic approach, although it assumes no previous experience with functional analysis. Hence, it fills a pedagogical gap in the literature. Furthermore, we also review the functional renormalization group and derive the Wetterich equation assuming a general field content that might include both bosonic and fermionic fields. Such a derivation is also hardly found in pedagogical introductions available in the high energy physics literature.

Question 558 multiple-choice

W-algebras are mathematical structures that extend the symmetries found in integrable models, particularly those derived from reductions of Lie algebras. Their connection to Toda systems and group-theoretic methods is central to modern mathematical physics and representation theory. In the context of finite W-algebras derived from principal embeddings of sl2 into sln, which feature distinguishes the resulting finite Toda equations from their infinite-dimensional counterparts? 1) The absence of conserved charges related to Cartan subalgebra elements 2) The non-existence of integrable structure in the finite case 3) The lack of quantization procedures for any finite-dimensional Toda system 4) The presence of non-commuting copies of W-algebras acting on solutions 5) The use of non-reductive Lie algebras in constructing the equations of motion 6) The involvement of the Cartan matrix of sln, resulting in ordinary finite Toda equations for the principal embedding 7) The irrelevance of group-theoretic methods in finding general solutions

✓ Correct Answer:

The correct answer is 6) The involvement of the Cartan matrix of sln, resulting in ordinary finite Toda equations for the principal embedding.

📚 Reference Text:

equations as will be shown i n a moment. Alternatively, one can reduce the action by writing down the following one-dimensional version of a gauged WZW model, S[g,A+,A−] =1 2/integraldisplay dtTr/parenleftigg g−1dg dtg−1dg dt/parenrightigg +/integraldisplay dtTr/parenleftig A−(J−t+) +A+(¯J−t−) +A−gA+g−1/parenrightig .(4.10) This action is invariant under the following transformatio ns g→h−gh+, A−→h−A−h−1 −−dh− dth−1 −, A+→h−1 +A+h+−h−1 +dh+ dt, (4.11) whereh±are arbitrary elements of G±. We can use the gauge invariance to put g+=g−=e(whereeis the unit element of the group G) in the Gauss decomposition ofg, thus we can take g=g0∈G0. Then from the equations of motion for A±we ﬁnd A+=g−1 0t+g0andA−=g0t−g−1 0. Substituting these back into the action it reduces to S[g0] =1 2/integraldisplay dtTr/parenleftigg g−1 0dg0 dtg−1 0dg0 dt/parenrightigg −/integraldisplay dtTr/parenleftig g0t−g−1 0t+/parenrightig . (4.12) The equations of motion for this action are indeed given by (4 .9), showing the equiva- lence of the two approaches. This generalized Toda action describes a particle moving on G0in some background potential. Two commuting copies of the ﬁnite Walgebra leave the action (4.12) in- variant and act on the space of solutions of the equations of m otion (4.9)†. This action is only given inﬁnitesimally, because we do not know how to ex ponentiate ﬁnite (or †More precisely, the symmetries of (4.12) form an algebra tha t is on-shell isomorphic to a ﬁnite W algebra 34 inﬁnite)Walgebras. One can, however, sometimes ﬁnd subspaces of the s pace of so- lutions that constitute a minimal orbit of the Walgebra, see for example [24] where this was worked out for the ordinary W3algebra. For the principal embeddings of sl2insln, the equations of motion reduce to ordi- nary ﬁnite Toda equations of the type d2qi dt2+ exp n−1/summationdisplay j=1Kijqj = 0, (4.13) wherei= 1,...,n −1,Kijis the Cartan matrix of sln, andg0= exp(qiHi). The general solution of the the equations of motion (4.9) can be constructed as follows. Let h(1) 0,h(2) 0be elements of G0. LetX0be an arbitrary element of g0. Ifg0(t) is deﬁned by the Gauss decomposition g−(t)g0(t)g+(t) =h(1) 0expt(X0+ (h(1) 0)−1t+h(1) 0+h(2) 0t−(h(2) 0)−1)h(2) 0, (4.14) theng0(t) is the most general solution of (4.9). The easiest way to ﬁnd the action of the ﬁniteWalgebra on these solutions, is to construct the conserved ch arges associ- ated to these ﬁnite Wsymmetries (which can be done via a time dependent Miura transformation), and to study the transformations they gen erate. This might provide a valuable tool in the study of the solutions (4.14). We leave a detailed investigation of this, as well as many other issues like the quantization of the action (4.1), to future study. 5. Quantization of ﬁnite WAlgebras In quantum mechanics, quantization amounts to replacing Po isson brackets by com- mutators. Since ﬁnite Walgebras are Poisson algebras, the question arises whether it is possible to quantize these Poisson algebras, to give ﬁn ite quantum Walgebras. In the inﬁnite dimensional case (i.e. the usual inﬁnite Walgebras), this is known to be possible for the standard Wnalgebras associated to the principal embeddings. The W3algebra constructed

Question 559 multiple-choice

In quantum chemistry, constructing wavefunctions for multi-electron systems requires careful consideration of symmetry and spin properties, often utilizing group theory and specialized operators. The symmetric group and Young diagrams play an important role in designing spin-adapted functions for these systems. Which statement most accurately describes the advantage of using standard spin functions derived from Young's orthogonal representation in the GI method? 1) They guarantee the lowest possible energy for the wavefunction in all cases. 2) They impose physical restrictions that correspond to observable quantum numbers. 3) They eliminate the need for optimization of spatial orbitals. 4) They ensure that only single-column Young diagrams are used for electron configuration. 5) They maximize the flexibility of spin function combinations without constraints. 6) They directly correspond to unique electronic states in atoms without ambiguity. 7) They provide computational ease in constructing spin-adapted functions, even though the choice is arbitrary and not physically required.

✓ Correct Answer:

The correct answer is 7) They provide computational ease in constructing spin-adapted functions, even though the choice is arbitrary and not physically required..

📚 Reference Text:

The quantity J.l. corresponds to a Young shape of one or two columns and i corresponds to a particular stand ard tableau. We will call the spin functions based on Young's orthogonal representation standard spin func tions. We showed in Paper I that, although thef" different operators Gi'' are linearly independent, the exact wave function can be written in the form Gtx using any *Partially supported by a grant (GP-6965) from the National Science Foundation. one Gl operator and a suitable product of spin functions in x if the spatial function is sufficiently general, (1) (The .pi.exact will be different for different values of i.) In Paper II,2 we considered a function of the form (2) where i.product is restricted to be a product of one-elec tron spatial orbitals and required that these spatial orbitals be the best possible ones. The result was a coupled set of integro-differential equations, k=l, 2, .. ·, N (3) for the best orbitals. (The integro-differential-permuta tional operator Ilk"' depends on p., i, and all the orbitals cp;, except cflk.) Since the orbitals {cflk} optimize the energy for the wavefunction War, they are referred to as the GI orbitals, and the equations (3) for these orbitals are called the GI equations. Even though the exact wavefunctions can be written in the form of (1) for any i, the constraint implied by a product may be more restrictive for some i's than for others; that is, our approximate wavefunction War depends on which of the spin-coupling schemes (de noted by i) we have selected. The standard spin func- tions are used in the GI method because Young's orthogonal representation of SN is easier to construct 157, 73 (1967), hereafter than other equivalent representations. This imposes t National Science Foundation Predoctoral Fellow. t Alfred P. Sloan Fellow. §Contribution No. 3807. 1 W. A. Goddard m, Phys. Rev. referred to as Paper I. 157, 81 (1967), hereafter an arbitrary restriction on the wavefunction because 2 W. A. Goddard m, Phys. Rev. referred to as Paper II. 3 E. P. Wigner, Group Theory (Academic Press Inc., New York, 1959)' p. 118. 4 D. E. Rutherford, S1tbstitutional Analysis (Edinburgh Uni versity Press, Edinburgh, Scotland, 1948). there is nothing physically special about the standard spin functions and because we can construct a new Gt-like operator, GtL, which yields a wavefunction in which the spin part is a general linear combination of 1073 Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp1074 R. C. LADNER AND W. A. GODDARD III 3.5~--,---.,----,----.----,----. 3.0 2.5 ~ 2.0 ::J f-- :::i Q._ :::;;: <( 1.5 .100 0 2.0 3.0 4.0 5.0 6.0 DISTANCE FIG. 1. Orb~tals for the 22S state of Li. The large drawing is of the SOGI orbitals (:;;; = 0.434 °); the core orbitals remain essentially uncp.anged wh~n A is within 10° of 0. The insert shows the valence orb!!al foro vanous values of:!;;; a: :2:=0, b: :2:=0.434°, c: 2:= 1.0° d: A=2.0, e:

Question 560 multiple-choice

Quantum channel capacities and decoherence times are central concerns in quantum information theory, as they determine how reliably quantum information can be transmitted and preserved in the presence of environmental noise. Advanced mathematical tools, such as representation theory and operator norms, are used to construct and analyze noise models. Which concept allows estimates of quantum channel properties to be transferred from classical Markov kernels using non-commutative functional inequalities and group representation theory? 1) Transference principle 2) Superposition theorem 3) Adiabatic mapping 4) Quantum error basis expansion 5) Decoherence symmetry method 6) Quantum detailed balance condition 7) Entanglement entropy transfer

✓ Correct Answer:

The correct answer is 1) Transference principle.

📚 Reference Text:

Title: Group Transference Techniques for the Estimation of the Decoherence Times and Capacities of Quantum Markov Semigroups Year: 2019 Paper ID: 5067b8a4044a718ea98a0f2f45e51e7717cccefd Source: semantic-scholar URL: https://www.semanticscholar.org/paper/5067b8a4044a718ea98a0f2f45e51e7717cccefd Abstract: Capacities of quantum channels and decoherence times both quantify the extent to which quantum information can withstand degradation by interactions with its environment. However, calculating capacities directly is known to be intractable in general. Much recent work has focused on upper bounding certain capacities in terms of more tractable quantities such as specific norms from operator theory. In the meantime, there has also been substantial recent progress on estimating decoherence times with techniques from analysis and geometry, even though many hard questions remain open. In this article, we introduce a class of continuous-time quantum channels that we called transferred channels, which are built through representation theory from a classical Markov kernel defined on a compact group. In particular, we study two subclasses of such kernels: Hörmander systems on compact Lie-groups and Markov chains on finite groups. Examples of transferred channels include the depolarizing channel, the dephasing channel, and collective decoherence channels acting on $d$ qubits. Some of the estimates presented are new, such as those for channels that randomly swap subsystems. We then extend tools developed in earlier work by Gao, Junge and LaRacuente to transfer estimates of the classical Markov kernel to the transferred channels and study in this way different non-commutative functional inequalities. The main contribution of this article is the application of this transference principle to the estimation of decoherence time, of private and quantum capacities, of entanglement-assisted classical capacities as well as estimation of entanglement breaking times, defined as the first time for which the channel becomes entanglement breaking. Moreover, our estimates hold for non-ergodic channels such as the collective decoherence channels, an important scenario that has been overlooked so far because of a lack of techniques.

Question 561 multiple-choice

Quantum algorithms for the hidden subgroup problem rely on preparing quantum states that encode information about subgroups of a group and performing measurements to identify them. The development of optimal single-copy measurement strategies plays a critical role in improving algorithm efficiency, especially for non-abelian groups with complex subgroup structures. Which measurement strategy provides the optimal single-copy distinction among equally probable subgroups in the hidden subgroup problem for general groups? 1) Measurement based solely on the Fourier transform over abelian groups 2) Measurement tailored for non-abelian groups with all subgroups conjugate 3) A hybrid measurement combining techniques from abelian and conjugate subgroup cases 4) Measurement using repeated collective measurements on multiple copies 5) Measurement based on random sampling of subgroup elements 6) Measurement that ignores prior probabilities of subgroups 7) Measurement using only classical post-processing of quantum data

✓ Correct Answer:

The correct answer is 3) A hybrid measurement combining techniques from abelian and conjugate subgroup cases.

📚 Reference Text:

Title: Optimal single-copy measurement for the hidden-subgroup problem Year: 2007 Paper ID: a1aa171180f45a6d806999bd3702392a75f2ba49 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a1aa171180f45a6d806999bd3702392a75f2ba49 Abstract: The optimization of measurements for the state distinction problem has recently been applied to the theory of quantum algorithms with considerable successes, including efficient new quantum algorithms for the non-abelian hidden subgroup problem. Previous work has identified the optimal single copy measurement for the hidden subgroup problem over abelian groups as well as for the non-abelian problem in the setting where the subgroups are restricted to be all conjugate to each other. Here we describe the optimal single copy measurement for the hidden subgroup problem when all of the subgroups of the group are given with equal a priori probability. The optimal measurement is seen to be a hybrid of the two previously discovered single copy optimal measurements for the hidden subgroup problem.

Question 562 multiple-choice

In the study of topological quantum computation, the Ising category is frequently used to model non-Abelian anyons and symmetry defects, with calculations often performed using graphical calculus and principles from representation theory. These calculations rely on properties such as Schur’s Lemma to simplify the analysis of matrix elements associated with braiding and fusion operations. When calculating diagonal matrix elements such as ⟨1|T|1⟩ in the Ising category using graphical calculus, why do off-diagonal elements vanish, allowing the focus to be only on diagonal terms? 1) The fusion rules prohibit off-diagonal transitions for all anyons. 2) The vacuum state is always orthogonal to excited states in anyonic models. 3) The graphical calculus enforces diagrammatic orthogonality of all strands. 4) Schur’s Lemma ensures that the operator is diagonal on simple objects, causing off-diagonal elements to vanish. 5) Quantum dimensions are zero for off-diagonal objects in the Ising category. 6) All symmetry defect charges fuse exclusively to the vacuum, eliminating off-diagonal contributions. 7) The G-crossed extension forbids nontrivial overlap between different basis states.

✓ Correct Answer:

The correct answer is 4) Schur’s Lemma ensures that the operator is diagonal on simple objects, causing off-diagonal elements to vanish..

📚 Reference Text:

the in dex on the symmetry defect, writing X:=X1, and since all anyons live in a single layer, we write a:=a1 for aa simple object in Ising . Moreover, since the total charge of the fusion trees are tri vial, we suppress the labeling of the tracial strand. As was mentioned in Section 4.3.2, the off diagonal matrix el ements vanish by a Schur’s Lemma argument, so it remains to calculate /an}bracke⊔le{⊔1|T|1/an}bracke⊔ri}h⊔and/an}bracke⊔le{⊔ψ|T|ψ/an}bracke⊔ri}h⊔. 24 yσ111X11X111 x σ111X11X1 11σ σ σ σ X X X X= σ σ σ σ X X X X σ x σ 1 X y Xσ 1 σ 1 X 1 X FIGURE 4. On the left is the diagram representing the matrix entries of the protocol. On the right is the shorthand notation we use in the following section. C.2. Calculation of /an}bracke⊔le{⊔1|T|1/an}bracke⊔ri}h⊔.We break down the calculation of the ﬁrst matrix entry (up to normalization) into ﬁve steps. Below is a formula that shows the contributions from each step, before simpliﬁcation. /an}bracke⊔le{⊔1|T|1/an}bracke⊔ri}h⊔=1 dX∑ c/radicalbig dc(RXX c)2 /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright Step 1/parenleftigSσ1,c S11,c/parenrightig2 /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright Step 3/vextendsingle/vextendsingle[FXXX X]c,11/vextendsingle/vextendsingle2d2 X dc/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright Step 4dσ1dX/radicalbig dc/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright Step 5 The dashed rectangles in each equation indicate the region o f the diagram where the graphical calculus is being applied. Step 1: Unbraid the symmetry defects using equation (1) twice. 25 σ σ σ σ X X X X σ1σ1X1Xσ1σ1X1X =∑c/radicalig dc dXdX/parenleftbig RXX c/parenrightbig2 σ σ σ σ X X X X σ1σ1X1Xσ1σ1X1X c (10) Note that in this case there is a loop labeled by σthat we can replace with dσright away. However, we have left it unsimpliﬁed so that the same picture applies for the calculation of/an}bracke⊔le{⊔ψ|T|ψ/an}bracke⊔ri}h⊔. Step 2: Slide the σloops under the ﬁrst defect charge line so that they encircle pairs of defect charge lines using equations (3)-(6). The U- and η- symbols this introduces all cancel, so that the anyons and defects can be moved past each o ther at no cost. σ σ σ σ X X X X σ1σ1X1Xσ1σ1X1X c= σ σ X X X X σ 1 X1Xσ 1 X1X σ σc (11) Step 3: Slide the loops over the trivalent vertices so that they enci rcle the fusion channel labeled by c, ﬂip the tilt of the bottom loop using (6), then use the loop re moval relation (7) twice. In general with loops encircling charge lines in the G-crossed graphical calculus we need to pay careful attention to the relative positions of charge lines, see equations (306) and (307) in [2]. However, in this case the subdiagram involves o nly anyons, so we can use the graphical calculus of a UMTC and ignore them. Neverthele ss, a careful account of the factors using the full G-extended calculus gives the same result: any factors of U- and η- 26 cancel. σ σ X X X X σ 1 X1Xσ 1 X1X cσ σ= σ σ X X X X σ 1 X1Xσ 1 X1X c σ σ(12) =/parenleftigSσ1,c S11,c/parenrightig2 σ σ X X X X

Question 563 multiple-choice

Quantum information theory utilizes mathematical concepts such as group theory and operator algebra to analyze and quantify entanglement properties in quantum systems. Efficient computation of entanglement measures like the Schmidt number often depends on exploiting symmetries within these systems. Which statement accurately describes the impact of leveraging orthogonal group symmetries in the characterization of k-positive maps and computation of Schmidt numbers for quantum states? 1) It restricts the class of quantum states that can be analyzed to only separable states. 2) It ensures all k-positive maps are also completely positive. 3) It eliminates the need for operator algebra in entanglement analysis. 4) It provides a method for computing Schmidt numbers only for non-invariant quantum states. 5) It makes the Schmidt number computation independent of the underlying group symmetry. 6) It limits entanglement detection to systems with unitary symmetry. 7) It enables systematic simplification and transfer of k-positivity results to efficiently compute Schmidt numbers for all orthogonally invariant quantum states.

✓ Correct Answer:

The correct answer is 7) It enables systematic simplification and transfer of k-positivity results to efficiently compute Schmidt numbers for all orthogonally invariant quantum states..

📚 Reference Text:

Title: k-Positivity and Schmidt number under orthogonal group symmetries Year: 2023 Paper ID: ccaf1c3269ed5a3e68aefc62d759d500e3fddaf1 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/ccaf1c3269ed5a3e68aefc62d759d500e3fddaf1 Abstract: In this paper, we present a new application of group theory to develop a systematical approach to efficiently compute the Schmidt numbers. The Schmidt number is a natural quantification of entanglement in quantum information theory, but computing its exact value is generally a challenging task even for very concrete examples. We exhibit a complete characterization of all orthogonally covariant k-positive maps. This result generalizes earlier results by Tomiyama (Linear Algebra Appl 69:169–177, 1985). Furthermore, we optimize duality relations between k-positivity and Schmidt numbers under group symmetries. This new approach enables us to transfer the results of k-positivity to the computation of the Schmidt numbers of all orthogonally invariant quantum states.

Question 564 multiple-choice

Quantum computation relies on manipulating qubits using various quantum gates to harness principles such as superposition and entanglement. Understanding the roles and properties of gates like Hadamard, CNOT, and Toffoli is essential for building quantum algorithms and circuits. Which sequence of gate operations creates a maximally entangled two-qubit Bell state starting from both qubits in the |0⟩ state? 1) Apply a Hadamard gate to the first qubit, then a CNOT gate with the first qubit as control and the second as target 2) Apply CNOT gates in parallel to both qubits 3) Apply a Toffoli gate with both qubits as controls and a third qubit as target 4) Apply Hadamard gates to both qubits, then a single-qubit Pauli-X gate 5) Apply two Hadamard gates in series to the first qubit 6) Apply a single-qubit Y gate to each qubit 7) Apply a CNOT gate with the second qubit as control and the first as target

✓ Correct Answer:

The correct answer is 1) Apply a Hadamard gate to the first qubit, then a CNOT gate with the first qubit as control and the second as target.

📚 Reference Text:

acting upon it by H, and measuring it in the computational basis is denoted by the circuit in Figure 2.1. Figure 2.1: A Simple Quantum Circuit j0i H NM 14 Application of a Hadamard gate to an arbitrary qubit jyi=aj0i+bj1iyields the state Hjyi=aHj0i+bHj1i=aj0i+j1ip 2 +bj0ij1ip 2 =a+bp 2 j0i+abp 2 j1i=jci: (2.41) If a Hadamard gate is now applied to jci, we get Hjci=a+bp 2 Hj0i+abp 2 Hj1i =a+bp 2j0i+j1ip 2+abp 2j0ij1ip 2(2.42) =2a+bb 2 j0i+aa+2b 2 j1i =aj0i+bj1i=jyi: Thus, two Hadamard gates applied in series restore a qubit to its original state. What if two Hadamard gates are applied in parallel, as shown in Figure 2.2? Figure 2.2: Hadamard Gates in Parallel j0i H1p 2(j0i+j1i) j0i H1p 2(j0i+j1i) In this case, (H H)j0ij0i=H 2j0i 2= (Hj0i)(Hj0i) =j0i+j1ip 2j0i+j1ip 2 (2.43) =1 2(j00i+j01i+j10i+j11i): 15 In general, for n2N H nj0i n=1p 2nå x2f0;1gnjxi: (2.44) 2.3.2 Multi-qubit gates Single-qubit gates cannot generate entanglement, as qubit interaction is required to produce an entangled state. An exponential speedup cannot be achieved by a quantum computer that does not exploit entanglement, so we now turn to exploration of multi-qubit gates. An important set of gates in quantum computation are controlled gates that implement an IF-THEN construct. An example is the 2-qubit controlled NOT, or CNOT gate, depicted in Figure 2.3, whereis the exclusive-OR operation (binary addition modulo 2). Figure 2.3: CNOT Gate controljxijxi targetjyi jyxi The CNOT gate ﬂips the state of the target qubit if the control qubit is in the state j1i, and acts trivially if the control is in the state j0i, i.e., it implements the following mapping CNOT :j00i7!j 00i;j01i7!j 01i;j10i7!j 11i;j11i7!j 10i (2.45) 16 with its transfer matrix given by CNOT : =j00ih00j+j01ih01j+j11ih10j+j10ih11j =j0ih0j I+j1ih1j X (2.46) =0 BBBBBBB@1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 01 CCCCCCCA(2.47) The CNOT gate can generate entangled states, i.e., states that cannot be decomposed into a tensor product of other states. Consider CNOT1p 2j0i+1p 2j1i j0i=1p 2j00i+1p 2j11i: (2.48) Suppose the state jyi=1p 2j00i+1p 2j11i (2.49) is separable, i.e., there exist a;b;g;d2Csuch that jyi=jy1i jy2i = (aj0i+bj1i) (gj0i+dj1i) =agj00i+bgj10i+adj01i+bdj11i (2.50) =1p 2j00i+1p 2j11i But then 8 >>< >>:ad =bg=0 ag =bd=1p 2(2.51) which is clearly impossible. 17 In general, a 2-qubit state jyi=0 BBBBBBB@a b g d1 CCCCCCCA(2.52) is separable if and only if ad=bg: (2.53) The effects of entanglement are clearly demonstrated when measurements are performed. If a measurement of the second qubit of the state jyiresults in 0, then the post-measurement state of the system isj00i. The measurement of one qubit of an entangled state affects the probability amplitude of the other qubits in the system. The CNOT gate can be extended to nqubits controlling the reversal of the target one. The CCNOT (Controlled-Controlled-NOT) gate, also known as the Toffoli gate, has two control qubits and ﬂips the target qubit only if both controls are in the state j1i. CCNOT : =j000ih000j+j001ih001j+j010ih010j+j011ih011j +j100ih100j+j101ih101j+j110ih111j+j111ih110j = (j00ih00j+j01ih01j+j10ih10j) I+j11ih11j X (2.54) =0 BBBBBBBBBBBBBBBBBBBB@1 0 0 0

Question 565 multiple-choice

Quantum phase estimation algorithms are essential for applications in quantum computing, and their performance is heavily affected by different types of errors in quantum circuits. Distributed implementations introduce unique challenges and opportunities for error mitigation, especially as the number of qubits increases. Which statement best explains why distributed quantum phase estimation (DQPE) demonstrates greater performance improvements over standard quantum phase estimation (QPE) as the number of control qubits increases? 1) DQPE eliminates all depolarization errors regardless of qubit count. 2) The number of single-qubit gates remains constant in DQPE as qubits increase. 3) DQPE requires fewer measurements per qubit compared to QPE. 4) Standard QPE benefits more from two-qubit gate errors than DQPE. 5) Measurement error in DQPE grows quadratically with qubit count. 6) DQPE’s dynamic circuit structure increases susceptibility to gate errors. 7) DQPE localizes error effects and mitigates impact from the increasing number of two-qubit gates as control qubits increase.

✓ Correct Answer:

The correct answer is 7) DQPE localizes error effects and mitigates impact from the increasing number of two-qubit gates as control qubits increase..

📚 Reference Text:

consider the following scenarios 1. Probability vs ϵg, where ϵg= 10ϵd,ϵm= 0 (Fig. 6a). 2. Probability vs ϵg, where ϵg= 10ϵd,ϵm=ϵg/4 (Fig. 6b). While each distributed quantum gate requires two local CNOT gates, the dy- namic quantum Fourier transform doesn’t suffer from two-qubit quantum gate errors, only measurement and depolarization errors, giving better results in Fig. 6. Since each non-local quantum gate requires measurement that is not present in the local implementation of a quantum circuit, we consider the measure- ment error in our analysis. Note that there are efficient methods to reduce the measurement error to ϵg/4 [46, 35]. 8 (a) Probability vs ϵg,ϵd=ϵg/10,ϵm= 0 (b) Probability vs ϵg,ϵd=ϵg/10,ϵm= ϵg/4 Figure 6: The probability measuring the required bit for the distributed QPE (DQPE) (blue) vs. QPE (red) in the presence of depolarization, gate, measure- ment errors ϵd,ϵg,ϵmrespectively. The improvement obtained with DQPE increases when considering a larger number of qubits. Let us consider the inverse QFT on five qubits (Fig. 7) that uses ten two-qubit and five Hadamard gates. q0H • • • • q1 P(−π/2)H • • • q2 P(−π/4)P(−π/2)H • • q3 P(−π/8)P(−π/4)P(−π/2)H • q4 P(−π/16) P(−π/8)P(−π/4)P(−π/2)H Figure 7: Quantum Fourier transform (inverse) on five qubits, where P(θ) refers to the phase gate The dynamic quantum circuit is given in Fig. 8, q0 H q1 P(−π/2) H q2 P(−π/4) P(−π/2) H q3 P(−π/8) P(−π/4) P(−π/2) H q4 P(−π/16) P(−π/8) P(−π/4) P(−π/2) H Figure 8: Dynamic quantum Fourier transform using five qubits 9 The quantum circuit used to implement the DQPE is shown in Fig. 9. |0⟩H • H |0⟩H • P(−π/2) H |0⟩H • P(−π/4) P(−π/2) H |0⟩H • P(−π/8) P(−π/4) P(−π/2) H |0⟩H • P(−π/16) P(−π/8) P(−π/4) P(−π/2) H |ψ⟩ U24U23U22U21U20 Figure 9: Distributed dynamic quantum phase estimation for five qubits where the qubits are in different nodes. The controlled unitary gates are applied using DQCS, followed by dynamic QFT We estimate the fraction ythat is calculated as ( y0y1y2y3y4)10/25. We prepare the eigenstate |ψ⟩, create the distributed QPE (DQPE) for the following qubit distributions: 1: [0, 1, 2, 3, 4], 2: [5, 6], and apply the dynamic QFT to the distributed quantum circuit. We consider the following scenarios: 1. Probability of obtaining the correct outcome vs error rate ϵ, where ϵg= ϵd=ϵ,ϵm= 0 (Fig. 10a). 2. Probability vs the two-qubit gate error ϵg, where ϵg= 10ϵd,ϵm= 0 (Fig. 10b). 3. Probability vs ϵ, where ϵg=ϵd=ϵ,ϵm=ϵg/4 (Fig. 10c). 4. Probability vs ϵg, where ϵg= 10ϵd,ϵm=ϵg/4 (Fig. 10d). In Fig. 10, we note that there is better improvement obtained using DQPE (using five qubits) than three qubits (Fig. 6). This could be understood the- oretically as follows: For quantum phase estimation using ncontrol qubits, suppose that we need Ntwo-qubit quantum gates to realize the controlled- unitary operations, and the quantum Fourier transform requires n(n−1)/2 two-qubit quantum gates. Theoretically, assuming the same error for the two- qubit (controlled-unitary and controlled phase) gates, the fidelity of the algo- rithm is approximated F≈(1−ϵg)N+n 2(n−1). For DQPE, where we have N

Question 566 multiple-choice

Implementing quantum algorithms on superconducting qubits requires careful consideration of experimental errors and protocol design, especially for algorithms with deep circuits like the Quantum Fourier Transform. Digital-Analog Quantum Computing (DAQC) protocols are emerging as promising alternatives to purely digital approaches in noisy intermediate-scale quantum devices. Which protocol variant most effectively minimizes fidelity loss due to error accumulation when implementing the Quantum Fourier Transform on superconducting qubits, and why? 1) Standard digital quantum computation (DQC), because it uses only discrete gate operations 2) Better digital-analog quantum computing (bDAQC), because it keeps the interaction Hamiltonian continuously "on" to reduce control errors 3) Simple digital-analog quantum computing (sDAQC), due to its lower analog block noise variance 4) DQC, because it has the smallest circuit depth for QFT 5) sDAQC, because it switches the interaction Hamiltonian off between gates 6) DQC, because material impurities affect only analog operations 7) bDAQC, because it uses only single-qubit gates

✓ Correct Answer:

The correct answer is 2) Better digital-analog quantum computing (bDAQC), because it keeps the interaction Hamiltonian continuously "on" to reduce control errors.

📚 Reference Text:

for n=3,5–7 qubits.IV . REALISTIC IMPLEMENTATION WITH EXPERIMENTAL ERRORS Impurities in the materials comprising superconductive circuits and spurious interactions among superconductingqubits (cross talk) and with two-level ﬂuctuators modify thedynamics of the system, directly affecting the results of anexperiment. Additionally, there are relevant control errors inthe pulses when applying the gates. In order to make a faircomparison among different methods, we must introduce theeffects of errors in the dynamics. In single-qubit gates, we have introduced a magnetic-ﬁeld noise/Delta1B γby adding to the Hamiltonian of the single-qubit gate a random variable taken from a uniform probabilitydistribution centered in 1, i.e., U(1−SQGN ,1+SQGN). We have chosen SQGN =0.0005. For the two-qubit gates, we add a Gaussian phase noise /epsilon1∈N(0,TQGN) with variance TQGN =0.2000 to the π/4 phases in the DQC protocol. Finally, to model the experimental control error on the analogblocks, we include a Gaussian coherent noise to the time thoseblocks are applied, this is t→t+δ, where δ∈N(0,ABN). The value of the variance ABN depends on which DAQCprotocol we are using. The value used on the sDAQC isdouble the value used for the bDAQC case. The values wehave considered are ABN s=0.0200 for the sDAQC case and ABN b=0.0100 for the bDAQC case. Thus, each ideal gate transforms as eiθkZ→eiθk/Delta1BZ, (20) ei(π/4)ZZ→ei(π/4)(1+/epsilon1)ZZ, (21) eitαHint→ei(tα+δ)Hint. (22) To test how the ﬁdelity behaves for each case, we have com- puted the Q FT of the family of states |ψ0/angbracketright=sinβ|Wn/angbracketright+ cosβ|GHZ n/angbracketright. We repeated the simulation 1000 times and calculate the average for a three-, ﬁve-, six-, and seven-qubitsystem. Both the sDAQC and the bDAQC perform betterthan the DQC protocol under realistic conditions as depictedin Fig. 4(b). The best result corresponds to the bDAQC, a completely different situation from the ideal case. The reasonis that turning on and off the interaction Hamiltonian ismuch more prone to suffer from experimental errors thankeeping it on until the end of the computation. Note that theﬁdelity of the DQC decreases faster than the ﬁdelity of theDAQC protocols with the number of qubits of the system. Anextrapolation of these results to larger systems might indicatethe convenience of this paradigm to foster near-term quantumcomputation. This shows the convenience of this paradigmto foster near-term quantum computation. The reason whythe ﬁdelity in Fig. 4(b) decays so fast with the number of qubits Nis the quadratic increment O(N 2) in the depth of the algorithm. This is an exponential speedup with respect to theclassical algorithm, but the errors accumulate very fast withthe system size. As a consequence, the implementation of theQFT, which is a key subroutine in many other quantum algo- rithms, is a challenge when compared against algorithms withlogarithmic scaling. This shows the importance of exploringalternative approaches, such as our DAQC, to optimize theimplementation of this quantum algorithm. 013012-4 DIGITAL-ANALOG QUANTUM ALGORITHM FOR THE … PHYSICAL REVIEW RESEARCH 2, 013012 (2020) FIG. 3. Implementation of the Q FT for a three-qubit system using three different protocols: DQC, sDAQC, and bDAQC. Digital implementation: We show the transformation between the usual DQC implementation of the Q

Question 567 multiple-choice

Quantum computing leverages principles such as superposition and measurement to solve complex search problems more efficiently than classical methods. The concept of elimination observables is central to identifying and discarding incorrect candidate quantum states in certain algorithmic contexts. In the context of quantum algorithms for hidden subgroup problems, which statement accurately characterizes the role and limitations of elimination observables when applied to the dihedral group DN? 1) Elimination observables for the dihedral group DN are universally efficient due to the group's Abelian structure. 2) The Fourier transform provides an efficient elimination observable for all non-Abelian groups, including DN. 3) The geometric properties of hidden subgroup states in DN allow efficient elimination of all candidate states. 4) The geometry of hidden subgroup states in the dihedral group DN prevents the existence of efficient elimination observables. 5) Elimination observables in DN are constructed by measuring in the computational basis only. 6) The dihedral group DN admits efficient elimination observables similar to those found in Abelian groups. 7) Efficient elimination observables in DN are realized through Grover's algorithm.

✓ Correct Answer:

The correct answer is 4) The geometry of hidden subgroup states in the dihedral group DN prevents the existence of efficient elimination observables..

📚 Reference Text:

Title: Quantum State Detection via Elimination Year: 1999 Paper ID: 3b821a1a068bdaf71803cca6444635f1bbca1bd7 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/3b821a1a068bdaf71803cca6444635f1bbca1bd7 Abstract: We present the view of quantum algorithms as a search-theoretic problem. We show that the Fourier transform, used to solve the Abelian hidden subgroup problem, is an example of an efficientelimination observable which eliminates a constant fraction of the candidate secret states with high probability. Finally, we show that elimination observables do not always exist by considering the geometry of the hidden subgroup states of the dihedral group DN.

Question 568 multiple-choice

In quantum computing and group theory, the Hidden Translation problem investigates identifying a secret shift between two injective functions over a finite group, and has connections to quantum algorithms and cryptography. Recent advances have analyzed the problem's reformulations and efficient solutions, especially in abelian groups and their extensions. Which of the following properties uniquely distinguishes the new efficient quantum algorithm for Hidden Translation in elementary abelian p-groups from previous approaches for abelian groups? 1) It utilizes quantum amplitude amplification to solve the translation directly. 2) It reduces the problem to factoring integers using Shor’s algorithm. 3) It applies the sieve technique to reach subexponential time complexity. 4) It generates orthogonal vectors to the translation, requiring exponential classical post-processing. 5) It relies solely on classical random walks for sample generation. 6) It requires the functions to be non-injective for solution uniqueness. 7) The quantum samples yield vectors nonorthogonal to the translation, enabling deterministic polynomial-time classical recovery via a system of linear inequations.

✓ Correct Answer:

The correct answer is 7) The quantum samples yield vectors nonorthogonal to the translation, enabling deterministic polynomial-time classical recovery via a system of linear inequations..

📚 Reference Text:

v e nt w o injective functions f 0andf1from a ﬁnite group Gto some ﬁnite set such that, for some group element u, the equality f1(xu)=f0(x)h o l d sf o re v e r y x. The task is to ﬁnd the translation u. In fact, whenever Gis abelian, Hidden Translation is an instance of Hidden Subgroup in the semidirect product G/multicloserightZ2, where the hiding function is f(x,b)=fb(x). The group action in G/multicloserightZ2is deﬁned as ( x1,b1)·(x2,b2)= (x1+(−1)b1x2,b1⊕b2), where + denotes the group operation in Gand⊕denotes the group operation in Z2.I nG/multicloserightZ2,fhides the subgroup H={(0,0),(u,1)}. Actually, there is an eﬃcient quantum reduction in the other direction as well, and the two problems arequantum polynomial time equivalent [13]. A nice consequence ofthis equivalence is that instead of dealing with Hidden Subgroup in the nonabelian groupG/multicloserightZ 2, we canaddress Hidden Translation in the abeliangroup G. Ettinger and Høyer [13] have shown that Hidden Translation c a nb es o l v e db yat w o - s t e p procedure when G=ZNis cyclic: a polynomial number of Fourier samplings over the abelian group ZN×Z2followed by an exponential time classicalstage without further queries. The best known quantum algorithm for Hidden Translation in cyclic HIDDEN TRANSLATION AND TRANSLATING COSET 3 (and, in general, abelian) groups is Kuperberg’s subexponential time method [31]. Its relation to certain lattice problems investigated by Regev [38] provides evidence that Hidden Translation in cyclic groups might in fact be diﬃcult. In a related work, van Dam, Hallgren, and Ip [44] gave eﬃcient solutions for three cases of what they call the hidden shift problem. They also deﬁne another problem called the hidden coset problem which generalizes hidden shift. Their hidden coset problem can be viewed as a generalization of our Hidden Translation to not necessarily injective functions. While their paper gives eﬃcient quantum algorithms for some speciﬁc hidden coset problems, in general the hidden coset problem is ofexponential query complexity even in Z n 2. Our ﬁrst result (Theorem 3.5) is an eﬃcient quantum algorithm for Hidden Translation in the case of elementary abelian p-groups, that is, groups Zn p, for any ﬁxed prime number p. The quantum part of our algorithm is the same as in Et- tinger and Høyer’s procedure [13]: it consists of performing Fourier sampling over the abelian group Zn p×Z2. But while their classical postpro cessing requires exponential time, here we are able to recover classically the translation in polynomial time from the samples. It turns out that Fourier sampling produces vectors ynonorthogonal to the translation u; that is, we obtain linear inequations for the unknown u.T h i si s diﬀerent from the situation in the standard algorithm for the abelian Hidden Sub- group, where only vectors orthogonal to the hidden subgroup are generated. We show that, after a polynomial number of samplings, the system of linear inequations has a unique solution with high probability, which we are able to determine in deter- ministic polynomial time. An immediate consequence of Theorem

Question 569 multiple-choice

Quantum arithmetic circuits leverage techniques such as the Quantum Fourier Transform (QFT), conditional rotation gates, and parity encoding to optimize resource usage and computation depth. Achieving efficient quantum addition and related operations is crucial for scalable quantum algorithms and practical hardware implementations. Which strategy allows the core quantum addition operation to be performed in a single time step, provided the circuit uses parity encoding and an appropriate qubit layout? 1) Using standard gate model arithmetic circuits without parity qubits 2) Mapping the addition step onto parity qubits with single-qubit operations 3) Employing only SWAP gates between registers R1 and R2 4) Utilizing controlled-NOT gates exclusively for addition 5) Performing QFT and inverse QFT on both registers simultaneously 6) Applying multicontrolled gates to all qubits in the register 7) Omitting conditional rotation gates in the addition process

✓ Correct Answer:

The correct answer is 2) Mapping the addition step onto parity qubits with single-qubit operations.

📚 Reference Text:

within register R2 (white) can be left out if no operations are necessary in register R2. space with conditional rotation gates. Reference [ 42] suggests extensions of the resulting circuit that allow for computing themean or weighted sum of a set of numbers, or even performing(modular) multiplication and exponentiation. Quantum addition (modulo 2 n)o ft w o n-qubit quantum states stored in registers R1 and R2 is realized by performinga QFT on one register (say, register R1), then performingcontrolled R zrotations between the two registers, and ﬁnally applying the inverse QFT to the previously Fourier trans-formed register [ 33,42]. The corresponding quantum circuit is given by U R1 QFT/bracketleftBiggn−1/productdisplay i=0n−1/productdisplay j=iCR(bn−j,an−i) j−i+1/bracketrightBigg UR1† QFT, (20) where aiandbidenote qubits of registers R1 and R2, respec- tively, and UR1 QFTdenotes a QFT on register R1. An illustration of the circuit is provided in Ref. [ 33]. On a chip with all-to-all connectivity, the quantum addition algorithm can be implemented in logarithmic depth, neglect-ing the gates required for performing the necessary QFT steps.In the parity encoding, we can perform the logarithmic-depthpart in a single time step and add the linear-depth QFT cir-cuits, without requiring any SWAP gates. As discussed in Sec. II, the controlled rotations can be implemented with single-qubit operations only, provided thenecessary parity qubits are available. In order to fulﬁll thatrequirement, we suggest the qubit layout depicted in Fig. 4. The amount of physical qubits compared to the previouslyintroduced architecture increases by n(n+1) such that there are 3 n(n+1)/2 qubits in total. If one of the registers does not require any computations within itself (for example, if it represents classical data), theparity qubits corresponding to interactions within that registerare not necessary and the data qubits of the register can beadded directly in continuation of the remaining logical lines,requiring n(n+2) qubits in total. Note that either way, the extensions of logical lines hardly affect the circuit depth ofthe QFT part, as they do not cross any lines correspondingto the other register. Our architecture therefore allows us toperform the core quantum addition circuit in a single time stepand the surrounding QFT parts with a depth linear in n,a sTABLE III. Required resources for the core step of the QFT- based quantum addition algorithm on an all-to-all connected deviceand parity mapped on a square lattice with nearest-neighbor con- nectivity. The resources required for the involved QFT circuits are not included in this table. All operations have been decomposed intosingle-body rotations R xandRz,H,a n d CNOT gates. Resource all-to-all LHZ qubits 2 nn (n+2)a CNOT n(n+1) 0 single-qubit1 2n(n+5) n(n+2) total gates1 2n(3n+7) n(n+2) circuit depth 3 log2(n)+11 aIf computations involving multiqubit gates are required within both registers, the number of qubits increases to3 2n(n+1). discussed in Sec. III A . A list comparing required resources for the quantum circuit in the standard gate model and in theparity scheme is given in Table III. C. Multicontrolled gates and Grover’s diffusion operator A difﬁculty arising in many quantum algorithms is the implementation of multicontrolled quantum gates,

Question 570 multiple-choice

Quantum Phase Estimation (QPE) algorithms play a crucial role in quantum computation, with various implementations optimized for noisy intermediate-scale quantum (NISQ) devices. Algorithmic choices regarding gate types and qubit usage directly impact accuracy and resource efficiency in practical quantum experiments. Which modification in Arbitrary Constant Precision Quantum Phase Estimation (ACP QPE) most directly contributes to improved accuracy on NISQ hardware? 1) Increasing the number of ancillary qubits to allow parallel phase estimation 2) Utilizing controlled rotation gates instead of single-qubit unitary rotations 3) Applying extensive classical post-processing after phase measurement 4) Replacing controlled rotation gates with unitary rotation gates and removing ancillary qubits 5) Employing error correction codes for each gate operation 6) Implementing multi-qubit entanglement throughout the circuit 7) Doubling the circuit depth to enhance phase distinguishability

✓ Correct Answer:

The correct answer is 4) Replacing controlled rotation gates with unitary rotation gates and removing ancillary qubits.

📚 Reference Text:

−π 2H•• ✌✌ q2:|0/angbracketright H U1(11π 4) • −π 4• −π 2H• ✌✌ q3:|0/angbracketright H U1(11π 8) • −π 4• −π 2H✌✌ Fig. 15: Arbitrary constant precision QPE gate without 1 anc illary qubit Arbitrary Constant Precision Quantum Phase Estimation 0000000100100011010001010110011110001001101010111100110111101111 Phase Phi = 0.x1x2x3x4 (Original Phi = 0.1011)00.050.10.150.20.25The probability of phase PhiIBM QX Fig. 16: Modiﬁed arbitrary constant precision QPE on IBMQX4 the correct phase ϕand even the probability (i.e., 0.335%) is completely distinguished from other estimated ϕvalues. Finally, we implemented the arbitrary constant precision (ACP) QPE algorithm based on the inverse quantum Fourier transform technique to ﬁnd the phase ϕon both the Qiskit Aer simulator and the IBMQX4 quantum computer. Fig.15 only shows the probability of ﬁnding ϕvalues from the IBMQX4 experiments because the probability of ﬁnding ϕvalue from the simulation results are exactly the same as the original ϕ. However, the highest probability of the phase ϕfrom the experimental results is when the ϕis 0.10110 instead of the correct ϕ=0.1011. To increase the accuracy, the ancillary control qubit was removed and the unnecessary controlled-rotation gates wer e replaced with unitary rotation gates for each qubit as descr ibed in Fig.16. The experimental results shows that our solution can ﬁnd the correct phase ϕand even the probability (i.e., 0.209%) is completely distinguished from other estimated ϕ values. However, the average accuracy of ithdigit on ACP QPE algorithm is around 95% so that the experimental results may vary with each experimental run. IV. D ISCUSSION This paper investigates methods to increase the accuracy of implementing different QPE algorithms on actual quantum computers. Although the paper presents several approaches to the QPE, this work mainly focuses on addressing practical 7 challenges of implementing QPE based on the inverse QFT. The theoretical results of such QPE algorithms from the simulator estimate the phase with almost 100 % accuracy. However, it is not feasible to estimate the correct phase for QPE with inverse QFT from the NISQ quantum computers due to the noise of the system. The Kitaev approach can estimate the phase of an actual system using two qubits. This approach provides a high ﬁdeli ty and low error rate. The disadvantage is that post-processin g is required and there must be a relatively large number of mea- surements performed relative to the other methods investig ated here for the determinations of the phase to be measured. The inverse QFT method does not require post processing. In addition, the binary digits of phase can be estimated separately. However, the inverse QFT method requires a larg e number of rotation gates to achieve a precise solution. The more gates in the system, the higher the level of noise. Highe r noise levels decrease the accuracy of ﬁnding the correct pha se. The constant phase approach has the same set of advantages and disadvantages as the inverse QFT approach. However, one of the relative merits of this approach is that the number of required rotation gates of the original inverse QFT can be

Question 571 multiple-choice

Quantum computing introduces unique challenges for ensuring the correctness and security of algorithms due to phenomena such as superposition and entanglement. Formal methods, especially those based on higher-order logic, are increasingly used to rigorously verify quantum algorithms and protocols. Which of the following statements accurately reflects the implications of the no-cloning theorem for quantum security? 1) It allows duplication of unknown quantum states, enabling safe backup of quantum information. 2) It prohibits copying unknown quantum states, ensuring that quantum information cannot be intercepted and duplicated. 3) It enables entanglement to be used for instantaneous communication between parties. 4) It allows for perfect error correction by creating multiple identical copies of a quantum state. 5) It restricts classical communication in quantum teleportation protocols. 6) It ensures that quantum algorithms can always be reversed to recover the input. 7) It makes quantum cryptographic keys vulnerable to cloning attacks.

✓ Correct Answer:

The correct answer is 2) It prohibits copying unknown quantum states, ensuring that quantum information cannot be intercepted and duplicated..

📚 Reference Text:

Title: Certified Quantum Computation in Isabelle/HOL Year: 2020 Paper ID: 781407c32f23a0afcf5210ee23c06c05fd81022a Source: semantic-scholar URL: https://www.semanticscholar.org/paper/781407c32f23a0afcf5210ee23c06c05fd81022a Abstract: In this article we present an ongoing effort to formalise quantum algorithms and results in quantum information theory using the proof assistant Isabelle/HOL. Formal methods being critical for the safety and security of algorithms and protocols, we foresee their widespread use for quantum computing in the future. We have developed a large library for quantum computing in Isabelle based on a matrix representation for quantum circuits, successfully formalising the no-cloning theorem, quantum teleportation, Deutsch’s algorithm, the Deutsch–Jozsa algorithm and the quantum Prisoner’s Dilemma. We discuss the design choices made and report on an outcome of our work in the field of quantum game theory.

Question 572 multiple-choice

In quantum information theory, locally maximally entangled (LME) states exhibit a high degree of symmetry and entanglement, and their classification leverages concepts from symplectic geometry and group theory. These states are closely related to the structure of reduced density matrices and group actions on quantum state spaces. Which mathematical condition precisely characterizes an LME state in a multipartite quantum system? 1) The reduced density matrix of each subsystem is diagonal but not necessarily proportional to the identity. 2) The total state is invariant under all global unitary transformations. 3) The norm infimum of every G-orbit is strictly zero for all states. 4) The function g → |gΨ|² achieves its minimum at every group element g ∈ G. 5) The moment map is non-vanishing for at least one subsystem. 6) The function g → |gΨ|² has an extremum at the identity element g = 1, equivalent to the vanishing of the moment map. 7) The reduced density matrices commute with all traceless matrices but are not maximally mixed.

✓ Correct Answer:

The correct answer is 6) The function g → |gΨ|² has an extremum at the identity element g = 1, equivalent to the vanishing of the moment map..

📚 Reference Text:

a state |Ψ/an}b∇acket∇i}ht, explicit calculation shows that Hk(x) =/an}b∇acketle{tΨ|1 1⊗···⊗ki⊗···⊗1 1|Ψ/an}b∇acket∇i}ht= tr(ρiki). (4.6) Thus, we have Hk(x) vanishing for all k(so thatµ(x) = 0) if and only if the trace of each reduced density matrix ρimultiplied by any traceless matrix equals zero. This will be true if andonlyifeach reduced density matrixisa multiple oftheidentity matr ixi.e. thesubsystem is maximally mixed. In summary, the space of locally maximally entangled states is the sam e as the inverse image of 0 under the moment map associated with K= SU(d1)× ···×SU(dn). In the classical mechanics language, it is the set of points on phase space where all c onserved quantities associated with inﬁnitesimal local unitary transformations vanish. The space of equivalence classes of these states up to local unitary transformations is the n the quotient µ−1(0)/K. A general result in symplectic geometry is that a quotient space deﬁn ed in this way (known as thesymplectic quotient ) is also a (possibly singular) symplectic manifold. 4.3SLME/Kas a complex manifold ByaremarkabledualityknownastheKempf-Nesstheorem, thesym plecticquotient µ−1(0)/K that we have identiﬁed with SLME/Kis equivalent to another quotient P(H)//G, known as thegeometric invariant theory (GIT) quotient of the full space of states P(H) by the larger groupG= SL(d1,C)× ··· ×SL(dn,C) that is the complexiﬁcation of K.14This group is the set of local invertible transformations (with unit determinant) , known in quantum infor- mation literature as the group of transformations under SLOCC (s tochastic local operations and classical communication). To explain the equivalence, consider the action of Gon the full vector space H=Cd1···dn before normalization. The space Hdecomposes into orbits of G, each of which contains vectors of diﬀerent norm (see ﬁgure 1). It is useful to distinguish between unstable orbits for which the inﬁmum of the norm is zero (i.e. the orbit contains states of arbitrarily small norm), and the re mainingsemistable orbits for which the inﬁmum of the norm is positive. A subset of semistable or bits known as the polystable orbits contain states of minimum norm on which the inﬁmum is achieved; the polystable orbits are those that are topologically closed. It turns out that these states of minimum norm on polystable Gorbits are precisely the LME states.15 14For a more complete discussion of geometric invariant theory, symp lectic geometry, and the Kempf-Ness theorem, see [MFK94], or see [Hos12] for a pedagogical introductio n. 15Here, we are including states with arbitrary normalization. 25 To see this, we will ﬁrst show that a state |Ψ/an}b∇acket∇i}ht ∈ His LME if and only if the function g∈G→ |g|Ψ/an}b∇acket∇i}ht|2(4.7) has an extremum at g= 1. The vanishing of the ﬁrst order variation of the norm-squared function at g= 1 is equivalent to /an}b∇acketle{tΨ|k|Ψ/an}b∇acket∇i}ht= 0 (4.8) wherekis an element of the Lie algebra of K= SU(d1,C)×···×SU(dn,C), the maximal compact subgroup of G. This is precisely the same condition we obtained from the vanishing of the moment map, so by the arguments above, the vectors satis fying this condition are exactly the maximally mixed ones. By looking at the second derivative of (4.7), it

Question 573 multiple-choice

In quantum Chern-Simons theory on the torus, the Hilbert space often admits a decomposition into continuous and discrete components, reflecting its topological and algebraic structure. Modular transformations play a key role in defining operators that act on this space. Which statement best describes the action of the modular group generators ˆS and ˆT on the Hilbert space in genus one Chern-Simons theory? 1) They act only on the continuous component L2, leaving the discrete component CZ invariant. 2) They act as tensor products of operators on L2 and CZ, with explicit formulas involving roots of unity. 3) They interchange the continuous and discrete components of the Hilbert space. 4) They only permute elements within the Weyl group invariant subspace. 5) Their action is limited to integrating over the torus without affecting the finite abelian group. 6) They commute with all elements of the mapping class group without generating a representation. 7) They project the Hilbert space onto its anti-invariant subspace under the Weyl group.

✓ Correct Answer:

The correct answer is 2) They act as tensor products of operators on L2 and CZ, with explicit formulas involving roots of unity..

📚 Reference Text:

Proposition 2. ˆS(f)(θ,γ) =F−1 Z◦FE(f)(θ,γ), (38) ˆT(f)(θ,γ) =GZ◦G−1 E(f)(θ,γ). (39) GENUS ONE COMPLEX QUANTUM CHERN-SIMONS REPRESENATION 11 Proof. Z−1◦˜S◦Z(f)(θ,γ) =1/radicalbig |Z|/summationdisplay ˆγ∈Z/an}b∇acketle{tγ,ˆγ/an}b∇acket∇i}ht1 VolΛ/integraldisplay FΛS(Z(f))(θ−ˆγ,˜θ)/angbracketleftBig θ+ ˆγ,˜θ/2/angbracketrightBig d˜θ =1/radicalbig |Z|/summationdisplay ˆγ∈Z/an}b∇acketle{tγ,ˆγ/an}b∇acket∇i}ht1 VolΛ/integraldisplay FΛZ(f)(˜θ,−θ+ ˆγ)/angbracketleftBig θ+ ˆγ,˜θ/2/angbracketrightBig d˜θ =1 VolΛ1 |Z|/summationdisplay ˜γ∈Z/summationdisplay λ∈Λ∗/integraldisplay FΛf(˜θ+λ,˜γ)/angbracketleftBig θ,˜θ/angbracketrightBig /an}b∇acketle{tλ,θ−˜γ/an}b∇acket∇i}htd˜θ ×/summationdisplay ˆγ∈Z/an}b∇acketle{tˆγ,γ−λ/an}b∇acket∇i}ht =1 VolΛ/summationdisplay ˜γ∈Z/summationdisplay ˜λ∈Λ/integraldisplay FΛf(˜θ+˜λ+γ,˜γ)/angbracketleftBig θ,˜θ+˜λ+γ/angbracketrightBig d˜θ/an}b∇acketle{tγ,−˜γ/an}b∇acket∇i}ht =1 VolΛ/summationdisplay ˜γ∈Z/integraldisplay Ef(˜θ,˜γ)/angbracketleftBig θ,˜θ/angbracketrightBig d˜θ/an}b∇acketle{tγ,−˜γ/an}b∇acket∇i}ht. Noticing that |Z|= (VolΛ)2we get the ﬁrst equation. Z−1◦˜T◦Z(f)(θ,γ) =1/radicalbig |Z|/summationdisplay ˜γ∈Z/an}b∇acketle{tγ,˜γ/an}b∇acket∇i}ht1 VolΛ/integraldisplay FΛT(Z(f))(θ−˜γ,˜θ)/angbracketleftBig θ+ ˜γ,˜θ/2/angbracketrightBig d˜θ =1/radicalbig |Z|/summationdisplay ˜γ∈Z/an}b∇acketle{tγ,˜γ/an}b∇acket∇i}ht1 VolΛ/integraldisplay FΛZ(f)(θ−˜γ,˜θ+θ−˜γ)/angbracketleftBig θ+ ˜γ,˜θ/2/angbracketrightBig d˜θ =1 |Z|1 VolΛ/summationdisplay ˆγ,˜γ∈Z/an}b∇acketle{tγ,˜γ/an}b∇acket∇i}ht/integraldisplay FΛ/summationdisplay λ∈Λ∗f(θ−˜γ+λ,ˆγ)/angbracketleftbigg −θ−˜γ 2,θ−˜γ+˜θ/angbracketrightbigg ×/angbracketleftBig −λ,θ−˜γ+˜θ/angbracketrightBig /an}b∇acketle{t−λ,ˆγ/an}b∇acket∇i}ht/angbracketleftBig θ+ ˜γ,˜θ/2/angbracketrightBig d˜θ =1 |Z|/summationdisplay ˆγ,˜γ∈Z/an}b∇acketle{tγ,˜γ/an}b∇acket∇i}ht/summationdisplay λ∈Λ∗f(θ−˜γ+λ,ˆγ)/angbracketleftbigg −θ−˜γ 2,θ−˜γ/angbracketrightbigg ×/an}b∇acketle{t−λ,θ−˜γ/an}b∇acket∇i}ht/an}b∇acketle{t−λ,ˆγ/an}b∇acket∇i}ht1 VolΛ/integraldisplay FΛ/angbracketleftBig ˜θ,˜γ−λ/angbracketrightBig d˜θ =1 |Z|/summationdisplay ˆγ,˜γ∈Z/an}b∇acketle{tγ,˜γ/an}b∇acket∇i}htf(θ,ˆγ)/angbracketleftbigg −θ−˜γ 2,θ−˜γ/angbracketrightbigg /an}b∇acketle{t−˜γ,θ−˜γ/an}b∇acket∇i}ht/an}b∇acketle{t−˜γ,ˆγ/an}b∇acket∇i}ht =/an}b∇acketle{t−θ/2,θ/an}b∇acket∇i}ht1/radicalbig |Z|/summationdisplay ˜γ∈Z/an}b∇acketle{tγ,˜γ/an}b∇acket∇i}ht/angbracketleftbigg˜γ 2,˜γ/angbracketrightbigg1/radicalbig |Z|/summationdisplay ˆγ∈Zf(θ,ˆγ)/an}b∇acketle{t−˜γ,ˆγ/an}b∇acket∇i}ht =G−1 E◦FZ◦GZ◦F−1 Z(f)(θ,γ) /square It is evident that both ˆSandˆThave a (unique up to a scalar) tensor product decomposition corresponding to L2(E)⊗CZ. We write this precisely in the next proposition Proposition 3. Letjandωbe complex numbers satisfying j4= 1, ω3=in/2j−1. (40) 12 JØRGEN ELLEGAARD ANDERSEN AND SIMONE MARZIONI The operators ˆSandˆTgenerate a representation of SL(2,Z)as endomorphisms ofL2(E)⊗CZ. For any possible choice of jandωthere is a tensor product decomposition ˆS=ˆS′⊗ˆS′′ ˆT=ˆT′⊗ˆT′′(41) ˆS′=jFEˆT′=ωG−1 E(42) ˆS′′=j−1FZˆT′′=ω−1GZ (43) whereˆS′andˆT′(resp.ˆS′′andˆT′′) generates a representation on L2(E)(resp. CZ) Proof. The proof is just a direct veriﬁcation of the relations of SL(2,Z). /square We now apply these computations to the genus 1Chern–Simons theory presented in the previous section (see also Appendix A for further nota tion). The Euclidean space is (t,(·,·)k)while the line bundle is π∗Lk→T×T. The integral lattice we consider is ΛRand its dual will be Λw (k). Their quotient is the ﬁnite abelian group Z(k). The operator Z◦FZ(k)deﬁnes an isometry between /tildewideH(k)andL2(t)⊗CZ(k). In this way, Proposition 3 provides an explicit description of a lift of the action of the mapping class group Γto/tildewideH(k). Consider the following action of the Weyl group WonL2(t)⊗CZ(k) ˆw·f(θ,γ)≡f(w(θ),w(γ)),∀w∈W. (44) The operator (Z◦FZ(k))is equivariant with respect to this action and the one on /tildewideH(k). Therefore it deﬁnes an isometry Z◦FZ(k):H(k)−→/parenleftbig S(t)⊗CZ(k)/parenrightbigW. (45) The action (44) behaves well with respect to the tensor produ ct decomposition L2(t)⊗CZ(k). Therefore it can be written as the tensor product ˆw= ˆw′⊗ˆw′′of the two actions ˆw′(f)(θ) =f(w(θ)) ˆ w′′(x)(γ) =x(w(γ)). (46) On aW–invariant vector f⊗x, i.e.ˆw·f⊗x=f⊗x, there will exist λw∈C∗such that ˆw′(f)(θ) =λwf(θ) ˆ w′′(x)(γ) =λ−1 wx(γ). (47) SinceWis generated by elements of order 2,λw∈ {1,−1}. In factλwhas to be either the identical character 1or the alternating character det(w). In the end, a vectorf⊗xisWinvariant if and only if fandxare both invariant or both anti-invariant for the actions of ˆw′andˆw′′respectively. With the notation from Appendix A we can write H(k)≃/parenleftbig L2(t)⊗CZ(k)/parenrightbigW =/parenleftbig L2 inv(t)⊗C(k)/parenrightbig/circleplusdisplay/parenleftbig L2 anti(t)⊗C(k)/parenrightbig (48) Moreover, all the operators involved in the description of t he action of the map- ping class group from proposition 3 are invariant with respe ct to the action (44), so they preserves the whole decomposition (48). GENUS ONE COMPLEX QUANTUM CHERN-SIMONS REPRESENATION 13 On tensors f⊗x, andg⊗y∈L2(t)⊗CZ(k)the inner product ⟪·,·⟫from (32) factor into two inner products ⟪f,g⟫t≡/integraldisplay tf(θ)g(θ)dvolk(θ),⟪x,y⟫Z(k)≡/summationdisplay γ∈Z(k)x(γ)y(γ) (49)

Question 574 multiple-choice

In computational group theory and quantum algorithms, solving systems of polynomial equations over finite fields often involves leveraging algebraic structures to improve efficiency. The symmetric power of a vector space over a finite field plays a crucial role in linearizing nonlinear problems. Which construction enables the transformation of nonlinear polynomial equations of degree (p−1) in n variables over Zp into linear equations, facilitating efficient solution via linear algebra? 1) The (p−1)th symmetric power of Zn_p, forming the space of degree-(p−1) homogeneous polynomials in n variables 2) The exterior (wedge) power of Zn_p, capturing alternating multilinear forms 3) The tensor product of Zn_p with itself, yielding bilinear maps 4) The dual space of Zn_p, consisting of all linear functionals 5) The group algebra Zp[Zn_p], encoding convolution operations 6) The quotient space of Zn_p modulo a subspace, reducing dimensions 7) The universal enveloping algebra of Zn_p, related to Lie algebras

✓ Correct Answer:

The correct answer is 1) The (p−1)th symmetric power of Zn_p, forming the space of degree-(p−1) homogeneous polynomials in n variables.

📚 Reference Text:

whence the statement follows as χy(u) = 1 if and only if y∈u⊥and/summationtext y∈G|1−χy(u)|2= 2|G|−2/summationtext y∈Gχy(u)=2|G|. WhenG=Zn p, the value χy(u)=e2πi py·udepends only on the inner product y·u overZp,a n dy∈u⊥exactly when y·u= 0. Therefore every ( y,1) generated satisﬁes y·u/negationslash= 0. Thus the output distribution is diﬀerent from the usual one obtained for the abelian Hidden Subgroup where only vectors orthogonal to the hidden subgroup are generated. We overcome the main obstacle, which is that we do not know the actual value of the inner product y·u, by raising these inequations to the power (p−1). They become a system of polynomial equations of degree at most ( p−1) sinceap−1= 1 for every nonzero a∈Zp. In general, solving systems of polynomial equations over any ﬁnite ﬁeld is NP-complete. But using the other special feature of our distribution, which is that the probability of sampling ( y,1) depends only on the inner product y·u, we are able to show that—for ﬁxed prime p—after a polynomial number of samplings, our system of equations has a unique solution with constant probability, and the solution can be found in deterministic polynomial time. To solve our system of polynomial equations, we linearize it in the ( p−1)th sym- metric power of Zn p. We think of Zn pas ann-dimensional vector space over Zp.F o r 10 FRIEDL, IVANYOS, MAGNIEZ, SANTHA, AND SEN ap r i m en u m b e r pa n da ni n t e g e r k≥0, letZ(k) p[x1,...,x n]b et h e kth symmetric power of Zn pwhich will be thought of as the vector space, over the ﬁnite ﬁeld Zp, of homogeneous polynomials of degree kin variables x1,...,x n. The monomials of degree(p−1)forma basis of Z(p−1) p[x1,...,x n], whose dimension is therefore/parenleftbign+p−2 p−1/parenrightbig , which is polynomial in nwhenpis constant. Z(1) p[x1,...,x n] is isomorphic to Zn pas a vector space. For two vectors Y1,Y2∈Z(p−1) p[x1,...,x n], we denote their standard inner product over the monomial basis by Y1·Y2. For every y=(a1,...,a n)∈Zn pand positive integer k, we deﬁne y(k)∈ Z(k) p[x1,...,x n] as the polynomial (/summationtextn j=1ajxj)k.F o r y=(a1,...,a n),z= (b1,...,b n)i nZn p, and positive integers k,l, we deﬁne the product y(k)z(l)∈ Z(k+l) p[x1,...,x n] as the polynomial (/summationtextn i=1aixi)k(/summationtextn j=1bjxj)l.N o wo b s e r v et h a ti f u=(u1,...,u n) is the hidden translationvector, then the vector u∗∈Z(k) p[x1,...,x n] which for every monomial xe1 1···xennhas coordinate ue1 1···uennsatisﬁesy(p−1)·u∗= (y·u)p−1. Therefore each linear inequation y·u/negationslash=0o v e r Zn pwill be trans- formed into the linear equation y(p−1)·U=1o v e r Z(p−1) p[x1,...,x n], where Uis ad i mZ(p−1) p[x1,...,x n]-sized vector of unknowns. We will see below that the vectors y(p−1)span the space Z(p−1) p[x1,...,x n]w h e n yranges over Zn p. Moreover, in what is the main part of our proof, we show in Lemma3.4thatwheneverthespanof y(p−1)forthesamples yisnotZ(p−1) p[x1,...,x n], our sampling process furnishes with probability at least 1 /pa vector z∈Zn psuch thatz(p−1)is linearly independent from

Question 575 multiple-choice

The Quantum Fourier Transform (QFT) is a key component of quantum algorithms that enables efficient frequency analysis by leveraging quantum mechanical principles. Understanding the structure and properties of QFT is essential for analyzing quantum circuits and their computational power. Which property allows the Quantum Fourier Transform to preserve quantum information and ensure reversibility in quantum algorithms? 1) The use of non-unitary matrices 2) Measurement of global phase differences 3) Transformation limited to classical basis states 4) Implementation with non-linear operators 5) Construction from unitary matrices 6) Randomization of qubit ordering 7) Encoding only real-number amplitudes

✓ Correct Answer:

The correct answer is 5) Construction from unitary matrices.

📚 Reference Text:

1ı−1−ı  (11) 2.1.3 Numerical examples: DFT To gain a better understanding of the inner workings of the DFT, we present several examples of manually computing the DFT in both it's regular and matrix forms. 1. N=2 and using (11) the DFT for the signal x= (1,2). WhenN= 2we havew=e−2πi/2=e−πi=−1andF2=H= 1 1 1−1 so the DFT is: X=F2x= 1 1 1−1 · 1 2 = 3 −1  (12) 6 SciPost Physics Codebases Submission 2. N=4, the DFT for the signal x= (1,2,0,0). WhenN= 4and using (10) we can obtain the DFT by matrix multiplication: X=F4x= 1 1 1 1 1−i−1i 1−1 1−1 1i−1−i · 1 2 0 0 = 3 1−2i −1 1 + 2i (13) 2.2 FROM THE DFT TO THE QFT In quantum computing, the QFT (16)is the quantum analogue of the DFT. The QFT can be performed eciently on a quantum computer using exponentially less gates than is required to compute classically. In QIP, the QFT is a Hadamard transform generalisation and both are quite similar, with the exception that the QFT introduces a phase 3.1. ◦The Quantum Fourier transformation : The QFT is a basis transformation described by the following unitary evolution. Let N= 2nand|k/angbracketright,|j/angbracketrightdenote computational basis vectors. Then: |k/angbracketrightQFT/mapsto−→1√ 2n2n−1/summationdisplay k=0ωjk|j/angbracketright (14) Whereωis theNth root of unity: ω:=e2πi/2n(15) We are going to adopt the notation used in [1]. Given a quantum state: ψ/angbracketright=N−1/summationdisplay j=0aj|j/angbracketright= a0 ... aN−1 (16) The QFT can be computed as follows: F|ψ/angbracketright=N−1/summationdisplay k=0bk|k/angbracketright (17) Where: bk=1√ NN−1/summationdisplay j=0aje2πijk/N(18) 2.2.1 Relative phase shift Imagine two states |ˆΘ/angbracketrightand|ˆΦ/angbracketrightwhich di er only by a relative phase shift and thus, if measured, the phases will be insigni cant and therefore the phase of the given state does not e ect the 7 SciPost Physics Codebases Submission probability of measuring that state . For instance, after a measurement, these two states are indistinguishable: |ˆΦ/angbracketright= β0 −iβ1 −β2 iβ3 (19) |ˆΘ/angbracketright= β0 β1 β2 β3 (20) However, as we shall see later, if we apply the QFT on the states |ˆΦ/angbracketrightand|ˆΘ/angbracketrightmeasure their Fourier Transform, this action will disclose the relative phases. QFT 41 2 1 1 1 1 = 1 0 0 0 (21) QFT41 2 1 i −1 −i = 0 1 0 0 (22) 2.2.2 Binary number representation It is critical to maintain the ordering of qubits during creation, access to results, and conversion of binary strings. When qubits are introduced to a circuit, they are added left to right (as in a binary string), beginning with the highest-order qubit and ending with the lowest-order qubit. Two tensor-ed states are denoted in Dirac notation by the symbol |x,y/angbracketright, for instance, |0,1/angbracketright. Additionally, the most important bit appears rst in this format. This type of state is also stated in decimals (23) . |0/angbracketright/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright MSB⊗···⊗⊗···⊗| 0/angbracketright/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright LSB(23) An example follows, wherein bit-strings are interpreted as binary integers: |1/angbracketright⊗|0/angbracketright⊗|1/angbracketright=|101/angbracketright (24) 8 SciPost Physics Codebases Submission In the QFT, as we shall shortly show, it is common practice to use a binary representation of decimal numbers. For

Question 576 multiple-choice

Quantum computing often explores problems involving hidden algebraic structures, such as those defined by functions over finite fields. The Hidden Polynomial Function Graph Problem investigates the limitations and advantages of quantum and classical approaches when dealing with multivariate polynomial functions. Which statement accurately describes a demonstrated quantum advantage in solving the Hidden Polynomial Function Graph Problem for functions over finite fields? 1) The quantum query complexity increases quadratically with the size of the field for all polynomial degrees. 2) Classical algorithms outperform quantum algorithms for hidden quadratic function graphs. 3) The quantum algorithm requires exponentially many measurements for univariate polynomials. 4) Classical query complexity remains constant regardless of the field size. 5) Quantum query complexity is independent of the size of the field for hidden polynomial function graphs, while classical query complexity grows polynomially with field size. 6) Only linear hidden functions admit efficient quantum algorithms in this context. 7) Success probability for quantum algorithms is always negligible for cubic polynomials.

✓ Correct Answer:

The correct answer is 5) Quantum query complexity is independent of the size of the field for hidden polynomial function graphs, while classical query complexity grows polynomially with field size..

📚 Reference Text:

Title: Efficient Quantum Algorithm for Hidden Quadratic and Cubic Polynomial Function Graphs Year: 2007 Paper ID: f7bafeb28f2b0da73069f7e7df5f16e355d94246 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/f7bafeb28f2b0da73069f7e7df5f16e355d94246 Abstract: We introduce the Hidden Polynomial Function Graph Problem as a natural generalization of an abelian Hidden Subgroup Problem (HSP) where the subgroups and their cosets correspond to graphs of linear functions over the finite field F_p. For the Hidden Polynomial Function Graph Problem the functions are not restricted to be linear but can also be multivariate polynomial functions of higher degree. For a fixed number of indeterminates and bounded total degree the Hidden Polynomial Function Graph Problem is hard on a classical computer as its black box query complexity is polynomial in p. In contrast, this problem can be reduced to a quantum state identification problem so that the resulting quantum query complexity does not depend on p. For univariate polynomials we construct a von Neumann measurement for distinguishing the states. We relate the success probability and the implementation of this measurement to certain classical problems involving polynomial equations. We present an efficient algorithm for hidden quadratic and cubic function graphs by establishing that the success probability of the measurement is lower bounded by a constant and that it can be implemented efficiently.

Question 577 multiple-choice

Quantum computing architectures require efficient preparation and manipulation of multi-qubit entangled states, with practical constraints on gate depth and connectivity. Parity encoding and compiler-driven circuit optimization are key strategies for scalable and error-resilient quantum algorithms. In parity-based quantum architectures, which optimization does parity encoding enable during the creation of arbitrary graph states of n qubits? 1) Preparation of cluster states using only long-range controlled-Z gates 2) Reduction of the required ancilla qubits to less than the number of graph edges 3) Implementation of multi-body driver Hamiltonians without additional gate depth 4) Efficient creation of arbitrary graph states with circuit depth at most n+3 and a CNOT count of 2n(n−1) 5) Elimination of the need for decoding circuits after measurement-based computation 6) Automatic correction of phase-flip errors using only nearest-neighbor interactions 7) Construction of graph states without the use of controlled gates or compiler support

✓ Correct Answer:

The correct answer is 4) Efficient creation of arbitrary graph states with circuit depth at most n+3 and a CNOT count of 2n(n−1).

📚 Reference Text:

J i,Jij, etc. Such optimization problems can be tackled on digital gate-based architectures using the quan-tum approximate optimization algorithm (QAOA) [ 44]. The QAOA variationally evolves the system with Hamiltonians H X andHZas |ψ/angbracketright=p/productdisplay j=1e−iβjHXe−iγjHZ, (22) where HX=/summationtext iσ(i) xis a driver term to explore the search space. While the problem Hamiltonian HZcan be imple- mented in the parity architecture in a single step, the driverHamiltonian can be realized with a gate sequence of depth7n−8 analogous to the circuit shown in Fig. 3(leaving out theR zrotations and replacing the physical Hadamard gates by parameterized Rxrotations). Such implementations and simi- lar variants have been analyzed thoroughly in recent literature[35,45,46]. Typically, an additional Hamiltonian containing parity constraints is added to give an energy penalty to stateswhich are not in the code space. These constraints simplify theoperations required for the driver Hamiltonian. In particular,whenever all parity constraints are energetically penalized,the driver Hamiltonian reduces to a sum of single-bodyterms. E. Preparation of graph states Graph states are an important resource for measurement- based quantum computing [ 8,9] as well as for several error correction protocols [ 47,48]. Cluster states, corresponding to square grid graphs, can be prepared efﬁciently using nearest-neighbor interactions. The preparation of more arbitrary graphstates, however, typically requires long-range interactions be-tween many different qubits, as it involves applying a CZ gate for every edge in the graph [ 34]. In the parity encoding, arbitrary graph states containing nqubits can be created with a circuit depth of at most n+3 and a CNOT -gate count of 2n(n−1) as follows. We start by preparing ndata qubits in the superposition state |+/angbracketright. We then apply the encoding sequence, in either the LHZ scheme or a reduced version of it,depending on the connectivity requirements of the respectivegraph, requiring a circuit of depth n+1 or less (for reduced versions). Subsequently, we apply CZgates to every pair of logical qubits to be connected in the graph, using parallelsingle-qubit gates on the physical qubits according to thedecomposition ( 6). The result of this is a graph state encoded in the parity architecture. For measurement-based quantumcomputing, which involves measurements of the logical qubits(in arbitrary axes), an additional decoding step of depth n+1 is required. The number of parity qubits required for this procedure is in principle equal to the number of edges in the desired graphstate. However, in some cases it can be useful to have addi-tional ancillary qubits to simplify the encoding and decodingcircuits. IV . CONCLUSION AND OUTLOOK In this work we have presented an approach to implement fundamental quantum algorithms for arbitrary system sizes,completely avoiding SWAP gates. A gate count analysis shows that our implementation has the potential to save multiqubitgates or reduce circuit depth for key constituents of Shor’salgorithm. In addition, tailored parity encodings for particularalgorithms can be constructed by utilizing the parity compiler[29]. On top of that, due to the redundant encoding of infor- mation, the parity encoding provides an intrinsic potential todetect and correct bit-ﬂip errors. One possible approach to error correction in the

Question 578 multiple-choice

The Hidden Subgroup Problem (HSP) is a fundamental challenge in quantum computing, and advances in quantum algorithms have extended solutions from abelian to certain non-abelian group classes. Nilpotent groups, especially those of nilpotency class 2 ("nil-2 groups"), are of particular interest due to their richer algebraic structure and tractability for efficient quantum algorithms. Which statement accurately describes the main result regarding efficient quantum algorithms for the Hidden Subgroup Problem in nil-2 groups? 1) Efficient quantum algorithms for HSP in nil-2 groups require the non-abelian Fourier transform as a central component. 2) For any nil-2 group, the quantum algorithm always finds hidden subgroups of arbitrary order. 3) The approach for nil-2 groups relies exclusively on classical matrix sum algorithms without quantum procedures. 4) The quantum algorithm for nil-2 groups does not generalize to nilpotent groups of higher class. 5) An efficient quantum procedure exists for finding hidden subgroups in nil-2 groups by reducing the problem to nil-2 p-groups of exponent p, leveraging both quantum hiding procedures and classical reductions. 6) Only extraspecial groups, not general nil-2 groups, admit efficient quantum algorithms for the HSP. 7) The quantum hiding procedure for nil-2 groups requires solving only single linear equations, making it equivalent in complexity to abelian cases.

✓ Correct Answer:

The correct answer is 5) An efficient quantum procedure exists for finding hidden subgroups in nil-2 groups by reducing the problem to nil-2 p-groups of exponent p, leveraging both quantum hiding procedures and classical reductions..

📚 Reference Text:

and Beth [19], Hallgren, Russell and Ta-Shma [8], Grigni, Schulman, Vaz irani and Vazirani [6] and Moore, Rockmore, Russell and Schulman [17]. For the Heisenberg group, Ba con, Childs and van Dam [1] used the pretty good measurement to reduce the HSP to some matrix sum pr oblem that they could solve classically. Ivanyos, Magniez and Santha [11] and Friedl, Ivanyos, Magniez, S antha and Sen [5] have eﬃciently reduced theHSPin some non-abelian groups to HSPinstances in abelian groups using classical and quantum group theoretical tools, but not the non-abelian Fourier transform. Th is latter approach was used recently by Ivanyos, Sanselme and Santha [12] for extraspecial groups. In this work we extend the class of groups where the HSP is eﬃciently solvable by a quantum algorithm to nilpotent groups of nilpotency class at most 2 (shortly nil-2 group s). These are groups whose lower (and upper) central series are of length at most 2. Equivalently, a grou p is nil-2 group if the derived group is a subgroup of the center. Nilpotent groups form a rich subclass of s olvable groups, they contain for example all (ﬁnite) p-groups. Extraspecial groups are, in particular, in nil-2 groups. O ur main result is: Theorem 1. LetGbe a nil-2group, and let us given an oracle fwhich hides the subgroup HofG. Then there is an eﬃcient quantum procedure which ﬁnds H. The overall structure of the algorithm presented here is closely re lated to the algorithm in [12] for extraspecial groups, but has also several additional features. The quantum part of the algorithm is restricted to speciﬁc nil-2 groups, which are also p-groups and are of exponent p. It consists essentially in the creation of a quantum hiding procedure(a natural quantum generalizationo fa hiding function) for the subgroup HG′ ofG. The procedure uses certain automorphisms of the groups to deﬁ ne some appropriate group actions, and is analogous to what have been done in [12] for extraspecial p-groups of exponent p. Whiledealingwith extraspecial p-groupsofexponent pbasicallysolvestheHSP forallextraspecialgroups (the case of remaining groups, of exponent p2, easily reduces to groups of exponent p), this is far from being true for nil-2 groups. Indeed, one of the main new features of the current algorithm is a classical reduction of the HSP in nil-2 groups to the HSP in nil-2 p-groups of exponent p, where moreover the hidden subgroup is either trivial or of cardinality p. In fact, our result is much more general: we prove an analogous re duction in nil-kgroups for any constant k. We believe that this general reduction might be useful for designin g eﬃcient quantum algorithms for the HSP in groups of higher nilpotency class. Our second main novel feature concerns the quantum hiding proce dure. While in extraspecial groups it was reduced to the eﬃcient solvability of a single quadratic and a single linear equation modulo p, here we look for a nontrivial solution of a homogeneous system of dquadratic and dlinear equations, where dcan be any integer. The reason for

Question 579 multiple-choice

Quantum algorithms are increasingly important for solving algebraic problems that underpin cryptography and computational mathematics. The hidden polynomial problem (HPP) and its bivariate case present significant challenges due to complex input structures like level set superpositions. Which of the following statements accurately describes a breakthrough achieved by a quantum algorithm for the bivariate hidden polynomial graph problem with constant-degree polynomials? 1) It solves univariate HPP instances in polynomial time for arbitrary-degree polynomials. 2) It provides an exponential speedup for factoring large integers using univariate polynomials. 3) It requires classical pre-processing of all level set superpositions before quantum computation. 4) It is only applicable to fields of prime order and cannot handle elliptic curves. 5) It achieves polylogarithmic time complexity with respect to field size for constant-degree bivariate polynomials, even when inputs are given as level set superpositions. 6) It is limited to solving linear hidden polynomial problems and not quadratic forms. 7) It demonstrates that classical algorithms outperform quantum algorithms in all cases for bivariate HPP.

✓ Correct Answer:

The correct answer is 5) It achieves polylogarithmic time complexity with respect to field size for constant-degree bivariate polynomials, even when inputs are given as level set superpositions..

📚 Reference Text:

Title: Polynomial time quantum algorithms for certain bivariate hidden polynomial problems Year: 2013 Paper ID: 05eb8abe2d380d5fff329f1d1386d54aac1e193d Source: semantic-scholar URL: https://www.semanticscholar.org/paper/05eb8abe2d380d5fff329f1d1386d54aac1e193d Abstract: We present a new method for solving the hidden polynomial graph problem (HPGP) which is a special case of the hidden polynomial problem (HPP). The new approach yields an efficient quantum algorithm for the bivariate HPGP even when the input consists of several level set superpositions, a more difficult version of the problem than the one where the input is given by an oracle. For constant degree, the algorithm is polylogarithmic in the size of the base field. We also apply the results to give an efficient quantum algorithm for the oracle version of the HPP for an interesting family of bivariate hidden functions. This family includes diagonal quadratic forms and elliptic curves.

Question 580 multiple-choice

Quantum computing leverages unitary operators to manipulate qubits and perform complex algorithms. Among these, the Controlled-NOT (CNOT) gate and the Quantum Fourier Transform (QFT) play vital roles in enabling reversible computation and efficient algorithmic transformations. Which statement accurately describes the structural relationship between the Quantum Fourier Transform (QFT) and quantum gate decompositions in multi-qubit systems? 1) The QFT always consists exclusively of CNOT gates arranged in sequence. 2) The QFT for n qubits is represented solely by diagonal matrices with real entries. 3) The QFT is implemented by decomposing its unitary matrix into a tensor product of Hadamard gates and controlled phase rotations. 4) The QFT cannot be defined for systems with more than two qubits. 5) The QFT matrix for n qubits is identical to the classical discrete Fourier transform matrix for any n. 6) The QFT implementation does not require consideration of gate count or circuit depth. 7) The QFT operator is inherently non-unitary for n>1 qubits.

✓ Correct Answer:

The correct answer is 3) The QFT is implemented by decomposing its unitary matrix into a tensor product of Hadamard gates and controlled phase rotations..

📚 Reference Text:

theCNOT operator is unitary and a reversible XOR function is achieved because thecontrol bit ∣q 1iis preserved from input to output. There exist powerful tools for the simulation of quantum operations (referred to as‘quantum gates ’) and for the rendering of multiple qubit quantum circuits [11]. Figure 1. Two qubit quantum circuit for H ⊗Z ðÞ ∣q1q0i. 5Quantum Fourier Operators and Their Application DOI: http:/ /dx.doi.org/10. 5772/intechopen.9 4902 Figure 2 shows a schematic representation of the CNOT circuit corresponding to Table 1 . In this circuit, the control bit is used to swap the target ∣q0i(using an X gate) if ∣q1i¼∣1i. For the sake of this work, we point out that an equally valid interpretation of the quantum CNOT function can be realized if the roles of the control and target areinterchanged where ∣q 1q0i¼∣tci(seeTable 2 ). In this case the CNOT operator becomes CNOT ¼1000 0001 001001002 66643 7775: (22) which is a permutation matrix that swaps the ∣11iand∣01iqubits and corresponds to the circuit in Figure 3 . We shall have more to say about this implementation in the following sections. For now, with this brief overview of quantum computation, we can now introduce the quantum Fourier transform.cin tin cout tout 0000 010110111110 Table 1. Controlled NOT. Figure 2.Two qubit CNOT quantum circuit swap of ∣11iand∣10iusing Qiskit [11]. tin cin tout cout 00 0 0 01 1 110 1 011 0 1 Table 2. Controlled NOT where ∣q1q0i¼∣tci. 6Real Perspective of Fourier Transforms and Current Developments in Superconductivity 3. The quantum Fourier transform It should be clear that the DFT matrix in Eq. (2) is unitary where F†F¼I (23) andF†is the Hermitian conjugate of F. Because of this unitarity, the potential for using the DFT within the context of quantum computation naturally follows. How-ever, such an application requires a decomposition involving tensor products ofunitary operations typically applied in quantum computation. As with the FFT, the choice of the decomposition dictates the algorithmic complexity. There is much introductory literature available regarding the QFT [3, 12 –14]. Given a specific quantum algorithm where the QFT is applied, current research endeavors reside inattempts to improve the computational complexity [4, 7, 9, 15, 16]. The QFT matrix is defined as Q¼ 1ﬃﬃﬃﬃ NpXN/C01 j¼0XN/C01 k¼0ei2π Njk∣kihj∣: (24) For example, with N¼2nand n = 1, we recover the Hadamard matrix Q¼1ﬃﬃﬃ 2p11 1/C01/C20/C21 , (25) or, for n = 2, Q¼1 211 1 1 1i/C01/C0i 11 1 /C01 1/C0i/C01i2 66666643 7777775: (26) As expected, this operator is unitary where, with Q †¼1ﬃﬃﬃﬃ NpXN/C01 j¼0XN/C01 k¼0e/C0i2π Njk∣jihk∣, (27) Figure 3. Two qubit CNOT quantum circuit swap of ∣11iand∣01i. 7Quantum Fourier Operators and Their Application DOI: http:/ /dx.doi.org/10. 5772/intechopen.9 4902 it should be clear that QQ†¼1ﬃﬃﬃﬃ NpXN/C01 j¼0XN/C01 k¼0ei2π Njkjkihjj ! 1ﬃﬃﬃﬃ NpXN/C01 j0¼0XN/C01 k0¼0e/C0i2π Nj0k0∣j0ihk0∣ ! ¼1 NXN/C01 k0¼0XN/C01 k¼0XN/C01 j0¼0XN/C01 j¼0ei2π Njk/C0j0k0ðÞ∣kijjj0hi h k0∣ ¼1 NXN/C01 k0¼0XN/C01 k¼0XN/C01 j0¼0XN/C01 j¼0ei2π Njk/C0j0k0ðÞδj0j∣kihk0∣ ¼1 NXN/C01 k0¼0XN/C01 k¼0XN/C01 j¼0ei2π Njk/C0k0ðÞ∣kihk0∣ ¼1 NXN/C01 k0¼0XN/C01 k¼0Nδk0k ðÞ ∣kihk0∣ ¼XN/C01 k¼0∣kihk∣ ¼I:(28) In general, given a state vector ∣ψi¼XN/C01 j¼0aj∣ji

Question 581 multiple-choice

In photonic quantum systems, photon loss is a fundamental challenge that affects the reliability of quantum information processing. Modeling and mitigating loss are essential for the design and simulation of optical quantum circuits. When simulating photon loss in optical quantum circuits, which method accurately captures the probabilistic redistribution of photons due to loss, and allows simplification by combining symmetric loss channels? 1) Using beamsplitter operations to model loss as redistribution between a system mode and a vacuum mode, with transmissivity linked to the loss rate 2) Modeling loss solely as a unitary transformation without introducing auxiliary modes 3) Treating photon loss as a deterministic process where photon number is simply reduced 4) Simulating loss by adding random phase shifts to each optical mode 5) Applying post-selection techniques without physically modeling the loss process 6) Representing loss channels as perfect photon-number-resolving detectors 7) Ignoring absorption and detection inefficiencies in all circuit elements

✓ Correct Answer:

The correct answer is 1) Using beamsplitter operations to model loss as redistribution between a system mode and a vacuum mode, with transmissivity linked to the loss rate.

📚 Reference Text:

Loss has several physical mechanisms, such as absorption in optical fiber and components, imperfect photon sources, and in photo-detection [14]. Loss is a non-unitary process, resulting in a mixed state over different lost photon number sectors [15]. We can write its action on a matrix element as Λη(|n1⟩⟨n2|) =ηn1+n2 2min{n1,n2}X k=0sn1 kn2 k1−η ηk |n1−k⟩⟨n2−k|. (3) Here 1−ηis the single photon loss probability. It can readily be observed that the process Eq. (3) is equivalent to applying a beamsplitter between the mode of interest and a fictitious mode containing 0 photons, where the transmisivity (probability for photon to transition) is 1−η[16]. Consider a beamsplitter that acts on modes a†, b†asa†→(sinθa†+ cos θb†).Then we can see |n,0⟩=1√ n!(a†)n|0,0⟩ →1√ n!(sinθa†+ cos θb†)n|0,0⟩=nX k=0sn k sinn−k(θ) cosk(θ)|n−k, k⟩. (4) 1a†|n⟩=√n+ 1|n+ 1⟩. 2In fact, any single non-trivial beamsplitter (with not all entries real) can generate all of linear optics [11] (with O(m2)of these still required). 4 ηηηηηηηηηηηηηηηη η3=η3η3η3 Figure 1: Example of circuit with symmetric losses that commute to the start of the circuit. Instead of applying 16 independent loss channels, it is equivalent to applying 4 loss channels with a modified parameter η→η3. In this diagram, the two qubit ‘gates’ are beamsplitters, and the boxes with enclosed ηrepresent a loss channel with loss rate 1−η. Upon tracing out the fictitious mode we get the same channel, with η= sin2θ. In practice this is simulated by sampling and keeping in memory only a single pure state. Consider for simplicity, an arbitrary nphoton pure state of mmodes, |ψn⟩. We append a vacuum mode and apply a beamsplitter between it and the lossy mode of interest, which splits the state into different photon number sectors: |ψn⟩ →nX k=0ck|ψn−k⟩|k⟩ →X k|ck|2|ψn−k⟩⟨ψn−k|. (5) Here the ckare complex amplitudes (P k|ck|2= 1) that depend on θof Eq. (4), and |ψn−k⟩is notation to indicate it is some n−kphoton state. After this step we can probabilistically sample |ψn−k⟩from the distribution {|ck|2}k(tracing out the fictitious mode). Another useful observation is that symmetric loss commutes through linear optics [15]. For example, Λ(1) ηΛ(2) ηB12= B12Λ(1) ηΛ(2) η, where B12is a beamsplitter between modes 1,2, and Λ(i) ηis a loss channel on mode i. Coupling this with fact3Λη1Λη2= Λ η1η2, certain circuits with symmetry can be simplified dramatically by commuting the losses. For an example, see Fig. 1, where all loss can be commuted to the beginning of the circuit. In such a case, following (5), we may simply sample input states with fewer photons rather than actually applying a loss channel. Further, if the probabilities |ck|2of (5) are known explicitly, we may often perform calculations for each possible input |ψn−k⟩⟨ψn−k|separately, then use linearity to calculate the desired quantities. (See Sec. II D 3 for this technique applied to the case of distinguishability.) In a low-order error model, where error rates are low and therefore multiple nearby errors are assumed to be rare, photon loss is generally heralded. Thus in the calculations below we will generally ignore loss; however, for a more

Question 582 multiple-choice

Three-dimensional Chern-Simons theory with complex gauge group SL(2,ℂ) is a topological quantum field theory deeply connected to knot invariants and the geometry of hyperbolic 3-manifolds. This framework provides insights into quantum gravity and the relationships between classical geometric structures and quantum observables. Which statement best captures the generalized relationship explored between knot invariants and quantum geometry in three-dimensional SL(2,ℂ) Chern-Simons theory? 1) The Alexander polynomial determines the metric properties of all hyperbolic 3-manifolds. 2) The colored Jones polynomial is independent of the representation used for Wilson loops. 3) The volume conjecture only applies to knots in spherical 3-manifolds. 4) The Melvin-Morton-Rozansky conjecture relates hyperbolic volume directly to Wilson loop expectation values. 5) The asymptotic behavior of the colored Jones polynomial encodes the hyperbolic volume of the knot complement and generalizes connections among quantum and classical invariants. 6) The A-polynomial is unrelated to representations of the knot group in SL(2,ℂ). 7) The partition function in Chern-Simons theory is solely determined by the genus of the underlying manifold.

✓ Correct Answer:

The correct answer is 5) The asymptotic behavior of the colored Jones polynomial encodes the hyperbolic volume of the knot complement and generalizes connections among quantum and classical invariants..

📚 Reference Text:

Title: Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial Year: 2003 Paper ID: debec40e9e09ee932a3f238d1cff08bbf39c050c Source: semantic-scholar URL: https://www.semanticscholar.org/paper/debec40e9e09ee932a3f238d1cff08bbf39c050c Abstract: We study three-dimensional Chern-Simons theory with complex gauge group SL(2,ℂ), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot. Using this approach, we find some new and rather surprising relations between the A-polynomial, the colored Jones polynomial, and other invariants of hyperbolic 3-manifolds. These relations generalize the volume conjecture and the Melvin-Morton-Rozansky conjecture, and suggest an intriguing connection between the SL(2,ℂ) partition function and the colored Jones polynomial.

Question 583 multiple-choice

Quantum computing and advanced mathematical frameworks such as supersymmetric quantum mechanics and random matrix models play a significant role in the modern study of number theory, including deep investigations of the Riemann hypothesis. The interplay between quantum algorithms and mathematical functions with non-trivial zeros is of growing interest for both theoretical insight and computational advances. Which statement accurately describes how ground state wave functions related to the Riemann hypothesis are treated in quantum computational approaches using the Quantum Fourier Transform? 1) They are encoded as classical bit strings and analyzed using integer factorization algorithms. 2) They are simulated exclusively in position space due to limitations of quantum hardware. 3) They are represented as quantum states in momentum space, constructed via supersymmetric quantum mechanics, and analyzed for non-trivial zeros using the Quantum Fourier Transform. 4) They are modeled as solutions to differential equations without reference to quantum algorithms or momentum space. 5) They are considered eigenstates of bosonic operators and only studied in the large N limit of random matrix theory. 6) They are interpreted as thermal states rather than ground states and do not utilize quantum error correction. 7) They are approximated by semiclassical wave packets and examined through classical numerical methods.

✓ Correct Answer:

The correct answer is 3) They are represented as quantum states in momentum space, constructed via supersymmetric quantum mechanics, and analyzed for non-trivial zeros using the Quantum Fourier Transform..

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Title: Quantum Computing and the Riemann Hypothesis Year: 2023 Paper ID: 0d20559eec7b9943fc9e61d53818db60043f1510 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0d20559eec7b9943fc9e61d53818db60043f1510 Abstract: Quantum computing is a promising new area of computing with quantum algorithms offering a potential speedup over classical algorithms if fault tolerant quantum computers can be built. One of the first applications of the classical computer was to the study of the Riemann hypothesis and quantum computers may be applied to this problem as well. In this paper we apply the Quantum Fourier Transform (QFT) to study three functions with non-trivial zeros obeying a version of the Riemann hypothesis. We perform our quantum computations with six qubits, but more qubits can be used if quantum error correction allows the QFT algorithm to scale. We represent these functions as ground state wave functions transformed to momentum space. We show how to obtain these functions as states in supersymmetric quantum mechanics. Finally we discuss the relation of these functions to (p,1) Random two matrix models at large N.

Question 584 multiple-choice

In quantum machine learning, understanding how symmetries and representations constrain operators is central to designing models for extracting information from quantum states. The relationship between Lie groups and Lie algebras provides a powerful framework for analyzing symmetry-related problems in this domain. Which statement most accurately explains why translating problems involving Lie group symmetries to their Lie algebras can simplify the analysis of quantum machine learning models? 1) Lie groups and Lie algebras always have identical structures, making their analysis interchangeable. 2) Lie algebras allow for the implementation of non-commuting measurement operators, which is not possible in Lie groups. 3) Lie group representations cannot be decomposed into irreducible components, while Lie algebras can. 4) Lie groups only describe discrete symmetries, whereas Lie algebras handle continuous symmetries. 5) Lie group symmetries are fundamentally non-linear, while Lie algebras provide a linear vector space framework suitable for linear algebraic methods. 6) Lie algebras capture the infinitesimal generators of symmetry in a tractable linear space, enabling complex group symmetry problems to be reformulated as linear algebra problems. 7) Lie algebras are used to classify quantum states, while Lie groups are limited to classical data analysis.

✓ Correct Answer:

The correct answer is 6) Lie algebras capture the infinitesimal generators of symmetry in a tractable linear space, enabling complex group symmetry problems to be reformulated as linear algebra problems..

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diagonal structure in thesymmetric-antisymmetric basis of U 2 gin Example 5. We will later learn in Example 9 that this representation can- not be further block diagonalized and the symmetric and antisymmetric spaces are irreducible—for now, take this for granted. Schur’s lemma will then imply that Mmust be of the form M=c013Lc11, where 13denotes the 33identity. This can be better visualized from the block structure of U 2and ofMin Fig. 6(b). For in- stance, the choice c0=c1= 1leads toM=1, whereas c0= 1andc1=1leads toM= SWAP , both of which an be trivially veriﬁed to commute with U 2 g. In partic- ular, writing i=1 2(1+r), and expanding 2 iin the symmetric-antisymmetric basis shows that the compo- nent in the antisymmetric space is1 4(1rr). Hence, we can readily see that the special choice c0= 0andc1= 1 leads to the outcome of the GQML model in Eq. (12) to be h(i) =1 4(1rr). Clearly, this GQML model can indeed classify the data in the purity dataset. The previous example is extremely rich and has quite a few points worth highlighting. First, we note that a QML model acting on k-copies ofiinduces ak-th order ten- sor representation of the symmetry group. For instance in the purity dataset where G=SU(2), a QML model with a single copy of ileads to the fundamental represen- tation which has no non-trivial block diagonal structure (the representation is irreducible), while a model working withk= 2copies leads to the tensor representation which admits a non-trivial block diagonal structure (the repre- sentation is reducible ).Understanding the block diagonal structure of the representation is crucial as its presence (or absence) imposes restrictions on the operators Mwe can use in a GQML model (see Fig. 6, were a block diagonal U 2 gimplies a block diagonal Mwith identities of diﬀerent sizes). Finally, since we compute the expectation value of Moveri,the block diagonal structure of Mdictates what is the information that we can access from the state . For a conventional experiment, we saw that the only accessi- ble information was the norm of i, which is not useful to solve the classiﬁcation task. On the other hand, in a quantum-enhanced experiment we found that 2 ialso ad- mits a block diagonal structure, and that Mcan precisely accesstheblockthatencodestheinformationaboutthepu- rity of the quantum state. This will be a running theme, theostinato if you will, for the rest of this work and in GQML:symmetries are all about block diagonalization. C. There and back again: Lie groups and algebras In this section, we unravel the mathematical correspon- dences between a Lie group Gand its Lie algebra g. While manysymmetriesariseasLiegroupsymmetries, Liegroups 15 can be somewhat unwieldy objects. Manifolds are compli- cated, and their often highly nonlinear structure requires sophisticated analysis to probe. But Lie algebras, which describe “derivatives of paths in the Lie group”, are ulti- mately vector spaces and thus vulnerable to the mighty hammers of linear algebra. A natural program arises from this observation: When we have a problem with Lie group symmetry, pass

Question 585 multiple-choice

In the study of finite 2-groups, properties of subgroup lattices and commutator relations play a critical role in classification. Group presentations often use specific generators and relations to encode structural information about these groups. Which subgroup is simultaneously the Frattini subgroup, the center, and the quotient of the commutator subgroup by ⟨c²⟩ in a finite 2-group G of order 2ⁿ⁺⁴ where each is abelian of type (2ⁿ⁻¹, 2, 2)? 1) Φ 2) Ω₁ 3) G′ 4) U/⟨c²⟩ 5) ⟨z⟩ 6) E₄ 7) G/Φ

✓ Correct Answer:

The correct answer is 1) Φ.

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n=m+1 (noting that Ω 1(G) =G′). Set /an}bracketle{tz/an}bracketri}ht= Ω1(/an}bracketle{tc/an}bracketri}ht) so that z=c2n−1andz∈G′. Then M=U/an}bracketle{tc2/an}bracketri}ht, Φ(G) = Z( G) =G′/an}bracketle{tc2/an}bracketri}htis abelian of type (2n−1,2,2) and |G|= 2n+4. Leta, b∈U−G′be such that U=G′/an}bracketle{ta, b/an}bracketri}ht, where a2, b2∈G′and G=/an}bracketle{ta, b, c /an}bracketri}ht. Since each maximal subgroup Hi(i= 1, ...,6) is nonmetacyclic and contains Φ( G) and zis a square in Φ( G), it follows that H′ i/ne}ati≀nslash=/an}bracketle{tz/an}bracketri}htfor all i= 1, ...,6. This implies that M′=/an}bracketle{tz/an}bracketri}htand therefore [ a, b] =z. Now, G//an}bracketle{tz/an}bracketri}hthas the unique maximal abelian subgroup M//an}bracketle{tz/an}bracketri}htand six minimal nonabelian maximal subgroups Hi//an}bracketle{tz/an}bracketri}ht,i= 1, ...,6, and so G//an}bracketle{tz/an}bracketri}ht is an A 2-group with the following properties. We have d( G//an}bracketle{tz/an}bracketri}ht) = 3 and soG//an}bracketle{tz/an}bracketri}htis nonmetacyclic of order 2n+3>24since n≥3, (G//an}bracketle{tz/an}bracketri}ht)′∼=E4, G′//an}bracketle{tz/an}bracketri}ht ≤Z(G//an}bracketle{tz/an}bracketri}ht) (since G′≤Z(G)) and G//an}bracketle{tz/an}bracketri}hthas a normal elementary abelian subgroup /an}bracketle{tG′,Ω2(/an}bracketle{tc/an}bracketri}ht)/an}bracketri}ht//an}bracketle{tz/an}bracketri}htof order 8. Hence G//an}bracketle{tz/an}bracketri}htis an A 2-group of Proposition 71.4(b) in [2] which implies the fact that /an}bracketle{tG′,Ω2(/an}bracketle{tc/an}bracketri}ht)/an}bracketri}ht//an}bracketle{tz/an}bracketri}ht= Ω1(G//an}bracketle{tz/an}bracketri}ht). Set a2=xandb2=yand consider the abelian group M//an}bracketle{tz/an}bracketri}ht. If the abelian subgroup U//an}bracketle{tz/an}bracketri}htof order 16 and exponent ≤4 has rank >2, then Ω 1(U//an}bracketle{tz/an}bracketri}ht)> G′//an}bracketle{tz/an}bracketri}htwhich contradicts the above fact. Hence U//an}bracketle{tz/an}bracketri}ht∼= C4×C4which implies that G′=/an}bracketle{tx, y, z /an}bracketri}ht. Since all Hiare minimal nonabelian (containing Φ( G) =/an}bracketle{tc2, x, y/an}bracketri}ht), we get [ a, c] =xαyzβand [b, c] =xyγzδ, where α, β, γ, δ = 0,1. From Φ(/an}bracketle{tab, c/an}bracketri}ht) =/an}bracketle{t(ab)2=xyz, c2,[ab, c] =xα+1yγ+1zβ+δ/an}bracketri}ht= Φ(G) and the fact that Ω 1(/an}bracketle{tc2/an}bracketri}ht) = /an}bracketle{tz/an}bracketri}htfollows that α+ 1 /ne}ati≀nslash=γ+ 1 which gives γ=α+ 1 and [ b, c] =xyα+1zδ. FINITE 2-GROUPS 83 Interchanging aandbandxandy, i.e., writing a′=b, b′=a, x′=y, y′= x, we get a′2=b2=y=x′, b′2=a2=x=y′,[a′, b′] = [b, a] =z, [a′, c] = [b, c] =xyα+1zδ= (x′)α+1y′zδ,[b′, c] = [a, c] =xαyzβ=x′y′αzβ. Writing again a, b, x, y instead of a′, b′, x′, y′, respectively, we get a2=x, b2=y,[a, b] =z,[a, c] =xα+1yzδ,[b, c] =xyαzβ, β, δ = 0,1, which are the old relations in which αis replaced with α+1. This shows that we may assume α= 0 and so we get [ a, c] =yzβand [b, c] =xyzδ,β, δ= 0,1. Finally, replacing cwithc′=caδbβ, we get c′2=c2(aδbβ)2[aδbβ, c] =c2l withl∈G′and so c′4=c4,/an}bracketle{tc′/an}bracketri}htcovers G/Uand/an}bracketle{tc′/an}bracketri}ht ∩G′=/an}bracketle{tz/an}bracketri}ht. In addition we have [a, c′] = [a, caδbβ] = [a, c][a, b]β=yzβzβ=y, [b, c′] = [b, caδbβ] = [b, c][b, a]δ=xyzδzδ=xy. Writing again cinstead of c′, we get the relations stated in our theorem. References [1] Y. Berkovich, Groups of prime power order, Vol. 1, Walter de Gruyter, Berlin-New York, 2008. [2] Y. Berkovich and Z. Janko, Groups of prime power order, Vo l. 2, Walter de Gruyter, Berlin-New York, 2008. [3] Y. Berkovich and Z. Janko, Groups of prime power order, Vo l. 3, in preparation. Z. Boˇ zikov Faculty of Civil Engineering and Architecture University of Split 21000 Split Croatia E-mail :Zdravka.Bozikov@gradst.hr Z. Janko Mathematical Institute University of Heidelberg 69120 Heidelberg Germany Received :20.2.2009. Revised :4.6.2009.

Question 586 multiple-choice

The dihedral hidden subgroup problem (DHSP) is a key challenge in quantum computing, closely connected to the computational hardness of the subset sum problem at various densities. Efficient algorithms for subset sum and optimal quantum measurements have significant implications for cryptography and quantum algorithm design. Which statement most accurately reflects the threshold for successful quantum state discrimination in DHSP using the pretty good measurement (PGM)? 1) The PGM is only optimal for distinguishing states of order greater than two in DHSP. 2) The threshold for non-negligible success probability using PGM occurs at density ν much less than 0.5. 3) A single hidden subgroup state always yields success probability greater than 0.5, regardless of density. 4) Achieving more than exponentially small success probability with PGM requires Ω(log N) hidden subgroup states at the critical density ν ≈ 1. 5) The PGM can efficiently distinguish DHSP states at all densities without sample complexity constraints. 6) Success probability for PGM does not depend on the number of states or problem density. 7) The optimality of PGM for DHSP is unproven, and its threshold behavior is unknown.

✓ Correct Answer:

The correct answer is 4) Achieving more than exponentially small success probability with PGM requires Ω(log N) hidden subgroup states at the critical density ν ≈ 1..

📚 Reference Text:

implement the measurement approxi- mately. In this case, an implementation of the measurement only implies a solution to the average case subset sum problem at density ν, which may be considerably easier. Conversely, to implement the measurement at density ν, it is suﬃcient to approximately quantum sample subset sum solutions at that density. The critical density ν∼1 for the success of the optimal measurement coincides with the critical density above which almost all subset sum in- stances are legal and below which almost all subsets have a distinct sum. OPTIMAL MEASUREMENTS FOR THE DIHEDRAL HSP 19 No eﬃcient algorithms are known for the subset sum problem at this crit- ical density. But for suﬃciently low or high density, the problem becomes tractable. For densities ν < 0.941, there is an eﬃcient algorithm assum- ing the ability to ﬁnd short vectors in lattices [5, 7, 18, 38], but this lattice problem seems to be diﬃcult. However, using known basis reduction algo- rithms, this approach can be used to eﬃciently solve subset sum problems with no computational assumptions for very low density, k < c√logNfor some constant c[18,38]. Since we require ν >1 for a solution to the dhsp , the high density regime is more interesting for our purposes. Until recently, the best known result was a poly( k)-time algorithm for the case k>cN for some constant c[6,19]. These results are not helpful since they yield algorithms whose running times are exponential in log N. Flaxman and Przydatek recently showed how to produce subset sum solutions in poly( k) time with k= 2O(√logN), using sim- ilar ideas as in Kuperberg’s quantum algorithm for the dhsp [16]. However, it is not immediately clear whether the Flaxman-Przydatek algorithm can be used to quantum sample (or even to randomly sample) from subset sum solutions, so it does not directly provide an implementation of the optimal measurement. It is not inconceivable that one could ﬁnd a quantum algorithm (or even a classical one) for the subset sum problem at still lower density, and thus ﬁnd an improved algorithm for the dhsp . Furthermore, we remark that our restriction of ﬁrst measuring xand then implementing the appropriate measurement conditional on x, while natural from a representation-theoretic viewpoint, is not necessarily the best way to implement the optimal measure- ment. A direct implementation of the measurement without ﬁrst measuring xcould in principle produce a quantum algorithm for the dhsp without solving the subset sum problem. 9.Discussion In this paper we have studied the optimal measurement for distinguish- ing dihedral hidden subgroup states for order two subgroups. Using a result of Holevo and Yuen, Kennedy, and Lax, we proved that the pretty good measurement is optimal for this problem. We showed that the success prob- ability of this measurement has a threshold around the critical density ν∼1, and in particular, that Ω(log N) hidden subgroup states are necessary for the measurement to succeed with more than an exponentially small success probability. We also demonstrated that the problem of

Question 587 multiple-choice

In the study of Abelian p-groups and their topological properties, the concept of pseudocompactness is closely linked to the algebraic structure and decomposition of these groups. Understanding the canonical presentations and associated cardinal parameters is essential for characterizing which groups admit certain topologies. Which of the following is both necessary and sufficient for an infinite Abelian p-group of bounded order to admit a pseudocompact group topology? 1) The group must be divisible and torsion-free. 2) Every element must have order exactly p. 3) The group must be countable and finitely generated. 4) The canonical presentation must use only finite cardinal numbers. 5) The socle must be trivial. 6) The group must have bounded order and its canonical presentation’s cardinal numbers must satisfy the cofinal admissibility condition. 7) There must exist a nontrivial compact subgroup.

✓ Correct Answer:

The correct answer is 6) The group must have bounded order and its canonical presentation’s cardinal numbers must satisfy the cofinal admissibility condition..

📚 Reference Text:

(cf. [18] (17.2);hintsto a proof of the second part are given in [18] (17.4) and [27] (§ 24). 3.16. Theorem. For every Abelian p-group G of bounded order, there exist an integer n > 0, distinct positive integer s r{ (l < / < n), and cardinal numbers yf (l < / < ri), such that n The presentation is unique up to isomorphism if the condition r± < r2 < · · * < rn is imposed. In this case the cardinal numbers yf are determinedby this rule: y{ is the rank of the quotient group Snpri~1G/SnpriG, with S the socle of G. D In what follows we refer to (with rv < r2 < ·' · < rn) s the canonical presentation of G. We say that G satisfies the cofinal admissibility condition if the cardinal numbers y,· (l < i < n) in the canonical presentation of G satisfy the cofinal admissibility condition of Definition 3.15. It is clear that if G has canonical presentation G = (J)"= ± @y. Z (pn) and G satisfies the cofinal admissibility condition, then yn is admissible and max{yf: l < / < n} is admissible. 3.17. Theorem. Lei p e P andlet G be an Abelianp-group. Then the following conditions are equivalent. (a) G admits a pseudocompact group topology. (b) G is of bounded order and G satisfies the cofinal admissibility condition. Proof. (a) => (b). That G is of bounded order is given by 1.3. Suppose now that G = (J)«= 4 ^j^ Ζ(ρΓί) with ^ < ... < ΓΛ but there is h < n such that every i with k < i < n has either yt < yk or y{ inadmissible. If k = l then | G \ is the inadmissible cardinal yl9 a contradiction, so we assume k>\. The continuous homomorphism χ -» prk~lx from G into G has kernel K such that G/K= 0"=k @yi2(pri~rk-1). Evidently G/K admits a pseudocompact group topology (the quotient topology), yet \GJK\ = £?=*}>,· is inadmissible. (b) => (a). We write G = φ?=ι @yiZ(pri) with rt < ··· < r„ and we argue by induction on n. When n = l we have G = @y.Z(/?ri) with y± admissible, and the required conclusion follows from 3.3. Bereitgestellt von | Technische Informationsbibliothek Hannover Angemeldet Heruntergeladen am | 27.03.18 14:50 Abelian torsion groups 333 Suppose the Statement true for n

Question 588 multiple-choice

Quantum algorithms utilize advanced techniques such as block-encoding and singular value transformation to efficiently process non-unitary matrices and extract information about eigenvalues. These methods underpin many breakthroughs in quantum linear algebra and eigenvalue estimation. Which statement correctly describes how block-encoding and singular value transformation can be used together in quantum algorithms for eigenvalue estimation? 1) Block-encoding is only applicable to strictly diagonal matrices, and singular value transformation cannot modify off-diagonal elements. 2) Block-encoding replaces the need for ancillary qubits in iterative phase estimation by performing all measurements coherently. 3) Singular value transformation enables classical error correction for Hadamard-tests without any quantum operations involved. 4) Block-encoding restricts quantum algorithms to probabilistic estimates unless all eigenvalues are integer multiples of π. 5) Block-encoding embeds a non-unitary matrix within a larger unitary, allowing quantum singular value transformation to apply polynomial functions to its singular values, thereby enabling modular manipulation of eigenvalue-related probabilities. 6) Singular value transformation requires that all input matrices be positive semi-definite and cannot be applied to block-encoded matrices. 7) Block-encoding and singular value transformation are only compatible in algorithms that use Grover's search as a subroutine.

✓ Correct Answer:

The correct answer is 5) Block-encoding embeds a non-unitary matrix within a larger unitary, allowing quantum singular value transformation to apply polynomial functions to its singular values, thereby enabling modular manipulation of eigenvalue-related probabilities..

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damage the input superposition and can no longer be considered coherent. Even worse, we show that any unitary quantum algorithm for estimation must perform a map in the form (2), and there always exist some values of λjwhere the neither of the ξj andζjare≈0. Therefore, theonlywaytogetanalgorithmthatperformsthedeterministic transformation (1) is to assume certain values of λjdo not appear. We refer to this as arounding promise, and show that if the rounding promise holds, then our algorithms perform the map (1). However, we also construct our algorithms in such a way that they give reasonable non-deterministic estimates as in (2) even when no rounding promise holds. In the following we give a brief outline of the method we use to construct the algo- rithms. They strongly resemble iterative phase estimation [Kit95], which works roughly like this: the estimate is computed one bit at a time, starting with the least signiﬁcant bit. A ‘Hadamard-test’ computes each bit with a decent success probability, which is then ampliﬁed to a high probability by taking the majority vote of many samples. This process could be made coherent naively by computing each sample into an ancilla, but this requires so many ancillae that any performance beneﬁt over the QFT- and median-based approach is lost. The key idea is to amplify without using any new ancillae. To manipulate the probabilities of the Hadamard-test, we use ‘block-encodings’. Block- encodings permit quantum computers to manipulate non-unitary matrices. In our case, the matrices’ eigenvalues encode our probabilities. A unitary matrix UAis a block-encoding ofAifAis in the top-left corner: UA=/bracketleftBigg A· · ·/bracketrightBigg (3) Block-encoded matrices can be manipulated in two ways. First, linear combinations of uni- taries[BCK15, CKS15] allow us to build block-encodings αA+βBgiven block-encodings ofAandB. Linear combinations of unitaries allow us to make block-encodings of Hamil- tonians presented as a sum of local terms, covering most practical applications. Second, singular value transformation [LC1610, LC1606, GSLW18] lets us apply certain polynomi- alsp(x)to the singular values of a block-encoded matrix A, using deg (p)many queries to controlled-UAand its inverse. Together, these two techniques permit the uniﬁcation of many quantum algorithms into a single framework. For an accessible introduction to these methods, along with a compre- hensive review of the most modern versions of these algorithms we refer to [MRTC21]. This work also presents the independent discovery of an algorithm very similar to our improved method of phase estimation, which is also sketched in [Chuang20]. Toobtainthe k’thbit( 0≤k

Question 589 multiple-choice

Topological quantum computation employs the mathematical framework of braid groups and knot polynomials to manipulate quantum information using anyons. Understanding the representations of braid groups is essential for constructing quantum gates in these systems. Which concept specifically enables the realization of braid group representations as physically implementable quantum gates in topological quantum computing platforms? 1) The Alexander polynomial derived from the Burau representation 2) The use of ordinary bosonic exchange statistics 3) The calculation of knot invariants using classical algorithms 4) The application of the Burau representation to non-abelian anyons 5) The existence of fermionic zero modes in superconductors 6) Localization of braid group representations 7) The decomposition of braid groups into cyclic groups

✓ Correct Answer:

The correct answer is 6) Localization of braid group representations.

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Title: Local unitary representations of the braid group and their applications to quantum computing Year: 2016 Paper ID: 106854e28fbc38dcd4346ef25e2d4e4f3a411bf3 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/106854e28fbc38dcd4346ef25e2d4e4f3a411bf3 Abstract: We provide an elementary introduction to topological quantum computation based on the Jones representation of the braid group. We first cover the Burau representation and Alexander polynomial. Then we discuss the Jones representation and Jones polynomial and their application to anyonic quantum computation. Finally we outline the approximation of the Jones polynomial and explicit localizations of braid group representations.

Question 590 multiple-choice

In quantum chemistry, the efficient and stable calculation of integrals involving Slater-type orbitals (STOs) is fundamental for electronic structure methods. Analytical techniques and mathematical operators are often employed to derive compact formulas that can be generalized to various atomic systems. Which feature most directly enables a derived formula for the Fourier transform of a product of two STOs to be applicable to atoms with arbitrary principal and angular momentum quantum numbers? 1) The use of Gaussian-type orbitals as a basis set 2) Restricting calculations to single-center integrals 3) Neglecting the electron-nucleus cusp condition 4) Limiting the method to fixed momentum values 5) Applying the Born-Oppenheimer approximation 6) Ignoring benchmark data and implementation details 7) Employing the shift-operator technique in the derivation

✓ Correct Answer:

The correct answer is 7) Employing the shift-operator technique in the derivation.

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Title: Efficient evaluation of the Fourier transform over products of Slater-type orbitals on different centers Year: 2008 Paper ID: 8276550de79b6e537e2060b3c4a7a17ffcdb79d8 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/8276550de79b6e537e2060b3c4a7a17ffcdb79d8 Abstract: Using the shift-operator technique, a compact formula for the Fourier transform of a product of two Slater-type orbitals located on different atomic centers is derived. The result is valid for arbitrary quantum numbers and was found to be numerically stable for a wide range of geometrical parameters and momenta. Details of the implementation are presented together with benchmark data for representative integrals. We also discuss the assets and drawbacks of alternative algorithms available and analyze the numerical efficiency of the new scheme.

Question 591 multiple-choice

Group Field Theory (GFT) is a quantum gravity approach that models spacetime as emerging from quantum fields defined on group manifolds. In cosmology, effective equations derived from GFT can describe universe evolution with quantum gravitational corrections. Which of the following features distinguishes the derivation of cosmological Friedmann equations from GFT when not relying on mean-field approximations? 1) Applicability to any quantum state, capturing quantum correlations beyond classical averages 2) Restriction to classical solutions with no quantum corrections 3) Necessity of fixed representation labels in all cases 4) Elimination of all singularity-resolving mechanisms 5) Dependence solely on Fock coherent states for semiclassical interpretation 6) Inability to include quartic interactions in the Hamiltonian 7) Exclusive use of numerical methods without analytical calculations

✓ Correct Answer:

The correct answer is 1) Applicability to any quantum state, capturing quantum correlations beyond classical averages.

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Title: Generalised effective cosmology from group field theory Year: 2019 Paper ID: c08000345b18786298c88ac9be21e2479bdac2f3 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c08000345b18786298c88ac9be21e2479bdac2f3 Abstract: We extend various recent results regarding the derivation of effective cosmological Friedmann equations from the dynamics of group field theory (GFT). Restricting ourselves to a single GFT field mode (or fixed values of Peter–Weyl representation labels), we first consider dynamics given by a quadratic Hamiltonian, which takes the form of a squeezing operator, and then add a quartic interaction that can be seen as a toy model for interactions in full GFT. Our derivation of effective Friedmann equations does not require a mean-field approximation; we mostly follow a general approach in which these equations in fact hold for any state. The resulting cosmological equations exhibit corrections to classical Friedmann dynamics similar to those of loop quantum cosmology, leading to generic singularity resolution, but also involve further state-dependent terms. We then specify these equations to various types of coherent states, such as Fock coherent states or Perelomov–Gilmore states based on the su(1, 1) structure of harmonic quantum cosmology. We compute relative uncertainties of volume and energy in these states, clarifying whether they can be interpreted as semiclassical. In the interacting case, both analytical and numerical approximations are used to obtain modified cosmological dynamics. Our results clarify how effective cosmological equations derived from GFT can provide reliable approximations to the full dynamics.

Question 592 multiple-choice

In quantum information theory and representation theory, distinguishing quantum states often involves analyzing probability distributions over combinatorial structures such as Young diagrams. Advanced techniques use bounds and measures like trace distance and total variation distance to assess the performance of quantum algorithms and statistical inference. Which mathematical result describes the Gaussian fluctuations of Young diagrams around their limiting shape under the Plancherel measure, thereby enabling precise bounds on asymmetry measures in high-dimensional settings? 1) The Peter-Weyl theorem 2) The Robinson-Schensted correspondence 3) The Central Limit Theorem for symmetric groups 4) The Littlewood-Richardson rule 5) The Markov inequality 6) Kerov’s fluctuation results 7) The Schur-Weyl duality

✓ Correct Answer:

The correct answer is 6) Kerov’s fluctuation results.

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ambient dimension, which is irrelevant by Lemma 5). Then (35) follows from the non-increase of trace distance under quantum operations. (It seems plausible that Lemma 8 should also hold for non-integer rwheneverrd1andrd2are integers, but we know of no proof of this claim.) Now we turn to the proof of Theorem 3. 10 Proof of Theorem 3. The case when k≤dis the simple one. Using ﬁrst the triangle inequality, and then Lemma 6, we ﬁnd /bardblSchur(k,d)−Schur(k,d/r)/bardbl1 ≤ /bardblSchur(k,d)−Planch(k)/bardbl1+/bardblSchur(k,d/r)−Planch(k)/bardbl1(37) ≤√ 2k+kr d≤O(kr/d). (38) This establishes the Ω( d/r) lower bound on the number of copies. Now, for any choice of dandr, we would like to show that k=d/rcopies is suﬃcient to obtain/bardblSchur(k,d)−Schur(k,d/r)/bardbl1≥Ω(1). The achievability part of Theorem 3 will then follow from the same sort of ampliﬁcation argument we used in the pro of of Corollary 7. Choose νto be the smallest positive integer satisfying rν≥72√ 2; note that ν≤7. Then we use both the upper and lower bounds of Lemma 6 to ﬁnd /vextenddouble/vextenddouble/vextenddouble/vextenddoubleSchur/parenleftbiggd r,d r/parenrightbigg −Planch/parenleftbiggd r/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble 1≥1 36(39) /vextenddouble/vextenddouble/vextenddouble/vextenddoubleSchur/parenleftbiggd r,drν−1/parenrightbigg −Planch/parenleftbiggd r/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble 1≤√ 2 rν≤1 72. (40) Next we abbreviate S k,d:= Schur(k,d) and P k:= Planch(k) and use the triangle inequality to bound/vextenddouble/vextenddouble/vextenddoubleSd r,d r−Sd r,d/vextenddouble/vextenddouble/vextenddouble 1+/vextenddouble/vextenddouble/vextenddoubleSd r,d−Sd r,dr/vextenddouble/vextenddouble/vextenddouble 1+···+/vextenddouble/vextenddouble/vextenddoubleSd r,drν−2−Sd r,drν−1/vextenddouble/vextenddouble/vextenddouble 1(41) ≥/vextenddouble/vextenddouble/vextenddoubleSd r,d r−Pd r/vextenddouble/vextenddouble/vextenddouble 1−/vextenddouble/vextenddouble/vextenddoubleSd r,drν−1−Pd r/vextenddouble/vextenddouble/vextenddouble 1≥1 72. According to Lemma 8, the ﬁrst term on the left hand side of (41) is t he largest. Thus, /vextenddouble/vextenddouble/vextenddouble/vextenddoubleSchur/parenleftbiggd r,d r/parenrightbigg −Schur/parenleftbiggd r,d/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble 1≥1 72ν≥1 504. (42) This concludes the proof of Theorem 3. Our only remaining task is to prove Lemma 6, establishing tight bounds on the distance be- tween Schur( k,d) and Planch( k). Here our primary tools are explicit formulas for the Plancherel averages of certain functions of Young diagrams over Sk. Proof of Lemma 6. Let∆k,ddenotethetotalvariationdistancebetweenSchur( k,d)andPlanch( k). According to (26) and (27), ∆ k,d:=/bardblSchur(k,d)−Planch(k)/bardbl1can be written ∆k,d=E λ⊢k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/productdisplay (i,j)∈λ/parenleftbigg 1+j−i d/parenrightbigg −1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(43) where the expectation is taken over Planch( k). We will nowderiveboth upper and lowerbounds on ∆ k,dfrom(43), startingwith the O(k/d) upper bound. By the Cauchy-Schwarz inequality (or concavity of t he square root and Jensen’s inequality), ∆2 k,d≤E λ⊢k /productdisplay (i,j)∈λ/parenleftbigg 1+j−i d/parenrightbigg −1 2 (44) =E λ⊢k /productdisplay (i,j)∈λ/parenleftbigg 1+j−i d/parenrightbigg2 −2/productdisplay (i,j)∈λ/parenleftbigg 1+j−i d/parenrightbigg +1  (45) =E λ⊢k/productdisplay (i,j)∈λ/parenleftbigg 1+j−i d/parenrightbigg2 −1, (46) 11 where in the last line we have used linearity of expectation and the fac t that Schur( k,d) is normalized. Now using the inequality 1+ x≤ex, we have ∆2 k,d≤E λ⊢kexp 2/summationdisplay (i,j)∈λj−i d −1 (47) =∞/summationdisplay m=12m m!dmE λ⊢kv1(λ)m(48) where v1(λ) :=/summationdisplay (i,j)∈λ(j−i) (49) can be viewed as a measure of the asymmetry of λ. Intuitively, we expect this to be small since typical Plancherel-distr ibuted Young diagrams areclosetoacertainlimitingshapethatissymmetric, andinwhichthele ngthofthelongestrow, λ1, is close to 2√ k[38,52] (in fact, λ1is exponentially tightly concentrated around this value [7]). Our proof will use results obtained by Kerov in the course of des cribing the asymptotically Gaussian ﬂuctuations about the limiting shape [35]. It turns out that v1can be expressed exactly in terms

Question 593 multiple-choice

In the study of topological groups, properties like pseudocompactness, total boundedness, and group completions are central to understanding their algebraic and topological structure. The relationship between these properties often leads to fundamental classification results. Which of the following is always true for a pseudocompact topological group? 1) It is necessarily compact. 2) Its Pontrjagin dual is finite. 3) It is totally bounded. 4) It must be a torsion group of bounded order. 5) Its group topology has weight strictly greater than its cardinality. 6) It cannot be densely embedded in any compact group. 7) The product with any totally bounded group is never pseudocompact.

✓ Correct Answer:

The correct answer is 3) It is totally bounded..

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satisfies χ(0, G) = χ(0, ΛΓ), so (a) gives · d(K) < X(G) · d(G) < vt;(G) · w(G) = w(G) , s required. D 1.2. Theorem, (a) The continuous Image of a pseudocompact space is pseudocompact. (b) Lei be an infinite pseudocompact Abelian group. Then there is a pseudocompact group topology ^U on G such that tfl ^ y and w(G, <%) < | G |. Proof. (a) is obvious, s is (b) in case w(G9^) < \G\. We assume then that w (G, F] > | G | . According to l . l (b) and [23] (24. 1 5, 22. 1 7), the Pontrjagin dual £ of the compact completion K of (G,^) satisfies (1) |£| = wK= w(G,^) and (2) £ separates points of K. Clearly there is a subgroup H of R such that | H \ < \ G \ and { h \ G : h e H} separates points of G. The (Hausdorff) topology m on G induced by {h | G : h e H} satisfies <% ^3?~ and w (G, *) < | H \ < \ G \ , and * is pseudocompact by part (a). D The proof of 1.2(b) is a special case of a construction given in [10]. It follows from [10], [23] (24.15) and 1.1 that w(G,^) = \H\. For compact Abelian groups, 1.3 is a consequence of the Baire category theorem noted in [23] (25.9); the generalization to the pseudocompact case, given in [9] (7.4), is routine. 1.3. Theorem ([9]). Every pseudocompact Abelian torsion group is of bounded order. D A topological group G = is said to be totally bounded (by some authors: pre-compacf) if for every non-empty U e 3~ there is a finite F e G such that G = FU. Bereitgestellt von | Technische Informationsbibliothek Hannover Angemeldet Heruntergeladen am | 27.03.18 14:50 326 W. W. Comfort, D. Remus It is a theorem of Weil [31] that there is a compact group G, unique in an obvious sense, in which G is dense; we call G the Weil completion of G. 1.4. Theorem ([11]). (a) Every pseudocompact group is totally bounded; (b) a totally bounded group G is pseudocompact ifand only ifG is G -dense in its Weil completion; and (c) if{G{ : i e 1} is a sei of pseudocompact groups, then Π *<=/ G, is pseudocompact. D The expression G « H indicates that G and H are isomorphic groups; the ambient topological structures (if any) are ignored. The symbols a, /?, y, and κ denote cardinal numbers, and ω is the least infinite cardinal. The symbol ξ denotes an ordinal number. For α > ω we write log α = min [ : 2 > a}. Clearly α = log α if and only if every β < α satisfies 2 < a; such a cardinal α is called a strong limit cardinal. The symbols N and [P denote respectively the set of positive integers and

Question 594 multiple-choice

In quantum physics and representation theory, the tensor product of group representations determines the possible ways subsystems can combine and the existence of invariant states. The structure and multiplicity of the trivial representation in these tensor products are crucial for applications such as quantum error correction and protected qubit formation. For three irreducible representations of a finite group G with dimensions (d₁, d₂, d₃), which condition must be satisfied for the tensor product of all three representations to contain the trivial (identity) representation? 1) The sum of the three dimensions equals the dimension of the group, d₁ + d₂ + d₃ = |G|. 2) The tensor product of any two contains the identity representation. 3) Each representation is self-dual. 4) All three representations are of equal dimension. 5) The tensor product of any two contains both the identity and the third representation. 6) The tensor product of any two representations contains the dual of the third. 7) The product of the three dimensions is less than the order of the group.

✓ Correct Answer:

The correct answer is 6) The tensor product of any two representations contains the dual of the third..

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the tensor prod uct of three representations at each vertex includes the identity. In some cases, when the tens or product of two represen- tations at a vertex contains multiple copies of the dual of the third r epresentation, there will be multiple independent ways to couple the three representations t o a trivial representation. In this case, we need to add an additional discrete label on the vert ex to indicate which structure we are choosing. Alternatively, we can work in the ungauged theory and add a term to the Hamiltonian which associates a large energy to states which are not in the trivial representation of H. For example, with H= SU(2), we can add E0J2for largeE0. If there are no other terms in the Hamiltonian, the model will have a ground state degeneracy eq ual to the multiplicity of the trivial representation in the tensor product of represent ationsRiassociated with the elementary subsystems. Since these systems have an energy gap to other states for which the local subsystems are not maximally mixed, quantum systems of t his type (if they can be realized) may have applications for quantum error correction or th e experimental realization of robust qubits/qudits. 3.3 Implications for representation theory The construction of Corollary 3.3 combined with our Theorem 1.1 on th e existence of LME states leads to an interesting implication for the representation th eory of ﬁnite and/or com- pact groups: Proposition 3.5. For dimensions (d1,...,dn), there can exist a (ﬁnite or compact) group Gwith unitary irreducible representations R1,R2, ...Rnof dimensions (d1,...,dn)whose tensor product contains the trivial representation only if the conditions in Theorem 1.1 for the existence of LME states are met. In the special case n= 3, the tensor product of three irreducible representations R1,R2, and 21 R3contains the trivial representation if and only if the tensor produc t of any two of the representations contains the dual of the third. Thus, we have Corollary 3.6. For dimensions (d1,d2,d3), there can exist a (ﬁnite or compact) group G with unitary irreducible representations of dimension d1andd2whose tensor product contains an irreducible representation of dimension d3only if d1d2d3>N(d2 1,d2 2,d2 3) (3.14) whereNis deﬁned in (1.5). It is interesting to ask whether the necessary conditions in Propos ition 3.5 and Corollary 3.6 might also be suﬃcient. Forn= 3, the ﬁrst ﬁve examples from section 3.1 show that suﬃciency hold s forn= 2 and forn= 3 withd1= 2. Using the GAP database of ﬁnite groups, it is possible to search f or groups with representations of particular dimensions and print the character tables to check whether the tensor product of two representation contains a th ird. Using this approach, we have also checked that for each case of the form ( d1,d2,d3) = (3,3,d3) admitting LME states (i.e. 3≤d3≤9), we can ﬁnd a group and irreducible representations of dimension s (3,3,d3) whose tensor product contains the trivial representation. Howe ver, this approach runs out of steam quickly due to the ﬁnite size of the

Question 595 multiple-choice

Quantum encoding of lattice structures is foundational for quantum algorithms in cryptography and computational number theory. One key property of such encodings is how the quantum states representing lattices respond to changes in the underlying lattices. Which property must a quantum encoding of lattices satisfy to ensure that small changes in the lattice result in small changes in the corresponding quantum state, and that very different lattices yield nearly orthogonal quantum states? 1) Lipschitz continuity of the encoding map 2) Unitary invariance of the encoding basis 3) Gaussian-weighted measurement of superpositions 4) Maximum likelihood estimation of lattice parameters 5) Strictly monotonic mapping of norms 6) Isometric embedding of ideals 7) Orthogonal projection onto lattice cosets

✓ Correct Answer:

The correct answer is 1) Lipschitz continuity of the encoding map.

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for BC. (c) Truncate the precision to qbits. Theorem 4.4. There is an algorithm that on input t∈ Qnandδcomputes a basis ˆb1,..., ˆbnthat isδ-close to a basis for etO, has bounded size, and runs in time polynomial inlog ∆,n,log||t||andlog 1/δ . Proof. We analyze the complexity of the algorithm given above. By Thiel [Thi95] we can take the initial precision as high as we need in Step 3 and Step 4. The main step in the algorithm consists of multiplying ideals BandC, so it is enough to show that each multiplication step BC can be done eﬃciently, and that we can bound the loss of precision. By Lemma 4.1 we can compute n2generators for BCfromngenerators for BandnfromCand bound the loss of precision. By Theorem 4.3, we can compute a basis approximating b1,...,bnforBCin polynomial time, with ||bj|| ≤(√n3+ 2)2n2−1 2·λj(BC). The loss of precision for the squaring step is bounded in Theorem 4.3. Therefore we may choose the initial number of precision bits qto be high enough to satisfy δat the end. 5. THE QUANTUM REPRESENTATION OF LATTICES The representation of a lattice by a basis is not unique, which makes it unsuitable for use in an algorithm that deals with quantum superpositions of lattices. To avoid this issue, we will represent each lattice by a unique quantum state, 298 namely, a superposition of the lattice points with certain weights. Let us ﬁrst discuss some desired properties of such a representation. We want a small deformation of the lattice to result in a small change in the quantum state, whereas substantially diﬀerent lattices should be mapped to almost orthogonal vectors. These requirements can be formalized as follows. Definition 5.1. Letdist(x,y)denote the distance bet- ween two points in a metric space X, and let Hbe some Hilbert space. A map f:X→ H is called an (a,r,ε) quan- tum encoding if the following conditions are met: 1./angbracketleftf(x)|f (x)/angbracketright= 1for allx∈X; 2.‚‚|f(x)/angbracketright − |f (y)/angbracketright‚‚/lessorequalslanta·dist(x,y)for allx,y∈X; 3. If dist(x,y)/greaterorequalslantr, then˛˛/angbracketleftf(x)|f (y)/angbracketright˛˛/lessorequalslantε. Given such an encoding, the vector |f(x)/angbracketright is called the sig- nature state forx. The number ain condition 2 is called a Lipschitz constant of the function f. WhenX=Rn(or, more generally, when Xis a Riemannian manifold), fcan be approximated by a smooth function to an arbitrary precision in the sup-norm at cost of an arbitrary small parameter change in the above deﬁnition. For smooth functions, ais simply an upper bound on the ﬁrst derivative of f. Lemma 5.2. Letf1:X1→ H 1andf2:X2→ H 2take values in unit vectors and satisfy the Lipschitz condition with constantsa1anda2, respectively. a) IfX1=X2=X, then the function f1⊗f2:X→ H1⊗ H 2is(a1+a2)-Lipschitz. b) IfX1andX2are Euclidean spaces, then the function g:X1×X2→ H 1⊗H 2deﬁned asg(x1,x2) =f1(x1)⊗ f2(x2)isa-Lipschitz, where a=pa2 1+a2 2. Example 5.3 (Straddle encoding). A representati- on of real numbers by quantum superposition of integers can be deﬁned as follows: ˛˛strν(x)¸ = cos`π 2t´ |k/angbracketright+ sin`π 2t´ |k+ 1/angbracketright, wherek=⌊x/ν⌋, t =x/ν−k. The map strν:R→CZis a`π 2ν,2ν,0´ quantum encoding. Applying this map to each coordinate on an n-dimensional real vector, we

Question 596 multiple-choice

Quantum computing has enabled new algorithms for arithmetic operations that can outperform classical methods by leveraging unique properties like quantum parallelism and the quantum Fourier transform (QFT). Efficient resource usage is critical, as quantum gates and qubits are expensive in current quantum hardware. Which algorithm is specifically optimized to add a classical constant to a quantum register using fewer operations and qubits than general-purpose register-by-register QFT-based addition methods? 1) Grover's search algorithm 2) Simon's algorithm 3) Quantum phase estimation algorithm 4) Standard QFT-based register-by-register addition 5) Quantum walk algorithm 6) Classical ripple-carry adder 7) Specialized QFT-based register-by-constant addition algorithm

✓ Correct Answer:

The correct answer is 7) Specialized QFT-based register-by-constant addition algorithm.

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Title: Quantum Algorithm based on Quantum Fourier Transform for Register-by-Constant Addition Year: 2022 Paper ID: da4762ad751bdc38d2aef72d46ff2c78f846990f Source: semantic-scholar URL: https://www.semanticscholar.org/paper/da4762ad751bdc38d2aef72d46ff2c78f846990f Abstract: Since Shor's proposition of the method for factoring products of prime numbers using quantum computing, there has been a quest to implement efficient quantum arithmetic algorithms. These algorithms are capable of applying arithmetic operations simultaneously on large sets of values using quantum parallelism. Draper proposed an addition algorithm based on the quantum Fourier transform whose operands are two quantum registers, which I refer to as register-by-register addition. However, for cases where there is the need to be added a constant value to a target register, Draper's algorithm is more complex than necessary in terms of number of operations and number of qubits used. In this paper, I present a more efficient addition algorithm than Draper's for cases where there needs to be added just a constant to a target register.

Question 597 multiple-choice

In quantum computing, the Hidden Subgroup Problem (HSP) is central to many algorithms, with the efficiency of these algorithms heavily influenced by the structure of the underlying group and the quantum measurements used. Entanglement across multiple quantum registers plays a critical role, especially for non-abelian groups. For efficient quantum algorithms solving the HSP in direct product groups of the form Gn, what is the minimum number of coset state registers across which entangled measurements must be performed? 1) Ω(log n) 2) Ω(n^2) 3) Ω(n) 4) Ω(√n) 5) Ω(log|G|) 6) n + 1 7) Ω(1)

✓ Correct Answer:

The correct answer is 3) Ω(n).

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coset state s that solves the hidden subgroup problem in G= PSL(2,Fq)in polynomial time has to make joint measurements on k= Ω(log|G|) = Ω(q)coset states. 4.3 Special andgeneral lineargroups Corollary 16. Anyalgorithm solving theHSPin SL(2,Fq)orGL(2,Fq)efﬁciently usingcoset statesneeds tomake measurements entangled across k= Ω(logq)registers. Proof.ByCorollary 15any algorithm solving theHSPin PSL(2,Fq)efﬁciently using coset states needs to makemeasurements entangled across k= Ω(logq)registers. Thestatement nowfollowsfromLemma1by using the facts that PSL(2,Fq)∼=SL(2,Fq)/ζ(SL(2,Fq))and thatSL(2,Fq)≤GL(2,Fq). Corollary 17. Any algorithm solving the HSP in GL(n,Fpm)efﬁciently using coset states needs to make measurements entangled across k= Ω(n(mlogp+logn))registers. Proof.SinceGL(n,Fpm)contains all n×npermutation matrices, a lower bound of k= Ω(nlogn)fol- lows from Corollary 14 and Lemma 1. Also, we can use the embedd ing ofGL(2,Fpnm)≤GL(2n,Fpm) via/parenleftbigga b c d/parenrightbigg /ma√st≀→/parenleftbiggMaMb McMd/parenrightbigg ,where for each x∈Fpnmthe matrixMx∈GL(n,Fpm)realizes mul- tiplication by xwith respect to a ﬁxed basis of FpnmoverFpm. Hence by Lemma 1 we obtain that for the HSP inGL(2n,Fpm)at least as much entanglement is necessary as in case of GL(2,Fpnm). The latter has been bounded by Ω(nmlogp)inCorollary 16. 4.4 Direct products ofthe form Gn In this section we show that for a large class of ﬁnite groups G, efﬁcient algorithms for HSP for direct products of the form Gn, wheren≥1, require entangled measurements on at least k= Ω(n)coset states. LetGbe a ﬁnite group and let /hatwideG={σ1,...,σ m}denote the irreducible representations of G. Recall that the centralizer C(g)of an element g∈Gis the subgroup C(g) :={c∈G:cg=gc}. Lethbe an involution in G, andletσ∈/hatwideG. Theneither |χσ(h)|=dσor|χσ(h)| dσ<1−2|C(h)| |G|holds [Gal94]. Wedeﬁne ε:= (1−2|C(h)| |G|)t,wheret=t(n)isafunction of ntobe determined later. Theirreps of Gn,wheren≥1, aregiven by σ:=σ1⊗...⊗σn,whereσi∈/hatwideG. WeletΛ :={σ∈/hatwideG: |χσ(h)|=dσ},λ:=/summationtext σ∈Λd2 σ, andµ:=/summationtext σ∈/hatwideG\Λd2 σ=|G|−λ. Thefollowing property of the set Sε:=/braceleftbigg σ∈/hatwideGn:|χσ(h,...,h)| dσ≥ε/bracerightbigg holdsforourchoiceoftheparameter ε: ifσ∈Sεthennecessarily atleast n−tpositionsσihavetobefrom Λ, i.e., have to satisfy |χσi(h)|=dσi. Indeed, otherwise we would have more than tpositionsσjin each of which|χσj(h)| dσj≤1−2|C(h)| |G|, making the product less than ε. We next give an estimate for the quantity /summationtext σ∈Sεd2 σappearing in Theorem 2. For that we require the following lem ma for estimating the tail of the binomial distribution. 18 Lemma18. Letα,β >0, letn≥1, and lett=n/c, wherec>α+β β. Then n/summationdisplay ℓ=n−t/parenleftbiggn ℓ/parenrightbigg αℓβn−ℓ≤/parenleftBigg α/parenleftbiggce(α+β) α/parenrightbigg1/c/parenrightBiggn . Proof.Wehave that n/summationdisplay ℓ=n−t/parenleftbiggn ℓ/parenrightbigg αℓβn−ℓ= (α+β)nn/summationdisplay ℓ=n−t/parenleftbiggn ℓ/parenrightbigg/parenleftbiggα α+β/parenrightbiggℓ/parenleftbiggβ α+β/parenrightbiggn−ℓ ≤(α+β)n/parenleftbiggn n−t/parenrightbigg/parenleftbiggα α+β/parenrightbiggn−t =αn/parenleftbiggn t/parenrightbigg/parenleftbiggα+β α/parenrightbiggt ≤αn/parenleftbiggne(α+β) tα/parenrightbiggt =/parenleftBigg α/parenleftbiggce(α+β) α/parenrightbigg1/c/parenrightBiggn , where the ﬁrst inequality follows from the union bound on pro babilities and the second one from/parenleftbign t/parenrightbig ≤/parenleftbigne t/parenrightbigt. Supposeweﬁx ℓ≥n−tlocations forputtinginirrepsfrom Λ. Thecontribution ofthisconﬁguration to/summationtext σ∈Sεd2 σisthe sum of products of squares of dimensions of ℓirreps from Λandn−ℓirreps from/hatwideG\Λ, which simpliﬁes to λℓµn−ℓ. Lettingα:=λ,β:=µ, andt=n/c, with some constant cto be determined later, weobtain the following bound from Lemma18: /summationdisplay σ∈Sεd2 σ≤n/summationdisplay ℓ=n−t/parenleftbiggn ℓ/parenrightbigg λℓµn−ℓ≤λn/parenleftBigg/parenleftbiggce|G| λ/parenrightbigg1/c/parenrightBiggn . Hence, for any given κ >0we can ﬁnd a constant c >0such that/summationtext σ∈Sεd2 σ≤λn(1+κ)nholds for alln≥c. Note that the same upper bound applies to/summationtext σ∈Sεdσ|χσ(h,...,h)|. Also, observe that /summationtext ρ∈/hatwideGndρ=/parenleftBig/summationtext ρ∈/hatwideGdρ/parenrightBign . Now,wecan bound the parameter δ1used inTheorem 2: δ1≤ε+1 |G|n/parenleftBigg/summationdisplay σ∈Sεdσ|χσ(h,...,h)|/parenrightBigg /summationdisplay ρ∈/hatwideGndρ  ≤/parenleftBigg/parenleftbigg 1−2|C(h)| |G|/parenrightbigg1/c/parenrightBiggn +λn(1+κ)n |G|n /summationdisplay ρ∈/hatwideGdρ n . For the following we make the

Question 598 multiple-choice

In representation theory, the Gelfand-Tsetlin basis provides a systematic way to label basis vectors for irreducible representations of unitary groups U(d), utilizing combinatorial structures such as Young tableaux. Operators associated with subgroups play a key role in understanding how these representations decompose along chains of unitary subgroups. When restricting an irreducible representation Vλ of U(d) to the subgroup U(d−1), which combinatorial operation on the corresponding semistandard Young tableau yields the labels µ for the U(d−1) irreducible components? 1) Adding a vertical strip labeled d 2) Transposing the entire tableau 3) Removing a horizontal strip labeled d 4) Filling the tableau with consecutive integers starting from 1 5) Swapping all instances of label d with label 1 6) Doubling the number of boxes in the first row 7) Rotating the tableau by 90 degrees

✓ Correct Answer:

The correct answer is 3) Removing a horizontal strip labeled d.

📚 Reference Text:

possible subsets of nletters from the alphabet {1,...,d}, the above set of vectors form the Gelfand-Tsetlin basis. Proof. In order to prove this, we will verify that this basis is the subgroup adapted basis of the tower of subgroupsU(d)⊃U(d−1)⊃···⊃U(1). Equivalently, we will show that this basis makes the following decomposition block diagonal. Vλ↓U(d) U(d−1)/similarequal/circleplusdisplay µVµ, (4.3) whereVλis the irrep of the unitary group U(d) that corresponds to the shape λandVµis an irrep of U(d−1) that corresponds to the shape µ. The direct sum runs over all the shapes µthat diﬀer from λby a horizontal strip. The shapes µare all obtained from λby considering the various SSYTs that constitute the basis of 17 the irrep space of the unitary group and removing the horizontal strips labelled d. This corresponds to all unitaries in U(d) that ﬁx the label d. The (unnormalized) basis states can be written in the form |λ,i,j/angbracketright=/radicalBigg dλ |G|/|H|/summationdisplay g[λ(gH)]i,j|gH/angbracketright, (4.4) whereGis the symmetric group Sn,His a Young subgroup of the form ( Sµ1×Sµ2×...Sµk) for somek≤d. To avoid clutter, we replace YTabove byH. We will call the tuple µ, which can be permuted to a valid Young diagram, the content corresponding to Hsince it is the content of the SSYTs that we obtain. The sum overgruns over elements of the transversal of HinG. With the embedding of the permutation module into the group algebra of Sndone in the previous section, we can interpret |H/angbracketrightas the computational basis state|11...1 2...2...k...k/angbracketrightwhere there are µ11s,µ22s etc. The state |gH/angbracketrightas the basis state which is permuted according to the transversal element g. In the proof of theorem 1, it was also shown the action of Hon the SYT jleads to a SSYT. In order to show the above block diagonalization, we need to show that the action of U(d−1) leaves the horizontal strip labeled dintact. The action of the Lie algebra of U(d−1) is generated by the operators J(l) 0,J(l) +andJ(l) −forl= 1,2,...,d−2. These operators were deﬁned in section 2.5. We show that these operators acting on any |λ,i,j/angbracketrightleave the horizontal strip labeled dintact. In other words, they give superpositions of states of the form |λ,i,j/prime/angbracketright, wherej/primeis a SSYT of shape λand whose horizontal strip labeled dis the same as the one in j. To show this, let us look at the action of J(l) +for somelin [d−2]. The action of J(l) −is the Hermitian conjugate and so it suﬃces to look at J(l) +. We work with the unnormalized states given above. J(l) +|λ,i,j/angbracketright=J(l) +/radicalBigg dλ |G|/|H|/summationdisplay g[λ(gH)]i,j|gH/angbracketright=/radicalBigg dλ |G|/|H|/summationdisplay g/primecg/prime|g/primeH/prime/angbracketright, (4.5) whereg/primeruns over the transversal of H/primeinGandH/primeis the Young subgroup that diﬀers from Hin the number ofls andl+ 1s with one more land one less l+ 1 thanHi.e., if the content corresponding to H isµ= (µ1,...,µk), then that of H/primeisµ/prime= (µ1,...,µl+ 1,µl+1−1,...,µk). The coeﬃcient cg/primecan be calculated to be cg/prime= [λ(g/primeJH)]i,j, (4.6) where the operator Jis deﬁned as J=σl−1+1/summationdisplay m=σl+1(σl+ 1,m), (4.7) whereσl=µ1+···+µland similarly σl+1. This shows that the new SSYT is a sum SSYTs obtained by taking a SYT

Question 599 multiple-choice

The Hidden Subgroup Problem (HSP) is a central challenge in computational group theory and quantum computing, concerning the identification of hidden subgroups within finite groups using function queries. Algorithmic approaches to HSP differ in their efficiency depending on whether the group is abelian or non-abelian and whether randomness is permitted. Which statement correctly characterizes the query complexity achieved by the latest deterministic algorithm for solving the HSP in abelian groups when the hidden subgroup has order m within a group of order n? 1) It matches the randomized algorithm's complexity, yielding $\Order(n/m)$ queries. 2) It provides a strictly exponential query complexity in both n and m. 3) It achieves $\Order(\log n)$ query complexity independent of m. 4) It achieves $\Order(\sqrt{n/m})$ query complexity, matching the optimal randomized approach. 5) It requires $\Order(n)$ queries for all cases regardless of subgroup size. 6) It matches the non-abelian case, requiring $\Order(\sqrt{(n/m)\log(n/m)})$ queries. 7) It always exceeds $\Order(\sqrt{n})$ query complexity even for abelian groups.

✓ Correct Answer:

The correct answer is 4) It achieves $\Order(\sqrt{n/m})$ query complexity, matching the optimal randomized approach..

📚 Reference Text:

Title: Deterministic Algorithms for the Hidden Subgroup Problem Year: 2021 Paper ID: 4b987daff67ecf1a04edb48240da649f860b9082 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/4b987daff67ecf1a04edb48240da649f860b9082 Abstract: We present deterministic algorithms for the Hidden Subgroup Problem. The first algorithm, for abelian groups, achieves the same asymptotic worst-case query complexity as the optimal randomized algorithm, namely~$ \Order(\sqrt{ n}\, )$, where~$n$ is the order of the group. The analogous algorithm for non-abelian groups comes within a~$\sqrt{ \log n}$ factor of the optimal randomized query complexity. The best known randomized algorithm for the Hidden Subgroup Problem has \emph{expected\/} query complexity that is sensitive to the input, namely~$ \Order(\sqrt{ n/m}\, )$, where~$m$ is the order of the hidden subgroup. In the first version of this article~\cite[Sec.~5]{Nayak21-hsp-classical}, we asked if there is a deterministic algorithm whose query complexity has a similar dependence on the order of the hidden subgroup. Prompted by this question, Ye and Li~\cite{YL21-hsp-classical} present deterministic algorithms for \emph{abelian\/} groups which solve the problem with~$ \Order(\sqrt{ n/m }\, )$ queries, and find the hidden subgroup with~$ \Order( \sqrt{ n (\log m) / m} + \log m ) $ queries. Moreover, they exhibit instances which show that in general, the deterministic query complexity of the problem may be~$\order(\sqrt{ n/m } \,)$, and that of \emph{finding\/} the entire subgroup may also be~$\order(\sqrt{ n/m } \,)$ or even~$\upomega(\sqrt{ n/m } \,) $.}We present a different deterministic algorithm for the Hidden Subgroup Problem that also has query complexity~$ \Order(\sqrt{ n/m }\, )$ for abelian groups. The algorithm is arguably simpler. Moreover, it works for non-abelian groups, and has query complexity~$ \Order(\sqrt{ (n/m) \log (n/m) }\,) $ for a large class of instances, such as those over supersolvable groups. We build on this to design deterministic algorithms to find the hidden subgroup for all abelian and some non-abelian instances, at the cost of a~$\log m$ multiplicative factor increase in the query complexity.

Question 600 multiple-choice

Quantum algorithms for group-theoretic problems often exploit subgroup structure using techniques from both group theory and quantum information. The quantum Fourier transform plays a pivotal role in manipulating quantum states associated with subgroups, especially when central subgroups are involved. In quantum algorithms addressing hidden subgroup problems, what key property allows the quantum Fourier transform over a central subgroup L to simplify the disentanglement of coset superpositions? 1) L being a maximal subgroup of G 2) L being a normal but not central subgroup 3) L having order coprime to that of G 4) L consisting of elements that commute with every element of G 5) L being isomorphic to the symmetric group S_n 6) L forming a cyclic subgroup of G 7) L acting freely on G

✓ Correct Answer:

The correct answer is 4) L consisting of elements that commute with every element of G.

📚 Reference Text:

⎠. Initially H0={I}and K=G. The commutator subgroup K/prime=G/prime consists of the matrices of the form⎛ ⎝1a13 1 1⎞ ⎠andK/K/prime∼=Z2 9, not elementary. Then Kis replaced by the group of matrices⎛ ⎝13a12a13 13 a23 1⎞ ⎠. (Here and in each step below, the notation aijshould be considered as independent of those used in the previous step(s).) It turns out that this Kis abelian ( K/prime={I}), but not elementary. It gets replaced by the group of matrices of the form⎛ ⎝13a123a13 13 a23 1⎞ ⎠, which is isomorphic toZ3 3and H∩Kconsists of the matrices of the form⎛ ⎝13a12 1 1⎞ ⎠and H0becomes this subgroup. The next Kwill be the normalizer of this H0, which is the group of matrices⎛ ⎝1a12a13 13a23 1⎞ ⎠, whose commutator subgroup K/primeconsists of the matrices of the form⎛ ⎝1a13 1 1⎞ ⎠and K/K/primeis isomorphic to Z3 3,s o K, and hence also K/H0, is semi-elementary. Therefore HSMC will ﬁnd H/H0inK/H0and the procedure terminates. 4 The main conversion LetLbe a subgroup of the center of Gisomorphic to Zn pwhere pis a prime. Then L is a normal subgroup of G. Our aim is to convert a copy of the subgroup state /Xi1G,H to a copy of /Xi1G/L,HL/L. In the light of the second conversion (“pushing”) described in Sect. 2.4, one could do it by converting ﬁrst to a copy of /Xi1G,HL. To this end, it would be desirable to have a procedure that converts the coset superpo- sition|aH/angbracketrightto|aHL/angbracketright. A possible approach would be computing |L/angbracketright=1√|L|/summationtext z∈L|z/angbracketright in a new register, multiplying |aH/angbracketrightwith it to obtain1√|HL|/summationtext z∈L/summationtext x∈H|azx/angbracketright|z/angbracketrightand 123 14 Page 12 of 23 M. Imran, G. Ivanyos then trying to “disentangle” |z/angbracketrightfrom|azx/angbracketright. The quantum Fourier transform of L almost does this job: If we apply it to the second register, we obtain the state 1√|L|/summationdisplay y∈Lω(y,z) √|HL|/summationdisplay z∈L/summationdisplay x∈H|azx/angbracketright|y/angbracketright, where ω=e2πi pand by (,)we denote the standard scalar product of Lmodulo p. For y∈L, let us denote by Pythe linear transformation of CGmapping |x/angbracketrightto 1√|L|/summationtext z∈Lω(y,z)|xz/angbracketright. With this notation, the state we have can be rewritten as 1√|L|/summationdisplay y∈L|Py(aH)/angbracketright|y/angbracketright. Using the assumption that Lis in the center of G, a direct calculation shows that for every x1,x2∈G,w eh a v e |Py(x1x2)/angbracketright=| x1Py(x2)/angbracketright=|(Py(x1)x2)/angbracketright. (1) It is also straightforward to see that for every x∈Gand for every w∈L,w eh a v e |wPy(x)/angbracketright=| Py(x)w/angbracketright=ω−(y,w)|Py(x)/angbracketright. (2) We deﬁne the support of an element |u/angbracketrightof CGas the set of elements appearing with nonzero coefﬁcient in the decomposition of |u/angbracketrightas a linear combination of group elements. Using equality ( 1), one can show that if x1and x2are not in the same left coset of LH, then the states |Py(aH)x−1 1/angbracketrightand|Py(aH)x−1 2/angbracketrightare orthogonal. This is because the support of |Py(aH)x−1 i/angbracketrightis contained in LHx−1 i=(xiLH)−1(i=1,2). On the other hand, if x1H=x2H, then Hx−1 1=Hx−1 2and the two states are equal. By the characterization given in Lemma 1, it follows that for any left coset aH,t h e state1√|G|/summationtext x∈G|x/angbracketright|P0(aHx−1)/angbracketrightis a puriﬁcation of /Xi1G,LH. Of course it is hopeless to enforce y=0i n|Py(aH)x−1 1/angbracketright.

Question 601 multiple-choice

In group theory, minimal non-abelian p-groups of odd prime order are studied to understand how non-abelian properties arise in finite groups. Subgroup enumeration in these groups is crucial for classifying their internal structure, especially when they are non-metacyclic. Which property distinguishes non-metacyclic minimal non-abelian p-groups of odd prime order from metacyclic groups in terms of their subgroup structure? 1) Every proper subgroup is non-abelian 2) All subgroups are cyclic 3) The group has a unique cyclic subgroup of maximal order 4) The group possesses a non-cyclic normal subgroup with a cyclic quotient 5) Every subgroup is normal 6) The group lacks a cyclic normal subgroup whose quotient is cyclic 7) The order of every subgroup divides the square of the prime

✓ Correct Answer:

The correct answer is 6) The group lacks a cyclic normal subgroup whose quotient is cyclic.

📚 Reference Text:

Title: On the number of subgroups of non-metacyclic minimal non-abelian p-groups Year: 2020 Paper ID: 68d95011e6d03b4eebe1980b2eeaa1298ad06b43 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/68d95011e6d03b4eebe1980b2eeaa1298ad06b43 Abstract: In this paper, we find the number of subgroups in non-metacyclic minimal non-abelian [Formula: see text]-groups, [Formula: see text] an odd prime. In particular, we study a problem introduced by Berkovich [Groups of Prime Power Order, Vol. III (Walter de Gruyter, Berlin, New York, 2011)].

Question 602 multiple-choice

Quantum walks are powerful tools in quantum computing, providing algorithmic speedups for complex search and graph-based problems. Circuit implementation efficiency is critical for executing quantum walks on present-day hardware. Which approach enables continuous-time quantum walks on circulant graphs to be simulated with a circuit whose size does not increase with evolution time, while leveraging graph symmetry for efficient implementation? 1) Employing discrete-time quantum walks with extra coin qubits 2) Using staggered quantum walk models on random graphs 3) Implementing quantum walks only on photonic processors 4) Applying classical random walk algorithms to circulant graphs 5) Constructing circuits based on Hamiltonian simulation with time-dependent gate counts 6) Utilizing coined quantum walks with additional ancilla qubits 7) Designing circuits for continuous-time quantum walks with the Quantum Fourier Transform and leveraging circulant graph spectral properties

✓ Correct Answer:

The correct answer is 7) Designing circuits for continuous-time quantum walks with the Quantum Fourier Transform and leveraging circulant graph spectral properties.

📚 Reference Text:

Quantum Walks in a Superconducting Quantum Computer Jaime Santos1;2, Bruno Chagas3;4, Rodrigo Chaves5 1Universidade de Aveiro, Aveiro, Portugal 2INESC TEC, Braga, Portugal 3Irish Centre for High-End Computing, Dublin, Ireland 4National University of Ireland, Galway, Ireland 5Universidade Federal de Minas Gerais, Brazil jaimepereirasantos123@gmail.com , bruno.chagas@ichec.ie , rodrigoogchaves@gmail.com Abstract. Quantum Walks are among the most widely used techniques with which we can construct new quantum algorithms. This paper aims to outline how to construct a circuit for the continuous-time quantum walk (CTQW) over circulant graphs using the Quantum Fourier Transform (QFT) due to the spec- tral properties of those graphs. Furthermore, we examine how the Approximate Quantum Fourier Transform (AQFT) allows us to shorten the size of the cir- cuit by reducing the number of controlled rotation gates. The contributions of this paper consist of the development of a general circuit implementation of the CTQW for an important class of graphs that does not scale up with time, and the study of the cases where the AQFT decreases the error by controlling the approximation. Finally, we provide a statistical analysis for several circulant graphs, running experiments in a IBM’s superconducting quantum computer, and we also explore the pretty good state transfer (PGST) for some graphs. 1. Introduction Quantum walks provide a versatile framework in which we can develop new quantum algorithms for problems such as searching [Childs and Goldstone 2004] and element dis- tinctness [Ambainis 2008]. This technique has been studied in the context of transport properties such as perfect state transfer [Coutinho 2014] and mixing time [Chakraborty et al. 2020]. The coined quantum walk model was implemented using the topological formulation in a cycle graph [Balu et al. 2018], and by using Quantum Fourier Trans- form (QFT) as the main tool in the construction of the circuit [Shakeel 2020]. One of the major drawbacks of the coined quantum walk model is that it requires an extra space for the coin, and for some graphs the usage of a Noisy Intermediate-Scale Quantum (NISQ) computer can be impractical. The staggered quantum walk model, which does not re- quire a coin space, was implemented in the cycle and two-dimensional lattice [Acasiete et al. 2020]. Those previous works were limited to the physical realization of discrete- time quantum walks, and there’s still a need for a larger class of graphs which require a minimum number of qubits, while having low circuit depth. In the literature we ﬁnd a physical realization of continuous-time quantum walk (CTQW) in a photonic quantum processor using circulant graphs, but this work is limited This work is ﬁnanced by the ERDF – European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT - Fundac ¸ ˜ao para a Ci ˆencia e a Tecnologia, within project POCI-01-0145-FEDER-030947. to a graph with four nodes [Qiang et al. 2016]. Our paper aims to revisit this implementa- tion in a superconducting quantum computer, constructing our quantum circuits with the

Question 603 multiple-choice

In the theory of finite groups, particularly those with rich subgroup structures, the conjugacy and centralizer properties of certain subgroups are essential for classification. The role of components isomorphic to SL3 and their centers often appear in the analysis of subgroup uniqueness and fusion patterns. Which statement accurately describes how the subgroup E is uniquely determined up to conjugacy within a group G0 featuring components X1, X2, X3 each isomorphic to SL3? 1) E is characterized by elements e1, e2, e3 in the centers of X1, X2, X3 such that their product is 1 and each equals a fixed element e, ensuring uniqueness up to conjugacy. 2) E is the intersection of all maximal tori of G0 containing elements of order 3. 3) E consists solely of elements conjugate to the identity in each SL3 component. 4) E is generated by elements from three distinct Sylow 2-subgroups within X1, X2, and X3. 5) E is defined as the normalizer of the extraspecial group K of order 37 in G0. 6) E is the kernel of the homomorphism from G0 to the symmetric group on three letters induced by the action on {X1, X2, X3}. 7) E comprises all central elements of order 3 in G0.

✓ Correct Answer:

The correct answer is 1) E is characterized by elements e1, e2, e3 in the centers of X1, X2, X3 such that their product is 1 and each equals a fixed element e, ensuring uniqueness up to conjugacy..

📚 Reference Text:

E is determined uniquely up to G0-conjugacy. Proof. Fix e E £#. By Lemmas 2.6 and 2.9(ii), we have C:= Co(e) = (X1X 2X3) (e, !) , so we may pick g = b 1 b2b3 E £\ ( e, !) . By the second paragraph, b; $ (a;, e) for i = 1, 2, 3. Replacing g by g2 if necessary, we may assume that [a;, b;) = e for i= 1, 2, 3 (since [a;, b;] = 1 implies that b; E (a;, e)). It follows that E= \e,f, g). Let K= (a1, a2, a3, b1, b2, b3), an extraspecial group of order 37 normalizing E. We have 32 A. M. COHEN, M. W. LIEBECK, J. SAXL AND G. M. SEITZ and Cc;(E) = CK(E)(t) where (t) is transitive on {X1, X2,

Question 604 multiple-choice

In quantum computing, Schur sampling circuits leverage group representation theory to efficiently extract symmetry-related information from quantum systems. These techniques are fundamental for tasks such as quantum data compression and superreplication, and involve procedures like block diagonalization and measurement of representation registers. Which register in a Schur sampling circuit is sufficient to measure in order to perform weak Schur sampling for most quantum information applications? 1) The irreducible representation (irrep) label register 2) The unitary group irrep register 3) The symmetric group irrep register 4) All three registers simultaneously 5) The phase estimation register 6) The Clebsch-Gordan decomposition register 7) The quantum superreplication register

✓ Correct Answer:

The correct answer is 1) The irreducible representation (irrep) label register.

📚 Reference Text:

20, 25]. It is also used in a scheme for optimal Hari Krovi: hkrovi@gmail.com, hari.krovi@raytheon.com 1Speciﬁcally, Harrow sketches a procedure to obtain the same labeling of irreps as the ones used for Gelfand-Tsetlin basis. Why this procedure leads to the Gelfand-Tsetlin basis itself (not just the same labeling scheme) could be explained in more detail. Accepted in Quantum2019-01-30, click title to verify 1arXiv:1804.00055v2 [quant-ph] 3 Feb 2019 compression [41], quantum superreplication [8, 7]. Recently, in [19], the authors present a classical algorithm to compute the transition amplitudes of the Schur sampling circuits. There are two main ways in which Schur sampling is used: weak and strong sampling. Recall that with any kind of Fourier sampling, we block diagonalize the group action to obtain the Fourier basis from the computational basis. Strong (resp. weak) sampling refers to measuring all the registers (resp. only the label of the irreducible representation) to obtain information from the Fourier basis. Similarly, a circuit for Schur sampling block diagonalizes the unitary group (or equivalently the symmetric group) representation on the n fold tensor power of a ddimensional space. In this block diagonalization, each block essentially contains three pieces of information: (a) the irrep label, which is common to both the unitary and symmetric groups, (b) the unitary group irrep register i.e., the register holding the state that transforms under the unitary group and (c) the symmetric group irrep register holding the state that transforms under the symmetric group. Strong Schur sampling refers to measuring all three registers and weak Schur sampling refers to measuring only the irrep label. One could always measure all three registers in some basis to obtain more information about the state being transformed. However, in most of the above applications, one only needs to measure the irrep label register. In other words, one only needs weak Schur sampling and the associated probability distribution over the irrep labels to get enough information for all of the above mentioned applications. Weak Schur sampling can be performed with a quantum circuit that is polynomial in logd,nandlog(1//epsilon1) using the so-called Generalized Phase Estimation (GPE) procedure. The construction of such a circuit for the weak Schur transform is described in [18]. In this paper, we present a circuit that can be used for strong Schur sampling using the representation theory of the symmetric group that runs in time polynomial in logd,nandlog(1//epsilon1). In the previous approach by Bacon, Chuang and Harrow (BCH) [5], the Schur transform is constructed using the unitary group representation theory. By contrast, our algorithm uses the symmetric group and, in fact, uses transforms such as Beals' algorithm for the Fourier transform over the symmetric group [6] and the GPE algorithm mentioned above. To give an intuition for how the dual Schur transform works, we should look at how the original algorithm by BCH for the Schur transform works. BCH developed a quantum circuit for the Clebsch-Gordan (CG) decomposition problem for the unitary group, which entails block diagonalizing a tensor product of two irreps of

Question 605 multiple-choice

Quantum groups at roots of unity and their categories of representations play a central role in modern algebra and geometry. Recent work explores deep connections between the centers of these representation categories and geometric objects associated with the Langlands program. In the study of hybrid quantum groups at a root of unity, which mathematical structure is shown to be isomorphic to the center of a subgenerically deformed category 𝒪? 1) The cohomology ring of the full affine Grassmannian 2) The center of the universal enveloping algebra of the Langlands dual group 3) The equivariant K-theory of the flag variety 4) The Hecke algebra associated to the quantum group 5) The endomorphism algebra of a simple object in category 𝒪 6) The cohomology of the nilpotent cone 7) The equivariant cohomology of the ζ-fixed locus in the affine Grassmannian attached to the Langlands dual group

✓ Correct Answer:

The correct answer is 7) The equivariant cohomology of the ζ-fixed locus in the affine Grassmannian attached to the Langlands dual group.

📚 Reference Text:

Title: On the category 𝒪 of a hybrid quantum group Year: 2024 Paper ID: 201c235d7a35211294f22564071c03d7a095f62c Source: semantic-scholar URL: https://www.semanticscholar.org/paper/201c235d7a35211294f22564071c03d7a095f62c Abstract:
We study the representation theory of a hybrid quantum group at root of unity ζ \zeta introduced by Gaitsgory. After discussing some basic properties of its category O \mathcal {O} , we study deformations of the category O \mathcal {O} . For subgeneric deformations, we construct the endomorphism algebra of big projective object and compute it explicitly. Our main result is an algebra isomorphism between the center of deformed category O \mathcal {O} and the equivariant cohomology of ζ \zeta -fixed locus on the affine Grassmannian attached to the Langlands dual group.

Question 606 multiple-choice

Quantum computation and knot theory intersect in studying how quantum algorithms approximate knot invariants, such as the Jones and HOMFLYPT polynomials, using representations of braid groups. These connections have deep implications for computational complexity and topological quantum computing. Which statement accurately describes a major computational complexity result concerning the Jones polynomial at primitive roots of unity? 1) Approximating the Jones polynomial is in the class BPP for all braid closures. 2) Computing the Jones polynomial at any value is NP-complete for trace closures. 3) Evaluating the Jones polynomial at arbitrary complex numbers is always PSPACE-hard. 4) The multiplicative approximation of the Jones polynomial corresponds to QCMA-complete problems exclusively. 5) Approximating the Jones polynomial for plat closures at roots of unity is #SAT-hard. 6) Evaluating the Jones polynomial for plat closures at most primitive roots of unity is #P-hard, and determining its most significant bit is PP-hard. 7) Additive approximation of the Jones polynomial is equivalent to factoring integers in polynomial time.

✓ Correct Answer:

The correct answer is 6) Evaluating the Jones polynomial for plat closures at most primitive roots of unity is #P-hard, and determining its most significant bit is PP-hard..

📚 Reference Text:

arXiv:quant-ph/0603069v3 3 Aug 2007Quantum Information and Computation, Vol. 0, No. 0 (2003) 00 0–000 c/circlecopyrtRinton Press THE JONES POLYNOMIAL: QUANTUM ALGORITHMS AND APPLICATIONS IN QUANTUM COMPLEXITY THEORY PAWEL WOCJANa Institute for Quantum Information, Caltech Pasadena, CA 91125, USA JON YARDb Institute for Quantum Information, Caltech Pasadena, CA 91125, USA Received (June 20, 2006) Revised (revised date) We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov e t al., we give eﬃcient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for app roximately evaluating a family of specializations of the HOMFLYPT two-variable po lynomial of trace closures of braids. We also give algorithms for approximating the Jon es polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al. that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight- strand braid group. We then give QCMA-complete and PSPACE-c omplete problems which are based on braids. We conclude with direct proofs tha t evaluating the Jones polynomial of the plat closure at most primitive roots of uni ty is a #P-hard problem, while learning its most signiﬁcant bit is PP-hard, circumve nting the usual route through the Tutte polynomial and graph coloring. Keywords : quantum algorithms, quantum complexity theory, topologi cal quantum com- putation Communicated by : to be ﬁlled by the Editorial 1 Introduction There is evidence that a computer which could manipulate qua ntum mechanical degrees of freedom would be more powerful than a classical computer. Si nce the discovery by Shor that a quantum computer could eﬃciently factor composite intege rs, a task which is believed to be hard on conventional computers, much eﬀort has been expen ded toward understanding the capabilities and limitations of quantum computers. In t his paper, we investigate ways in which the Jones polynomial invariant of knots and links, t ogether with its generalizations, contribute to this understanding. Witten discovered [40] t hat the Jones polynomial could aSchool of Electrical Engineering and Computer Science, Uni versity of Central Florida, Orlando, FL 32816, USA. Electronic address: wocjan@cs.ucf.edu bElectronic address: jtyard@caltech.edu 1 2The Jones Polynomial: Quantum Algorithms and Applications in Quantum Complexity Theory be understood via tools from topological quantum ﬁeld theor ies. Freedman, Kitaev, Larsen and Wang [11] established its connection to computer scienc e in the context of topological quantum computation. In a topological quantum computer, th e trajectories of particles which are restricted to a plane are braided in order to manipulate t he internal state of the computer. They showed that such computers can be simulated on conventi onal quantum computers, and also that when the underlying physics of a computer is descri bed by a suitable topological quantum

Question 607 multiple-choice

In representation theory and quantum physics, the topology of Lie groups plays a critical role in determining which representations of their Lie algebras can be extended to the entire group. This distinction has significant implications for physical phenomena such as spin and angular momentum. Which statement correctly explains why half-integer spin representations are possible for SU(2) but not for SO(3)? 1) SO(3) and SU(2) are both simply connected, allowing only integer spin representations. 2) SU(2) is not a matrix Lie group, so it cannot represent spin-1/2 systems. 3) The Lie algebra so(3) admits only integer spin representations for all groups sharing it. 4) The exponential map fails to be a diffeomorphism near the identity for SO(3), preventing half-integer spins. 5) SO(3) is simply connected, so all its representations correspond to those of its Lie algebra. 6) SU(2) and SO(3) have different Lie algebras, which restricts the kinds of spin representations possible. 7) SU(2) is simply connected and allows all representations of its Lie algebra, including half-integer spins, while SO(3) is not simply connected and only supports integer spin representations.

✓ Correct Answer:

The correct answer is 7) SU(2) is simply connected and allows all representations of its Lie algebra, including half-integer spins, while SO(3) is not simply connected and only supports integer spin representations..

📚 Reference Text:

gebra onV. Then there is a unique representation RofG with the property R(eX) =er(X)for allX2g: Corollary 1 (Lie algebra reps locally lift to Lie group reps).LetGbe a matrix Lie group, and let rbe a rep- resentation of the corresponding Lie algebra on V. Then using Theorem 1, we can always locally deﬁne a represen- tationRonGby the mapping R(g) =er(X)deﬁned for all g=eXnearby 1: Here, by “nearby” we mean “wherever the exponential map is a diﬀeomorphism”. Indeed, in this region, all gcan be written asg=eX. Put together, Theorems 2 and 3 mean that if Gis simply connected, there is a one-to-one correspondence between their representations. If we relax the simply connected assumption, the power of this theorem weakens, but not too terribly much: the corollary guarantees the existence of a locally deﬁned representation. An important example of this distinction between the cases for simply connected and non simply connected groups arises naturally in physics: the group of 3D ro- tationsSO(3)is not simply connected, but the spin group 8Lie’s third theorem assures us that every ﬁnite dimensional Lie algebra is the Lie algebra of a Lie group (not necessarily a matrix Lie group, but often this is the case), so this handles most cases.SU(2)is simply connected. They have the same Lie al- gebra so(3)=su(2), and we can take as a basis the or- bital angular momentum operators Lx;Ly;Lz(of course, the Pauli operators X;Y;Zalso form a basis for this Lie algebra). It can be shown that this Lie algebra has, up to isomorphism, exactly one irreducible representation each (see Deﬁnition 13) on C2(spin-1/2) and on C3(spin-1).9 Theorem 3 guarantees that both of these representations ofsu(2)lift to representations on SU(2), called spin rep- resentations. But only the spin-1 representation lifts to a representation of SO(3)—the spin-1/2 representation does not yield a representation of SO(3). This justiﬁes the pre- viously mysterious (at least to us) undergraduate physics statement: “orbital angular momentum can only have in- teger quantum numbers, but spin angular momentum can have half-integer quantum numbers”, where quantum num- bersreallyjustmeanlabelsofirreduciblerepresentations.10 Now, returning to our focus: many interesting symmetry groups in quantum machine learning are not simply con- nected! How do we deal with these? There are two general approaches: 1. UseCorollary1toworknearby 1. Forinstance, ifthe only physical symmetries that are realized in a model are perturbations of 1, as may occur in some noisy classiﬁcation tasks, we do not need the full represen- tation and a local one will suﬃce. Indeed, this is a common situation, especially in physically motivated tasks. 2. There is a topological construction called the “univer- sal cover” ~Gof a Lie group G. The universal cover is simply connected by deﬁnition, and the quotient group ~G=Gis always a discrete group. In practice, this group is often ﬁnite and small: in the spin ex- ample, we have ~G=G =SU(2)=SO(3)=Z2. The good news about the universal cover is that every representation of Gcomes from a representation of ~G. This means that the only thing that could go wrong is that a Lie algebra

Question 608 multiple-choice

In abstract algebra, the group UT(3,p) consists of 3x3 upper triangular matrices with ones on the diagonal and entries from the finite field Z_p, and it is often studied alongside its extensions and automorphisms. The analysis of its irreducible representations and conjugacy classes is central to understanding its algebraic structure. For the group UT(3,p), how many nontrivial one-dimensional irreducible representations does it have? 1) p 2) p² - 1 3) p - 1 4) p² 5) p + 1 6) 2p - 1 7) p³ - 1

✓ Correct Answer:

The correct answer is 2) p² - 1.

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are given by  1a b 0 1c 0 0 1 →e2πi p(xa+yc)x,y∈Zp (B.2) Here, we have the trivial representation for x=y= 0 andp2−1 nontrivial one-dimensional irreducible representations. 44 pdimensional irreducible representations of UT(3,p) Thep−1p-dimensional irreducible representations canbelabeled by a parame tery∈/parenleftbig Zp/parenrightbig×. We can describe these explicitly by Ry:UT(3,p)→GL(p,C), 1a b 0 1c 0 0 1 /ma√sto→M M jk=δj[k−a]e2πi py(c(k−a)+b)(B.3) where the matrix index in square brackets is interpreted modulo p. B.2 The group H=UT(3,p)⋊φZ2 The group used in our construction is a semidirect product UT(3,p)⋊φZ2. To deﬁne this, let{1,s}be the elements of Z2withs2= 1, and label the elements of UT(3,p)⋊φZ2by (h,t) whereh∈UT(3,p) andt∈Z2. To specify the multiplication rule for H, we deﬁne the group homomorphism φ:Z2→Aut(UT(3,p)) given by φ(1) = 1 1, φ(s) =φswith φs: 1a b 0 1c 0 0 1 → 1−a b 0 1−c 0 0 1 . (B.4) It is straightforward to verify that φsis indeed an automorphism of UT(3,p) that preserves elements in the center of UT(3,p). In terms of φ, the group multiplication rule for His then speciﬁed as (h, t)·(h′, t′) = (hφt(h′), tt′). (B.5) Conjugacy class structure for UT(3,p)⋊Z2 The group UT(3,p)⋊ Z2has some conjugacy classes ‘inherited’ from the smaller group, UT(3,p) and then plarge conjugacy classes outside of this normal subgroup. All toge ther there arep2−1 2classes of size 2 pandpclasses of size 1 making up the UT(3,p) subgroup, and pclasses of size p2making up its complement. In order to understand conjugation, we note that for g= (h,t)∈UT(3,p)⋊ Z2, we have that g−1= (φt(h−1), t), (B.6) using the fact that φtis an involution. 45 The general conjugation formula of an element g= (h, t)∈Hby the element g′= (h′, t′)∈ His then (h′, t′)(h, t)(h′, t′)−1= (h′φt′(h)φt(h′−1), t). (B.7) Forgin theUT(3,p) subgroup, this gives g′gg′−1= (h′φt′(h)h′−1, e). (B.8) Thus, it is clear that conjugacy classes of elements in the normal subgroup UT(3,p) are almost unchanged with respect to those of UT(3,p) as an independent group, except that now elements become conjugate to their images under the automor phismφs. From the structure of φsit is clear that only central elements ( a=c= 0) are ﬁxed under this automorphism. We thus ﬁnd that the elements of the subgroup UT(3,p) split intop2−1 2 conjugacy classes (labelled by ( a, c)/ne}ationslash= (0,0)∈(Zp×Zp)/{±1}) of size 2pcomprising the sets:/braceleftbigg/parenleftbigg 1±a b 0 1±c 0 0 1 , e/parenrightbigg |b∈Zp/bracerightbigg , (B.9) andpclasses (labelled by b∈Zp) of size one comprising the sets: /braceleftbigg/parenleftbigg 1 0b 0 1 0 0 0 1 , e/parenrightbigg/bracerightbigg . (B.10) In the case that the element which we are conjugating lies in the comp lement of the UT(3,p) subgroup, equation (B.7) tells us that we remain in this set. Taking h∈UT(3,p) of the form (B.1) we obtain the following result for the UT(3,p) piece: h′φt′(h)φs(h′−1) = 1x y 0 1z 0 0 1  1±a b 0 1±c 0 0 1  1x−y+xz 0 1z 0 0 1  = 1±a+2x b+2xz±az±cx 0 1 ±c+2z 0 0 1 ,(B.11) with the upper

Question 609 multiple-choice

Quantum algorithms for group-theoretic problems, such as the dihedral hidden subgroup problem (DHSP), have broad implications for cryptography and computational complexity. The relationship between DHSP and classical problems like subset sum is particularly relevant for understanding quantum computational limits. Which statement accurately describes the significance of quantum algorithms for implementing the optimal measurement in DHSP with respect to the subset sum problem? 1) Efficient quantum algorithms for DHSP guarantee polynomial-time solutions for all NP-complete problems. 2) The optimal measurement for DHSP can be implemented classically if subset sum has a classical randomized algorithm. 3) The number of state copies required for DHSP is independent of the group order when linked to subset sum. 4) Quantum sampling from all subset sum solutions is unnecessary for implementing DHSP measurements. 5) Efficient quantum algorithms for the optimal DHSP measurement imply efficient quantum algorithms for average-case subset sum, and vice versa. 6) Solving subset sum efficiently yields direct classical solutions for DHSP without quantum resources. 7) The threshold phenomenon in DHSP measurement does not affect its computational connection to subset sum.

✓ Correct Answer:

The correct answer is 5) Efficient quantum algorithms for the optimal DHSP measurement imply efficient quantum algorithms for average-case subset sum, and vice versa..

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Title: Optimal measurements for the dihedral hidden subgroup problem Year: 2005 Paper ID: 0f845b07bf00f121022977a4b54d77afee48015d Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0f845b07bf00f121022977a4b54d77afee48015d Abstract: We consider the dihedral hidden subgroup problem as the problem of distinguishing hidden subgroup states. We show that the optimal measurement for solving this problem is the so-called pretty good measurement. We then prove that the success probability of this measurement exhibits a sharp threshold as a function of the density nu=k/log N, where k is the number of copies of the hidden subgroup state and 2N is the order of the dihedral group. In particular, for nu 1 the optimal measurement identifies the hidden subgroup with a probability of order unity. Thus the dihedral group provides an example of a group G for which Omega(log|G|) hidden subgroup states are necessary to solve the hidden subgroup problem. We also consider the optimal measurement for determining a single bit of the answer, and show that it exhibits the same threshold. Finally, we consider implementing the optimal measurement by a quantum circuit, and thereby establish further connections between the dihedral hidden subgroup problem and average case subset sum problems. In particular, we show that an efficient quantum algorithm for a restricted version of the optimal measurement would imply an efficient quantum algorithm for the subset sum problem, and conversely, that the ability to quantum sample from subset sum solutions allows one to implement the optimal measurement.

Question 610 multiple-choice

Quantum optimization problems often leverage semidefinite programming (SDP) hierarchies and swap matrices to approximate solutions and enforce symmetric constraints. These approaches can be formulated using either operator programs or *-algebraic frameworks, each with distinct mathematical foundations. Which mathematical construction serves as the bridge between positive linear functionals on *-algebras and quantum states in Hilbert spaces, establishing the equivalence between abstract algebraic and concrete operator formulations of quantum SDP hierarchies? 1) Jordan decomposition 2) Schur complement 3) GNS (Gelfand–Naimark–Segal) construction 4) Stone–Weierstrass theorem 5) Singular value decomposition 6) Hahn–Banach extension 7) Spectral gap analysis

✓ Correct Answer:

The correct answer is 3) GNS (Gelfand–Naimark–Segal) construction.

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Question 611 multiple-choice

Quantum optimization frequently utilizes algebraic and group-theoretic methods to exploit symmetries and reduce computational complexity, particularly in problems involving Hamiltonians and permutation-invariant quantum states. Semidefinite programming (SDP) relaxations are a powerful tool for approximating solutions to these problems. In quantum optimization algorithms for the max-cut problem, which mathematical structure allows constraints to be enforced that target specific eigenspaces corresponding to irreducible representations of the symmetric group? 1) Swap variables and relations derived from group theory 2) Tensor networks representing entanglement entropy 3) Boolean variables encoding graph partitions 4) Lagrange multipliers for probability normalization 5) Differential equations modeling time evolution 6) Linear constraints corresponding to vertex degrees 7) Pauli matrices acting on individual qubits

✓ Correct Answer:

The correct answer is 1) Swap variables and relations derived from group theory.

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relaxation d, GBn=Gr¨ obner basis for Iswap n Output : Solution to the dth relaxation of quantum max-cut for hG FindVd(n); Replace each entry of Vd(n) by its remainder upon division with GB n; Compute a basis for the span of all the entries of Vd(n); Form the vector Vd(n) with the basis as its entries; FindMd(n) :=Vd(n)Vd(n)∗; Replace each entry of Md(n) by its remainder upon division with GB n; Replace each distinct term appearing in Md(n) with a new (scalar) variable; call the resulting matrix Md(L); Solve the SDPν d(hG) = sup{⟨Md(L),ΓG⟩|Md(L)⪰0,Md(L)1,1= 1}, (5.7) where ΓGis any matrix satisfying hG=Vd(n)∗ΓGVd(n); returnν d(hG) This algorithm, like Algorithm 2 runs in finite time with similar justifications. However, this algorithm requies as input a Gr¨ obner basis for Iswap n. Finding such a Gr¨ obner basis for finitedand arbitrary nmay not be possible in polynomial time. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 45 5.5 Irrep Swap Relaxations As seen in Corollary 2.9, eigmax(HG) =⌊n 2⌋ max k=0/parenleftig eigmax(H[n−k,k] G)/parenrightig . In this section we explain how to form SDP relaxations to find eigmax(H[n−k,k] G ) for a given graph Gonnvertices andk≤n/2. From the proof of Lemma 2.12 (in particular Eq. (2.46)) we have that the following constraint is satisfied inside the [ n−k,k] irrep /summationdisplay (i,j)∈E(Kn)ρ[n−k,k]((i j)) =/hatwideη[n−k,k]I=/parenleftigg/parenleftigg n 2/parenrightigg +k2−k(n+ 1)/parenrightigg I. (5.8) Enforcing this constraint is equivalent to enforcing that hKn=ηn−k,k⇐⇒hKn−ηn−k,k= 0. (5.9) and so, by Proposition 3.10 the algebra formed by the sijvariables satisfying this extra relation is isomorphic to the relevant Irrep Symbolic Swap Algebra. Stated differently, Proposition 3.10 shows that all constraints in the [ n−k,k] irrep can be derived from Equation (5.8) above along with the relations given in Definition 3.1. We can force the swap variables to satisfy the same constraint by requiring that one of them, namely sn−1n, (the last one in our chosen monomial order) can be written in terms of the others: sn−1n=−/summationdisplay i

Question 612 multiple-choice

In noncommutative ring theory, ideals are often generated by specific sets of elements, and commutator identities play a key role in understanding their structure. Inductive proofs are valuable tools for establishing that entire families of elements belong to a given ideal. Which method is most effective for proving that all multilinear elements of forms (12) and (15) are contained in a two-sided ideal generated by elements of forms (13) and (14) in a free associative ring? 1) Direct computation for every possible degree and element 2) Construction of a Gröbner basis for the ideal 3) Application of the Wedderburn-Artin theorem 4) Use of the Chinese Remainder Theorem 5) Employing tensor product decompositions 6) Inductive proof on the degree of the elements, using commutator identities to relate forms (13), (14), and (15) 7) Verification through explicit enumeration of generators

✓ Correct Answer:

The correct answer is 6) Inductive proof on the degree of the elements, using commutator identities to relate forms (13), (14), and (15).

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ideal) by all elements of the forms (13) (d h,dh,dh) (ii, 12,13 6 I), (14) (4,, 4)(4-3,4 4) + (dh, duKdh, 4) d"i, «2, is, t4 e I). PROOF. Assume that & is the free associative ring on a free generating set [dt \ i e 1} (it clearly suffices to prove Lemma under this assumption). Let &' be the two-sided ideal in 3) generated by all elements (13)—(14). Using the identity (uv, w) = u(v, w) + (u, w)v, one can check easily that 3, U 2, U 4) = M,(M 3, U 2, U 4) + (Mi, M 2, U4)U3 + (u u U 2)(u^, U 4) + («i, M 4)(«3, so all elements of the form (15) (ui, u 2)(u3, u4) + (uu u4)(u3, u2) (ui,u 2,u3,u4e @) https://doi.org/10.1017/S1446788700002056 Published online by Cambridge University Press 342 C. K.Gupta and A. N. Krasil'nikov [14] are contained in ST. Therefore, &' c &. Since elements (12) and (15) are multilinear with respect to «,-, one can assume that all u t in (12) and (15) are monomials (on {d, | i e I}). We shall prove that &' contains all elements of the forms (12) and (15) (and so & = &') by induction on the degree of such an element. Let/ 6 S> be of the form (12) or (15). Note that every element/ of the form (12) of degree 3 is contained in &' and so is every element/ of the form (15) of degree 4 (because such / is of the form (13) or (14)). Suppose that / is of degree k (k > 4) and all elements of the forms (12) and (15) of degrees less than k are contained in S'. Consider / = (MI, M 2)(M 3, M 4) + (u\, M 4)(M 3, M 2) of degree k. Suppose that M4 = « 4«4', where M4, H4 are monomials of degree at least 1. Then = («,, H 2)(«3, «4M4') + («1, M 4 = (Mi, M 2)U4(M 3, M 4) + (Mi, H 2)(«3, « 4)«4 Since (Ml, M 2)M 4(M 3, «4) + U' 4(UU M 4)(M 3, U 2) = M 4[(«l, U2)(U 3, M4) + («i, M 4)(M 3, M 2)] + (M,, M 2, M4)(M 3, <) is contained in &' and so is (by the same argument) (Mi, M 2)(M 3, U4)U4' + («[, M 4)M 4'(M 3, U 2), we have/ 6 5". Note that if M 4 is of degree 1 but for some i (1 < i < 3) u t = M';M", where u\, u" are monomials of degree at least 1 then one can prove / 6 &' in a quite similar way. Thus, under the inductive assumption each element / of the form (15) of degree k is contained in &'. Consider/ = (M^ M 2, M 3) of degree k. If w3 = M 3M3', where M 3, M 3' are monomials of degree at least 1 then /

Question 613 multiple-choice

In spectral graph theory and quantum algorithms, efficiently computing eigenvalues of Hamiltonians associated with graphs often involves techniques like clique tree decomposition and the analysis of irreducible representations (irreps). Algorithmic scalability and error propagation are critical when dealing with large or complex graph structures. Which property is guaranteed when all leaf graphs in a clique tree decomposition are cliques? 1) All eigenvalues of the original graph are integers and exactly computable. 2) The runtime for eigenvalue computation reduces to linear time in the number of vertices. 3) The algorithm requires approximate eigenvalues for the leaf graphs. 4) Only the lowest irrep Hamiltonian eigenvalues are computable. 5) The error in eigenvalue computation can be arbitrarily large. 6) The spectral properties of the original graph become independent of the clique decomposition. 7) The ncSoS algorithm cannot be applied to bound eigenvalues.

✓ Correct Answer:

The correct answer is 1) All eigenvalues of the original graph are integers and exactly computable..

📚 Reference Text:

m/2∈O(m) different two row irreps corre- sponding to any m-vertex graph ˜G. Additionally, we have/uniontext ˜G∈Gc=Gcby construction and|V(Gc)|=nby assumption. So the inner loop of Algorithm 7 involves maximiz- ing/minimizing over partitions of at most nelements and then summing over the results of that maximization/minimization, all of which can be done in time O(n). The graph Ghas at most nvertices, hence at most O(n) two row irreps, and so the outer loop of Algorithm 7 is repeated at most O(n) times. The runtime of O(n2) follows. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 58 Correctness of Algorithm 8 follows from the observation that the assert statement in the inner loop of the algorithm is always satisfied, since we know the max and min eigenvalues of the irrep Hamiltonians for graphs corresponding to leaf vertices by assumption, and all other graph irrep Hamiltonians are computed from the “bottom up” by construction. To bound the runtime of Algorithm 8 note that, by Item (2) of Theorem 6.3 and the bound on the runtime of Algorithm 7 given above, for any i∈{d−1,...,1}, Algorithm 8 requires time at most O(n2) to compute the max and min irrep eigenvalues of all graphs G(w) corresponding to vertices wat depthiin the tree clique decomposition of G. Then Item (1) of Theorem 6.3 gives that d≤nand the upper bound of O(n3) follows. Remark 6.11. A straightforward generalization of Algorithms 7 and 8 using Corollary 6.9 gives us an algorithm computing the rhighest and lowest eigenvalues of all irrep Hamilto- nians ofGgiven therhighest and lowest eigenvalues of all irrep Hamiltonians of all graphs indexed by leaves in the clique tree decomposition of G. The runtime of this algorithm remains polynomial in randn. Remark 6.12. If all the graphs indexed by leaves in the clique tree decomposition of G are cliques, then we can compute all eigenvalues of all irrep Hamiltonians of those graphs in exact arithmetic by Lemmas 2.11 and 2.12. In this case all the eigenvalues of Gwill be obtained by taking sums and differences of clique eigenvalues ηλfor different values of λand, in particular, all eigenvalues will be integer. We give an example of a graph Gadmitting such a decomposition in the next section. Remark 6.13. If we apply the same procedure as described in Algorithm 8 but take as input approximate eigenvalues of the irrep Hamiltonians of the graphs indexed by leaves in the clique tree decomposition of Gwe obtain approximate eigenvalues of G. It is straightforward to show that the error in this approximation is at worst the sum of the approximation errors for the eigenvalues of each input graph. Additionally, if the algorithm is given upper/lower bounds on the minimum/maximum eigenvalues of all irrep Hamiltonians of graphs indexed by leaves in the tree clique decomposition of G, it produces upper/lower bounds on the maximum/minimum eigenvalues of G. One way to obtain these bounds is by running the ncSoS algorithm inside all two row irreps of the graphs indexed by leaves in

Question 614 multiple-choice

In quantum computing, exploiting symmetries such as those of the SU(2) group and the symmetric group Sn is crucial for efficient state preparation, particularly in quantum algorithms that address many-body systems. Quantum Schur transforms and Young tableaux play significant roles in representing and manipulating these symmetric quantum states. Which approach enables the efficient initialization of quantum states in arbitrary Sn irreducible representations by generalizing from the balanced Young diagram, and what key mathematical duality does it utilize? 1) Applying the Casimir operator to directly diagonalize the Hamiltonian in the computational basis using Schur-Weyl reciprocity 2) Using Clebsch-Gordan coefficients to iteratively couple single-qubit spins in a sequential circuit 3) Implementing the Jordan-Wigner transformation to map fermionic modes to qubit states in the symmetry-adapted basis 4) Expanding initial states solely in the standard computational basis and projecting onto target subspaces 5) Employing the Wigner-Eckart theorem to construct basis states in irreducible subspaces 6) Leveraging permutation modules and the SU(d)-Sn duality to inductively construct initialization algorithms for arbitrary irreps 7) Utilizing Fourier transform techniques on the symmetric group to generate all possible Young tableau states

✓ Correct Answer:

The correct answer is 6) Leveraging permutation modules and the SU(d)-Sn duality to inductively construct initialization algorithms for arbitrary irreps.

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each site i. Without having to employ the Casimir operator, Sn z,(Sn z)2and a problem Hamiltonian HPalready form a SU (2)-CQA model. More precisely, as follows. Corollary 1: Let H Pbe any path-connected Hamilto- nian on the SU(2)-irrep basis, the CQA-ansatz gener- ated by H P,Sn z,(Sn z)2is dense in each SU(2)-irrep block U(dimWλ). B. State preparation for Sn-CQA ansätze To investigate evaluation of the matrix elements of Sn Fourier coeﬃcients, we are conﬁned to the Young basis, which requires the implementation of quantum Schur transform [ 47,51,52]. However, for a wide variety of quan- tum machine learning and optimization tasks, such as determining the ground-state sign structure of frustratedmagnets, it is often advantageous to relax the constraints and ask how easy it is to initialize the states that live in any given S nirrep. An algorithm to initialize a state in the Snirrep with Young diagram being (n/2,n/2), is given in Ref. [ 46]. We generalize this result to an arbitrary Snirrep in general SU (d)−Snduality. The key is to utilize dif- ferent permutation modules and multiplicities of Snirreps as in Fig. 2(c). Similar to the previous subsection, we construct the algorithm inductively: we ﬁrst consider the SU(2)−Snduality in which a (λ1,λ2)-Snirrep is dual to a spin-(λ1−λ2)/2 irrep. Let |/Psi1init/angbracketright=| 0···0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright kmany/angbracketright⊗| s/angbracketright⊗···⊗| s/angbracketright/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright [(n−k)/2] many,( 5 ) where |s/angbracketright=1√ 2(|01/angbracketright−| 10/angbracketright)is one of the Bell states and we assume n−kis even. Then we have the following.Lemma 4: Letμ=λ=[(n+k)/2,(n−k)/2]. The ini- tialized state |/Psi1init/angbracketrightis contained in Sλand belongs to the permutation module Mμ. Proof. Acting by the spin operator Sz=/summationtext iSz i, it is easy to check that the spin component of |/Psi1init/angbracketrightequals j=k/2 hence it belongs to Mμ. By Theorem 4 and discussion from the previous subsection, we have the expansion |/Psi1init/angbracketright=/summationtext TcT|αT;k/2/angbracketright. Since by deﬁnition J+|/Psi1init/angbracketright=0, the Young diagram underlying each αTfrom the summation must be the same and equals λ. /squaresolid We now illustrate by several examples how to expand |/Psi1init/angbracketrightas a linear combination of Young-basis elements: (a) let |/Psi1init/angbracketright=| s/angbracketright⊗(n/2). This is the state used in Ref. [ 46]. One can check by YJM elements that it is a single Young- basis element. (b) For a more involved case, consider the (4, 2)irrep of S6and write |/Psi1init/angbracketright=| 00/angbracketright⊗| s/angbracketright⊗| s/angbracketright =2 3|αT1;1/angbracketright−√ 2 3|αT2;1/angbracketright−√ 2 3|αT3;1/angbracketright +1 3|αT4;1/angbracketright, where αTicorresponds to GZ paths in Fig. 6. The ﬁrst two boxes in Fig. 6correspond to trivial irrep ofS2acting on the subsystem formed by the ﬁrst two qubits. Indeed, S2acts trivially on |00/angbracketrightfrom|/Psi1init/angbracketright.A s it would be more apparent to see how to get αTiin the SU(2)picture, we use the spin label-content vector dual- ity and trace the path of spin coupling. As the total spin of|00/angbracketrightand|00/angbracketright⊗| s/angbracketrightare identical (Lemma 4), there are two ways to add two more boxes: putting the third box on the rhs of the ﬁrst two and then putting the fourth on the bottom or conversely. Tensoring again with the singlet |s/angbracketright, we retrieve four branching paths in total. Moreover, by

Question 615 multiple-choice

Efficient simulation of quantum algorithms on classical hardware is vital for validating and benchmarking algorithms while quantum devices remain inaccessible or immature. Field Programmable Gate Arrays (FPGAs) are increasingly used in this domain for their customizable parallel processing capabilities. Which approach enables significant speedup and scalability when simulating the Quantum Fourier Transform (QFT) on FPGAs while maintaining high numerical fidelity? 1) Utilizing only floating-point arithmetic with fixed precision 2) Relying on multi-core CPUs for parallel simulation 3) Employing general-purpose GPUs with standard libraries 4) Implementing customized-precision arithmetic via high-level synthesis on FPGAs 5) Simulating QFT exclusively with variable-length integer representations 6) Using analog circuit emulation for quantum gates 7) Applying software-based error correction in post-processing

✓ Correct Answer:

The correct answer is 4) Implementing customized-precision arithmetic via high-level synthesis on FPGAs.

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Title: Precision Analysis and Hardware Acceleration for Large-Scale Quantum Fourier Transformation on Modern FPGAS Year: 2025 Paper ID: b2d483daa6dc55bcb6c98df1ac54c5b0d2672acc Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b2d483daa6dc55bcb6c98df1ac54c5b0d2672acc Abstract: With the development of quantum algorithms and practical quantum computing devices, how to efficiently simulate emerging quantum algorithms via quantum circuit simulators on classical parallel-computing hardware platforms has raised more and more interest. The superior customized parallelism and pipeline efficiency over general-purpose processors make FPGAs ideal for large scale quantum algorithms/circuits simulation. Since the quantum Fourier transform (QFT) has been used in many quantum algorithms, such as quantum linear algebra, quantum phase estimation and quantum machine leaning, this paper presented a novel precision analysis and an efficient customized-precision QFT simulator accelerated on AMD Alveo U55C FPGAs. Customized-precision representations via high-level synthesis are employed for QFT implementation with up to 16 qubits, demonstrating up-to 8.3× speedup versus implementations in floating-point arithmetic within an error tolerance of 0.1%.

Question 616 multiple-choice

The Hidden Subgroup Problem (HSP) is a central challenge in quantum computing and group theory, where efficient algorithms often depend on the existence and structure of subgroups within finite groups. Understanding which group-theoretic properties enable efficient solutions is key to algorithmic advancements. Which property guarantees that a finite group has a subgroup of every order dividing the group’s order, thereby making it particularly suitable for certain HSP algorithms? 1) Simplicity 2) Being abelian 3) Satisfying the Converse of Lagrange’s Theorem (CLT) 4) Nilpotency 5) Having a cyclic center 6) Being non-abelian 7) Solvability

✓ Correct Answer:

The correct answer is 3) Satisfying the Converse of Lagrange’s Theorem (CLT).

📚 Reference Text:

and correctness follows as in Corollary 4.2. If Ghas a proper subgroup of order n1with n/m≤n1≤κn/mfor some κ, the algorithm ﬁnds a collision as in the abelian case in an iteration with k≥ℓ≥0, where ℓis such that κn/m∈/bracketleftbig n/2ℓ+1,n/(2ℓ+1)/bracketrightbig . It thus outputs the correct answer. Further, we have ℓ=⌈log2(m/κ)⌉−1, and initially, k= ⌈log2n⌉−1≥0. By Theorem 4.1, we have |S1|,|S2|≤/radicalbig |G1|ln|G1|+1 in every iteration with queries. So the query complexity of the algorithm is at most ⌈log2n⌉−1 ∑ k=ℓ3/parenleftbigg 1+/parenleftBign 2klnn 2k/parenrightBig1/2/parenrightbigg ≤ 3(⌈log2n⌉−⌈ log2(m/κ)⌉+1)+3/parenleftBign 2ℓlnn 2ℓ/parenrightBig1/2∑ i≥01√ 2i ≤ 6+3 log2(κn/m)+3(2+√ 2)/parenleftbigg2κn mln2κn m/parenrightbigg1/2 , asℓ≥log2(m/κ)−1. The bound on the query complexity when m=1 is 2⌈√ nlnn⌉more than 9 that for m=2, as the algorithm executes all the iterations until k=−1. Part 2 of the theorem thus follows. Unlike abelian groups, a non-abelian group of order nmay not have a subgroup of order n/m when it has a proper subgroup of order m. For example A4, the alternating group of degree 4, has order 12, has several subgroups of order 2, but does not have a subgroup of order 6. However, in large classes of instances of the Hidden Subgroup Problem, s ubgroups of suitable size exist. An immediate example is the class of CLT groups. (A group Gis called a converse Lagrange Theorem (CLT) group if it contains a subgroup of order dfor every positive divisor dof|G|.) CLT groups include supersolvable groups; see, e.g., Ref. [9]. For such instances, Find-Collision achieves query complexity O (/radicalbig (n/m)ln(n/m)). 6 Finding the hidden subgroup Unlike the algorithms in Corollary 3.6 and Corollary 4.2, Al gorithm 1 may ﬁnd only one non- trivial element from the hidden subgroup. In this section, w e show how to extend Algorithm 1 to ﬁnd the entire hidden subgroup. When the group Gis abelian, the problem of ﬁnding the entire subgroup may be re duced to that of ﬁnding one non-trivial element of the subgroup. This a llows us to identify the subgroup by repeatedly using Algorithm 1 to ﬁnd a set of generators. Algorithm 2: Find-Abelian-Subgroup( G,f) Input : group Gof order n, with n>1; oracle for f:G→Sthat hides a subgroup Output : injective , orgenerators S⊂Gof the hidden subgroup 1S←∅; 2repeat 3 H1←/a\}⌊ra⌋k⌉tl⌉{tS/a\}⌊ra⌋k⌉tri}ht, the subgroup generated by S; 4 G1←G/H1; 5 if|G1|>1then 6 Letf1be the function deﬁned by fand H1in the proof of Theorem 6.1 ; 7 outcome←Find-Collision( G1,f1) ; 8 ifoutcome=collision a,bthen S←S∪/braceleftBig g−1 1g2/bracerightBig , where a=g1H1and b=g2H1; 9 else 10 outcome←injective 11until outcome=injective ; 12ifS=∅then return injective ; 13else return generators S; Theorem 6.1. There is a deterministic algorithm that solves the Hidden Su bgroup Problem over any ﬁnite abelian group G, and ﬁnds the hidden subgroup with O((logm)√ n/m)queries when the order of G is n and that of the hidden subgroup is m. Proof. Suppose the oracle is f, the hidden subgroup is H, and we know a set of generators for a subgroup H1≤H. Deﬁne a function f1on the quotient group G/H1asf1(gH1):=f(g)for any g∈G. The function f1is well-deﬁned

Question 617 multiple-choice

Quantum algorithms can dramatically outperform classical algorithms for certain decision and search problems, especially those involving hidden structures and periodicity. These breakthroughs have profound implications for cryptography and computational complexity. Which quantum algorithm utilizes the quantum Fourier transform to efficiently factor large integers, threatening the security of widely-used cryptographic protocols? 1) Deutsch-Jozsa algorithm 2) Grover's algorithm 3) Simon's algorithm 4) Bernstein-Vazirani algorithm 5) Shor's algorithm 6) Quantum counting algorithm 7) Quantum walk algorithm

✓ Correct Answer:

The correct answer is 5) Shor's algorithm.

📚 Reference Text:

oolean function of nvariables (i.e. a function of all nbit strings with one-bit values). It is promised that the fun ction is either a constant function or ‘balanced’ in the sense that ex actly half of the values are 0 and 1 half are 1. We wish to determine with certainty whether the gi ven function is balanced or constant, using the least number of queries to the box. Thus w e are asking for one bit of information about the 2nvalues of the function. Classically 2n−1+ 1 queries are necessary in the worst case (if the problem is to be solved with certainty ) but quantumly the problem can be solved in all cases with just one query [10]. However if we tolerate any arbitrarily small probability of error in the answer then there is also a c lassical algorithm using only a constant number of queries. Inspired by these results, Simon [6] considered a more compl icated situation of a class of functions from nbits tonbits and developed a computational task displaying an expon ential gap between the classical and quantum query complexities, e ven if (in contrast to Deutsch’s algorithm) the algorithm is required to work only with bound ed error probability of 1 /3 i.e. we allow probabilistic algorithms and in any run the answer m ust be correct with probability at least 2/3. In retrospect (c.f. below) Simon’s problem turns out to be an example of a “generalized periodicity” or hidden subgroup problem, for the group of nbit strings under binary bitwise addition. Shor recognized the connection with periodicity determination and generalized the constructions to the group of integers modulo N, showing signiﬁcantly that the associated dis- crete Fourier transform may be eﬃciently implemented in tha t context as well. Finally using known reductions of the tasks of integer factorisation and e valuation of discrete logarithms to periodicity determinations, he was able to give polynomi al time quantum algorithms for these computational tasks too. 2 The quantum Fourier transform and periodicities We begin with an account of how a quantum computer may eﬃcient ly determine the peri- odicity of a given periodic function. Consider the followin g basic example. Suppose that we have a black box which computes a function f:ZN→ Zthat is guaranteed to be periodic with some period r: f(x+r) =f(x) for all x (1) HereZNdenotes the additive group of integers modulo N. We also assume that fdoes not take the same value twice within any single period. Note that eq. (1) can hold only if r dividesNexactly. Our aim is to determine r. Classically (in the absence of any further information abo ut f) we can merely try diﬀerent values of xin the black box hoping for two equal results which will then give information about r. Generally we will require O(N) random tries to hit two equal values with high probability. Using quantum eﬀects we will be able to ﬁnd rusing onlyO((logN)2) steps, which represents an exponential speedup over any kn

Question 618 multiple-choice

The Hidden Subgroup Problem (HSP) plays a central role in computational group theory and quantum algorithms, with practical ramifications for cryptography and complexity theory. Algorithms for HSP often rely on properties of group structure, such as normality and product decompositions, to efficiently identify hidden subgroups. Which of the following statements accurately describes a key advantage provided by bicrossed (Zappa-Szép) product structures in the context of classical algorithms for the Hidden Subgroup Problem? 1) They guarantee the existence of a normal subgroup in every finite group. 2) They restrict the application of subgroup-finding algorithms to only abelian groups. 3) They ensure that all finite groups can be expressed as semidirect products. 4) They generalize semidirect products, enabling subgroup-finding algorithms to work in more group types, such as symmetric groups. 5) They eliminate the need for oracle functions in all subgroup-finding algorithms. 6) They always lead to exponential speedup of classical algorithms for HSP. 7) They require that every subgroup be cyclic.

✓ Correct Answer:

The correct answer is 4) They generalize semidirect products, enabling subgroup-finding algorithms to work in more group types, such as symmetric groups..

📚 Reference Text:

(n/m)log(n/m))queries when G is non-abelian. Proof. The existence of a subgroup G0as in the statement of the theorem implies that all the hy- potheses of Theorem 6.2 are satisﬁed for every subgroup H1ofH. So, starting with H1={e}, where eis the identity element of G, Find-New-Collision( G,H1,f) returns a collision a,bin each 12 Algorithm 4: Find-Subgroup( G,f) Input : group Gof order n, with n>1; oracle for f:G→Sthat hides a subgroup Output : injective , orgenerators S⊂Gof the hidden subgroup 1S←∅; 2repeat 3 H1←/a\}⌊ra⌋k⌉tl⌉{tS/a\}⌊ra⌋k⌉tri}ht; 4 outcome←Find-New-Collision( G,H1,f) ; 5 ifoutcome=collision a,bthen S←S∪/braceleftbig a−1b/bracerightbig ; 6until outcome=injective ; 7ifS=∅then return injective ; 8else return generators S; iteration of Algorithm 4 in which H1/\⌉}atio\slash=H. Since a−1b/\⌉}atio\slash∈H1, the set S∪/braceleftbig a−1b/bracerightbig generates a larger subgroup of H. Thus, the size of the subgroup H1increases by a factor of at least 2 in every itera- tion in which a collision is found, and the algorithm termina tes after at most log2miterations. The query complexity follows from Theorem 6.2. A subgroup as in the statement of Theorem 6.3 exists if Gis the semidirect product of Hwith another subgroup, i.e., His normal and there is a subgroup Ksuch that G=H⋊K, or there is a normal subgroup Ksuch that G=H⋉K. We may then take G0:=K. Not all groups have a semidirect product structure, even if the hidden subg roup is normal. For example, every proper subgroup Hof the abelian group Zpkwith k>1 is normal, but Zpkcannot be expressed as a semidirect product of Hwith another subgroup. On the other hand, a group need not hav e a semidirect product structure for a subgroup with the prope rties in Theorem 6.3 to exist. For example, consider Sn, the symmetric group of degree n, and Sn−1as its subgroup consisting of permutations that map nto itself. Then Sn=Sn−1H, where His the subgroup generated by the cycle(1 2 3···n), and neither Sn−1norHis normal in Snforn≥4. Such groups are known as thebicrossed products (also as Zappa-Sz´ ep orknitproducts); see, e.g., [1, 4]. Thus, Algorithm 4 ﬁnds the hidden subgroup Hwith query complexity as in Theorem 6.3 whenever Gis the bicrossed product of Hwith another group. A description of groups arising as bicro ssed products is a matter of ongoing research [1]. 7 Open problems We conclude with a few open problems. The query complexity of the algorithm designed by Ye and Li [14] for ﬁnding the hidden subgroup in abelian instan ces may be smaller than that of Algorithm 2. The lower query complexity hinges on an intrica te analysis of the structure of the hidden subgroup. Can we establish the same query complexity through simpler means? There are a number of variants of the Hidden Subgroup Problem , for example, when the un- derlying group is speciﬁed implicitly. These may also admit d eterministic algorithms with optimal classical query complexity. The precise characterization of the deterministic query complexity of the Hidden Subgroup Problem for explicitly speciﬁed groups, especially in the non-abelian case, is perhaps the most interesting problem left open by this

Question 619 multiple-choice

In the theory of abelian p-groups and their automorphisms, understanding the structure and nilpotency class of groups under the action of automorphisms of finite order is crucial. Commutator identities and subgroup decompositions play an essential role in determining explicit bounds for nilpotency. In an abelian p-group A with automorphism b of order pⁿ, suppose b centralizes the subgroup A^p and the basic subgroup L is sparse. What is a key consequence for the group (A, b) when n = 1 regarding its nilpotency? 1) The subgroup generated by all commutators [A, p b] is trivial, establishing that (A, b) is nilpotent of class 1. 2) The nilpotency class of (A, b) is unbounded and depends on the Ulm invariants of A. 3) The automorphism b acts freely on every direct summand of A, so (A, b) cannot be nilpotent. 4) The group (A, b) is simple since b centralizes A^p. 5) The basic subgroup L must be homocyclic of rank 1, making (A, b) abelian. 6) The exponent of every commutator subgroup [A, b^k] is greater than p for all k. 7) The quotient group A/A^p is always of class at least p + p² under b.

✓ Correct Answer:

The correct answer is 1) The subgroup generated by all commutators [A, p b] is trivial, establishing that (A, b) is nilpotent of class 1..

📚 Reference Text:

b) ) ^ nni A^" = 1 and (A, b) is of class at most p 4- 1 in all cases. However, we have to reduce this/? 4- 1 top, and we proceed as follows. First we show that b centralizes A p . For all a in A we have (") 1 = [A,pb] p[a, (p + \)b]\ 2' • • • [a, (2p - \)b], so that [a, pb]p = 1. However this gives that (P) ( " ) 1 = [a, bf[a, 2b]P^ 2' ...[<>AP- W^ -1 ' so that commutation (p — 2) times with b gives [a, (p - \)bf = 1, and the last term of drops out. Repeating this commutation process we successively eliminate the last factor of until we reach the conclusion that [a, bf = \ for all a. In other words, [Ap\ b] = 1. Since L = A/AS ) we know from Theorem 4.2 that (A/A^ \ b) has class at most r x + 2 where r x is the rank of the homocyclic component L, of exponent/». Thus [A, (r, + 2)/?] ^ A {]) ^ ^ 2 so that [A, (r, + 3)6] = 1. If rj + 3 = p we are done. Otherwise r x + 2 ^ /? so that, as L is sparse, r, + r 2 = p — 2; so rj = p — 2 and r 2 = 0. Thus L 2 = 0. From properties of basic subgroups there is a decomposition of A as A = L x X L 2X K. where a basic subgroup of K begins with a homocyclic component of exponent at least p 3. Since L 2 = 0 here, this decomposition gives AIA^ = L,^"/^" X Jt4< ,V^(,); the fact that the two factors intersect trivially follows since L x has no elements of infinite height. Since L = A/A^ is sparse with KA^/A {X) having its homocyclic components of exponent at least p 3 and b acts as an automorphism on A/A { \ it follows from Step 1 that [v4/^4 (1), bA(])] is of https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press NILPOTENT EXTENSIONS 1049 exponent /?, and hence that [A/A^\ bA {l)] =g LXA{X)/A^ X KP 2A{X)/A^\ that is, [A,b] ^ L xAp\ Thus [A, 2ft] ^ [L], ft] since b centralizes A p . But [L 1? ft] is elementary abelian and has rank at most r x. Thus so is [A, 2b] and so, since it is ft-invariant, [A, (r, + 2)/)] = 1. So [A,pb] = 1 and Step 2 is complete in the case n = 1; and as we said at the beginning, the rest of the proof is straightforward. Step 3. Let A be a countable reduced abelian p-group with eventually sparse basic subgroup L, and let b be an automorphism of order pn on A. Then {A, ft) is nilpotent of class c ^ p + p 2 +

Question 620 multiple-choice

In abstract algebra, group presentations using generators and relations are used to define finite groups, particularly p-groups. These presentations can involve parameters such as prime powers and arithmetic constraints, which influence the group's structure and classification. Which constraint would require the use of quadratic non-residues in the construction of a family of group presentations, potentially impacting the group's automorphism structure? 1) Imposing that all generators have order dividing p 2) Requiring that all commutators are trivial 3) Setting p = 2 for all presentations 4) Restricting all relations to power relations only 5) Limiting parameters to positive integers 6) Specifying s ∈ F_p for some s 7) Demanding ν be a quadratic non-residue modulo p

✓ Correct Answer:

The correct answer is 7) Demanding ν be a quadratic non-residue modulo p.

📚 Reference Text:

= [d,b] = 1/an}bracketri}ht; (N15)/an}bracketle{ta,b,c,d/vextendsingle/vextendsinglea8=b4=d2= 1,[a,b] =c,[c,a] =c2=a4,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht; (N16)/an}bracketle{ta,b,c,d/vextendsingle/vextendsinglea8=d2= 1,[a,b] =c,[c,a] =c2=a4=b4,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht; (O1)/an}bracketle{ta,b,c,d,e/vextendsingle/vextendsinglea3=b3=c3=d3=e3= 1,[a,b] =c,[c,a] =d,[c,b] =e,[d,a] = [d,b] = [e,a] = [e,b] = 1/an}bracketri}ht; (O2)/an}bracketle{ta,b,c,d/vextendsingle/vextendsinglea3=b9=c3=d3= 1,[a,b] =c,[c,a] =d,[c,b] =b3,[d,a] = [d,b] = 1/an}bracketri}ht; 29 (O3)/an}bracketle{ta,b,c,d/vextendsingle/vextendsinglea9=c3=d3= 1,b3=a3,[a,b] =c,[c,a] =d,[c,b] =a3,[d,a] = [d,b] = 1/an}bracketri}ht; (O4)/an}bracketle{ta,b,c,d/vextendsingle/vextendsinglea9=b3=c3=d3= 1,[a,b] =c,[c,a] =d,[c,b] =a−3,[d,a] = [d,b] = 1/an}bracketri}ht; (O5)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea9=b9=c3= 1,[a,b] =c,[c,a] =a3,[c,b] =b3/an}bracketri}ht; (O6)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea9=b9=c3= 1,[a,b] =c,[c,a] =b3,[c,b] =a3,[a3,b] = 1/an}bracketri}ht; (O7)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea9=b9=c3= 1,[a,b] =c,[c,a] =b−3,[c,b] =a3,[a3,b] = 1/an}bracketri}ht; (P1)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpn+1=cp= 1,[a,b] =c,[c,a] =apn,[c,b] =bpn/an}bracketri}ht, where p >2andn≥2forp= 3; (P2)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpn+1=cp= 1,[a,b] =c,[c,a] =apnbνpn,[c,b] =bpn/an}bracketri}ht, where p >2,n≥2forp= 3,ν= 1or a ﬁxed quadratic non-residue modular p; (P3)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpn+1=cp= 1,[a,b] =c,[c,a] =bνpn,[c,b] =a−pn,[apn,b] = 1/an}bracketri}ht, wherep >2,n≥2forp= 3,ν= 1or a ﬁxed quadratic non-residue modular p; (P4)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpn+1=cp= 1,[a,b] =c,[c,a]1+r=apnbpn,[c,b]1+r= a−rpnbpn,[apn,b] = 1/an}bracketri}ht, wherep >2,n≥2forp= 3,r= 1,2,...,p−2; (P5)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea2n+1=b2n+1=c2= 1,[a,b] =c,[c,a] =b2n,[c,b] =a2n,[a2n,b] = 1/an}bracketri}ht, wheren≥3; (P6)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea2n+1=b2n+1=c2= 1,[a,b] =c,[c,a] =a2n,[c,b] =b2n/an}bracketri}ht, where n≥3; (P7)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea2n+1=b2n+1=c2= 1,[a,b] =c,[c,a] =a2nb2n,[c,b] =a2n,[a2n,b] = 1/an}bracketri}ht, wheren≥3; (P8)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn+1=bpn=cp=dp= 1,[a,b] =c,[c,a] =apn,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3forp= 2andn≥2forp= 3; (P9)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn=bpn+1=cp=dp= 1,[a,b] =c,[c,a] =bνpn,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3forp= 2andn≥2forp= 3,ν= 1or a ﬁxed quadratic non-residue modular p; (P10)/an}bracketle{ta,b,c,d,e/vextendsingle/vextendsingleapn=bpn=cp=dp=ep= 1,[a,b] =c,[c,a] =d,[c,b] = e,[d,a] = [d,b] = [e,a] = [e,b] = 1/an}bracketri}ht, wheren≥3forp= 2andn≥2for p= 3; (Q1)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpn+1= 1,[a,b] =c,[c,a] =cp=bspn,[c,b] =a−νpncνtpn/an}bracketri}ht, wheren≥3forp= 2andn≥2,ν= 1or a ﬁxed quadratic non-residue modular p,s∈F∗ p,t= 0,1,...,p−1 2; (Q2)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpn=cp2= 1,[a,b] =c,[c,a] =cp,[c,b] =a−νpnctνp/an}bracketri}ht, where n≥3forp= 2andn≥2,ν= 1or a ﬁxed quadratic non-residue modular p,t= 0,1,...,p−1 2; (Q3)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpn+1= 1,[a,b] =c,[c,a] =cp=apn,[c,b] =aspnbpn/an}bracketri}ht, where n≥3forp= 2andn≥2,s∈Fp; (Q4)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpn+1= 1,[a,b] =c,[c,a] =cp=apn,[c,b] =bspn/an}bracketri}ht, where n≥3forp= 2andn≥2,s= 2,3,...,p−1; 30 (Q5)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn+1=bpn=dp= 1,[a,b] =c,[c,a] =cp=apn,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3forp= 2andn≥2; (Q6)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn=bpn+1=cp2= 1,[a,b] =c,[c,a] =cp,[c,b] =bpn/an}bracketri}ht, wheren≥3 forp= 2andn≥2; (Q7)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn=bpn+1=dp= 1,[a,b] =c,[c,a] =cp=bspn,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3forp= 2andn≥2,s∈F∗ p; (Q8)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn=bpn=cp2=dp= 1,[a,b] =c,[c,a] =cp,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3forp= 2andn≥2; (R1)/an}bracketle{ta,b,c,d/vextendsingle/vextendsinglea4=b2=c4=d2= 1,[a,b] =c,[c,a] =d,[c,b] =c2,[d,a] = [d,b] = 1/an}bracketri}ht; (R2)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea4=b4=c4= 1,[a,b] =c,[c,a] =b2,[c,b] =c2/an}bracketri}ht; (R3)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8=b2=c4= 1,[a,b] =c,[c,a] =a4,[c,b] =c2/an}bracketri}ht; (R4)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8=c4= 1,b2=a4,[a,b] =c,[c,a] =a4,[c,b] =c2/an}bracketri}ht; (R5)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8=b2=c4= 1,[a,b] =c,[c,a] =a4c2,[c,b] =c2,[c2,a] = 1/an}bracketri}ht; (R6)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8=b4= 1,c2=a4b2,[a,b] =c,[c,a] =b2,[c,b] =c2/an}bracketri}ht; (R7)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8=b4= 1,c2=b2,[a,b] =c,[c,a] =a4b2,[c,b] =c2/an}bracketri}ht; (S1)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpm+1=cp= 1,[a,b] =c,[c,a] =apn,[c,b] =bspm/an}bracketri}ht, where n≥3andm≥2forp= 2andn > m,s∈F∗ p; (S2)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpm+1=cp= 1,[a,b] =c,[c,a] =bν1pm,[c,b] =a−ν2pn,[a,bpm] = 1/an}bracketri}ht, wheren≥3andm≥2forp= 2andn > m,ν1,ν2= 1or a ﬁxed quadratic non-residue modular p; (S3)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn+1=bpm=cp=dp= 1,[a,b] =c,[c,a] =d,[c,b] =a−νpn,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3andm≥2forp= 2andn > m,ν= 1or a ﬁxed quadratic non-residue modular p; (S4)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn=bpm+1=cp=dp= 1,[a,b] =c,[c,a] =bνpm,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3andm≥2forp= 2andn > m,ν= 1or a ﬁxed quadratic non-residue modular p; (S5)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn=bpm+1=cp=dp= 1,[a,b] =c,[c,a] =d,[c,b] =bpm,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3andm≥2forp= 2andn > m; (S6)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn+1=bpm=cp=dp= 1,[a,b] =c,[c,a] =apn,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3andm≥2forp= 2andn > m; (S7)/an}bracketle{ta,b,c,d,e/vextendsingle/vextendsingleapn=bpm=cp=dp=ep= 1,[a,b] =c,[c,a] =d,[c,b] =e, [d,a] = [d,b] = [e,a] = [e,b] = 1/an}bracketri}ht, wheren≥3andm≥2forp= 2and n > m; (T1)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea2n+1=b4= 1,c2=b2,[a,b] =c,[c,a] =a2n,[c,b] =c2/an}bracketri}ht, wheren≥3; (T2)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea2n+1=b4= 1,c2=a2n,[a,b] =c,[c,a] =b2,[c,b] =c2/an}bracketri}ht, wheren≥3; (T3)/an}bracketle{ta,b,c,d/vextendsingle/vextendsinglea2n+1=b2=d2= 1,c2=a2n,[a,b] =c,[c,a] =d,[c,b] =c2,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3; (T4)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea2n=b4=c4= 1,[a,b] =c,[c,a] =b2,[c,b] =c2/an}bracketri}ht, wheren≥3; 31 (T5)/an}bracketle{ta,b,c,d/vextendsingle/vextendsinglea2n=b4=d2= 1,c2=b2,[a,b] =c,[c,a] =d,[c,b] =c2,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3; (T6)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea2n+1=b2=c4= 1,[a,b] =c,[c,a] =a2n,[c,b] =c2/an}bracketri}ht, wheren≥3;

Question 621 multiple-choice

Property testing in group theory enables the inference of group properties through randomized algorithms, which is instrumental for applications in cryptography and coding theory. Recent advances link the existence of certain test subsets within groups to generalizations of classical results in error correction. Which fundamental concept is generalized when proving the existence of test subsets in non-commutative groups that can distinguish proper subgroups from the whole group using randomized queries? 1) Lagrange's theorem for finite groups 2) Cayley’s theorem on group representations 3) The classification of finite simple groups 4) Shannon's theorem on reliable communication and error correcting codes 5) Burnside's lemma in group actions 6) The Sylow theorems in group structure 7) The Chinese Remainder Theorem in algebraic systems

✓ Correct Answer:

The correct answer is 4) Shannon's theorem on reliable communication and error correcting codes.

📚 Reference Text:

Title: Non-commutative error correcting codes and proper subgroup testing Year: 2024 Paper ID: a82946f70b89e10359682471fbeca6bab7501ad3 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a82946f70b89e10359682471fbeca6bab7501ad3 Abstract: Property testing has been a major area of research in computer science in the last three decades. By property testing we refer to an ensemble of problems, results and algorithms which enable to deduce global information about some data by only reading small random parts of it. In recent years, this theory found its way into group theory, mainly via group stability. In this paper, we study the following problem: Devise a randomized algorithm that given a subgroup $H$ of $G$, decides whether $H$ is the whole group or a proper subgroup, by checking whether a single (random) element of $G$ is in $H$. The search for such an algorithm boils down to the following purely group theoretic problem: For $G$ of rank $k$, find a small as possible test subset $A\subseteq G$ such that for every proper subgroup $H$, $|H\cap A|\leq (1-\delta)|A|$ for some absolute constant $\delta>0$, which we call the detection probability of $A$. It turns out that the search for sets $A$ of size linear in $k$ and constant detection probability is a non-commutative analogue of the classical search for families of good error correcting codes. This paper is devoted to proving that such test subsets exist, which implies good universal error correcting codes exist -- providing a far reaching generalization of the classical result of Shannon. In addition, we study this problem in certain subclasses of groups -- such as abelian, nilpotent, and finite solvable groups -- providing different constructions of test subsets for these subclasses with various qualities. Finally, this generalized theory of non-commutative error correcting codes suggests a plethora of interesting problems and research directions.

Question 622 multiple-choice

In quantum computing, entanglement and non-locality are essential phenomena that contribute to the performance of quantum algorithms. Mermin polynomials and the Cayley hyperdeterminant are mathematical tools used to detect and quantify these quantum correlations during algorithm execution. When monitoring non-locality in the Quantum Fourier Transform (QFT) algorithm, which procedure must be followed to optimally track entanglement at each computational step? 1) Use a single fixed Mermin operator throughout the entire QFT execution 2) Apply the Cayley hyperdeterminant exclusively at each step 3) Monitor only the initial state with a maximally violating Mermin operator 4) Select a different optimal Mermin operator for each step of the QFT 5) Alternate between Mermin polynomials and Bell inequalities at every other step 6) Disregard operator choice and measure entanglement only at the final output 7) Use the same operator sequence as in Grover’s algorithm regardless of algorithm structure

✓ Correct Answer:

The correct answer is 4) Select a different optimal Mermin operator for each step of the QFT.

📚 Reference Text:

Title: Mermin polynomials for non-locality and entanglement detection in Grover’s algorithm and Quantum Fourier Transform Year: 2020 Paper ID: f732aef37aec9f016ff91e6d99be4846949ac121 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/f732aef37aec9f016ff91e6d99be4846949ac121 Abstract: The non-locality and thus the presence of entanglement of a quantum system can be detected using Mermin polynomials. This gives us a means to study non-locality evolution during the execution of quantum algorithms. We first consider Grover’s quantum search algorithm, noticing that states during the execution of the algorithm reach a maximum for an entanglement measure when close to a predetermined state, which allows us to search for a single optimal Mermin operator and use it to evaluate non-locality through the whole execution of Grover’s algorithm. Then the Quantum Fourier Transform is also studied with Mermin polynomials. A different optimal Mermin operator is searched for at each execution step, since in this case nothing hints us at finding a predetermined state maximally violating the Mermin inequality. The results for the Quantum Fourier Transform are compared to results from a previous study of entanglement with Cayley hyperdeterminant. All our computations can be repeated thanks to a structured and documented open-source code that we provide.

Question 623 multiple-choice

In measurement-based quantum computing, graph states serve as key resources due to their entanglement properties and algebraic structure. Efficient representation of these states can impact quantum error correction and the scalability of quantum algorithms. For an n-qubit star graph state, how many stabilizer generators are minimally required to describe the state within its stabilizer group? 1) n 2) n - 1 3) ⌈n/2⌉ 4) log₂(n) 5) 1 6) n/2 7) n + 1

✓ Correct Answer:

The correct answer is 5) 1.

📚 Reference Text:

Title: An efficient algebraic representation for graph states for measurement-based quantum computing Year: 2022 Paper ID: 473ecd42002150a029bea7893d6504c6386b1396 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/473ecd42002150a029bea7893d6504c6386b1396 Abstract: Graph states are the main computational building blocks of measurement-based computation and a useful tool for error correction in the gate model architecture. The graph states form a class of quantum states which are eigenvectors for the abelian group of stabilizer operators. They own topological properties, arising from their graph structure, including the presence of highly connected nodes, called hubs. Starting from hub nodes, we demonstrate how to efficiently express a graph state through the generators of the stabilizer group. We provide examples by expressing the ring and the star topology, for which the number of stabilizers reduces from $n$ to $\lceil\frac{n}{2}\rceil$, and from $n$ to 1, respectively. We demonstrate that the graph states can be generated by a subgroup of the stabilizer group. Therefore, we provide an algebraic framework to manipulate the graph states with a reduced number of stabilizers.

Question 624 multiple-choice

In quantum information theory, analyzing the overlap between entangled states and product (separable) states provides insight into the degree of entanglement and the complexity of quantum computations. The maximum overlap serves as a useful metric in distinguishing between different classes of quantum states. For a balanced generalized W state of 2n qubits (with n zeros and n ones in each basis state), what is the asymptotic value of the maximum overlap (Pmax) with the nearest product state as n becomes large? 1) Pmax ≈ 1/2n 2) Pmax ≈ 1/n 3) Pmax ≈ 1/2 4) Pmax ≈ 1/√πn 5) Pmax ≈ 2/n 6) Pmax ≈ 1/4n 7) Pmax ≈ 2/√πn

✓ Correct Answer:

The correct answer is 4) Pmax ≈ 1/√πn.

📚 Reference Text:

order to ﬁnd Pmax: fq S(x1,...,x q;θ1,... θq) =/an}brack⌉tl⌉{tϕ|ψq S/an}brack⌉tri}ht=1/radicalbig |S|/summationdisplay j1...jq∈Sei/summationtextq m=1jmθmq/productdisplay m=1Cm jm(11) where Cm j=  √1−xmj= 0 √xmj= 1.(12) To be more precise, we need to maximize P=|fq S|2, which is equivalent to maximizing the magnitude of the complex function fq S, ignoring its phase. In fact, we can ﬁx the relative phasesθmto zero, since Eq. (11) then becomes: fq S(x1,...,x q) =1/radicalbig |S|/summationdisplay j1...jq∈Sq/productdisplay m=1Cm jm, (13) which is clearly not smaller in magnitude. This reduces the d imension of the search space from2qtoq(this is actually a special case of an observation already ma de in Ref. [6]). Another simpliﬁcation arises in an ES state for which there i s one qubit, 1≤m≤q, that all the basis states in the superposition “agree” on (that is ,jmis the same for all k∈S). This qubit is not entangled with the rest of the qubits and can be factored out. We shall call such an ES state reducible , since we can ﬁx xm=jmand reduce the dimension of the search space by one. 2. Examples: Special high symmetry states We have so far reduced our problem to ﬁnding the values of x1,...,x qfor the nearest product state |ϕ/an}brack⌉tri}ht, to aq-qubit nonreducible ES state |ψq S/an}brack⌉tri}ht, by maximizing the overlap functionfq Sin Eq. (13). Let us consider two special cases of this problem , which were already analyzed in Ref. [6]. The ﬁrst one is the q-qubit GHZ state: 8 |GHZ/an}brack⌉tri}ht=|0...0/an}brack⌉tri}ht+|1...1/an}brack⌉tri}ht√ 2. (14) For the GHZ state, the nearest product state is any one of the t wo basis states that comprise it: |ϕ/an}brack⌉tri}ht=|0...0/an}brack⌉tri}htor|ϕ/an}brack⌉tri}ht=|1...1/an}brack⌉tri}ht, (15) and for both states: Pmax(GHZ) = 1 /2. The second special case is the 2n-qubit balanced generalized Wstate, denoted by |φ(n,2n)/an}brack⌉tri}ht. It consists of all the basis states of 2nqubits that havenzeros andnones: |φ(n,2n)/an}brack⌉tri}ht=/parenleftbigg2n n/parenrightbigg−1 2/summationdisplay /summationtext2n m=1jm=n|j1...j2n/an}brack⌉tri}ht. (16) In this case the nearest product state is the complete ES stat e |ϕ/an}brack⌉tri}ht=|η/an}brack⌉tri}ht, (17) andPmaxis given by Pmax[φ(n,2n)] = 2−2n/parenleftbigg2n n/parenrightbigg ≈1√πn. (18) 3. The approximated formula Given a general ES state |ψq S/an}brack⌉tri}ht, our aim is to ﬁnd the nearest product states |ϕ/an}brack⌉tri}ht. The results presented previously motivate us to examine two typ es of product states as candidate states: (i)|ϕ/an}brack⌉tri}ht=|k/an}brack⌉tri}htfor somek∈S, which means that each xmis either 0or1. This yields Pq S= 1/|S|, wherePq Sis the estimated value of P. (ii)|ϕ/an}brack⌉tri}ht=|η/an}brack⌉tri}ht(the complete ES state), which means that xm= 1/2for allm’s. This yieldsPq S=|S|/Q. Clearly, the ﬁrst guess is better for a small S, and the second guess is better for a large S. They become equally good for |S|=√Q. Thus, we can combine them into one improved guess: 9 Pq S=  1 |S||S| ≤√Q |S| Q|S|>√Q(19) We note that Pq Sis a lower bound on Pmax(ψq S). We can also present a crude argument to support the claim that Pq Sis a good approximation for Pmax(ψq S). For a general product state|ϕ/an}brack⌉tri}ht, consider expanding it to a superposition of basis states. I n this expansion, we would like to maximize the

Question 625 multiple-choice

Second-order holomorphic linear differential equations on Riemann surfaces are central in complex analysis and mathematical physics, with their monodromy groups encoding crucial information about solution spaces and symmetries. The characterization of these monodromy groups, particularly regarding unitarity, involves algebraic and geometric criteria with implications for spectral theory. For an irreducible second-order holomorphic linear differential equation on a Riemann surface, which algebraic criterion determines whether its monodromy group is unitary? 1) Existence of a nontrivial abelian subgroup within the monodromy group 2) Satisfaction of trace conditions on the local monodromy matrices 3) Conjugacy of the monodromy group to a model subgroup of $\operatorname{GL}(2,\mathbb{C})$ 4) Preservation of a real quadratic form by the monodromy group 5) The dimension of the algebra generated by the rescaled group being equal to 1 6) Diagonalizability of all monodromy matrices over $\mathbb{R}$ 7) Vanishing determinant for every monodromy matrix

✓ Correct Answer:

The correct answer is 2) Satisfaction of trace conditions on the local monodromy matrices.

📚 Reference Text:

Title: On Unitary Monodromy of Second-Order ODEs Year: 2024 Paper ID: 15a19b5a879e5b443a203d1023481e150fbeac12 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/15a19b5a879e5b443a203d1023481e150fbeac12 Abstract: Given a second-order, holomorphic, linear differential equation $Lf=0$ on a Riemann surface, we say that its monodromy group $G\subset\operatorname{GL}(2,\mathbb{C})$ is \emph{unitary} if it preserves a non-degenerate (though not necessarily positive) Hermitian form $H$ on $\mathbb{C}^2$ under the action $g\circ H\doteq g^\dagger H g$. In the present work, we give two sets of necessary and sufficient conditions for a differential operator $L$ to have a unitary monodromy group, and we construct the form $H$ explicitly. First, in the case that the natural representation of $G$ on $\mathbb{C}^2$ is irreducible, we show that unitarity is equivalent to a set of easily-verified trace conditions on local monodromy matrices; in the case that it is reducible, we show that $G$ is unitary if and only if it is conjugate to a subgroup of one of two model subgroups of $\operatorname{GL}(2,\mathbb{C})$. Second, we show that unitarity of $G$ is equivalent to a criterion on the real dimension of the algebra $A$ generated by a rescaled group $G'\subset\operatorname{SL}(2,\mathbb{C})$: that $\dim(A)=1$ if $G\subset S^1$ is scalar, $\dim(A)=2$ if $G$ is abelian, $\dim(A)=3$ if $G$ is non-abelian but its action on $\mathbb{C}^2$ is reducible, and $\dim(A)=4$ otherwise. We leverage these results to extend a conjecture of Frits Beukers on the spectrum of Lam\'e operators -- namely, we give asymptotic and numerical evidence that his conjecture should apply similarly to a wider class of Heun operator. Our work makes progress towards characterizing the spectra of second-order operators on Riemann surfaces, and in particular, towards answering the accessory parameter problem for Heun equations.

Question 626 multiple-choice

Quantum lattice models are essential for studying the behavior of fundamental particles and fields, but computational challenges persist, especially at finite density and in real-time dynamics. Tensor network approaches have emerged as promising alternatives to traditional statistical sampling methods, providing new formulations and tools for simulating complex quantum systems. Which primary advantage do tensor network methods offer in reformulating lattice quantum field theories compared to conventional statistical sampling techniques? 1) They automatically solve the sign problem for all gauge theories. 2) They require no discretization of spacetime, enabling continuous simulations. 3) They eliminate the need for renormalization group analysis. 4) They make all field integrations analytically tractable for any model. 5) They directly enforce unitarity without approximations. 6) They replace field integrations with discrete sums, enabling efficient coarse-graining and practical computation. 7) They guarantee the preservation of all topological features without truncation.

✓ Correct Answer:

The correct answer is 6) They replace field integrations with discrete sums, enabling efficient coarse-graining and practical computation..

📚 Reference Text:

Title: Tensor lattice field theory for renormalization and quantum computing Year: 2020 Paper ID: 2c4dfa821487b7b83fa3f3fb399dac7e7a517dc7 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/2c4dfa821487b7b83fa3f3fb399dac7e7a517dc7 Abstract: We discuss the successes and limitations of statistical sampling for a sequence of models studied in the context of lattice QCD and emphasize the need for new methods to deal with finite-density and real-time evolution. We show that these lattice models can be reformulated using tensorial methods where the field integrations in the path-integral formalism are replaced by discrete sums. These formulations involve various types of duality and provide exact coarse-graining formulas which can be combined with truncations to obtain practical implementations of the Wilson renormalization group program. Tensor reformulations are naturally discrete and provide manageable transfer matrices. Combining truncations with the time continuum limit, we derive Hamiltonians suitable to perform quantum simulation experiments, for instance using cold atoms, or to be programmed on existing quantum computers. We review recent progress concerning the tensor field theory treatment of non-compact scalar models, supersymmetric models, economical four-dimensional algorithms, noise-robust enforcement of Gauss's law, symmetry preserving truncations and topological considerations. We discuss connections with other tensor network approaches.

Question 627 multiple-choice

In computational knot theory and quantum algorithms, estimating knot invariants such as the Jones polynomial often relies on randomized sampling and probabilistic bounds to ensure algorithmic accuracy. The handling of generalized closures of braids enables approximation methods to extend beyond standard knot cases. Which statement correctly describes how generalized closures of braids are efficiently handled in randomized algorithms for approximating the absolute value of the Jones polynomial? 1) Each generalized closure requires exponential time due to the ambiguity in orientation. 2) The phase of the Jones polynomial is always well-defined for generalized closures, so only real sampling is necessary. 3) The union bound is not applicable when estimating the Jones polynomial for generalized closures. 4) Generalized closures can be represented as plat closures of related braids with more strands using a polynomially longer braid, allowing existing estimation methods to be applied. 5) Sampling is infeasible for generalized closures because complex random variables cannot be generated efficiently. 6) Only canonical oriented closures can be approximated; generalized closures lack efficient algorithms. 7) Additive error bounds for generalized closures scale linearly, not quadratically, in the inverse error tolerance.

✓ Correct Answer:

The correct answer is 4) Generalized closures can be represented as plat closures of related braids with more strands using a polynomially longer braid, allowing existing estimation methods to be applied..

📚 Reference Text:

exp/parenleftbigg −Mδ2 4/parenrightbigg . Suppose the above algorithm is run Mtimes, obtaining random variables {Xj,Yj}M j=1. Setting Zj=Xj+iYj, the union bound and the above lemma imply Pr  /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 MM/summationdisplay j=1Zj−EZ1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>δ  ≤Pr  /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 MM/summationdisplay j=1Xj−EX1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>δ√ 2  + Pr  /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 MM/summationdisplay j=1Yj−EY1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>δ√ 2   ≤4 exp/parenleftbigg −Mδ2 8/parenrightbigg . 22The Jones Polynomial: Quantum Algorithms and Applications in Quantum Complexity Theory b/d95 /d95 /d95 /d95 /d95 /d95 /d95 /d31/d31/d31/d31/d31/d31/d31/d31/d31 /d95 /d95 /d95 /d95 /d95 /d95 /d95/d31/d31/d31/d31/d31/d31/d31/d31/d31 Fig. 6.1. The trace closure of b∈B3is isotopic to the plat closure of the modiﬁed braid (σ5 3σ3 2)−1bσ5 3σ3 2=σ−1 2σ−1 4σ−1 3bσ3σ4σ2. Note that we identify bwith its image ι◦ι◦ι(b)∈B6. Thus, for any δ >0, the probability that Z≡e2πi(k−1)e(b) 2ℓ1 M/summationtextZjdeviates from the desired value of the normalized one-variable HOMFLYPT polynomial b y more than δis smaller than the desired error of 1 /4, provided that the number of samples satisﬁes M >32 ln(2)/δ2. Since each sample is obtained in O/parenleftbig poly(m)/parenrightbig time, the desired additive approximation is obtained inO/parenleftbig poly(m,1/δ)/parenrightbig time as required. Approximating the Jones polynomial of generalized closure s In this section we show how to approximate the absolute value of the Jones polynomial of generalized closures of a braid. As mentioned in Section 2, w e focus on the absolute value because there is no unambiguous way to assign an orientation to a generalized closure. Recall the setup for Problem 2.2. We are given a length- mbraidb∈Bn, two nonnegative integers p,rfor which 2p+r=nand a positive integer ℓ. We lose no generality in assuming that the top and bottom braids which deﬁne the generalized closure ar ex,y= 1n. As we showed in (5.38), the largest absolute value of the Jones polynomial i s obtained when the closure yields p+rtrivial knots. Therefore we focus on estimating the normali zed quantity 1 [2]p+r−1 ℓ/vextendsingle/vextendsingle/vextendsingleJ/parenleftbig ξp,r 1,1(b),e2πi/ℓ/parenrightbig/vextendsingle/vextendsingle/vextendsingle. We show two diﬀerent ways of achieving this goal, as each cons titutes a diﬀerent generalization of the previous algorithm which is interesting in its own rig ht. The ﬁrst makes use of the following proposition [19], which shows that the generaliz ed closureξp,r(b) can be written as the plat closure of a related braid on 2 p+ 2rstrands. Proposition 6.3 Letb∈B2p+r. Thenξp,r 1,1(b) =/tildewiderc−1bc, wherec=σ2p+2r−1 2p+r···σ2p+5 2p+3σ2p+3 2p+2 and fori

Question 628 multiple-choice

Representation theory is a cornerstone in quantum information science, underpinning techniques for classifying quantum states, optimizing algorithms, and leveraging symmetries in quantum systems. Its applications extend from entanglement theory to quantum machine learning and efficient computation in large-scale quantum systems. Which mathematical construct is specifically used to classify equivalence classes of quantum states under stochastic local operations and classical communication (SLOCC), providing robust entanglement measures invariant under the action of the special linear group SL(d)? 1) Symmetric polynomials 2) Matrix Product States (MPS) 3) Schur polynomials 4) Quantum property testing algorithms 5) Semidefinite programming dual variables 6) Projected Entangled Pair States (PEPS) 7) SL-invariant polynomials (SLIPs)

✓ Correct Answer:

The correct answer is 7) SL-invariant polynomials (SLIPs).

📚 Reference Text:

of matrix product states(MPS)liketheAKLTchainandmoregenerally projected entangled pair states (PEPS) [107, 108]. It should not be surprising that representation theory plays a critical role in the classiﬁcation of symmetry- protected topological phases of matter. •A large class of optimization problems in quantum information can be phrased as semideﬁnite programs with aU p (U) qsymmetry. Leveraging this symmetry allows one to prove signiﬁcantly better timescalingthanstandardsemideﬁniteprogramming guarantees [109]. •Representations of the symmetric group in turn are closely tied to symmetric polynomials (aka Schur polynomials). While it is not obvious to us how these will appear in QML contexts, it is still worth keep- ing them in mind, as their ubiquity in representation theory makes use cases feel inevitable. •Classifying states under so-called "stochastic local operations and classical communication" (SLOCC) is a major goal of entanglement theory. Two states are equivalent under SLOCC if one can convert be- tween them with some non-zero probability. In fact, these classes can be formalized through the orbits of the special linear group SL(d). One can classify and measure entanglement through SL-invariant polyno- mials (SLIPs) [110–115]. 35 D. Variable rate loans: other dangerous traps Say that we want to check if a given problem, deﬁned in terms of a Hamiltonian Hexhibits symmetry to a given groupG. Thiscanbecastintothetaskofcheckingwhether Hcommutes with every representation Ugof the elements gofG. Despite its innocuous look, this task can quickly become intractable if His very large (for n-qubit systems, the dimension of Hgrows as 2n). However, one can tackle this problem from the optics of quantum property testing, whichquiteliterallystudiesthetaskof“givenalargeobject that either has a certain property or is somehow far from having that property, a tester should eﬃciently distinguish between these two cases”. We refer the reader to Ref. [116] for a nice review on quantum property testing. Recent eﬀorts have been put forward towards develop- ing symmetry-testing algorithms. For instance, the work in [117] proposes a quantum algorithm to check whether a Hamiltonian exhibits symmetry with respect to a group, and the work in [118] sets forth a variety of quantum al- gorithms with corresponding numerics which test symme- tries of states and channels. These point towards an ex- citing new era of the symmetry program in physics and computer science, wherein we may algorithmically test for symmetries without requiring analytic expressions of mod- els and data. IX. OUTLOOK Representation theory is one of the most powerful tools that any quantum computing or quantum information sci- entist can possess under their belt. For the particular context of QML, representation theory allows for the ma- nipulation and understanding of symmetries in the data, as well as to study how physical processes can be built to respect those symmetries. Despite its tremendous im- portance, many key results and insights in representation theoryarehiddenbehindmathematicalandalgebraicnota- tionsthatmayseeminsurmountablefornon-experts. How- ever, we hope that after reading this article, the reader will ﬁnd themselves comfortable enough with the notational and conceptual basics to dig deeper into the rich litera- ture of representation theory (we again cannot recommend enough Refs. [51–54]) and its fundamental importance

Question 629 multiple-choice

In quantum computing, the Hidden Subgroup Problem (HSP) involves identifying a hidden subgroup of a group given access to a quantum state encoding information about the subgroup. The design of optimal quantum measurements for solving single-copy instances of HSP relies heavily on group representation theory and the symmetry properties of subgroup probability distributions. Which property of subgroup probability distributions specifically allows group-theoretic symmetry arguments to be used in constructing optimal measurement operators for the single-copy Hidden Subgroup Problem? 1) Uniform probability within conjugacy classes of subgroups 2) Independence of probability from subgroup structure 3) Arbitrary assignment of probabilities to all subgroups 4) Uniform probability for all irreducible representations 5) Probability proportional to subgroup order 6) Probability confined to a single subgroup 7) Random probability distribution without constraints

✓ Correct Answer:

The correct answer is 1) Uniform probability within conjugacy classes of subgroups.

📚 Reference Text:

in a ﬁxed basis, the inequalities turns into an inequality between the constants appearing before t hese operators. Since |Hmax|is larger than all |H| for which Pµ,H/ne}ationslash= 0, it follows that these constants in front of the operators obey the inequality. Thus we have shown that this measurement is the optimal measurement for the single co py hidden subgroup problem. It is useful to rephrase the optimal measurement we have derived . The optimal measurement consists of measure- ment operators of the form EH=/circleplusdisplay µIdµ⊗eµ,HPµ,H (36) where eµ,H=/braceleftBigg |Hmax| |Cmax(µ)|dµP h∈Hmaxχµ(h)ifH ∈Cmax(µ) 0 if H/∈Cmax(µ). (37) It is easy to see that in our derivation of the optimal single copy meas urement, had we restricted ourselves to only one set of conjugate subgroups, we would have obtained the P GM, while if we had restricted ourselves to only subgroups belonging to diﬀerent sets of conjugate subgroups we would have obtained a measurement like that of Ip [15]. Thus the optimal measurement for a single copy of the HSP is a hybrid between the previous two known single copy optimal measurements for the HSP. 7 C. Strengthening the Result It is also possible to strengthen our result beyond the case where a ll subgroups are given with equal probability. In particular consider the case where the probability of a subgroup is u niform across diﬀerent conjugate subgroups, but allowed to vary between the diﬀerent conjugate subgroups. In su ch a setting, the a prior probability of a subgroup pHdepends only on which set of conjugate subgroups Hbelongs to. In other words, consider a probability pHsuch thatpH=pH′ifHandH′are conjugate to each other. Keep pHthroughout the calculations performed above leads to a set of conditions similar to Eqs. (33) and (34). In particular the condition that the generalized measurement is a valid measurement remains unchanged, /summationdisplay C∈C(G)sµ(HC)cµ,C= 1, cµ,C≥0 (38) but the optimality condition becomes /summationdisplay C∈C(G)pHC|HC|sµ(HC)cµ,CIdµ≥pH′|H′|Pµ,H′ (39) for allH′∈Sub(G). From this expression it is clear that instead of choosing the larges tHsuch that Pµ,H/ne}ationslash= 0, the above condition can be satisﬁed by choosing a subgroup H∗withPµ,H∗/ne}ationslash= 0 such that pH∗|H∗|is maximal (breaking ties arbitrarily.) Let C∗denote the set of conjugate subgroups to which such a H∗belongs. Given this choice the optimal measurement will have the form EH=/circleplusdisplay µIdµ⊗eµ,HPµ,H (40) where eµ,H=/braceleftBigg |H∗| |C∗(µ)|dµP h∈H∗χµ(h)ifH ∈C∗(µ) 0 if H/∈C∗(µ). (41) Thus we have solved the slightly more general problem of the optimal measurement when the a priori probability of a subgroup is required to be constant only across conjugate subg roups. Optimal measurements for arbitrary a priori probability distributions are likely to be more diﬃcult to obtain, since th e symmetry arguments we have used do not hold in such a setting. IV. CONCLUSION AND OUTLOOK We have determined the optimal measurement for a single copy of th e HSP when all possible subgroups of a group are given with equal a priori probability. We have also presented a slig htly stronger version of this problem where the a priori probabilities are uniform over

Question 630 multiple-choice

In group theory, the congruence subgroup problem explores the relationship between subgroups defined by congruence conditions and those defined abstractly, particularly in the context of groups acting on combinatorial structures such as rooted trees. Branch groups, which often act recursively on such trees, reveal rich subgroup structures through their interaction with concepts like branch and rigid kernels. Which property characterizes the branch kernel of a regular branch group acting on a rooted tree? 1) It is always non-abelian and infinite. 2) It consists solely of elements with infinite order. 3) It is trivial for all regular branch groups. 4) It has a simple structure and is cyclic. 5) It is isomorphic to the whole automorphism group of the tree. 6) It is abelian. 7) It is necessarily nilpotent of class greater than one.

✓ Correct Answer:

The correct answer is 6) It is abelian..

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Title: The congruence subgroup problem for branch groups Year: 2009 Paper ID: e8b7e9a960c57b1ed15ab445163bbe3c32b664b5 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/e8b7e9a960c57b1ed15ab445163bbe3c32b664b5 Abstract: We state and study the congruence subgroup problem for groups acting on rooted trees, and for branch groups in particular. The problem is reduced to the computation of the congruence kernel, which we split into two parts: the branch kernel and the rigid kernel. In the case of regular branch groups, we prove that the first one is abelian while the second has finite exponent. We also establish some rigidity results concerning these kernels.We work out explicitly known and new examples of non-trivial congruence kernels, describing in each case the group action. The Hanoi tower group receives particular attention due to its surprisingly rich behaviour.

Question 631 multiple-choice

Quantum algorithms are increasingly being explored to solve signal processing problems such as linear prediction, which estimates future signal values from past data. These approaches leverage quantum phenomena and specialized circuits to potentially outperform classical methods for large-scale tasks. Which key technique enables exponential speedup in estimating signal autocorrelation within quantum linear prediction algorithms? 1) Quantum phase estimation 2) Quantum amplitude amplification 3) Quantum Fourier transform 4) Quantum error correction 5) Quantum random walk 6) Quantum entanglement swapping 7) Quantum teleportation

✓ Correct Answer:

The correct answer is 3) Quantum Fourier transform.

📚 Reference Text:

Title: Quantum Linear Prediction for System Identification and Spectral Estimation Applications Year: 2023 Paper ID: c2b023b415f2edeb680d1f12525258ff63e3e623 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c2b023b415f2edeb680d1f12525258ff63e3e623 Abstract: This paper presents the design and implementation of quantum algorithms and circuits for linear prediction. The intended applications are system identification, spectral estimation, and speech processing. A frequency-domain method that uses the quantum Fourier transform is developed to estimate the autocorrelation sequence of the signal and a quantum circuit is designed to estimate the linear prediction parameters. The quantum linear prediction performance is evaluated in system identification, autoregressive (AR) spectral estimation, and speech analysis applications. More specifically, we evaluate the performance of the quantum linear predictor in terms of quantum noise effects, qubit precision, and overall computational requirements. Comparisons of quantum versus classical linear prediction are presented.

Question 632 multiple-choice

In quantum information theory, mutually unbiased bases (MUBs) are crucial for optimal state discrimination and tomography, while the Hidden Subgroup Problem (HSP) plays a central role in quantum algorithms over finite groups. The Heisenberg group $ H_p $ serves as an important example of a non-abelian group with rich representation theory. Which property of the quantum Fourier sampling approach over the Heisenberg group $ H_p $ fundamentally limits its ability to efficiently solve the Hidden Subgroup Problem in this context? 1) The measurement outcome distributions are uniform and do not depend on the hidden subgroup index. 2) Quantum Fourier sampling always collapses the state into an abelian subgroup. 3) The projections used are rank greater than one, causing ambiguity in measurements. 4) Weil sums introduce non-uniformity that makes subgroup identification impossible. 5) The existence of more than $ p+1 $ mutually unbiased bases obscures subgroup structure. 6) The subgroup structure of $ H_p $ is trivial, so no information is encoded. 7) Classical randomized algorithms outperform quantum Fourier sampling for $ H_p $.

✓ Correct Answer:

The correct answer is 1) The measurement outcome distributions are uniform and do not depend on the hidden subgroup index..

📚 Reference Text:

Section 3.1). For i,j∈Fp, we obtain by a straightforward computation that Pk;i,j= 1 p/summationtext µ,ν∈Fpωk((µ 2)i+µj−(ν 2)i−νj) p |ν/an}bracketri}ht/an}bracketle{tµ|. Hence,Pk;i,j=|ψk;i,j/an}bracketri}ht/an}bracketle{tψk;i,j|, where |ψk,i,j/an}bracketri}ht=1√p/summationdisplay µ∈Fpω−k((µ 2)i+µj)|µ/an}bracketri}ht, i,j∈Fp,k∈F∗ p. In the casei=∞,j∈Fp, we getPk;∞,j=|ψk;∞,j/an}bracketri}ht/an}bracketle{tψk;∞,j|, where|ψk;∞,j/an}bracketri}ht=|−j/an}bracketri}htj∈Fp,k∈F∗ p. Thus for all k∈F∗ p,i∈Fp∪ {∞},j∈Fp,rank(Pk;i,j) = 1andPk;i,jis an orthogonal projection onto |ψk;i,j/an}bracketri}ht. Forj,j′∈Fp,weget/bardblPk;∞,jPk;∞,j′/bardbl=δj,j′. Fori,i′,j′∈Fp, weget/bardblPk;i,jPk;∞,j′/bardbl=1√p. For i,i′,j,j′∈Fp,weget /bardblPk;i,jPk;i′,j′/bardbl=|/an}bracketle{tψk;i,j|ψk;i′,j′/an}bracketri}ht|=1 p/summationdisplay µ∈Fpωk((µ 2)(i−i′)+µ(j−j′)). Toevaluate thelast term above, weneed thefollowing fact ab out quadratic Weil sums in Fp. 6 Fact 1([LN94,Theorem 5.37]) Leth(X)∈Fp[X]beadegreetwopolynomial. Then,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay x∈Fpωh(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=√p. By Fact 1, if i/ne}ationslash=i′,|/an}bracketle{tψk;i,j|ψk;i′,j′/an}bracketri}ht|=1√pirrespective of jandj′. Ifi=i′, it is easy to see that |/an}bracketle{tψk;i,j|ψk;i′,j′/an}bracketri}ht|=δj,j′. Tosummarise, wehave shown the following result: Lemma1 Supposepis an odd prime. Let i,i′∈Fp∪ {∞},j,j′∈FpandAi,j,Ai′,j′be two order p subgroups of Hpother than the centre ζ(Hp). Letρk, wherek∈F∗ p, be an irreducible representation of Hpof degreep. LetPk;i,jbedeﬁned by Pk;i,j:=1 p/summationtext a∈Ai,jρk(a)and letPk;i′,j′bedeﬁned similarly. Then Pk;i,j,Pk;i′,j′are rank one orthogonal projections, and their overlap is gi ven by /bardblPk;i,jPk;i′,j′/bardbl=/braceleftBigg1√p:i/ne}ationslash=i′,for allj,j′, δj,j′:i=i′. Thus, for any k∈F∗ p, thevectors |ψk;i′,j′/an}bracketri}htform aset of (p+1)mutually unbiased bases for Cp. 2.4 Failureofexistingmethods to solvethe HSPover Hp A straightforward classical randomised algorithm for the H SPoverHpis as follows: Query f:Hp→Sat O(p)random elements of Hp. If wedo not ﬁnd a1,a2∈ Hp,a1/ne}ationslash=a2such thatf(a1) =f(a2), wedeclare {1}to be the HSP of f. Suppose we do ﬁnd such a pair a1,a2. Then there is a unique order psubgroupA ofHpsuch thata−1 1a2∈A.fcan now be thought of as a function on Hp/A. QueryfatO(√p)random elements of Hp/A. If we do not ﬁnd b1,b2∈ Hp/A,b1/ne}ationslash=b2such thatf(b1) =f(b2), we declare Ato be the HSP off. Suppose we do ﬁnd such a pair b1,b2. LetB=/an}bracketle{tA,b−1 1b2/an}bracketri}ht. If|B|=p3, declare the HSP to beHp. If|B|=p2, queryfat an element c∈ Hp,c/ne}ationslash∈B. Iff(c) =f(B), declare the HSP to be Hp, else declare the HSP to be B. The correctness of the algorithm follows from the subgroup structure of Hpand the birthday paradox. A matching lower bound of Ω(p)for classical randomised algorithms can be proved using the subgroup structure of Hpand Yao’s minimax principle. Suppose the HSP is Ai,j, for somei∈F∗ p,j∈Fp. It can be shown (see Section 3.3 for details) that Fourier sampling gives a p-dimensional representation with probability 1−1 p, and eachp-dimensional representation hasequal probability toshowup. Supposeon esuchrepresentation ρk,k∈F∗ pshowsup. The natural representation basis |a/an}bracketri}ht,a∈Fpis the basis |ψk;∞,j/an}bracketri}ht, wherej∈Fp. TheZp-Fourier transform of thenatural representation basisisthebasis |ψk;0,j/an}bracketri}ht,wherej∈Fp. ByLemma1,theprobability distribution obtainedbymeasuringthecolumnsof ρkinthenaturalrepresentationbasisorinthe Zp-Fouriertransformof the natural representation basis is the uniform distributi on. This shows that weak Fourier sampling, strong Fourier sampling in the natural representation basis of Hpas well as strong Fourier sampling in the Zp- Fourier transform of the natural representation basis give exponentially small information about the index i ofAi,j. Recall that Hp=A0,0⋉N∞∼=Zp⋉(Zp×Zp). Suppose wetry toperform Fourier sampling over the abelian group A0,0×N∞∼=Zp×Zp×Zp(theforgetful abelian method ) instead of the non-abelian group Hp. LetFdenote the Fourier transform over Zp×Zp×Zp,i.e., F=p−3/2/summationdisplay a,b,c,x,y,z ∈Fpωax+by+cz|a,b,c/an}bracketri}ht/an}bracketle{tx,y,z|. For abelian groups G, the probability distributions obtained by Fourier sampli ng overGare independent of the actual coset of the hidden subgroup that arises on measur ing the function value; however, they depend 7 of course on thehidden subgroup. But since

Question 633 multiple-choice

In computational quantum chemistry, the analysis of wavefunctions and molecular orbitals is essential for understanding reaction mechanisms and energy profiles, particularly in simple exchange reactions such as H + H₂. The role of symmetry-adapted orbitals and the effect of different computational methods are central to predicting accurate barrier heights and electronic structures. In the H + H₂ exchange reaction, what is the primary feature of the optimum SOGI wavefunction at the saddle point that distinguishes its description of the reaction mechanism? 1) Inclusion of explicit bond breaking and formation at the transition state 2) Orbitals adopt symmetry functions, making all bonds equivalent without requiring bond breaking 3) All three orbitals become nonbonding, eliminating bonding interactions 4) The wavefunction localizes electrons exclusively on the lone hydrogen atom 5) Symmetry is broken, favoring localized bonding on either side of the molecule 6) Only one bonding orbital persists throughout the reaction pathway 7) SOGI wavefunction predicts a lower barrier height than experiment due to delocalized electrons

✓ Correct Answer:

The correct answer is 2) Orbitals adopt symmetry functions, making all bonds equivalent without requiring bond breaking.

📚 Reference Text:

ergy of -1.59978 a.u., whereas the optimum SOGI wavefunction using symmetry functions yields an en ergy of -1.62382 a.u. Thus at the saddle point the optimum orbitals are symmetry functions. Figures 4-6 are plots of the SOGI orbitals for nu clear configurations 2-4. The G1, SOGI, and CI ener gies, the optimum values of 'Z, and the SOGI orbital energies are given in Table IV. The barrier height from the SOGI calculations is "-'17 kcal as compared to an experimental value27 of 9.8 kcal and a value from the CI calculations24 of 11 kcal. The predicted barrier height might be slightly lower if basis functions and saddle-point configurations appropriate for the SOGI wavefunction were used rather than those from the CI calculations. However, the decrease would prob ably not be large. From Figs. 4, 5, and 6 we see that the best inde- 27 I. Shavitt, J. Chern. Phys. 49, 4048 (1968). Esoor z Ecr• -1.651526 oo -1.66959 -1.642236 3.5° -1.663036 -1.626282 16.4° -1.653359 -1.623820 20.9° -1.652073 b This calculation was restricted so that orbitals 1 and 2 were symmetry related (see Sec. III.A.2); this is not necessarily the best G1 wavefunction. pendent particle description of the reaction is as fol lows: For large separations between H2 and H, the orbitals 1 and 2 which are localized on HA (the H farther from the lone H) and HB (the H nearer the lone H) form a bond between HA and HB and are also weakly bonding to He (the lone H). Orbital 3 which is localized on He has a node midway between HA and HB and is weakly bonding to HB and anti bonding to HA, with these two effects roughly cancel ling. As we move along the minimum-energy path, orbitals 1 and 2 remain approximately equally bonding by strengthening the Hs-He bond and weakening the HrHB bond. At the saddle point, these two orbitals are CT0 functions and the HA-HB and Hs-He bonds are equivalent. In response to these changes in orbitals 1 and 2, the negative lobe of orbital 3 gets larger and larger until at the saddle point it becomes the CTu or bital. Thus, even at the saddle point, this orbital is roughly nonbonding. As we pass over the saddle point in the exchange reaction the O'u orbital begins to localize on HA and the u0 orbitals begin to localize on HB and He and eventually form H+H 2• We see that through out the reaction there are two strongly bonding orbitals and one nonbonding orbital and that it is not necessary to break a bond during the chemical reaction. Thus the SOGI orbitals yield a clear and reasonable descrip- ion of the H + H2 reaction. An alternate independent-particle description of the saddle point is for orbitals 1 and 2 to localize the right and left, but we have shown above that this is possible only if the G 1 spin coupling is used. The best ( G 1) localized orbitals are shown in

Question 634 multiple-choice

Computational group theory deals with algorithms for fundamental tasks in finite groups, especially when group elements are accessed via black-box operations. Parameters related to group structure, such as ν, play a crucial role in determining the complexity and feasibility of these algorithms. Which of the following statements is true regarding the parameter ν for a finite group G? 1) ν is always equal to the number of Sylow subgroups in G. 2) ν gives the minimal number of generators required for G. 3) ν equals the order of the center of G for non-Abelian groups. 4) ν is defined only for Abelian groups and is always zero. 5) ν is the smallest natural number such that every non-Abelian composition factor of G has a faithful permutation representation of degree at most ν. 6) ν measures the maximal order of elements in G. 7) ν is always infinite for non-solvable groups.

✓ Correct Answer:

The correct answer is 5) ν is the smallest natural number such that every non-Abelian composition factor of G has a faithful permutation representation of degree at most ν..

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3 Group algorithms In [5] Beals and Babai described probabilistic Las Vegas alg orithms for several important tasks related the structure of ﬁnite black-box groups. In order to state their result, we will need some deﬁnitions, in particular the deﬁnition of the parameter ν(G), where Gis any group. Let us recall that a composition series of a group Gis a sequence of subgroups G=G1✄G2✄ ...✄Gt= 1 such that each Gi+1is a proper normal subgroup in Gi,and the factor groups Gi/Gi+1 are simple. The factors Gi/Gi+1are called the composition factors ofG.It is known that the composition factors of Gare – up to order, but counted with multiplicities – uniquely determined byG. Beals and Babai deﬁne the parameter ν(G) as the smallest natural number νsuch that every non-Abelian composition factor of Gpossesses a faithful permutation representation of degree at mostν. Bydeﬁnition, forasolvablegroup Gtheparameter ν(G)equals1. Also, representation-theoretic results of [11] and[18] imply that ν(G) is polynomially boundedinthe inputsize in many important special cases, such as permutation groups or even ﬁnite matr ix groups over algebraic number ﬁelds. Theconstructive membership test in Abelian subgroups is the following problem. Given pairwise commuting group elements h1,...,h r,gof a non necessarily commutative group, either express gas a product of powers of the hi’s or report that no such expression exists. Babai and Szemer ´ edi have shown in [2] that under some group operations oracle this pro blem cannot be solved in polynomial time by classical algorithms. This test is usually required only forelementary Abelian groups , that is groups which are isomorphic to Zn pfor some prime pand integer n. We can now quote part of the main results of [5]. Theorem 4. (Beals and Babai [5], Theorem 1.2) LetGbe a ﬁnite black-box group with not necessarily unique encoding. Assume that the followings ar e given: (a) a superset of the primes dividing the order of G, 4 (b) an oracle for taking discrete logarithms in ﬁnite ﬁelds o f size at most |G|, (c) an oracle for the constructive membership tests in eleme ntary Abelian subgroups of G. Then the following tasks can be solved by Las Vegas algorithms of running time polynomial in the input size +ν(G): (i) test membership in G, (ii) compute the order of Gand a presentation for G, (iii) ﬁnd generators for the center of G, (iv) construct a composition series G=G1✄G2✄...✄Gt= 1forG, together with nice repre- sentations of the composition factors Gi/Gi+1, (v) ﬁnd Sylow subgroups of G. Apresentation ofGis a sequence g1,...,gsof generator elements for G, together with a set of group expressions in variables x1,...,x s, called the relators, such that g1,...,gsgenerate Gand the kernel of the homomorphism from the free group F(x1,...,x s) ontoGsendingxitogiis the smallest normal subgroup of F(x1,...,x s) containing the relators. We remark that the generators in the presentation may diﬀer from the original generators of G. Anice representation of a factor Gi/Gi+1means a homomorphism from Giwith kernel Gi+1 to either a permutation group of degree polynomially bounde d in the input size

Question 635 multiple-choice

In knot theory and quantum algebra, the Markov trace on Hecke algebras plays a fundamental role in constructing knot invariants such as the Jones and HOMFLYPT polynomials. Its properties and normalization are crucial for ensuring invariance under manipulations of braid representations of links. Which property or modification of the Markov trace is specifically required to ensure invariance under both type I and type II Markov moves in the construction of knot invariants? 1) Multiplication by the determinant of the braid matrix 2) Restriction to abelian subalgebras 3) Extension to the quantum double of the symmetric group 4) Addition of a constant term for each braid strand 5) Replacement of all generators with involutions 6) Use of unnormalized trace directly on closed braids 7) Normalization involving powers of q and combinatorial coefficients

✓ Correct Answer:

The correct answer is 7) Normalization involving powers of q and combinatorial coefficients.

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follows. It is a theorem of Ocneanu [16] (see [19] for a simple proof) that for every 1 ≤k<ℓ and for each n, there is a linear function tr kℓ:Hn(q)→Cuniquely determined by 1. tr kℓ(1n) = 1 for 1 n∈Hn(q) 2. tr kℓ(wv) = tr kℓ(vw) forw,v∈Hn(q) 3. tr kℓ(enι(w)) =[k−1]ℓ [k]ℓ[2]ℓtrkℓ(w) for allw∈Hn(q) for alln. Pawel Wocjan and Jon Yard 15 b /squigglerightb Fig. 5.4. Trace closure after the second Markov move on some b∈B3. The dotted box on the left represents the braid σ3ι(b)∈B4. We have omitted the strands which close the two leftmost strands for visibility. Using (3.7), a direct calculation shows that 3. implies that forw∈Hn(q), trkℓ(gnι(w)) =/parenleftbigg q−(1 +q)[k−1]ℓ [k]ℓ[2]ℓ/parenrightbigg trkℓ(w) =qk+1 2 [k]ℓtrkℓ(w). (5.17) Such a function is known as a positive Markov trace on the inductive limit H∞(q) of the Hecke algebras Hn(q). This trace has a representation-theoretic formula which we recall below in (5.21). But ﬁrst, we show how it can be used to express the Jones and one-variable HOMFLYPT polynomials. One ﬁrst extends this trace to Bnvia the identiﬁcation σ± i/ma√sto→g± i. Cyclicity of the trace implies that tr kℓis invariant under type I Markov moves. However, it is not invariant under the second move because (5.17) implies t hat multiplication of w∈Hn(q) by eithergnorg−1 naﬀects the trace nontrivially. We will therefore construct , for eachn, a normalized version trkℓ:Hn(q)→Cof the Markov trace which is also invariant under type II Markov moves, i.e. trkℓ(σ±1 nι(b)) =trkℓ(b) (5.18) for eachb∈Bn. For this, we deﬁne trkℓ(b) = [k]n−1 ℓq−k+1 2e(b)trkℓ(b). (5.19) Here,e(b) is the sum of the exponents of the generators in the expressi on ofb. Now ifb∈Bn andb′=σ±1 nι(b)∈Bn+1, trkℓ(b′) = [k]n ℓq−k+1 2e(b′)trkℓ(b′) = [k]n ℓq−k+1 2(e(b)±1)q±k+1 2 [k]ℓtrkℓ(b) = [k]n−1 ℓq−k+1 2e(b)trkℓ(b) =trkℓ(b) as required. To see that this normalized trace is equal to the one-variable HOMFLYPT of the trace closure of b, we follow [19] by considering an arbitrary oriented link di agramL0 described as the trace closure of some braid. Choose a crossi ng in the braid arbitrarily – we assume it to be positive without loss of generality. By a ty pe I Markov move, one can then express L0as the trace closure of a braid where the given positive cross ing is at the bottom, i.e. L0=/hatwiderσib. Inserting a positive or negative crossing adjacent to the o ne speciﬁed 16The Jones Polynomial: Quantum Algorithms and Applications in Quantum Complexity Theory above results in the oriented links L+=/hatwidestσ2 ibandL−=/hatwideb. Using the formula (5.19) to express the normalized trace in terms of the linear trace tr kℓ, and replacing σ2 iusing the quadratic relation (3.4), it is straightforward to check that qk/2trkℓ(σ2 ib) =q−k/2trkℓ(b) + (q1/2−q−1/2)trkℓ(σib). Therefore, the normalized trace satisﬁes the deﬁnition (2. 2) of the one-variable HOMFLYPT polynomial from the skein relation (2.1) with H(k) cσib(q) =trkℓ(σib), so that H(k) bb(q) = [k]n−1 ℓq−k+1 2e(b)trkℓ(b). (5.20) Forw∈Hn(q), the Markov trace has the following representation-theor etic formula trkℓ(w) =/summationdisplay λ∈Λ(k,ℓ) ns(k,ℓ) λTrπ(k,ℓ) λ(w). (5.21) Here, the Markov

Question 636 multiple-choice

The Learning with Errors (LWE) problem is central to post-quantum cryptography and underpins many lattice-based cryptographic schemes. Advances in algorithms for solving LWE, such as variations of the BKW algorithm and statistical distinguishers, directly affect the practical security of these systems. Which statement most accurately describes a recent advancement in the use of FFT distinguishers within BKW-style algorithms for LWE? 1) FFT distinguishers require more samples than the optimal distinguisher when using the same number of hypotheses. 2) FFT distinguishers have been found to increase the memory requirements compared to standard distinguishers. 3) The sample complexity of FFT distinguishers is worse than previously modeled in theoretical predictions. 4) FFT distinguishers are less efficient at detecting correlations in the data than traditional linear algebraic methods. 5) The performance of FFT distinguishers is highly sensitive to sample dependencies introduced by sample amplification. 6) FFT distinguishers show reduced efficiency when LF2 improvements are applied to the algorithm. 7) FFT distinguishers can match the sample complexity of the optimal distinguisher and perform better in practice than earlier theoretical predictions indicated.

✓ Correct Answer:

The correct answer is 7) FFT distinguishers can match the sample complexity of the optimal distinguisher and perform better in practice than earlier theoretical predictions indicated..

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Title: Modeling and simulating the sample complexity of solving LWE using BKW-style algorithms Year: 2022 Paper ID: be694277feb558025f6ccdeaffc635ebb65f12dd Source: semantic-scholar URL: https://www.semanticscholar.org/paper/be694277feb558025f6ccdeaffc635ebb65f12dd Abstract: The Learning with Errors (LWE) problem receives much attention in cryptography, mainly due to its fundamental significance in post-quantum cryptography. Among its solving algorithms, the Blum-Kalai-Wasserman (BKW) algorithm, originally proposed for solving the Learning Parity with Noise (LPN) problem, performs well, especially for certain parameter settings with cryptographic importance. The BKW algorithm consists of two phases, the reduction phase and the solving phase. In this work, we study the performance of distinguishers used in the solving phase. We show that the Fast Fourier Transform (FFT) distinguisher from Eurocrypt’15 has the same sample complexity as the optimal distinguisher, when making the same number of hypotheses. We also show via simulation that it performs much better than previous theory predicts and develop a sample complexity model that matches the simulations better. We also introduce an improved, pruned version of the FFT distinguisher. Finally, we indicate, via extensive experiments, that the sample dependency due to both LF2 and sample amplification is limited.

Question 637 multiple-choice

Quantum algorithms for the hidden subgroup problem (HSP) in finite groups rely on exploiting group-theoretic structure to reduce computational complexity, especially for p-groups with bounded nilpotency class. Techniques such as polycyclic presentations and recursive subgroup analysis play a crucial role in making these reductions efficient. In the quantum algorithmic reduction of HSP for a p-group of constant nilpotency class, which structural subgroup is specifically constructed to capture all elements of order p or 1, enabling reduction to exponent p cases? 1) The Frattini subgroup 2) The derived subgroup 3) The commutator subgroup 4) The subgroup G* consisting of all elements of order p or 1 5) The socle of the group 6) The lower central series subgroup 7) The subgroup of fixed points under an automorphism

✓ Correct Answer:

The correct answer is 4) The subgroup G* consisting of all elements of order p or 1.

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p|G|) calls to P. Lemma 3. LetCbe a class of p-groups of constant nilpotency class that is closed under ta king subgroups and factor groups. Then the instances of the hidden subgroup problem in members of C, when the subgroup is either trivial or of cardinality p,can be reduced to groups in Cof exponent p. Proof.Ifpis not larger than the class of G, the algorithm of [5] is applicable. Otherwise the elements of orderpor 1 form a subgroup G∗, see Chapter 12 of [7]. The hidden subgroup His also a subgroup of G∗since|H| ≤p. The function hiding HinGalso hides it in G∗, therefore the reduction will consist in determining G∗. We design an algorithm that ﬁnds G∗by induction on the length of reﬁned polycyclic presentations. If |G|=pthenG∗=G. Otherwise, let G=G1⊲G2⊲...⊲Gs⊲{1}be a reﬁned polycyclic presentation with s≥2. It is easy to construct a presentation where Gsis a subgroup of the center of G, which we suppose from now on. For the ease of notation we set M=G2andN=Gs. We ﬁrst describe the inductive step in a simpliﬁed case, with the additio nal hypothesis ( G/N)∗=G/N. Observe that the hypothesis is equivalent to saying that the map φ:x/ma√sto→xpsends every element of Ginto N. From this it is also clear that the hypothesis carries over to M, that is ( M/N)∗=M/N. We further claim that either G∗=GorG∗is a subgroup of Gof index p. In fact this follows Theorem 12.4.4 of [7] which states that the map φis constant on cosets of G∗and distinct on diﬀerent cosets. ¿From a polycyclic presentation of Git can be read oﬀ whether or not G=G∗. IfG∗=Gwe are done. Otherwise we compute inductively M∗. IfM∗=MthenG∗=M. IfM∗is a proper subgroup of MthenM∗has index p2inG. 6 Pick an arbitrary u∈M\M∗andy∈G\M. By the assumptions, up=gjusfor some integer 0 < ju< p, andyp=gjysfor some integer 0 ≤jy< p. Recall that in the polycyclic presentation model, computing normal words for upandyp– using fast exponentiation – amounts to computing juandjy. Setx=ujyj−1 u. For thisxwe have xp=yp, and therefore xy−1∈G∗. Sincexy−1∈G∗\M∗, we have G∗=/a\}bracketle{tM∗,xy−1/a\}bracketri}ht. In the general case ﬁrst ( G/N)∗is computed inductively. If ( G/N)∗=G/Nthen one proceeds as in the simpliﬁed case. Otherwise we set K= (G/N)∗N. We claim that G∗=K∗. For this we will show that. G∗⊆K. To see this, let xbe an element of G∗. Thenx=yzwherey∈G/Nandz∈N. We show that yis in (G/N)∗which implies that x∈K. Indeed, yp=ypzp= (yz)p= 1, where the ﬁrst equality follows from|N|=p, the second from N≤Z(G) and the third from x∈G∗. Finally observe that ( K/N)∗=K/N sinceK/N= (G/N)∗. Therefore one can determine K∗inductively as in the simpliﬁed case. Letc(s) denote the number of recursive calls when the length of a present ation iss. In the simpliﬁed case the number of calls is s−1. Therefore in the general case we have c(s) =c(s−1)+s−2, whose solution isc(s) =O(s2). 4 The quantum algorithm The quantum part of our algorithm, up to technicalities, follows the s ame lines as the algorithm given in [12] for extraspecial groups. The proof in these section are included h

Question 638 multiple-choice

Topological quantum computing leverages exotic phases of matter, such as those found in fractional quantum Hall systems, to achieve robust and fault-tolerant quantum information processing. Mathematical frameworks like super-modular categories and braid group representations play a crucial role in describing the fusion and braiding of anyons and symmetry defects. Which property of the ν=1/3 fractional quantum Hall liquid specifically enables direct theoretical analysis of bilayer SU(3)₁ topological order? 1) Its excitations are non-abelian anyons 2) It lacks Z₂ bilayer-exchange symmetry 3) Its modular category does not split under stacking 4) Its modular category splits, simplifying the study of bilayer SU(3)₁ order 5) The phase difference in its gate protocol is always trivial 6) It cannot be described using super-modular categories 7) Universal quantum gates cannot be generated from its symmetry defects

✓ Correct Answer:

The correct answer is 4) Its modular category splits, simplifying the study of bilayer SU(3)₁ order.

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the presence of a magnetic ﬁeld [16]. But it is within the realm of possibilit y that engineering a topological phase with symmetry defects could present a shorter path to a universal quantum computer. On this note, bilayer systems are interesting to study becau se any topological phase can be stacked into a bilayer phase that then possesses a Z2bilayer-exchange symmetry. The best understood examples of topological phases are fractio nal quantum Hall liquids, so it is realistic to consider what experiments can be performed to p robe symmetry defects in bi- layer fractional quantum Hall systems. Since fractional qu antum Hall systems are electron systems, strictly speaking one needs a theory of G-extensions of super-modular categories SYMMETRY DEFECTS AND THEIR APPLICATION TO TOPOLOGICAL QUAN TUM COMPUTING 19 in order to formulate the problem mathematically. One impor tant exception is the ν=1 3 fractional quantum Hall liquid, where the associated super -modular category splits into SU(3)1and the physical electron. In this case the system can be stud ied via a bilayer SU(3)1topological order. An interesting question is then how to us e symmetry defects to generate a universal gate set for quantum computing from thi s abelian topological phase, which is available in almost every fractional quantum Hall l ab. Apart from the physical motivations, symmetry defects in th e context of TQC are inter- esting for mathematical reasons alone. Going forward, it is highly desirable to understand the precise mathematical relationship between the (projec tive) braid group representations afforded by symmetry defects and the braid group representa tions afforded by the under- lying anyons. Relatedly, it is an open question to determine the precise relationship of the braid group representations arising from a UMTC Cand its gauging (C× G)G. An answer to this question could help resolve the Property-F conjecture [21], one of most well-known open problems in modular tensor category theory. A systematic and detailed study of TQC with symmetry defects is to appear in [12]. ACKNOWLEDGEMENTS We would like to thank Maissam Barkeshli, Parsa Bonderson, a nd Meng Cheng for con- tributing ﬁgures, Xingshan Cui for sharing Mathematica cod e, as well as Eric Samperton and the referees for helpful feedback on the draft. 20 COLLEEN DELANEY AND ZHENGHAN WANG Appendices In the appendices we provide a derivation of the matrix entri es of the T-gate protocol from Section 4.2, with data and graphical formulas for reference . σ σ σ σ X X X X In Appendix A we list all the data needed to compute the matrix entries, including the conjectured G-crossed braiding and F-symbols for the Z2-extension of Ising⊠Ising . In Appendix B we provide a list of helpful moves in the graphical calculus for UGxBFCs (and the UMTCs they contain) for reference. In Appendix C, we demonstrate a sequence of moves in the graphical calculus that produces the formula s for the matrix entries. We ﬁnd that the phase difference induced on the basis states is i ndeed eπi/4as anticipated by the physical

Question 639 multiple-choice

In lattice-based cryptography, attacks on the Learning With Errors (LWE) problem often utilize lattice reduction and statistical techniques to recover part or all of a secret vector from noisy equations. Understanding the impact of various error sources and algorithmic steps is crucial for analyzing attack success and complexity. Which of the following factors is described as being polynomial in its contribution to the complexity of dual attacks on LWE, specifically relating to the misidentification of candidate secret vectors? 1) Deq, associated with LWE error 2) Dround, arising from modulus switching rounding errors 3) The required number of input vectors for FFT 4) Dfpf, accounting for false positives and false negatives 5) The dimension of the secret vector 6) The threshold parameter C for statistical testing 7) The size of the constructed basis B

✓ Correct Answer:

The correct answer is 4) Dfpf, accounting for false positives and false negatives.

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FL(senum,˜sﬀt)with˜sﬀt/n⌉}a⊔ionslash=sﬀtmorediﬃcult.Consequently,itissimplertoassume that we can only recover senumwith certainty. We can therefore reformulate the algorithm of [MAT22] as looking for ˜senumsuch that there exists ˜sﬀtsuch that FL(˜senum,˜sﬀt)>C. The claimed relationships hold with high probability over the choice of th e elements in L, assuming suﬃciently many vectors. Here “suﬃciently many” de- pends on the input LWE parameters as well as parameters of the du al attack algorithm. In particular, this magnitude depends on (a) Deqwhich is an expo- nential factor coming from the LWE error, (b) Droundwhich is an exponential factor coming from the rounding when modulus switching and (c) Dfpfnis a polynomial factor controlling false positives and negatives. The follo wing lemma formally states the required magnitudes. Lemma 2 (Adapted from Theorem 5.2 in [MAT22]). Let(n,m,q,χ s,χe) be LWE parameters and (β0,β1,kenum,kﬀt,klat,p)be parameters for Algorithm 2. Let0< µ <1be the targeted failure probability. Let σebe the standard devia- tion ofχe,σsbe the standard deviation of χsandα=σe/σs. Denote by ℓthe expected Euclidean length of the vectors returned by Algori thm 1. Then, Algo- rithm 2 succeeds with probability at least 1−µfor C=φfp(µ)·/radicalbig Darg·DandD≥Deq·Dround·Darg·Dfpfn(µ) where φfp(µ), φfn(µ) =Φ−1/parenleftbigg 1−µ 2·Nenum(senum)·pkfft/parenrightbigg , Φ−1/parenleftig 1−µ 2/parenrightig 12 Deq= exp/parenleftigg 4/parenleftbiggπτ q/parenrightbigg2/parenrightigg forτ2=α−2·/⌊ard⌊le/⌊ard⌊l2+/⌊ard⌊lslat/⌊ard⌊l2 m+klatℓ2, Dround= kfft−1/productdisplay t=0 st/negationslash=0 sin/parenleftig πst p/parenrightig πst p  −2 forsﬀt= (s0,...,s kfft−1), Darg=1 2+exp/parenleftigg −8/parenleftbiggπτ q/parenrightbigg2/parenrightigg andDfpfn(µ) = (φfp(µ)+φfn(µ))2. Remark 1. In [MAT22] two theorems are given establishing costs: Theorem 5.2 (essentially reproduced above), which is with respect to a ﬁxed tup le (s,e) and Theorem 5.9 which is with respect to distributions χs,χeand which will be restated in Lemma 6. Algorithm 2: Dual Attack of [MAT22] Input:LWE parameters ( n,m,q,χs,χe), integersβ0,β1/lessorequalslantd, integers kenum,kﬀt,klatsuch thatkenum+kﬀt+klat=n, an integer p/lessorequalslantq, an integerD, a real number C, and an LWE pair ( A,b)∈Zm×n q×Zm q. Output: (Guess of) the ﬁrst kenumcoordinates of sor⊥. 1Decompose Aas/bracketleftbig AenumAﬀtAlat/bracketrightbig of respective dimensions m×kenum, m×kﬀtandm×klat. 2Compute the matrix B=/bracketleftbiggαIm0 AT latqIklat/bracketrightbigg whereα=σe σs 3Run Algorithm 1 on the basis Bwith parameters β0,β1,Dto get a list LofD short vectors. 4forevery value ˜senumin decreasing order of likelihood according to the secret distribution do 5Initialise a table Tof dimensions p×p×···p/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright kffttimes 6forevery short vector ( αxj,ylat) inLdo 7 Compute yj,ﬀt=xT j·Aﬀt. 8 Compute yj,enum=xT j·Aenum. 9 Add exp/parenleftBig (xT j·b−yT j,enum·˜senum)·2iπ q/parenrightBig to cell⌊p qyj,ﬀt⌉ofT. 10Perform FFT on T 11iffor any˜sﬀt, the real part of1 ψ(˜sfft)T[˜sﬀt] is larger than Cthen 12 return˜senum. 13return⊥ Remark 2. In Line 4 the algorithm asks to enumerate candidates for ˜senumin decreasing order of likelihood. A technique for achieving this enumer ation is discussed in [MAT22, Section 5.4]. Roughly, this is accomplished by enum erating with increasingboundsonlog-likelihoodsforcandidates,see[MAT22] fordetails. 13 3 Quantum Guessing In this section, we give a more rigorous analysis of the classical and q uantum guessing complexity for targets following a discrete Gaussian distrib ution. In more detail, let Xbe a random variable on a ﬁnite or countable set. We consider the problem of guessing the value taken by Xby asking questions of the form “IsXequal tox?” until the answer is yes. This

Question 640 multiple-choice

Quantum hypothesis testing involves determining which of several possible quantum states a system is in, often by applying measurements optimized for distinguishability. The rate at which error probabilities decrease and the mathematical objects used to quantify this process are central to quantum information theory. In quantum hypothesis testing, which statement accurately describes the exponential error rate for distinguishing between convex combinations of quantum states and a reference state σ? 1) The exponential error rate is always lower for convex combinations than for individual hypotheses. 2) The exponential error rate for convex combinations is the same as for individual hypotheses. 3) The error rate is determined solely by the trace distance between the states. 4) Convex combinations cannot be distinguished optimally in the asymptotic limit. 5) The error rate depends on the purity of the reference state, not on the mixture. 6) Only nonsingular reference states allow for exponential error rates. 7) The error rate is independent of the spectra of the quantum states involved.

✓ Correct Answer:

The correct answer is 2) The exponential error rate for convex combinations is the same as for individual hypotheses..

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in the case of a nonsingular σ, since the singular case can in dependence of the set {ρs}s∈Seither be treated identically by reduction of the dimension of the und erlying system or because an optimal test is already given by projecting onto the usual frequ ency typical subspaces corresponding toσ. This reasoning applies to the case of the AVQS as well. Our technical formulation of the quantum Sanov theorem for i.i.d. hy potheses reads as follows. Theorem 2. LetS={ρs}s∈S⊂ S(Cd)andσ=/summationtextd i=1ti|˜ei/an}bracketri}ht/an}bracketle{t˜ei| ∈ S(Cd)where all ti>0be given. We will use the abbreviations rs:=rρsfor the spectra of the states of our set and ˜rs:= ˜rρsfor the distributions that are induced by pinching them to the eigenbasis of σ. Let a sequence (Pn)n∈Nof projections deﬁned by Pn:=/summationdisplay (¯f,¯λ)∈ΛεPn f,λ, (21) where the frequency typical subspaces are deﬁned with respe ct to{˜ei}d i=1and Λε:={(p,q)∈P([d]×[d])| ∃s∈S:/bardblp−˜rs/bardbl ≤εand/bardblq−rs/bardbl ≤ε} (22) and the˜rsare the probability distributions deﬁned by ˜rs(i) :=/an}bracketle{t˜ei,ρs˜ei/an}bracketri}ht. For this sequence, both inf s∈Str{Pnρ⊗n s} ≥1−2−n(αε2−d2 nlog(2n))(23) and 1 nlogtr{Pnσ⊗n} ≤ −inf s∈SD(ρs/bardblσ)+Θ(n,ε,d,σ) (24) are true. In addition, for every 0≤A≤ /BD⊗nsatisfying infs∈Str{Anρ⊗n s} ≥1−ν(for some ν∈(0,1)) it holds that −inf s∈SD(ρs/bardblσ)−Θ′(n,ν,d)≤1 nlogtr{Anσ⊗n}. (25) The functions ΘandΘ′are deﬁned by Θ(n,ε,d,σ) :=d2 nlog(2n)+ε|log(ε/d)|+dε·max i|logti|, (26) Θ′(n,ν,d) := Θ(n,n−1 4,d)−2d6 nlog(2n)+1 nlog(1−ν−2−α√n (2n)2d2). (27) This proves the quantum Stein’s Lemma by replacing εwith a suitably chosen sequence (εn)n∈Nthat satisﬁes εn=1√α/radicalBig 1 nlog(ν)+d2 nlog(2n)for alln∈N. Remark 2. By linearity of the trace, this implies that we can distingui sh the hypotheses conv({ρ⊗n s}s∈S) andσ⊗nat the same rate. Using Theorem 2, we can prove the following result that relates the K ostka numbers to quantum states: Theorem 3. First, if Kf,λ>0then there exists a quantum state with spectrum ¯λand pinching ¯f. Second, if ρis a quantum state with spectrum rρand pinching ˜rρ, then for every ε >0there exists N∈N and for all n≥Nthere isλ∈Yd,nand a corresponding frequency such that /bardbl¯λ−rρ/bardbl ≤n−1/4and /bardbl¯f−˜rρ/bardbl ≤n−1/4andKf,λ>0. 7 Our next result concerns the arbitrarily varying quantum source. The proof of this theorem rests on the proof of Theorem 2, so that the functions and objects deﬁne d in that theorem are used here as well: Theorem 4. Under the same preliminaries as above we have: For a ﬁnite set S′={ρs}s∈Sand with the projections Pndeﬁned as in Theorem 2 using the set S:= conv(S′), the estimates tr{Pnρsn} ≥1−2−nαε2+2d2+|S| nlog(2n)(28) and 1 nlogtr{Pnσ⊗n} ≤ − min ρ∈conv(S′)D(ρ/bardblσ)+Θ(n,ε,d,σ)+|S| nlog(2n) (29) and for every operator 0≤An≤ /BD⊗nsatisfying minsn∈Sntr{Anρsn} ≥1−νfor some ν∈(0,1)we have −min ρ∈conv(S′)D(ρ/bardblσ)−Γ(n,ν,d,σ)≤1 nlogtr{Anσ⊗n} (30) where Γ(n,ν,d,σ) := Θ(n,n−1/4,d,σ)+Γ′(n,ν,d)+8d6 nlog(2n) (31) Γ′(n,ν,d) :=1 nlog[(1−ν−2−n[α(n−1/4−2|S|n−1)−|S|−2d2 nlog(2n)])(2n)−8d2]. (32) This proves Theorem 1 by replacing εwithεn:=1√α/radicalBig 1 nlog(ν)+d2+|S| nlog(2n). In case that |S|=∞, the following is true: For the sequence (δn)n∈Ndeﬁned by δn:= 12n−1/4d2(n∈N) and the depolarizing channel Nδn:B(Cd)→ B(Cd),a/ma√sto→(1−δn)a+δn1 d /BDwe have inf sn∈Sntr{N⊗n δn(Pn)ρsn} ≥1−2−n(αε2−d2+√n nlog(2n))(33) and tr{N⊗n δn(Pn)σ⊗n} ≤2−n(infρ∈conv(S′)D(ρ/bardblσ)+d·log(1−δn)−Θ(n,ε,d,σ))(34) as well as, for every ν∈[0,1)and0≤An≤ /BD⊗n: Ifinfsn∈Sntr{Anρsn} ≥1−ν, then tr{Anσ⊗n} ≤2−n(infρ∈conv(S′)D(ρ/bardblσ)−Γ(n,ν,d,Nδn(σ),√n)−c(δn)). (35) whereτlog(d) +h(τ) +τlog(1/td) =:c(τ,d,σ). Note that, for an arbitrary ν∈(0,1), we have limn→∞Γ(n,ν,d,Nδn(σ),√n) = 0and that also limn→∞c(δn,d,σ) = 0. This proves Theorem 1

Question 641 multiple-choice

Quantum algorithms can offer significant speedups over classical algorithms, particularly in cryptanalysis applications like solving the Learning With Errors (LWE) problem. Efficient quantum memory access, such as using qRAM, is crucial for exploiting these speedups, but practical circuit complexity imposes limitations. Which statement most accurately describes the practical limitation of implementing quantum random-access memory (qRAM) for quantum algorithms processing large datasets? 1) qRAM gates universally achieve constant-time access regardless of dataset size due to quantum parallelism. 2) QRACM can be implemented with linear resources but achieves exponential speedup in memory access time. 3) Even the most efficient QRACM implementations require at least linear time in the size of the dataset, making ideal polylogarithmic time complexity unattainable without highly parallel circuits. 4) QRAQM models are always less efficient than QRACM due to increased error rates in quantum data. 5) Quantum circuits inherently allow unrestricted superposition-based access to classical memory with no resource constraints. 6) Classical memory access times do not affect the overall complexity of quantum algorithms using qRAM. 7) Partitioning the input data removes all memory access bottlenecks in quantum algorithms.

✓ Correct Answer:

The correct answer is 3) Even the most efficient QRACM implementations require at least linear time in the size of the dataset, making ideal polylogarithmic time complexity unattainable without highly parallel circuits..

📚 Reference Text:

constant probability, AO(N) =n0. The algorithm runs in expected time T=O(√n0)(orO(√ N)if n0=⊥), uses a polynomial number of qubits and makes an expected nu mberT of calls to O. Furthermore, if the algorithm returns i∈[0,N−1], then it only queriesOon values in [0,min(N−1,2i)]. Memory Access. “Baseline” quantum circuits are simply built using a univer- sal quantum gate set. A requirement for many quantum algorithms to process data eﬃciently is to be able to access classical data in quantum super position. Such algorithms use quantum random-access memory, often deno ted as qRAM, and require the circuit model to be augmented with the so-called “qR AM gate”. These qRAM gates are assumed to have a time complexity polylogarith mic in the amount of classical data stored, so that each call is not time co nsuming. This model is inspired by the classical RAM model where we usually assume m emory access in time O(1). Given an input integer 0 ≤i≤r−1, which represents the index of a memory cell, and many quantum registers |x0,...xr−1/an}⌊rack⌉⊔ri}h⊔, which represent stored data, the qRAM gate fetches the data from register xi, possibly in superposition: |i/an}⌊rack⌉⊔ri}h⊔|x0,...xr−1/an}⌊rack⌉⊔ri}h⊔|y/an}⌊rack⌉⊔ri}h⊔ /ma√s⊔o→ |i/an}⌊rack⌉⊔ri}h⊔|x0,...xr−1/an}⌊rack⌉⊔ri}h⊔|y⊕xi/an}⌊rack⌉⊔ri}h⊔. Following the terminology of [Kup13], there are three types of qRAMs : –If the input iis classical, then this is the plain quantum circuit model. We can implement it using a universal quantum gate set. –If thexjareclassical,we have classical memory with quantum random access (QRACM). The qRAM gate becomes |i/an}⌊rack⌉⊔ri}h⊔|y/an}⌊rack⌉⊔ri}h⊔ /ma√s⊔o→ |i/an}⌊rack⌉⊔ri}h⊔|y⊕xi/an}⌊rack⌉⊔ri}h⊔. 9 –Ingeneral,wehave quantummemory with quantumrandom access (QRAQM). This is the most powerful quantum memory model where the data ar e also in superposition. In our algorithm for the dual attack, we will be using QRACM. It is pos sible to implement a QRACM using a universal quantum gate set, albeit at a c onsid- erable cost. Given a classical data set {x0,···,xr−1}, one can construct, in time ˜O(r), a circuit using ˜O(r) qubits that implements a QRACM for this data set. The obtained circuit then allows query in the form |i/an}⌊rack⌉⊔ri}h⊔|y/an}⌊rack⌉⊔ri}h⊔ /ma√s⊔o→ |i/an}⌊rack⌉⊔ri}h⊔|y⊕xi/an}⌊rack⌉⊔ri}h⊔and has circuit depth O(polylog(r)) [GLM08, KP20, MGM20, HLGJ20]. Note that even low depth implementation of QRACM has at least Ω(r) gates, hence has time complexity at least Ω(r) by our deﬁnition. Therefore, the assumption that the qRAM gates have time complexity polylog( r) is very strong and corresponds to parallelevaluationofthecircuit.Thefeasibilityofconstructingeﬃc ientQRACM is further discussed in e.g. [AGPS20]. 2.5 The Classical Algorithm of [GJ21, MAT22] In this section, we give an overview of the algorithm in [MAT22], reprod uced in Algorithm 2. Our quantum algorithm will be a modiﬁed version that relie s essentially on the same analysis for the correctness but a new analy sis for the quantum complexity. We give an overviewof the involved parameters in Table 1. We are given a sample from LWE n,m,χ s,χe, whereχsandχehave small varianceσ2 sandσ2 erespectively. We partition sinto three components: s= senum sﬀt slat  wheresenumhaskenumcoordinates, sﬀthaskﬀtcoordinates, and slathasklat= n−kenum−kﬀtcoordinates. We split Ainto three components accordingly as well: A= [AenumAﬀtAlat] so thatA·s=Aenum·senum+Aﬀt·sﬀt+Alat·slat. We deﬁne the matrix: B=/parenleftbigg αIm0

Question 642 multiple-choice

In finite group theory and algebraic topology, the poset of nontrivial elementary abelian $p$-subgroups, denoted $\mathcal{A}_p$ for a finite group $G$, plays a crucial role in understanding group cohomology and homological properties. Quillen’s dimension property links the maximal degree of non-zero homology of this poset to the group's $p$-rank. For a finite group $G$ with trivial $p$-core, which of the following correctly describes Quillen’s dimension property at a prime $p$? 1) The poset $\mathcal{A}_p$ has non-zero homology in degree equal to the $p$-rank of $G$ minus $1$ 2) The poset $\mathcal{A}_p$ is always contractible for all primes $p$ 3) The poset $\mathcal{A}_p$ has non-zero homology only in degree 0 4) The poset $\mathcal{A}_p$ is empty whenever $G$ is simple 5) The poset $\mathcal{A}_p$ always has trivial homology in all degrees 6) The poset $\mathcal{A}_p$ has non-zero homology in degree equal to the order of $G$ 7) The poset $\mathcal{A}_p$ is always disconnected for $p=2$

✓ Correct Answer:

The correct answer is 1) The poset $\mathcal{A}_p$ has non-zero homology in degree equal to the $p$-rank of $G$ minus $1$.

📚 Reference Text:

Title: Maximal subgroups of exceptional groups and Quillen’s dimension Year: 2023 Paper ID: 0c85508c283492d7e085b10449a5a38a2b349b6a Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0c85508c283492d7e085b10449a5a38a2b349b6a Abstract: Given a finite group $G$ and a prime $p$, let $\mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. The group $G$ satisfies the Quillen dimension property at $p$ if $\mathcal{A}_p(G)$ has non-zero homology in the maximal possible degree, which is the $p$-rank of $G$ minus $1$. For example, D. Quillen showed that solvable groups with trivial $p$-core satisfy this property, and later, M. Aschbacher and S.D. Smith provided a list of all $p$-extensions of simple groups that may fail this property if $p$ is odd. In particular, a group $G$ with this property satisfies Quillen's conjecture: $G$ has trivial $p$-core and the poset $\mathcal{A}_(G)$ is not contractible. In this article, we focus on the prime $p = 2$ and prove that the $2$-extensions of the exceptional finite simple groups of Lie type in odd characteristic satisfy the Quillen dimension property, with only finitely many exceptions. We achieve these conclusions by studying maximal subgroups and usually reducing the problem to the same question in small linear groups, where we establish this property via counting arguments. As a corollary, we reduce the list of possible components in a minimal counterexample to Quillen's conjecture at $p = 2$.

Question 643 multiple-choice

Quantum algorithms frequently utilize group theory to solve problems like period finding, with special attention to both abelian and non-abelian groups. The quaternion group Q8 exemplifies a non-abelian group with distinctive subgroup properties relevant in mathematics and quantum computing. Which of the following statements about the quaternion group Q8 is true? 1) Q8 has only one nontrivial proper subgroup. 2) Q8 is an abelian group of order 8. 3) All subgroups of Q8 are cyclic but not normal. 4) The Cayley table for Q8 includes only commutative element combinations. 5) Q8 contains exactly two nontrivial proper normal subgroups. 6) Every subgroup of Q8 is normal, making Q8 a Hamiltonian group. 7) No subgroup of Q8 can be generated by -1, i, j, or k.

✓ Correct Answer:

The correct answer is 6) Every subgroup of Q8 is normal, making Q8 a Hamiltonian group..

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H2=f0;2g=h2i H3=f0;1;2;3g=h1i Since a periodic function fwith period ron a cyclic grouphxihides a subgrouphxri, this case is treated as a special case of period ﬁnding, a pivotal part of Shor’s algorithm. LetG=Z4;and suppose the black-box function f:Z4!Z4is such that f(x) =f(x+r)for all x2Z4:The algorithm is implemented by the circuit in Figure 4.2. Figure 4.2: Quantum Circuit for Period Finding over Q4 j00i H 2 UfQFT† NM j00i 44 The Hadamard gate puts the qubits in the top register into an equally-weighted superposi- tion: 1p 22(j00i+j01i+j10i+j11i) =1 2(j0i+j1i+j2i+j3i): (4.19) Application of the unitary transformation Ufproduces the state jy1i=Uf1 2å x2Z4jxij0i=1 2å x2Z4jx;f(x)i=1 2(j0;f(0)i+:::+j3;f(3)i): (4.20) TheQFT† jxi7!1 23 å y=0e2pixy=4jyi (4.21) applied to the top register results in the state jy2i=1 4å x;ye2pixy=4jy;f(x)i =1 4j0i [jf(0)i+jf(1)i+jf(2)i+jf(3)i] +1 4j1i h jf(0)i+e2pi=4jf(1)i+e2pi2=4jf(2)i+e2pi3=4jf(3)ii +1 4j2i h jf(0)i+e2pi2=4jf(1)i+e2pi4=4jf(2)i+e2pi6=4jf(3)ii (4.22) +1 4j3i h jf(0)i+e2pi3=4jf(1)i+e2pi6=4jf(2)i+e2pi9=4jf(3)ii : Ifr=2, the hidden subgroup is H2=h2i, and f(0) =f(2) =g1,f(1) =f(3) =g2,g16=g2. 45 Hence, the statejy2ibecomes jy2i=1 4å x;ye2pixy=4jy;f(x)i =1 2j0i [jg1i+jg2i] +1 4j1i h jg1i 1+epi +jg2i epi=2+epi3=2i +1 4j2i jg1i 1+e2pi +jg2i epi+e3pi (4.23) +1 4j3i h jg1i 1+e3pi +jg2i epi3=2+epi9=2i and reduces to jy2i=1 2(j0;g1i+j0;g2i+j2;g1ij2;g2i): (4.24) Measurement of the top register now yields, with probability1 2, either 0 or 2. If 0 is measured, the algorithm must be repeated. 4.3 The Group Q8 4.3.1 Group Presentation The group Q8, usually referred to as the quaternions, is given by Q8=hx;yjx4=1;x2=y2;y1xy=x1i: (4.25) Following Hamilton’s notation, the group is often written as Q8= 1;i;j;kji2=j2=k2=i jk=1 (4.26) However, it was Arthur Cayley who ﬁrst observed that the quaternions form a group, so it is ﬁtting to display its full Cayley table. 46 Table 4.1: Cayley Table for Q8 11 ii jj kk 1 11 ii jj kk 11 1i i -j jk k i ii1 1 kkj j ii i 11k k jj j jjk k1 1 ii jj j kk 11i i k kk jjii1 1 kk kj j ii 11 Letqbe a quaternion given by q=a+jb (4.27) where a;b2C. A matrix representation of qinC22is given by 0 B@a b ¯b¯a1 CA: (4.28) Then the group elements are represented in terms of matrices as follows: i7!0 B@i0 0i1 CA;j7!0 B@0 1 1 01 CA;k=i j7!0 B@0i i01 CA (4.29) 47 A quaternion q=a+ib+jc+kd,a;b;c;d2Rcan be represented by the following matrix inR44: 0 BBBBBBB@abcd b ad c c d ab dc b a1 CCCCCCCA(4.30) 4.3.2 Subgroup Classiﬁcation Theorem 4.3.1. The only nontrivial proper subgroups of Q 8are H1=f1;1g=h1i H2=f1;1;i;ig=hii (4.31) H3=f1;1;j;jg=hji H4=f1;1;k;kg=hki Proof. Suppose a subgroup of Q8contains i. Then it must include i, since i1=i. Similarly, any subgroup of Q8that contains j(ork) must includej(ork). Moreover, if a subgroup containsi, it must include1= (i)2. Similarly, any subgroup that contains j(ork) must contain1= (j)2= (k)2. Finally, if a subgroup contains any two of i;j;k, it must be Q8itself since i j=k;jk=iand ik=j. Lemma 4.3.2. All subgroups of Q 8are normal. Proof. The group Q8has a unique element of order 2 and it generates H1=h1i. Then every subgroup of Q8, except forf1g, possess H1as its

Question 644 multiple-choice

Modular tensor categories (MTCs) provide the mathematical foundation for topological quantum computation, encoding information in anyons and their braiding statistics. Understanding the properties and classification of MTCs is essential for assessing their potential in quantum algorithms. Which property of a modular tensor category is directly linked to the ability to achieve universality in topological quantum computation by approximating any unitary transformation using anyon braiding? 1) Density of irreducible unitary braid representations in the unitary group 2) Existence of a non-degeneracy condition for simple objects 3) Classification of categories by fixed rank 4) Compatibility of braiding, twist, and duality 5) Finiteness of the number of simple objects 6) Definition of dual objects and morphisms 7) Provision of generating functions for ranks of categories

✓ Correct Answer:

The correct answer is 1) Density of irreducible unitary braid representations in the unitary group.

📚 Reference Text:

studied related to the quantum com puting applications. One question is whether the images of the irreducible unitar y braid representations (see Remark 2.1) aﬀorded by a unitary MTC are dense in the unit ary group. This is related to a sine qua non of quantum computation known as universality . Progress towards answering this question has been made in [ FLW] and was extended by FROM QUANTUM GROUPS TO UNITARY MODULAR TENSOR CATEGORIES 3 Larsen, Wang, and the author in [ LRW ]. Another problem is to prove the con- jecture of Z. Wang: There are ﬁnitely many MTCs of a ﬁxed rank (see Subsection 2.2). This has been veriﬁed for ranks 1,2,3 and 4: see [ O1] and [ O2] for ranks 2 and 3 respectively, and [ BRSW ] for both ranks 3 and 4. It is with this conjecture in mind that we provide generating functions for ranks of cat egories in Subsection 4.7. Acknowledgements. The author wishes to thank the referees for especially careful readings of previous versions of this article and fo r comments leading to a much-improved exposition. Special thanks also to Z. Wang fo r many useful discus- sions on topological quantum computation. 2. General Deﬁnitions We give the basic categorical deﬁnitions for modular tensor categories, remark on some consequences and describe the crucial condition of u nitarity. 2.1. Axioms. In this subsection we outline the axioms for the categories w e are interested in. We follow the paper [ T1], and refer to that paper or the books by Turaev [ T2] or Kassel [ K] for a complete treatment. LetObe a category deﬁned over a subﬁeld k⊂C. A modular tensor category is a semisimple ribbon Ab-category Owith ﬁnitely many isomorphism classes of simple objects satisfying a non-degeneracy condition. We u nravel these adjectives with the following deﬁnitions. 1. Amonoidal category is a category with a tensor product ⊗and an iden- tity object 1 1 satisfying axioms that guarantee that the ten sor product is associative (at least up to isomorphism) and that 1 1⊗X∼=X⊗1 1∼=X for any object X. See [Mc] for details. 2. A monoidal category has duality if there is a dual object X∗for each object Xand morphisms bX: 1 1→X⊗X∗, dX:X∗⊗X→1 1 satisfying (IdX⊗dX)(bX⊗IdX) = Id X, (dX⊗IdX∗)(IdX∗⊗bX) = Id X∗. The duality allows us to deﬁne duals of morphisms too: for any f∈Hom(X, Y) we deﬁne f∗∈Hom(Y∗, X∗) by: f∗= (dY⊗IdX∗)(IdY∗⊗f⊗IdX∗)(IdY∗⊗bX). 3. Abraiding in a monoidal category is a natural family of isomorphisms cX,Y:X⊗Y→Y⊗X satisfying cX,Y⊗Z= (Id Y⊗cX,Z)(cX,Y⊗IdZ), cX⊗Y,Z= (cX,Z⊗IdY)(IdX⊗cY,Z). 4 ERIC C. ROWELL 4. Atwist in a braided monoidal category is a natural family of isomorp hisms θX:X→X satisfying: θX⊗Y=cY,XcX,Y(θX⊗θY). 5. In the presence of a braiding, a twist and duality these str uctures are com- patible if θX∗= (θX)∗. A braided monoidal category with a twist and a compatible dua lity is a ribbon category. 6. An Ab-category is a monoidal category in which all morphism spaces are k-vector spaces and the composition and

Question 645 multiple-choice

Quantum error correction in physical systems often involves tradeoffs between code size, symmetry properties, and the ability to correct different types of errors. Theoretical bounds help determine the limits of error correction when codes are only approximately covariant and resources are finite. Which statement accurately describes a fundamental limitation on quantum error-correcting codes with approximate covariance and finite subsystem resources? 1) An adversary can irreversibly decohere logical information by extracting partial global charge from any single subsystem, preventing perfect error correction. 2) Arbitrary unknown-location errors can always be perfectly corrected if the code is covariant, regardless of subsystem size. 3) Entanglement infidelity after erasure does not depend on the number of subsystems or the magnitude of charge fluctuations. 4) Perfect transversality guarantees exact error correction even in codes with approximate covariance. 5) The three-qubit repetition code can correct any type of single-qubit error, including erasures, without restriction. 6) If some subsystems are immune to noise, lower bounds on residual errors are independent of the noise model. 7) Codes with exact covariance and finite size can always prevent adversaries from accessing logical state information in any scenario.

✓ Correct Answer:

The correct answer is 1) An adversary can irreversibly decohere logical information by extracting partial global charge from any single subsystem, preventing perfect error correction..

📚 Reference Text:

version of the Eastin-Knill theorem that applies iferasure is only approximately correctable and/or the code is only approximately covariant. We focus especially on how accurately erasure can be corrected when the number ofphysical subsystems is large but finite or the dimension ofeach subsystem is large but finite. In Theorem 1, we consider codes that can correct erasure of a subsystem only approximately, and we derive a lowerbound on the worst-case entanglement infidelity ϵ worstthat can be achieved by the best recovery map after an erasure. In keeping with the findings of Ref. [12], this lower bound approaches zero when the number nof subsystems approaches infinity or when the fluctuations of the local charge of individual subsystems grow without bound. The idea behind the lower bound is that, if the number ofsubsystems and the local charge fluctuations are both finite,then some information about the value of the global logical charge is available to an adversary who takes possession of a single physical subsystem, resulting in irreversibledecoherence of the logical state. In Theorem 2 andCorollary 3, we extend the result by relaxing several assumptions. This more general theorem applies when the code is not exactly covariant, when the logical chargeoperator is not exactly transversal, and when more than one subsystem is erased. The focus on erasures at known locations as a noise model in Theorems 1 and 2 is motivated by the following considerations. First, commonly encountered quantumerror-correcting codes are designed to correct any errorswhich affect a sufficiently small number of subsystems; this design implies, in particular, that erasures at known locations can be corrected. Second, if a code can correcterrors at unknown locations, it can certainly also correct errors at known locations, and, therefore, our results apply. Third, to derive general lower bounds on the accuracy oferror correction in covariant codes, it makes sense to choose a noise model in which any one of the physical subsystems could potentially be damaged by the noise.Otherwise, the physical charge operators fT igmight be supported on subsystems that are unaffected by the noise. Even in that case, we could study how the channelcomplementary to the noise channel leaks information to the environment as in the proof of Theorem 2. But the corresponding lower bound on the residual error wouldthen depend on detailed features of the noise model,beyond merely specifying how many subsystems can be simultaneously affected (see, e.g., Refs. [84,85] ). To appreciate how the accuracy of error correction depends on the noise model, consider the three-qubit repetition code αj0iþβj1i→αj000iþβj111i, which is covariant with respect to the logical charge T L¼3Zand TA¼Z1þZ2þZ3, where Z¼j0ih0j−j1ih1j. This code can correct an Xerror on any of the three qubits, but it cannot correct a Zerror on any qubit, nor can it correct erasure of any qubit. Nevertheless, under a noise model where only Xerrors occur, it is a covariant code that corrects single-qubit errors. On the other hand, if a Zerror occurs at a random location, then a measurement on the environment can reveal information about the

Question 646 multiple-choice

In quantum field theory and quantum information, the notion of entanglement entropy must be reconsidered in systems with local gauge symmetry due to the special structure of gauge-invariant observables. Understanding how entanglement is defined and computed in gauge theories is fundamental for analyzing quantum correlations in such systems. Which of the following best explains why the Hilbert space in a gauge theory does not generally factorize as a tensor product across spatial regions A and B? 1) The presence of superselection sectors prevents any algebraic decomposition of the Hilbert space. 2) All physical observables are strictly local in gauge theories, forbidding any nonlocal correlations. 3) Gauge symmetry requires assigning independent gauge degrees of freedom to each region. 4) The absence of entanglement in gauge theories eliminates the need for factorization. 5) The boundary conditions imposed by gauge invariance cause the Hilbert space to become trivial. 6) Certain gauge-invariant operators, such as Wilson loops, can cross between regions, preventing a clean tensor product factorization. 7) The extended Hilbert space always allows for a tensor product structure, making non-factorization irrelevant.

✓ Correct Answer:

The correct answer is 6) Certain gauge-invariant operators, such as Wilson loops, can cross between regions, preventing a clean tensor product factorization..

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the gauge-invariant observables associated to region A. In particular, we deﬁne the operators in Aas (the restriction to Hphysof) all operators in L(H)that are local to Aand commute with all gauge transformations. In general, given a region A, although we can deﬁne the local gauge-invariant observables A, the full algebra of gauge-invariant operators L(Hphys)will not factorize as L(Hphys)?=A⊗Bfor any complementary factor B.14Therefore, the usual deﬁnitions of partial trace, reduced density matrix, and entanglement entropy must be modiﬁed. One option is to work entirely within the “extended” or “unphysical” Hilbert space Hand calculate entanglement entropy according to the usual deﬁnition. However, this deﬁnition may be undesirable in the context of gauge theory, where one often wants to view the theory as intrinsic toHphys, without reference to some embedding. A second option for deﬁning entanglement entropy in gauge theories is to use the “algebraic” deﬁnition of entanglement entropy, applied to the gauge-invariant Hilbert space and observables. In this algebraic framework [8, 21], one deﬁnes a reduced density matrix and entanglement entropy for any state, taken with respect to a sub-algebra such as A. The following brief review follows Section 3 of [18]. We will assume a ﬁnite-dimensional Hilbert space throughout. To deﬁne the entanglement entropy, ﬁrst note that for general Hilbert space Vand algebra A⊂L(V), there exists a decomposition of the Hilbert space as a direct sum of tensor products, V=/circleplusdisplay iVi⊗/tildewideVi (84) such that the operators OA∈Aare precisely those which take the form OA=/summationdisplay iOA,i⊗1/tildewideVi(85) for some OA,i∈L(Vi). (This follows from an elementary version of the Artin-Wedderburn theo- rem.) Schematically, we can also write A=/circleplusdisplay iL(Vi)⊗1/tildewideVi. (86) Deﬁne Πias the projectors onto the sectors Vi⊗/tildewideViin equation (84). For any state|ψ/angbracketright∈V, note Πi|ψ/angbracketright/angbracketleftψ|Πi∈L(Vi)⊗L(/tildewideVi), and deﬁne ρA,i= Tr/tildewideVi(Πi|ψ/angbracketright/angbracketleftψ|) pi= Tr(Πi|ψ/angbracketright/angbracketleftψ|)(87) Then one deﬁnes the entanglement entropy S(|ψ/angbracketright,A)of|ψ/angbracketrightwith respect to sub-algebra Aas S(|ψ/angbracketright,A) =−/summationdisplay ipilog(pi) +/summationdisplay ipiS(ρA,i). (88) An equivalent deﬁnition is S(|ψ/angbracketright,A) =S(PA(|ψ/angbracketright/angbracketleftψ|)), (89) where the superoperator PAprojects the density matrix |ψ/angbracketright/angbracketleftψ|into the algebra A. (However, when using this shorthand deﬁnition, the image of the projection must be considered to lie in 14For a paradigmatic example demonstrating that no factorization exists, consider a Wilson loop that straddles the boundary of A, crossing from Ato¯Aand back. Such an operator cannot be written as a sum of products of gauge-invariant operators on Aand ¯A, so evidently the tensor product of the gauge-invariant algebras of observables onAand ¯Ais a proper subspace of the entire gauge-invariant algebra. Accepted in Quantum2020-09-17, click title to verify. Published under CC-BY 4.0. 23 the abstract algebra A, rather than inside its embedding A⊂L(V); the latter has extra identity factors, and the entropy of these factors is not included in the algebraic deﬁnition of the entropy.) The ﬁrst and second term of (88)are sometimes referred to as the “classical” and “quantum” pieces, respectively. In the context of gauge theory, or more generally in an operational context where observers associated to Aare restricted to operations in A, the second term alone quantiﬁes the distillable entanglement [22, 24, 25]. For this reason, the entanglement

Question 647 multiple-choice

Quantum algorithms often exploit group-theoretic properties, such as hidden subgroups and normality, to achieve computational speedups. The quaternion group Q8 serves as a classic example in group theory and quantum computation due to its non-abelian structure and notable subgroup properties. Which of the following statements is true about the subgroup structure of the quaternion group Q8? 1) Q8 has at least one non-normal subgroup. 2) All proper subgroups of Q8 are cyclic of order 2. 3) Q8’s only normal subgroup is the center {1, -1}. 4) Q8 contains no subgroups generated by j or k. 5) The subgroup generated by i in Q8 is not normal. 6) Every subgroup of Q8 is normal and generated by ±1, ±i, ±j, or ±k. 7) Q8 is abelian and all its subgroups are trivially normal.

✓ Correct Answer:

The correct answer is 6) Every subgroup of Q8 is normal and generated by ±1, ±i, ±j, or ±k..

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H2=f0;2g=h2i H3=f0;1;2;3g=h1i Since a periodic function fwith period ron a cyclic grouphxihides a subgrouphxri, this case is treated as a special case of period ﬁnding, a pivotal part of Shor’s algorithm. LetG=Z4;and suppose the black-box function f:Z4!Z4is such that f(x) =f(x+r)for all x2Z4:The algorithm is implemented by the circuit in Figure 4.2. Figure 4.2: Quantum Circuit for Period Finding over Q4 j00i H 2 UfQFT† NM j00i 44 The Hadamard gate puts the qubits in the top register into an equally-weighted superposi- tion: 1p 22(j00i+j01i+j10i+j11i) =1 2(j0i+j1i+j2i+j3i): (4.19) Application of the unitary transformation Ufproduces the state jy1i=Uf1 2å x2Z4jxij0i=1 2å x2Z4jx;f(x)i=1 2(j0;f(0)i+:::+j3;f(3)i): (4.20) TheQFT† jxi7!1 23 å y=0e2pixy=4jyi (4.21) applied to the top register results in the state jy2i=1 4å x;ye2pixy=4jy;f(x)i =1 4j0i [jf(0)i+jf(1)i+jf(2)i+jf(3)i] +1 4j1i h jf(0)i+e2pi=4jf(1)i+e2pi2=4jf(2)i+e2pi3=4jf(3)ii +1 4j2i h jf(0)i+e2pi2=4jf(1)i+e2pi4=4jf(2)i+e2pi6=4jf(3)ii (4.22) +1 4j3i h jf(0)i+e2pi3=4jf(1)i+e2pi6=4jf(2)i+e2pi9=4jf(3)ii : Ifr=2, the hidden subgroup is H2=h2i, and f(0) =f(2) =g1,f(1) =f(3) =g2,g16=g2. 45 Hence, the statejy2ibecomes jy2i=1 4å x;ye2pixy=4jy;f(x)i =1 2j0i [jg1i+jg2i] +1 4j1i h jg1i 1+epi +jg2i epi=2+epi3=2i +1 4j2i jg1i 1+e2pi +jg2i epi+e3pi (4.23) +1 4j3i h jg1i 1+e3pi +jg2i epi3=2+epi9=2i and reduces to jy2i=1 2(j0;g1i+j0;g2i+j2;g1ij2;g2i): (4.24) Measurement of the top register now yields, with probability1 2, either 0 or 2. If 0 is measured, the algorithm must be repeated. 4.3 The Group Q8 4.3.1 Group Presentation The group Q8, usually referred to as the quaternions, is given by Q8=hx;yjx4=1;x2=y2;y1xy=x1i: (4.25) Following Hamilton’s notation, the group is often written as Q8= 1;i;j;kji2=j2=k2=i jk=1 (4.26) However, it was Arthur Cayley who ﬁrst observed that the quaternions form a group, so it is ﬁtting to display its full Cayley table. 46 Table 4.1: Cayley Table for Q8 11 ii jj kk 1 11 ii jj kk 11 1i i -j jk k i ii1 1 kkj j ii i 11k k jj j jjk k1 1 ii jj j kk 11i i k kk jjii1 1 kk kj j ii 11 Letqbe a quaternion given by q=a+jb (4.27) where a;b2C. A matrix representation of qinC22is given by 0 B@a b ¯b¯a1 CA: (4.28) Then the group elements are represented in terms of matrices as follows: i7!0 B@i0 0i1 CA;j7!0 B@0 1 1 01 CA;k=i j7!0 B@0i i01 CA (4.29) 47 A quaternion q=a+ib+jc+kd,a;b;c;d2Rcan be represented by the following matrix inR44: 0 BBBBBBB@abcd b ad c c d ab dc b a1 CCCCCCCA(4.30) 4.3.2 Subgroup Classiﬁcation Theorem 4.3.1. The only nontrivial proper subgroups of Q 8are H1=f1;1g=h1i H2=f1;1;i;ig=hii (4.31) H3=f1;1;j;jg=hji H4=f1;1;k;kg=hki Proof. Suppose a subgroup of Q8contains i. Then it must include i, since i1=i. Similarly, any subgroup of Q8that contains j(ork) must includej(ork). Moreover, if a subgroup containsi, it must include1= (i)2. Similarly, any subgroup that contains j(ork) must contain1= (j)2= (k)2. Finally, if a subgroup contains any two of i;j;k, it must be Q8itself since i j=k;jk=iand ik=j. Lemma 4.3.2. All subgroups of Q 8are normal. Proof. The group Q8has a unique element of order 2 and it generates H1=h1i. Then every subgroup of Q8, except forf1g, possess H1as its

Question 648 multiple-choice

Quantum algorithms often manipulate and estimate properties of quantum systems using specialized circuits, but managing ancillary information ("garbage") and error metrics is crucial for reliable computation. The diamond norm is a widely used tool for quantifying the accuracy of quantum operations, especially in scenarios involving quantum channels and ancillary outputs. Which property makes the diamond norm particularly suitable for error analysis in quantum algorithms that may introduce garbage or are applied to subsystems entangled with external environments? 1) It measures only the average-case error over all pure input states. 2) It ignores the presence of ancillary qubits in the output. 3) It quantifies the overlap between eigenstates of the input operator. 4) It captures the worst-case difference between quantum channels, including their action on external systems via an extended Hilbert space. 5) It restricts error measurement to computational basis states. 6) It is equivalent to the spectral norm for all quantum channels. 7) It assumes perfect uncomputation of all ancillary information.

✓ Correct Answer:

The correct answer is 4) It captures the worst-case difference between quantum channels, including their action on external systems via an extended Hilbert space..

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each algorithm, it is thus more eﬃcient to leave the decision of when to uncompute to the user of the subroutine. Deﬁnition 2. Aphase estimator is a protocol that, given some n∈Z+andα∈(0,1), and any error target δ>0, produces a quantum circuit involving controlled UandU†calls to some unitary U. IfU=/summationtext je2πiλj|ψj/angbracketright/angbracketleftψj|satisﬁes an (n,α)-rounding promise then this circuit implements a quantum channel that is δ-close in diamond norm to the map: |0n/angbracketright|ψj/angbracketright→|ﬂoor(λj2n)/angbracketright|ψj/angbracketright (9) Similarly, an energy estimator is such a protocol that instead involves such calls to UHwhich is a block-encoding (see Deﬁnition 9) of a Hamiltonian H=/summationtext jλj|ψj/angbracketright/angbracketleftψj|that satisﬁes an (n,α)-rounding promise. Thequery complexity of an estimator is the number of calls to UorUHin the resulting circuit, as a function of n,α,δ. An estimator is said to ‘ have garbage ’ or ‘havemqubits of garbage ’ if it is instead close to a map that produces some j-dependent m-qubit garbage state in another register: |0n/angbracketright|0...0/angbracketright|ψj/angbracketright→|ﬂoor(λj2n)/angbracketright|garbagej/angbracketright|ψj/angbracketright (10) (Note that the quantum circuit can allocate and discard ancillae, but that does not count as garbage.) 1[Ambainis10] makes use of such a map in Algorithm 4. [KP17] states this as Theorem II.2. [KLLP18] sketches but does not analyze a protocol for this after Claim 4.5. Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 7 An estimator is said to ‘ have phases ’ if the map introduces a j-dependent phase ϕj: |0n/angbracketright|ψj/angbracketright→eiϕj|ﬂoor(λj2n)/angbracketright|ψj/angbracketright (11) Ifanestimatorisboth‘withphases’and‘withgarbage’, thenwecanjustabsorbthe eiϕj into|garbagej/angbracketrightso the ‘with phases’ is technically redundant. However, in our framework it makes more sense to treat phases and garbage independently since some of our algorithms are just ‘with phases’. The reason we measure errors in diamond norm has to do with uncomputation of approximate computations with garbage. Consider for example a transformation Vacting on an answer register and a garbage register: V|0/angbracketright|0...0/angbracketright=√ 1−ε|0/angbracketright|garbage0/angbracketright+√ε|1/angbracketright|garbage1/angbracketright (12) for some small nonzero ε. We copy the answer register into the output register: →√ 1−ε|0/angbracketright⊗|0/angbracketright|garbage0/angbracketright+√ε|1/angbracketright⊗|1/angbracketright|garbage1/angbracketright (13) and then we project the answer and garbage registers onto V|0/angbracketright|0...0/angbracketright: →√ 1−ε|0/angbracketright·√ 1−ε+√ε|1/angbracketright·√ε (14) The resulting state (1−ε)|0/angbracketright+ε|1/angbracketrightis not normalized, meaning that the projection V|0/angbracketright|0...0/angbracketrightsucceeds with some probability <1. Thus the uncomputed registers are not always returned to the |0/angbracketright|0...0/angbracketrightstate. At this point our options are either to postselect these registers to |0/angbracketright|0...0/angbracketrightor to discard them. Postselection improves the accuracy, but also implies that the algorithms do not always succeed. Since many applications of coherent energy estimation demand repeating this operation many times, it is important that the algorithm always succeeds. Thus, we need to discard qubits, so we need to talk about quantum channels. Therefore, the diamond norm is the appropriate choice for an error metric. Recall that the diamond norm is deﬁned in terms of the trace norm [AKN98]: |Λ|/diamondmath:= sup ρ|(Λ⊗I)(ρ)|1 (15) whereIis the identity channel for some Hilbert space of higher dimension than Λ. The following analysis shows how to remove phases and garbage from an estimator, even in the approximate case. Lemma 3. Getting rid of phases and garbage. Given a phase/energy estimator with phases and/or garbage

Question 649 multiple-choice

Translational tilings in the d-dimensional integer lattice $\mathbb{Z}^d$ are central objects of study in harmonic analysis and discrete mathematics. These tilings have deep connections to periodicity, group theory, and applications in fields such as crystallography and coding theory. Which statement characterizes a fundamental open conjecture regarding translational tilings by finite sets in higher-dimensional integer lattices $\mathbb{Z}^d$? 1) Every level-$k$ tiling in $\mathbb{Z}$ must be aperiodic. 2) Any tiling by infinite sets in $\mathbb{Z}^d$ is necessarily non-periodic. 3) Level-$k$ tilings in $\mathbb{Z}^2$ cannot exist for $k > 1$. 4) There are no tilings in $\mathbb{Z}^d$ that cover each point exactly once. 5) If there exists any tiling by translates of a finite set in $\mathbb{Z}^d$, then a periodic tiling also exists. 6) In one dimension, non-periodic tilings by finite sets occur frequently. 7) The existence of aperiodic tilings in $\mathbb{Z}^d$ requires that the set be infinite.

✓ Correct Answer:

The correct answer is 5) If there exists any tiling by translates of a finite set in $\mathbb{Z}^d$, then a periodic tiling also exists..

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Title: The structure of translational tilings in $\mathbb{Z}^d$ Year: 2021 Paper ID: 7ffc86cdca55db2ab1c2a0060b25a8c605084bc6 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/7ffc86cdca55db2ab1c2a0060b25a8c605084bc6 Abstract: The structure of translational tilings in $\mathbb{Z}^d$, Discrete Analysis 2021:16, 28 pp. A significant theme in harmonic analysis, already represented by three previous papers in this journal, is that of tiling. In general, a tiling of a group $G$ by a function $f$ is a collection of translates of $f$ that add up to a constant function, and there are many interesting questions about their existence and properties. An important special case, which explains the word "tiling", is when $f$ is the characteristic function of a finite set $A$ and the constant function takes the value 1 everywhere -- this is a partition of $G$ into translates of $A$. (In the non-Abelian case one specifies whether one is taking left translates or right translates.) This paper concerns tilings of $\mathbb Z^d$, and in particular $\mathbb Z^2$, by characteristic functions of finite sets, but as well as the special case just discussed, they also look at _level-$k$_ tilings, which are tilings for which the constant function takes the value $k$, or in other words every point is covered exactly $k$ times by the translates of the set in question. The paper is partly motivated by a conjecture of Jeffrey Lagarias and Yang Wang from 1996. Among other results, they showed that every tiling of $\mathbb R$ of any level $k$ is periodic. The corresponding statement for $\mathbb Z$, which was proved by Donald Newman in 1977 and is broadly similar but simpler, can be seen as follows. Note first that if an interval $r,r+1,\dots,s$ of length greater than the diameter of $A$ can be covered $k$ times by translates of $A$, then any extension of that covering to a level-$k$ tiling of $\mathbb Z$ must be unique: at each stage the smallest not fully covered integer bigger than $s$ must be covered by the smallest element of a translate of $A$ and the largest not fully covered integer smaller than $r$ must be covered by the largest element of a translate of $A$. But by the pigeonhole principle there must be two such intervals that are covered in exactly the same way, and from that the periodicity follows. A higher-dimensional tiling $A$ is said to be periodic if the set of translates used is a lattice $\Lambda$ (that is, a subgroup of $\mathbb Z^d$ of finite index). Easy examples show that higher-dimensional tilings do not have to be periodic: for instance one can cover $\mathbb Z^2$ by translates of $\{0,1,\dots,n-1\}^2$ in the obvious way and then shift each "column of squares" vertically by an arbitrary amount. However, for this example one sees that even though there exists a non-periodic tiling by $A$, there also exists a periodic tiling. Lagarias and Wang conjectured that this is always the case: that is, if there is a tiling of $\mathbb R^d$ by translates of a finite set $A$, then there must also be a periodic tiling

Question 650 multiple-choice

In quantum computing, tensor network representations such as Matrix Product Operators (MPOs) are widely used to optimize simulations of quantum circuits, including the Quantum Fourier Transform (QFT). These methods employ concepts like bond dimension, tensor decomposition, and isometric tensors to manage computational complexity and error. When representing the sequence of controlled phase gates in a Quantum Fourier Transform circuit as a tensor network MPO, what is the minimum bond dimension required to express all controlled phase gates as a single MPO using copy tensor decompositions? 1) 1 2) 2 3) n 4) χ 5) √n 6) 3 7) n²

✓ Correct Answer:

The correct answer is 2) 2.

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we introduce a tensor network repre- sentation of the QFT derived from the quantum circuit and describe how to turn it into a bond-dimension- χMPO in O(χ3n2)time with error O(ne−χlog(χ/3)/√χ). We start by reviewing how quantum circuit diagrams can be exactly reinterpreted as tensor network diagrams. Consider the deﬁnition of a controlled phase gate, (49) One can rewrite it as an inner product of two operator- valued vectors: /parenleftbig|0/angbracketright/angbracketleft0||1/angbracketright/angbracketleft1|/parenrightbig/parenleftbigg I P(θ)/parenrightbigg . (50) This decomposition can be diagrammatically represented as breaking the controlled phase gate into two order-3 tensors, and thus forming a two-site MPO: (51) The tensor on the top is an order-3 copy tensor (52) where δa,b,c=δa,bδb,candδa,bis the Kronecker δ.T h e tensor on the bottom is an order-3 phase tensor (53)In Eqs. (52) and(53), the horizontal indices x1and x/prime 1run over the indices of operators |0/angbracketright/angbracketleft0|,|1/angbracketright/angbracketleft1|,I,a n d P(θ), and the vertical indices x/prime/prime 1and x2run over the operator- valued vectors. More generally, vector or matrix indices (forming the tensor network) are the vertical indices in the tensor diagram and the horizontal indices are for opera- tors. Note that here we are reinterpreting the vertical lines: in a quantum circuit diagram, the vertical lines are used merely for indicating the control-target relations, but here we consider them as tensor indices, just like the horizontal tensor indices for operators acting on qubits. Such reinter- pretation is often used when one is considering quantum circuits as tensor networks [ 40]. We then show how a series of controlled phase gates can be turned into a single MPO with bond dimension 2. To start with, we introduce an identity of copy tensors: (54) In general, an order- ncopy tensor can be decomposed into an arbitrary number of copy tensors contracted together in arbitrary order, as long as the total number of open indices remains the same. For details, see Appendix F. This allows us to deform the tensor diagram of two adjacent controlled phase gates as follows: (55) If we contract the copy tensor and order-3 phase tensor in the center into one tensor, (56) for which we deﬁne as the order-4 phase tensor (57) 040318-9 JIELUN CHEN, STOUDENMIRE, and WHITE PRX QUANTUM 4,040318 (2023) The series of controlled phase gates in the QFT circuit can be then turned into the MPO (58) corresponding to the decomposition /parenleftbig|0/angbracketright/angbracketleft0||1/angbracketright/angbracketleft1|/parenrightbig/parenleftbiggI 0 0P/parenleftbigπ 2/parenrightbig/parenrightbigg/parenleftbiggI 0 0P/parenleftbigπ 4/parenrightbig/parenrightbigg ··· ×/parenleftBiggI P/parenleftBig π 2n−1/parenrightBig/parenrightBigg , (59) which we refer to as the “phase MPOs.” It is easy to verify that the copy tensor on the ﬁrst (i.e., top) site is an isometric tensor (60) and the order-4 phase tensor is isometric up to a scalar 2 in both upward and downward directions: (61) However, the order-3 phase tensor at the last site is not isometric, i.e., (62) Thus, up to a scaling factor of 2n−2, we can view the phase MPO as having the orthogonality center on the last site. We also introduce the Htensor (63) which is identical to an Hgate. The QFT circuit

Question 651 multiple-choice

In code-based cryptography and representation theory, analyzing the distinguishability of group substructures is essential for assessing security properties. Techniques involving irreducible representations and their projections play a crucial role in bounding this distinguishability. Which approach is used to bound the expectation EH(ρ) associated with a subgroup H in a group G for a fixed irreducible representation ρ? 1) Restricting to real-valued characters only 2) Applying the centralizer algebra of the subgroup 3) Separating irreducible components of ρ⊗ρ* by dimension and applying different bounding techniques 4) Counting fixed points of the subgroup action 5) Using only one-dimensional representations in the analysis 6) Estimating the determinant of the representation matrix 7) Relying solely on Schur’s lemma without further decomposition

✓ Correct Answer:

The correct answer is 3) Separating irreducible components of ρ⊗ρ* by dimension and applying different bounding techniques.

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factor was recommended to be (n=1024,k/prime≥644,t=38)(see Note 8.32 in [14]). Bernstein et al. [1] suggested two other sets of parameters, (n=2048,k/prime=1751,t=27) and(n=1632,k/prime=1269,t=34), that achieve the standard security against all known (classical) attacks. All of these parameters meet our requirement. Well, of course, these parameters were recommended f or the original McEliece, or for the equivalent Neider- reiter system that uses parity check matrices over the subﬁeld F2with n−k/prime=/lscriptkrows. However, if we view each element in Fq/lscriptas a vector of dimension /lscriptover the subﬁeld Fq,t h e na k×ncanonical parity check matrix over Fq/lscriptcan be viewed as a /lscriptk×nparity check matrix over Fq. 5 Bounding Distinguishability We now sketch the proof for the main theorem (Theorem 1). Fixing a nontrivial sub- group H0, and we let Πρ⊗ρ∗ σ denote the projection operator whose image is aσσ, that is, the subspace spanned by all copies of σ. Our upper bound given in Lemma 4 below generalizes the bound given in Lemma 4.3 of [15], which only applies to subgroups Hof order 2. Lemma 4. (DECOUPLING LEMMA )L e t ρbe an irrep of G. Then for any vector b∈Vρ, Var g/bracketleftbig /bardblΠρ Hgb/bardbl2/bracketrightbig ≤∑ σ∈I(ρ⊗ρ∗) χσ(H)/vextenddouble/vextenddouble/vextenddoubleΠρ⊗ρ∗ σ (b⊗b∗)/vextenddouble/vextenddouble/vextenddouble2 . Back to our goal of bounding EH(ρ)using the bound in Lemma 4, the strategy will be to separate irreps appearing in the decomposition of ρ⊗ρ∗into two groups, those with small dimension and those with large d imension, and treat them differently. If dσ is large, we shall rely on bounding χσ(H).I fdσis small, we shall control the projection given by Πρ⊗ρ∗ σ using the following lemma which was proved implicitly in [15]: Lemma 5. For any irrep σ, we have ∑b∈Bρ/vextenddouble/vextenddouble/vextenddoubleΠρ⊗ρ∗ σ (b⊗b∗)/vextenddouble/vextenddouble/vextenddouble2 ≤d2 σ. The method discussed above for bounding EH(ρ)is culminated into Lemma 6 below. Lemma 6. Letρ∈/hatwideG be arbitrary and S ⊂/hatwideG be any subset of irreps that does not contain ρ.T h e n EH(ρ)≤4|H|2/parenleftbigg χ S(H)+|S∩I(ρ⊗ρ∗)|d2 S dρ/parenrightbigg . To apply this lemma, we should choose the subset Ssuch

Question 652 multiple-choice

In quantum algorithms such as Shor’s algorithm, signal processing concepts play a vital role in accurately extracting periodicity from measurements on quantum registers. Spectral leakage poses a significant challenge to this process when the signal’s period does not align perfectly with the sampled data window. Which technique from classical signal processing is commonly applied to minimize spectral leakage caused by window discontinuities in discrete transforms used for quantum period-finding? 1) Windowing with smooth attenuation functions like Hamming or Hann windows 2) Increasing the sampling rate beyond the Nyquist frequency 3) Applying zero-padding to the data before transformation 4) Using non-unitary transforms instead of unitary ones 5) Employing linear phase shifts to realign data boundaries 6) Discarding samples at the boundaries of the data window 7) Implementing randomization of phase in the signal prior to transformation

✓ Correct Answer:

The correct answer is 1) Windowing with smooth attenuation functions like Hamming or Hann windows.

📚 Reference Text:

impulse train perfectly to a reciprocally spaced comb function9-12. Further similarities to the DFT are also realized, in particular, translations of the comb function will also generate a centred comb function i n the reciprocal space but with 0 or π rotations of the appropriate individual impulse terms (rather than a mor e finely quantized linear phase term which the DFT produces). However, t his, as required by the unitary nature of the HT, allows recovery of the original displaced comb function when the inverse HT is applied. Thus the HT appears to have the desired prope rties for the very specific requirement of the Shor algorithm, i.e. the transformation of the comb functio n resulting from the partia l collapse of the first quantum register in r esponse to a measurement made on the output register which before measurement holds in a superposition sta te the results of the modular exponentiation operations, xa mod N , a = 0 to q-1, as detailed in Section 2 . There must, therefore, be an additional disrupting factor that prevents results much better than those predicated by the simulation work reported by Fowler and Hollenberg15. This factor appears to be due to a spectral leakage phenomenon affecting the accuracy of the HT transform. As has been explained in Section 2 , the modular exponential function is periodic. However, and most importantly, this period will not , in the general case, divide the overall period of the q-1 samples of the data window. Since the modular exponential is effectively a sampled function (from a = 0 to q-1) it must be considere d, as a well known consequ ence of the data sampling effect , to be periodically extended beyond the bounds of the data cell9. Thus, if a whole number of periods of the modular exponential function are not contained within this data window, a discontinuity will result at the data ce ll boundaries which, in turn, will lead to spectral leakage effects being apparent in the transformed data array , an effect that is well known in signal processing9. This will result in the reciprocal comb function (produced by the HT from the first register comb function ) being disrupted . Thus rather than being a sequence of precisely located impulse functions there will be a spreading of spectral energy around each impulse which will in turn adversely affect the probability of the wavefunction coll apsing to the precise locations of the data impulses comprising the reciprocal comb function which is required to determine the impulse train periodicity and hence factor the integer N. A well -known signal processing technique to reduce spectral leakage is the application of a windowing function to the signal prior to transformation to the spectral domain9-11. Window functions smoothly attenuate the signal samples in the data window towards its boundaries , usually to zero, to force a periodicity of the data within the sampling window. In this way, the data samples at the cell extremities are

Question 653 multiple-choice

In the study of finite p-groups, subgroup structure and element orders are critical to classifying and understanding group properties. Tools such as commutator calculations and subgroup intersections reveal essential relationships that determine possible group types. Which of the following statements correctly describes the group of order 25 in terms of its abelian and nonabelian structure? 1) It is necessarily nonabelian and metacyclic. 2) Every group of order 25 is abelian, either cyclic or elementary abelian. 3) It is a simple group with nontrivial center. 4) The group of order 25 must contain a subgroup of order 5 that is nonabelian. 5) There exists a nonabelian group of order 25 with a trivial Frattini subgroup. 6) The group of order 25 is always noncyclic and nonabelian. 7) It is a minimal nonabelian group with maximal subgroups of order 5.

✓ Correct Answer:

The correct answer is 2) Every group of order 25 is abelian, either cyclic or elementary abelian..

📚 Reference Text:

so Φ(/an}bracketle{tt, c/an}bracketri}ht) =/an}bracketle{tc2, u/an}bracketri}htwhich gives Ω 1(/an}bracketle{tc/an}bracketri}ht)∈ {/an}bracketle{tz/an}bracketri}ht,/an}bracketle{tuz/an}bracketri}ht}. We have [ t, t′c] =zuand so Φ( /an}bracketle{tt, t′c/an}bracketri}ht) =/an}bracketle{t(t′c)2=c2, zu/an}bracketri}htwhich implies that Ω 1(/an}bracketle{tc/an}bracketri}ht)∈ {/an}bracketle{tu/an}bracketri}ht,/an}bracketle{tz/an}bracketri}ht}which together with our previous result gives Ω 1(/an}bracketle{tc/an}bracketri}ht) =/an}bracketle{tz/an}bracketri}ht. We have [ t′, tc] =zand so Φ( /an}bracketle{tt′, tc/an}bracketri}ht) =/an}bracketle{t(tc)2=c2u, z/an}bracketri}ht. If o( c)≥8, then ( c2u)2=c4and/an}bracketle{tc4/an}bracketri}ht ≥ /an}bracketle{tz/an}bracketri}ht since Ω 1(/an}bracketle{tc/an}bracketri}ht) =/an}bracketle{tz/an}bracketri}ht. In this case Φ( /an}bracketle{tt′, tc/an}bracketri}ht)/ne}ati≀nslash≥Φ(G), a contradiction. Hence o(c) = 4 and so c2=z. We have obtained a uniquely determined group of order 25given in part (b) of our theorem. From now on we may assume that Ω 1(G) =E∼=E8. (i) Assume that Ω 1(G) =E/ne}ati≀nslash≤Z(G) = Φ( G) and E/ne}ati≀nslash≤A, where Ais the unique abelian maximal subgroup of G. Then A∩E=G′,Acovers G/EandAis metacyclic. Since there are three maximal subgroups of Gcontaining E, there is at least one of them, denoted with H, which is minimal nonabelian. If H/E is noncyclic, then there are two distinct maximal subgroups X1/ne}ati≀nslash=X2ofHcontaining E. In that case E≤X1∩X2= Φ(H) = Φ( G) = Z( G), a contradiction. Hence H/E is cyclic. Since d( G/E) = 2, G/Eis abelian of type (2m,2),m≥1. Therefore A/G′∼=G/Eis of type (2m,2), where A∩H/G′∼=C2m. Let abe an element in A∩Hsuch that /an}bracketle{ta/an}bracketri}htcovers A∩H/G′. Noting that Ω 1(G) =E, we have o( a) = 2m+1and Ω 1(/an}bracketle{ta/an}bracketri}ht) =/an}bracketle{tz/an}bracketri}ht ≤G′, where z=a2m. Ift∈E−G′, then [ t, a] =u∈G′− /an}bracketle{tz/an}bracketri}htbecause His nonmetacyclic and therefore uis not a square in H. Since A/G′∼=C2m×C2, there is an element b∈A−Hsuch that 1/ne}ati≀nslash=b2∈G′andb2/ne}ati≀nslash=z. Indeed, if b2=z, then taking an element vof order 4 in/an}bracketle{ta/an}bracketri}ht, we get ( bv)2=b2v2=z2= 1, where bv∈A−H, a contradiction. Hence we get b2∈ {u, uz }. We have Φ( H) = Φ( G) =/an}bracketle{ta2, u/an}bracketri}htandG=/an}bracketle{ta, b, t /an}bracketri}ht. If [b, t]∈ /an}bracketle{tu/an}bracketri}ht, then G//an}bracketle{tu/an}bracketri}htis abelian, a contradiction. Hence [ b, t]∈ {z, uz}, o(b) = 4 and Ais abelian of type (4 ,2m+1). We set [ b, t] =zuǫandb2=uzη, ǫ, η= 0,1. First suppose that o( a)>4 so that /an}bracketle{ta4/an}bracketri}ht ≥ /an}bracketle{tz/an}bracketri}ht. In that case Φ( /an}bracketle{tb, t/an}bracketri}ht) = /an}bracketle{tb2,[b, t]/an}bracketri}ht ≤G′<Φ(G) = /an}bracketle{ta2, u/an}bracketri}htand so M=/an}bracketle{tb, t/an}bracketri}htΦ(G). The fact that Φ(/an}bracketle{tab, t/an}bracketri}ht) =/an}bracketle{t(ab)2=a2uzη,[ab, t] =zuǫ+1/an}bracketri}ht= Φ(G) gives ǫ= 0. We may assume that b2=u, i.e., η= 0. Indeed, if b2=uz=u′, then we replace H=/an}bracketle{ta, t/an}bracketri}htwithH1=/an}bracketle{ta′=ab, t/an}bracketri}ht, where o( a′) = o( a),/an}bracketle{ta′/an}bracketri}ht ≥ /an}bracketle{t z/an}bracketri}htand [a′, t] =uz=u′and so writing again aanduinstead of a′andu′, respectively, we have obtained the relations for groups Gof order 2m+4given in part (c) of our theorem. 78 Z. BO ˇZIKOV AND Z. JANKO It remains to examine the case o( a) = 4. In this case m= 1,a2=z,G is a special group of order 25, where Φ( G) =/an}bracketle{tu, z/an}bracketri}ht. We have A=/an}bracketle{ta, b/an}bracketri}ht∼= C4×C4and/an}bracketle{ta, t/an}bracketri}ht=His the nonmetacyclic minimal nonabelian group of order 24. We have [ b, t] =zuǫwith Φ( /an}bracketle{tb, t/an}bracketri}ht) =/an}bracketle{tuzη, zuǫ/an}bracketri}ht, [ab, t] =zuǫ+1with Φ(/an}bracketle{tab, t/an}bracketri}ht) =/an}bracketle{tuzη+1, zuǫ+1/an}bracketri}ht, [b, at] =zuǫwith Φ( /an}bracketle{tb, at/an}bracketri}ht) =/an}bracketle{tuzη, uz, zuǫ/an}bracketri}ht, and [ab, at] =zuǫ+1with Φ( /an}bracketle{tab, at /an}bracketri}ht) =/an}bracketle{tuzη+1, uz, zuǫ+1/an}bracketri}ht. Ifǫ=η= 0, then Φ(/an}bracketle{tab, t/an}bracketri}ht)

Question 654 multiple-choice

In relativistic quantum mechanics, the behavior of spinor fields under rotations and their representation in polar coordinates is crucial for understanding fermionic systems with rotational symmetry. The periodicity properties of spinors directly impact the allowed quantum modes in angular momentum decompositions. Which property of spinor components in polar coordinates ensures that only half-integer angular momentum modes appear in their Fourier expansion? 1) Their coefficients are invariant under 2π rotation. 2) They satisfy Dirichlet boundary conditions at ϕ = 0 and ϕ = 2π. 3) Their metric components are periodic in ϕ. 4) Their expansion uses only integer multiples of ϕ. 5) They are 2π-anti-periodic, acquiring a minus sign after a full rotation. 6) Their transformation under rotations preserves bosonic statistics. 7) Their coupling to the electromagnetic field enforces periodicity in ϕ.

✓ Correct Answer:

The correct answer is 5) They are 2π-anti-periodic, acquiring a minus sign after a full rotation..

📚 Reference Text:

i 1a b@t+ i ~ 2()a b@r +i r ~ 3()a b@+1 2 ~ 2()a b m: (13) Note that the operators iand ~ iare represented by the same matrices, but in di erent bases. Note also that thechange of basis, being unitary, conserves the components of the metric i.e. ab=1 ifa=b 0 otherwise: (14) In this new basis, the at-spacetime Dirac equation readsDa b b= 0. This can be abbreviated into i1@t2@r1 r 3@+1 22 m! = 0;(15) which we call the Polar Dirac equation (PDE). We will use this compact form in the remainder of this article when no confusion with the CDE seems possible. As usual, the coupling of the Dirac fermion with an electromagnetic eld with 3-potential ( A) = (At;Ar;A) is achieved by adding +i qAto@for= 0;1;2. We choose to set the charge qto1 and get: i1(@tiAt)2(@riAr) 1 r 3(@iA) +1 22 m! = 0:(16) Let us conclude this section by pointing out a very important property of the PDE. The second polar coor- dinateis an angle. Thus, the components Land Rof in the Cartesian spin basis, when written as functions ofrand, are 2-periodic functions of . So are the time component At, the cartesian components Ax,Ay and the polar components ArandAof the potential. The components and +of in the polar spin basis are linear combinations of Land Rwith coecients cos =2 and sin =2 . These two coecients are 2 - anti-periodic in i.e.they obeyf(+2) =f() for all 2[0;2[. It follows that the polar components and +are also 2-anti-periodic in . This expresses the fact that spinors belong to representations of the double cover of the rotation group SO(2 ;R) and, thus get an extra mi- nus sign after a rotation by 2 . Thus, the PDE is de ned overf(r;);r2R +;2[0;4[gand should only be used with initial conditions which are 2 -anti-periodic in . By construction, the PDE conserves this anti-periodicity over time. Finally, only half integer modes k=p+ 1=2, p2Zenter the decomposition of the polar spinor com- ponents in terms of Fourier modes exp(i k). Another method leading to the PDE is to use the so- called curved spacetime Dirac equation, which is actually valid for any spacetime, at or curved, and in arbitrary coordinates, and particularize the treatment to (1 + 2)D at Minkowski spacetime equipped with polar coordi- nates in 2D physical space. This derivation is presented in Appendix A. The presentation retained above in the main part of this article has three distinct advantages: it is computationally simpler, it requires less geometry, and 3 it highlights the global 2 -anti-periodicity of spinor com- ponents in the polar basis, which is not readily apparent from the purely local derivation given in Appendix A. However, things are di erent for quantum automata. There is indeed no way to obtain from a standard DQW de ned on a cartesian grid a DQW approximating the PDE. For DQWs, the easiest route is to adapt the pro- cedure presented in Appendix A for the

Question 655 multiple-choice

Quantum circuit optimization leverages graph representations and gradient-based methods to design efficient algorithms using both discrete and continuous gate parameters. Techniques such as the Frobenius norm and automatic differentiation are commonly used in this field. When optimizing a quantum circuit to implement an 8-dimensional Quantum Fourier Transform (QFT), which of the following configurations correctly reflects the minimum number of qubits and gates required when using phase shift and Hadamard gates? 1) Four qubits and twelve gates 2) Five qubits and eight gates 3) Two qubits and six gates 4) Three qubits and twelve gates 5) Four qubits and seven gates 6) Three qubits and seven gates 7) Two qubits and eight gates

✓ Correct Answer:

The correct answer is 6) Three qubits and seven gates.

📚 Reference Text:

1. Figure 3 summarizes the basic steps of the MCGS Algorithm. Please note that such a graph can also be reused for different kinds of optimizations and 6 FIG. 1. Example graph of quantum circuits. The edges are labelled by possible elementary gates. The vertices are identified with the unitary operator built by taking the product of the gates along the shortest path. A trajectory on this graph directly corresponds to a quantum circuit. FIG. 2. Example graph of quantum circuits it can be analyzed very generally to identify cy- cles, clusters and other structural properties on the effect of quantum gates.A. Optimization of Continuous Variables Several gates can contain continuous vari- ables for optimization, e.g. phase shift gates P(ϕ) = 1 0 0 exp( iϕ) (4) 7 FIG. 3. Basic steps of the Quantum MCGS Algorithm with corresponding Jacobians: ∂P(ϕ) ∂ϕ= 0 0 0 iexp(iϕ) (5) Since the involved functions are smooth and dif- ferentiable, the Jacobian of such a matrix and the Jacobian for a chain of operations is easy to compute via the product rule. Thus, given a quantum circuit of length Lwhich is described by an ordered tuple ( O(1), O(2), . . . , O (L)) of quantum gates and Φ = ( ϕ1. . . ϕ k) continu- ous variables within this chain, the resulting unitary operation is then a function U(Φ) = U(ϕ1, . . . , ϕ k). This function is typically used for the optimization of a loss function, for ex- ample the distance from a target matrix O, e.g., a DFT matrix. For numerical and implementa- tion convenience the loss function was chosen to be the Frobenius norm between OandU(Φ), L(Φ) = ∥O−U(Φ)∥F= q tr [(O−U(Φ))†(O−U(Φ))].(6)Although the Frobenius norm does not have a simple operational interpretation, it is easy to compute both numerically and also experimen- tally via a swap test. The Jacobian of the loss function is given by ∇ΦL(Φ) =∂L(Φ) ∂ϕ1, . . . ,∂L(Φ) ∂ϕk .(7) This vector can be numerically computed and used for optimization by using automatic differ- entiation. Optimization of the involved param- eters can be then carried out with a gradient descent iteration using Φt+1= Φt−η∇ΦL(Φ). Here, ηdenotes a damping factor which has been set to 0 .2 for all our experiments. III. EXPERIMENTS Since the MCGS algorithm we describe here is capable of optimizing both discrete and mixed discrete and continuous settings, the experi- ments are divided in two parts. In the first 8 part only results for discrete optimizations are presented. In the second part we also describe some experiments involving mixed discrete and continuous variables. A. Discrete quantum architecture search The first experiment we conducted targeted the optimization of the quantum Fourier trans- form. It is well known that the steps of the radix-2 FFT can be realised as a quantum cir- cuit using primarily phase shift and Hadamard gates. E.g., for an 8-dimensional FFT, three qubits are sufficient and 7 gates can be used to compute the FFT matrix.

Question 656 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) often utilize group theory and representation-theoretic tools to efficiently extract information about hidden subgroups. The Clebsch-Gordan transform plays a key role in organizing quantum states according to group symmetries and facilitating optimal measurements. In the quantum algorithm for the Heisenberg Hidden Subgroup Problem, what is the primary function of the Clebsch-Gordan transform with respect to extracting hidden subgroup information from a two-copy state? 1) It maps every solution directly to its unique subgroup element without ambiguity. 2) It converts the quadratic equation into a purely linear equation for measurement. 3) It reorganizes the state in terms of irreducible representation labels, allowing basis change and measurement that concentrate the subgroup information into specific registers. 4) It prepares the state for classical post-processing by collapsing all superpositions. 5) It measures subgroup indices by performing a Quantum Fourier Transform over the cyclic group Z₂ₚ. 6) It eliminates the need for Pretty Good Measurement by producing orthogonal states. 7) It discards indices that do not correspond to valid solutions through probability filtering.

✓ Correct Answer:

The correct answer is 3) It reorganizes the state in terms of irreducible representation labels, allowing basis change and measurement that concentrate the subgroup information into specific registers..

📚 Reference Text:

reveal iandjwith probability1 2(since only half of the values of uandvwill arise in solving the quadratic equations.) Exactly such a transform which can take the superposition over two solutions to t heuandvvalues which can be enacted using the same basic idea as that used to construct U2eﬃciently above, but instead of taking the two square roots to the square, a more comp licated quadratic equation must be solved. We refer the reader to [14] for details of this const ruction. From the above description of the PGM derived quantum algorithm fo r the Heisenberg HSP we can use the insights we have gained in solving this using a Clebsch -Gordan transform to explain the structure of the above algorithm. First note that by performing a QFT over Z2 p, measuring the outcome and producing the state in Eq. (80) we are, in eﬀect, doing something 26How a Clebsch-Gordan Transform Helps to Solve ... very near to that of the QFT over the Heisenberg group. Notice ho wever that in the original PGM state, we keep around two indices yandz. However it is easy to see that if we average the pure states in Eq. (80) over y, we obtain another pure state |i,j,y/an}bracketri}ht=1√p/summationdisplay b∈Zpω2−1b(b−1)iy+bjy|b/an}bracketri}ht. (85) In other words the yregister contains no information about the hidden subgroup. This is like the register containing the left regular representation not conta ining information about the hidden subgroup. Thus in the two copy state we could have equally we ll have dealt with the state 1 p/summationdisplay b1,b2∈Zpωuj+wi|b1,b2/an}bracketri}ht (86) where u=b1y1+b2y2 w= 2−1b1(b1−1)y1+2−1b2(b2−1)y2. (87) We see that y1andy2serve as the k1andk2irreps labels (with an exception occurring when k1=k2= 0.) Next we note that, following our observation about how to perform the Clebsch-Gordan transform on the Heisenberg group, we can perform a basis chang e /summationdisplay s,t∈Zp|s−t/an}bracketri}ht/an}bracketle{ts|⊗|(sy1+ty2)(y1+y2)−1/an}bracketri}ht/an}bracketle{tt|, (88) assuming y1+y2/ne}ati≀nslash= 0. This transforms |i,j,y1/an}bracketri}ht|i,j,y2/an}bracketri}htinto 1 p/summationdisplay b1,b2∈Zpωuj+wi|b1−b2,(y1b1+y2b2)(y1+y2)−1/an}bracketri}ht. (89) IntheClebsch-Gordanmethodwecanmeasuretheregisterofthe irreplabel. InthePGMcase this corresponds to performing the above transform and then me asuring the second register. Let this second register measurement outcome yield value m. This will produce the state 1√p/summationdisplay r∈Zpωwi|r/an}bracketri}ht⊗|m/an}bracketri}ht, (90) wherer=b1−b2and we can express was w= 2−1/bracketleftbiggy1y2 y1+y2r2+m2(y1+y2)+m/bracketrightbigg , (91) which, factoring out the global phase dependent on mis 1√p/summationdisplay r∈Zpωy1y2 2(y1+y2)ir2 |r/an}bracketri}ht⊗|m/an}bracketri}ht. (92) D. Bacon 27 Which we see is exactly the state we obtained after the Clebsch-Gor dan transform, Eq. (75). Thus we see that the eﬀect of the Clebsch-Gordan transform is to transform the quadratic equation in Eq. (82) into a form in which the resulting quadratic equat ion contains only a quadratic term. So, in quite a real sense, our derivation of solving the HSCP using the Clebsch-Gordan transform leads directly to at least part of the me asurement for the HSP used in the PGM approach. Thus it is best to view our Clebsch-Gordan derivation as giving a representation theoretic derivation of the PGM algorithm. Note ho wever that it does not give quite the same derivation, since in the PGM based algorithm one obtain s the

Question 657 multiple-choice

In advanced group theory, the concept of verbally Noetherian groups explores whether verbal subgroups are always finitely generated. Relatively free groups and their subgroup structures play a key role in understanding the limitations of Noetherian-type finiteness in infinite groups. Which statement accurately describes a key method for proving that the verbal ideal in the group algebra K(F/U) is not finitely generated? 1) Utilizing torsion elements in F/U to construct infinite chains of cyclic subgroups 2) Demonstrating the nonexistence of endomorphism-invariant subgroups in F/U 3) Showing that every subgroup of F/U is maximal under inclusion 4) Constructing an algebra over a field of characteristic zero and applying automorphism identities 5) Using the abelianization of F/U to identify infinite verbal ideals 6) Building specific elements v_m from commutators of free generators whose associated verbal ideal cannot be generated by finitely many elements 7) Proving the failure of ascending chains of normal subgroups to stabilize in F/U

✓ Correct Answer:

The correct answer is 6) Building specific elements v_m from commutators of free generators whose associated verbal ideal cannot be generated by finitely many elements.

📚 Reference Text:

verbally Noetherian. Let M be the set of all positive integers and let $ be the set of all functions : N -> N such that a(f> < b when a < b. We also write O for the corresponding sets of endomorphisms of F (such that Xj = xi for all i) and of F/ U k. A subgroup L in F (in F/ U k) is called a ^-subgroup if L is closed under all endomorphisms In fact, rather than Proposition 1 we shall prove the following stronger assertion. PROPOSITION 2. For every positive integer k the relatively free group F / U k satisfies the maximal condition on normal *l) = 1. Indeed, let h \,... , h 2M be elements of H* such that v k {h i,... , h 2M) ^ 0 and let X be the map of the set (y, | / e N] into H* such that v,x = h,^ for i = 1,... , 2k + 3, y:X = 1 for / > 2k + 3. By Lemma 1, x can be extended to a homomorphism of FI U into H* which, in its turn, can be extended to a homomorphism (also denoted by x) of the algebra K(F/ U) into

Question 658 multiple-choice

Quantum algorithms for eigenvalue estimation, such as phase estimation, must carefully handle discontinuities in the estimation function to avoid errors that can irreversibly affect quantum states. One important technical condition is the rounding promise, which restricts the locations of eigenvalues with respect to these discontinuities. Which statement correctly characterizes the (n, α)-rounding promise in coherent quantum phase estimation protocols for Hermitian or unitary matrices? 1) It guarantees that all eigenvalues are exactly at the critical points of discontinuity. 2) It allows eigenvalues to be distributed arbitrarily within the interval [0,1). 3) It mandates that the total number of eigenvalues is limited by α. 4) It excludes eigenvalues from small intervals around critical regions, such that the total measure of excluded intervals within [0,1) is α. 5) It requires the estimation function to be continuous everywhere within [0,1). 6) It is satisfied automatically in any quantum system with more than n eigenvalues. 7) It ensures the amplitude values of estimated states are either 0 or 1 everywhere.

✓ Correct Answer:

The correct answer is 4) It excludes eigenvalues from small intervals around critical regions, such that the total measure of excluded intervals within [0,1) is α..

📚 Reference Text:

that the amplitude cannot be 0 or 1 everywhere, there must exist points where it crosses inter- mediate values in order to interpolate in between the two. In both the ‘ n-bit estimate’ case and the additive-error case, this causes a problem for coherent quantum algorithms since the estimate cannot always be uncomputed. For some eigenvalues λj, the algorithm will yield an output state of the form α|ˆλ/angbracketright+β|ˆλ/prime/angbracketright. Even if ˆλ,ˆλ/primeare good estimates of λj, the Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 5 uncompute trick [BBBV97] cannot be used for such an output state. Thus, the damage to the input superposition over |λj/angbracketrightis irreparable. The only way to deal with this issue is to assume that the λjdo not take certain values. Then the amplitudes can interpolate between 0 and 1 at those points. Observe that the discontinuities in the function ﬂoor (2nλj)occur at multiples of 1/2n. A ‘rounding promise’ simply disallows eigenvalues near these regions. Deﬁnition 1. Letn∈Z+andα∈(0,1). A hermitian matrix Hsatisﬁes an (n,α)- rounding promise if it has an eigendecomposition H=/summationtext jλj|ψj/angbracketright/angbracketleftψj|, all the eigenval- uesλjsatisfy 0≤λj<1, and for all x∈{0,...,2n}: λj/negationslash∈/bracketleftbiggx 2n,x 2n+α 2n/bracketrightbigg (8) Similarly, a unitary matrix Usatisﬁes an (n,α)-rounding promise if it can be written asU=/summationtext je2πiλj|ψj/angbracketright/angbracketleftψj|and the phases λjsatisfy the same assumptions. We haveλj∈[0,1), and we have disallowed λjfrom certain sub-intervals of [0,1). The deﬁnition above is chosen such that the total length of these disallowed subintervals isα, regardless of the value of n. We have essentially cut out an α-fraction of the allowed eigenvalues. Guaranteeing a rounding promise demands an enormous amount of knowledge about the eigenvalues. We do not expect many Hamiltonians or unitaries in practice to actually provably satisfy such a promise. However, we expect that the estimation algorithms dis- cussed in this paper will still perform pretty well even if they do not satisfy such a promise. One can increase the chances of success by setting αto be very small, so that the vast majority of the eigenvalues do not fall into a disallowed region. Then, if the input state is close to a uniform distribution over the eigenstates, then one can be fairly conﬁdent that only anαfraction of the eigenvalues in the support will be disallowed. The need for a rounding promise can be also sidestepped entirely by avoiding the use of energy estimation as a subroutine and approaching the desired problem directly. This has been accomplished for ground state ﬁnding [LT21]. The rounding promise is not just a requirement of our work in particular - it is a requirement for any coherent phase estimation protocol. The fact that coherent phase estimation does not work for certain phases is largely disregarded in the literature: for example, works like [KP16] neglect this issue entirely. However, there are some works that have observed this problem and attempt to mitigate it. Such methods are called ‘consistent phase estimation’ [TaShma13] since the error of their estimate is supposedly independent of the phase being estimated, thus allowing ampliﬁcation

Question 659 multiple-choice

Quantum Markov semigroups are mathematical objects used to model time evolution in open quantum systems, drawing on analogies with classical Markov processes and incorporating operator algebra techniques. Transference principles allow inequalities and spectral properties from classical Markov semigroups to be extended to quantum settings, providing valuable tools for quantum information theory. Which statement accurately characterizes the relationship between the spectral gap of a quantum Markov semigroup generator and its classical counterpart, as established by transference principles? 1) The spectral gap of the quantum generator is always less than that of the classical generator. 2) The spectral gap of the quantum generator equals the logarithmic Sobolev constant of the classical generator. 3) The spectral gap of the quantum generator is independent of the classical generator's properties. 4) The spectral gap of the quantum generator is determined by the completely bounded norm of the classical semigroup. 5) The spectral gap of the quantum generator cannot be compared to that of the classical generator. 6) The spectral gap of the quantum generator is at least as large as that of the classical generator. 7) The spectral gap of the quantum generator is always zero for non-commutative spaces.

✓ Correct Answer:

The correct answer is 6) The spectral gap of the quantum generator is at least as large as that of the classical generator..

📚 Reference Text:

By Stein’s interpolation theorem, /parallel.alt1VG;f(1/slash.leftp)/parallel.alt12p≤1, completing the proof by homogeneity. It is straightforward to extend the p=1proof to the case of Lie groups, but p>1requires a more technical use of interpolation theory. We refer to Section A for the details. We note that in the context of a QMS with kernel k≡kt, a variation of the inequality above is straightforward to prove given a bound on t()=inf{t/divides.alt0/parallel.alt1kt−1/parallel.alt1∞≤}: Indeed, it is easy to see that it follows from the deﬁnition oft()that: kt(g)UgXU† g≤(1+)UgXU† g holds almost everywhere. By integrating the inequality above and conjugating with −1/slash.left2^pwe have −1/slash.left2^pTt()−1/slash.left2^p≤(1+)−1/slash.left2^pEfix()−1/slash.left2^p: It is then easy to see that this gives for any p>1 Dp(Tt()/divides.alt0/divides.alt0)≤log(1+)+Dp(Efix()/divides.alt0/divides.alt0) and the statement for p=1immediately follows by taking the appropriate limit. Thus, we see that a variation of the transference statement is easy to obtain if we start from a t()bound and the bound above is only advantageous if we can control /parallel.alt1k/parallel.alt1Lp(G)or the relativeentropy ofk. This is for instance the case when one has access to the so-called modiﬁed logarithmic Sobolev constant for the underlying group. C. Application to quantum Markov semigroups In this subsection, we show how the machinery developed in Section III-B provides estimates on the norms and entropies at the output of a quantum Markov convolution semigroup of Section II in terms of the kernel of their associated classical semigroup. We start by recalling the notations of Section II: (St)t≥0, St=e−tLis a Markov semigroup on the compact group G(either Lie or ﬁnite), with right-invariant kernel (kt)t≥0. The QMS (Tt)t≥0,Tt=e−tLis the transferred QMS on B(H)deﬁned by Equation (3) through the projective representation g/uni21A6u(g)ofGon the ﬁnite dimensional Hilbert space H. The next theorem regroups all the transference techniques which will be frequently used in this paper (see [6]). We recall that the spectral gap of the symmetric Lindblad generator L, denoted by min(L)(resp. ofL, denoted bymin(L)) is the smallest non-zero eigenvalue ofL(resp. ofL) and is a quantity that palys a central role in the theory of decoherence times . Theorem III.2 (Transference) .LetTt=e−tL,St=e−tL as above. Then i) The spectral gap for Lis bigger than the spectral gap forL:min(L)≥min(L); ii) For any 1≤p;q≤∞, we have /parallel.alt1Tt∶Lp s(Nfix⊂ B(H))→Lq s(Nfix⊂B(H))/parallel.alt1cb≤/parallel.alt1St∶Lp(G)→ Lq(G)/parallel.alt1cb; iii) For any 1≤p;q≤∞, we have /parallel.alt1Tt−Efix∶ Lp s(Nfix⊂B(H))→Lq s(Nfix⊂B(H))/parallel.alt1cb≤ /parallel.alt1St−EG∶Lp(G)→Lq(G)/parallel.alt1cb; We conclude this section by brieﬂy explaining how this theorem can be applied to get estimates for the QMS. First we recall a result on the cb norm in commutative C∗-algebra (see e.g. Theorem 3.9 of [22]). Lemma III.3. LetSbe an operator on the commutative Lp()space. Assume that either p=1orq=+∞. Then /parallel.alt1S∶Lp(G)→Lq(G)/parallel.alt1cb=/parallel.alt1S∶Lp(G)→Lq(G)/parallel.alt1: Let us furthermore mention that for a classical Markov semigroup (St)t≥0acting on a compact group Gwith kernel (kt)t≥0, a convexity argument yields /parallel.alt1St∶L1(G)→L∞(G)/parallel.alt1=sup g∈G/divides.alt0kt(g)/divides.alt0 (27) and similarly /parallel.alt1St−EG∶L1(G)→L2(G)/parallel.alt1 (28) =/parenleft.alt3/integral.disp/divides.alt0kt(g)−1/divides.alt02dG/parenright.alt31/slash.left2 =/parallel.alt1kt−1/parallel.alt12: 10 This gives us “for free” the following estimates on the norm of the transferred QMS between certain non-commutative Lpspaces. We will see in the next sections how to apply these estimates to concrete situations arising in quantum information theory. Corollary III.4. LetTtandStbe as above,

Question 660 multiple-choice

Quantum simulation platforms are increasingly used to study topological phases, where the stability of topological invariants under experimental noise is critical for robust realization. The Haldane model and engineered flat Chern bands provide key testbeds for investigating these phenomena in controlled cold atom systems. In quantum simulations of topological flat bands, which condition ensures that the Bott index remains quantized even when Gaussian or colored noise is present in the Hamiltonian? 1) The energy band gap remains open under noisy perturbations 2) The lattice size exceeds the correlation length of the noise 3) The noise strength is strictly zero throughout the experiment 4) The Chern number is calculated for each individual experimental run 5) The QQFT is performed only in one dimension 6) The band width remains constant regardless of noise strength 7) The model employs a triangular lattice instead of square or honeycomb

✓ Correct Answer:

The correct answer is 1) The energy band gap remains open under noisy perturbations.

📚 Reference Text:

The real-space lattice in the experimental realization corresponds to discretizing the Brillouin zone of the model in our scheme. We impose a square lattice grid of size LLon the Bril- louin zone, for which the required 2D QQFT in experimentalrealization is separable simply containing two successive 1D QQFTs, as explained in Sec. II D. The realized Chern band with our approach is com- pletely ﬂat by design. To investigate its robustness against experimental imperfections, we add Gaussian noise to the Hamiltonian sequence as in the previous example, and examine the band ﬂatness and the topological prop- erty. For a concrete demonstration, we consider the Hal- dane honeycomb lattice model83, which consists of two sublattices Aand B. The Hamiltonian has a form in Eq. (36), with d1(k)=t11+cos(kg2)+cos(kg3), d2(k)=t1sin(kg2)sin(kg3), and d3(k)=M 2t2sin P3 j=1sin kgj . Here, the primitive lattice vectors g1=e2e3,g2=e3e1, and g3=e1e2, where e1,e2, ande3are the three unit vectors pointing from one Blattice site to its three neighboring Asites on the honeycomb lattice. We choose t2=t1=1=p 3 here. In our ﬂatband engineering protocol, we normalize the dvector, i.e., d!d=jdj. The energy spectrum and the topological property are obtained by decomposing the dynamical evolution operator UCI=eiHCIT, i.e.,UCIjn; i=expiT En; jn; i, with = +() and n2[1;L2] indexing the upper (lower) band, and the eigenstates within each band, respectively. To demonstrate the band ﬂatness, we calculate the band gap G=minfEn;+g maxfEn;gand the band width W=maxfEn;gminfEn;gin presence of noise. We ﬁnd that the band gap (band width) decreases (increases) with the strength of the Gaussian white noise, and conﬁrm that the QQFT protocol is reasonably robust—the band ﬂatness ratio W=Gremains below 10% even at=2:5103(Fig. 6(a,b)). The required noise level is below 102, and is close to the quantum control preci- sion demonstrated in experiments30,84, and is anticipated to be reachable in near terms with the recently developed cali- bration protocol38. For the topological property, we compute Bott index (B)85–87, which is a topological invariant irrespec- tive of the noise induced translation symmetry breaking (see Fig. 6 (c)). The Bott index of our topological ﬂatband is equal to the Chern number C83in the clean limit. In presence of noise, the Bott index remains quantized and una ected as long as the band gap remains open. For the experimental realization with Li atoms, the QQFT based quantum engineering of the topological Chern band takes about 100 ms to simulate lattice with size 16 16. We show a topological ﬂat band can be achieved with bandgap and bandwidth reaching 2 2 kHz and 2 0:2 kHz, as- suming a noise level =2:5103(Fig. 6). We expect this realization to be accessible to cold atom experiments. The simulation of ﬂat-band Chern insulators is also robust against the colored noise that has an exponential correlation (see Eq. (35)). For the colored noise, the energy gap also grad- ually closes with increasing . But once if the gap is open, the noises do not change the Bott index, which always equals the ground-state Chern number of the Haldane model. In

Question 661 multiple-choice

Quantum algorithms have shown promise in solving computational group theory problems more efficiently than classical approaches, particularly for specific classes of finite groups. The implementation of these algorithms relies on the structure of the group and the capabilities of the underlying quantum circuit model. Which condition allows an exact quantum algorithm to efficiently solve the zero sum subsequence problem in sequences over Zn_p, thereby enabling polynomial-time solutions to certain group-theoretic problems? 1) The group is non-nilpotent and the prime p is arbitrary 2) The prime p is bounded by a constant 3) The group is abelian and the group order is composite 4) The quantum circuit uses infinite gate sets 5) The group elements are encoded as polynomials 6) The lower central series terminates at the trivial subgroup 7) The quantum algorithm simulates classical subset sum algorithms

✓ Correct Answer:

The correct answer is 2) The prime p is bounded by a constant.

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subgroup state in the abelian factor of the group. As an application of the technique, we prove Proposition 2which tells us that we can solve by a polynomial time exact quantum algorithm the problem HSMC in a semi-elementary p-group of constant nilpotency class provided that we can ﬁnd zero sum subsequences of sequences consisting of poly (nlogp)vectors from Z n pin time poly (nlogp). In Sect. 5, we prove Theorem 2on efﬁciently solvability the latter task in the case when pis bounded by a constant. Propositions 1and 2, together with Theorem 2, immediately imply Theorem 1. Section 6is devoted to concluding remarks. 2 Preliminaries 2.1 On exact quantum computations To obtain sufﬁciently general intermediate results, we use the model of uniform circuit families described by Nishimura and Ozawa [ 23]. This is because some of the exact methods of [ 16] as well as our main conversion technique work under the assump- tion that the quantum Fourier transforms and their inverses modulo the prime factors of|G|can be exactly implemented. As it is pointed out in [ 24], this task cannot be 123 14 Page 6 of 23 M. Imran, G. Ivanyos accomplished using a ﬁxed ﬁnite gate set. For the sake of transparency, we state our intermediate result using assumptions on availability of the quantum Fourier trans- forms rather than on gates required by the exact implementations of them. (See theimplementation of the Fourier transform modulo general numbers proposed by Mosca and Zalka [ 15].) Note however that for the case of our Theorem 1, where these primes are assumed to be bounded by a constant, a constant number of gates are sufﬁcientand hence, by [ 25], the theorem remains valid in the quantum Turing machine model of Bernstein and V azirani [ 26]. 2.2 Groups For standard notations and concepts from group theory such as subgroups, normal subgroups, cosets, conjugates, commutators, commutator subgroup, center, etc., we refer the reader to the textbooks, e.g., to [ 27]. For subsets Uand VofGwe denote byUV the set {uv:u∈U,v∈V}. If both Uand Vare subgroups and either Uor Vis normal in Gthen UV is a subgroup. For subgroups U,V,b y[U,V]we denote thesubgroup generated by the commutators [u,v](u∈U,v∈V). Recall that the lower central series of a ﬁnite group Gis the sequence G=G 0>G1>...> Gcof normal subgroups Gi/triangleleftGrecursively deﬁned as Gi=[G,Gi−1].H e r ew ea s s u m e that cis the smallest index isuch that Gi=[G,Gi]. The group Gis nilpotent if Gc={1}and then cis called the (nilpotency) class of G. A ﬁnite group is nilpotent if and only if it is the direct product of its Sylow subgroups. To obtain sufﬁciently general results, we work over black-box groups with unique encoding of elements. The concept captures various “real” groups such as permu-tation groups and matrix groups over ﬁnite ﬁelds. Elements of a black-box group are represented by binary strings of a certain length /lscriptand the group operations are given by oracles and as input, a generating set for the group is given. Subgroups will

Question 662 multiple-choice

Quantum algorithms have demonstrated significant speedups over classical algorithms for specific problem classes, particularly in the analysis of algebraic and combinatorial structures. The study of Boolean functions with distinct spectral properties has become increasingly relevant in both theoretical computation and cryptography. Which property enables quantum algorithms to efficiently solve hidden shift problems for bent functions, resulting in an exponential separation from classical algorithms? 1) The ability to factor large integers using Grover's algorithm 2) Bent functions having linear Fourier spectra 3) Non-abelian group structure of the underlying problem 4) Classical algorithms' ability to exploit hidden shift efficiently 5) The vulnerability of bent functions to linear cryptanalysis 6) The duality between bent functions and their Fourier transforms, which quantum algorithms can exploit 7) The presence of a unique solution to every hidden subgroup problem

✓ Correct Answer:

The correct answer is 6) The duality between bent functions and their Fourier transforms, which quantum algorithms can exploit.

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Title: Quantum algorithms for highly non-linear Boolean functions Year: 2008 Paper ID: 13f0c54cd6e8c575e4972ccce405c955f01c9215 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/13f0c54cd6e8c575e4972ccce405c955f01c9215 Abstract: Attempts to separate the power of classical and quantum models of computation have a long history. The ultimate goal is to find exponential separations for computational problems. However, such separations do not come a dime a dozen: while there were some early successes in the form of hidden subgroup problems for abelian groups-which generalize Shor's factoring algorithm perhaps most faithfully-only for a handful of non-abelian groups efficient quantum algorithms were found. Recently, problems have gotten increased attention that seek to identify hidden sub-structures of other combinatorial and algebraic objects besides groups. In this paper we provide new examples for exponential separations by considering hidden shift problems that are defined for several classes of highly non-linear Boolean functions. These so-called bent functions arise in cryptography, where their property of having perfectly flat Fourier spectra on the Boolean hypercube gives them resilience against certain types of attack. We present new quantum algorithms that solve the hidden shift problems for several well-known classes of bent functions in polynomial time and with a constant number of queries, while the classical query complexity is shown to be exponential. Our approach uses a technique that exploits the duality between bent functions and their Fourier transforms.

Question 663 multiple-choice

Quantum algorithms have shown significant advantages in solving algebraic problems involving polynomials over finite fields, particularly through the use of input substitutions and reductions to hidden subgroup problems. These approaches are crucial in efficiently determining unknown coefficients in multivariate polynomials. When recovering the coefficients of a bivariate quadratic polynomial over a finite field using quantum algorithms, which strategy allows efficient determination of the coefficients' ratios up to a scalar factor? 1) Randomly selecting inputs and collecting output values to form a system of nonlinear equations 2) Using Grover's algorithm to search for nonzero coefficients directly 3) Substituting zeros into both variables to eliminate all cross terms 4) Applying the Fourier transform to the polynomial's output space 5) Evaluating the polynomial at a complete set of field elements and performing classical interpolation 6) Making linear substitutions such as (x, x) and (x, αx) to isolate and compute ratios of unknown coefficients 7) Factoring the polynomial and equating coefficients of identical monomials

✓ Correct Answer:

The correct answer is 6) Making linear substitutions such as (x, x) and (x, αx) to isolate and compute ratios of unknown coefficients.

📚 Reference Text:

easily all coeﬃcients of Pup to a common factor. If a11+a22=0 ,t h e nw e also can compute all coeﬃcients easily. •a11/negationslash=0,a22=0 :I fa12= 0, then we use the inputs ( x1,x2): =(x,x)a n d obtain HQPPforthepolynomial a11x2+(b1+b2)x. WithRwecandetermine the quotient ( b1+b2)/a11and together with the already known value b1/a11 we obtain the missing b2/a11.I fa12/negationslash= 0, then we pick α∈Fq\{0}such that 1+ αa12/a11/negationslash= 0 and we use the inputs ( x1,x2): =(x,αx). We obtain HQPPfor (a11+αa12)x2+(b1+αb2)x, which can be used to ﬁnd r∈Fqsuch that (b1+αb2)=r(a11+αa12). This gives us the missing fraction b2/a11. The case a22/negationslash=0a n d a11= 0 can be treated in a similar way. •a11=a22=0,q/negationslash= 2: We use the inputs ( x1,x2): =(x,x)a n do b t a i n HQPP for the polynomial a12x2+(b1+b2)xthat can be used to decide whether a12= 0 or not. If it is nonzero, then we compute ( b1+b2)/a12.F u r t h e r m o r e , we can choose α∈F× q,α/negationslash= 1, and we use the inputs ( x1,x2): =(x,αx) to compute the fraction ( b1+αb2)/(αa12). From these two fractions we can determine b1/a12andb2/a12.I fa12= 0, then we have the polynomial b1x1+b2x2andcandeterminetheratiobetween b1andb2bythealgorithmfor the abelian HSP over the additive group of F2 q. Note that we use a quantum computer for an eﬃcient implementa tion of this step of the reduction. HIDDEN SYMMETRY SUBGROUP PROBLEMS 1999 •a11,a22=0,q= 2: We use the inputs ( x1,x2): =(x,0) and obtain HPPfor the polynomial b1x. We can easily test whether it is constant, i.e., b1=0 , or not. The coeﬃcient b2can be computed in a similar way. The input (x1,x2): =(x,1) give us a12+b1. This shows that we can ﬁnd all coeﬃcients of a bivariate polynomial up to a common factor when we use Ronly a constant number of times and some additional operations, which can be performed eﬃciently on a quantum computer. Nextweconsiderthecase n= 3. Substituting zeroin x3, wecanusethealgorithm for the bivariate case to test whether a11=0 . I f a11/negationslash=0 ,w ec a nd e t e r m i n et h e quotient of the remaining coeﬃcients (except for a23)a n da11by substituting zero in x2orx3and using the algorithm for the bivariate case. For a23we can substitute (x1,x2,x3)=(x,y,y) and obtain the polynomial a11x2+(a12+a13)xy+(a22+a23+a33)y2+b1x+(b2+b3)y. Then the algorithm for the bivariate case gives us ( a22+a23+a33)/a11from which we can compute a23/a11. The cases where any of the coeﬃcients b1,a22,b2,a33, orb3is nonzero can be treated in a similar way. It remains to handle the case of a polynomial of the form a12x1x2+a13x1x3+a23x2x3. Then substituting 1 in x3gives the polynomial a12x1x2+a13x1+a23x2, and the ratio between the three coeﬃcients can be found by the bivariate algorithm. Thecase n= 4canbehandledasfollows. Weappl ythealgorithmofthepreceding paragraphto the four polynomials obtained by substituting zero in x1,x2,x3,a n dx4, respectively. Observethat the se steps determine the ratiobetween pairsofcoeﬃcients that have indices that ﬁt in a three-element subset of {1,2,3,4}. By transitivity, we are done unless our polynomial is of the form a12x1x2+a34x3x4,a13x1x3+a24x2x4, ora14x1x4+a23x2x3.I fi ti so ft h

Question 664 multiple-choice

Morava K-theory is a sophisticated cohomology theory used to study the algebraic and topological properties of finite groups, particularly 2-groups. Spectral sequences and characteristic classes play central roles in describing the structure of cohomology rings for these groups. Which of the following groups of order 16 is a central product considered "almost extraspecial" and is known to have even Morava K-theory concentrated in even degrees? 1) D8 × C2 2) Q8 × C2 3) C4 ⋉ D8 4) SD16 (semidihedral group) 5) D16 (dihedral group of order 16) 6) Q16 (quaternion group of order 16) 7) QD16 (quasidihedral group of order 16)

✓ Correct Answer:

The correct answer is 3) C4 ⋉ D8.

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simple). More precisely, if K.n/.BHH/ K.n/.BH/ K.n/K.n/.BH/DFTis the decomposition into free and trivial summands, then FC2andTare in the image of restriction from Gand thus consist of permanent cycles. 3 Groups of smaller order The later sections require a few results about smaller groups, in particular concerning the behaviour of certain spectral sequences. The ﬁrst observation is that all groups of order at most 16have even Morava K–theory; this follows from the results quoted in the previous section: the only nonabelian groups of order 8 are D8andQ8, and there are 11 nonabelian groups of order 16, namely Algebraic & Geometric Topology , Volume 11 (2011) 510 Björn Schuster (a)D8C2,Q8C2, (b)D16,Q16, the semidihedral group SD16, the quasidihedral group QD16, which are metacyclic, (c) the central product C4D8, also known as almost extraspecial group, (d)H1Dhg1;g2;g3jg4 1Dg2 2Dg2 3Dg1;g2Dg2;g3D1;g3g1g3Dg1g2i .C4C2/ÌC2, (e)H2Dhg1;g2jg4 1Dg4 2D1;g1 2g1g2Dg1 1iC4ÌC4. The groups H1;H2(numbers 9 and 10 in the Hall–Senior list) are minimal nonabelian 2–groups. Although we already know they are both “good”, we shall later use speciﬁc calculations for their Morava K–theories. H1(also known as 162c) has a rank 3 elementary abelian subgroup EDhg2 1;g2;g3i, and we consider the corresponding extension (3) 1!E!H1!C2!1 with associated Serre spectral sequence E2DH.BC2IK.n/.BE//. Lemma 3.1 The Serre spectral sequence for the extension (3)is simple. Proof This is very similar to the wreath product calculation in Lemma 2.5. Let 1;2;3be the linear characters of Ewith1.g2 1/D2.g2/D3.g3/D1and iD1on the other generators. Then g1 iDiforiD1;3andg1 2D12. Furthermore, 3extends to a character ofHDH1. Set yiDc1.i/, then g1acts trivially on y1;y3, whereas g 1.y2/Dy1CK.n/y2. Thus K.n/.BE/M K.n/y3=.y2n 3/ where Mis the module for the switch action as in the wreath product calculation. Clearly y3DResH Ec1. /is in the image of restriction from H, but so are.1Cg 1/yk 2D ResH ETrH E.yk 2/and y2g 1.y2/kDResH E.c2.IndH E2/k/. Thus all invariants are permanent cycles, as they are in Im .ResH E/. The group H2has the index two subgroup BDhg2 1;g2iC2C4with corresponding extension (4) 1!B!H2!C2!1 and associated Serre spectral sequence E2DH.BC2IK.n/.BE//. Lemma 3.2 The Serre spectral sequence for the extension (4)is simple. Algebraic & Geometric Topology , Volume 11 (2011) Morava K–theory of groups of order 32 511 Proof Instead of using the switch action of C2onC2C2, one can use the action ofC2onC4by inversion to calculate K.n/.BD8/; the invariants of the C2–action onK.n/.BC4/DWMare again restrictions of Chern classes. If ydenotes the Euler class of the representation ofBwith.g2 1/D1and.g2/D1, then E2M K.n/y=.y2n/. Sinceextends to a complex character of H2, the class yis in the image of restriction, whence all invariants are. 4 Groups of order 32– the easy part There are 51 groups of order 32, which we shall denote as G1–G51, the index referring to the Hall–Senior number of the group. 4.1 Groups 1–15 The ﬁrst 7 groups are abelian, and the next 8 have an abelian factor. Thus their Morava K–theories are generated by Euler classes, and in particular concentrated in even degrees, 4.2 Groups 16–22 G16–G22have a central quotient isomorphic to C2C2. Thus the technical Lemma 2.3 gives the

Question 665 multiple-choice

In advanced representation theory and mathematical physics, Lie algebras like sl(n) are often studied by decomposing them using gradings induced by elements of their Cartan subalgebra. Such gradings play a crucial role in the structure of constraints and the construction of finite W-algebras. Which of the following statements most accurately describes the decomposition of a Lie algebra g using a grading induced by an element δ in the Cartan subalgebra, as it relates to the construction of finite W-algebras? 1) The grading by δ always coincides with the grading by any other Cartan subalgebra element, so all decompositions produce identical subalgebras. 2) The decomposition via δ yields only nilpotent subalgebras and omits any block diagonal components. 3) The algebra decomposes into g₋, g₀, and g₊ subspaces, where g₀ forms block diagonal matrices as a direct sum of sl(m_k) subalgebras (modulo u(1) factors), and g₊ and g₋ are spanned by specific off-diagonal elements. 4) The δ grading leads exclusively to second class constraints, preventing the construction of any gauge group. 5) The subalgebra g₀ generated by δ consists of only abelian elements with no relation to sl(m_k) structure. 6) In this framework, projections π₊ and π₋ map onto the Cartan subalgebra rather than the nilpotent components. 7) Constraints constructed using the δ grading cannot generate any gauge symmetry or relate to W-algebras.

✓ Correct Answer:

The correct answer is 3) The algebra decomposes into g₋, g₀, and g₊ subspaces, where g₀ forms block diagonal matrices as a direct sum of sl(m_k) subalgebras (modulo u(1) factors), and g₊ and g₋ are spanned by specific off-diagonal elements..

📚 Reference Text:

If the fundamental representation of slnis spanned by vectors v1,...v n, on which slnacts viaEij(vk) =δjkvi, then the irreducible rep- resentations nlofsl2to which it reduces are spanned by {vl+sk−1}1≤k≤nl. Previously we decomposed the lie algebra ginto eigenspaces of adt0. However, for our present purposes it is convenient to introduce a diﬀerent grading of the Lie algebra g, that we need to describe the generalized Miura map and the BRST quant ization. The grading is deﬁned by the following element of the Cartan subalgebra o fsln: δ=/summationdisplay k≥1mk/summationdisplay j=1/parenleftigg/summationtext llml/summationtext lml−k/parenrightigg Esk−1+j,sk−1+j. (3.70) This leads to the alternative decomposition g=g−⊕g0⊕g+ofginto spaces with negative, zero and positive eigenvalues under the adjoint a ction ofδrespectively (note that in case the grading of adt0is an integral grading then we have t0=δso in those cases nothing happens. In general however t0/\e}atio\slash=δand alsog(m)/\e}atio\slash=gm). The subalgebra g0consists of matrices whose nonzero entries are in square blo cks of dimensions m1×m1, m2×m2, etc. along the diagonal of the matrix and is therefore a dire ct sum of slmksubalgebras (modulo u(1) terms). The nilpotent subalgebra g+is spanned by {El+sk−1,r+sk}l≥1; 1≤k≤nl−1;r>0, and the nilpotent subalgebra g−by the transpose of these. Let π±denote the projections onto g±. Then the following theorem describes the replacement of the mixed system of ﬁrst and second class c onstraints by a system of ﬁrst class constraints only. Theorem 1 The constraints {Jl+sk−1,r+sk−δr,l}l≥1; 1≤k≤nl−1;r>0are ﬁrst class. The gauge group they generate is ˆH= exp(g−)acting via the adjoint representation on g. The resulting ﬁnite Walgebra is the same as the one obtained by imposing the constraints (3.18). Proof: decompose gin eigenvalues of ad δ,g=⊕kgk. Note that ad δhas only integral eigenvalues. Using the explicit form of t+in (3.69), one easily veriﬁes that [ δ,t+] =t+. Thus,t+∈g1. Again since [ g+,g+] = [g≥1,g≥1]⊂g≥2it follows (exactly like in lemma 3) that the constraints {Jl+sk−1,r+sk−δr,l}l≥1; 1≤k≤nl−1;r>0are ﬁrst class. The gauge group can be determined similarly as in section 3.2, and the a nalogue of theorem 1 of section 3.2 can be proven in the same way with the same choice o fgfix, if one uses the decomposition of gin eigenspaces of ad δrather than ad t0. Therefore the resulting ﬁniteWalgebra is the same, because theorem 2 and 3 of section 3.2 sho w that it only depends on the form of gfix. ∗The commutation relations are [ t0, t±] =±t±, and [ t+, t−] = 2t0. 26 Let us explain this theorem in words. The number of second cla ss constraints in any system is necessarily even. If one switches to the adδgrading the set of second class constraints is split into half: one half gets grade 1 w.r.t. adδwhile the other half gets grade 0. Now what one does is impose only that half that has obt ained grade 1 w.r.t. theadδgrading. These constraints are then ﬁrst class. The gauge tr ansformations they generate can be completely ﬁxed by imposing the constraints that were in the other half. Having done that we are back in exactly the same situati

Question 666 multiple-choice

Matrix and tensor scaling algorithms are fundamental tools in computational mathematics, especially for normalizing data to prescribed row and column sums. Advanced versions of these algorithms rely on group theory, optimization techniques, and careful selection of basis to ensure efficient convergence in high-dimensional settings. Which condition specifically requires an initial random or clever change of basis to guarantee convergence of scaling algorithms in non-uniform, high-dimensional matrix or tensor scaling problems? 1) When the permanent of the matrix is zero 2) When all row and column sums are equal 3) When the matrix is already doubly stochastic 4) When the scaling factors are restricted to diagonal matrices 5) When alternating minimization fails to reduce the potential function 6) When prescribed marginals are non-uniform and orbit spaces lack a canonical basis 7) When group actions commute for all scaling matrices

✓ Correct Answer:

The correct answer is 6) When prescribed marginals are non-uniform and orbit spaces lack a canonical basis.

📚 Reference Text:

or column normalized. This three-step analysis shows that the scaling algorithm is able to solve the doubly stochastic scaling problem in polynomial time. The diﬃcult part of the analysis is coming up with a potential function satisfying the properties above. This is the role played by invariant theory later. A source ofgoodpotentialfunctionswillturnouttobe highestweightvectors ,whichare(informallyspeaking) “eigenvectors”oftheactionofcertainsubgroupsofthemaingroupaction. Notethatthepermanent isaneigenvectoroftheactionof TpnqTpnqsincePermpRXCqequals± iRi;i± jCj;j PermpXq forpR;CqPTpnqTpnq. Ifwewanttosolvethemoregeneralscalingproblem, wherewearegivenaprescribedvaluefor therowandcolumnsums,sayasannon-negativeintegervector pr;cqpr1;:::rn;c1;:::;cnq,the same natural algorithm can be applied. The only change one needs to make in the algorithm above is that we will now normalize the rows of Ato have sums pr1;:::;rnqand the columns to have sum pc1;:::;cnq. Theanalysisisalsoquitesimilar: onecanchoosethepotentialfunction,forexample, tobethepermanentofmatrix Bobtainedfrom Abyrepeating ithrowritimesandjthcolumncj times. However,thedistinctionbetweentheuniformandthenon-uniformversionsoftheproblems is much starker in our higher dimensional non-commutative setting, as we will see next. 1.6 Techniques and proof overview Our algorithm and its analysis generalize two recent works [ BGO17,Fra18], which in turn generalize the analysis of matrix scaling in Section 1.5. The paper [ BGO17] studies the special case whenppiqis the uniform distribution (over a set of size ni) for alliwhile the paper [ Fra18] studies the special case d2. Our algorithm is a natural common generalization of the algorithms in [Fra18,BGO17] while our analysis generalizes the analysis in [ BGO17] replacing the use of invariants with highest weight vectors (we will explain what these are later). Letusdevelopsomeintuitionforthealgorithm. Itisusuallythecasewithscalingproblems,aswe sawwithmatrixscaling,andmoregenerallyintheframeworkofalternatingminimization,thatone oftheconstraintsiseasytosatisfybyscaling. Thesameistruefortheproblemwehaveathand. We aregivenatensor XPTenpn0;n1;:::;ndq. Supposewewant spec piq X ppiq. Withtheshorthand ppiq Ò:pppiq ni;:::;ppiq 1q, we act on Xbyg I;I;:::; diag ppiq Ò 1{2 piq X 1{2 ;:::;I , where the non-trivialelementisinthe ithlocation. Thisiswillsatisfythe ithconstraint. Orindeed,onecan /one.superior⁹There is some dependency on the bit complexity of the input that we are ignoring. 10 chooseanymatrix Rs.t.RR:piq XandactonXbyg I;I;:::; diag ppiq Ò 1{2 R1;:::;I . This willalsosatisfythe ithconstraint. Bychoosingeachtimetoﬁxtheindexwhichis“farthest”from its target spectrum, we have deﬁned an iterative algorithm (up to the choice of Rat each step) that keeps on alternately ﬁxing the constraints. It turns out that this algorithm works (for any choice of Rat each step!) when ppiq’s are all uniform and converges in a polynomial number of iterations [BGO17]. Interestingly, the choice of Rthat works for general ppiq’s is that of upper triangular matrices! /two.superior⁰ This was the choice made in [ Fra18] as well. This restriction on scaling factors will make the analysis more complicated as we shall soon see. One intuitive reason for the diﬀerence between the uniform and the general case is the following: in the general case, we made an arbitrary decision to try to scale Xto have marginals diag ppiq while we could have chosen to scale it to any piqs.t. specppiqqdiag ppiq . Thischoiceofbasisisnotpresentintheuniformcasesinceallbasesarethe same! Thisrestrictiononscalingfactorscreatesanotherproblem: itdisconnectstheorbitspace(see example below). Thus, we need to initialize the algorithm with a random basis change of the given input, and only then resume the restricted scaling. This idea is used as well in [ Fra18]. We explain, via an example, why this random basis change (or at least a “clever” basis change) is needed at the start of the algorithm. Consider the diagonal unit tensor XPTenp1; 2;2;2q, whereXj;k;`1iﬀ

Question 667 multiple-choice

Accurate modeling of liquid-phase halogenated compounds in computational chemistry often relies on benchmarking force fields and quantum methods, followed by detailed simulations to predict thermodynamic properties. Techniques such as Monte Carlo and Molecular Dynamics are used to assess properties like density and viscosity under controlled ensembles. Which computational method serves as the high-level quantum chemistry reference for benchmarking DFT-D functionals in the validation of force fields for halogenated compounds? 1) CCSD@cbs 2) MP2 3) Hartree-Fock 4) B3LYP 5) Møller–Plesset perturbation theory (MP3) 6) PBE-D3 7) M06-2X

✓ Correct Answer:

The correct answer is 1) CCSD@cbs.

📚 Reference Text:

Ain the Supporting Information. For a restricted number of the investigated species, namely C1,C2,N1andB3, the performances of each of the considered DFT-D functional s are benchmarked over a large set (50) of dimer arrangements against high level QM da ta. As reference, the so called ”gold standard”73of quantum chemistry is here adopted, i.e.the Coupled Cluster method, considering single, double and (perturbatively) t riple excitations, estimated at the complete basis set limit (CCSD(T)@cbs). Additional deta ils regarding QM calcula- tions can be found in the Supporting Information. 2.3 Simulations All MC simulations are carried out on systems of 216 molecules with an ”in house” code, namedMCgblj .74All runs are performed in the isothermal isobaric ensemble (NP T), and the shape and the volume of the computational box is chang ed, by attempting to vary arandomlyselectededgeofthebox. Onlytrialdisplacement sofmolecularcentersofmass androtationsofmaininertiaaxesareattemptedduringeach MCrun, inordertomaintain the starting internal geometry of each monomer. All the afore mentioned trial moves are selected randomly and not sequentially, to preserve the detailed balance condition. During all runs, the short range intermolecular interactio ns were truncated at R c= 12 ˚A, employing the standard energy correction. Charge-charge long range interactions are handled with the Ewald method, using a convergence paramete rαof 5.36/2R c. MD simulations are performed resorting to the Gromacs5.1 engine.75Systems of 11 1000moleculesareﬁrstequilibratedfor1nsintheNPTensemb leatachosentemperature and at a pressure of 1 atm, keeping them constant through the B erendsen thermostat and barostat, respectively. Thereafter, 5 ns NPT production run s are carried out for each species, employing the velocity rescale with a stochastic term76and Parrinello-Rahman77 schemes, setting τTandτPto 0.1 and 5.0 ps. In all simulations, given the larger number of molecules (hence larger simulation box) considered in MD with respect to MC runs a cut-oﬀ of 13 ˚A was employed for both charge-charge and LJ terms, whereas l ong-range electrostatics was accounted through the particle mesh Ewa ld (PME) procedure. In most runs, bond lengths are constrained at their equilibrium val ue using the Lincsalgorithm, allowing for a 1 fs time step. Finally, two additional 1 ns runs are carried out for each com pound, saving the trajec- tories every 4 fs, one in the NPT and one in the NVE ensemble, in ord er to compute the speciﬁc heat and the viscosity, respectively, as detailed i n the Supporting Information. In both simulations, the constraints on the bond lengths ere removed, and the time step consequently decreased to 0.1 fs. 2.4 Thermodynamic Properties The quality of the QMD-FFs is assessed by computing bulk dens ity (ρ), vaporization enthalpy (∆Hvap), volumetric expansion coeﬃcient ( α), isothermal compressibility ( κT), speciﬁc heat at constant pressure (c P) and viscosity ( η) for each of the investigated sys- tems. Details on the calculations of the above quantities fr om the MD trajectories can be found in the Supporting Information. 3 Results 3.1 Preliminary CAM-B3LYP-D results The ﬁrst attempt to parameterize halogenated compounds was performed on the two smallest molecules of the benchmark set displayed in Figure 1, namely C1(CHCl 3) and C2(CH2Cl2).

Question 668 multiple-choice

Quantum computing leverages phenomena such as interference and entanglement to achieve computational advantages over classical methods. Signal processing in areas like image analysis often relies on various transforms, each with unique computational requirements and limitations. Which transform can be computed in constant time, independent of input size, using a quantum qudit architecture with only one gate, while also being suitable for arbitrary-length data? 1) Discrete Fourier Transform (DFT) 2) Fast Fourier Transform (FFT) 3) Fractional quantum Kravchuk-Fourier Transform (KT) 4) Hadamard Transform 5) Quantum Fourier Transform (QFT) 6) Discrete Cosine Transform (DCT) 7) Walsh Transform

✓ Correct Answer:

The correct answer is 3) Fractional quantum Kravchuk-Fourier Transform (KT).

📚 Reference Text:

Title: Quantum interference enables constant-time quantum information processing Year: 2018 Paper ID: 238296c7b01e071e8203a585d53709871d821ec8 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/238296c7b01e071e8203a585d53709871d821ec8 Abstract: We found that quantum interference allows processing large sets of data faster and more accurately than computer algorithms. It is an open question how fast information processing can be performed and whether quantum effects can speed up the best existing solutions. Signal extraction, analysis, and compression in diagnostics, astronomy, chemistry, and broadcasting build on the discrete Fourier transform. It is implemented with the fast Fourier transform (FFT) algorithm that assumes a periodic input of specific lengths, which rarely holds true. A lesser-known transform, the Kravchuk-Fourier (KT), allows one to operate on finite strings of arbitrary length. It is of high demand in digital image processing and computer vision but features a prohibitive runtime. Here, we report a one-step computation of a fractional quantum KT. The quantum d-nary (qudit) architecture we use comprises only one gate and offers processing time independent of the input size. The gate may use a multiphoton Hong-Ou-Mandel effect. Existing quantum technologies may scale it up toward diverse applications.

Question 669 multiple-choice

Fishnet quantum field theories in four dimensions exhibit remarkable integrable structures, allowing the use of advanced mathematical techniques to compute operator scaling dimensions. These theories are closely related to gamma-deformed N=4 Super Yang-Mills and feature connections to spin chains and Quantum Spectral Curve methods. In the context of planar four-dimensional fishnet-type quantum field theories, which mathematical tool is crucial for constructing equations that determine anomalous dimensions of local operators, specifically enabling high-precision calculations for operators characterized by charge J=3? 1) Bethe-Salpeter equation 2) Lax pair formalism 3) Operator product expansion (OPE) 4) Renormalization group equations 5) Conformal bootstrap constraints 6) Quantum Spectral Curve (QSC) combined with Baxter equation techniques 7) Faddeev–Popov ghost analysis

✓ Correct Answer:

The correct answer is 6) Quantum Spectral Curve (QSC) combined with Baxter equation techniques.

📚 Reference Text:

Title: Integrability of conformal fishnet theory Year: 2017 Paper ID: 210f9c4607171c1bd622b32a8af3383615e4996e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/210f9c4607171c1bd622b32a8af3383615e4996e Abstract: A bstractWe study integrability of fishnet-type Feynman graphs arising in planar four-dimensional bi-scalar chiral theory recently proposed in arXiv:1512.06704 as a special double scaling limit of gamma-deformed N$$ \mathcal{N} $$ = 4 SYM theory. We show that the transfer matrix “building” the fishnet graphs emerges from the R−matrix of non-compact conformal SU(2, 2) Heisenberg spin chain with spins belonging to principal series representations of the four-dimensional conformal group. We demonstrate explicitly a relationship between this integrable spin chain and the Quantum Spectral Curve (QSC) of N$$ \mathcal{N} $$ = 4 SYM. Using QSC and spin chain methods, we construct Baxter equation for Q−functions of the conformal spin chain needed for computation of the anomalous dimensions of operators of the type tr(ϕ1J) where ϕ1 is one of the two scalars of the theory. For J = 3 we derive from QSC a quantization condition that fixes the relevant solution of Baxter equation. The scaling dimensions of the operators only receive contributions from wheel-like graphs. We develop integrability techniques to compute the divergent part of these graphs and use it to present the weak coupling expansion of dimensions to very high orders. Then we apply our exact equations to calculate the anomalous dimensions with J = 3 to practically unlimited precision at any coupling. These equations also describe an infinite tower of local conformal operators all carrying the same charge J = 3. The method should be applicable for any J and, in principle, to any local operators of bi-scalar theory. We show that at strong coupling the scaling dimensions can be derived from semiclassical quantization of finite gap solutions describing an integrable system of noncompact SU(2, 2) spins. This bears similarities with the classical strings arising in the strongly coupled limit of N$$ \mathcal{N} $$ = 4 SYM.

Question 670 multiple-choice

Quantum algorithms for the hidden subgroup problem often exploit the structure of group representations to optimize measurement strategies, particularly in non-Abelian cases like the dihedral group. The measurement focuses on extracting subgroup information using the quantum Fourier transform and analysis of irreducible representations. When solving the dihedral hidden subgroup problem using quantum algorithms, why does measurement of the row index after the quantum Fourier transform fail to yield information about the hidden subgroup? 1) Because the row index only labels group elements, not subgroups 2) Because the row index is entangled with noise from quantum gates 3) Because the row index is always collapsed to zero after the Fourier transform 4) Because the row index corresponds to an observable that commutes with all other operators 5) Because the measurement of the row index projects onto eigenstates unrelated to subgroup structure 6) Because the row index is discarded before measurement in standard algorithms 7) Because the row state is maximally mixed, making the measurement outcome completely random and uninformative about the hidden subgroup

✓ Correct Answer:

The correct answer is 7) Because the row state is maximally mixed, making the measurement outcome completely random and uninformative about the hidden subgroup.

📚 Reference Text:

in the new basis ρH=1 |G|/circleplusdisplay x∈ˆGIdx⊗/parenleftBigg/summationdisplay h∈HΓx(h)/parenrightBigg (75) =/summationdisplay x∈ˆGp(x)Idx dx⊗ρH,x⊗ |x/angbracketright/angbracketleftx|, (76) a classical mixture over the irrep label x∈ˆGwith probabilities p(x):=dx |G|/summationdisplay h∈Hχx(h) (77) (whereχx:= tr Γ xdenotes the character of the xth irrep) of a maximally mixed row state Idx/dxand the column state ρH,x:=1/summationtext h∈Hχx(h)/summationdisplay h∈HΓx(h). (78) Since the row state is maximally mixed, it is clear that one learns nothing by measuring the row index [21]. Thus, it is natural to work in this basis and to discard the row state, focusing on the column state ρH,x. This procedure corresponds exactly to the particular form of the optimal povm considered in Section 8. To see this correspondence in detail for the dhsp , consider the irreducible representations of the dihedral group [48]. These irreps are all either one- or two-dimensional. The two-dimensional irreps may be conveniently labeled by an integer 1 ≤x≤ ⌈N/2⌉ −1, and are given by (79) Γ x(sk) =/parenleftbigg ωxk0 0ω−xk/parenrightbigg and Γ x(rsk) =/parenleftbigg 0ω−xk ωxk0/parenrightbigg whereω:= exp(2πi/N). Notice that the irreps satisfy Γ −x(g) =XΓx(g)X whereXis the Pauli matrix (80) X:=/parenleftbigg 0 1 1 0/parenrightbigg . OPTIMAL MEASUREMENTS FOR THE DIHEDRAL HSP 25 WhenNis odd, there are two one-dimensional irreps, the trivial irrep Γτ(sk) = Γ τ(rsk) = 1 (81) and the alternating irrep Γσ(sk) = 1 and Γ σ(rsk) =−1. (82) WhenNis even, there are two additional one-dimensional irreps, the even irrep Γe(sk) = Γ e(rsk) = (−1)k(83) and the odd irrep Γo(sk) = (−1)kand Γ o(rsk) =−(−1)k. (84) Now consider the approach to the dhsp of ﬁrst performing a quantum Fourier transform over G=DNand then measuring the irrep index. If the result obtained corresponds to a two-dimensional irrep, then we can measure the row index and will randomly obtain one of two outcomes. We associate one of these outcomes with the irrep label xand the other with the irrep label −x, performing the Xoperation on the column in the latter case. Furthermore, we group the trivial irrep and the alternating irrep together into a two-dimensional space, and similarly for the even and odd irreps. Labeling these two spaces 0 and N/2, respectively, it is now easy to see that this procedure is equivalent to the measurement procedure outlined in Section 8, where the irrep label xcorresponds to the measurement of x∈ZN. Department of Computer Science and Engineering, University of Washing- ton, Seattle, WA 98195, USA E-mail address :dabacon@cs.washington.edu Institute for Quantum Information, California Institute of Technology, Pasadena, CA 91125, USA E-mail address :amchilds@caltech.edu Departments of Computer Science and Physics, University of California, Santa Barbara, Santa Barbara, CA 93106, USA E-mail address :vandam@cs.ucsb.edu

Question 671 multiple-choice

In algebraic combinatorics and representation theory, the Robinson-Schensted-Knuth (RSK) algorithm is central for relating sequences to Young tableaux, with row insertion ensuring certain tableau properties. Semi-standard Young tableaux (SSYTs) play a key role in decomposing representations and constructing symmetric functions. Which operation in the RSK algorithm is responsible for maintaining the semi-standard property of a Young tableau by bumping larger entries downward when inserting a new integer? 1) Column removal 2) Diagonal shifting 3) Row insertion 4) Reverse bumping 5) Strip deletion 6) Tableau merging 7) Box promotion

✓ Correct Answer:

The correct answer is 3) Row insertion.

📚 Reference Text:

2,33 3. (2.12) In terms of horizontal strips, the above decomposition of an irrep λofU(d)into irrepsµofU(d−1)can be rephrased as the set of all µthat can be obtained from λby removing a horizontal strip in all possible ways. 2.4 RSK algorithm and composition of Young tableaux The discussion in this section is taken from the book by Fulton [11]. This procedure is used in the algorithm DualSchur (step 4). For a more detailed explanation of how to obtain SYTs with permuted content, see [11] chapter 4. The RSK (Robinson-Schensted-Knuth) algorithm establishes a correspondence between pairs of words and pairs of tableaux. The main part of the RSK is a procedure called row insertion that lets one insert a letter into a tableau such that the resulting tableau is semi-standard with one more box. This correspondence has several applications, but the primary application here is to produce semi-standard Young tableaux where the content is permuted. In this subsection, we brie y describe this algorithm. The row insertion procedure takes as input a tableau Tand an integer xand outputs a tableau with one more box thanT. The procedure is as follows 1. Find the number in the rst row of Tthat is greater than x. 2. If there is none, then place xin a box at the end of the rst row. 3. If there are numbers greater than xin the rst row, let ybe the smallest among them. Place xiny's position (x`bumps'y). 4. Repeat the previous steps with y(in the place of x) and starting with the second row. 6 The RSK algorithm uses the above row bumping procedure. It takes a pair of words, say u=u1u2...urand v=v1v2...vrthat has the following two properties as input. The rst is that uis weakly increasing and second ifuk−1=uk, thenvk−1≤vk. Given such pairs as input the procedure produces a pair of tableaux (P,Q) iteratively as follows. Start with the base tableaux xandy, wherex=v1andy=u1. Then from any pair (Pk−1,Qk−1), row insert vkintoPk−1gettingPk. Then add a box to Qk−1in the position where the new box is in Pkand putukin this box. This procedure allows us to de ne a product or composition of tableaux mentioned in the previous subsection. Suppose SandTare two tableaux, then S·Tis de ned as follows. If Tconsists of only one box, then the product of SandTis the result of row insertion into S. IfTcontains more than one box, then we row insert them one by one into Sstarting from the bottom left box and moving left to right along each row and upwards along the rows. In this paper, we will need this procedure (in the algorithm DualSchur step 4) to create an SSYT V from another SSYT Uwith the content permuted. In order to describe it for any permutation, we only need to show it for a single transposition. As noted above, an SSYT consists of horizontal strips each having the same number. Now suppose that the transposition is (k,k+1) i.e., if the original SSYT Ucontainsnkboxes numberedkandnk+1boxes numbered k+ 1, then the new

Question 672 multiple-choice

In knot theory and quantum computation, the calculation of knot invariants such as the Jones and HOMFLYPT polynomials is central to classifying links and understanding their properties. Quantum algorithms and additive approximation frameworks are increasingly used to efficiently estimate these invariants, particularly for cases where classical computation is infeasible. Which property allows the absolute value of the Jones polynomial evaluated at primitive roots of unity to serve as an invariant for unoriented links? 1) Its definition only depends on the number of braid strands. 2) The HOMFLYPT polynomial is always orientation-independent at any value. 3) The trace closure of braids always produces oriented links. 4) The Jones polynomial is always real-valued regardless of orientation. 5) The absolute value at roots of unity is invariant under link orientation. 6) The Kauffman bracket is unaffected by orientation changes in all closures. 7) Quantum algorithms can exactly compute the Jones polynomial for any link.

✓ Correct Answer:

The correct answer is 5) The absolute value at roots of unity is invariant under link orientation..

📚 Reference Text:

top with the corresponding rstrands at the bottom as in the trace closure. Unlike with the trace closure, there is no unambiguous way to assign an orientation to the plat or generalized closure of a braid (unless n=t). While this poses a potential problem for deﬁning the Jones and HOMFLYPT polynomials of such closures of braids, it is known [26] (and easily proved using the Kauﬀmann bracket [21] formula) that if− →Land− →L′are two oriented links that are isotopic as unoriented links, their Jones pol ynomials satisfy J− →L(q) =qmJ− → L′(q) for some easily computable m∈Z. This implies the absolute value of the Jones polynomial at roots of unity is independent of the orientation of a link, an d is thus an invariant of unoriented links. In Section 5, we provide a representation-theoretic formula for the absolute value of the Jones polynomial of any generalized closure of a braid at primitive roots of unity, while in Section 6, we present a quantum algorithm which approximate s these invariants of unoriented links. Additive approximations The notion of approximation which we will require was recent ly formalized in [7]; we review that material here. Let Xbe a set of problem instances, and suppose we are given a non- negative function f:X → Rwhich is potentially diﬃcult to evaluate exactly. The main i dea of [7] is to approximate the function fwith respect to some positive normalization function g:X → R. Departing slightly from the deﬁnition given in [7], an additive approximation for the normalized function f/gassociates a random variable Z(x) to any problem instance Pawel Wocjan and Jon Yard 7 x∈ Xandδ>0 satisfying Pr/braceleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(x) g(x)−Z(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤δ/bracerightbigg ≥3/4. In addition, it is required that such an approximation be ach ieved in time which is polynomial in the size of the problem instance and in 1 /δ. In Section 5 we provide tight upper bounds on the evaluations of the invariants we wish to approximate. We will see in (5.24) that if b∈Bn, the one-variable HOMFLYPT of the trace closure obeys/vextendsingle/vextendsingleH(k) bb(e2πi/ℓ)/vextendsingle/vextendsingle≤[k]n−1 ℓ. Similarly, we show in (5.38) that if n= 2p+t andx,y∈Bn, the Jones of the generalized closure satisﬁes/vextendsingle/vextendsingleJ(ξp,r x,y(b),e2πi/ℓ)/vextendsingle/vextendsingle≤[2]p+t−1 ℓ. These bounds are achieved by closing the identity braid for t he one-variable HOMFLYPT of the trace closure, or otherwise the braid x−1y−1for the Jones of the generalized closure. We use these normalizations for our additive approximation s, so that the absolute values of the resulting normalized polynomials always lie between 0 a nd 1, regardless of the size of the braid, number of strands, or particular primitive root of un ity. We therefore consider the following two problems: Problem 2.1 (Approximate HOMFLYPT Trace Closure) Given is a braid b∈Bnof lengthm,δ >0and positive integers k,ℓsatisfying 2≤k < ℓ, wherekis a constant. The task is to sample from a random variable Z∈C,|Z| ≤1, which is an additive approximation of the one-variable HOMFLYPT polynomial of the trace closur e, evaluated at e2πi/ℓ, in the sense that Pr/braceleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 [k]n−1 ℓH(k) bb(e2πi/ℓ)−Z/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤δ/bracerightbigg ≥3/4. Problem 2.2 (Approximate

Question 673 multiple-choice

Algorithms for tensor scaling under group actions are fundamental in computational mathematics, intersecting areas such as representation theory, numerical analysis, and complexity theory. Ensuring efficient termination and practical feasibility requires careful analysis of probabilistic, algebraic, and computational properties. In high-dimensional tensor scaling algorithms that employ group actions, which technique is used to guarantee termination by demonstrating that continued iteration would violate established bounds? 1) Applying spectral norm minimization to the tensor at each step 2) Utilizing randomized rounding to approximate group elements 3) Imposing a fixed iteration limit independent of input parameters 4) Relying on the central limit theorem for convergence guarantees 5) Proving a contradiction by showing that ongoing execution violates derived lower and upper bounds 6) Enforcing sparsity constraints on the scaling matrices 7) Using deterministic sampling of highest weight vectors

✓ Correct Answer:

The correct answer is 5) Proving a contradiction by showing that ongoing execution violates derived lower and upper bounds.

📚 Reference Text:

that, with probability at least 1/2, Algorithm 1 terminates in step 3 by outputting an appropriate scaling. In step 1 we select a tuple of random matrices gaccording to the parameters explained in Corollary 2.7. ThusitfollowsfromCorollary2.7that,withprobabilityatleast1/2, gX0andrgXsPGrXs and there exists a highest weight vector PPHWVpCrVspkqqof degree 0 k¤K, wherekp, such thatPpgXq0. We may condition on this event. By Proposition 2.9, we may further assume that Phas integer coeﬃcients and that it satisﬁes the bound jPpYqj¤pn1:::ndqkkYk (17) for all tensors YPV. Forstep2,notethat PpgXq0,sopPpgXq. Ontheotherhand,theranksoftheone-body marginalspiqare invariant under scaling by the group action G[dur00,BGO17]. Therefore, since the target spectra ppiqhave full rank by assumption, this means that the piq gXnecessarily must have full rank also. It follows that the algorithm does not halt and instead proceeds to step 3. This also implies that gis not singular and is hence actually inG, becausepiq gXgpiqAgpiq;:for some matrix A, and this would be singular if gpiqwere singular. We now move to the scaling step 3. Let us denote by grtsPGthe value of the group element g at the beginning of the t-th iteration, and by Yrts:grtsXthe corresponding tensor. Suppose for sake of ﬁnding a contradiction that the algorithm has not terminated after Tsteps but instead proceeds to 4. We will prove the following three statements: •Lower bound: jPpYr1sqj¥2kp2°d i0log2pniqbdlog2pMqq, •Progress per step: jPpYrt1sqj¡2k 32 ln 2"2jPpYrtsqjfort1;:::;T, •Upper bound: jPpYrtsqj¤2k°d i1log2pniq. 28 Forthelowerbound,notethat gr1spgp1q{kgXk;gp2q;:::;gpdqqandsoYr1s:gX{kgXk. Now, jPpYr1sqjjPpgXqj kgXkk¥1 kgXkk; sincePis homogeneous and both PandgXhave integer coeﬃcients only. On the other hand, kgXk¤kXkd¹ i1kgpiqkop¤kXkd¹ i1kgpiqkF¤?2n0n1:::nd2bd¹ i1pniMq 21 21 2log2pn0n1:::ndqblog2pn1:::ndqdlog2pMq¤22°d i0log2pniqbdlog2pMq; where kkopdenotes the operatornorm (assuming n0n1nd¡1). Thus thelower bound follows. Theprogressperstepfollowsdirectlyfromtheanalysisintheprecedingsection(Proposition3.1) andthefactthat Yrtsremainunitvectorsthroughout,whichweprovebelow. Theupperbound also follows from the fact that Yrtsremain unit vectors throughout and Eq. (17). The unit norm condition is clear for Yr1s, and fort1;:::;Twe have that kYrt1sk2trrpiq Yrtss trrdiagpppiq Òq1{2pRpiqq1piq Yrt1sppRpiqq1q:diagpppiq Òq1{2s trrdiagpppiq Òqs1; whereiis the index of the marginal that we selected in the t-th scaling step and Rpiqthe Cholesky factor ofpiq Yrts. Thus we have proved all three statements. Together, they imply that Tk 32 ln 2"2 kd¸ i1log2pniq k 2d¸ i0log2pniqbdlog2pMq Thekappears on both sides (which is indeed crucial since kcould be exponential in the input parameters!) and we get that T 32 ln 2"2 d¸ i1log2pniq2d¸ i0log2pniqbdlog2pMq¤3d¸ i0log2pniqbdlog2pMq and so T 32 ln 2 "2 3d¸ i0log2pniqbdlog2pMq ; which is the desired contradiction. 3.3 Bit complexity analysis of Algorithm 1 In this section, we give a sketch on how to implement Algorithm 1 so that all the intermediate computationsaredoneonnumberswithpolynomialnumberofbits. Theanalysisweprovideseems simpler than the analysis in [ GGOW16 ,Fra18]. We use notation from the proof of Theorem 1.13 in 29 Section3.2. Wewill maintainthegroup elements grts’s. Supposethe algorithm,given grts,would have chosen to normalize the ithcoordinate. That is, it would have multiplied Ato theithelement ofgrts. Wewilltruncate AtoobtainB,andsetgrt1stobeBmultipliedtothe ithcoordinateof grts(we will also normalize by an appropriate factor close to 1to be speciﬁed later). The truncation will be done to a ﬁxed number of bits after the binary point. Suppose we have a tensor YYrtsgrtsX(at some point t) and a marginal piq Ythat is at least"-far

Question 674 multiple-choice

In computational group theory, algorithms for permutation groups rely on structural concepts such as coset representatives, stabilizer chains, and strong generator sets. These foundations enable efficient solutions to problems involving group actions on sets and canonical labeling. Which statement accurately describes the role of a strong generator set in the context of permutation groups and stabilizer chains? 1) It consists of all elements fixed by every subgroup in the chain. 2) It is the set of all possible coset representatives for the group. 3) It is formed by selecting random elements from each subgroup in the stabilizer chain. 4) It contains only the identity permutation and its inverse. 5) It is the minimal set of generators that commute with every group element. 6) It is built by taking unions of coset representatives at each level of the stabilizer chain, allowing efficient generation of the entire group. 7) It consists of permutations that only generate abelian subgroups.

✓ Correct Answer:

The correct answer is 6) It is built by taking unions of coset representatives at each level of the stabilizer chain, allowing efficient generation of the entire group..

📚 Reference Text:

equal to |H|and the set {ϕ1,ϕ2,...,ϕ k}is a set of coset representatives of HinG. As developed by Sims [20], pointwise stabilizers are fundamental in the de- sign of algorithms for permutation group problems. The structure used is the chain of stabilizers subgroups in Ggiven by: 1=G(n)≤G(n−1)≤...≤G(1)≤ G(0)=G.LetCibe a complete set of right coset representatives of G(i)in G(i−1),1≤i≤n. Then/uniontextn−1 i=1Ciforms a generator set for G. Such a genera- tor set is called a strong generator set forG[20, 9]. Any g∈Ghas a unique factorization g=g1g2...g n, withgi∈Ci. We now recall two basic algorithmic results concerning permutation groups. Given as input the generator set Sfor a permutation group G≤Sn, the fol- lowing two basic algorithmic tasks can be implemented in time polynomial in n(see e.g. [20, 9] for these and other results and [15, 11] for a comprehensive treatment). Theorem 2.4 1. For each element i∈[n]the orbit of i, deﬁned as {ig|g∈G}, can be computed in polynomial time. 4 2. The tower of subgroups 1=G(n)≤G(n−1)≤...≤G(1)≤Gcan be computed in time polynomial in n. (I.e. the right coset representative sets Cifor the groups G(i)inG(i−1),1≤i≤ncan be computed in polynomial time giving a strong generator set for each G(i)includingG). 3 Computing the least element of a right coset In this section we describe a simple polynomial-time algorithm that takes as input a permutation group /angbracketleftA/angbracketright=G≤Snand a permutation σ∈Snand computes the lexicographically least element of the right coset GσofGinSn. Here, we use the standard lexicographic ordering of permutations in Sngiven by the ordering of the set [ n] ={1,2,...,n }. This algorithm is a crucial ingredient in the proof of the main theorem in the next section. Theorem 3.1 There is a polynomial-time algorithm that takes as input a per- mutation group /angbracketleftA/angbracketright=G≤Snand a permutation σ∈Snand computes the lexicographically least element of the right coset Gσ. Proof . We describe the easy algorithm and then argue its correctness. Input :G≤Sn,σ∈Sn Output : Lexicographically least element in Gσ LetG(n)≤G(n−1)≤...≤G(1)≤Gbe the tower of subgroups of G where, by Theorem 2.4, the generator set for each G(i)and the strong generator set for Gcan be computed in polynomial time; π0=σ; fori:= 0ton−1do letx:=i+ 1; ﬁnd the element yinxG(i)such thatyπiis minimum; {This can be done in polynomial time as the entire orbit xG(i)ofxin G(i), which is a set of size at most n−i, can be computed in polynomial time by applying Theorem 2.4, and ﬁnding the minimum in the orbit takes linear time. }; Letgi∈G(i)be such that xgi=y; πi+1:=giπi; end Result :πnAlgorithm 1 :Lexicographically least in a Right Coset Sinceπ0=σandG(n−1)={1}, it suﬃces to prove the following claim in order to show that the algorithm computes the lexicographically least element ofGσ. Claim 3.2 For all 0≤i

Question 675 multiple-choice

Quantum optimization leverages advanced algebraic techniques to solve problems involving local Hamiltonians, which are fundamental in quantum computing and physics. Representation theory and operator algebras offer alternative methods to traditional approaches for analyzing these quantum systems. Which technique utilizes the algebra generated by qubit swap operators, introduces a non-commutative Sum of Squares (ncSoS) hierarchy for quantum optimization, and achieves numerically exact solutions for the Quantum Max Cut Hamiltonian on small graphs with uniform edge weights? 1) Quantum Approximate Optimization Algorithm (QAOA) 2) Classical Lasserre Hierarchy using Pauli matrices 3) Matrix Product State (MPS) variational optimization 4) Semidefinite programming over commuting observables 5) Tensor network contraction algorithms 6) Variational quantum eigensolver (VQE) with Pauli operators 7) Non-commutative Sum of Squares hierarchy based on swap operators

✓ Correct Answer:

The correct answer is 7) Non-commutative Sum of Squares hierarchy based on swap operators.

📚 Reference Text:

Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators Adam Bene Watts1, Anirban Chowdhury1, Aidan Epperly2, J. William Helton2, and Igor Klep3,4 1University of Waterloo 2University of California San Diego 3Faculty of Mathematics and Physics, University of Ljubljana 4Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for local Hamiltonian problems. In this paper we attack this problem using the algebraic structure of QMC, in particular the relationship between the quantum max cut Hamiltonian and the representation theory of the symmetric group. The first major contribution of this paper is an extension of non-commutative Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of relaxations to Quantum Max Cut. The hierarchy we present is based on op- timizations over polynomials in the qubit swap operators. This is in contrast to the “standard” quantum Lasserre Hierarchy, which is based on polynomials expressed in terms of the Pauli matrices. To prove correctness of this hierarchy, we exploit a finite presentation of the algebra generated by the qubit swap op- erators. This presentation allows for the use of computer algebraic techniques to manipulate and simplify polynomials written in terms of the swap opera- tors, and may be of independent interest. Surprisingly, we find that level-2 of this new hierarchy is numerically exact (up to tolerance 10−7) on all QMC instances with uniform edge weights on graphs with at most 8 vertices. The second major contribution of this paper is a polynomial-time algorithm that computes (in exact arithmetic) the maximum eigenvalue of the QMC Hamiltonian for certain graphs, including graphs that can be “decomposed” as a signed combination of cliques. A special case of the latter are complete bi- partite graphs with uniform edge-weights, for which exact solutions are known from the work of Lieb and Mattis [LM62]. Our methods, which use represen- tation theory of the symmetric group, can be seen as a generalization of the Lieb-Mattis result. Adam Bene Watts: adam.benewatts1@uwaterloo.ca Anirban Chowdhury: anirban.chnarayanchowdhury@uwaterloo.ca Aidan Epperly: aepperly@ucsd.edu J. William Helton: helton@math.ucsd.edu Igor Klep: igor.klep@fmf.uni-lj.si Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 1arXiv:2307.15661v3 [quant-ph] 24 Apr 2024 Contents 1 Intro 3 1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Overview of Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Reader’s Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Quantum Max Cut and The Swap Matrix Algebra 9 2.1 Pauli

Question 676 multiple-choice

Semidirect discrete logarithm problems (SDLP) over nonabelian groups are being evaluated for their cryptographic hardness, especially in relation to signature schemes and quantum security. The parameter choices for group order and exponent significantly impact the vulnerability of these cryptographic constructions. Under which parameter condition does an efficient method exist to solve SDLP over finite nonabelian groups of order $p^3$ and exponent $p^2$, directly compromising the security of SPDH-Sign? 1) When SDLP instances are parameterized by $n \leq \text{poly}(\log p) \cdot p$ 2) When SDLP instances have $n \geq p^3$ 3) When the group order is less than $p$ 4) When SDLP is considered over abelian groups of order $p^2$ 5) When $n$ is approximately $p^2$ or larger 6) When the exponent is less than $p$ 7) When the group has a trivial center

✓ Correct Answer:

The correct answer is 1) When SDLP instances are parameterized by $n \leq \text{poly}(\log p) \cdot p$.

📚 Reference Text:

Title: A small serving of mash: (Quantum) algorithms for SPDH-Sign with small parameters Year: 2025 Paper ID: 7dcba5f653426a5080bb6fc4e2d8917224f7cf8e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/7dcba5f653426a5080bb6fc4e2d8917224f7cf8e Abstract: Abstract We find an efficient method to solve the semidirect discrete logarithm problem (SDLP) over finite nonabelian groups of order p 3 {p}^{3} and exponent p 2 {p}^{2} for certain exponentially large parameters. This implies an attack on SPDH-Sign, Pronounced “SPUD-Sign”. a signature scheme based on the SDLP, for such parameters. In particular, SDLP instances over such groups are parameterised by an n < ( p − 1 ) p 6 n\lt \left(p-1){p}^{6} : we develop a method to solve instances when n ≤ poly ( log p ) ⋅ p n\le {\rm{poly}}\left(\log p)\hspace{0.25em}\cdot p . Letting λ \lambda be the security parameter of SPDH-Sign, which is taken p = exp λ p=\exp \lambda , we find we may solve instances of SDLP corresponding to SPDH-Sign instances with exponentially large p p . However, for n ≈ p 2 n\approx {p}^{2} and larger, our method no longer completely solves the SDLP instances. We also study the linear hidden shift problem for a group action corresponding to SDLP and take a step towards proving the quantum polynomial time equivalence of SDLP and the semidirect computational Diffie–Hellman problem.

Question 677 multiple-choice

In quantum information theory, the overlap between product states and highly entangled multi-qubit states is crucial for analyzing entanglement and optimizing state preparation. The W state, defined for q qubits, exhibits unique properties regarding its nearest product state and overlap behavior. For a q-qubit W state, what is the exact maximum overlap Pmax between the W state and its optimal product state, and how does this value behave asymptotically as q becomes large? 1) Pmax = 1/2 and it approaches zero as q increases 2) Pmax = 1/q and it remains constant for large q 3) Pmax = (1/q)^q and it approaches one as q increases 4) Pmax = 1/q and it approaches zero as q increases 5) Pmax = (q−1)/q and it approaches one as q increases 6) Pmax = [(q−1)/q]^(q−1) and it approaches 1/e as q increases 7) Pmax = 2^(-q) and it approaches zero as q increases

✓ Correct Answer:

The correct answer is 6) Pmax = [(q−1)/q]^(q−1) and it approaches 1/e as q increases.

📚 Reference Text:

number of basis states |k/an}brack⌉tri}htthat have corresponding basis states in|ψq S/an}brack⌉tri}ht(namely, for which k∈S). The more such basis states, the larger the overlap will be (ignoring, for now, the amplitudes of the states). In the c ase of a small S, a single basis state is a good guess for |ϕ/an}brack⌉tri}ht, since trying to vary any of the xm’s away from the edges of their range will add a lot of basis states to the expansion, and most of them will not be members ofS, consequently decreasing P. In the case of a large S, a product state is desired with a lot of basis states in its expansion, since there are a lot of m embers inS, and in that case the complete ES state is hard to beat. 4. A counterexample It turns out that the approximated formula presented above i s not valid for all the ES states. A state that violates this formula was analyzed in Re f. [6]. This is the q-qubitW state, which consists of all the basis states that have q−1zeros and a single 1: |W/an}brack⌉tri}ht=1√qq−1/summationdisplay m=0|2m/an}brack⌉tri}ht. (20) Since for the simple Wstate|S|=q, andq≤√Qfor a large enough q(practically q≥4is suﬃcient), our guess yields one of the basis states as the nea rest product state and Pq S= 1/q. However, it turns out that the real nearest product state is |ϕ/an}brack⌉tri}ht=/parenleftbigg/radicalbigg q−1 q|0/an}brack⌉tri}ht+/radicalbigg 1 q|1/an}brack⌉tri}ht/parenrightbigg⊗q , (21) which yields Pmax(W) = [(q−1)/q]q−1. This is not only diﬀerent from Pq S; it is also asymp- totically diﬀerent, as 10 Pq S− −− → q→∞0 Pmax(W)− −− → q→∞1 e.(22) What property of the simple Wstate makes it violate the validity of the approximation? Note that for each qubit, the GHZ and balanced generalized Wstates have an equal number of zeros and ones across all the basis states. The simple Wstate obviously does not have this property. To understand why this is important, let us recons ider the maximization problem of ﬁnding the nearest product state |ϕ/an}brack⌉tri}ht. We can explore the possible product states by taking a small v ariation around some basis state|k0/an}brack⌉tri}ht. For two binary numbers k1andk2, let us denote the Hamming distance between them, which is the number of bits they diﬀer on, as d(k1,k2). We can then divide the set S to disjoint subsets according to the Hamming distance from k0: S=S0·∪...·∪Sq Sm={k∈S:d(k,k0) =m}.(23) Without loss of generality we can ﬁx k0= 0(this can be arranged by applying local NOT gates which do not aﬀect the entanglement nor the Hamming dis tances). The Hamming distanced(k,k0)is then equal to the number of ones in k, and a small variation around k0 means that the xm’s are small. The terms in Eq. (13) can then be grouped accordi ng to the subsets ofS: fq S(x1,...,x q) =1/radicalbig |S|q/summationdisplay n=0/summationdisplay j1...jq∈Sn jm1,...,jmn=1√xm1·····√xmn/productdisplay m/negationslash=mi√1−xm. (24) Thenth term in this expansion has nmultiplicands of the form√xm, so it is dominant in respect to the (n+1)th term. Through this expansion we see that the nearest produ ct state

Question 678 multiple-choice

The quantum double D of a finite group is an important algebraic structure in topological quantum field theory and quantum computing, where its irreducible representations classify quasiparticle excitations. These representations can be understood geometrically and are closely related to advanced concepts in noncommutative geometry and braided algebraic structures. Which labeling uniquely specifies an irreducible representation of the quantum double D for a finite group G in the context of topological quantum computation? 1) A pair (C, π), where C is a conjugacy class of G and π is an irreducible representation of the isotropy group of an element in C 2) A single irreducible character of G 3) A triple (G, H, χ), where H is a subgroup and χ is a character of H 4) The dimension of the representation and the order of G 5) A sequence of eigenvalues associated with the center of D 6) The tensor product of two irreducible representations of G 7) The Frobenius-Schur indicator and the conjugacy class of G

✓ Correct Answer:

The correct answer is 1) A pair (C, π), where C is a conjugacy class of G and π is an irreducible representation of the isotropy group of an element in C.

📚 Reference Text:

Title: Quantum geometric Wigner construction for D(G) and braided racks Year: 2025 Paper ID: 6c73933e5961fcb686b3e2129f50bff92bcf1317 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/6c73933e5961fcb686b3e2129f50bff92bcf1317 Abstract: The quantum double of a finite group plays an important role in the Kitaev model for quantum computing, as well as in associated topological quantum field theories, as a kind of Poincaré group. We interpret the known construction of its irreps, which are quasiparticles for the model, in a geometric manner strictly analogous to the Wigner construction for the usual Poincaré group of R1,3. Irreps are labeled by pairs (C,π), where C is a conjugacy class in the role of a mass-shell, and π is a representation of the isotropy group CG in the role of spin. The geometric picture entails as a quantum homogeneous bundle where the base is G/CG, and D∨(G)→C(G) as another homogeneous bundle with base the group algebra CG as noncommutative spacetime. Analysis of the latter leads to a duality whereby the differential calculus and solutions of the wave equation on CG are governed by irreps and conjugacy classes of G respectively, while the same picture on C(G) is governed by the reversed data. Quasiparticles as irreps of D(G) also turn out to classify irreducible bicovariant differential structures ΩC,π1 on D∨(G) and these in turn correspond to braided-Lie algebras LC,π in the braided category of G-crossed modules, which we call “braided racks” and study. We show under mild assumptions that U(LC,π) quotients to a braided Hopf algebra BC,π related by transmutation to a coquasitriangular Hopf algebra HC,π.

Question 679 multiple-choice

In quantum machine learning (QML), understanding and exploiting the symmetry properties of data and models is essential for designing efficient and generalizable algorithms. Symmetry groups can be discrete or continuous, and their correct implementation affects model behavior and performance. Which condition is necessary to guarantee that predicted labels from a quantum machine learning model remain invariant under the action of a symmetry group G? 1) The quantum neural network must be non-linear with respect to group actions. 2) Both the quantum channel and the measurement operator must be equivariant with respect to the symmetry group. 3) The data must be labeled using only discrete symmetry groups. 4) The model must ignore measurement operator symmetry. 5) Training must use only example-driven tutorials. 6) The symmetry group must be continuous and parameterized by real numbers. 7) Feature engineering must rely exclusively on classical machine learning principles.

✓ Correct Answer:

The correct answer is 2) Both the quantum channel and the measurement operator must be equivariant with respect to the symmetry group..

📚 Reference Text:

groups Gi. Further, endowing ﬁxed data with diﬀerent labels will often change the symmetry groups at play. This could lead, for instance, to a scenario where the data in diﬀerent classes is associated with diﬀer- ent symmetry groups. In all cases, it is up to the wisdom of the QML practitioner to both discover the symmetries in a given task and decide which of them to leverage. As usual in science, good examples will cultivate this wisdom, and we have thus tried to make this tutorial as example-driven as possible. Recently, there have been several proposals within the nascent ﬁeld of GQML to create QML models hthat re- spect the symmetries and label invariances of the prob- lem at hand, as this can leads to models with less data- requirements, simpler training landscapes, and better gen- eralization (see Refs. [24–32, 57]). While there are many ways to ensure that a QML model has in Eq. (1) is invariant under G, here we invoke a strategy requiring the following two conditions: equivari- anceunderGof the parameterized quantum channel W(U k k i(Uy) k) =U k0W((i) k)(Uy) k0;8U2G; (3) and equivariance of the measurement operator [M;U k0] = 0;8U2G: (4) One can readily verify that if Eqs. (3) and (4) are satisﬁed, thenthemodelproducespredictedlabelsthatare invariant 5 under the action of G: h(UiUy) = Tr W((UiUy) k)Mi = Trh U k0W((i) k)(Uy) k0Mii = Trh W((i) k)(Uy) k0MiU k0i = Tr W((i) k)Mi =h(i);8U2G: (5) Conceptually, we can think of equivariant quantum neu- ral networks as “ passing” the action of the symmetry from their input, to their output, while equivariant measure- ments lead to models that “ absorb” the action of the sym- metry. WereferthereadertotheexistingGQMLliterature of Refs. [24–32, 57] for additional details on the importance of equivariance and invariance, and their crucial role on im- proving the performance of QML models. IV. EXAMPLES OF DISCRETE AND CONTINUOUS SYMMETRIES IN QML Discrete groups, as the name implies, have underlying discrete sets, often ﬁnite sets or the set of integers Z. For instance, consider a QML task of where we want to classify one-dimensional real-valued data. Namely, as shown in Fig. 2(a), we want to distinguish blue circles (with label yi= 0) from orange triangles (with label yi= 1). Note that here the labels of the data are invariant under the transformation x!x. To translate this into a quantum problem, we encode the data in a single qubit state (see Fig. 2(a)), with a data-dependent rotation about the Y axis of the Bloch sphere acting on the j+istate. One can verify that now the labels are invariant, for example, under a bit-ﬂip transformation, corresponding to the group Gb ip=f11;Xg; (6) withXthe Pauli-xoperator. Clearly this group is discrete and contains two elements. Continuous groups, on the other hand, are also man- ifolds, which means that they are locally homeomorphic to an Euclidean space, and as such we can parameterize regions of the group by tuples of real numbers called “co- ordinates”. For example, consider the binary QML

Question 680 multiple-choice

In knot theory, the fundamental group of the knot complement can be described using generators and relations derived from a knot diagram, particularly through the Wirtinger presentation. The interaction between group-theoretic concepts such as conjugacy classes and topological constructions like bands and crossings provides a powerful method for encoding algebraic properties into knot invariants. When constructing a knot diagram to encode conjugacy relations from a finite group G with a generating conjugacy class C, what is the effect of a band crossing beneath another stroke in terms of the group-theoretic mapping? 1) The band condition is lost, and the strokes become independent under homomorphisms. 2) The band condition is preserved, with the values related by conjugation in the group. 3) The crossed stroke is forced to equal the band value, regardless of orientation. 4) The band induces an abelian relation among all involved strokes. 5) Both the incoming and outgoing values of the crossed stroke are set to the identity. 6) The band imposes a commutator relation on the strokes. 7) The band condition forces all bands to map to the same group element.

✓ Correct Answer:

The correct answer is 2) The band condition is preserved, with the values related by conjugation in the group..

📚 Reference Text:

the Wirtinger present ation. Generating groups constants As mentioned above, we assume that Gis a ﬁnite group and that C= {hg|g∈G}is a conjugacy class that generates G. We begin by showing how to construct a knot that—with controlled error—permits us to distinguish a particular se t of generators of G(up to automorphism). Letc= (c1,...,c t) denote a sequence of elements of Cthat generate Gand [c] ={φ(c) = (φ(c1),...,φ (ck))|φ∈Aut(G)}. Our goal is to construct a knot kc, together with a distinguished collection of strokes ( s1,...,s k), with the property that uniform selection of a homomorphism ψfromH(k,C), with overwhelming probability, yields one for which ψ(s) = (ψ(s1),...,ψ (sk))∈[c]. (Observe that Aut( G) acts on the collection of legal homomorphisms, so our demand—that the resulting knot disti nguish a particular orbit under the Aut( G) action—is the strongest requirement of this type on which we can insist upon.) The construction begins with a collection of k(oriented) circles; see Figure 5a. Without further con- straints, the knot group of this link is free: legal maps ψmay assign arbitrary elements of Cto each circle. We introduce constraints among the circles with band connections ; see Figure 5b. Aband is a pair of strands that are constrained to be equal (under any homomorphism int oC), but are oppositely oriented. They are convenient because of the simple eﬀect they have upon crossi ng other strokes: 1. Should a band cross beneath another stroke, as in Figure 4b, the constraints of ( 3a) preserve the band condition: if the pair of strokes on the left side form a band ( are constrained to take equal values under any homomorphism), the same is true of the two strokes on the r ight. Furthermore, the values assumed by these two bands are related by conjugation of the value of t he stroke under which they cross. 2. Should a band cross over another stroke, eﬀectively divid ing it into three strokes as in Figure 4b, the constraints of ( 3a) demand that the “incoming” and “outgoing” values assumed b y the stroke so crossed over are the same, as though the crossing had never happened. The small interior stroke will take on another (conjugate) value, but we shall never allow these to interact with other strokes. Sincec1,...,c mare conjugates, we can write each ciasg−1c1g, for a group element g. As thecigenerate the group, however, gmay be written as a word in the ciand we have, for each ian equation ci=cvi 1, (5.2) whereviis a word in the ciand their inverses. Introducing a family of indeterminates x1,...,x kand replacing each appearance of ciin (5.2) with the variable xi, we conclude that the cisatisfy the equations xi=xwi 1, (5.3) 19 x x (a) A band.x xy yxy−1 yxy−1 (b) A band crosses beneath a stroke.x xy y (c) A band crosses over a stroke. Figure 4: Bands and band crossings. wherewiis the word in the xiand their inverses obtained from vi. We now adopt a construction of Johnson [20], who

Question 681 multiple-choice

In quantum algorithms for hidden subgroup problems, representation theory of finite groups and the quantum Fourier transform are essential tools for analyzing measurement outcomes. Concepts such as Plancherel distribution, frame vectors, and character ratios play a critical role in distinguishing subgroup structures. Which property of an irreducible representation τ most directly determines whether it is classified as a "good" or "bad" irrep in the context of second moment analysis of measurement statistics? 1) The order of the group G associated with τ 2) The number of tensor factors in τ's decomposition 3) The multiplicity of τ in the Clebsch-Gordan decomposition 4) The dimension of the frame vector space corresponding to τ 5) The character ratio |χτ(h)| divided by the dimension dτ 6) The orthogonality of τ's matrix elements under group averaging 7) The rank of the projection operator onto τ's subspace

✓ Correct Answer:

The correct answer is 5) The character ratio |χτ(h)| divided by the dimension dτ.

📚 Reference Text:

Mﬁrstapplies QFT⊗k G toσ⊗k Hg, measures the name of an irrep of Gk,ρ∗together with a row index of ρ∗, and then measures the resulting reduced state in the column space of ρ∗using a frame BofVρ∗=Vρ. Ifρ=⊗k i=1ρi,ρi∈/hatwideG, thenρ∗=⊗k i=1ρ∗ i. Theframe Bused depends onthe observed ρ∗but not on the observed row indices. 7 Suppose the hidden subgroup is Hgfor someg∈G. It is easy to see that the probability that M measures ρ∗isgiven by MHg(ρ∗) =dρ∗|Hg|k·rρ∗((Hg)k) |G|k=2kdρrρ(Hk) |G|k. Notice that MHg(ρ∗) =MH(ρ). LetB={ab,b}, where0≤ab≤1and/summationtext bab|b/a\}bracketri}ht/a\}bracketle{tb|=11Vρ. Then the reduced state in the column space of ρ∗isρ((Hg)k) rρ(Hk), ifrρ/\e}ati≀\slash= 0. Hence, the probability of observing a particular bconditioned on having observed ρ∗is MHg(b|ρ∗) =ab/angbracketleftbig b/vextendsingle/vextendsingleρ((Hg)k)/vextendsingle/vextendsingleb/angbracketrightbig rρ(Hk), ifrρ(Hk)/\e}ati≀\slash= 0, and0otherwise. Similarly, if the hidden subgroup is the identit y subgroup then M{1}(ρ∗) =d2 ρ |G|k=P(ρ), whereP(·)is thePlancherel distribution on irreps of Gk. Also M{1}(b|ρ∗) =ab dρ=N(b|ρ∗), whereN(· |ρ∗)is thenatural distribution corresponding tothe frame B. For a non-empty subset I⊆[k], deﬁneρI:= (⊗i∈Iρi)⊗(⊗i′∈[k]\I11dρi′), where11dρi′denotes the identity representation of Gof degree equal to that of ρi′. For non-empty subsets I1,I2⊆[k], deﬁne ρI1,I2:=ρI1⊗ρI2. For a representation θ=⊗n i=1θiofGn,θirepresentation of G, we use θ(g)as a shorthand for ⊗n i=1θi(g). For an irrep τ∈/hatwideG, we useaθ τto denote the multiplicity of τin the Clebsch- Gordan decomposition of θ, i.e. the number of times τoccurs in θwhenθis viewed asa representation of Gembedded as thediagonal subgroup of Gn. WeletΠθ τdenote theorthogonal projection from Vθonto the homogeneouscomponentof τintheabovedecomposition. Weusethefollowingshorthandf orexpectations: Eρ[·],Eb[·]andEg[·]denote expectations over the Plancherel distribution on ir reps, natural distribution on frame vectors and uniform distribution on elements of Grespectively. Wedeﬁne afunction X:/hatwideG⊗k×B×G→[−1,1]as X(ρ,b,g) :=/angbracketleftBig b/vextendsingle/vextendsingle/vextendsingleρ((Hg)k)/vextendsingle/vextendsingle/vextendsingleb/angbracketrightBig −1 2k, whereBis a frame for Vρ. The importance of Xwill become clear in Lemma 11 below, which shows that Eρ,b,g[|X(ρ,b,g)|]isclosely related tothe total variation distance between MHgandM{1}. We start by proving the following lemma, which is similar to [ MR05, Lemma 11]. The lemma gives us a way to express the second moment of Xin terms of projections of ‘coupled’ frame vectors b⊗b onto homogeneous components corresponding to irreps τ∈/hatwideG. The advantage of doing this is that we can now distinguish between ‘good’ irreps, namely those wit h|χτ(h)| dτsmall, and ‘bad’ irreps, namely those where|χτ(h)| dτis large. The contribution of ‘good’ irreps to the second mom ent ofXis small. This idea of distinguishing between ‘good’ and ‘bad’ irreps goes back to [MRS05]. Lemma4. Eg[X(ρ,b,g)2] =1 4k/summationdisplay I1,I2/\e}atio\slash={}/summationdisplay τ∈/hatwideGχτ(h) dτ/vextenddouble/vextenddouble/vextenddoubleΠρI1,I2 τ(b⊗b)/vextenddouble/vextenddouble/vextenddouble2 . 8 Proof.Since X(ρ,b,g) =1 2k /angbracketleftbig b/vextendsingle/vextendsingle11dρ/vextendsingle/vextendsingleb/angbracketrightbig +/summationdisplay I/\e}atio\slash={}/angbracketleftbig b/vextendsingle/vextendsingleρI(ghg−1)/vextendsingle/vextendsingleb/angbracketrightbig −1 2k =1 2k/summationdisplay I/\e}atio\slash={}/angbracketleftbig b/vextendsingle/vextendsingleρI(ghg−1)/vextendsingle/vextendsingleb/angbracketrightbig , weget Eg[X(ρ,b,g)2] = Eg 1 4k/summationdisplay I1,I2/\e}atio\slash={}/angbracketleftbig b/vextendsingle/vextendsingleρI1(ghg−1)/vextendsingle/vextendsingleb/angbracketrightbig ·/angbracketleftbig b/vextendsingle/vextendsingleρI2(ghg−1)/vextendsingle/vextendsingleb/angbracketrightbig  =1 4k/summationdisplay I1,I2/\e}atio\slash={}Eg[/angbracketleftbig b⊗b/vextendsingle/vextendsingleρI1,I2(ghg−1)/vextendsingle/vextendsingleb⊗b/angbracketrightbig ] =1 4k/summationdisplay I1,I2/\e}atio\slash={}Eg /angbracketleftBigg b⊗b/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/circleplusdisplay τ∈/hatwideGaρI1,I2 ττ(ghg−1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleb⊗b/angbracketrightBigg  =1 4k/summationdisplay I1,I2/\e}atio\slash={}/angbracketleftBigg b⊗b/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/circleplusdisplay τ∈/hatwideGaρI1,I2 τEg[τ(ghg−1)]/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleb⊗b/angbracketrightBigg =1 4k/summationdisplay I1,I2/\e}atio\slash={}/angbracketleftBigg b⊗b/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/circleplusdisplay τ∈/hatwideGaρI1,I2 τχτ(h) dτ11Vτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleb⊗b/angbracketrightBigg =1 4k/summationdisplay I1,I2/\e}atio\slash={}/summationdisplay τ∈/hatwideGχτ(h) dτ/vextenddouble/vextenddouble/vextenddoubleΠρI1,I2 τ(b⊗b)/vextenddouble/vextenddouble/vextenddouble2 . Theﬁfthequality above follows from Schur’s lemma. Lemma4takes careof the‘good’ irreps. However for ‘bad’ irr epsτ,wehavetodosomething tobound/vextenddouble/vextenddouble/vextenddoubleΠρI1,I2 τ(b⊗b)/vextenddouble/vextenddouble/vextenddouble2 . The papers [MRS05, MR05] tried to bound it using the followi ng simple geometric argument: If Bisanorthonormal basisfor Vρ,then{b⊗b}b∈Bisanorthonormal setin Vρ⊗Vρ. Hencethe expectation,overtheuniformdistributionon B,oftheabovequantityisupperboundedbyrank(ΠρI1,I2 τ)

Question 682 multiple-choice

Quantum algorithms are increasingly investigated for their potential to revolutionize computer vision tasks, offering advantages in speed and accuracy for applications such as real-time object detection. Understanding the strengths of different quantum techniques and hybrid quantum-classical models is critical for leveraging their benefits in complex visual data analysis. Which quantum approach is specifically noted for enabling faster and more accurate real-time object detection through the use of amplitude amplification? 1) Quantum Fourier Transform 2) Quantum walk algorithms 3) Quantum Approximate Optimization Algorithm (QAOA) 4) Grover’s Algorithm and Quantum Counting 5) Variational Quantum Eigensolver (VQE) 6) Quantum Convolutional Neural Networks (QCNNs) 7) Quantum phase estimation

✓ Correct Answer:

The correct answer is 4) Grover’s Algorithm and Quantum Counting.

📚 Reference Text:

Title: Quantum Algorithms and Hybrid Solutions for Real-Time Vision Applications: Evaluation and Comparison Year: 2024 Paper ID: 5b76392443116f01861a0495fa9a93233f7b63c1 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/5b76392443116f01861a0495fa9a93233f7b63c1 Abstract: This paper presents an evaluation of various quantum algorithms for their potential application in real-time vision applications. By analyzing the underlying principles and mechanisms of key quantum algorithms, including those based on the Quantum Fourier Transform (QFT), amplitude amplification, quantum walks, and quantum optimization, the study aims to assess their suitability for enhancing vision tasks. The computational advantages offered by these quantum algorithms, such as speed and efficiency, are highlighted, particularly in comparison to classical approaches. Algorithms based on amplitude amplification, like Grover’s Algorithm and Quantum Counting, demonstrate significant promise for improving the accuracy and speed of object detection, especially in scenarios requiring real-time processing. Similarly, quantum walk algorithms show potential in navigating structured data efficiently, making them relevant for rapid detection tasks. The paper further explores the integration of quantum algorithms with classical techniques, emphasizing the development of hybrid models, such as Quantum Convolutional Neural Networks (QCNNs). Experimental comparisons between traditional Convolutional Neural Networks (CNNs) and QCNNs reveal that quantum-enhanced models may offer advantages in initial training phases, suggesting enhanced capabilities in feature extraction and pattern recognition. The results indicate that while quantum algorithms hold considerable promise for advancing the field of object detection, particularly in real-time applications, further research is needed to fully realize their potential. This study contributes to the growing field of quantum computing by providing a detailed analysis of how various quantum algorithms can be applied to complex data processing tasks in computer vision, offering new directions for the development of more efficient detection systems.

Question 683 multiple-choice

Quantum algorithms for hidden subgroup problems often utilize advanced measurement strategies to distinguish subtle group structures, with success depending on parameters such as group order and the number of quantum states prepared. The matrix sum problem plays a crucial role in determining the effectiveness of these algorithms. In the context of distinguishing hidden shifts using the pretty good measurement (PGM), what is the expected number of solutions to the matrix sum problem for random choices of x and w, given groups of orders M and N and k quantum states? 1) N^k/M 2) M/N^k 3) kM/N 4) M^k/N^2 5) M^(k-1)/N 6) N/M^k 7) M^k/N

✓ Correct Answer:

The correct answer is 7) M^k/N.

📚 Reference Text:

ZM×ZNfollowed by a computational basis measurement also fails forM≪N. (Note that poly(log N) such classical samples information theoretically determine the answer even for M= 2 [9], but it is not known how to process this data eﬃciently.) Instead, wewilluse k >1statesandapplytheprettygoodmeasurement. To see the connection to the matrix sum problem, it is helpful to write these states in a diﬀerent basis. Fourier transforming the second r egister over ZN, we ﬁnd ˜ρ⊗k s=1 (MN)k/summationdisplay x∈Zk N/summationdisplay b,c∈{0,...,M−1}kω(b·x−c·x)s|b,x/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tc,x|.(6) QUANTUM ALGORITHM FOR A GENERALIZED HIDDEN SHIFT PROBLEM 5 Now introduce the states |Sx w/an}⌊ra⌋k⌉tri}ht:=1√ηxw/summationdisplay b∈Sxw|b/an}⌊ra⌋k⌉tri}ht (7) which are uniform superpositions over the solutions of the m atrix sum prob- lem, Sx w:={b∈ {0,...,M−1}k:b·x=wmodN}. (8) Here the number of solutions is ηx w:=|Sx w|. If there are no solutions (i.e., if ηx w= 0), then we deﬁne |Sx w/an}⌊ra⌋k⌉tri}ht:= 0. In terms of these states, we can rewrite (6) as ˜ρ⊗k s=1 (MN)k/summationdisplay x∈Zk N/summationdisplay w,v∈ZNω(w−v)s/radicalbig ηxwηxv|Sx w,x/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tSx v,x|.(9) Given such a state, we would like to determine the value of s. 3. Pretty good measurement approach In this section, we review the pgmapproach to distinguishing hidden subgroup states [2,3] as applied to the generalized hidden s hift states (9). Thepretty good measurement (also known as the square root measure- ment or least squares measurement) is a measurement that oft en does well at distinguishing the members of an ensemble of quantum stat es [13] (and in fact is sometimes optimal in a certain sense). For an ensem ble of states {σj}with equal a priori probabilities, the pretty good measurem ent is the povmwith elements Ej:= Σ−1/2σjΣ−1/2(10) where Σ:=/summationdisplay jσj (11) and where the inverse is taken over the support of the ensembl e. For the states (9), the pgmnormalization matrix is Σ =N (MN)k/summationdisplay x∈Zk N/summationdisplay w∈ZNηx w|Sx w,x/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tSx w,x|, (12) givingpovmelements Ej=1 N/summationdisplay x∈Zk N/summationdisplay w,v∈ZNω(w−v)j|Sx w,x/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tSx v,x|. (13) The probability of successfully identifying the hidden shi ftsis indepen- dent ofs, and is given by Pr(success) := trEs˜ρ⊗k s (14) =1 MkNk+1/summationdisplay x∈Zk N/parenleftbigg/summationdisplay w∈ZN/radicalbig ηxw/parenrightbigg2 . (15) 6 ANDREW M. CHILDS AND WIM VAN DAM Usingthisexpression, wecanshowthatthesuccessprobabil ityisappreciable when the matrix sum problem is likely to have a solution. Spec iﬁcally, we have Lemma 1 (Lemma 2 of [3]) .IfPr(ηx w≥α)≥βfor uniformly random x∈Zk Nandw∈ZN(i.e., if most instances of the matrix sum problem have many solutions), then αβ2N/Mk≤Pr(success) ≤Mk/N. Proof. For the upper bound, we have Pr(success) ≤M Nk+1/summationdisplay x∈Zk N/parenleftbigg/summationdisplay w∈ZNηx w/parenrightbigg2 =Mk N(16) since the η’s are integers and/summationtext w∈ZNηx w=Mkfor anyx. For the lower bound, we have Pr(success) ≥N Mk/parenleftbigg1 Nk+1/summationdisplay x∈Zk N/summationdisplay w∈ZN/radicalbig ηxw/parenrightbigg2 (17) by Cauchy’s inequality applied to (15). Now 1 Nk+1/summationdisplay x∈Zk N/summationdisplay w∈ZN/radicalbig ηxw≥√αPr(ηx w≥α), (18) so by the hypothesis, Pr(success) ≥αβ2N/Mkas claimed. /square For uniformly random x∈Zk Nandw∈ZN, the expected number of matrix sum solutions is µ:=E x,w[ηx w] =Mk N, (19) where we have again used the fact that/summationtext wηx w=Mkfor anyx. Thus, we expect the matrix sum problem to typically have no solutio ns fork≪ logN/logM, many solutions for

Question 684 multiple-choice

Quantum groups at roots of unity and their representation theory play a central role in modern algebra, with Ext-groups and Kazhdan-Lusztig polynomials providing key tools for computing cohomological invariants. Singular blocks in this setting involve intricate symmetries, requiring refined combinatorial techniques for analysis. In the representation theory of Lusztig quantum enveloping algebras at roots of unity, which mathematical object determines the dimensions of Ext-groups between irreducible modules in singular blocks? 1) Standard Kazhdan-Lusztig polynomials associated to the finite Weyl group 2) Graded dimensions of affine Hecke algebras 3) Parabolic Kazhdan-Lusztig polynomials associated to a parabolic subgroup of the affine Weyl group 4) Characters of the corresponding algebraic group 5) Intersection cohomology of the flag variety 6) Quantum dimensions of the modules 7) Affine root system multiplicities

✓ Correct Answer:

The correct answer is 3) Parabolic Kazhdan-Lusztig polynomials associated to a parabolic subgroup of the affine Weyl group.

📚 Reference Text:

Title: Cohomology in Singular Blocks for a Quantum Group at a Root of Unity Year: 2016 Paper ID: 98858fc6ce29fab882e5e019cabb63345c2e1d88 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/98858fc6ce29fab882e5e019cabb63345c2e1d88 Abstract: Let Uζ be a Lusztig quantum enveloping algebra associated to a complex semisimple Lie algebra 𝔤$\mathfrak {g}$ and a root of unity ζ. When L, L′ are irreducible Uζ-modules having regular highest weights, the dimension of ExtUζn(L,L′)$\text {Ext}^{n}_{U_{\zeta }}(L,L^{\prime })$ can be calculated in terms of the coefficients of appropriate Kazhdan-Lusztig polynomials associated to the affine Weyl group of Uζ. This paper shows for L, L′ irreducible modules in a singular block that dimExtUζn(L,L′)$\dim \text {Ext}^{n}_{U_{\zeta }}(L,L^{\prime })$ is explicitly determined using the coefficients of parabolic Kazhdan-Lusztig polynomials. This also computes the corresponding cohomology for q-Schur algebras and many generalized q-Schur algebras. The result depends on a certain parity vanishing property which we obtain from the Kazhdan-Lusztig correspondence and a Koszul grading of Shan-Varagnolo-Vasserot for the corresponding affine Lie algebra.

Question 685 multiple-choice

In lattice gauge theory, the characterization of quantum states and their equivalence under gauge transformations is foundational for understanding physical observables like Wilson loops. The relationship between holonomies, conjugacy classes, and gauge equivalence is central to classifying these states. Which statement accurately describes when two group basis states in a lattice gauge theory are gauge-equivalent with respect to their holonomies at a fixed base-point? 1) They are gauge-equivalent if their holonomies differ by group automorphisms that permute group elements. 2) They are gauge-equivalent if and only if their holonomies at the base-point are simultaneously conjugate by a single group element. 3) They are gauge-equivalent if their Wilson loop expectation values are equal for all loops. 4) They are gauge-equivalent only if their holonomies are identical for every possible base-point. 5) They are gauge-equivalent if their holonomies are related by conjugacy class-preserving maps. 6) They are gauge-equivalent if their group elements assigned to edges are related by arbitrary group multiplication. 7) They are gauge-equivalent only if their holonomies are invariant under all gauge transformations at every vertex.

✓ Correct Answer:

The correct answer is 2) They are gauge-equivalent if and only if their holonomies at the base-point are simultaneously conjugate by a single group element..

📚 Reference Text:

to normalize |ψ/angbracketright. Gauge transformations at most change holonomies by conjugation, so |ψ/angbracketrighthas the same Wilson loops as the state |g/angbracketrightfrom which it was constructed. We can do the same for the state |φ(g)/angbracketright=/circlemultiplytext e∈E(Λ)|φ(ge)/angbracketright, and deﬁne |χ/angbracketright=Nχ/summationdisplay |g/prime/angbracketright∼|φ(g)/angbracketright|g/prime/angbracketright (45) Sinceφpreserves conjugacy classes, |φ(g)/angbracketrightand|g/angbracketrighthave the same Wilson loops, and so it follows that|ψ/angbracketrightand|χ/angbracketrighthave the same Wilson loops as well. It remains only to show that these two states are orthogonal. We will do this by showing that |g/angbracketrightis gauge-inequivalent to |φ(g)/angbracketright, from which 9Note thatBγ(h)is a special case of the ribbon operator F(1,h)(γ)associated to a closed ribbon γ, as deﬁned in equation (24) of [13]. Therefore, Wilson loop operators Bγ([h])are particular linear combinations of ribbon operators. 10As we will see, this constitutes a suﬃcient condition on Gso that, for Λlarge enough, (Σ,Λ,G)will satisfy Proposition 3.2. Accepted in Quantum2020-09-17, click title to verify. Published under CC-BY 4.0. 13 it follows that every term in the sum in equation (44)is orthogonal to every term in the sum in equation (45). For this we need the following lemma. Lemma 3.3. Fix an arbitrary base-point v0inV. Two states|g/angbracketright,|g/prime/angbracketrightin the group basis are gauge equivalent if and only if their holonomies based at v0agree up to simultaneous conjugation, i.e. if and only if there is a single hinGsuch thatg/prime γ=hgγh−1for every loop γwhich starts and ends atv0. Proof.In the forward direction, if |g/angbracketrightand|g/prime/angbracketrightare gauge-equivalent, they are by deﬁnition related by a product of gauge-transformations at diﬀerent vertices, |g/prime/angbracketright=/producttext v∈VAv(hv)|g/angbracketrightfor somehv inG. The gauge-transformations away from v0do not change the holonomies based at v0, while the single gauge-transformation Av0(hv0)atv0changes all holonomies by conjugation by hv0, i.e. g/prime γ=hv0gγh−1 v0. In the reverse direction, assume that the holonomies based at v0are simultaneously conjugate by an element hinG. We will specify a sequence of gauge transformations which transforms |g/angbracketright into|g/prime/angbracketright. First, we specify the gauge transformation needed at the base-point. Since acting with a gauge transformation at v0conjugates all holonomies based at v0, we act with Av0(h−1), so chosen becauseAv0(h−1)|g/angbracketrightwill have the same based holonomies as |g/prime/angbracketright. Lay down a maximal spanning tree TofΛwhich contains v0. Recall that gauge transformations away from v0do not change holonomies based at v0, and note that states in the group basis are fully determined by their holonomies based at v0as well as the group elements assigned to the edges ofT. With this in mind, we will apply our remaining gauge transformations to the vertices ofTin order to make the states agree on the edges of T(and therefore on all of Λ). We proceed inductively. Choose a path from v0to any leaf, and label the vertices which arise as[v0,v1,...,vr]and the edges between them as [e1,...,er]. Compare the group element assigned toe1byAv0(h−1)|g/angbracketrightand|g/prime/angbracketright, and act with the unique gauge transformation Av1(h1)which makes Av1(h1)Av0(h−1)|g/angbracketrightand|g/prime/angbracketrightagreeattheedge e1. Inductivelywalkthroughthepath, andatthe nth step, apply the unique gauge transformation Avn(hn)which makes Avn(hn)···Av1(h1)Av0(h−1)|g/angbracketright agree with|g/prime/angbracketrightat the edge en, noting that application of Avn(hn)does not interfere with any of the previously assigned edges e1,...,en−1since by assumption Tis a tree. The two states one

Question 686 multiple-choice

In computational number theory, lifting solutions modulo composite numbers and ensuring properties like quadratic residues across all divisors are fundamental techniques, especially when working with numbers that have multiple prime factors. The structure and smoothness of numbers, as well as iterative lifting procedures, play a critical role in algorithmic efficiency and feasibility. If N is a composite integer with k distinct prime factors, what is the probability that a randomly chosen residue modulo N is a quadratic residue modulo N? 1) 1/k 2) 1/2 3) 1/2^k 4) 1/k^2 5) 1/4 6) 1/(k+1) 7) 1/N

✓ Correct Answer:

The correct answer is 3) 1/2^k.

📚 Reference Text:

not just powersmoo th)divisors ofNand this also works for powers of 2 (which is not handled in the main bod y of the paper as Nis assumed to be odd). This does not handle all cases as for example N≈2ncould have√nprime divisors all of size 2√n. D.2 Fixing n(C),n(D) modulo N Acompletelydiﬀerentdirectioncouldbetousespecial γstoensurethat4 q(4p2n(ABCD)− (n(AC)−pn(AD))2) is a square. The idea here is that this discriminant only de- pends onn(C) andn(D) moduloNand is also homogeneous, hence it is enough to specifyn(C) andn(D) moduloNup to an invertible scalar. This is more less a problem of ﬁnding a powersmooth number in a residue class, thus a p urely elementary number theory problem Title Suppressed Due to Excessive Length 35 D.3 Improved lifting idea Our most promising idea to handle the general case is the following one . Assume that Nhaskdiﬀerent prime factors. Then a residue modulo Nis a square if and only if it is a square modulo every divisor of N. A random number has thus 1 /2kchance of being a square. However, a random number is usually a square modulo half of the prime divisors of N(assuming that Nis odd). Letσ0be the quaternion we would like to lift. Thus one can do the ﬁrst step in the lifting procedure which will result in a σ1which is not perfect lift but σ0≡σ1modN1OwhereN1dividesNandN1≈√ N. The lifting procedure ensures that σ1is invertible modulo Nsoσ0/σ1≡1 modN1O. This Now we have reduced the original problem to lifting special elements, namely ones that are congruent to 1 modulo N1O. If we can lift such an element, then we can multiply it with σ1(which is powersmooth) and get a lift of σ0. Now for elements congruent to 1 modulo N1Owe will try to write them as a productα1γα2γα2where every component is in Z+N1O(as we only care about lifting modulo a scalar). We can also assume that the αi∈Ras those elements can be lifted similarly to elements in Rj(elements in Rcan be written as the product of two elements in Rjbut also a more direct lifting procedure is providedin[ KMPW21 ]).Whendoingthisliftingconditionmodulo Nweruninto a similar discriminant issue but now the mod N1part is automatically satisﬁed. So then one now expects that the quadratic residue condition is sat isﬁed for half of the prime divisors of N/N1. So thus we obtain a lift os σ0/σ1moduloN1N2 whereN2dividesN/N1andN2≈/radicalbig N/N1. We can thus iterate this procedure and inO(logN) steps we potentially arrive at a complete lift. Then the product ofO(log(N)) powersmooth numbers is still powersmooth.

Question 687 multiple-choice

In quantum algorithms for the hidden subgroup problem, representation theory and Fourier sampling play crucial roles in distinguishing subgroups of finite groups. Understanding how projection operators and subspace overlaps affect distinguishability is essential for analyzing algorithmic efficiency. Which measure provides a lower bound on the trace distance between probability distributions from strong Fourier sampling that distinguishes two subgroups H₁ and H₂, by aggregating contributions across all irreducible representations of a finite group G? 1) The rank of the projection operator Π₁ associated with H₁ only 2) The parameter δ representing overlap between invariant subspaces 3) The number of subgroups s of the finite group G 4) The dimension of the representation space V_{ρ}^{H₁} 5) The parameter ˆh, which aggregates sizes and ranks of the subgroups 6) The trace of the intersection projection Π₁,₂ 7) r(G; H₁, H₂), the aggregated distinguishability measure over irreducible representations

✓ Correct Answer:

The correct answer is 7) r(G; H₁, H₂), the aggregated distinguishability measure over irreducible representations.

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unitaryrepresentation of G. Let Πidenote the orthogonal projection onto Vρ Hii.e.Πi=1 |Hi|/summationtext h∈Hiρ(h). LetΠ1,2denote the orthogonal projection onto Vρ H1∩Vρ H2. It is easy to check that Vρ H1∩Vρ H2=Vρ /angbracketleftH1,H2/angbracketright, where/an}bracketle{tH1,H2/an}bracketri}htdenotes the subgroup of Ggenerated by H1andH2. Thus,Π1,2=1 |/angbracketleftH1,H2/angbracketright|/summationtext h∈/angbracketleftH1,H2/angbracketrightρ(h). DeﬁneΠ′ i= Πi−Π1,2. Π′ iis the orthogonal projection onto the subspace V′ ideﬁned as the orthogonal complement of Vρ H1∩Vρ H2 inVρ Hi.V′ 1andV′ 2have trivial intersection. Deﬁne ri= rank(Π i)andr′ i= rank(Π′ i). Deﬁne ˆh= max{|H1|r1,|H2|r2},˜h=|(|H1|r1−|H2|r2)|andδ=/bardblΠ′ 1Π′ 2/bardbl. Recallthat δ= ovlap(V′ 1,V′ 2). Consider the following three cases: 1. When√ dρ log|G|= Ω((r1+r2)3/2). Loosely speaking, r1,r2areboth small. In this case, deﬁne r(G;H1,H2;ρ) = max/braceleftBiggˆh 2/parenleftBigg Ω/parenleftBigg/radicalbig r′ 1 r1+/radicalbig r′ 2 r2/parenrightBigg −2δ1/2(1−δ2)−1/4/parenrightBigg ,˜h/bracerightBigg . 2. When√ dρ log|G|= Ω(r1)andr2 r1= Ω(log2|G|). Loosely speaking, r1is small and r2is relatively large with respect to r1. In this case, deﬁne r(G;H1,H2;ρ) = max/braceleftBiggˆh 2·Ω/parenleftbigg1√r1/parenrightbigg ,˜h/bracerightBigg . 14 3. Otherwise, deﬁne r(G;H1,H2;ρ) =˜h. Deﬁnition 2( r(G;H1,H2),r(G))LetH1,H2≤G. Deﬁner(G;H1,H2) =/summationtext ρdρ |G|·r(G;H1,H2;ρ)and r(G) = min H1,H2r(G;H1,H2). From the above deﬁnition, it iseasy tosee that r(G;H1,H2)≥w(G;H1,H2). Deﬁnition 3( PG H,B)LetBbe a set of orthonormal bases for the irreducible unitary rep resentations of G. SupposeH≤G.PG,B Hdenotestheprobability distribution ontherepresentatio n namesandcolumnindices (ρ,j)got by strong Fourier sampling the state |H/an}bracketri}htaccording to B. Thesigniﬁcance of r(G;H1,H2)arises from the following theorem. Theorem 1 Withprobabilityatleast 1−exp(−Ω(log2|G|))overthechoiceofrandomrepresentationbases Bfor Fourier sampling, /bardblPG,B H1−PG,B H2/bardbltr≥r(G;H1,H2). Using this theorem, wecan apply a‘minimum-ﬁnding-like’ al gorithm toidentify thehidden subgroup. Corollary 1 Lets(G)denote the number of distinct subgroups of G. With probability at least 2/3over the choice of random bases for representations of G, Fourier sampling O/parenleftBig logs(G) r2(G)/parenrightBig times in a random basis gives enough classical information toidentify ahidd en subgroup in G. In particular, O/parenleftbigg/parenleftBig log|G| r(G)/parenrightBig2/parenrightbigg random strong Fourier samplings sufﬁce. Proof: From Theorem 1, we get that for all pairs of subgroups H1,H2≤G, with probability at least 1−exp(−Ω(log2|G|))over the choice of random bases Bfor representations of G,/bardblPG,B H1−PG,B H2/bardbltr≥ r(G). Call a set of representation bases Bgoodif/bardblPG,B H1−PG,B H2/bardbltr≥r(G)for all pairs of subgroups H1,H2≤G. Bythe union bound on probabilities, arandom choice of repr esentation bases gives a good B with probability at least 1−s(G)exp(−Ω(log2|G|)) = 1−exp(−Ω(log2|G|)). Suppose we have such a goodB. Under the promise that the hidden subgroup is either H1orH2,Brecognises which one it is with conﬁdence atleast 1/2+r(G)/4usingBayes’s rule. UsingFact5,theconﬁdence canbebooste d toatleast 1−1 4s(G)by Fourier sampling O/parenleftBig logs(G) r2(G)/parenrightBig times with B. We can now run a classical ‘minimum-ﬁnding- like’ algorithm on the measured samples, comparing two subg roupsH1,H2≤Gat a time, to discover the actual hidden subgroup HinGwith conﬁdence at least 1−s(G) 4s(G)= 3/4. The overall conﬁdence bound becomes (1−exp(−Ω(log2|G|)))·3 4≥2 3. The second bound follows from the fact that s(G)≤2log2|G|, since anygroup of size ahas at most logagenerators. TherestofthesectionisdevotedtoprovingTheorem1. Weﬁrs tprovesomenecessarytechnicallemmas. Lemma2 LetW={a1,...,ap}∪{b1,...,bq}∪{c1,...,cr}be arandom orthonormal set of p+q+r vectors in Cd. Letai jdenote thejth coordinate of vector ai; similar notations will be used for the vectors bi,citoo. Deﬁne two d-dimensional probability vectors S,Tasfollows: Sj=1 p+r/parenleftBiggp/summationdisplay i=1|ai j|2+r/summationdisplay i=1|ci j|2/parenrightBigg ,Tj=1 q+r/parenleftBiggq/summationdisplay i=1|bi j|2+r/summationdisplay i=1|ci j|2/parenrightBigg . 15 Thenthereexists δ=θ((p+q+r)−3/2)suchthat

Question 688 multiple-choice

Calabi-Yau threefolds are central in algebraic and symplectic geometry, especially in the study of mirror symmetry and enumerative invariants. Modern methods employ tropical geometry and toric degenerations to construct and analyze Lagrangian submanifolds within these spaces. Which statement best describes the relationship between the multiplicity of a tropical curve and the properties of its corresponding Lagrangian rational homology sphere in the mirror quintic Calabi-Yau threefold? 1) The multiplicity determines the genus of the associated Lagrangian submanifold. 2) The multiplicity fixes the dimension of the homology of the Lagrangian. 3) The multiplicity equals the Joyce’s weight assigned to the Lagrangian. 4) The multiplicity specifies the number of Dehn twists associated with the Lagrangian. 5) The multiplicity defines the Hamiltonian isotopy class of the Lagrangian. 6) The multiplicity sets the intersection number between Lagrangians. 7) The multiplicity corresponds to the number of rational homology spheres in the threefold.

✓ Correct Answer:

The correct answer is 3) The multiplicity equals the Joyce’s weight assigned to the Lagrangian..

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Title: Tropically constructed Lagrangians in mirror quintic threefolds Year: 2019 Paper ID: 701c0d89aaecd22dcd0b2e4b016e2db60cac10ee Source: semantic-scholar URL: https://www.semanticscholar.org/paper/701c0d89aaecd22dcd0b2e4b016e2db60cac10ee Abstract: Abstract We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. The homology spheres are mirror dual to the holomorphic curves contributing to the Gromov-Witten (GW) invariants. In view of Joyce’s conjecture, these Lagrangians are expected to have special Lagrangian representatives and hence solve a special Lagrangian enumerative problem in Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2,875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and the Joyce’s weight of each of these Lagrangians equals the multiplicity of the corresponding tropical curve. As applications, we show that disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians and we check in an example that $>300$ mutually disjoint curves (and hence Lagrangians) arise. Dehn twists along these Lagrangians generate an abelian subgroup of the symplectic mapping class group with that rank.

Question 689 multiple-choice

Quantum algorithms are increasingly being explored for their ability to accelerate complex optimization tasks, especially those involving algebraic structures and error-correcting codes. Decoded quantum interferometry (DQI) is a recent approach that leverages quantum Fourier transforms and decoding methods to solve such problems efficiently. Which key feature enables decoded quantum interferometry (DQI) to achieve superpolynomial speed-ups in approximating optimal polynomial fits over finite fields? 1) Use of Grover’s search to identify optimal solutions directly 2) Exploitation of algebraic structure to map optimization problems to efficiently solvable decoding tasks 3) Reliance on quantum random walks to sample solution spaces rapidly 4) Employment of variational quantum eigensolvers for polynomial fitting 5) Application of amplitude amplification for clause satisfaction 6) Transformation of optimization problems into Hamiltonian minimization routines 7) Use of quantum error correction to maximize solution accuracy

✓ Correct Answer:

The correct answer is 2) Exploitation of algebraic structure to map optimization problems to efficiently solvable decoding tasks.

📚 Reference Text:

Title: Optimization by decoded quantum interferometry Year: 2024 Paper ID: f75654bd952322e4b3f715c41452531e0af3e9d0 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/f75654bd952322e4b3f715c41452531e0af3e9d0 Abstract: Achieving superpolynomial speed-ups for optimization has long been a central goal for quantum algorithms1. Here we introduce decoded quantum interferometry (DQI), a quantum algorithm that uses the quantum Fourier transform to reduce optimization problems to decoding problems. When approximating optimal polynomial fits over finite fields, DQI achieves a superpolynomial speed-up over known classical algorithms. The speed-up arises because the algebraic structure of the problem is reflected in the decoding problem, which can be solved efficiently. We then investigate whether this approach can achieve a speed-up for optimization problems that lack an algebraic structure but have sparse clauses. These problems reduce to decoding low-density parity-check codes, for which powerful decoders are known2,3. To test this, we construct a max-XORSAT instance for which DQI finds an approximate optimum substantially faster than general-purpose classical heuristics, such as simulated annealing. Although a tailored classical solver can outperform DQI on this instance, our results establish that combining quantum Fourier transforms with powerful decoding primitives provides a promising new path towards quantum speed-ups for hard optimization problems. Decoded quantum interferometry is a quantum algorithm that uses the quantum Fourier transform to reduce optimization problems to decoding problems.

Question 690 multiple-choice

Quantum algorithms for group-theoretic problems often rely on structural properties of groups, such as nilpotency and the ability to reduce complex problems to computations over simpler factor groups. The hidden subgroup problem (HSP) is a central challenge in this area, with solutions impacting fields from cryptography to coding theory. In the context of nilpotent groups and quantum algorithms, which property of the factor groups Gi−1/Gi in the lower central series is crucial for enabling efficient purification and exact hidden subgroup algorithms? 1) They are always non-abelian simple groups. 2) They can be embedded into permutation groups of arbitrary degree. 3) They are elementary abelian and efficiently isomorphic to direct products of Zp. 4) They have order divisible by every prime dividing the order of the original group. 5) They possess unique minimal normal subgroups. 6) They exhibit maximal nilpotency class in all cases. 7) They correspond to irreducible representations over complex numbers.

✓ Correct Answer:

The correct answer is 3) They are elementary abelian and efficiently isomorphic to direct products of Zp..

📚 Reference Text:

of the group oracle, calls to the oracle computing the puriﬁcation of the subgroup state; and the inverses of these. (The elements of G are assumed to beuniquely encoded by strings of length /lscript.) Proof We compute the lower central series G=G 0>G1> ... > Gc={ 1} using the method presented in Subsection 2.2.A s G/G/primeis elementary abelian, so are the factors Gi−1/Gi(i=1,..., c). This is because the factor groups Gi−1/Giare homomorphic images of tensor powers (as Z-modules) of the G/G/prime, see Theorem 5.2.5 123 14 Page 14 of 23 M. Imran, G. Ivanyos of [27]. Also, isomorphisms of Gi−1/Giwith Znipcan be efﬁciently computed using the method of [ 16]. Iteration of Lemma 2gives a procedure to convert/producttextc i=2S(ni)copies of a puriﬁcation of /Xi1G,Hto a copy of a puriﬁcation of /Xi1G/G/prime,HG/prime/G/prime. The composition of instances of the original subgroup state creating procedure (the calls to the oracle) with the conversion gives a procedure for creating a puriﬁcation of /Xi1G/G/prime,HG/prime/G/prime.W e can use this as the oracle input for the exact hidden subgroup algorithm of [ 16]i n Zn1p. Fori=1,..., c,w eh a v e S(p,ni)≤S(p,n)and T(p,ni)≤T(p,n)because Znip can be embedded in Zn pas a subgroup. /intersectionsq/unionsq As an example, consider the group Gof unitriangular 4 by 4 matrices modulo 3. We can visualize Gas⎛ ⎜⎜⎝1∗∗∗ 1∗∗ 1∗ 1⎞ ⎟⎟⎠Here and below in the example, a matrix with asterisks denotes the set of matrices obtained by substituting the asterisks by integers modulo 3, independently at each place. Assume that His the subgroup⎛ ⎜⎜⎝1∗ 1 1 1⎞ ⎟⎟⎠. Gis a semi-elementary matrix group. The lower central series of GisG0=G, G1=⎛ ⎜⎜⎝1∗∗ 1∗ 1 1⎞ ⎟⎟⎠,G2=⎛ ⎜⎜⎝1∗ 1 1 1⎞ ⎟⎟⎠, and G3={I}, thus the nilpotency class is 3. In the ﬁrst round, we convert copies subgroup states for Hto those for HG 2/G2=⎛ ⎜⎜⎝1∗∗ 1 1 1⎞ ⎟⎟⎠/slashbig⎛ ⎜⎜⎝1∗ 1 1 1⎞ ⎟⎟⎠and next to HG 1/G1=⎛ ⎜⎜⎝1∗∗∗ 1∗ 1 1⎞ ⎟⎟⎠/slashbig⎛ ⎜⎜⎝1∗∗ 1∗ 1 1⎞ ⎟⎟⎠in the elementary abelian group G/G1. There we can solve the HSMC for Gby using the exact abelian hidden subgroup algorithm from [ 16]i nG/G1. In the non-exact setting, essentially the same proof gives the following. Proposition 3 Let G be a semi-elementary black-box group with unique encod- ing of order pn. Assume that there exists a quantum (or a randomized) algorithm that ﬁnds zero sum subsequences of sequences of S (p,n)elements of Zpnin time T (p,n)≥S(p,n)with high probability. Then the problem HSMC can be solved by a quantum algorithm that uses poly(T(p,n)O(c)/lscript)elementary operations, poly(T(p,n)O(c)log|G|)applications of the group oracle and calls to the oracle computing the puriﬁcation of the subgroup state. 5 Zero sum subsequences in Zn p In this section, we assume that our input is a sequence of vectors from Zn p.W ea l s o assume that pis an odd prime as for p=2 a zero sum subsequence can be obtained from n+1 vectors in the form of a zero linear combination. As subsequences can be 123 Zero sum subsequences and hidden subgroups

Question 691 multiple-choice

The Quantum Fourier Transform (QFT) is a pivotal operation in quantum computing, central to algorithms like Shor’s for integer factorization. Understanding its entangling properties is crucial for assessing both quantum and classical computational approaches. Which of the following statements correctly characterizes the core Quantum Fourier Transform (QFT) operation in terms of entanglement and classical simulation efficiency? 1) It produces only a small, system-size-independent amount of entanglement due to exponentially decaying Schmidt coefficients, enabling efficient classical simulation via matrix product states with low bond dimension. 2) It generates maximal entanglement proportional to the number of qubits, making classical simulation exponentially difficult. 3) Its entanglement properties are dominated by the bit reversal operation, which is the main bottleneck for classical simulation. 4) The entangling power of QFT is analogous to Hamiltonians with algebraically decaying interactions, requiring high bond dimension for classical simulation. 5) QFT can only be simulated efficiently classically for small qubit systems because entanglement grows rapidly with system size. 6) The QFT’s Schmidt coefficients decay linearly, causing moderate entanglement and limited classical simulation. 7) Classical simulations using matrix product states become less efficient for QFT as the system size increases due to growing entanglement.

✓ Correct Answer:

The correct answer is 1) It produces only a small, system-size-independent amount of entanglement due to exponentially decaying Schmidt coefficients, enabling efficient classical simulation via matrix product states with low bond dimension..

📚 Reference Text:

Title: Quantum Fourier Transform Has Small Entanglement Year: 2022 Paper ID: 5b93732a3655b0b91123def022a94f6b94bda787 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/5b93732a3655b0b91123def022a94f6b94bda787 Abstract: The Quantum Fourier Transform (QFT) is a key component of many important quantum algorithms, most famously as being the essential ingredient in Shor's algorithm for factoring products of primes. Given its remarkable capability, one would think it can introduce large entanglement to qubit systems and would be difficult to simulate classically. While early results showed QFT indeed has maximal operator entanglement, we show that this is entirely due to the bit reversal in the QFT. The core part of the QFT has Schmidt coefficients decaying exponentially quickly, and thus it can only generate a constant amount of entanglement regardless of the number of qubits. In addition, we show the entangling power of the QFT is the same as the time evolution of a Hamiltonian with exponentially decaying interactions, and thus a variant of the area law for dynamics can be used to understand the low entanglement intuitively. Using the low entanglement property of the QFT, we show that classical simulations of the QFT on a matrix product state with low bond dimension only take time linear in the number of qubits, providing a potential speedup over the classical fast Fourier transform (FFT) on many classes of functions. We demonstrate this speedup in test calculations on some simple functions. For data vectors of length $10^6$ to $10^8$, the speedup can be a few orders of magnitude.

Question 692 multiple-choice

Quantum algorithms such as the Quantum Fourier Transform (QFT) are increasingly being explored to address computational challenges in modern power systems, especially for analyzing the dynamics of microgrids under transient conditions. Understanding the relative computational complexity and applicability of QFT compared to classical methods is crucial in this domain. What is the primary advantage of applying the Quantum Fourier Transform (QFT) via quantum circuits to analyze transient responses in islanded microgrids with droop control? 1) QFT achieves exponential speedup in computing frequency spectra, reducing computational complexity from $\mathcal{O}(N \log_2 N)$ to $\mathcal{O}(\log_2^2 N)$. 2) QFT ensures absolute immunity to noise and decoherence in quantum hardware. 3) QFT eliminates the need for droop control settings in microgrid operation. 4) QFT enables direct physical measurement of microgrid frequency without sensors. 5) QFT guarantees perfect accuracy regardless of input data quality. 6) QFT provides instantaneous real-time feedback to all microgrid controllers. 7) QFT allows seamless integration with classical FFT algorithms for hybrid processing.

✓ Correct Answer:

The correct answer is 1) QFT achieves exponential speedup in computing frequency spectra, reducing computational complexity from $\mathcal{O}(N \log_2 N)$ to $\mathcal{O}(\log_2^2 N)$..

📚 Reference Text:

Title: Transient Stability Assessment of Islanded Microgrids Through Quantum Computing Year: 2023 Paper ID: da24b16d86b6eafb1eb93f59f7d57ce84969a0d4 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/da24b16d86b6eafb1eb93f59f7d57ce84969a0d4 Abstract: Quantum Fourier Transform (QFT) is introduced in this paper, which is a quantum implementation of discrete Fourier transform (DFT) that is a cornerstone of power system dynamics analysis. The essential idea of QFT and quantum circuit implementation have been developed to compute the frequency spectrum of the transient responses of microgrids under islanded operation with droop control. QFT reduces the complexity of DFT from $\mathcal{O}(N \log_{2} N$) for classical computing to $\mathcal{O}(\log_{2}^{2} N$) for quantum computing. Thus, DFT for transient analysis can be performed exponentially faster than the fast Fourier transform in classical computers. Numerical examples of a typical networked microgrid system have tested and verified the effectiveness of QFT. It casts light on solving challenges through quantum algorithms that are hard for classical computers.

Question 693 multiple-choice

Quantum field theory in de Sitter space employs advanced mathematical frameworks to address the behavior of massless and higher-spin fields in curved backgrounds. The ambient space formalism and group representation theory are central to understanding field propagation and symmetry constraints in cosmological settings. Which statement best characterizes the propagation of massless fields with spin s ≥ 3 in the ambient space formalism for de Sitter quantum field theory? 1) They propagate freely and admit well-defined gauge invariant equations. 2) Their propagation is allowed only for conformally coupled scalar backgrounds. 3) They cannot propagate due to incompatibility with gauge and causality constraints. 4) Their dynamics are fully described by analytic two-point functions. 5) They propagate only in flat spacetime, not in de Sitter space. 6) Their field operators require fermionic vacuum states for consistency. 7) They are equivalent to massless spin-2 fields under de Sitter group representations.

✓ Correct Answer:

The correct answer is 3) They cannot propagate due to incompatibility with gauge and causality constraints..

📚 Reference Text:

Title: Quantum Field Theory in de Sitter Universe: Ambient Space Formalism Year: 2014 Paper ID: 832ac820135838831a75fb801468a02092a41fd0 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/832ac820135838831a75fb801468a02092a41fd0 Abstract: Quantum field theory in the 4-dimensional de Sitter space-time is constructed in the ambient space formalism in a rigorous mathematical framework. This work is based on the group representation theory and the analyticity of the complexified pseudo-Riemannian manifolds. The unitary irreducible representations of de Sitter group and their corresponding Hilbert spaces are reformulated in the ambient space formalism. Defining the creation and annihilation operators, quantum field operators and their corresponding analytic two-point functions for various spin fields have been constructed. The various spin massless fields can be constructed in terms of the massless conformally coupled scalar field in this formalism. Then the quantum massless minimally coupled scalar field operator, for the first time, is also constructed on Bunch-Davies vacuum state which preserve the analyticity. We show that the massless fields with s ≥ 3 cannot propagate in de Sitter ambient space formalism. The massless gauge invariant field equations for s = 1, 3 ,2 are studied. The gauge spin- 3 fields satisfy the Grassmannian algebra, and hence provoke one to couple them with the gauge spin-2 field and the super-algebra is naturally appeared.

Question 694 multiple-choice

The Hidden Subgroup Problem (HSP) is a foundational challenge in quantum computing and algebra, particularly for vector spaces over finite fields and congruence distributive algebras. Understanding the interplay between group structure and computational algorithms is key to determining tractability. In the context of products of simple, congruence distributive algebras, which property ensures that the kernel of a function can always be described as the kernel of a projection onto a subset of coordinates, thereby allowing HSP to be solved classically in linear time? 1) Existence of nontrivial center in each algebra 2) Every congruence is the kernel of a projection 3) Closure under non-associative operations 4) Maximal subgroups have order equal to the group 5) Each algebra admits only trivial congruences 6) Absence of any distributive lattice structure 7) The group is non-abelian

✓ Correct Answer:

The correct answer is 2) Every congruence is the kernel of a projection.

📚 Reference Text:

and only if Sg (X0)/nequalHθ. The setX0fails to generateHθif and only if X0is contained in somemaximalsubgroup MofHθ.SinceHθisabeliananda subgroup ofZn 2,these are all ofsize |Hθ|/2.Thus, P/parenleftbigCg(X)/nequalθ/parenrightbig=P/parenleftbigSg(X0)/nequalHθ/parenrightbig =P/parenleftbig∃maximalM≤HθwithX0⊆M/parenrightbig ≤/summationdisplay.1 maximalM≤HθP/parenleftbigX0⊆M/parenrightbig =/summationdisplay.1 maximalM≤Hθ/productdisplay.1 /afii10069.ital∈X0P/parenleftbig/afii10069.ital∈M/parenrightbig =/summationdisplay.1 maximalM≤Hθ/productdisplay.1 /afii10069.ital∈X01 2=L 2|X0|, whereListhenumberofmaximalsubgroupsof Hθ.Wenow endeavor to calculate L. SinceZn 2is an elementary abelian 2-group, Hθis as well. ItfollowsthatHθcanberegardedasa Z2-vectorspace.Max- imalsubgroupscorrespondtomaximalsubspaces,andeach maximal subspace can be uniquely characterized as the or- thogonalcomplementofa1-dimensionalsubspace.Thenum- berof1-dimensionalsubspacesis |Hθ|−1,andsoisatmost 2n−1.Combiningeverything, we have P/parenleftbigCg(X)/nequalθ/parenrightbig≤L 2|X0|≤2n−1 2|X0|. The Hidden SubgroupProblem forUniversalAlgebras LICS’20,July8–11,2020,Saarbrücken,Germany Itfollowsthat Xgeneratesθwithprobability1 −1/τ,where |X| ∈Θ(n+lg(τ)),asclaimed. /square Theorem 4.5 (Simon [19]).LetA=Zn 2. The quantum cir- cuitsφ(n)of Deﬁnition 3.2solvesHKP(A)with probability 1−1/τusingΘ(n+lg(τ))iterations. Proof.Instead of calculating the pre-measurement output (callit|ψ/a\}⌊ra⌋ketri}ht),itwillbemoreconvenientforustocalculatethe density matrix ofthe pre-measurement output, ρ=|ψ/a\}⌊ra⌋ketri}ht /a\}⌊ra⌋ketle{tψ|, and takethe partial trace. Thecircuitoperatesontworegisters,initiallysetto |ψ0/a\}⌊ra⌋ketri}ht= |0n/a\}⌊ra⌋ketri}ht ⊗|0m/a\}⌊ra⌋ketri}ht.Thisstatehasdensity matrix ρ0=|ψ0/a\}⌊ra⌋ketri}ht/a\}⌊ra⌋ketle{tψ0|.Af- ter the ﬁrst set ofHadamardgates, thedensity matrix is ρ1≔(H⊗n⊗I)ρ0(H⊗n⊗I) =/parenleftbigH⊗n|0n/a\}⌊ra⌋ketri}ht ⊗ |0m/a\}⌊ra⌋ketri}ht/parenrightbig/parenleftbig/a\}⌊ra⌋ketle{t0n|H⊗n⊗ /a\}⌊ra⌋ketle{t0m|/parenrightbig =/parenleftBig1 2n/2/summationdisplay.1 x∈{0,1}n|x/a\}⌊ra⌋ketri}ht ⊗ |0m/a\}⌊ra⌋ketri}ht/parenrightBig/parenleftBig1 2n/2/summationdisplay.1 /y.alt∈{0,1}n/a\}⌊ra⌋ketle{t/y.alt| ⊗ /a\}⌊ra⌋ketle{t0m|/parenrightBig =1 2n/summationdisplay.1 x,/y.alt∈{0,1}n|x/a\}⌊ra⌋ketri}ht/a\}⌊ra⌋ketle{t/y.alt| ⊗ |0m/a\}⌊ra⌋ketri}ht/a\}⌊ra⌋ketle{t0m|. We now describe the /hatwideφgate. A homomorphism φ:A→ Bis given as input as an oracle from which we may make queries. As a function, φcan be regarded as mapping bit strings from {0,1}nto{0,1}m.Deﬁne the function /hatwideφby /hatwideφ:{0,1}n+m→ {0,1}n+m, (a,b) /ma√sto→ (a,φ(a)+b). Observe that /hatwideφis invertible and that the original value of φ(a)can be recovered — /hatwideφ(a,0)=(a,φ(a)). The function /hatwideφcan be regarded as a permutation of basis vectors of the spaceB⊗(n+m),andthuscanbeextendedtoaunitarytrans- formation /hatwideφonthe space. Continuingwiththeevaluationofthecircuit,afterapply- ing the/hatwideφgate thedensity matrix is ρ2≔/hatwideφρ1/hatwideφ†=1 2n/summationdisplay.1 x,/y.alt∈{0,1}n/parenleftbig/hatwideφ|x/a\}⌊ra⌋ketri}ht ⊗ |0m/a\}⌊ra⌋ketri}ht/parenrightbig /parenleftbig/hatwideφ|/y.alt/a\}⌊ra⌋ketri}ht ⊗ |0m/a\}⌊ra⌋ketri}ht/parenrightbig† =1 2n/summationdisplay.1 x,/y.alt∈{0,1}n/parenleftbig|x/a\}⌊ra⌋ketri}ht ⊗ |φ(x)/a\}⌊ra⌋ketri}ht/parenrightbig /parenleftbig|/y.alt/a\}⌊ra⌋ketri}ht ⊗ |φ(/y.alt)/a\}⌊ra⌋ketri}ht/parenrightbig† =1 2n/summationdisplay.1 x,/y.alt∈{0,1}n|x/a\}⌊ra⌋ketri}ht/a\}⌊ra⌋ketle{t/y.alt| ⊗ |φ(x)/a\}⌊ra⌋ketri}ht/a\}⌊ra⌋ketle{tφ(/y.alt)|. Applying the last set of Hadamard gates yields the ﬁnal pre-measurement density matrix, ρ≔1 2n/summationdisplay.1 x,/y.alt∈{0,1}nH⊗n|x/a\}⌊ra⌋ketri}ht/a\}⌊ra⌋ketle{t/y.alt|H⊗n⊗ |φ(x)/a\}⌊ra⌋ketri}ht/a\}⌊ra⌋ketle{tφ(/y.alt)| =1 22n/summationdisplay.1 a,b∈{0,1}n, x,/y.alt∈{0,1}n(−1)a·x+b·/y.alt|a/a\}⌊ra⌋ketri}ht /a\}⌊ra⌋ketle{tb| ⊗ |φ(x)/a\}⌊ra⌋ketri}ht/a\}⌊ra⌋ketle{tφ(/y.alt)|.Thedensitymatrixfortheﬁrstregisterisgivenbytaking thepartialtraceof ρoverthesecondregister.Doingthis,we have Tr2(ρ)=1 22n/summationdisplay.1 a,b∈{0,1}n, x,/y.alt∈{0,1}n(−1)a·x+b·/y.alt|a/a\}⌊ra⌋ketri}ht /a\}⌊ra⌋ketle{tb|Tr/parenleftbig|φ(x)/a\}⌊ra⌋ketri}ht/a\}⌊ra⌋ketle{tφ(/y.alt)|/parenrightbig. It isn’t hard to see that Tr (|φ(x)/a\}⌊ra⌋ketri}ht/a\}⌊ra⌋ketle{tφ(/y.alt)|)=1 ifφ(x)=φ(/y.alt) (i.e. if(x,/y.alt) ∈ker(φ))and is0 otherwise. Thus, Tr2(ρ)=1 22n/summationdisplay.1 a,b∈{0,1}n/summationdisplay.1 (x,/y.alt)∈ker(φ)(−1)a·x+b·/y.alt|a/a\}⌊ra⌋ketri}ht /a\}⌊ra⌋ketle{tb|. The congruence ker (φ)is a subgroup of (Zn 2)2and we havea·x+b·/y.alt=(a,b) · (x,/y.alt). Lemma 4.3thus applies tothe inner summation. Applyingit yields Tr2(ρ)=|ker(φ)| 22n/summationdisplay.1 (a,b)∈ker(φ)⊥|a/a\}⌊ra⌋ketri}ht/a\}⌊ra⌋ketle{tb|, where ker(φ)⊥=/braceleftbig(a,b) | (a,b)·(x,/y.alt)=0 forall(x,/y.alt) ∈ker(φ)/bracerightbig. The congruence ker (φ)is aZ2-subspace ofA2, and this is its orthogonalcomplement.Itfollowsthat ker (φ)⊥is also a congruenceofAand|ker(φ)||ker(φ)⊥|=|A|2=22n.Thus Tr2(ρ)=1 |ker(φ)⊥|/summationdisplay.1 (a,b)∈ker(φ)⊥|a/a\}⌊ra⌋ketri}ht /a\}⌊ra⌋ketle{tb|. This corresponds to a uniform distribution over elements in ker(φ)⊥. By Lemma 4.4, iterating this procedure Θ(n+ lg(τ))times will produce a generating set for ker (φ)⊥with probability 1 −1/τ. As ker(φ)=/parenleftbigker(φ)⊥/parenrightbig⊥, we have succeeded in calculat- ingker(φ)with probability 1 −1/τusingΘ(n+lg(τ))itera- tionsof acircuit ofsize Θ(n). /square Atthispoint,wehaveestablishedaquantumpolynomial- timealgorithmfortheHKPforpowersofalgebrasabovethe cloneAP0inPost’sLattice.Asidefromthegroup Z2,thisin- cludesonly3otherstructures(seeFigure 3).Thenextlemma andtheoremgreatlyexpandthislistbydeducingaclassical algorithm for powers of certain congruence distributive al- gebraswhichincludes theinﬁnite “wings” of Post’sLattice. Lemma 4.6. LetB1,...,Bnbe similar algebras which are simpleandletA=/producttext.1n i=1Bibecongruencedistributive. Every congruence ofAis the kernel of a projection homomorphism projI:A→/producttext.1 i∈IBiforsomeI⊆ [n]. Proof.Foreachi∈ [n]deﬁneηi=ker(proji).Letθbeacon- gruenceofA.EachalgebraBiissimple,soeachcongruence ηiis maximalin Con (A).Itfollowsthat θ∨ηi=/braceleftBigg ηiifθ≤ηi, 1otherwise. LICS’20,July8–11,2020,Saarbrücken,Germany Ma/t_thew Moore and TaylorWalenczyk Using congruencedistributivity, we now have θ=θ∨0=θ∨/logicalanddisplay.1 i∈[n]ηi=/logicalanddisplay.1 i∈[n]/parenleftbigθ∨ηi/parenrightbig=/logicalanddisplay.1 i∈Iηi=ker(projI), whereI={i|θ≤ηi}. /square Theorem4.7. LetB1,...,Bnbesimilar algebraswhich are simpleandletA=/producttext.1n i=1Bibecongruencedistributive. There isaclassical O(n)algorithm solving HKP(A). Proof.Letφ:A→Dbe a homomorphism. Lemma 4.6im- plies that ker (φ)=ker(projI)for someI⊆ [n]. Thus, spec- ifyingIuniquely determines ker (φ). Without loss of gener- ality, we may assume that |Bi| ≥2

Question 695 multiple-choice

In the study of quantum machine learning models that respect group symmetries, representation theory provides crucial insights into how data and operations are structured in Hilbert spaces. Understanding the implications of block-diagonalization and the commutant of a unitary representation is essential for designing effective symmetric models. Which of the following best explains why a quantum machine learning model with built-in group symmetry cannot access information in the off-diagonal blocks of input quantum states? 1) The neural network architecture lacks sufficient depth to analyze off-diagonal elements. 2) Measurement operators in quantum mechanics are insensitive to coherence between irreducible representations. 3) Quantum states with off-diagonal elements violate the unitary evolution required by the model. 4) The presence of entanglement prevents access to mixed symmetry information. 5) Off-diagonal blocks are eliminated during model training by regularization techniques. 6) The underlying Hilbert space does not support off-diagonal elements between multiplicity spaces. 7) Both the model and measurement operators inherit a block-diagonal structure dictated by representation theory, restricting access to information only within symmetry-respecting irrep blocks.

✓ Correct Answer:

The correct answer is 7) Both the model and measurement operators inherit a block-diagonal structure dictated by representation theory, restricting access to information only within symmetry-respecting irrep blocks..

📚 Reference Text:

means that Uappears two times, a statement which which is reﬂected in the matrix forms of the following families of operators. Theorem 7 (Commutant structure [52]) .LetUbe a uni- tary representation of a compact group Gon the Hilbert spaceHand its decomposition into irreps be U=KM k=11mk Uk; wheremkis known as the multiplicity of the irrep Uk. Then the elements of its commutant are of the following form C(U) =KM k=1Bmk 1dim(Uk); (30) whereBmkdenotes bounded operators in a mk-dimensional Hilbert space. To connect this with our earlier block-diagonalization statement while discussing complete reducibility (Deﬁni- tion 14), we observe that we are saying the matrix repre- sentatives for Uand the elements of the commutant can be simultaneously block diagonalized as 19The more precise version of this is that the commutant of a unitary representation Udecomposes into minimal central projections, and these are in one-to-one correspondence with irreducible represen-tations ofU. This is reﬂected in Theorem 7. 20It would be irresponsible to state this theorem without saying Schur’s lemma is crucial to the proof: it guarantees that inequiva- lent representations do not “mix”. 29 U=0 BBBBBBBBBBBBBBBBBBBBBBBBBBBB@U1 0 ... 0U10 ... Uk 0 ... 0Uk |{z} mkdim(Uk) ... 0UK 0 ... 0UK1 CCCCCCCCCCCCCCCCCCCCCCCCCCCCA: (31) C(U) =0 BBBBBBBBBBBBBBBBBBBBBBBB@Bm1 1dim(U1) 0 ... Bmk 1dim(Uk) |{z} mkdim(Uk) ... 0BmK 1dim(UK)1 CCCCCCCCCCCCCCCCCCCCCCCCA: (32) Morally, this means that the elements of the commutant can really only act upon irreps among multiplicities. The proof of this theorem rests fundamentally upon Schur’s lemma (Theorem6), whichensuresthatequivariantmapsbetween inequivalent irreps are zero maps, and equivariant maps between equivalent reps are scalar multiples of 1. Let us now show how the block diagonal structure of the problem can signiﬁcantly help us understand the way in which GQML models with equivariant quantum neural networks and measurements process information. In particular, let us consider the schematic diagram of Fig. 10. We analyze a QML task where the dataset is S=fi;yigN i=1, the GQML model is of the form h(i) = Tr W()iWy()Mi as in Eq. (18), and the symmetry group that leaves the labels invariant is G. We know that any representation Ugwill admit some block diago- nal structure as in Eqs. (31), and that both the equivari- ant quantum neural network W()and the measurementoperatorMwill also have a block-diagonal structure as in Eq. (32) (as they both belong to the commutant, see Eqs. (3) and (4)). These block structures are schemati- cally pictured in Fig. 10. Then, we note that, as previously mentioned, the states need not be preserved under the action of a representa- tionG. In this case, UgiUy g6=i, and hence, icannot be simultaneously diagonalized as in Eqs. (31) and (32). That is,idoes not necessarily admit a block diagonal structure in the same bases where Ug,W(), andMare block diagonal. As shown in as Fig. 10, the previous im- plies that there is information in the quantum state ithat the modelh(i)cannot access (i.e., the oﬀ-diagonal infor- mation outside of the sub-blocks of iassociated to the diﬀerent irreps). Hence, the QML model can only process, extract,andlearnfromtheinformationassociatedtotheir- rep sub-blocks.

Question 696 multiple-choice

In computational complexity theory, the Graph Isomorphism (GI) problem concerns determining whether two graphs are structurally identical, while SPP is a specialized counting complexity class defined using GapP functions. The classification of GI within various complexity classes has significant implications for algorithm design and theoretical computer science. Which of the following complexity classes is the smallest gap-definable class that contains the Graph Isomorphism problem and is low for several other counting classes such as ⊕P and PP? 1) #P 2) NP 3) coNP 4) Mod kP 5) UP 6) LWPP 7) SPP

✓ Correct Answer:

The correct answer is 7) SPP.

📚 Reference Text:

Graph Isomorphism is in SPP V. Arvind and Piyush P Kurur Institute of Mathematical Sciences, C.I.T Campus Chennai 600113, India email: {arvind,ppk }@imsc.ernet.in Abstract We show that Graph Isomorphism is in the complexity class SPP, and hence it is in ⊕P (in fact, it is in Mod kP for each k≥2). We derive this result as a corollary of a more general result: we show that a generic problem FIND-GROUP has an FPSPPalgorithm. This general result has other consequences: for example, it follows that the hidden subgroup problem for permutation groups, studied in the context of quantum algorithms, has an FPSPPalgorithm. Also, some other algorithmic problems over permutation groups known to be at least as hard as Graph Isomorphism (e.g. coset intersection) are in SPP, and thus in Mod kP for each k≥2. 1 Introduction The Graph Isomorphism problem —of testing if two graphs are isomorphic— is a well-studied algorithmic problem in the class NP. Formally, the decision problem GI can be deﬁned as: GI = {/angbracketleftX1,X2/angbracketright |X1andX2are isomorphic graphs }. Graph Isomorphism has attracted a lot of research because there is no known polynomial-time algorithm for it and on the other hand there is strong evidence that it is not NP-complete. In [6] it was shown that Graph Nonisomorphism is in AM implying that GI is in NP ∩coAM. It follows that GI cannot be NP-complete unless the polynomial hierarchy collapses to Σp 2[6, 18]. Sch¨ oning, who introduced the notion of lowness in complexity theory, pointed out in [18] that GI is lowfor Σp 2. I.e. GI is powerless as oracle for Σp 2. Subsequently, it was shown in [13] that GI is also lowfor the class PP (the language class corresponding to #P). This result is proven using the machinery of GapP functions introduced in the seminal paper of [7]. In [7] the languages classes SPP and LWPP are introduced as generalizations of Valiant’s class UP. It is shown in [7] that UP ⊆SPP⊆LWPP, and LWPP is low for PP. 1 SPP is perhaps the most important and natural of the counting classes. It is known that SPP is in and low for the classes ⊕P, C =P, PP etc. SPP has a host of other nice properties as well (see [7] for details). For instance, SPP is characterized exactly as the class of languages low for GapP. In summary, SPP can be seen as the GapP analogue of UP. In [7] it is also shown that SPP is the smallest reasonable gap-deﬁnable class. Coming back to [13], the result that GI is low for PP is shown in that paper by proving that GI is in LWPP. It is also shown in [13] that GA (testing if a given graph has a nontrivial graph automorphism) is in SPP. It is known that GA is polynomial-time reducible to GI, but the converse is open. 1.1 Summary of new results In this paper, we show that GI is in the class SPP. This was left as an open

Question 697 multiple-choice

Q-Schur rings are algebraic structures that provide a framework for analyzing symmetries and group actions in combinatorial and algebraic settings. The concept of CI-groups addresses when Cayley graphs are uniquely determined by the underlying group structure. Which principle is essential for proving that explicitly constructed elements belong to a Q-Schur ring, particularly by ensuring their distinctness within the ring's structure? 1) Lagrange's theorem 2) Sylow's theorems 3) Orbit-stabilizer theorem 4) Maschke's theorem 5) Schur-Wielandt principle 6) Burnside's lemma 7) Frobenius reciprocity

✓ Correct Answer:

The correct answer is 5) Schur-Wielandt principle.

📚 Reference Text:

2l+e2l−1))+(−e2l,H(e 2l))=(e2l−1,V) is a set of A for 2≤l≤p−1. Therefore (e2l−1,H(e 2l−1))=(e2l−1,V)◦(((2l)−1)U4)∈Afor 2≤l≤p−1. J Algebr Comb (2007) 26: 343–355 355 So far we have proved that (ei,H(e i))∈Afori≥2. In particular, by the deﬁnition ofT7, we get q=(e1,H(e 1)+{ ±ε})∈A.W eh a v e q+q=2pn−1(2e1,H(e 1))/unionmultipn−1(2e1,H(e 1)+{ ± 2ε}). Therefore, arguing as usual, we get (e1,H(e 1))∈A. Now (ei,H(e i))+(ej,H(e j))=(ei+ej,V) is a set of A.S o (ei+ej,H(e i+ej))=(ei+ej,V)◦S∈Afor any i,jwithi/negationslash=j. This proves that Ais aQ-Schur ring with the required properties and so W×Vis not a CI-group. Once again using the ﬁrst paragraph of Sect. 5 .1i n[ 5] we have that any elementary Abelian p-group of rank n≥4p−2 is not a CI-group. Case p =3.We have (S+S)◦S=0T/prime/unionmulti2pn−1T∗/unionmulti2pn−2T2/unionmulti2(n−2)T3/unionmulti(2(n−2)/unionmulti2pn−2)T4 /unionmulti(2(n−2)+4pn−2)T5/unionmulti2(n−1)T6/unionmulti(2(n−1)+6pn−2)T7 where T/prime=/summationtext i∈X(0,εi),T∗=(e−en,H(e−en))andT2,...,T 7are deﬁned as in the case p≥5. Note that 0 ,2p4,2pn−2,2(n−2),(2(n−2)+2pn−2),(2(n−2)+4pn−2),2(n− 1),(2(n−1)+6pn−2)are all distinct. So, by Schur-Wielandt principle, the elements T/prime,T∗,T2,...,T 7lie inA. Now the computations are exactly the same as in the case p≥5. /square Acknowledgements The author thanks Dr. Joy Morris for drawing his attention to the isomorphism problem of ﬁnite Cayley digraphs and for her many helpful suggestions during the preparation of this paper. Furthermore the author is in debt to the GAP group [ 2] and mainly to Prof. L.Soicher’s package GRAPE. References 1. Babai, L. (1977). Isomorphism problem for a class of point-symmetric structures. Acta Math. Acad. Sci. Hung. ,29(3–4), 329–336. 2. The GAP Group (2005). GAP—groups, algorithms, and programming, version 4.4, http://www. gap-system.org. 3. Hirasaka, M., & Muzychuk, M. (2001). An elementary Abelian group of rank 4 is a CI-group. J. Comb. Theory Ser. A ,94(2), 339–362. 4. Lakatoš, P., & Suš ˇcans’ki ˘ı, V .¯I. (1976). The coincidence of central series in wreath products, I. Russian. Publ. Math. Debrecen ,23(1–2), 167–176. 5. Li, C. H. (2002). On Isomorphisms of ﬁnite Cayley graphs—a survey. Discrete Math. ,256(1–2), 301– 334. 6. Muzychuk, M. (2003). An elementary Abelian group of large rank is not a CI-group. Discrete Math. , 264(1–3), 167–185. 7. Nowitz, L. A. (1992). A non-Cayley-invariant Cayley graph of the elementary Abelian group of order 64.Discrete Math. ,110(1–3), 223–228. 8. Robinson, D. J. S. (1980). A course in the theory of groups . Berlin: Springer (pp. 304–305). 9. Wielandt, H. (1964). Finite permutation groups . Berlin: Academic.

Question 698 multiple-choice

Geometric Complexity Theory (GCT) is an approach that applies advanced concepts from algebraic geometry and representation theory to major problems in computational complexity, such as the P vs. NP question. This framework leverages deep mathematical properties, including positivity conditions and connections to classical results in number theory. Which of the following statements accurately describes a central theoretical link in GCT’s approach to resolving the P vs. NP problem in characteristic zero? 1) It relies on finding negative formulae for plethysm constants using classical group theory. 2) It demonstrates that all decision problems in complexity are undecidable by reduction to elliptic curves. 3) It shows that quantum group theory provides a direct algorithm for NP-complete problems. 4) It proves that lower bounds in complexity can be established by disproving the Riemann hypothesis for integers. 5) It connects the tractability of certain decision problems to the existence of positive formulae for structural constants, whose proofs depend on the Riemann hypothesis over finite fields. 6) It establishes that Kazhdan-Lusztig polynomials always have zero coefficients in representation theory. 7) It asserts that the P vs. NP question can be resolved without any input from algebraic geometry or number theory.

✓ Correct Answer:

The correct answer is 5) It connects the tractability of certain decision problems to the existence of positive formulae for structural constants, whose proofs depend on the Riemann hypothesis over finite fields..

📚 Reference Text:

Title: Geometric Complexity Theory VI: the flip via saturated and positive integer programming in representation theory and algebraic geometry Year: 2007 Paper ID: 34b3f300e694953cca88a64cd4a37c1c362b14d6 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/34b3f300e694953cca88a64cd4a37c1c362b14d6 Abstract: This article belongs to a series on geometric complexity theory (GCT), an approach to the P vs. NP and related problems through algebraic geometry and representation theory. The basic principle behind this approach is called the flip. In essence, it reduces the negative hypothesis in complexity theory (the lower bound problems), such as the P vs. NP problem in characteristic zero, to the positive hypothesis in complexity theory (the upper bound problems): specifically, to showing that the problems of deciding nonvanishing of the fundamental structural constants in representation theory and algebraic geometry, such as the well known plethysm constants--or rather certain relaxed forms of these decision probelms--belong to the complexity class P. In this article, we suggest a plan for implementing the flip, i.e., for showing that these relaxed decision problems belong to P. This is based on the reduction of the preceding complexity-theoretic positive hypotheses to mathematical positivity hypotheses: specifically, to showing that there exist positive formulae--i.e. formulae with nonnegative coefficients--for the structural constants under consideration and certain functions associated with them. These turn out be intimately related to the similar positivity properties of the Kazhdan-Lusztig polynomials and the multiplicative structural constants of the canonical (global crystal) bases in the theory of Drinfeld-Jimbo quantum groups. The known proofs of these positivity properties depend on the Riemann hypothesis over finite fields and the related results. Thus the reduction here, in conjunction with the flip, in essence, says that the validity of the P vs. NP conjecture in characteristic zero is intimately linked to the Riemann hypothesis over finite fields and related problems.

Question 699 multiple-choice

Quantum algorithms for hidden subgroup problems often utilize quantum Fourier transforms, controlled unitary operations, and measurements to extract hidden structure from functions. Efficient implementation and resource optimization are crucial for practical quantum computing applications in areas like cryptanalysis. In a quantum algorithm designed to find hidden periods or generators, which component allows for the extraction of hidden parameter t from a superposition by transforming encoded phase information back into computational basis states? 1) Application of controlled-NOT gates on target registers 2) Inverse quantum Fourier transform on control registers 3) Measurement of ancillary qubits in the Pauli-Z basis 4) Preparation of control registers using Hadamard gates 5) Classical post-processing via Gaussian elimination 6) Initialization of the target register in the |Ψt⟩ state 7) Use of swap gates to reorder qubit states

✓ Correct Answer:

The correct answer is 2) Inverse quantum Fourier transform on control registers.

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that the probability of error is suﬃciently small. By estimating tj/pmjforj= 1,2, . . ., l , we determine t. The algorithm starts by preparing lcontrol registers in the state |0/an}brack⌉tri}htand one target orauxiliary register in the state |Ψt/an}brack⌉tri}ht, applies the appropriate Fourier transforms to produce /parenleftBiggpm1−1/summationdisplay x1=0|x1/an}brack⌉tri}ht/parenrightBigg · · ·/parenleftBiggpml−1/summationdisplay xl=0|xl/an}brack⌉tri}ht/parenrightBigg |Ψt/an}brack⌉tri}ht (12) followed by Uf(xej)forj= 1,2, . . ., n , using the jth register as the control and |Ψt/an}brack⌉tri}htas the target, to produce /parenleftBiggpm1−1/summationdisplay x1=0exp(2πix1t1 pm1)|x1/an}brack⌉tri}ht/parenrightBigg · · ·/parenleftBiggpml−1/summationdisplay xl=0exp(2πixltl pml)|xl/an}brack⌉tri}ht/parenrightBigg |Ψt/an}brack⌉tri}ht. (13) Then apply F−1 pmjto the jth control register for each jto yield |t1/an}brack⌉tri}ht |t2/an}brack⌉tri}ht. . .|tl/an}brack⌉tri}ht |Ψt/an}brack⌉tri}ht (14) 9 from which we can extract t. As in the previous section, we do not know how to construct |Ψt/an}brack⌉tri}ht, but we do know that |f(0)/an}brack⌉tri}ht=/summationdisplay t∈T|Ψt/an}brack⌉tri}ht. So we start with |0/an}brack⌉tri}ht |0/an}brack⌉tri}ht · · · |0/an}brack⌉tri}ht |f(0)/an}brack⌉tri}ht=/summationdisplay t∈T|0/an}brack⌉tri}ht |0/an}brack⌉tri}ht · · · |0/an}brack⌉tri}ht |Ψt/an}brack⌉tri}ht apply Fourier transforms to get /summationdisplay t∈T/parenleftBiggpm1−1/summationdisplay x1=0|x1/an}brack⌉tri}ht/parenrightBigg · · ·/parenleftBiggpml−1/summationdisplay xl=0|xl/an}brack⌉tri}ht/parenrightBigg |Ψt/an}brack⌉tri}ht (15) then apply Uf(xej)using the jth register as a control register, for j= 1,2, . . ., n , and the last register as the target register to produce /summationdisplay t∈T/parenleftBiggpm1−1/summationdisplay x1=0exp(2πix1t1 pm1)|x1/an}brack⌉tri}ht/parenrightBigg · · ·/parenleftBiggpml−1/summationdisplay xl=0exp(2πixltl pml)|xl/an}brack⌉tri}ht/parenrightBigg |Ψt/an}brack⌉tri}ht.(16) We ﬁnally apply F−1 pmjto the jth control register for j= 1,2, . . ., l , to produce /summationdisplay t∈T|t/an}brack⌉tri}ht |Ψt/an}brack⌉tri}ht. (17) Observing the ﬁrst register lets us sample the t’s uniformly at random, and thus with O(n) repetitions we will, by (10), have enough independent line ar relations for us to determine a generating set for K. For example, in the case of Simon’s problem, the |t/an}brack⌉tri}htall satisfy t·s=/summationtextl j=1tjsjmod 2 = 0 mod 2, where K={0,s}. We could also guarantee that each new non-zero element of Twill increase the span by a technique discussed in the appendix. This analysis of eigenvectors and eigenvalues is based on th e work in [Ki]. The problem is that, unlike in [Ki], we do not always have the o perator Uf(xej). However, note that, like in Sect. 4.2, going from (15) to (16) maps  /summationdisplay 0≤xj≤pmj|x/an}brack⌉tri}ht |f(0)/an}brack⌉tri}ht to /summationdisplay 0≤xj≤pmj|x/an}brack⌉tri}ht |f(x)/an}brack⌉tri}ht =/summationdisplay t∈T/parenleftBiggpm1−1/summationdisplay x1=0exp(2πix1t1 pm1)|x1/an}brack⌉tri}ht/parenrightBigg · · ·/parenleftBiggpml−1/summationdisplay xl=0exp(2πixltl pml)|xl/an}brack⌉tri}ht/parenrightBigg |Ψt/an}brack⌉tri}ht. 10 We can create state (16) by applying Uf, which we do have, to the starting state /summationdisplay 0≤xi

Question 700 multiple-choice

In quantum computing and information theory, the problem of distinguishing hidden subgroups within group-theoretic frameworks often relies on analyzing measurement distributions from high-dimensional representations. Advanced probabilistic techniques such as concentration of measure and Chernoff bounds play a key role in establishing hardness results for these problems. When using strong Fourier sampling to distinguish conjugate subgroups within a group representation, which factor most directly limits the ability to differentiate the outcome distributions, requiring exponentially many measurements for success? 1) The presence of low-rank projection operators in the representation 2) A block-diagonal structure in all group representations 3) Exponential concentration of measure resulting in distributions close to uniform 4) The existence of non-abelian subgroups in every finite group 5) The use of classical measurement techniques instead of quantum ones 6) An absence of independent Gaussian variables in the representation basis 7) The dimension of the representation always being polynomially bounded

✓ Correct Answer:

The correct answer is 3) Exponential concentration of measure resulting in distributions close to uniform.

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In particular, there exists β>0, depending only on /epsilon1, such that /summationdisplay v|Pb(v)−Pb/prime(v)|δr d/bracketrightBig <4e−rδ2/48. Proof. We use an argument similar to [10]. We can think of a random d- dimensional complex vector vas a random 2 d-dimensional real vector of the same length, and we can think of this in turn as vi=wi/summationtext2d i=1w2 i, where the wiare independent Gaussian variables with zero mean and unit variance. By choosing a basis in which πprojects onto the ﬁrst r(complex) components of v, we have |π·v|2 2=/summationtext2r i=1w2 i/summationtext2d i=1w2 i=r d(1/2r)/summationtext2r i=1w2 i (1/2d)/summationtext2d i=1w2 i. Now we use the following Chernoﬀ bound, which can be derived from the moment generating function. For any t,w eh a v e Pr/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBigg 1 tt/summationdisplay i=1w2 i/parenrightBigg −1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>/epsilon1/bracketrightBigg <2/bracketleftBig (1 +/epsilon1) 1/2e−/epsilon1/2/bracketrightBigt . For|/epsilon1|<1/2, we have ln(1 + /epsilon1)/epsilon1/bracketrightBigg <2e −t/epsilon12/6. Now, for any a, b,i f|a/b−1|>δ, where δ<2, then either |a−1|>δ /4o r |b−1|>δ /4. Taking the union bound over these events where a=( 1/2r)/summationtext2r i=1w2 i andb=( 1/2d)/summationtext2d i=1w2 i, setting /epsilon1=δ/4 and t=2r≤2din (5) gives the stated bound. Setting d=dρandr=rkπ, Lemma 2 and the union bound imply that, for any constant A>√ 48, if (6) δ=A/radicalbigg logdρ rkπ, then, with high probability, for all dρbasis vectors vwe have /vextendsingle/vextendsingle/vextendsingle/vextendsingleP b(v)−1 dρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle<δ dρ. Summing over all v, this implies that the /lscript1distance between Pb(v) and the uniform Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE POWER OF STRONG FOURIER SAMPLING 953 distribution is at most δ. Now recall that rkπ=(p−1)/q.I f qp/epsilon1, and (6) gives δ

Question 701 multiple-choice

The LLL algorithm is widely used in computational number theory and cryptography for lattice basis reduction, with efficiency improvements possible when leveraging arithmetic over algebraic number rings and their symmetries. Exploiting automorphisms and advanced multiplication techniques can lead to substantial computational speed-ups in lattice reduction tasks. Which strategy offers the greatest theoretical polynomial speed-up for the LLL algorithm when reducing lattices over algebraic number rings of degree n? 1) Using the Gram-Schmidt process to orthogonalize basis vectors over real numbers 2) Applying standard integer arithmetic for all lattice calculations 3) Utilizing the Karatsuba multiplication method for complex number operations 4) Increasing the dimension of the lattice basis without exploiting algebraic structure 5) Exploiting automorphisms of the algebraic number ring to reduce the effective computational complexity 6) Randomizing the lattice basis at each step of the reduction process 7) Employing solely real-number arithmetic without leveraging underlying symmetries

✓ Correct Answer:

The correct answer is 5) Exploiting automorphisms of the algebraic number ring to reduce the effective computational complexity.

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For any vector (/tildewidec,m)∈ ⊕k j=1Z(˜cj,mj)such that m= (m(1),...,m(k))is not a relation ofb1,...,b k, i.e.m/n⌉}ationslash∈L∗, i.e./summationtextk j=m(j)/tildewidebj/n⌉}ationslash= 0.Then, the arguments in the proof of [13, Th 3.1] can be copied in a verbatim manner to o btain: /⌊ard⌊l/tildewidecj/⌊ard⌊l>2(k−1)/2˜M1≤j≤k. (5) Since Equations (4) and (5) are contradictory for 1≤j≤k−m, the only possibility is that m∈L∗. We do the same transformations on the matrix with bjinstead of/tildewidebjand callcjthe vectors which replace /tildewidecjin this case. What we have proved is that c1=c2=·=ck−m= 0. Since the transformations done in the LLL algorithm on the co lumns of the matrix are reversible, they preserve the rank of any subset o f rows of the matrix. The rank of the ﬁrst mrows of this matrix is dimL=m. Sincec1=c2=·= ck−m= 0, the vectors ck−m+1,...,c kform a basis of L. LLL over Z[i√ d]as a practical improvement The constant hidden in the big Oh of implementation is (inverse) proportional to logδ, which (inverse) proportional tolog(δ−MK). Hence the diﬀerence is of only a few percentages. The depend ence of the time in the rank is quartic, so we replace a number of ope rations of high- precision real numbers by 24= 16 times less complex numbers at the same level of precision. Finally, using the Karatsuba trick, a mu ltiplication of complex numbers costs 3multiplications of real numbers, so the overall gain is a fac tor 16/3≈5.33. An implementation [23] of Z[i√ d]-LLL shows that the reduction 16 Razvan Barbulescu and Adrien Poulalion of anOK-lattice is ≈5times faster when its algebraic structure is used when compared to forgetting it and using only the underlying stru cture of Z-lattice: they used OK=Z[i], the coordinates of the basis vectors have 512bits and δ= 0.99both over ZandZ[i].11 Consequences in cryptography. Given our current contribution, the speed-up due to automorphisms is 5.33. It is an open question to extend it to a larger class of OK-rings (e.g. [37] investigate if LLL can be extended to arbit raryOKby solving CVP instances in dimension [OK:Z].). If Buchmann-Pohst can be extended, the best speed-up that one can target is [OK:Z]3. Indeed, the dependence of LLL in the dimension is quartic whereas the cost of the arithm etic over OKis quadratic or quasi-linear if fast arithmetic is used The speed-up due to the automorphisms is polynomial in the nu mber of au- tomorphisms. This is a familiar situation because a similar speed-up happens for automorphisms in the case of classical algorithms for fa ctorization, discrete logarithms in ﬁnite ﬁelds and discrete logarithms on ellipt ic curves (see e. g. [30, Sec 4.3],[5, Sec 5.3]). For instance, the attacks in [20] aga inst ECDSA using endomorphisms of the curve achieved a polynomial speed-up i n an algorithm of exponential complexity. The community reacted by sugges ting to use elliptic curves with endomorphisms as their eﬀect was now considered to be benign, e.g. [25]. In a similar manner, if the security of the class gr oup of a given degree and determinant is considered suﬃcient, we suggest to use

Question 702 multiple-choice

Quantum process tomography (QPT) is used to experimentally reconstruct and analyze quantum operations, especially in platforms like nuclear magnetic resonance (NMR) quantum processors. Understanding sources of error and their quantitative contributions is critical for improving gate fidelity in quantum computing experiments. In an experimental implementation of quantum process tomography for the quantum Fourier transform on a three-qubit NMR system, which error source was attributed the largest percentage of fidelity loss? 1) Tomography (measurement) errors 2) Decoherence 3) Coherent control errors 4) Calibration drift 5) Crosstalk between qubits 6) State preparation errors 7) Environmental magnetic field fluctuations

✓ Correct Answer:

The correct answer is 2) Decoherence.

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the experimental supermatrix Mobsfor comparison. Fromthis we see that the eigenvalues of MCPTPare closer to being cocircular, indicating that it is closer to an attenuated unitary than was Mobs, but that only after taking the unitary part did they become perfectly cocircular. TABLE IV: The correlation coeﬃcients between Theoretical, Simulate d and Experimental supermatrices (see text) and the optimum completely positive and trace-pr eserving approximation MCPTPto the experimentally determined supermatrix Mobs, as well as the best unitary approximation to its largest Kraus operator ¯U1,CPTP⊗U1,CPTPandMobsitself for comparison. SupermatrixCorrelation with ¯U1,CPTP⊗U1,CPTPCorrelation with MCPTPCorrelation with Mobs Theoretical 0.86 0.89 0.82 Simulated 0.82 0.82 0.79 Experimental 0.95 0.97 1.00 25 IX. CONCLUSIONS Inconclusion, wehave implemented quantumprocesstomographyo fthequantumFourier transform on a three-qubit NMR quantum information processor. The overall gate ﬁdelity (attenuated correlation between superoperator matrices) was 0.64, whereas the (unattenu- ated) supermatrix correlation was 0 .82 (see Table II). Judging by the fact that making the unitary part of the largest Kraus operator correspond as closely as possible to the theoreti- cal QFT gave a correlation of 0 .95, we conclude that the loss of ﬁdelity due to incoherence and/or measurement errors during state tomography was of ord er 5%. The loss of mag- netization due to incoherence and decoherence, on the other han d, was 0.64/0.82 = 0.78, i.e. about 22%. This implies that the cummulative eﬀects of coherent e rrors reduced the ﬁdelity by about 0 .82/0.95 = 0.86, or 14%, consistent with the fact that the correlation between the largest Kraus operator and the theoretical QFT unit ary was 0 .925≈√ 0.86. More importantly, QPT of the QFT has enabled us to validate the esse ntial correctness of our model of the system-plus-apparatus used for the experim ents in great detail, and to isolate its remaining shortcomings. It has further prompted us t o develop a range of data analysis and visualization techniques for quantum process tom ography, which should be broadly applicable in quantum information processing. The experim ents described here demonstrate the precision with which complex quantum dynamics can be controlled, and highlightsthesigniﬁcanceofliquid-stateNMRasatestbedforachiev ing suchcontrol. While full QPT on larger quantum systems will never be practical, the analy sis done here should serve as an initial guide as to how information about the errors in qua ntum information processors can be extracted, and perhaps someday, how to deb ug a quantum computer. 26 ACKNOWLEDGEMENTS The authors thank J.P. Paz for helpful discussions. This work was s upported by ARDA/ARO grants DAAD19-01-1-0519 & DAAD19-01-1-0654, DAR PA grant MDA972- 01-1-0003, NSF grant EEC 0085557, the Air Force Oﬃce of Sponso red Research, and by the Cambridge-MIT Institute, Ltd. [1] B. Schumacher, Phys. Rev. A, 54, 2614, (1996). [2] G. Teklemariam, E. M. Fortunato, M. A. Pravia, T. F. Havel , and D. G. Cory, Phys. Rev. Lett.86, 5845 (2001). [3] E. M. Fortunato, M. A. Pravia, N. Boulant, G. Teklemariam , T. F. Havel, D. G. Cory, J. Chem. Phys. 116, 7599-7606, (2002). [4] N. Boulant, K. Edmonds, J. Yang, M. A.

Question 703 multiple-choice

Advances in computational technology have led to the exploration of quantum computing, which leverages principles like superposition and entanglement to tackle complex problems more efficiently than classical computers. A key mathematical operation in many fields, convolution, has motivated the development and application of both classical and quantum Fourier transforms for improved computational performance. Which statement accurately describes the computational complexity advantage of the Quantum Fourier Transform (QFT) over the classical Fast Fourier Transform (FFT) for specific problems? 1) The QFT and FFT both require O(N²) operations for all input sizes. 2) The QFT is always slower than the FFT due to quantum decoherence. 3) The FFT outperforms the QFT in processing large datasets for all known algorithms. 4) For certain problems, the QFT can reduce computational complexity to O((log N)²), providing an exponential speedup over the FFT's O(N log N). 5) The QFT is only applicable to analog signal processing, not discrete signals. 6) The FFT eliminates the need for any form of quantum computation in all cases. 7) The QFT and FFT have identical performance across all types of computations.

✓ Correct Answer:

The correct answer is 4) For certain problems, the QFT can reduce computational complexity to O((log N)²), providing an exponential speedup over the FFT's O(N log N)..

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INTRODUCTION Classical computing progressed as a means of rapidly tackling real-world problems. The speed and capabilities of these computers have continually increased over the past few decades, enabling them to be used in virtually every aspect of our lives. However, as transistors shrink to the size of individual atoms [13], Moore's Law's projected exponential growth may stall. Quantum computing is one proposed answer to this challenge. Benio [14] was the rst to propose the use of quantum systems for information processing. Deutsch described quantum computers in 1985, which leverage the superposition of many particle states to achieve massive parallelism [4]. Researchers have also investigated the prospect of tackling speci c classes of problems more e ectively than conventional computers can [4]. These theoretical possibilities have piqued the curiosity of many researchers in the realisation of quantum hardware [4] [2] [3] [7] [1]. The quantum world is fundamentally di erent from the classical one. A particular, signi - cant implication is that, for certain classes of problems, computations can be performed more eciently if the information is processed using quantum rather than conventional methods. In mathematics, convolution is an operation acting on two functions that produces a third function expressing how the shape of one is modi ed by the other. In classical computing, convolution is known to be explicitly dicult with a computational complexity that is ex- ponential to the number of input functions [4]. In linear, time-invariant systems where the input and output relationship is described by a convolution, this complexity can be reduced by using the Fourier transform. The Fourier transform of the convolution of two functions is simply the product of the Fourier Transforms of the functions. On the importance of the QFT one can learn from the work of Mastriani's Fourier's quantum information processing [3]: "We demonstrate that quantum information processing (QIP) completely rests on quantum Fourier transform (QFT), and 190 years after his death, the work of Jean-Baptiste Joseph Fourier is more present than ever in Physics, constituting the heart of QIP, and showing the spectral nature of quantum entanglement, quantum 3 SciPost Physics Codebases Submission teleportation, and quantum secret sharing." [3] The Fast Fourier Transform (FFT) is a frequently used technique for calculating the dis- crete Fourier transforms (DFT) of complex or real-valued data sequences [2]. The transformed data can be partitioned into the numerous pure frequencies that comprise it (1), a technique that is applicable to a wide variety of applications, including computational physics as well as machine and deep learning. Using the FFT in this case reduces computational complexity fromO/parenleftbigN2/parenrightbigtoO(NlogN)[3]. This speedup is powerful and crucial to real-time processing of data. 1.1 Motivation Due to the fact that the QFT is fundamental to a large number of quantum algorithms [2], the motivation of this paper is to o er a proof-of-concept by utilizing the QFT for discrete signal processing and ultimately showing how a quantum computer can provide reduced complexity over classical computers. Figure 1 Time and frequency domain representations of a

Question 704 multiple-choice

In computational group theory and quantum algorithms, the structure and properties of groups such as abelian, solvable, and smoothly solvable groups play a crucial role. The quantum Fourier transform (QFT) is a key tool for efficiently solving problems related to hidden subgroups in finite abelian groups. Which of the following groups is a typical example of a smoothly solvable group, having constant derived length and factor groups that are (e,s)-smooth for polynomially bounded s and constant e? 1) The symmetric group $ S_n $ for large $ n $ 2) Unitriangular matrix groups over finite fields of constant characteristic 3) The alternating group $ A_n $ with $ n \geq 5 $ 4) The infinite cyclic group $ \mathbb{Z} $ 5) The general linear group $ GL_n(\mathbb{F}_q) $ for large $ n $ 6) The dihedral group $ D_{2n} $ for large $ n $ 7) The quaternion group $ Q_8 $

✓ Correct Answer:

The correct answer is 2) Unitriangular matrix groups over finite fields of constant characteristic.

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G0G1 ···Gmsuch that G=G0,Gm={1G}and the factors Gi/Gi+1(i=0,1,...,m−1) are abelian. A natural way of constructing a subnormal series of the solvable group Gis to consider its derived subgroups. For any group H, let us ﬁrst deﬁne and denote thecommutator subgroup H/primeofHbyH/prime=/angbracketleft{h−1k−1hk:h,k∈H}/angbracketright. Then the derived subgroups G(i)(i=0,1,2,...) are deﬁned by induction: G(0)=G;a n dt h e 6 FRIEDL, IVANYOS, MAGNIEZ, SANTHA, AND SEN (i+ 1)th derived subgroup G(i+1)is deﬁned as the commutator ( G(i))/primeofG(i).A l l the subgroups G(i)are normal subgroups of G(j)for 0≤j

Question 705 multiple-choice

Quantum computing relies on a set of fundamental gates that can be composed to perform any computation, with efficiency and accuracy being central concerns. The theoretical framework surrounding these gate sets guarantees the feasibility of constructing arbitrary quantum circuits. Which theorem ensures that any unitary operation can be efficiently approximated to arbitrary precision using a finite universal gate set, with the number of gates required scaling polylogarithmically with the inverse of the approximation error? 1) No-Cloning Theorem 2) Gottesman-Knill Theorem 3) Quantum Threshold Theorem 4) Bernstein-Vazirani Theorem 5) Quantum Error Correction Theorem 6) Holevo's Theorem 7) Solovay-Kitaev Theorem

✓ Correct Answer:

The correct answer is 7) Solovay-Kitaev Theorem.

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0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 01 CCCCCCCCCCCCCCCCCCCCA: 18 The Toffoli gate is shown in Figure 2.4, with the AND operation ^deﬁned in Table 2.1. Figure 2.4: The Toffoli Gate controljxijxi controljyijyi targetjzi jz(x^y)i Table 2.1: The AND Operation xyx^y 11 1 10 0 01 0 00 0 The phase shift can be implemented as a controlled transformation CP=0 BBBBBBB@1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 eiq1 CCCCCCCA: (2.55) where the phase shift is applied to the target only when the control qubit is in the state j1i. An important ingredient in many quantum circuits is the controlled discrete phase gate CRk described by CRk:=j00ih00j+j01ih01j+j10ih10j+e2pi 2kj11ih11j =0 BBBBBBB@1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e2pi 2k1 CCCCCCCA(2.56) 19 2.3.3 Universal Quantum Gates In classical computation, complicated transformations are implemented as a sequence of simpler operations selected from a set of elementary gates. For example, any logical circuit can be simulated using only the AND and the NOT gate, and so these two gates form a set of universal logical gates. In quantum computation, a circuit designer can similarly rely on universal quantum gates. However, due to the probabilistic nature of quantum computation, a unitary transformation Vapproximates a desired unitary transformation Uto a speciﬁed level of accuracy [22], with the approximation error deﬁned by E(U;V)max jyik(UV)jyik (2.57) where kjcikp hcjci: (2.58) Theorem 2.3.1. The errors of approximating a series of unitary gates U 1;:::;Unby unitary trans- formations V 1;:::;Vnadd, so that E(UnUn1:::U1;VnVn1:::V1)E(Un;Vn)+E(Un1;Vn1)++E(U1;V1) (2.59) Proof. Suppose the desired unitary transformation U2U1is implemented by a unitary transforma- tionV2V1. Then E(U2U1;V2V1) =max jyik(U2U1V2V1)jyik =max jyik(U2U1V2U1+V2U1V2V1)jyik (2.60) =max jyik(U2V2)U1jyi+V2(U1V1)jyik: 20 Since U1andV2are unitary, E(U2U1;V2V1)E(U2;V2)+E(U1;V1): (2.61) and the result follows by induction. If a unitary transformation is to be implemented to a precision ewith Ngates, it is only necessary to implement each gate to a precision ofe N. It has been shown that all quantum circuits can be constructed by using only CNOT and single-qubit gates [27], which brings us to a deﬁnition of a universal quantum gate set. Deﬁnition 2.3.2. A set of gates Gis a universal quantum gate set if for every e>0,n2N, any nqubit unitary operator can be approximated to an accuracy eby a ﬁnite sequence of gates from setG. The universal gate sets commonly used in quantum computation are: 1. The set G=fCNOT ;H;Tg[23] 2. The set G=fCNOT ;H;S;Toffolig[24] The Solovay-Kitaev Theorem [25, 26] establishes that gates from a universal gate set approximate any unitary transformation to an arbitrary accuracy ewith the efﬁciency polylogarithmic in1 e. 2.4 Probabilistic Nature of Quantum Algorithms The nature of the quantum world dictates that quantum computers are

Question 706 multiple-choice

Quantum error correction codes are essential for protecting quantum information and enabling fault-tolerant quantum computing. The interplay between continuous symmetries and error correction has significant implications for both practical code construction and theoretical physics, particularly in systems with symmetries such as unitary groups. Which statement correctly describes the resource scaling required to achieve a universal set of transversal gates with fixed accuracy in quantum error correction codes covariant under continuous symmetry groups? 1) The required number of physical qubits per logical qubit remains constant regardless of tolerated error. 2) The required number of physical qubits per logical qubit increases inversely with the tolerated error. 3) The required number of physical qubits per logical qubit decreases as the tolerated error becomes smaller. 4) The required number of physical qubits per logical qubit depends only on the logical qubit dimension and not on error tolerance. 5) The required number of physical qubits per logical qubit decreases linearly with the subsystem dimension. 6) The required number of physical qubits per logical qubit is independent of the specific continuous symmetry group. 7) The required number of physical qubits per logical qubit is determined purely by the number of subsystems without regard to code accuracy.

✓ Correct Answer:

The correct answer is 2) The required number of physical qubits per logical qubit increases inversely with the tolerated error..

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Continuous Symmetries and Approximate Quantum Error Correction Philippe Faist,1,2,*,†Sepehr Nezami,3,†Victor V. Albert ,1,4Grant Salton,1,3Fernando Pastawski,2 Patrick Hayden,3and John Preskill1,4 1Institute for Quantum Information and Matter, Caltech, Pasadena, 91125 California, USA 2Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany 3Stanford Institute for Theoretical Physics, Stanford University, Stanford, 94305 California, USA 4Walter Burke Institute for Theoretical Physics, Caltech, Pasadena, 91125 California, USA (Received 12 April 2019; revised 27 July 2020; accepted 7 September 2020; published 26 October 2020) Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here, westudy the compatibility of these two important principles. If a logical quantum system is encoded into n physical subsystems, we say that the code is covariant with respect to a symmetry group Gif aG transformation on the logical system can be realized by performing transformations on the individual subsystems. For a G-covariant code with Ga continuous group, we derive a lower bound on the error- correction infidelity following erasure of a subsystem. This bound approaches zero when the number ofsubsystems nor the dimension dof each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with nordas the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem for quantum computation: If a code admits auniversal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, weneed a number of physical qubits per subsystem that is inversely proportional to the error parameter. Weconstruct codes covariant with respect to the full logical unitary group, achieving good accuracy for large d (using random codes) or n(using codes based on Wstates). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knilltheorem, and our five-rotor code can be stacked to form a covariant holographic code. DOI: 10.1103/PhysRevX.10.041018 Subject Areas: Quantum Information, String Theory I. INTRODUCTION Quantum error-correcting codes protect fragile quantum states against noise [1]. If quantum information is cleverly encoded in a highly entangled state of many physical subsystems, then damage inflicted by local interactions with the environment can be reversed by a suitable recovery operation. Aside from their applications to resilient quan- tum computing, quantum error-correcting codes appear in awide variety of physical settings where quantum states are delocalized over many subsystems, such as topological phases of matter [2–5]and the AdS/CFT correspondence in holographic quantum gravity [6,7].On the other hand, naturally occurring physical systems often respect symmetries. For instance, phases of matter can be classified according to how these symmetries are realized in equilibrium states. Likewise, quantum error-correctingcodes often have approximate or exact symmetries with important implications. In the case of a time-translation- invariant many-body system, for example, certain energysubspaces

Question 707 multiple-choice

In quantum information theory, constructing maximally entangled states often relies on group representation theory and the properties of symmetry groups. Special cases arise in multipartite systems where system dimensions and group structures dictate the uniqueness and classification of entangled states. In the classification of locally maximally entangled (LME) quantum states for systems with dimensions (2, N, N), which mathematical property determines the dimension of the manifold of such states when N ≥ 4? 1) The parity of N as an even or odd integer 2) The rank of the SU(2) Lie algebra 3) The number of irreducible representations of UT(3, p) 4) The order of the product group SU(2) × SU(2) 5) The formula 2(N−3), describing the real manifold dimension of the space of LME states 6) The minimal eigenvalue of the entanglement Hamiltonian 7) The determinant of upper triangular matrices over Zp

✓ Correct Answer:

The correct answer is 5) The formula 2(N−3), describing the real manifold dimension of the space of LME states.

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⊗/parenleftbigg 0eiθ eiχ0/parenrightbigg ⊗/parenleftbigg 0e−iα−iθ e−iβ−iχ0/parenrightbigg (3.8) Thus, instead of S3, we could have started with this group or any subgroup of this grou p acting irreducibly on all the factors. Example 3: LME states from SU(2)3j-symbols. As a more general example, when the dimensions satisfy triangle ineq ualities and sum to an odd number, the tensor product of SU(2) representations wit h dimensions A,B, andC containsthetrivial representation. Theexplicit representation o fthistrivial stateisprecisely described by the 3 j-symbols (closely related to Clebsch-Gordon coeﬃcients): |0/an}b∇acket∇i}ht=/summationdisplay mA,mB,mC/parenleftbiggA−1 2B−1 2C−1 2 mAmBmC/parenrightbigg |mA/an}b∇acket∇i}ht⊗|mB/an}b∇acket∇i}ht⊗|mC/an}b∇acket∇i}ht. (3.9) It is easy to verify that ρA,ρB, andρCare all proportional to the identity matrix us- ing standard orthogonality relations for 3 jsymbols. For A= 2, this gives examples with (d1,d2,d3) = (2,N,N+1). As we have seen above, for each N, there is a unique LME state with these dimensions up to local unitary transformations, so our r epresentation theory construction gives the only example. Example 4: states from product groups We can obtain more examples by considering product groups. For ex ample, if a group Hhas irreducible representations {R1,R2,R3}whose tensor product contains the identity and group H′has irreducible representations {R′ 1,R′ 2,R′ 3}whose tensor product contains the identity, then group H×H′has representations {(R1,R′ 1),(R2,R′ 2),(R3,R′ 3)}whose tensor product contains the identity. This allows us to build examples with dimensions (d1d′ 1,d2d′ 2,d3d′ 3). The states constructed in this way are simply tensor products |Ψ/an}b∇acket∇i}ht=|ψ/an}b∇acket∇i}ht⊗|ψ′/an}b∇acket∇i}ht (3.10) As an example, we can consider the group SU(2) ×SU(2), and representations (2 ,1), (N,K), and (N+1,K) (labeled by dimension) to construct examples with dimensions (2 ,NK,(N+ 1)K). Again, we have seen above that there is a unique LME state up to lo cal unitary transformations with these dimensions, so our construction prov ides the only example. 16 Example 5: states for (d1,d2,d3) = (2,N,N) For dimensions (2 ,d2,d3) we have seen how to construct all possible LME states using rep- resentation theory except for the cases (2 ,N,N) withN >2. We can construct examples with these dimensions so long as we can ﬁnd a group Hpand representations {R2,Rp,ˆRp} giving a construction for the case (2 ,p,p) withpodd. In this case, we can use the group Hp×SU(2) to construct examples with dimensions (2 ,N,N) withpa prime factor of Nby choosing representations {(R2,1),(Rp,N/p),(ˆRp,N/p)}(we can use H2=S3as above). In appendixBwe show that a group with the desired properties is Hp=UT(3,p)⋊Z2a certain semidirect product of UT(3,p) (the group of upper triangular matrices with elements in Zp), andZ2. This is a ﬁnite group with 2 p3elements. Thus, for every case with dimensions (2 ,d2,d3) for which LME states exist, there is a con- struction based on representation theory that provides example s. In most cases, the LME state is unique up to local unitary transformations. For the cases (2,N,N) withN≥4 we found a 2( N−3) real dimensional space of such states; in this case, we expect t hat the representation theory construction gives special states in t his space which are invariant under a larger subgroup of SU(2)

Question 708 multiple-choice

Deliberate self-harm (DSH) among adolescents and young adults is a complex phenomenon involving various psychological, behavioral, and family-related factors. Understanding the mechanisms and motivations behind different types of self-injurious behavior is crucial for effective prevention and intervention. Which of the following statements best reflects the relationship between the severity of self-harm methods and underlying psychological distress in adolescents? 1) Major self-harm methods always indicate the presence of severe psychiatric disorders. 2) Minor self-harm behaviors are typically unrelated to psychological problems. 3) Only adolescents with borderline personality disorder engage in severe self-harm. 4) Both minor and major self-harm behaviors can be associated with significant underlying psychological problems such as impulsivity and low self-esteem. 5) The type of self-harm directly determines the urgency of clinical intervention. 6) Major self-harm is only seen in those with a history of trauma. 7) Minor self-harm methods are generally harmless and do not require attention.

✓ Correct Answer:

The correct answer is 4) Both minor and major self-harm behaviors can be associated with significant underlying psychological problems such as impulsivity and low self-esteem..

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into wellsTable 4 Prevalence of deliberate self-harm (DSH) behavior from various studies Author Population Prevalence 1. Yates179th to 12 grade (USA)26-37% (self-cutting) 2. Li 18Teens & Adults (Taiwan)1% of all injuries; 80% self-cutting 3. Nixon 1914-21 year olds (Canada) 17% DSH; 83%: cutting, scratch, self-hitting 4. Morey 203,881 Irish Teenagers9.1% DSH; 66%: self-cutting 5. De Leo 223,754 Teens (Australia)6.2% DSH past 12 months of survey 6. Ystgaard 234.060 Teens (Norway)6.6% DSH; 74%: self-cutting 7. Matsumoto 27Teenagers (Japan)9.8% DSH with self-cutting Keio J Med 2009; 58 (3): 144 －151 147 there is a steady increase of tension until a threshold is reached (the “spring” path), or as a result of the “switch - ing on” of the impulse (the “switch-path”). Dissociation during cutting and a craving for cutting were more often associated with switching to this impulse.17, 41 Further - more, two main personal characteristics in adolescent self-harmers are having overwhelmingly negative emo - tion and self-derogation. Reasons noted for the non-sui - cidal self-injury include efforts to resist thoughts of sui - cide, expression of self-anger or disgust, resolve times of dissociation, influence others, or seek help from oth - ers.42, 43 Research findings also showed that deliberate self- harm was related to individual psychological well-being. A study of 201 adolescents in a juvenile detention center in Japan noted that 16.4% had cut themselves at least once and 35% had burned themselves at least once.44 Those who had both self-cutting and self-burning behav - iors had increased rates of depression and dissociation while those who had self-burning without cutting had less psychopathology. Studies also demonstrated that factors underlying deliberate self-harm included differ - ent forms of psychopathologies,45 such as schizophrenia, major depression, and substance abuse disorders;28, 46, 47 there is an association of drug abuse (including alcohol abuse) and eating disorders with some self-injurers.48 Deliberate self-harm is also associated with personality disorders.49,50 However, self-cutting can also arise de novo without any overt history of trauma or borderline personality disorder.51 Nevertheless, the type of self-harm does not necessari - ly identify the seriousness of the underlying psychopa - thology. Cryole, for example, studied 280 college under - graduates and noted that 31% had “minor” self-harming behavior characterized by skin picking or fingernail bit - ing over the previous three years; 20% had “major” self- injurious behavior with self-cutting and self-burning.52 Both groups were associated with underlying psychopa - thology, including disordered eating patterns (with nega - tive self-esteem),53 negative thinking,53 impulsivity, so - matic issues, and obsessive-compulsive issues.31 From the family perspective, different family processes are postulated to be related to adolescent suicidal behav - ior. Richman54 proposed that communication disturbanc -es and excessive secretiveness were family factors that might lead to heightened suicidal behavior. Research findings demonstrated that parent-adolescent communi - cation was associated with adolescent suicidal ideation and adolescent deliberate self-harm.55, 56 Syed and Khan showed that family conflict was the main factor contrib - uting to deliberate self-harm in young people in Paki - stan.57 Besides, a major

Question 709 multiple-choice

Quantum algorithms such as Shor’s utilize signal processing concepts to enhance computational outcomes, particularly when employing transforms like the quantum Fourier transform (QFT) and Hadamard transform. Windowing techniques are adapted to quantum contexts to address challenges like spectral leakage and unitarity requirements. When implementing Shor’s algorithm for integer factorization on quantum hardware, which technique can be used to mitigate spectral leakage when using the Hadamard transform instead of the full quantum Fourier transform, while preserving unitarity? 1) Multiplying the comb function by a Hamming window before transformation 2) Applying a Hann window after measurement 3) Attenuating central terms in the q-point cell via zero-padding 4) Replacing the Hadamard transform with a classical discrete Fourier transform 5) Multiplying all data samples by a rectangular window 6) Adding a Bartlett (triangular) window function to the comb before the Hadamard transform 7) Normalizing the output spectrum by dividing by the number of q-points

✓ Correct Answer:

The correct answer is 6) Adding a Bartlett (triangular) window function to the comb before the Hadamard transform.

📚 Reference Text:

matched to zero with the -1 and qth samples i.e. the last and f irst samples of the preceding and following data cells . Thus the data samples are contained within an overall envelope function that is now periodic within the q-point data cell. Thus the window function would pre -multiply the comb function prior to transformation . A suitable windowing function might be the B artlett window as this is a triangular function10: 𝑤(𝑚)= { 2𝑚 𝑞−1, 0≤𝑚<𝑞−1 2 2−2𝑚 𝑞−1 , 𝑞−1 2 ≤𝑚≤𝑞−1 (12) that is a function that has a linear dependence on index q and so could be applied to the register contents with a S ingle Instruction Multiple Data (SIMD) instruction (in a similar manner to the modular exponentiation operations to produce xa mod N). However, the operation must be unitary and so the comb functio n impulse train can not be atten uated. Thus, rather than attenuating the impulse train by multiplying by w(m), we instead add w(m) to the comb function. This boosts the central terms in the data window , increasing their relative weight, and so will ameliorate the effect of a non -periodic fit of the data in the window. Since the comb function will not fall to zero at the window edges, this will not be as effective in reducing spectral leakage as multiplying by w(m) but will still aid in r educing its deleterious effects . This, in turn, will increase the probability of detecting the reciprocal comb function impulse locations since they will be less spread and be more sharply peaked due to the reduction in spectral leakage effects. Thus the p robability of the collapse of the output wavefunction, |𝜓3> , at one of the correct locations will be increased and a more reliable result achieved. 4. CONC LUSION This paper has described the Shor quantum algorithm for large integer factorisation from a signal processing perspective, emphasising the importance of the quantum Fourier transform as an essential component of the algorithm. A problem will arise in the implementation of the QFT for large integer sizes as the phase control gates necessary for i ts exact implementation require , nominally, an exponentially fine phase control. Approximations to the QFT that would allow a much coarser phase control are reviewed . A quantum version of the Hadamar d transform , requiring only a 0 or π phase control, is considered as this may provide an effective solution for the sp ecific requirement of the Shor algorithm, namely the transfor mation of a comb function to a reciprocal comb function . The reasons for a possible degradation of performance of the Hadamard trans form have been discussed and a possible means to amelior ate this degradation consider ed by applying a triangular windowing function prior the Hadamard transformation to effect a reduction in spectral leakage effects and so increasing the probability of accurate location of the output impulse function train upon measurement of the output wavefunction. REFERENCES 1. Young

Question 710 multiple-choice

Quantum Hidden Subgroup (QHS) problems play a fundamental role in the development of quantum algorithms, providing the framework for breakthroughs such as quantum factoring and phase estimation. Traditionally, QHS algorithms have focused on finite or discrete groups, but recent research extends these techniques to continuous and infinite-dimensional mathematical structures. Which of the following groups is a functional space relevant to constructing QHS algorithms for quantum processes involving trajectories in n-dimensional real space? 1) The discrete cyclic group ℤₙ 2) The multiplicative group of nonzero real numbers ℝ* 3) The additive group of integers ℤ 4) The symmetric group Sₙ 5) The additive group of rational numbers ℚ 6) The additive group of L2 paths from [0,1] to ℝⁿ 7) The finite abelian group ℤ/pℤ

✓ Correct Answer:

The correct answer is 6) The additive group of L2 paths from [0,1] to ℝⁿ.

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Title: Continuous quantum hidden subgroup algorithms Year: 2003 Paper ID: 98a1b6d75ad449ada642af751347eb431866fe94 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/98a1b6d75ad449ada642af751347eb431866fe94 Abstract: In this paper we show how to construct two continuous variable and one continuous functional quantum hidden subgroup (QHS) algorithms. These are respectively quantum algorithms on the additive group of reals R, the additive group R/Z of the reals R mod 1, i.e., the circle, and the additive group Paths of L2 paths x:[0,1] → Rn in real n-space Rn.

Question 711 multiple-choice

In the study of topological abelian groups, Pontryagin duality relates a group to its group of continuous characters, and properties such as reflexivity and compactness play crucial roles. The Baer-Specker group, its homomorphism group, and their topologies provide important examples highlighting subtle distinctions in infinite group theory. Which property holds for the group $ G = \text{Hom}(\mathbb{Z}^{\mathbb{N}}, \mathbb{Z}) $ endowed with the topology of pointwise convergence? 1) It is a locally compact group. 2) It is reflexive under Pontryagin duality. 3) It contains no infinite compact subsets. 4) It is a finite abelian group. 5) It is a discrete group under pointwise convergence. 6) It is not a closed subgroup of any Tychonoff product of copies of $ \mathbb{Z} $. 7) Its Pontryagin double dual is non-discrete.

✓ Correct Answer:

The correct answer is 3) It contains no infinite compact subsets..

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Title: A countable free closed non-reflexive subgroup of ℤ^{} Year: 2015 Paper ID: d1af1ece1647e39b206e544285fe802d712e9ffe Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d1af1ece1647e39b206e544285fe802d712e9ffe Abstract: We prove that the group G=Hom(P,Z) of all homomorphisms from the Baer-Specker group P to the group Z of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that the second Pontryagin dual of G is discrete. As G is non-discrete, it is not reflexive. Since G can be viewed as a closed subgroup of the Tychonoff product of continuum many copies of the integers Z, this provides an example of a group described in the title, thereby answering Problem 11 from [J.Galindo, L.Recorder-N\'{u}\~{n}ez, M.Tkachenko, Reflexivity of prodiscrete topological groups, J. Math. Anal. Appl. 384 (2011), 320--330.] It follows that an inverse limit of finitely generated (torsion-)free discrete abelian groups need not be reflexive.

Question 712 multiple-choice

In topological quantum computing, the mathematical modeling of anyons and symmetry defects plays a crucial role in determining the types of quantum gates that can be realized. These models utilize advanced algebraic and topological structures to describe particle exchanges and system symmetries. Which mathematical construction is specifically used to model both anyons and symmetry defects in systems with underlying symmetries, thereby enabling projective unitary representations of the braid group for quantum gate implementation? 1) Group cohomology extensions 2) Hopf algebra modules 3) Spin network models 4) G-crossed braided extensions of unitary modular tensor categories 5) Representation theory of Lie algebras 6) Quantum error-correcting codes 7) Clifford algebras

✓ Correct Answer:

The correct answer is 4) G-crossed braided extensions of unitary modular tensor categories.

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Title: Symmetry defects and their application to topological quantum computing Year: 2018 Paper ID: bf95350fe678ee235e311269a742696e9d31f3bc Source: semantic-scholar URL: https://www.semanticscholar.org/paper/bf95350fe678ee235e311269a742696e9d31f3bc Abstract: We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions of unitary modular tensor categories. The algebraic data of these categories afford projective unitary representations of the braid group. Elements in the image of such representations correspond to quantum gates arising from exchanging anyons and symmetry defects in topological phases of matter with symmetry. We provide some small examples that highlight features of practical interest for quantum computing. In particular, symmetry defects show the potential to generate non-abelian statistics from abelian topological phases and to be used as as a tool to enlarge the set of quantum gates accessible to an anyonic device.

Question 713 multiple-choice

Quantum algorithms such as Deutsch-Jozsa and Bernstein-Vazirani have been generalized to operate on qudits, which are quantum systems with more than two levels, allowing for broader computational applications. These generalizations leverage tools like the quantum Fourier transform and phase kick-back to efficiently solve complex problems in higher-dimensional spaces. In the qudit generalization of the Bernstein-Vazirani algorithm, which operation enables the extraction of an unknown integer string encoded in an oracle when dealing with modulo d arithmetic? 1) Measurement in the computational basis without prior transformations 2) Applying the Hadamard transform to each qudit 3) Using Grover’s search algorithm to iteratively amplify the correct state 4) Initializing all qudits in the |0⟩ state and directly querying the oracle 5) Performing a swap test between input and reference states 6) Executing the quantum Fourier transform followed by its inverse after oracle application 7) Replacing qudits with multiple entangled qubit registers

✓ Correct Answer:

The correct answer is 6) Executing the quantum Fourier transform followed by its inverse after oracle application.

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qubit version of the Deutsch- Jozsa algorithm. If the function Ufis constant, the ﬁnal state after the measurement is |0〉⊗r|n−1〉as for j≠0 every states in the x-register have null amplitudes. Therefore, if every x-register qudit yields |0〉, it is a constant function; otherwise the function is balanced. The Deutsch-Jozsa algorithm i n the qudit system shares the same idea while enabling more applications such asdetermining the closed form of an af ﬁne function. Although this algorithm is mainly of theoretical interest, the n-nary version of it may have applications in image processing. It has the potential to distinguish between maps of texture in a Marquand chart since the images of which are encoded byafﬁne functions [ 121]. This algorithm can also be modi ﬁed to set up a secure quantum key-distribution protocol [ 121]. Other proposed Deutsch-Jozsa algorithms exist such as a method thatmakes use of the arti ﬁcially allocated (subsystems) as qudits [88] and a generalized algorithm on the virtual spin representation [ 86]. 3.1.3 Qudit Generalization of the Bernstein-Vazirani Algorithm InSection 3.1.2 we have discussed an application of a qudit Deutsch-Jozsa algorithm (DJA): verify a closed expression of an afﬁne function. This application is closely related to the Bernstein-Vazirani algorithm discussed in this section. Givenan input string and a function that calculates the bit-wize inner-product of the input string with an unknown string, theBernstein-Vazirani algorithm determines the unknown string[12]. This algorithm can be treated as an extension of the Deutsch-Jozsa algorithm. The qudit generalization of the Bernstein –Vazirani algorithm can determine a number string of integers modulo dencoded in the oracle function [ 95, 119 ]. First we introduce a positive integer dand consider the problem in modulo dthroughout. Given an N-component natural number string g(a):/equals⎥parenleftbigg(a 1),g(a2),g(a3),... ,g(aN)⎥parenrightbig,g(aj)∈{0,1,... ,d−1}, (82) we de ﬁne Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504 12Wang et al. Qudits and High-Dimensional Quantum Computing f(x):/equalsg(a)·xmod d /equalsg(a1)x1+g(a2)x2+...+g(aN)xNmod d, (83) for x/equals(x1,x2,... ,xN)∈{0,1,... ,d−1}N. (84) The oracle in the algorithm applies f(x)to the input string xand computes the result, namely, the number string g(a)encoded in the function f(x). The input state xis chosen to be |ψ0〉/equals|0〉⊗N|d−1〉, where |0〉⊗Nmeans initialization of the Ncontrol-qudits into their |0〉 states and |d−1〉means the target qudit is in its d−1 state. Quantum Fourier transforms of the pertinent input states are |0〉QFT→⎥summationdisplayd−1 y/equals0|y〉⎥radicaltpext⎥radicaltpext d√ (85) and |d−1〉QFT→⎥summationdisplayd−1 y/equals01⎥radicaltpext⎥radicaltpext d√ωd−y|y〉, for ωa root of unity Eq. 14 . The component-wize Fourier transform of a string encoded in the state |x1x2...xN〉is |x1x2...xN〉QFT→⎥summationdisplay z∈Kωx·z|z〉⎥radicaltpext⎥radicaltpext⎥radicaltpext dN√ , (86) where K/equals{0,1,... ,d−1}N,z:/equals(z1,z2,... ,zN). (87) We denote the Fourier transform of the |d−1〉state as |ϕ〉and the input state after the Fourier transform is |ψ1〉/equals⎥summationdisplay x∈K|x〉⎥radicaltpext⎥radicaltpext⎥radicaltpext dN√ |ϕ〉 (88) Now we introduce the oracle as the Of(x)gate such that |x〉⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉Of(x)→|x〉|⎥parenleftbigf(x)+j⎥parenrightbigmod d〉, (89) where f(x)/equalsg(a)·xmod d. (90) By applying the Of(x)gate to |ψ1〉and following the formula by phase kick-back, we obtain the output state Of(x)|ψ1〉/equals|ψ2〉/equals⎥summationdisplay x∈Kωf(x)|x〉⎥radicaltpext⎥radicaltpext⎥radicaltpext dN√ |ϕ〉. (91) Finally, obtain the |ψ3〉which is the state after inverse Fourier transform of the ﬁrstNqudits

Question 714 multiple-choice

Quantum algorithms have revolutionized the way circulant linear systems are solved by leveraging matrix structures and specialized transforms. Hybrid quantum-classical and quantum-inspired classical algorithms offer new strategies for achieving efficient and accurate solutions in scientific computing. Which key innovation allows the hybrid quantum-classical algorithm for circulant linear systems to estimate inner products efficiently, while only requiring one extra ancilla qubit? 1) Utilizing amplitude amplification for eigenvalue inversion 2) Applying Tikhonov regularization to stabilize solutions 3) Using unitary decomposition and the HHL algorithm 4) Diagonalizing permutation operators via the classical fast Fourier transform 5) Employing sample and query access for classical emulation 6) Performing the Hadamard test with a single ancilla qubit 7) Implementing cyclic permutations with multi-qubit entanglement

✓ Correct Answer:

The correct answer is 6) Performing the Hadamard test with a single ancilla qubit.

📚 Reference Text:

Given the unique properties of circulant matrices and the close relationship between circulant systems and the discrete Fourier transform (DFT), it is interesting to see what further speedups could be achieved by utilizing the quantum Fourier transform (QFT) [17] to solve circulant systems via quantum algorithms. Zhou and Wang [18] showed an efficient implementation of circulant matrices through unitary decomposition and QFT, allowing the problemt to be solved by HHL algorithm. Wan et al. [19] on the other hand, directly constructed the inverse of the circulant matrix by diagonalizing with QFT and employing amplitude amplification [20] to invert all eigenvalues. In this work, we propose an efficient algorithm for solving circulant linear systems by studying the properties of circulant matrices. Following Huang et al. [7], instead of explicitly diagonalizing the matrix with DFT, we first decompose the circulant matrix into a sum of cyclic permutation operators. These operators can then be diagonalized individually by DFT. Using QFT in our algorithm requires quantum hardware slightly beyond the capacities of current near-term systems [21]. Our theoretical results show that such a decomposition provides a much tighter upper bound ∗huangpowei22@u.nus.edu †lixiufan@u.nus.edu ‡cqtkjkk@nus.edu.sg §cqtfpr@nus.edu.sgarXiv:2309.11451v1 [quant-ph] 20 Sep 2023 2 for convergence guarantees, allowing an experimentally achievable guarantee without invoking gradient expansion Ansatz trees and other heuristical methods, or additional regularization such as Tikhonov regularized regression on the solution vector. This gives rise to a hybrid quantum-classical algorithm that consumes one additional ancilla qubit via Hadamard test [22] to achieve fast inner product estimation. Alternatively, noting that circulant matrices are simply weighted sums of cyclic permutation operators, we propose a quantum-inspired classical algorithm that uses sample and query access [8] to achieve the same performance as the hybrid algorithm. When cyclic permutation operators are applied to a vector, sample and query access to the original vector suffices to achieve the same access to the permuted vector as only a shift of indexes is required. The efficacy of our algorithms is demonstrated through numerical solutions1to one-dimensional time-dependent heat transfer problems discretized by finite difference methods [23, 24]. Through classical simulations, we show that both the hybrid algorithm and the quantum-inspired algorithm provide decent approximations to exact solutions obtained by matrix multiplications. Furthermore, we implement the hybrid algorithm on IBM quantum hardware with an error of less than 0.05 in terms of the mean-square-error (MSE) loss, showcasing the feasibility of solving physical problems on practical quantum computers. II. PRELIMINARIES Notations. Let [n] :={0,1, . . . , n −1}indicate the range of integers from 0 to n−1. Let [ a..b] be the range of integers from atob, inclusive of endpoints. Given a field Fof real or complex numbers, for vectors u, v∈FN, we denote its inner product as ⟨u, v⟩=u†v, where u†is the adjoint of u. For quantum states, we use the Dirac notation for inner product representation. We denote a vector’s ℓ2norm by ∥v∥:=qPN i=1|vi|2=⟨v, v⟩. LetMN(F) indicate the space of square matrices of size Non the field F. For a matrix A∈ M N(C), let

Question 715 multiple-choice

In quantum chemistry, the symmetry of atomic systems can be exploited to optimize wavefunctions for multi-electron atoms, improving computational efficiency and accuracy. For three-electron atoms, such as lithium and beryllium ions, variational approaches often use symmetry-adapted wavefunctions parameterized by phase angles. Which statement is true regarding the phase angle parameter (:E:) in optimizing wavefunctions for three-electron atoms? 1) The optimal range for :E: is always 0° to 360° due to the lack of orbital symmetry. 2) Wavefunctions at :E:, :E:+90°, and :E:+180° are equivalent with proper orbital interchange. 3) The optimized wavefunction is always labeled GF, regardless of the value of :E:. 4) Only a 120° range for :E: (typically -60° to +60°) needs to be considered because wavefunctions for :E:, :E:+120°, and :E:+240° are equivalent when orbitals are interchanged. 5) For transition states, the optimal value of :E: remains near zero. 6) SOGI wavefunctions cannot be connected to Gl or GF by varying :E:. 7) The electron density at the nucleus is always independent of the choice of :E:.

✓ Correct Answer:

The correct answer is 4) Only a 120° range for :E: (typically -60° to +60°) needs to be considered because wavefunctions for :E:, :E:+120°, and :E:+240° are equivalent when orbitals are interchanged..

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+aa,B( -!+! cos:E:-iVJ sinS:) +.Baa( -i-cos:E:) Jl. (29) In (27) we have that L(:E:+360°) = -L(:E:); however in (29) we see that :E:, :E:+120°, and :E:+240° lead to equivalent wavefunctions if we interchange the orbit als appropriately. Thus we need consider only a 120° range of :E:, which we will usually take as -60° to +60°. If :E:=O we have (30a) and by optimizing the orbitals we get the Gl wave function. If :a;= 60°, and by optimizing the orbitals we get the GF wave function [note that in (30b) the roles of the second and third orbitals are interchanged]. If :E:= -60° we obtain (?Lcf>acf>llbca.Ba = !GJ''cf>ccf>bcf>aaa,B ( 30c) and by optimizing the orbitals we again get the GF wavefunction (here the roles of the first and third orbitals have been interchanged). Thus by varying :E: we can go continuously from Gl to GF. We will find below that in most cases the optimum angle :E: is near zero and the Gl wavefunction is nearly as good as the SOGI wavefunction. However, for Ha near the transi tion states the optimum angle :E: is significantly differ ent from zero. Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jspQUANTUM THEORY OF MANY-ELECTRON SYSTEMS. v 1081 TABLE II. Properties for three-electron atoms (entries for GF, HF, UHF, CI, and experimental are taken directly from Ref. 19). G1 SOGI GF HF CI Exptl A. Energies z•sLi -7.44756038 -7.447 56516 -7.432813 --7.432725 -7.4779 -7.47807 32SLi -7.32517867 -7.32517894 -7.310216 -7.310210 -7.35410 22PLi -7.38011191 -7.38011631 -7.365091 -7.365069 -7.40838 -7.41016 32PLi -7.30819798 -7.30819844 -7.293189 -7.293186 -7.33715 22S Be+ -14.29162398 -14.29163664 -14.27762 22SB++ -23.38990151 -23.38991969 -23.37632 G1 SOGI GF HF UHF B. Density at the nucleus, (1: o(r;)) i 22S Li 13.8646 13.8646 13.8159 13.8160 13.8159 32SLi 13.7594 13.7594 13.7067 13.7080 13.7067 22PLi 13.7052 13.7052 13.6534 13.6534 13.6535 32PLi 13.7179 13.7180 13.6661 13.6660 13.6661 22S Be+ 35.1392 35.1392 35.111 22S B+ + 71.4976 71.4975 71.493 Gl SOGI GF HF UHF CI Exptl C. Spin density at the nucleus, Q(O) =2 (1: o(r;)S,(u;)) i 22SLi 0.20957 0.22654±0.0001 0.2406 0.1667 0.2248 0.2249 0.2313 32SLi 0.04873 0.0529 0.05622 0.03864 0.05253 22P Li 0.0 -0.0172 -0.02304 0.0 -0.01747 -0.02222 32PLi 0.0 -0.00547±0.0001 -0.007318 0.0 -0.005531 22S Be+ 0.94671 0. 9938±0.0015 1.008 22S B++ 2.4303 2.516±0.001 2.521 G1 SOGI GF HF UHF CI D. Orbital dipole constant, (1: [l,(r;) /r;3]) i 22PLi 0.058693 0.058725 0.05861 0.05848 0.05852 a 32PLi 0.0176467 0.017649 0.01760 0.01759 0.01760 G1 SOGI GF HF E. Second moment of r, (2:: rl·) i 22S Li 18.6642 18.6626 18.6090 18.6376 32SLi 119.571 119.569 119.447 119.480 22PLi 28.690 28.685 28.692 28.716 32PLi 172.203 172.198 172.511 172.536 2•s Be+ 6.559 6.559 6.545 22S B+ + 3.4136 3.4135 3.4094 • (1/r• )orb =0.05974, (1/r• )dip =0.05923. Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp1082 R. C. LADNER AND W. A. GODDARD III TABLE III. Optimum values of :S for three-electron atoms. Li 2'S Li 32S Li 22P Li 32P Be+

Question 716 multiple-choice

The Hidden Subgroup Problem (HSP) is a foundational challenge in quantum computing, with quantum algorithms leveraging the structure of coset states to extract information about hidden subgroups in a given group. The efficiency and feasibility of measurement strategies are critical factors in solving HSP for various types of groups. Which statement best characterizes why efficient quantum algorithms exist for the Hidden Subgroup Problem in abelian groups but are much harder to construct for general non-abelian groups? 1) Abelian groups have larger hidden subgroups, making measurements easier to implement. 2) Non-abelian groups lack periodic structure, preventing quantum advantage entirely. 3) Efficient entangled measurements on multiple coset states are always possible in abelian groups. 4) The quantum Fourier transform enables efficient single-state measurements in abelian groups, while efficient measurement strategies for general non-abelian groups are unknown. 5) In non-abelian groups, function values can always be exploited for extra information, unlike abelian groups. 6) Abelian groups require exponentially many coset states for classical information extraction. 7) Only non-abelian groups admit uniform superpositions over cosets, limiting abelian algorithms.

✓ Correct Answer:

The correct answer is 4) The quantum Fourier transform enables efficient single-state measurements in abelian groups, while efficient measurement strategies for general non-abelian groups are unknown..

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polynomial speedup [Gro96, BBBV97]. On the other hand, the quantum states resulting from HSP instances have far more structure since they captur e some periodicity aspects of the function f. Evaluating the function fin superposition and ignoring the function value results in a random coset state. Coset states are quantum states of the form |gH/a\}bracketri}ht=1√ |H|/summationtext h∈H|gh/a\}bracketri}ht, in other words, a coset state is a uniform superposition over the elements of the left cose tgH. The challenge in using coset states lies in the fact that gis a random element of the group, beyond our control, that is, we only have the mixed stateσG H=1 |G|/summationtext g∈G|gH/a\}bracketri}ht/a\}bracketle{tgH|and we have to determine Hfrom it. Though it is conceivable that some advantage can be had by making use of the function values, cur rently there are no proposals for using function values in any meaningful way. How much information can be extracted from coset states? The most general way to extract classical information fromquantumstatesarePOVMs[NC00]. AﬁxedPOV Moperatesonaﬁxednumber kofcoset states at once. This induces a probability distribution ove r the set of classical outcomes associated with the POVM. A potential source of power with no classical analog is that the distribution induced by a POVM onkcoset states may have signiﬁcantly more information than a P OVM that acts on just one coset state at a time. In other words, the resulting distribution when th e POVM is applied to kcoset states can be far from a product distribution. In this case we say that the POVM is an entangled measurement. The goal of this paper is to determine how small kcan be made such that a polynomial amount of information abou t Hcan be obtained from a POVM on kcoset states. More precisely, we want to know how small kcan be so that there exists a POVMon kcoset states that gives polynomially large total variation distance between every pair of candidate hidden subgroups. Note that this POV M can have many classical outcomes, and it may have to be repeated several times if we want to identify th e actual hidden subgroup Hwith constant probability. Inthispaper, weshowthatformanygroups Gthisnumber khastobequitelarge, sometimesaslargeas Ω(log|G|). Thismatchestheinformationtheoreticupperboundof O(log|G|)forgeneralgroups[EHK99b]. Our result can be viewed as a negative result because highly e ntangled measurements seem hard to imple- ment in general. Note that the time required to perform a gene ric measurement entangled across kstates increases exponentially with k. For abelian groups the picture simpliﬁes dramatically. Ind eed, in this case a POVM operating on one coset state (i.e., k= 1) exists that gives a polynomial amount of information about the hidden subgroup. Moreover, this measurement is efﬁciently implementable us ing the quantum Fourier transform over the group. The Fourier based approach extends to some non-abeli an groups as well, e.g., dihedral, afﬁne and Heisenberg groups, and shows that for these groups there are measurements on single coset states that give polynomially large information about the hidden subgroup [ EH00,MRRS04,RRS05]. Other than the general information-theoretic upper bound, only a few examples

Question 717 multiple-choice

Quantum hidden subgroup algorithms leverage group representation theory and quantum Fourier transforms to efficiently identify hidden structures within groups. Measurement strategies such as projective and POVM approaches play a critical role in the success probability of extracting parameters. In the context of quantum hidden subgroup algorithms involving the group Z_p^*, what is the probability n_ρ of observing the (p−1)-dimensional irreducible representation ρ after a measurement, and how does this probability behave as p becomes large? 1) n_ρ = 1/p, which decreases as p increases 2) n_ρ = p/(p−1), which increases without bound as p increases 3) n_ρ = 1/2, remaining constant regardless of p 4) n_ρ = (p+1)/p, approaching 1 as p increases 5) n_ρ = p−1, which grows linearly with p 6) n_ρ = p/(p+1), which approaches 1 as p increases 7) n_ρ = (p−1)/p, which approaches 1 as p increases

✓ Correct Answer:

The correct answer is 7) n_ρ = (p−1)/p, which approaches 1 as p increases.

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(2). The associated projection operator is πHb(ρ)j,k=1 p−1ωb(j−k) p for 1≤j, k < p . This is a circulant matrix of rank 1. More speciﬁcally, every column is some root of unity times the vector (ub)j=1 p−1ωbj p, Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 946 MOORE, ROCKMORE, RUSSELL, AND SCHULMAN 1≤j

Question 718 multiple-choice

Schur duality is a powerful tool in quantum information theory for decomposing quantum states into subspaces associated with partitions, and is relevant in the study of quantum algorithms and measurement strategies for problems like the hidden subgroup problem (HSP). Measurement techniques such as weak Schur sampling and weak Fourier sampling play a critical role in determining how much information about hidden structures can be extracted from quantum states. Which statement best describes the effectiveness of weak Schur sampling in identifying hidden subgroups in non-abelian groups given coset states? 1) Weak Schur sampling yields very little information about the hidden subgroup and is generally insufficient for subgroup identification. 2) Weak Schur sampling efficiently distinguishes all hidden subgroups in non-abelian groups. 3) Weak Schur sampling provides complete information about the underlying group structure. 4) Weak Schur sampling is as powerful as strong Fourier sampling for solving the hidden subgroup problem. 5) Weak Schur sampling requires no bounds on the number of coset states for successful subgroup identification. 6) Weak Schur sampling is only effective when combined with weak Fourier sampling. 7) Weak Schur sampling always outperforms entangled measurement strategies for HSP.

✓ Correct Answer:

The correct answer is 1) Weak Schur sampling yields very little information about the hidden subgroup and is generally insufficient for subgroup identification..

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Title: Weak Fourier-Schur Sampling, the Hidden Subgroup Problem, and the Quantum Collision Problem Year: 2006 Paper ID: a12cd06a2ae38a0267b0bcbbc9d004b7ea2b3d68 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a12cd06a2ae38a0267b0bcbbc9d004b7ea2b3d68 Abstract: Schur duality decomposes many copies of a quantum state into subspaces labeled by partitions, a decomposition with applications throughout quantum information theory. Here we consider applying Schur duality to the problem of distinguishing coset states in the standard approach to the hidden subgroup problem.We observe that simply measuring the partition (a procedure we call weak Schur sampling) provides very little information about the hidden subgroup. Furthermore, we show that under quite general assumptions, even a combination of weak Fourier sampling and weak Schur sampling fails to identify the hidden subgroup. We also prove tight bounds on how many coset states are required to solve the hidden subgroup problem by weak Schur sampling, and we relate this question to a quantum version of the collision problem.

Question 719 multiple-choice

Quantum algorithms for hidden subgroup problems have revolutionized computational complexity, especially in cryptography and group theory. Extending these algorithms from abelian to non-abelian groups introduces new challenges in both representation theory and algorithmic efficiency. Which factor is most critically linked to both the efficiency of the non-abelian Fourier transform and the overall success of generalizing Shor's algorithm to non-abelian groups? 1) The use of one-dimensional irreducible representations only 2) The choice of subgroup generators 3) The selection of an appropriate transversal 4) The order of the non-abelian group 5) The implementation of classical post-processing 6) The structure of the group’s center 7) The existence of normal subgroups

✓ Correct Answer:

The correct answer is 3) The selection of an appropriate transversal.

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produce an eﬃcient algorithm. On t he other hand, a bad choice will lead to an ineﬃcient algorithm. Unfortunately, Shor’s original algorithm g ives no hint as to how to choose the transversal τ:Q−→Ggenerically. This, of course, is only one of the diﬃculties encountere d in trying to generalize Shor’s algorithm to non-abelian groups. But it appears to be a crucial one that has, for the most part, been ignored. The signiﬁcance of the correct choice of a tra nsversal is made transparent by Lomonaco and Kauﬀman,11where the correct transversal for the abelian QHS algorithm is fou nd to be what is called in their paper a Shor transversal. Another obstacle to the development of new non-abelian QHS algorit hms is the time complexity of the non-abelian Fourier transform. 4. THE NON-ABELIAN FOURIER TRANSFORM The Fourier transform on non-abelian groups is deﬁned as follows: LetGbe a ﬁnite non-abelian group, and let π(1),π(2),...,π(k)be a complete set of distinct irreducible representations of the group G. Each irreducible representation is a morphism π(i):G−→Aut(Wi) fromGto the group of automophisms of the representation space Wi. Letπ(i)#:G−→Aut(W∗ i), denote the corresponding contragradient repre- sentation, where W∗ iis the dual Wi. Extend the representation π(i)#:G−→Aut(W∗ i) to the group ring, i.e., toπ(i)#:CG−→End(W∗ i), where CGdenotes the group ring of Gover the complex numbers C, and where End(W∗ i) denotes the ring of endomorphisms of W∗ i. Then the non-abelian Fourier transform FGon the group Gis deﬁned as FG=k/circleplusdisplay i=1π(i)#:CG−→k/circleplusdisplay i=1End(W∗ i) , where/circleplusdisplay denotes the direct sum. The above Fourier transform can be expressed more explicitly, but less transparently, in terms of matrices. The algorithmic time complexity of the non-abelian Fourier transform depends to a large extent on the basis chosen to express the Fourier transform in terms of matrice s. But this complexity will depend on which transversal τ:Q−→Gis chosen for the non-abelian QHS algorithm! 5. CONCLUSION In this paper, we conjecture that one of the main obstacles to cre ating new non-abelian QHS algorithms for the hidden subgroup problem f:G−→His the correct choice for the transversal τ:Q−→G. 6. ACKNOWLEDGEMENTS This work was supported by the Defense Advanced Research Proj ects Agency (DARPA) and Air Force Research Laboratory, Air Force Materiel Command, USAF, under agreemen t F30602-01-2-05022. Some of this eﬀort was also sponsored by the National Institute for Standards and T echnology (NIST). The U.S. Government is authorized to reproduce and distribute reprints for Governmen t purposes notwithstanding any copyright annotations thereon. The views and conclusions contained herein a re those of the authors and should not be interpreted as necessarily representing the oﬃcial policies or endo rsements, either expressed or implied, of the Defense Advanced Research Projects Agency, the Air Force Res earch Laboratory, or the U.S. Government. (Copyright 2004.) The ﬁrst author would also like to thank the Mathe matical Sciences Research Institute (MSRI) at Berkeley, California for its support of this work. REFERENCES 1. Ettinger, Mark and Peter Hoyer, On Quantum Algorithms for Noncommutative Hidden Subgroups , http://xxx.lanl.gov/abs/quant-ph/9807029 2. [Ettinger2]Ettinger, Mark, PeterHoyer,

Question 720 multiple-choice

Quantum networking leverages photonic systems to interconnect quantum devices, enabling secure communication and distributed quantum computation. Understanding the properties of photonic circuits and the mathematical frameworks behind quantum gate sets is fundamental for achieving universality and robust performance in these systems. Which statement accurately characterizes the relationship between linear optical unitaries, unitary t-designs, and the SNAP gate in photonic quantum circuits for dimension V ≥ 2? 1) Linear optical unitaries form a universal gate set and naturally constitute a 2-design in U. 2) The addition of a phase shifter to linear optical unitaries is sufficient to achieve universality in photonic quantum computing. 3) Linear optical unitaries form a 2-design but require error correction to achieve universality. 4) Unitary 2-designs are always closed and non-universal in the unitary group U for V ≥ 2. 5) Linear optical unitaries form a 1-design but not a 2-design, and augmenting with a SNAP gate achieves universality. 6) SNAP gates alone can generate all possible unitary transformations in photonic circuits without linear optics. 7) The combination of linear optical unitaries with measurement-only protocols guarantees 2-design properties in U.

✓ Correct Answer:

The correct answer is 5) Linear optical unitaries form a 1-design but not a 2-design, and augmenting with a SNAP gate achieves universality..

📚 Reference Text:

Advancing Quantum Networking: Some Tools and Protocols for Ideal and Noisy Photonic Systems Jason Saied,1,∗Jeffrey Marshall,1, 2Namit Anand,1, 3Shon Grabbe,1and Eleanor G. Rieffel1 1Quantum Artificial Intelligence Laboratory, NASA Ames Research Center, Moffett Field, CA 94035, USA 2USRA Research Institute for Advanced Computer Science, Mountain View, CA 94043, USA 3KBR, Inc., 601 Jefferson St., Houston, TX 77002, USA (Dated: March 6, 2024) Quantum networking at many scales will be critical to future quantum technologies and experiments on quan- tum systems. Photonic links enable quantum networking. They will connect co-located quantum processors to enable large-scale quantum computers, provide links between distant quantum computers to support distributed, delegated, and blind quantum computing, and will link distant nodes in space enabling new tests of fundamental physics. Here, we discuss recent work advancing photonic tools and protocols that support quantum networking. We provide analytical results and numerics for the effect of distinguishability errors on key photonic circuits; we considered a variety of error models and developed new metrics for benchmarking the quality of gener- ated photonic states. We review a distillation protocol by one of the authors that mitigates distinguishability errors. We also review recent results by a subset of the authors on the efficient simulation of photonic circuits via approximation by coherent states. We study some interactions between the theory of universal sets, unitary t-designs, and photonics: while many of the results we state in this direction may be known to experts, we aim to bring them to the attention of the broader quantum information science community and to phrase them in ways that are more familiar to this community. We prove, translating a result from representation theory, that there are no non-universal infinite closed 2-designs in U(V)when dimV≥2. As a consequence, we observe that linear optical unitaries form a 1-design but not a 2-design. Finally, we apply a result of Oszmaniec and Zimborás to prove that augmenting the linear optical unitaries with any nontrivial SNAP gate is sufficient to achieve universality. CONTENTS I. Introduction 2 II. Distinguishability and loss errors 3 A. Simulation of Loss 3 B. Simulation of Distinguishability 4 C. Distinguishability distillation 5 D. Impact of errors on notable circuits 6 1. Distinguishability error analysis for (generalized) fusion 6 2. Distinguishability error analysis for GHZ state generation 9 3. Numerical distinguishability error analysis for Bell state generation 10 III. Coherent-rank framework 12 IV . Unitary Designs, Universality, and Linear Optics 14 A. Unitary designs 14 B. The group of linear optical unitaries 15 C. SNAP gates 16 D. Result: Continuous 2-designs 16 E. Results on linear optical t-designs and universality 17 V . Discussion 17 ∗jason.saied@nasa.govarXiv:2403.02515v1 [quant-ph] 4 Mar 2024 2 Acknowledgments 18 References 18 A. Resource state generation 19 1. Proof sketches for Sec. II D 1 19 2. GHZ generation 20 a. Proof sketch for Lemma 3 20 b. Proof sketch for Theorem 4 20 3. Numerical calculations 21 B. Representation theory 22 1. Representation theory background 22 2. Representation theory and Proposition 3 22 3. Proof of Theorem

Question 721 multiple-choice

Highly entangled subspaces in tensor products of finite-dimensional Hilbert spaces play a vital role in quantum information theory, particularly in the study of quantum channels and entanglement detection. The representation theory of free orthogonal quantum groups provides tools for constructing such subspaces and analyzing their mathematical properties. Which property is established by using representation theory of free orthogonal quantum groups to construct highly entangled subspaces, leading to results about the minimum output entropy of associated quantum channels? 1) The subspaces maximize the trace distance between quantum states 2) The subspaces minimize the rank of the reduced density matrices 3) The subspaces admit a decomposition into separable pure states 4) The subspaces have minimal von Neumann entropy for all inputs 5) The subspaces yield channels with zero classical capacity 6) The subspaces enable lower bounds for minimum output entropy of quantum channels 7) The subspaces ensure that all output states are maximally mixed

✓ Correct Answer:

The correct answer is 6) The subspaces enable lower bounds for minimum output entropy of quantum channels.

📚 Reference Text:

Title: Highly Entangled, Non-random Subspaces of Tensor Products from Quantum Groups Year: 2016 Paper ID: 9c51784472298da15f9950b6ad25d288d1c4e85f Source: semantic-scholar URL: https://www.semanticscholar.org/paper/9c51784472298da15f9950b6ad25d288d1c4e85f Abstract: In this paper we describe a class of highly entangled subspaces of a tensor product of finite-dimensional Hilbert spaces arising from the representation theory of free orthogonal quantum groups. We determine their largest singular values and obtain lower bounds for the minimum output entropy of the corresponding quantum channels. An application to the construction of d-positive maps on matrix algebras is also presented.

Question 722 multiple-choice

Quantum sensing with superconducting qubits utilizes microwave pulse sequences and adaptive feedback to measure magnetic flux with high precision, often leveraging empirical calibration techniques. Nondestructive readout and advanced phase estimation algorithms are key components in optimizing measurement sensitivity beyond classical limits. Which of the following best explains how operating a transmon qubit away from the "sweet spot" affects its performance in quantum magnetometry? 1) It decreases the qubit’s transition frequency sensitivity to magnetic flux, improving coherence but reducing sensor responsiveness. 2) It causes the readout resonator frequency to become highly flux-dependent, complicating the measurement process. 3) It increases the qubit’s transition frequency sensitivity to magnetic flux, enhancing measurement range but introducing more flux noise. 4) It forces the use of destructive measurement techniques, limiting repeated sensing cycles. 5) It renders quantum phase estimation algorithms ineffective, reducing overall measurement precision. 6) It eliminates the need for empirical calibration, as theoretical models become perfectly accurate. 7) It prevents the creation of superposition states required for phase estimation.

✓ Correct Answer:

The correct answer is 3) It increases the qubit’s transition frequency sensitivity to magnetic flux, enhancing measurement range but introducing more flux noise..

📚 Reference Text:

sequence of two π/2 microwave pulses at a carrier frequency ωd=2π× 7.246 GHz separated by a time delay τ. The sequence drives the transmon into a superposition of ground and excited states where the state amplitudes depend on the accumulated phase ϕ=[ωd−ω01(Φ)]τ. The qubit state is read out nondestructively using a probe pulse sent to the CPW resonator; the re ﬂected signal is downconverted (not shown in the ﬁgure), digitized, and analyzed by a computer. Next, the computer updates a ﬂux distribution function PðΦÞstored in its memory, determines the next optimal Ramsey delay time, and feeds it back into the AWG. aQubit transition frequency ω01(Φ) as a function of magnetic ﬂuxΦ(parabolic curve). The bottom inset shows the CPW resonator ’s spectrum. The red circles indicate the bias point of our transmon sensor: we operate far away from the ‘sweet spot ’ in a regime where the transmon ’s frequency ω01(Φ) is an approximately linear function of the ﬂuxΦwithin the entire ﬂux range ΔΦ. For the ﬂuxes around the point considered here, the frequency ωrof the readout CPW resonator remains approximately constant. bA pre-measured sample-speci ﬁc Ramsey interference fringes pattern de ﬁnes the “passport ”function of our sensor. This can be regarded as a non-normalized probability function Pp(τ,Φ) to observe the qubit in the excited state after a Ramsey sequence with a delay τfor a speci ﬁc value of the magnetic ﬂuxΦ. The largest ﬂux value used to obtain the Ramsey interference fringes pattern Φ=0.1394 Φ0corresponds to a frequency detuning Δω=ωd−ω01(Φ)=2π× 15.8 MHz between the drive and the qubit transition frequencies. The ﬂux range of the “passport ”ΔΦ~2.5 × 10−3Φ0corresponds to a range 2 π× 13.8 MHz in frequency detuningQuantum-enhanced magnetometry by phase estimation algorithms S Danilin et al. 3 Published in partnership with The University of New South Wales npj Quantum Information (2018) 29 present in actual experiments. For brevity, we will still refer to these protocols as the Kitaev and Fourier phase estimationalgorithms. We demonstrate the superiority of these algorithms over the standard technique and show that we can beat the standard quantum limit. Instead of relying on the ideal theoreticalprobability function P[τ,Δω(Φ)] of Eq. ( 1) these modi ﬁed Kitaev and Fourier protocols exploit the empirical probability P p(τ,Φ), the so-called passport, which we measure by a set of Ramsey sequences at various magnetic ﬂuxes Φ, representing the result on a discrete equidistant grid in the form Pp(τj,Φi), see Fig. 1b. Here, Φi=(i−1)[δΦ]step+Φ1with the index ichosen from the ﬂux-index set I0=[1, 161] [ δΦ]step≃1.59 × 10−5Φ0,Φ1≃0.137 Φ0, and discrete time delays τj=(j−1) × 2 ns, j=1,…241 quantify- ing the time separation between the two π/2 rf-pulses of the Ramsey sequence. In order to increase the signal-to-noise ratio, weaverage over 65000 Ramsey experiments at each discrete point ( τ j, Φi). The resulting pattern is only approximately described by Eq. (1) due to the fact that the resonator frequency changes slightly with the applied ﬂux, thus modifying our calibration (see Methods and SI 2). In principle, one can change

Question 723 multiple-choice

Quantum error correction in systems constrained by continuous symmetries is a key area at the intersection of quantum information theory and quantum gravity. The compatibility between error-correcting codes and symmetry operations, such as those related to charge conservation, imposes fundamental limits on code performance. Which of the following expressions gives the lower bound on the worst-case residual error (ε_worst) for a quantum error-correcting code that is covariant with respect to a continuous symmetry U(1), when encoding information across n subsystems with individual charge operators? 1) ε_worst ≥ (ΔT_L) / [2 ΔT_total] 2) ε_worst ≥ (ΔT_L) / [n * min_i ΔT_i] 3) ε_worst ≥ (ΔT_L) / [max_i ΔT_i] 4) ε_worst ≥ (ΔT_L) / [2n * max_i ΔT_i] 5) ε_worst ≥ (ΔT_L) / [2n * min_i ΔT_i] 6) ε_worst ≥ (ΔT_L) / [n^2 * max_i ΔT_i] 7) ε_worst ≥ (ΔT_L) / [2n^2 * min_i ΔT_i]

✓ Correct Answer:

The correct answer is 4) ε_worst ≥ (ΔT_L) / [2n * max_i ΔT_i].

📚 Reference Text:

results in the context of quantum gravity and, in particular, the AdS/CFTcorrespondence. Time evolution itself provides an example of a symmetry that must be reconciled with the error- correcting properties of the system. The remainder of the manuscript is organized as follows (Fig. 2). In Sec. II, we summarize our main results in an intuitive fashion (this section may be skipped for readers who prefer more precise explanations). The notation is laid out in Sec. III, and in Sec. IVwe prove a bound on the performance of codes covariant with respect to a Uð1Þ symmetry. A criterion certifying code performance is derived in Sec. V. In Sec. VI, we apply our bounds andcriterion to the following examples of Uð1Þ-covariant encodings: an infinite-dimensional rotor extension of the qutrit ½½3;1;2/C138/C138and qubit ½½5;1;3/C138/C138codes as well as a many- body Dicke-state code. We apply our bound to codesadmitting universal transversal gates in Sec. VII, discussing aUðdÞ-invariant encoding based on Wstates in Sec. VII C . Erasure-correcting codes whose transversal gates form a general group Gare introduced in Sec. VIII. In Sec. IX,w e relate our work to recent observations about global sym-metries in quantum gravity. We conclude with a discussion in Sec. X. II. SUMMARY OF MAIN RESULTS This section presents a high-level overview of our main results using intuitive arguments. The reader may safely skip to Sec. IIIif they prefer a more precise exposition. A. Bound on the accuracy of codes covariant with respect to a continuous symmetry Our first main result is a bound on the accuracy of any approximate quantum error-correcting code that is covar- iant with respect to a continuous symmetry (Sec. IV). We consider an encoding map from a logical system Lto a physical system Aconsisting of nsubsystems denoted A1;A2;…;An. A one-parameter family of continuous unitary symmetries acting on Lis generated by the logical charge observable TL, which corresponds to the physical charge observable TAacting on A. We assume that the symmetry acts transversally, so that TA¼Pn i¼1Ti, where Tiacts on subsystem Ti. How well does this code protect the logical system against erasure of one of the subsystems? To quantify the code’s performance, we may use the worst-case entangle- ment fidelity , where “worst case ”means the minimum fidelity for any entangled state shared by the logical system and a reference system. (See Sec. IIIfor a precise definition.) Then we consider the value fworst of this worst-case entanglement fidelity which is achieved bythe best possible recovery map applied after an erasure error. A measure of the residual error after recovery is ϵ worst ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1−f2 worstq : ð1Þ Our result is a lower bound on ϵworst which limits the performance of any covariant quantum code: ϵworst≥ΔTL 2nmax iΔTi; ð2Þ where ΔTdenotes the difference between the maximum and minimum eigenvalue of T. That is, the code ’s accuracy is constrained by the range of charges one wishes to be able to encode, by the range of possible charge values that each subsystem can accommodate, and by the number ofPHILIPPE

Question 724 multiple-choice

In quantum computing, group actions and their associated structures play a foundational role in designing efficient algorithms for problems involving symmetry and hidden information. Concepts such as stabilizers, cosets, and quantum Fourier sampling are key tools in solving the Hidden Subgroup and Hidden Translation problems. Which statement accurately describes the algebraic structure formed by the set of group elements mapping one quantum state to another under a group action, provided this set is nonempty? 1) It always forms a normal subgroup of the group. 2) It is equivalent to the intersection of the stabilizers of the two states. 3) It is a union of disjoint left cosets. 4) It forms a cyclic subgroup generated by a single element. 5) It forms a left or right coset in the group, and the stabilizers of the two states are conjugate subgroups. 6) It is necessarily a singleton set containing only the identity element. 7) It consists of all elements commuting with the hidden translation.

✓ Correct Answer:

The correct answer is 5) It forms a left or right coset in the group, and the stabilizers of the two states are conjugate subgroups..

📚 Reference Text:

u i v a l e n t l y , |G·φ/angbracketright=1 √ |G|/summationtext x∈G|x·φ/angbracketright.T h e translating coset of two states |φ0/angbracketrightand|φ1/angbracketrightof Γ is the set {u∈G:|u·φ1/angbracketright= |φ0/angbracketright}. The translating coset of |φ0/angbracketrightand|φ1/angbracketrightis either empty or a left coset uG|φ1/angbracketright (or equivalently a right coset G|φ0/angbracketrightu)f o rs o m e u∈G. If the latter case occurs, |φ0/angbracketright and|φ1/angbracketrighthave conjugate stabilizers: G|φ0/angbracketright=uG|φ1/angbracketrightu−1.Translating Coset is a generalization of Stabilizer : Translating Coset (G,α,t) Input:tcopies of two quantum states |φ0/angbracketright,|φ1/angbracketright∈Γ. Output: •reject ifG(|φ0/angbracketright)∩G(|φ1/angbracketright)=∅; •u∈Gsuch that |u·φ1/angbracketright=|φ0/angbracketrightand O(log |G|) generators for G|φ1/angbracketright otherwise. For a function fonG, deﬁne the superposition |f/angbracketright=1 √ |G|/summationtext g∈G|g/angbracketright|f(g)/angbracketright,a n d forx∈G, deﬁne the function x·f:g/mapsto→f(gx). Let Γ( f)={|x·f/angbracketright:x∈G}.T h e n a group element xacts naturally on |f/prime/angbracketright∈Γ(f) by mapping it to the superposition |x·f/prime/angbracketright. We call this group action the translation action . The mapping |x/angbracketright|f/prime/angbracketright/mapsto→ |x/angbracketright|x·f/prime/angbracketrightis realized by right multiplying the ﬁrst register of |f/prime/angbracketrightbyx−1. Proposition 2.2. Suppose Gis a ﬁnite group and let t= poly(log |G|).T h e n Hidden Subgroup (G)(resp., Hidden Translation (G)) can be solved with a call HIDDEN TRANSLATION AND TRANSLATING COSET 9 toStabilizer (G,τ,t)(resp., Translating Coset (G,τ,t)), where τdenotes the translation action. Proof.L e tfbe an instance of Hidden Subgroup . Then the stabilizer of |f/angbracketrightis the group hidden by f.L e t(f0,f1) be an instance of Hidden Translation .T h e n the translating coset of |f0/angbracketrightand|f1/angbracketrightis a singleton whose element is the translation of (f0,f1). 3. Hidden translation in Zn p.Inthissection, weshowthat Hidden Transla- tion(G) can be solved in quantum polynomial time in the special case when G=Zn p for any ﬁxed prime number p>2. In this section we use the additive notation for the group operation, and x·y∈Zpstands for the standard inner product for x,y∈Zn p. SinceZn 2/multicloserightZ2is isomorphic to the abelian group Zn 2×Z2, one already has a quantum polynomialtimealgorithmfor Hidden Translation inZn 2by reducing it to Hidden Subgroup inZn+1 2by the method of [13]. For the convenience of the reader we present our method using intermediate mea- surements. However, the measurements can always be eliminated (see [2]), giving a unitary and therefore reversible algorithm, possibly with errors. The quantum part of our algorithm consists of performing FourierSampling over the abelian group Zn p×Z2. It turns out that from the samples we will only use elements of the form ( y,1). The important property of these elements yis that they arenotorthogonal to the hidden translation. Some properties of the distribution of the samples are stated for general abelian groups in the following lemma. Lemma 3.1. LetGbe a ﬁnite abelian group. Let f=(f0,f1),f:G×Z2→S be an instance of Hidden Translation (G)having a translation u/negationslash=0.T h e n FourierSamplingf(G×Z2)outputs an element in G×{1}with probability 1/2.M o r e - over, the probability of sampling the element (y,1)depends only on χy(u),a n di s0if and only if y∈u⊥. Proof. The state vector of FourierSamplingf(G×Z2) before the ﬁnal observa- tion is 1 2|G|/summationdisplay x∈G/summationdisplay y∈G/summationdisplay c=0,1χy(x)/parenleftbig 1+(−1)cχy(u)/parenrightbig |y/angbracketright|c/angbracketright|f0(x)/angbracketright. Therefore the probability of sampling ( y,1) is proportional to |1−χy(u)|2,

Question 725 multiple-choice

Quantum circuit partitioning is an essential strategy for managing the limitations of current quantum hardware, especially in distributed and sequential computation scenarios. Advanced mathematical frameworks and classical partitioning algorithms are adapted to optimize sub-circuit creation for scalable quantum computing. Which partitioning algorithm is specifically recognized for its efficiency in hypergraph partitioning and is commonly applied to optimize quantum circuit cuts involving multi-qubit interactions? 1) Stoer-Wagner algorithm 2) Kernighan-Lin algorithm 3) Spectral clustering algorithm 4) Fiduccia-Mattheyses algorithm 5) Louvain community detection 6) Maximum flow-minimum cut algorithm 7) Greedy coloring algorithm

✓ Correct Answer:

The correct answer is 4) Fiduccia-Mattheyses algorithm.

📚 Reference Text:

Title: Hypergraphic Partitioning for Spatial and Temporal Quantum Circuit Cutting Year: 2024 Paper ID: e6ae87637d82995fb0528cc806066c0e3e55ce4e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/e6ae87637d82995fb0528cc806066c0e3e55ce4e Abstract: Quantum computing holds immense promises for solving complex problems by leveraging quantum mechanics in a computational paradigm. However, challenges around the computational limitations of current noisy intermediate-scale quantum computers make it difficult to widely apply quantum advantages. These machines, with a limited number of qubits and high error rates, struggle to execute large quantum circuits, with higher dimensions of depth, size or width. To address those challenges, distributing circuits among processing agents or cutting circuits in small segments to be run in sequence are considered viable approaches to enhance scalability and computational power. The main goal for this paper is to present a hypergraphic representation approach for quantum circuit cutting, applicable to spatial and temporal scenario, using hypergraphic theory and heuristics. For each approach, the target is to reduce communication costs in the spatial scenario, and the initialization cost in the temporal scenario. To accomplish this, we apply heuristics for hypergraphic partitioning, considering the high level representation of the quantum circuit with different combinations of quantum gates. We compared results against different heuristics, such as Stoer-Wagner, Fiduccia-Mattheyses, and Kernighan-Lin, creating graphics based in a new circuit dimension called, coupling ratio for the circuit. The observed results from this method indicate the hypergraphic partitioning process as a critical step as part of a new reference architecture for quantum computers, in distributed quantum computing environments.

Question 726 multiple-choice

Quantum computing leverages quantum mechanical principles to achieve speed-ups over classical algorithms for specific tasks, such as factoring and signal processing. Quantum algorithms often utilize efficient quantum operations and error correction techniques to outperform their classical counterparts. Which statement correctly describes a key quantum advantage of the quantum Fourier transform (QFT) over the classical discrete Fourier transform (DFT) for processing signals of length N? 1) The QFT allows direct measurement of quantum state amplitudes with polynomial efficiency. 2) The QFT can be implemented in O(log^2 N) elementary operations, offering exponential speed-up compared to the classical DFT's O(N log N) time. 3) The QFT is inherently non-unitary, enabling more diverse transformations than the DFT. 4) The QFT eliminates the need for quantum error correction during execution. 5) The QFT is limited to applications in quantum communication and cannot be used for computational algorithms. 6) The QFT requires exponentially more quantum gates as N increases, unlike the DFT. 7) The QFT provides exact solutions to all hidden subgroup problems regardless of noise or decoherence.

✓ Correct Answer:

The correct answer is 2) The QFT can be implemented in O(log^2 N) elementary operations, offering exponential speed-up compared to the classical DFT's O(N log N) time..

📚 Reference Text:

1 Algorithms for Quantum Systems — Quantum Algorithms Th. Beth, M. Grassl, D. Janzing, M. Rötteler, P . Wocjan, and R. Zeier Institut für Algorithmen und Kognitive Systeme Universität KarlsruheGermany 1.1 Introduction Since the presentation of polynomial time quantum algorithms for discrete log and factoring[Sho94] it is generally accepted that quantum computers may—at least for some problems—outperform classical ones. But the ﬁeld of quantum information processing does not onlyhave implications for computational problems. Using quantum mechanical systems for infor- mation processing naturally leads to the problem of ﬁnding efﬁcient ways to control quantum mechanical systems. Here we address several algorithmic aspects of quantum information processing in differ- ent areas. First, the state of the art of quantum signal transforms which are at the core of a huge class of quantum algorithms, namely hidden subgroup problems, is presented. Second, we discuss aspects of quantum error-correcting codes, in particular an interesting view onstabilizer codes which relates them to simple interaction Hamiltonians. The efﬁcient imple-mentation of unitary operations by given Hamiltonians is investigated next. Finally, results on the simulation of one quantum mechanical system by another are discussed. 1.2 Fast Quantum Signal Transforms A basic task in classical signal processing is to ﬁnd fast algorithms which compute the matrix- vector-product of a given transformation with an arbitrary input vector. Suppose that the inputis a vector of length Nwith complex entries. A transformation is said to have a fastalgorithm if the number of arithmetic operations needed to compute the matrix vector product—i. e., the number of additions and multiplications—is bounded by O(Nlog cN), for some constant c. Amongst the most useful algorithms in computer science, physics, and engineering is the discrete Fourier transform DFT Nwhich is given by the unitary matrix FN:=1 √ N·/bracketleftbig ωi·j/bracketrightbig i,j=0,...,N −1, where ω=e2πi/Ndenotes a primitive N-th root of unity. The computational complexity of computing the product FN·xfor an input vector x∈CNisO(NlogN). In quantum computing the unitary transformation FNis used in the following way. Sup- pose that a system consisting of nqubits holds a normalized state vector |ψ/angbracketright=/summationtextN i=1xi|i/angbracketright∈ Quantum Information Processing, 2nd Edition. Edited by Thomas Beth and Gerd Leuchs Copyright © 2005 Wiley-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 3-527-40541-0 2 1 Algorithms for Quantum Systems — Quantum Algorithms CN,w h e r e N:= 2n. Note that here the information is encoded into the amplitudes of the basis states. To this state the unitary transformation FNcan be applied resulting in a state FN|ψ/angbracketright. Unlike the situation in classical signal processing the components of |ψ/angbracketrightandFN|ψ/angbracketright are not directly accessible; they merely can be ex tracted by (POVM) measurements. However, for feature detecting purposes like the location of spectral peaks this approach is well-suited. A quantum Fourier transform can be computed using O(log2N)elementary operations. This exponential speed-up compared to the classical complexity of the DFT , which surpris- ingly enough is obtained by a direct adaptation of the classic Cooley-Tukey algorithm to thequantum circuit model, is an essential indication for the power of quantum computing. Thisbecomes manifest in

Question 727 multiple-choice

In the representation theory of quantum affine algebras, combinatorial structures like quivers and Cartan matrices play a central role in classifying and analyzing finite-dimensional modules. Recent developments involve the use of Q-datum and generalized Coxeter elements to extend these frameworks beyond the simply-laced types. Which mathematical innovation provides a unified combinatorial formula for the inverse of the quantum Cartan matrix of a complex finite-dimensional simple Lie algebra, applicable to both simply-laced and non-simply-laced cases? 1) Twisted Auslander–Reiten quivers 2) Q-datum 3) Kirillov–Reshetikhin modules 4) Denominators of normalized R-matrices 5) Block decomposition 6) Simple module invariants $\Lambda(V,W)$ 7) Height functions on Dynkin diagrams

✓ Correct Answer:

The correct answer is 2) Q-datum.

📚 Reference Text:

Title: Q-data and Representation Theory of Untwisted Quantum Affine Algebras Year: 2020 Paper ID: a3654c1f481a689127185f1597c046a22d07c68f Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a3654c1f481a689127185f1597c046a22d07c68f Abstract: For a complex finite-dimensional simple Lie algebra g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {g}}$$\end{document}, we introduce the notion of Q-datum, which generalizes the notion of a Dynkin quiver with a height function from the viewpoint of Weyl group combinatorics. Using this notion, we develop a unified theory describing the twisted Auslander–Reiten quivers and the twisted adapted classes introduced in Oh and Suh (J Algebra 535(1):53–132, 2019) with an appropriate notion of the generalized Coxeter elements. As a consequence, we obtain a combinatorial formula expressing the inverse of the quantum Cartan matrix of g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {g}}$$\end{document}, which generalizes the result of Hernandez and Leclerc (J Reine Angew Math 701:77–126, 2015) in the simply-laced case. We also find several applications of our combinatorial theory of Q-data to the finite-dimensional representation theory of the untwisted quantum affine algebra of g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {g}}$$\end{document}. In particular, in terms of Q-data and the inverse of the quantum Cartan matrix, (i) we give an alternative description of the block decomposition results due to Chari and Moura (Int Math Res Not 5:257–298, 2005) and Kashiwara et al. (Block decomposition for quantum affine algebras by the associated simply-laced root system, 2020. arXiv:2003.03265), (ii) we present a unified (partially conjectural) formula of the denominators of the normalized R-matrices between all the Kirillov–Reshetikhin modules, and (iii) we compute the invariants Λ(V,W)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda (V,W)$$\end{document} and Λ∞(V,W)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda ^\infty (V, W)$$\end{document} introduced in Kashiwara et al. (Compos Math 156(5):1039–1077, 2020) for each pair of simple modules V and W.

Question 728 multiple-choice

In algebraic topology, spectral sequences are essential tools for calculating (co)homology groups, and their differentials play a crucial role in determining module structures. Morava K-theory is a generalized cohomology theory where computations frequently utilize spectral sequences, especially for finite group extensions. For a finite group G formed as a semidirect product G = H ⋊ C2, with H having even Morava K-theory, which statement correctly describes the behavior of differentials in the associated Serre spectral sequence for Morava K-theory? 1) All differentials past d2 vanish, so the spectral sequence collapses at E2. 2) The first nontrivial differential occurs at d5, and all higher differentials are zero. 3) Both d3 and d5 are nontrivial and control the extension problems in every case. 4) Differentials d3, d5,.., d_{2n+1} are all nonzero for every n. 5) There is exactly one nonzero differential in the spectral sequence, namely d_{2n+1}. 6) The spectral sequence does not contain any nonzero differentials for this group structure. 7) Every differential is determined by the action of the fundamental class in H.

✓ Correct Answer:

The correct answer is 5) There is exactly one nonzero differential in the spectral sequence, namely d_{2n+1}..

📚 Reference Text:

sees that d3is either trivial, or has image .w1w2/modvn. Ifd3is trivial, we are done by the assumption on E4and the fact that the Atiyah– Hirzebruch spectral sequence for the dihedral quotient produces a ﬁnite and even module, since Dis a “good” group. Otherwise, we have E4K M00H M0 where KandHare deﬁned as before as kernel and homology of d3and.w1w2/1d3 onK.n/.BN/, respectively. Here M0DMfw1w2gandM00DM=.w1w2/, where MDH.DIF2/(note thatw1w2is not a zero divisor in M). As next differential is d2nC11DvnQn, we need to calculate the Qn–homology of M0andM00. We claim H.MIQn/DF2x2 1;x2 2=.x2 1x2 2;x2nC1 2;x2nC1 2/ F2w2 2=.w2n 2/F2w2 2fw2n 2;g withDPn rD0x2nC12rC2C1 1w2r 22MI H.M00IQn/DF2w2f1;x1w2gF2x2 1;x2 2=.x2 1x2 2;x2nC1 2;x2nC1 2/; H.M0IQn/DF2w2 2fx2nC1 1;Qn.x1w2/;Qnw2g F2fx2i 1w2k 2;x2j 2w2l 2j1i;j<2n;0k;l<2n1g: Assuming this calculation, one ﬁnds that the E2nC1–page E2nC1K H.M00IQn/H H.M0IQn/ Algebraic & Geometric Topology , Volume 11 (2011) 508 Björn Schuster is still inﬁnite, and there must be further differentials. The only way to arrive at a ﬁnite E1–page is when x1w2andQnw2support differentials, and we obtain an E1–page concentrated in even degrees. It remains to prove the claim on the Qn–homologies. Recall Qn.xi/Dx2nC1 i; using Sq1w2Dw1w2one sees inductively Qnw2DnX rD0w2nC12rC2C1 1w2r 2: Since x1w1Dx2 1, replacingw1in this sum with x1yields a cycle forQn. Com- paring coefﬁcients one readily proves that the Qn–cycles are given by F2x2 1;x2 2;w2 2=..x1x2/2/f1;Qnw2gF2w2 2fg (note x2 1Dx2 1Qnw2andx2 2D0). Clearly the image of Qnin odd degrees is given by F2x2 1;x2 2;w2 2=..x1x2/2/fQnw2g. Furthermore, all classes x2kC2 iw2m 2with kC2m2nC1are also in the image; the ﬁrst claim follows. ForM00the calculation is much simpler: modulo w1w2, one has Qn.w2/DnX rD0w2nC12rC1C1 1w2r 2D0;Qn.x1w2/D0; giving H.M00IQn/DF2w2f1;x1w2gF2x2 1;x2 2=.x2 1x2 2;x2nC1 2;x2nC1 2/: ForM0, one could either do this directly, or, since we have calculated H.MIQn/, by means of the short exact sequence of Qn–modules 0!M0!M!M00!0and the associated long exact sequence(s) ! Hs.M0IQn/ !Hs.MIQn/ !Hs.M00IQn/ !HsCjQnj.M0IQn/!: We need to determine and. The formulas for the action of Qngive .x2k i/D.w2l 2/D0; .w2kC1 2/Dw2k 2Qnw2(note that Qnw2is not a boundary in M0), .x1w2k 2/Dx2nC1 1w2k 2; .x1w2kC1 2/DQn.x1w2/w2k 2DnX rD1x2nC12rC1C2 1w2rC2k 2: Algebraic & Geometric Topology , Volume 11 (2011) Morava K–theory of groups of order 32 509 Reduction modulo w1w2gives Ker./DF2fx2i 1w2k 2;x2j 2w2l 2j1i;j<2n;0k;l<2n1g: Splitting the long exact homology sequences into short exact sequences thus yields an additive isomorphism H.M0IQn/F2w2 2fx2nC1 1;Qn.x1w2/;Qnw2g F2fx2i 1w2k 2;x2j 2w2l 2j1i;j<2n;0k;l<2n1g: Corollary 2.4 Suppose in addition that all elements in E0; 4are restrictions of good elements of K.n/.BG/. Then K.n/.BG/is good. Proof This is a consequence of the following facts: (i) the x2 iare clearly represented by Euler classes of one-dimensional complex representations of G; (ii) there is an extension problem identifying w2as a polynomial in elements of E0; 4: this can be seen either by restriction to subgroups, or by appealing to the extension class in (ordinary) cohomology. Finally, we need one more fact from the wreath product calculation. Lemma 2.5 [2]LetGDHoC2where Hhas even Morava K–theory. The Serre spectral sequence of the extension (2) 1!HH!G!C2!1 has only one differential d2nC11(ie, is

Question 729 multiple-choice

In abelian group theory, properties like quasi-injectivity, divisibility, and the structure of endomorphism rings play a central role in classifying groups and understanding their homomorphisms. The behavior of fully invariant subgroups and automorphism extensions is particularly significant for $ p $-groups and torsion-free groups. Which of the following statements is true regarding torsion-free abelian groups and quasi-injectivity? 1) Every torsion-free abelian group is quasi-injective regardless of divisibility. 2) Only finite torsion-free abelian groups can be quasi-injective. 3) A torsion-free abelian group must be divisible to be quasi-injective. 4) Torsion-free abelian groups are always projective if they are quasi-injective. 5) Quasi-injectivity in torsion-free abelian groups implies they are totally projective. 6) All torsion-free abelian groups possess fully invariant subgroups that are quasi-injective. 7) The endomorphism ring of a torsion-free abelian group is always a division ring if the group is quasi-injective.

✓ Correct Answer:

The correct answer is 3) A torsion-free abelian group must be divisible to be quasi-injective..

📚 Reference Text:

height and let H be fully invariant in G. Then Ylom E(H, G) = Hom#(i7, H). https://doi.org/10.4153/CJM-1972-056-6 Published online by Cambridge University Press ABELIAN GROUPS 621 To get our main theorem we remark that it follows from results in [6] that if G has no elements of infinite height, B is basic in G and H is B-large, then Hom E(H, H) is simply the center of E(H). Thus by Proposition 4.1 each map in Hom E(H, G) is induced by multiplication by some £-adic integer, an operation defined on G as well. Finally, if G is a direct sum of cyclic ^-groups, every fully invariant subgroup of G is .3-large with B taken to be G itself. This proves THEOREM 4.2. Direct sums of cyclic p-groups are quasi-injective over their endomorphism rings. Free abelian groups are not quasi-injective. In fact if G is torsion-free and quasi-injective, then G must be divisible since the E-map H is an E(i7)-homomorphism. / is also an £-homomor- phism and so there exists an .E-endomorphism/' which extends/. If /* = f'\H, then /* is in E(H). To show /* is an £(i7)-homomorphism, let a £ E(H) and h G H. From [2, Corollaries 4.3 and 4.5] we can write a = U + ^ where II, ^ are in the automorphism group of H. From [2, Corollary 4.4], II and ^ can be extended to automorphisms 11' and ^' of G. Let a* = n/ + ^f r which belongs to E. Then This shows that H is quasi-injective over K(H). REFERENCES 1. L. Fuchs, Infinite abelian groups, Vol. 1 (Academic Press, New York, 1970). 2. P. Hill, Endomorphism rings generated by units, Trans. Amer. Math. Soc. 141 (1969), 99-105. 3. I. Kaplansky, Infinite abelian groups (University of Michigan Press, Ann Arbor, 1969). 4. J. Lambek, Lectures on rings and modules (Blaisdell Publishing Company, Waltham, 1966). 5. R. J. Nunke, Homology and direct sums of countable abelian groups, Math. Z. 101 (1967), 182-212. 6. J. D. Reid, On the endomorphism rings of abelian p-groups (to appear). 7. F. Richman and E. A. Walker, Modules over p.i.d.'s that are infective over their endomorphism rings (to appear in Proceedings of the Park City Ring Theory Conference). Texas Tech University, Lubbock, Texas', Wesley an University, Middletown, Connecticut https://doi.org/10.4153/CJM-1972-056-6 Published online by Cambridge University Press

Question 730 multiple-choice

In the classification of finite p-groups, algebraic invariants such as group presentations and matrix forms are used to distinguish non-isomorphic group structures, particularly for groups of small order. Techniques involving the Frattini subgroup and congruence of associated matrices over finite fields are central to this process. When classifying finite p-groups constructed with generators and commutator relations, which condition guarantees that two groups G and Ḡ are isomorphic if and only if there exists an invertible matrix Y over the finite field Fp such that w(Ḡ) = Y w Yᵗ? 1) The orders of G and Ḡ are equal and both groups are abelian 2) The parameters n and m satisfy n ≠ m and p is prime 3) The generators of G and Ḡ have the same orders but different commutator relations 4) The Frattini subgroup Φ(G′) is trivial, i.e., Φ(G′) = 1 5) Both groups have nilpotency class greater than two 6) The derived subgroups of G and Ḡ are cyclic of maximal order 7) The group presentations involve a quadratic non-residue parameter ν

✓ Correct Answer:

The correct answer is 4) The Frattini subgroup Φ(G′) is trivial, i.e., Φ(G′) = 1.

📚 Reference Text:

(T7)/an}bracketle{ta,b,c,d/vextendsingle/vextendsinglea2n=b2=c4=d2= 1,[a,b] =c,[c,a] =d,[c,b] =c2,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3; (U1)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpm+1= 1,[a,b] =c,[c,a] =apn,[c,b] =c−p=bspm/an}bracketri}ht, where n > m≥2,s∈F∗ p; (U2)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpm+1= 1,[a,b] =c,[c,a] =bνpm,[c,b] =c−p=aspn/an}bracketri}ht, where n > m≥2,s∈F∗ p,ν= 1or a ﬁxed quadratic non-residue modular p; (U3)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn+1=bpm=dp= 1,[a,b] =c,[c,b] =c−p=aspn,[c,a] = d,[d,a] = [d,b] = 1/an}bracketri}ht, wheren > m≥2,s∈F∗ p,ν= 1or a ﬁxed quadratic non-residue modular p; (U4)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn=bpm+1=cp2= 1,[a,b] =c,[c,a] =bνpm,[c,b] =c−p/an}bracketri}ht, where n > m≥2,ν= 1or a ﬁxed quadratic non-residue modular p; (U5)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn=bpm+1=dp= 1,[a,b] =c,[c,a] =d,[c,b] =c−p=bpm,[d,a] = [d,b] = 1/an}bracketri}ht, wheren > m≥2; (U6)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpm=cp2= 1,[a,b] =c,[c,a] =apn,[c,b] =c−p/an}bracketri}ht, where n > m≥2; (U7)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn=bpm=cp2=dp= 1,[a,b] =c,[c,a] =d,[c,b] =c−p,[d,a] = [d,b] = 1/an}bracketri}ht, wheren > m≥2; (V1)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpm+1= 1,[a,b] =c,[c,a] =cp=bspm,[c,b] =a−νpncνtp/an}bracketri}ht, wheren > m≥2,ν= 1or a ﬁxed quadratic non-residue modular p,s∈F∗ p, t= 0,1,...,p−1 2; (V2)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpm=cp2= 1,[a,b] =c,[c,a] =cp,[c,b] =a−νpnctνp/an}bracketri}ht, wheren > m≥2,ν= 1or a ﬁxed quadratic non-residue modular p,t= 0,1,...,p−1 2; (V3)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpm+1= 1,[a,b] =c,[c,a] =cp=apn,[c,b] =bspm/an}bracketri}ht, where n > m≥2,s=F∗ p; (V4)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn+1=bpm=dp= 1,[a,b] =c,[c,a] =cp=apn,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht, wheren > m≥2; (V5)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn=bpm+1=cp2= 1,[a,b] =c,[c,a] =cp,[c,b] =bpmcsp,[cp,b] = 1/an}bracketri}ht, wheren > m≥2,s∈Fp; (V6)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn=bpm+1=dp= 1,[a,b] =c,[c,a] =cp=bspm,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht, wheren > m≥2,s∈F∗ p; (V7)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn=bpm=cp2=dp= 1,[a,b] =c,[c,a] =cp,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht, wheren > m≥2. Proof Case 1. n=m. In this case, m≥2 forp= 2. Ifp=n=m= 2, then |G|= 27. By checking the list of groups of order 27, we get the groups of Type (N1)–(N16). If p= 3 and n=m= 1, 32 then|G|= 35. By checking the list of groups of order 35, we get the groups of Type (O1)–(O7). In the following, we may assume that n >1 forp= 3 and n >2 forp= 2. Subcase 1.1 Φ(G′) = 1. In this subcase, v(G) = (0,0)t. Suppose that Gand¯Gare two groups described in the theorem. By Theorem 6.2 and 6.4, G∼=¯Gif and only if there exists Y=/parenleftbiggy11y12 y21y22/parenrightbigg , an invertible matrix over Fp, such that w(¯G) =Yw(G)Yt. That is, w(¯G) andw(G) are mutually congruent. By Lemma 2.4, 2.5 and 2.6, we get the groups of Type (P1)–(P10). Subcase 1.2. Φ(G′)/ne}ationslash= 1. Ifn=m= 1 and p >3, then [a,bp] = [a,b]p= 1 and hence Φ( G′) = 1. Thus, in this subcase, we have n≥2. Since v(G) =/parenleftbiggw31 w32/parenrightbigg /ne}ationslash=/parenleftbigg0 0/parenrightbigg , there exist w41,w42such that/parenleftbiggw41w31 w42w32/parenrightbigg is invertible. Let Y=/parenleftbiggw41w31 w42w32/parenrightbigg−1 . ThenYv(G) =/parenleftbigg0 1/parenrightbigg . In the following, we assume that Gand¯Gbe two groups described in the theorem with v(G) =v(¯G) =/parenleftbigg0 1/parenrightbigg . By Theorem 6.2 and 6.4, G∼=¯Gif and only if there exists Y=/parenleftbiggy110 y211/parenrightbigg , an invertible matrix over Fp, such that w(¯G) =Yw(G)Yt. By suitably choosing y21, we can simplify w(G) to be one of the following types: (a)/parenleftbiggw11w12 0w22/parenrightbigg wherew11/ne}ationslash= 0, (b)/parenleftbigg0w12 −w12w22/parenrightbigg wherew12/ne}ationslash= 0, (c)/parenleftbigg0w12 w210/parenrightbigg wherew12/ne}ationslash= 0 and w21/ne}ationslash=−w12, (d)/parenleftbigg0 0 w210/parenrightbigg wherew21/ne}ationslash= 0 and (e)/parenleftbigg0 0 0w22/parenrightbigg . In the following, we assume that both w(G) andw(¯G) are such matrices. It is easy to check that (i) diﬀerent types give non-isomorphic groups, (ii)G∼=¯Gif and only if there exists y11∈F∗ psuch that w(¯G) =Yw(G)YtwhereY=

Question 731 multiple-choice

Isogenies between elliptic curves are central to certain cryptographic protocols, with computational hardness based on arithmetic properties of curves and their endomorphism rings. Reductions between group action problems and isogeny evaluation play a key role in analyzing security and efficient computation. Which of the following best describes the relationship between stabilizer subgroups and Borel subgroups in the context of group actions on algebraic structures? 1) Stabilizer subgroups are always equal to the subgroup of diagonal matrices. 2) Stabilizer subgroups are conjugates of the subgroup of upper triangular matrices, known as Borel subgroups. 3) Stabilizer subgroups are intersection points of all simple normal subgroups. 4) Stabilizer subgroups only exist for finite groups, while Borel subgroups require infinite groups. 5) Stabilizer subgroups correspond to the set of lower triangular matrices in all cases. 6) Stabilizer subgroups and Borel subgroups are unrelated in algebraic group theory. 7) Stabilizer subgroups form the center of the group in algebraic contexts.

✓ Correct Answer:

The correct answer is 2) Stabilizer subgroups are conjugates of the subgroup of upper triangular matrices, known as Borel subgroups..

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of the codomain of φ(which is isomorphic to the right order of Iφ). Proposition 4.6. The stabilizer subgroups are conjugates of the subgroup of upper triangular matrices (i.e., a Borel subgroup). Proof.Follows from [ Voi18, 23.1.3]. For an elementary proof see Appendix B. ⊓ ⊔ 4.2 The main reductions. In this section, we prove a quantum polynomial-time equivalence betw een the the Group Action Evaluation Problem and the IsERP. Theorem 4.7. The Group Action Evaluation Problem reduces to the IsERP in classical polynomial-time. Proof.Assume we have an eﬃcient algorithm to solve the IsERP. Let us take an instance of the Group Action Evaluation Problem. So we have N,E,End(E), a representation for φand a matrix M, and we want to compute a representation forM∗φ. The ﬁrst step of the reduction is to compute the ideal Iφassociated to φ. There are several ways to do that, but to keep this proof short, we will use some of the results proven in [ Ler22a]. Thus, our ﬁrst step is to build a suborder rep- resentation for the isogeny φas in [Ler22a]. The suborder representation is made of endomorphisms of Z+NEnd(E)֒→End(E′) of powersmooth norm. Since we know End( E) and End( E′), the algorithms of the Deuring correspondence can be used to compute their kernels in polynomial time (their norm be ing pow- ersmooth implies that their kernels are deﬁned overa small extensio n). Then, we can compute the suborder representation using V´ elu’s formulas. Once we have the suborder representation, we can apply the equivalence betwe en the SOIP and the SOERP [ Ler22a, Proposition 13] to ﬁnd the ideal Iφ. OnceIφhas been computed, we need to compute the ideal IM⋆φ. For that we are going to use a Title Suppressed Due to Excessive Length 19 σsuch thatMσ=M. We can build such a σin polynomial time from End( E) using Proposition 4.1. Onceagood σisknown,wegetthe ideal IM⋆φasσ(Iφ∩Oσ)σ−1+NO(where we takeO∼=End(E)). Since the ideal IMσ⋆φis a valid isogeny representation, this proves the result. ⊓ ⊔ Theorem 4.8. The IsERP reduces to the Group Action Evaluation Problem in quantum polynomial time. Proof.Assume we can solve the Group Action Evaluation Problem. Let us take an input of the IsERP, we have a curve E, its endomorphism ring End(E), an integer Nand the isogeny representation associated to an isogeny φ of degreeN. The algorithm to solve the Group Action Evaluation Problem allows us to compute eﬃciently the groupactionintroducedin Section 4.1.Using P roposition 4.6 and Theorem 3.4 one can compute the stabilizer subgroup associa ted toφ. As the stabilizer subgroups of φgive us matrices corresponding to some σin the Eichlerorder Z+Iφ, we cancompute the embedding ofthis orderin O∼=End(E) in polynomial time using the algorithm of Proposition 4.1. Then, we can e xtract the idealIφand compute OR(Iφ) which is isomorphic to End( E′) and this gives the result. ⊓ ⊔ When the degree Nis smooth, we can modify the proof of Theorem 4.8 to get a classical reduction by using Theorem 3.3 instead of Theorem 3.4 . Theorem 4.9. Suppose that

Question 732 multiple-choice

In computational group theory and complexity theory, algorithms for constructing generator sets and solving group-related problems are closely linked to specific complexity classes. The relationship between the Graph Isomorphism problem and broader frameworks such as the hidden subgroup problem is of particular interest for both classical and quantum algorithm design. Which of the following statements is true regarding the computational complexity and group-theoretic relationships among Graph Isomorphism (GI), Automorphism (AUTO), and the hidden subgroup problem (HSP)? 1) The Graph Isomorphism problem is currently known to be NP-complete and cannot be formulated as an instance of the hidden subgroup problem. 2) AUTO is polynomial-time equivalent to GI, and since AUTO is an instance of FIND-GROUP, GI is contained within SPP and Mod kP for every k ≥ 2. 3) There is no known relationship between the complexity classes SPP and Mod kP for group-theoretic algorithms. 4) Efficient quantum algorithms for the hidden subgroup problem have no implications for integer factorization. 5) The FPSPP and UPSVSPP algorithms are fundamentally distinct and not computationally equivalent for FIND-GROUP. 6) Strong generator sets for groups cannot be constructed using coset representatives or prefix search techniques. 7) Problems like Graph Isomorphism cannot be represented as instances of the hidden subgroup problem.

✓ Correct Answer:

The correct answer is 2) AUTO is polynomial-time equivalent to GI, and since AUTO is an instance of FIND-GROUP, GI is contained within SPP and Mod kP for every k ≥ 2..

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an element in G(i−1)that mapsitoj. We will ﬁnd it by a preﬁx search that extends the partial permutation π/prime } fork:=i+ 1tondo ﬁnd the element lnot in the range of π/primesuch that /angbracketleftx,0n,Di,i,j,π/prime[k:=l]/angbracketright ∈L; π/prime:=π/prime[k:=l]; end {At this point π/primewill be a permutation in Sn} Ci−1:=Ci−1∪ {π/prime}; end end {At this point Ci−1is a complete set of coset representatives of G(i) inG(i−1)} Di−1=Di∪Ci−1 end Result :D0 Algorithm 2 : FPLalgorithm CONSTRUCT( /angbracketleftx,0n/angbracketright) We claim that a call to the FPLalgorithm CONSTRUCT( /angbracketleftx,0n/angbracketright) outputs a strong generator set D0for the group G=Gx. We show this by induction. Initially,Dn−1= 1 clearly generates G(n−1)=1. Suppose at the beginning of theithiteration it holds that Diis a strong generator set for G(i). It suﬃces to show that at the end of the ithiterationDi−1=Di∪Ci−1is a strong generator set forG(i−1). For eachj:i+1≤j≤n, the query /angbracketleftx,0n,Di,i,j,π/prime/angbracketright ∈Lchecks if there is an element in G(i−1)that mapsitoj. The subsequent preﬁx search with queries to Lcomputes the lexicographically least element in G(i−1)that mapsitoj. Furthermore, by Claim 4.2, as DigeneratesG(i), all queries made toLare UP-like. Thus, at the end of the ithiterationCi−1is a complete set of coset representatives for G(i)inG(i−1)and henceDi−1is a strong generator set forG(i−1). Thus at the end D0is a a strong generator set for G. Therefore, 9 we have an FPLalgorithm problem for FIND-GROUP . Finally, since the FPLalgorithm makes only UP-like queries to the NP oracle L, it follows from Lemma 2.3 that FIND-GROUP has an FPSPPalgorithm. Remark. We note that there is alternative way to conceive of an FPSPPal- gorithm for the FIND-GROUP problem: we can ﬁrst design an UPSVSPPalgo- rithm, where the preﬁx search that we do in CONSTRUCT( /angbracketleftx,0n/angbracketright) is replaced by directly guessing a permutation in the right coset (consisting of elements that ﬁx 1 to i−1 and map itoj) and rejecting along all paths on which we do not guess the lexicographically least element of the coset. Then, by a general preﬁx search argument we can see that FPSPPand UPSVSPPare the same and hence conclude that FIND-GROUP is in FPSPP. As we already noted, GI and AUTO are polynomial-time equivalent and AUTO , being an instance of FIND-GROUP has an FPSPPalgorithm by Theo- rem 4.1. Since SPPSPP= SPP and SPP ⊆Mod kP for each k≥2, the next corollary is an immediate consequence. Corollary 4.3 Graph Isomorphism is in SPP and hence in Mod kPfor every k≥2. 5 Hidden subgroup problem and other applica- tions We recall the general deﬁnition of the hidden subgroup problem. Deﬁnition 5.1 The hidden subgroup problem HSP has an input instance a ﬁnite group G(presented by a ﬁnite generator set) and we are given (in the form of an oracle) a function ffromGto some ﬁnite set Xsuch thatfis constant and distinct on diﬀerent right cosets of a hidden subgroup HofG. The problem is to determine a generator set for H. Many natural problems like Graph Isomorphism, integer factorization etc, can be cast as a special case of HSP. An eﬃcient quantum algorithm for the general problem will result in

Question 733 multiple-choice

Quantum computing leverages principles of quantum mechanics and advanced mathematics to solve certain computational problems much more efficiently than classical computers. A central concept in the design of quantum algorithms is the hidden subgroup problem, which has broad implications for cryptography and computational complexity. Which of the following statements best explains why quantum algorithms can efficiently solve the abelian hidden subgroup problem, whereas classical algorithms struggle? 1) Classical algorithms lack the ability to perform arithmetic in finite groups. 2) Quantum algorithms exploit quantum parallelism and interference, enabling efficient extraction of subgroup structure via the Fourier transform on finite abelian groups. 3) Quantum algorithms avoid the need for any mathematical analysis in their operation. 4) Classical computers cannot represent group elements in memory. 5) Quantum measurement always reveals detailed information about each individual input. 6) Quantum computers require exponentially more resources for pattern recognition than classical computers. 7) Classical algorithms can efficiently solve all hidden subgroup problems by brute force.

✓ Correct Answer:

The correct answer is 2) Quantum algorithms exploit quantum parallelism and interference, enabling efficient extraction of subgroup structure via the Fourier transform on finite abelian groups..

📚 Reference Text:

arXiv:quant-ph/0012084v1 17 Dec 2000Quantum factoring, discrete logarithms and the hidden subgroup problem Richard Jozsa Department of Computer Science, University of Bristol, Woodland Road, Bristol BS8 1UB U.K. Abstract Amongst the most remarkable successes of quantum computati on are Shor’s eﬃcient quantum algorithms for the computational tasks of integer f actorisation and the evalu- ation of discrete logarithms. In this article we review the e ssential ingredients of these algorithms and draw out the unifying generalization of the s o-called abelian hidden sub- group problem. This involves an unexpectedly harmonious al ignment of the formalism of quantum physics with the elegant mathematical theory of g roup representations and fourier transforms on ﬁnite groups. Finally we consider the non-abelian hidden subgroup problem mentioning some open questions where future quantu m algorithms may be ex- pected to have a substantial impact. 1 Introduction Quantum algorithms exploit quantum physical eﬀects to prov ide new modes of computation which are not available to “conventional” (classical) comp uters. In some cases these modes provide eﬃcient (i.e. polynomial time) algorithms for comp utational tasks where no eﬃcient classical algorithm is known. The most celebrated quantum a lgorithm to date is Shor’s algorithm for integer factorisation [7, 8, 10]. It provides a method for factoring any integer of ndigits in time (i.e. in a number of computational steps) that grows less rapidly than O(n3). Thus it is a polynomial time algorithm in contrast to the best known classical algorithm for this fundamental problem, which runs in superpolynomial ti me of order exp( n1 3(logn)2 3). At the heart of the quantum factoring algorithm is the discre te Fourier transform and the remarkable ability of a quantum computer to eﬃciently deter mine periodicities. This in turn rests on the mathematical formalism of fast Fourier transfo rms combined with principles of quantum physics. In this article we will review these issues including further applications such as the evaluation of discrete logarithms. We will outli ne a unifying generalization of these ideas: the so-called hidden subgroup problem which is just a natural group theoretic generalization of the problem of periodicity determinatio n. Finally we will consider some interesting open questions related to the hidden subgroup p roblem for non-abelian groups, where future quantum algorithms may be expected to have a sub stantial impact. We may think of periodicity determination as a particular ki nd of pattern recognition. Quantum computers are able to store and process large volume s of information, represented compactly in the identity of an entangled quantum state, but quantum measurement theory severely restricts our access to the information. Indeed on ly a relatively small amount of the information may be read out but this may be of a “global” na ture, such as a few broad features of a large intricate pattern, which may be impossib le to extract eﬃciently by classical means. This intuition is exempliﬁed in the earliest quantum algorithm, known as Deutsch’s algorithm [10]. Here we are given a black box that computes a B

Question 734 multiple-choice

In the theory of Hopf algebras and quantum groups, structures such as centers, q-characters, and module actions play crucial roles in understanding invariants and symmetries. The Drinfeld double construction and associated maps provide deep connections between dual spaces, representation theory, and quantum algebraic invariants. Which statement correctly describes the function of the map f: D∗ → D in the context of the Drinfeld double D and its invariants? 1) It sends group-like elements to primitive elements without regard to module structure. 2) It maps the center Z(D) isomorphically onto the space of q-characters C(D). 3) It acts as an automorphism interchanging left and right module actions. 4) It is an isomorphism of D-modules with respect to adjoint actions and maps q-characters C(D) isomorphically onto the center Z(D). 5) It identifies integrals with canonical elements related to the square of the antipode. 6) It intertwines R-matrices with M-matrices in a way that solves the Yang-Baxter equation. 7) It defines the freeness of Hopf modules in terms of Radford's relations.

✓ Correct Answer:

The correct answer is 4) It is an isomorphism of D-modules with respect to adjoint actions and maps q-characters C(D) isomorphically onto the center Z(D)..

📚 Reference Text:

for the left action of DonD∗and ⊳for the corresponding right action. We also use the adjoint actions ofDon itself given byad(a)(y) =a′yS(a′′) and on D∗given by ad∗(a)(λ) =a′′⊲λ⊳S(a′). The invariance in Dunder the adjoint action is precisely the centerZ(D) and the invariance in D∗are the q-characters C(D) ={λ∈D∗:λ(xy) =λ(S2(y)x)}, which were introduced in [Dr1]. In [Dr1] it is shown that these two spaces are related to each other by the map f:D∗→D:λ→λ⊗1(M). (2.17) HereM∈D⊗2is the element M=τ(R)R=/summationtext ijfjei⊗ejfi=/summationtext kmk⊗nk. In the case of a double D=D(A){mk}and{nk}are diﬀerents basis of D(A) so that Mis nondegenerate. The following is a slightly extended version of a lemma in [Dr1]. Lemma 2 1. The map f:D(A)∗/tildewidest−→D(A)is an isomorphism of D(A)-modules with respect to the adjoint actions. 2.f:C(D(A))/tildewidest−→Z(D(A))is an isomorphism of algebras. 3.f:Ch(D(A))/tildewidest−→G(D(A))is the group isomorphism (g,γ)/ma√sto→(γ,g) 11 4. We have f∗◦S∗=S−1◦f Proof: The fact that fintertwines the actions of D(A) follows from basic Hopf algebra re- lations, (2.14) and the identity ( S(y′)⊗1)M(y′′⊗1) = (S(y′)⊗1)M(y′′⊗(y′′′S(y′′′′))) = (S(y′)⊗1)(y′′⊗y′′′)M(1⊗S(y′′′′)) = (1⊗y′)M(1⊗S(y′′)). It is clear that fis an isomorphism. In particular we can write it as the composition: D(A)∗(.⊲⊳)∗ /tildewidest−→(A∗⊗A)∗=A⊗A∗. /tildewidest−→D(A) Clearly, the invariances are mapped isomorphically to each other and a computation in [Dr1] shows that this is a homomorphism. In fact we have f(χλ) =f(χ)f(λ) for χ∈C(D(A)) and any λ∈D(A)∗.3.)follows from Lemma 1 and the form of M. Finally, (2.13) implies S⊗S(M) =τ(M) and thereby 4.). ✷ It follows from (2.12) that the Rmatrices satisfy the Yang Baxter equation R12R13R23 =R23R13R12. For later computations of modular relation we derive here an analog ous equation for the Mmatrices. Lemma 3 ForMand bases{ei},{fi},{nk},{mk}as above we have (τ(M)⊗1)(1⊗M) =/summationdisplay kjn′ kfj⊗mk⊗S−1(e′′ j)n′′ ke′ j or equivalently 1⊗λ⊗1/parenleftBig (τ(M)⊗1)(1⊗M)/parenrightBig =/summationdisplay ijf(λ)′fifj⊗S−1(ej)f(λ)′′ei. Proof:If we multiply Rmatrices from the left and right to the Yang Baxter equation and permute the ﬁrst and third factor we obtain ( R−1)31(R−1)32R21R31=R21(R−1)32. Applying 1⊗1⊗S−1to this equation and using (2.13) we ﬁnd /summationdisplay ijfi⊗eifj⊗ej=/summationdisplay i,j,k,lfifjfk⊗flej⊗S−1(ek)elei Multiplication with R⊗1 from the left and 1 ⊗Rfrom the right yields (τ(M)⊗1)(1⊗M) =/summationdisplay tsijkletfifjfk⊗ftflejes⊗S−1(ek)eleifs by(2.12)=/summationdisplay ijkle′ lfif′′ jfk⊗flej⊗S−1(ek)e′′ leif′ j by(2.14)=/summationdisplay ijkle′ lf′ jfifk⊗flej⊗S−1(ek)e′′ lf′′ jei. 12 The formulas follow now from ∆( n(lj)) =e′ lf′ j⊗e′′ lf′′ jand again (2.12). ✷ Let us also record here the canonical elements from [Dr1] and [Ly1] implementing the square of the antipode. They are deﬁned by u:=/summationdisplay iS(fi)eiand ˆ u:=/summationdisplay iS2(ei)fi (2.18) and satisfy the relations S2(y) = uyu−1= ˆuyˆu−1,ˆu=S(u)−1 and M=u⊗u∆(u−1) = ˆu−1⊗ˆu−1∆(ˆu) (2.19) Fromuand ˆuone has two further elements of a quasitriangular Hopf algebra Dwith special properties: g:=uˆu ,withg∈G(D) and S4(y) =gyg−1(2.20) z:=uˆu−1,withz∈Z(D) and M2=z⊗z∆(z−1) (2.21) 2.) Integrals, Moduli and Radford’s Relations : We start this section with a review of basic facts fromHopf algebra theory anda summary of the formula e in[Rd], we will use in thispaper. The analysis ofintegralsofHopf algebrasin[LSw] isbased onthefundamental Theorem of Hopf modules. It asserts that a Hopf module Mof a Hopf algebra Dis free in the sense that Mcov⊗D/tildewidest−→Mis an isomorphism of Hopf modules. Here Dacts on itself by multiplication and comultiplication, Mcovis the

Question 735 multiple-choice

In group theory, the classification of finite p-groups often depends on the structure of their subgroups, such as the commutator subgroup and the Frattini subgroup. Parameters like quadratic non-residues and subgroup indices play a key role in distinguishing non-metacyclic p-groups with specific subgroup properties. Which of the following group characteristics most directly determines whether the subgroup Φ(G')G₃ is cyclic of order p or elementary abelian of order p² in finite p-group classification? 1) The order of the center Z 2) Whether the group is abelian or non-abelian 3) The structure and interaction of the commutator subgroup G' and the third lower central subgroup G₃ 4) The total order of the group 5) The number of generators in the group's presentation 6) The automorphism group of the p-group 7) The presence of a minimal non-abelian subgroup of index p

✓ Correct Answer:

The correct answer is 3) The structure and interaction of the commutator subgroup G' and the third lower central subgroup G₃.

📚 Reference Text:

1/an}bracketri}ht; (20)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea9=b9=c3= 1,[a,b] =c,[c,a] =a3,[c,b] =b3/an}bracketri}ht; (21)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea9=b9=c3= 1,[a,b] =c,[c,a] =b3,[c,b] =a3,[a3,b] = 1/an}bracketri}ht; (22)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea9=b9=c3= 1,[a,b] =c,[c,a] =b−3,[c,b] =a3,[a3,b] = 1/an}bracketri}ht; (23)/an}bracketle{ta,b,c/vextendsingle/vextendsingleap2=bp2=cp= 1,[a,b] =c,[c,a] =apbνp,[c,b] =bp/an}bracketri}ht, wherep >3, ν= 1or a ﬁxed quadratic non-residue modular p; (24)/an}bracketle{ta,b,c/vextendsingle/vextendsingleap2=bp2=cp= 1,[a,b] =c,[c,a] =bνp,[c,b] =a−p/an}bracketri}ht, wherep >3, ν= 1or a ﬁxed quadratic non-residue modular p; (25)/an}bracketle{ta,b,c/vextendsingle/vextendsingleap2=bp2=cp= 1,[a,b] =c,[c,a]1+r=apbp,[c,b]1+r=a−rpbp/an}bracketri}ht, wherep >3,r= 1,2,...,p−2; (26)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleap2=bp=cp=dp= 1,[a,b] =c,[c,a] =ap,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht, wherep >3; (27)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleap=bp2=cp=dp= 1,[a,b] =c,[c,a] =bνp,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht, wherep >3,ν= 1or a ﬁxed quadratic non-residue modular p; (28)/an}bracketle{ta,b,c,d/vextendsingle/vextendsinglea4=b2=c4=d2= 1,[a,b] =c,[c,a] =d,[c,b] =c2,[d,a] = [d,b] = 1/an}bracketri}ht; (29)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea4=b4=c4= 1,[a,b] =c,[c,a] =b2,[c,b] =c2/an}bracketri}ht; (30)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8=b2=c4= 1,[a,b] =c,[c,a] =a4,[c,b] =c2/an}bracketri}ht; (31)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8=c4= 1,b2=a4,[a,b] =c,[c,a] =a4,[c,b] =c2/an}bracketri}ht; (32)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8=b2=c4= 1,[a,b] =c,[c,a] =a4c2,[c,b] =c2,[c2,a] = 1/an}bracketri}ht; 38 (33)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8=b4= 1,c2=a4b2,[a,b] =c,[c,a] =b2,[c,b] =c2/an}bracketri}ht; (34)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8=b4= 1,c2=b2,[a,b] =c,[c,a] =a4b2,[c,b] =c2/an}bracketri}ht; (35)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bp2=cp= 1,[a,b] =c,[c,a] =apn,[c,b] =bsp/an}bracketri}ht, wherep >2 andn >1,s∈F∗ p; (36)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bp2=cp= 1,[a,b] =c,[c,a] =bν1p,[c,b] =a−ν2pn/an}bracketri}ht, where p >2andn >1,ν1,ν2= 1or a ﬁxed quadratic non-residue modular p; (37)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn+1=bp=cp=dp= 1,[a,b] =c,[c,a] =d,[c,b] =a−νpn,[d,a] = [d,b] = 1/an}bracketri}ht, wherep >2andn >1,ν= 1or a ﬁxed quadratic non-residue modular p; (38)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn+1=bp=cp=dp= 1,[a,b] =c,[c,a] =apn,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht, wherep >2andn >1; (39)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea2n+1=b4= 1,c2=b2,[a,b] =c,[c,a] =a2n,[c,b] =c2/an}bracketri}ht, where n≥3; (40)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea2n+1=b4= 1,c2=a2n,[a,b] =c,[c,a] =b2,[c,b] =c2/an}bracketri}ht, where n≥3; (41)/an}bracketle{ta,b,c,d/vextendsingle/vextendsinglea2n+1=b2=d2= 1,c2=a2n,[a,b] =c,[c,a] =d,[c,b] = c2,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3; (42)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea2n=b4=c4= 1,[a,b] =c,[c,a] =b2,[c,b] =c2/an}bracketri}ht, wheren≥3; (43)/an}bracketle{ta,b,c,d/vextendsingle/vextendsinglea2n=b4=d2= 1,c2=b2,[a,b] =c,[c,a] =d,[c,b] =c2,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3; (44)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea2n+1=b2=c4= 1,[a,b] =c,[c,a] =a2n,[c,b] =c2/an}bracketri}ht, wheren≥3; (45)/an}bracketle{ta,b,c,d/vextendsingle/vextendsinglea2n=b2=c4=d2= 1,[a,b] =c,[c,a] =d,[c,b] =c2,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3; ProofByLemma2.2(2), wehaveΦ( G′)G3≤Z(G), Φ(G′)G3≤C2 pandG/Φ(G′)G3∈ A1. SinceGis not metacyclic, G/Φ(G′)G3∼=Mp(n,m,1) where n >1 forp= 2 and n≥m. Case 1. Φ(G′)G3∼=Cp. IfG3≤Φ(G′), then, by Theorem 3.1, Ghas either 0 or 1 A1-subgroup of index p. Hence Φ( G′) = 1 and G3∼=Cp. By Theorem 4.1, m= 1. By Theorem 4.6, we get the groups (1)–(16). Case 2. Φ(G′)G3∼=C2 p. IfG3∼=Cp, then, by Theorem 5.1 and 5.5, Ghas noA1-subgroup of index p. Hence Φ(G′)≤G3∼=C2 p. By Theorem 6.1 and 6.5, we get the groups (17)–(44). /square 39 (1) (F) (13) (J3) where m= 1 (25) (P4) where n= 1 (37) (S3) where m= 1 (2) (G1) where m= 1 (14) (J4) where m= 1 (26) (P8) where n= 1 (38) (S6) where m= 1 (3) (G2) where m= 1 (15) (J5) where m= 1 (27) (P9) where n= 1 (39) (T1) (4) (G3) where m= 1 (16) (J6) where m= 1 (28) (R1) (40) (T2) (5) (H1) (17) (O1) (29) (R2) (41) (T3) (6) (H2) (18) (O3) (30) (R3) (42) (T4) (7) (H3) (19) (O4) (31) (R4) (43) (T5) (8) (I1) (20) (O5) (32) (R5) (44) (T6) (9) (I2) (21) (O6) (33) (R6) (45) (T7) (10) (I3) (22) (O7) (34) (R7) (11) (J1) where m= 1 (23) (P2) where n= 1 (35) (S1) where m= 1 (12) (J2) where m= 1 (24) (P3) where n= 1 (36) (S2) where m= 1 Table 7.2: Correspondence for Theorem 7.1 References [1] L.J. An, L.L. Li, H.P. Qu and Q.H. Zhang, Finite p-groups with a minimal non- abelian subgroup of index p(II),Sci. China Math.

Question 736 multiple-choice

In group theory, graphs constructed from group elements or their conjugacy classes can reveal deep structural properties of groups. The completeness of such graphs often relates to specific subgroup configurations and their interactions. Which property must hold for a finite non-abelian group to have a complete KN-graph with respect to its irreducible conjugacy classes? 1) All normal subgroups must be abelian. 2) The group must have a trivial center. 3) Every minimal non-central normal subgroup must satisfy a necessary and sufficient condition related to KN-completeness. 4) The group must be simple. 5) The group must have only one conjugacy class. 6) All elements must commute. 7) The group must be cyclic.

✓ Correct Answer:

The correct answer is 3) Every minimal non-central normal subgroup must satisfy a necessary and sufficient condition related to KN-completeness..

📚 Reference Text:

Title: On the class of KN-complete groups Year: 2021 Paper ID: 15d40504f122d1203af919bd6fc1bdf72aec39db Source: semantic-scholar URL: https://www.semanticscholar.org/paper/15d40504f122d1203af919bd6fc1bdf72aec39db Abstract: Kayal and Nezhmetdinov introduced in [Factoring groups efficiently, Int. Colloquium on Automata[Formula: see text] Languages[Formula: see text] and Programming (Springer, Berlin, Heidelberg, 2009), pp. 585–596] the concept of an irreducible conjugacy class and defined, for any finite non-abelian group, a simple graph on the set of all such classes. This graph is a key ingredient in their proposed algorithm for solving the group decomposition problem. This paper considers the class of finite groups for which the KN-graph is complete (KN-complete groups). The main result is a necessary and sufficient condition for a given group to be KN-complete, formulated in terms of its minimal non-central normal subgroups. This condition is used to obtain a more detailed description of the rather rich class of KN-complete groups.

Question 737 multiple-choice

In quantum error correction and holographic codes, the structure and properties of logical operators play a crucial role in determining which symmetries can be implemented within a code. In particular, the distinction between standard quantum codes and holographic codes has significant implications for the presence of global symmetries in quantum gravity models. Which property of holographic (AdS/CFT) codes leads to the exclusion of both discrete and continuous global symmetries, distinguishing them from typical quantum error-correcting codes? 1) The ability to implement any symmetry operation as a non-local physical operator 2) The requirement that each factor in the tensor product decomposition of a symmetry operator is itself a logical operator that preserves the code space 3) The existence of non-correctable erasure regions within the code 4) The capacity for logical operators to increase the energy of states in the boundary theory 5) The presence of universal transversal implementation for discrete symmetries 6) The reliance on high-energy excitations to encode logical information 7) The allowance for local operators to move states out of the code space

✓ Correct Answer:

The correct answer is 2) The requirement that each factor in the tensor product decomposition of a symmetry operator is itself a logical operator that preserves the code space.

📚 Reference Text:

bulk global symmetry operator UL, when reconstructed on the boundary, can be expressed as a tensor product ⊗ kWk, where Wkis supported on Ak. (Here, we ignore a correction factor supported only where the regions touch, which is inessential to the argument.) This step is just the property that we have assumed throughout this paper and which is exemplified by the three-qutrit code discussed above. The second step of the argument (also explained further below) is the crucial one, which invokes a property of theAdS/CFT code which is not shared by the typical quantum codes which arise in work on fault-tolerant quantum computation. Harlow and Ooguri argue that each Wkis itself a logical operator; that is, each Wkmaps the code space to the code space. The essence of this part of the argument is that the code space is the span of low-energystates in the CFT, and the W k’s, perhaps after suitable smoothing, can be chosen so that they do not increase the energy of the CFT by very much. As we have already emphasized, this property does not apply to the three-qutritcode, where X Lis a logical operator, yet its weight-one factors X⊗I⊗IandI⊗X−1⊗Iare not logical. Indeed, because X⊗I⊗Ichanges the eigenvalue of the unitary operator SZby the multiplicative factor ω, it maps the code space (the simultaneous eigenspace of SZandSXwith eigenvalue 1) to a subspace orthogonal to the code space(the eigenspace of S Zwith eigenvalue ω). A logical operator supported on a region A, where erasure of Ais correctable, must be the logical identity. We can easily see that is true, because otherwise an adversary could steal region Aand apply a nontrivial logical operator, altering the encoded state and, therefore, introducing an uncorrectable error. Now the conclusion of Harlow and Ooguri follows easily. The bulk global sym-metry operator U Lis a product of logical operators, each of which is trivial; therefore, ULmust be the identity. As Harlow and Ooguri note (footnote 69 in Ref. [16]), their argument, which excludes discrete symmetries of the AdS/CFT code as well as continuous symmetries, is quitedifferent than the Eastin-Knill argument, which excludesonly continuous symmetries of a code. Both arguments apply in a framework where the symmetry of the code can be applied transversally, as a product of local operators.But, for the Eastin-Knill argument, there is no need to assume that these local operators individually preserve the code space, and, therefore, the argument applies to generalcodes. In contrast, Harlow and Ooguri assert that for the AdS/CFT code, in particular, the local operators do individually preserve the code space. Therefore, theirargument excluding discrete symmetries applies to the AdS/CFT code but not to the typical codes studied by quantum information theorists. For completeness, we now sketch the two key steps of the Harlow-Ooguri argument in slightly greater detail,starting with the step which shows that a bulk globalsymmetry acts transversally on the boundary. By definition, a global symmetry in the bulk maps a bulk local operator to another bulk local operator at the same position.Furthermore, a local

Question 738 multiple-choice

Floating-point arithmetic is essential for representing real numbers in both classical and quantum computing systems. Modern standards like IEEE 754 define how these numbers are encoded, handled, and rounded in hardware and software implementations. Which statement most accurately describes how subnormal numbers are represented and handled in the 32-bit single precision IEEE 754 floating-point format? 1) Subnormal numbers are encoded using all ones for the exponent and a zero mantissa. 2) Subnormal numbers use a nonzero exponent field and a fractional mantissa. 3) Subnormal numbers use a sign bit of one and a zero exponent. 4) Subnormal numbers are represented by an exponent of all ones and a nonzero mantissa. 5) Subnormal numbers are encoded with a zero mantissa and a nonzero exponent. 6) Subnormal numbers are represented by an exponent field of all zeros and a nonzero fractional mantissa. 7) Subnormal numbers require additional guard and sticky bits for their encoding.

✓ Correct Answer:

The correct answer is 6) Subnormal numbers are represented by an exponent field of all zeros and a nonzero fractional mantissa..

📚 Reference Text:

later. In the binary representation, thebits following the binary point are the fractional part of the mantissa. Once floating-point numerical computation on classical computers became commonplace, theindustry standard IEEE 754 was introduced [31]. A similar standard for floating-point representations on a quantum computer does not yet exist, but is desirable [30]. A key feature of the IEEE standard is that it requires correctly rounded operations: correctly rounded arithmetic operations, correctly rounded remainderand square root operations and correctly rounded format conversions. Typically,rounding to the nearest floating pointing number available in the destination (out-put register) is used. In the quantum circuits in the present work, rounding down tonearest is used, for reasons of simplicity. Detailed analysis of quantum-circuitsdeveloped here for squaring and multiplication operations shows that ‘correctly ’ rounding to nearest involves a significant increase in circuit complexity (i.e. using quantum equivalents of guard and sticky bits, that are well established in arithmetic 7Quantum Algorithms for Nonlinear Equations in Fluid Mechanics DOI: http:/ /dx.doi.org/10. 5772/intechopen.9 5023 on classical computers [31]). A key aspect of the IEEE 754 that has been incorpo- rated in the present work is the definition of sub-normal numbers . To illustrate the concept of subnormal numbers, the IEEE 754 standard representation of single format numbers using a 32-bit word is considered. The first bit is the sign bit,followed by 8 bits representing the exponent. Then, 23 bits are used to store a 24-bitrepresentation of the mantissa, i.e. b 0is not stored. Numbers with exponent bits 00000000ðÞ2¼0ðÞ10and 11111111ðÞ2¼255ðÞ10are defined special cases. The smallest normalized number is 1 :0…0 ðÞ2/C22/C0126¼2/C0126. Sub-normal numbers are used to represent smaller numbers, i.e. in this case the exponent field has a zero bit string but the fraction field has a nonzero bit string. Zero is represented with azero bit string for the fractional field. For all subnormal numbers, the 00000000 ðÞ used for the exponent represents 2 /C0126and by using the 23 fractional field bits, equally-spaced numbers in the range 0 :00…01 ðÞ2/C22/C0126(with 22 zero bits after the binary point) to 0 :11…11 ðÞ2/C22/C0126(with 23 one bits after the binary point) are encoded. 5.3 Quantum floating-point format used in present work Based on the floating point representation defined in the IEEE standard, the present work introduces a floating-point system with fewer bits (i.e. qubits in this case) than the 32 used for single format numbers. This is the direct result of the limited number of qubits available on current and near-term quantumcomputers. To optimize the range of floating-point numbers that can berepresented with the approach used here, the following key aspects of the IEEEstandard were adopted: For the mantissa only the fractional part is stored, Exponent bit strings 00 …00 ðÞ 2and 11 …11 ðÞ2are used for special cases, i.e. dealing with 0, subnormal numbers as well as cases with overflow, The remaining range of exponent bit strings is used for a range of exponential centred around 20¼01…11 ðÞ2, Sub-normal numbers are used to extend the range of small numbers, Rounding down to nearest

Question 739 multiple-choice

Quantum computing poses significant risks to traditional cryptographic schemes, motivating the development of quantum-safe cryptographic primitives. The Hidden Subgroup Problem (HSP) is a foundational problem with deep connections to the security of group-based cryptography in the quantum era. Which of the following statements most accurately describes the relationship between the Hidden Subgroup Problem (HSP) and the security of group-based cryptographic platforms? 1) HSP is only relevant to classical cryptography and does not affect quantum resistance. 2) The security of group-based cryptography is guaranteed regardless of the solvability of HSP. 3) HSP is a minor consideration; most group-based schemes rely solely on hash functions for security. 4) Efficient quantum algorithms for HSP in any group ensure the absolute security of group-based cryptography. 5) Only abelian groups are affected by HSP in the context of quantum computing. 6) If quantum algorithms efficiently solve HSP for the underlying group, the cryptosystem based on that group may become insecure. 7) The complexity of HSP is unrelated to the choice of group in group-based cryptography.

✓ Correct Answer:

The correct answer is 6) If quantum algorithms efficiently solve HSP for the underlying group, the cryptosystem based on that group may become insecure..

📚 Reference Text:

arXiv:1805.04179v2 [cs.CR] 21 May 2018The Hidden Subgroup Problem and Post-quantum Group-based Cryptography Kelsey Horan1and Delaram Kahrobaei2,3 1The Graduate Center, CUNY, USA khoran@gradcenter.cuny.edu 2The Graduate Center and NYCCT, CUNY 3New York University, USA dk2572@nyu.edu⋆ Abstract. InthispaperwediscusstheHiddenSubgroupProblem(HSP) in relation to post-quantum cryptography. We review the rel ationship between HSP and other computational problems discuss an opt imal so- lution method, and review the known results about the quantu m com- plexity of HSP. We also overview some platforms for group-ba sed cryp- tosystems. Notably, eﬃcient algorithms for solving HSP in t he proposed inﬁnite group platforms are not yet known. Keywords: HiddenSubgroupProblem,QuantumComputation,Post-Quantum Cryptography, Group-based Cryptography 1 Introduction InAugust2015theNationalSecurityAgency(NSA)announcedpla nstoupgrade security standards; the goal is to replace all deployed cryptogra phic protocols with quantum secure protocols. This transition requires a new secu rity standard to be accepted by the National Institute of Standards and Techn ology (NIST). Proposals for quantum secure cryptosystems and protocols hav e been submitted for the standardization process. There are six main primitives curr ently pro- posed to be quantum-safe: (1) lattice-based (2) code-based (3 ) isogeny-based (4) multivariate-based (5) hash-based, and (6) group-based cryptographic schemes. One goal of cryptography,as it relates to complexity theory, is to analyze the complexityassumptionsusedasthebasisforvariouscryptograph icprotocolsand schemes. A central question is determining how to generate intrac tible instances of these problems upon which to implement an actual cryptographic scheme. The candidates for these instances must be platforms in which the h ardness assumption is still reasonable. Determining if these group-based cr yptographic ⋆Research of Delaram Kahrobaei was partially supported by a P SC-CUNY grant from the CUNY research foundation. Research of Delaram Kahr obaei was also sup- ported by the ONR (Oﬃce of Naval Research) grant N0001415121 64. We thank E. Kasheﬁ and L. Perret for discussions and hospitality in summ er 2017 at University of Sorbonne. schemes are quantum-safe begins with determining the groups in wh ich these hardness assumptions are invalid in the quantum setting. In what follows we address the quantum complexity of the Hidden Subgroup Problem (HSP) to determine the groups in which the hardness assumption still stands.The Hidden Subgroup Problem (HSP) asks the following: given a descrip- tion of a group Gand a function f:G→Xfor some ﬁnite set Xis guaranteed to be strictly H-periodic, i.e. constant and distinct on left (resp. right) cosets of a subgroup H≤G, ﬁnd a generating set for H.It is important to note that Simon’s problem of computing a XOR-mask, Shor’s algorithm for facto ring and ﬁnding the discrete log, Boneh’s algorithm for ﬁnding a hidden linear fu nction, and Kitaev’s algorithm for the abelian stabilizer problem are all special cases of HSP. Therefore, the HSP is directly related to problems such as b reaking one-time pad, discrete logarithm problem, graph isomorphism proble m (which is now known to be in quali-polynomial), lattice-based problems, the pr oblem for factoring for RSA. The classical complexity of HSP is known [1]: Suppose that Ghas a set H ofNsubgroups, such that H1∩H2∩...∩HH=eG. Then a classical computer must make

Question 740 multiple-choice

Fusion categories are mathematical structures used to model symmetries and particle-like excitations in topological phases of matter, with applications in condensed matter physics and quantum computation. The classification and extension of these categories often involve advanced concepts from group cohomology and categorical algebra. In the construction of a G-crossed braided fusion category, which specific cohomology group determines whether a group homomorphism from G to the group of braided-tensor autoequivalences can be lifted to a monoidal functor, thereby ensuring the existence of a categorical global symmetry? 1) H¹(G, U(1)) 2) H²(G, U(1)) 3) H⁴(G, A) 4) H²ₚ(G, A) 5) H³(G, A) 6) H³(G, U(1)) 7) H⁴(G, U(1))

✓ Correct Answer:

The correct answer is 5) H³(G, A).

📚 Reference Text:

why this is true, observe that the tensor unit (the vacuum) ha s quantum dimension one and the quantum dimension function d−: Obj(C)→R≥0is linear and multiplicative, in the sense that a⊗b=/circleplusdisplay cNab cc=⇒da·db=∑ cNab cdc. In particular, if da=db=1, then a⊗b=cwhere dc=1. It also follows that the dual a∗ of an abelian anyon ais also abelian, since a⊗a∗=1 implies d∗ a=1. It follows that the subset of abelian anyon types contains the vacuum and is clos ed under multiplication and inverses. Since fusion is associative on the level of isomor phism classes, Aforms a group. SYMMETRY DEFECTS AND THEIR APPLICATION TO TOPOLOGICAL QUAN TUM COMPUTING 3 We denote by Aut⊗ br(C)the group of braided-tensor autoequivalences of Cup to natural isomorphism under composition. We will also need the catego rical group Aut⊗ br(C), with group elements of Aut⊗ br(C), tensor product given by composition, and morphisms given by natural isomorphisms of functors. 2.1.2. From group symmetry to group extension theory of MTCs. To construct a UGxBFC, we start with a UMTC Cand a group homomorphism ρ:G→Aut⊗ br(C), called a global symmetry . When this group homomorphism ρcan be promoted to a monoidal functor ρ:G → Aut⊗ br(C), it is called a categorical global symmetry ofC. The existence of such a liftρdepends on the vanishing of a certain cohomological obstruc tion in H3(G,A), where Ais the subgroup of abelian anyons under fusion. When this obs truction vanishes, the different extensions are “classiﬁed" by H2 ρ(G,A)in the sense that the distinct liftings of the global symmetry ρto the categorical global symmetry ρform a torsor over H2 ρ(G,A). With a categorical global symmetry in hand, the next ingredi ent for constructing a G-extension of Cis an isomorphism of the categorical groups Aut⊗ br(C)and Pic(C), the categorical Picard group consisting of invertible bimodul e categories of C[14]. This as- sociates to each categorical braided tensor autoequivalen ceρgan invertible C-bimodule categoryCg, a correspondence which is at the heart of the relationship b etween symmetry in the abstract and symmetry defects. When an additional coh omological obstruction in H4(G;U(1))vanishes, consistent fusion of invertible bimodule catego riesCg⊠Chcan be deﬁned so that elements of Cgsatisfy the pentagon equations. The different solutions to the pentagon equations then form a torsor over H3(G,U(1)). Each such solution corresponds noncanonically to a unitary fusion category denoted by C× G=/circleplustext g∈GCg, whereC0=C. The inequivalent G-crossed braided extensions of a UMTC Care classiﬁed by the data (ρ,α,β), where α∈H2 ρ(G,A)andβ∈H3(G,U(1))[2, 14]. We will see in the next subsection that C× Ghas the structure of a unitary G-crossed braided fusion category . Now the anyons of Care identiﬁed as simple objects in C0and the symmetry defects as simple objects in/circleplustext g/ne}a⊔ionslash=0Cg: Deﬁnition 2.1. A symmetry defect with ﬂux g is a simple object in the invertib leC0- bimodule category Cg. We will sometimes call the bimodule category Cgtheg-ﬂux sector, or a g-sector. 2.1.3. Physical interpretation of UGxBFCs. We note that in the physics literature, espe- cially in the early development of TQC, the word “anyon" was

Question 741 multiple-choice

In quantum field theory, modular theory provides insights into the dynamics of operator algebras and their symmetries, particularly relevant in the study of conformal field theories and holographic dualities such as AdS/CFT. The interplay between algebraic structures and emergent geometric symmetries has profound implications for our understanding of spacetime and black hole physics. Which mathematical property of time interval operator algebras guarantees the existence of a representation of the universal cover of the conformal group PSL(2, ℝ) in any quantum system? 1) The presence of integer scaling dimensions in all field operators 2) Strict geometric modular flow for all time intervals 3) Absence of the Generalized Hilbert Transform in modular dynamics 4) Symmetry under discrete time translations 5) Invariance of operator algebras under all local spacetime diffeomorphisms 6) Commutativity of modular operators for disjoint intervals 7) Twisted modular inclusion and twisted modular intersection

✓ Correct Answer:

The correct answer is 7) Twisted modular inclusion and twisted modular intersection.

📚 Reference Text:

Title: Modular intersections, time interval algebras and emergent AdS2 Year: 2024 Paper ID: 4559fb73367809a980c24dc7916ae59b348d0f97 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/4559fb73367809a980c24dc7916ae59b348d0f97 Abstract: We compute the modular flow and conjugation of time interval algebras of conformal Generalized Free Fields (GFF) in 0 + 1-dimensions in the vacuum. For non-integer scaling dimensions, for general time intervals, the modular conjugation and the modular flow of operators outside the algebra are non-geometric. This is because they involve a Generalized Hilbert Transform (GHT) that treats positive and negative frequency modes differently. However, the modular conjugation and flows viewed in the dual bulk AdS2 are local, because the GHT geometrizes as the local antipodal symmetry transformation that pushes operators behind the Poincaré horizon. These algebras of conformal GFF satisfy a Twisted Modular Inclusion and a Twisted Modular Intersection property. We prove the converse statement that the existence of a (twisted) modular inclusion/intersection in any quantum system implies a representation of the (universal cover of) conformal group PSL(2, ℝ), respectively. We discuss the implications of our result for the emergence of AdS2 geometries in large N theories without a large gap. Our result applied to higher dimensional eternal AdS black holes explains the emergence of two copies of PSL(2, ℝ) on future and past Killing horizons.

Question 742 multiple-choice

In fault-tolerant quantum computing, efficient implementation of the Quantum Fourier Transform (QFT) and its approximations is essential for algorithms like Shor’s factoring and phase estimation. Optimizing gate counts, especially for costly non-Clifford gates such as T gates, is a critical challenge in scalable quantum circuit design. Which architectural strategy enables a logarithmic-depth adder circuit to be both fast and spatially efficient for small register sizes when implementing the Approximate Quantum Fourier Transform (AQFT)? 1) Using only 1D Linear Nearest Neighbor connectivity with no additional circuit layout optimization 2) Employing a simple sequential arrangement of gates without spatial partitioning 3) Relying exclusively on global qubit interactions in a fully connected architecture 4) Adopting an H-tree layout in a 2D Square Lattice for compact depth and efficient locality 5) Implementing ring-based qubit connectivity with distributed CNOT gates 6) Applying a star topology for all-to-one communication among qubits 7) Utilizing randomized SWAP circuits to minimize depth overhead

✓ Correct Answer:

The correct answer is 4) Adopting an H-tree layout in a 2D Square Lattice for compact depth and efficient locality.

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the adder circuit, except forcontrolled-Z1∕2and controlled-Z1∕4gates; the two require 7 T gates to implement and 1 T gate to correct for the phase in the partial gradient operation. The construction of the special state ψdþ1;bþ1/C12/C12 irequires implementation of O(b)Zarotations, and we approximate each rotation with Oðlogðb=εÞÞT gates33to achieve accuracy ε∕bper rotation. There are two sources of approximation errors in our construction. Our circuit differs from the ideal AQFT circuit onlyin the preparation of the special state ψdþ1;bþ1/C12/C12 i. Therefore, the spectral norm distance between our AQFT circuit and the ideal AQFT is O(b⋅ε∕b)=O(ε). This ensures that, with 1 −O(ε2) probability, regardless of how many operations to follow from theψdþ1;bþ1/C12/C12 istate preparation stage, our circuit implements the ideal AQFT. If we choose b¼Oðlogðn=εÞÞ, the spectral norm error of the ideal AQFT circuit will be O(ε). Due to the triangle inequality, the total error can be upper bounded by adding the error of the Clifford +T synthesis and the error of AQFT, which is still O(ε). The above error analysis shows that for all effective purposes (speci ﬁcally, when ε≻n∕2n) we can drop the dependence on the approximation error ε, resulting in the claimed T-count of OðnlognÞ. Future work Future lines of inquiry may include laying out our circuit in restricted architectures and the optimization of depth. To address former, both the basic QFT22,23and the adder31we rely on (being the long adder) can be laid out in the Linear Nearest Neighbor architecture with a constant SWAP overhead. Thus, the increase in the CNOT gate count due to SWAP operations will remain under control, and the overall cost of the implementation is expected to continue being dominated by the cost of the T gates (note that the introduction of SWAP gates does not increase the number of T gates), although the cost of the CNOTS will start to matter more. To address depth, we ﬁrst note that everything but the adder is already parallelized. To optimize depth, one may choose to rely on a fast logarithmic-depth adder and lay it out in 2D Square Lattice (a natural architecture for superconducting circuit quantum information processors) using the H-tree –an H-tree, popular in VLSI design, is a fractal tree, embedded in a 2D square lattice, constructed from a repeating pattern that resembles the letter H. This will introduce additional gates and require more space, but it may reduce the depth. Note that for small numbers such as those used in our result ( b=13) the H-tree remains compact and requires few SWAP operations. Conclusion Before our contribution, the best known coherent approximation of the n-qubit QFT to an error εby a quantum fault-tolerant Clifford +T circuit featured the T-count of Oðnlogðn=εÞlogðnlogðn=εÞ εÞÞ, with the term Oðnlogðn=εÞÞoriginat- ing from the standard AQFT construction using controlledTable 1. Quantum resource counts for implementing an n-qubit AQFT with b=13.nqdenotes the number of qubits required to execute the corresponding circuit. Columns CNOT and T report the number of respective gates in the circuits. All circuits are available in ref.38. Our

Question 743 multiple-choice

In the classification of finite groups, especially p-groups, matrix methods over finite fields are commonly used to distinguish isomorphism classes by analyzing the properties of associated structural matrices. The form and rank of these matrices play a crucial role in determining the type and uniqueness of group structures. Which condition guarantees that two groups with structural matrices w and w(Ḡ) over the finite field Fp are isomorphic, assuming both matrices are invertible? 1) There exists an invertible matrix X over Fp such that w(Ḡ) = X w Xᵗ 2) The matrices w and w(Ḡ) have the same trace 3) The determinant of w equals the determinant of w(Ḡ) 4) Both matrices can be diagonalized by the same invertible matrix 5) Their ranks are equal and both are full rank 6) w and w(Ḡ) are similar via conjugation by a scalar matrix 7) The entries of w and w(Ḡ) differ only by a permutation of rows

✓ Correct Answer:

The correct answer is 1) There exists an invertible matrix X over Fp such that w(Ḡ) = X w Xᵗ.

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and n=m≥2. Suppose that Gand¯Gare two groups described in the theorem. By Theorem 3.3 and 3.4, G∼=¯Gif and only if there exists X=/parenleftbiggx11x12 x21x22/parenrightbigg , an invertible matrix over Fp, such that/parenleftbigg¯w11 ¯w21/parenrightbigg =|X|−1X/parenleftbiggw11 w21/parenrightbigg 7 and /parenleftbigg¯w12 ¯w22/parenrightbigg =X/parenleftbiggw12 w22/parenrightbigg . That is,/parenleftbigg¯w11¯w12 ¯w21¯w22/parenrightbigg =X/parenleftbiggw11w12 w21w22/parenrightbigg/parenleftbigg|X|−10 0 1/parenrightbigg (3.7) Ifw(G) is invertible, then, letting X= diag(1 ,|w(G)|)(w(G))−1, we have w(¯G) = diag(1,|w(G)|). Hence we get the group of Type (D1) where t=|w(G)|. By Equation 3.7,|w(¯G)|=|w(G)|. Hence diﬀerent tgive non-isomorphic groups. Ifw(G) = 0, then Gis the group of Type (D5). In the following, we assume that w(G) is of rank 1. By suitably choosing Xsuch that |X|= 1, that is, using some row operations, we can simplify w(G) to be/parenleftbiggw11w12 0 0/parenrightbigg where (w11,w12)/ne}ationslash= (0,0). Letw(¯G) and w(G) be such matrices. Then, by Equation 3.7, it is easy to check t hatG∼=¯Gif and only if there exists X= diag(x11,x22), an invertible matrix over Fp, such that w(¯G) =Xw(G)diag(|X|−1,1). By Table 3.1, we get the groups of (D2)–(D4). It is Case X w (¯G) Group w11/ne}ationslash= 0 and w12/ne}ationslash= 0 diag( w−1 12,w11)/parenleftbigg1 1 0 0/parenrightbigg (D2) w11/ne}ationslash= 0 and w12= 0 diag(1 ,w11)/parenleftbigg1 0 0 0/parenrightbigg (D3) w11= 0 and w12/ne}ationslash= 0 diag( w−1 12,w12)/parenleftbigg 0 1 0 0/parenrightbigg (D4) Table 3.1: Case 3 in Theorem 3.5 where w(G) is of rank 1 easy to check that diﬀerent types give non-isomorphic groups . Case 4: n > m≥2. Suppose that Gand¯Gare two groups described in the theorem. By Theorem 3.3 and 3.4,G∼=¯Gif and only if there exists X=/parenleftbiggx11x12 x21pn−mx22/parenrightbigg , an invertible matrix overFp, such that /parenleftbigg¯w11 ¯w21/parenrightbigg =|X|−1/parenleftbiggx110 x21x22/parenrightbigg/parenleftbiggw11 w21/parenrightbigg (3.8) and /parenleftbigg¯w12 ¯w22/parenrightbigg =/parenleftbiggx11x12 0x22/parenrightbigg/parenleftbiggw12 w22/parenrightbigg (3.9) By suitably choosing x21, that is, using an elementary row operation, we can simplify/parenleftbiggw11 w21/parenrightbigg to be one of the following three types: (a1)/parenleftbiggw11 0/parenrightbigg wherew11/ne}ationslash= 0, (b1)/parenleftbigg0 w21/parenrightbigg wherew21/ne}ationslash= 0, (c1)/parenleftbigg0 0/parenrightbigg . 8 In the following, we assume that both/parenleftbiggw11 w21/parenrightbigg and/parenleftbigg¯w11 ¯w21/parenrightbigg are such matrices. By Equation 3.8 and 3.9, it is easy to check that (i) diﬀerent type s give non-isomorphic groups, and (ii) G∼=¯Gif and only if there exists X=/parenleftbiggx11x12 0x22/parenrightbigg , an invertible matrix over Fp, such that /parenleftbigg¯w11 ¯w21/parenrightbigg =|X|−1/parenleftbiggx110 0x22/parenrightbigg/parenleftbiggw11 w21/parenrightbigg (3.10) and /parenleftbigg¯w12 ¯w22/parenrightbigg =/parenleftbiggx11x12 0x22/parenrightbigg/parenleftbiggw12 w22/parenrightbigg . (3.11) By suitably choosing x12, that is, using an elementary row operation, we can simplify/parenleftbiggw12 w22/parenrightbigg to be one of the following three types: (a2)/parenleftbigg0 w22/parenrightbigg wherew22/ne}ationslash= 0, (b2)/parenleftbiggw12 0/parenrightbigg wherew21/ne}ationslash= 0, (c2)/parenleftbigg0 0/parenrightbigg . In the following, we assume that both/parenleftbiggw12 w22/parenrightbigg and/parenleftbigg¯w12 ¯w22/parenrightbigg are such matrices. By Equation 3.10 and 3.11, it is easy to check that (i) diﬀerent ty pes give non-isomorphic groups, and (ii) G∼=¯Gif and only if there exists X= diag(x11,x22), an invertible matrix over Fp, such that /parenleftbigg¯w11 ¯w21/parenrightbigg =|X|−1X/parenleftbiggw11 w21/parenrightbigg and/parenleftbigg¯w12 ¯w22/parenrightbigg = X/parenleftbiggw12 w22/parenrightbigg .(3.12) By Equation 3.12, G∼=¯Gif and only if there exists X= diag(x11,x22), an invertible matrix over Fp, such that w(¯G) =Xw(G)diag(|X|−1,1). By Table 3.2, we get the /parenleftbiggw11 w21/parenrightbigg /parenleftbiggw12 w22/parenrightbigg

Question 744 multiple-choice

Quantum algorithms for group theory problems leverage advanced techniques such as Fourier sampling and group-theoretic state manipulation to efficiently analyze algebraic structures. These methods enable the extraction of subgroup generators and facilitate the construction of series like polycyclic sequences, which are pivotal for understanding solvable groups. Which procedure specifically enables the incremental construction of a polycyclic series by testing generator orders and halting if certain divisibility criteria are not met in quantum algorithms for group decomposition? 1) Applying Grover's search to discover commutator subgroups 2) Utilizing Simon's algorithm to detect hidden subgroups in non-Abelian groups 3) Implementing quantum phase estimation to find eigenvalues of group elements 4) Using quantum walks to traverse group Cayley graphs 5) Employing amplitude amplification to optimize membership testing 6) Executing Shor's algorithm to factor group orders directly 7) Testing the orders of subgroup generators and incrementally building the series, stopping when a generator fails to satisfy the divisibility requirement

✓ Correct Answer:

The correct answer is 7) Testing the orders of subgroup generators and incrementally building the series, stopping when a generator fails to satisfy the divisibility requirement.

📚 Reference Text:

to be prepared to handle some "degenerate" cases that occur with small probability. (Watrous discarded these and used repetition. However, the possibility of a variant that also works in the degenerate cases is mentioned in [19].) Notice thatjKi=1pmP x2ZmjuxNi. We apply the Fourier sampling described in Subsec- 786 An exact quantum hidden subgroup algorithm and applications to solvable groups tion 2.3 for the function f:Zm!CGgiven asf(x) =juxNi. This time we are not interested which subgroup is hidden by f, our objectives are the states j yiin the second register of j i in (2) and in (3). In this special case, j yi=1pmX x2Zm!xyjuxNi: (Here we go back to (1) and do not use the decomposition of j yiby the cosets of the hidden subgroup.) Another minor di erence from Subsection 2.3 is that we compute fin a register initiallyjNirather thanj0i. By applying Fourier sampling on the available scopies ofjNi, we obtain (m)s=2m1X y1;:::;ys=0jy1i:::jysij y1i:::j ysi: We consider the terms of this sum. For each s-tuple (y1;:::;ys) we will change the rst s1 states toj 0i=jKiusing Watrous's trick. The key fact behind is that j yiis an eigenstate with eigenvalue !yfor the action of uby multiplication. Based on this, it is easy to show that if we have a pair j yij ziof such states then multiplying the content of the rst part by the tth power of the content of the second part, the e ect can be interpreted as the rst part j yi remains unchanged while the second part becomes j ztyi, whereztyis understood modulo m. We use this rst to arrange that the index ysof the last state ysbecomes a generator for the subgroup LofZmgenerated by y1;:::;ys. To this end, using the method of Lemma 1, we nd integers u1;:::;us1such that gcd( u1y1+:::+us1ys1+ys;m) = gcd(y1;:::;ys;m). Thenu1y1+:::+us1ys1+ysmodulomis a generator for L. We multiply (the content of) j y1iby theu1th power ofj ysi, thenj y2itheu2th power of by an appropriate power of (the new)j ysi, and so on. Eventually ysbecomes the generator u1y1+:::+us1ys1+ys modulom. Thenyitiysmodulomfor some integer ti(i= 1;:::;s1), and multiplying j ysiwith thetith power ofj yiichangesj yiitoj yitiysi=j 0i=jKi. 4.3 Constructing a polycyclic series We start with testing if the orders of the generators are divisors of some power of m. (The proof of Corollary 4 includes a straightforward classical method doing this.) If one of the tests fails, then no further computation is needed: we can return that the order of Gdoes not satisfy the required property. When a new element occurs, we can perform this test and exit in case of failure. We build a polycyclic series from the bottom. Assume that NCGand we already have a polycyclic series of Nwith factors of order dividingm. Initially, Nis the trivial subgroup. By the previous subsection, we have an ecient method for Task (i) of Theorem 2 with Nin place ofGand one can use Corollary 3 (or Corollary 4) to test membership in N. Our rst goal is to nd a normal subgroup KCG such thatK=N is

Question 745 multiple-choice

In computational number theory and lattice-based cryptography, efficiently computing an accurate lattice basis from approximate generators is crucial, especially when working under finite precision and error constraints. Algorithms like LLL reduction are foundational, but their effectiveness depends on how well they control error propagation from input to output. When computing a lattice basis from approximate generators using the Buchmann-Pohst algorithm, which key technique is used to ensure linear independence and allow the LLL algorithm to operate effectively on the constructed lattice? 1) Applying Gram-Schmidt orthogonalization directly to the approximated generators 2) Concatenating each generator with a unit vector to form higher-dimensional vectors 3) Scaling each generator by its norm before reduction 4) Projecting generators onto a lower-dimensional subspace 5) Replacing real-valued entries with their nearest integer values without augmentation 6) Using random perturbations to break dependencies among generators 7) Performing QR decomposition prior to basis reduction

✓ Correct Answer:

The correct answer is 2) Concatenating each generator with a unit vector to form higher-dimensional vectors.

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in Section 4.4. A real number xis typically approximated by a rational number ˜ x=p 2msuch that |˜x−x|/lessorequalslantε. The param- eterεis called the absolute error and the ratio γ=ε/|x| is called the relative error. A vector x= (x(1),...,x(n)) is approximated by another ex= (˜x(1),..., ˜x(n)) such that /bardblex−x/bardbl/lessorequalslantε; the relative error is deﬁned as γ=ε//bardblx/bardbl. We start with a lemma bounding the error in the ideal multiplication step. Lemma 4.1. Suppose a basis b1,...,bnfor a lattice et1O and a basis c1,...,cnforet2Oare given with relative preci- sionγ, and each etihas norm 1, then the resulting products bicjhave relative precision 4γ√n||bi||n. In the next section we show how to compute a basis from a generating set. 4.3 Computing a basis of a lattice from an ap- proximate generating set To compute in polynomial time a basis ˆb1,..., ˆbnthat is bounded in size and δ-close to a basis for etOwe need to compute a bounded-length basis from a generating set. This is also used in the hidden subgroup problem algorithm when computing the basis for the unit lattice from a generating set for its dual. The input and output vectors for this are approximate. We need an algorithm that can ﬁnd integer dependencies among the rounded vectors and also to bound the errors that result from the transformation to the reduced basis. An algorithm for computing a lattice basis from a set of generators was given by Buchmann and Pohst [BP87] and Buchmann and Kessler [BK93]. They do not bound all of the errors, though. In order to bound the errors that result from the transformation to the reduced basis we need better bounds on the coeﬃcient sizes in the transformation than what they give. For that reason we will present their algo- rithm and improve their analysis. Both [BP87] and [BK93] analyze the same algorithm for computing a basis, so we give an outline of their algorithm and bounds before giving our analysis and proving the error bounds. The setup is as follows. Suppose that vectors a1,...,ak∈ Rngenerate an r-dimensional lattice L. We are given ap- proximations ˆa1,..., ˆak∈Znsuch that ||ai−ˆai/2q|| ≤√n/2q+1. We want to compute a set of vectors ˆc1,..., ˆcr which approximate a basis c1,...,crofLwith some pre- cisionq/prime, which will necessarily be smaller than q. The al- gorithms in [BP87] and [BK93] compute such a basis via a reduction to LLL. We will use their algorithm to solve this problem, but we need a better analysis to make sure that the output accuracy q/primeis not too much smaller than q. This is done in Theorem 4.3 below. 4.3.1 The Buchmann-Pohst algorithm The reduction to LLL takes the input vectors ˆa1,..., ˆak and creates a new lattice with basis deﬁned from the con- catenated vectors ˜aj= (e j,ˆaj) (1 ≤j≤k), where ejdenotes the j-th unit vector in Zk. These vectors ˜a1,...˜ak∈Zn+kare clearly linearly independent and form a basis of the lattice ˜L=Lk j=1Z˜aj.Note that with this setup, the bottom of the matrix has vectors ⌊2qai⌉as the basis vectors, where ⌊·⌉rounds each

Question 746 multiple-choice

In quantum computing, the Quantum Fourier Transform (QFT) and its approximate variant (AQFT) are essential for algorithms that detect periodicity in quantum states, such as Shor’s factoring algorithm. AQFT reduces computational resources by sacrificing some precision, leading to a tradeoff between execution time and success probability. When using the AQFT instead of the exact QFT in a period-finding algorithm, which of the following statements accurately describes the impact on success probability and required computational runs as register size L grows large? 1) The probability of success per run approaches 1, reducing the need for repeated computations. 2) The AQFT yields a lower probability of success per run, requiring more repeated runs to match the QFT's reliability, but this increase is only polynomial in L. 3) The AQFT always produces exact phase information regardless of precision parameter m. 4) The number of required runs for AQFT increases exponentially compared to QFT as L increases. 5) AQFT and QFT have identical success probabilities for all values of m and L. 6) The probability of success per run in AQFT is independent of the precision parameter m. 7) AQFT cannot detect any periodicity in quantum states for large register sizes.

✓ Correct Answer:

The correct answer is 2) The AQFT yields a lower probability of success per run, requiring more repeated runs to match the QFT's reliability, but this increase is only polynomial in L..

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the form exp( iǫ) with |ǫ| ≤2πL/2m. The execution time of the AQFT grows as ∼Lm. 4 4 Estimating Periodicity The quantum Fourier transform, like the ordinary Fourier tr ansform, is a powerful tool for uncovering periodicities. Any periodicity in prob ability amplitudes describing a quantum state of a register in a computational basis can be e stimated (with some probability of success) by performing the QFT followed by a m easurement of the register in the computational basis. The result is obtained by reading the qubits of the register in the reversed order. For example, an interesting periodic state which plays an im portant role in Shor’s quantum factoring algorithm can be written as |Ψ/an}bracketri}ht=1√ N2L−1/summationdisplay a=0f(a)|a/an}bracketri}ht, (11) where Nis an appropriate normalisation factor and f(a) =δl, amod r. (12) It is the state of a quantum register of size Lin which the probability amplitudes f(a) occur with periodicity rand oﬀset l. If this oﬀset is unknown, a measurement performed in the computational basis cannot reveal ror any of its integer multiples directly. This is illustrated in Fig. 3(a). However, if we pe rform the QFT on the register ﬁrst and subsequently measure its state we obtain n umber ¯ cwhich, with probability greater than 4 /π2, is a multiple of 2L/rregardless the oﬀset l(cf. Ap- pendix). The probability is not equal to unity, because the ﬁ nite size of the register leads to a “broadening” of the Fourier–transformed data, as illustrated in Fig. 3(b). (This is because 2L/ris not necessarily an integer, and the quantum register can have only integer values; this is discussed in detail in the A ppendix). For the AQFT the corresponding probability, in the limit of l argeL, satisﬁes Prob A≥8 π2sin2/parenleftbiggπ 4m L/parenrightbigg . (13) This result is derived in the Appendix. The eﬀect of the appro ximation is illustrated in Fig. 4 where we plot the modulus of the amplitude of the tran sformed state |Ψ/an}bracketri}ht (with l= 9 and r= 10) for the AQFT of diﬀerent orders m. Fig. 5 shows how the phase of the transformed state becomes corrupted when mbecomes smaller. If the quantum Fourier transform forms a part of a randomised algorithm then the computation can be repeated several times in order to amp lify the probability of the correct result. In such cases the performance of the AQFT is only polynomially less eﬃcient than that of the QFT. For example, consider Shor ’s quantum factoring algorithm and substitute the AQFT for the QFT. In order to obt ain a correct factor with a prescribed probability of success, we have to repeat t he computation several times. Let kandk′be the number of runs respectively with the QFT and the AQFT so that we obtain the same probability of getting at least one correct result, i.e., 1−(1−p)k= 1−(1−p′)k′. (14) 5 Herep= 4/π2andp′=8 π2sin2/parenleftig 4 πm L/parenrightig are the corresponding probabilities of success in a single run. The ratio k′/kscales as k′ k=log/parenleftig 1−4 π2/parenrightig log/parenleftig 1−8 π2sin2/parenleftig 4 πm

Question 747 multiple-choice

Quantum shadow tomography is a technique that enables efficient estimation of multiple properties of a quantum state using a limited number of measurements. The efficiency of these protocols depends on properties of the observables and the structure of the associated operators. Which property of an observable operator ensures that the variance of its estimator depends only on its traceless part, and why is this significant for quantum measurement protocols? 1) The variance relies solely on the traceless part because the estimator and expectation value differ only by terms orthogonal to the identity, simplifying variance calculations and sample complexity bounds. 2) The variance is determined by the full operator because both diagonal and off-diagonal elements contribute equally to quantum fluctuations. 3) The variance depends exclusively on the trace of the operator since it encodes all information about measurement outcomes. 4) The traceless part is irrelevant; only the commutator with the density matrix matters for the variance. 5) The variance is minimized when the observable is proportional to the identity, making the traceless part unnecessary. 6) The dependence on the traceless part arises only in classical measurement protocols, not quantum protocols. 7) The variance depends on the operator's eigenvalues, regardless of whether they are associated with the identity component.

✓ Correct Answer:

The correct answer is 1) The variance relies solely on the traceless part because the estimator and expectation value differ only by terms orthogonal to the identity, simplifying variance calculations and sample complexity bounds..

📚 Reference Text:

To simplify the calculations, we can utilize a trick exploited in [1], namely that the variance Var (ˆ oi) only depends on the traceless part of the operator Oi, which is defined as O(0) i:=Oi−Tr(Oi)I d. This follows by using that Var (ˆ oi) =E/bracketleftig (ˆoi−Tr(Oiρ))2/bracketrightig and observing that ˆ oi−Tr(Oiρ) only depends on the traceless part of Oi. In fact, by denoting ˆ ρ=M−1/parenleftbig U†|b⟩⟨b|U/parenrightbig , we have: ˆoi−Tr(Oiρ) = Tr(Oiˆρ)−Tr(Oiρ) = Tr/parenleftig O(0) iˆρ/parenrightig −Tr/parenleftig O(0) iρ/parenrightig , (323) where we used that Tr(ˆ ρ) = 1, sinceM−1is trace-preserving (because Mis trace-preserving). Therefore, by substituting OiwithO(0) iin Eq.(322), the right-hand side simplifies nicely: Tr/parenleftigg/parenleftig ρ⊗M−1/parenleftig O(0) i/parenrightig ⊗M−1/parenleftig O(0) i/parenrightig/parenrightig/parenleftigg/summationdisplay π∈S3Vd(π)/parenrightigg/parenrightigg (324) = 0 + Tr(ρ) Tr/parenleftbigg M−1/parenleftig O(0) i/parenrightig2/parenrightbigg + 0 + Tr/parenleftbigg ρM−1/parenleftig O(0) i/parenrightig2/parenrightbigg + Tr/parenleftbigg ρM−1/parenleftig O(0) i/parenrightig2/parenrightbigg + 0 (325) = (d+ 1)2Tr/parenleftig O(0)2 i/parenrightig + 2(d+ 1)2Tr/parenleftig ρO(0)2 i/parenrightig , (326) where we used the fact, according to Eq.(315), that M−1/parenleftig O(0) i/parenrightig = (d+ 1)O(0) iis traceless. Sub- stituting this expression back into Eq.(319), we have: Var (ˆoi) =(d+ 1)2 (d+ 1)(d+ 2)/parenleftig Tr/parenleftig O(0)2 i/parenrightig + 2 Tr/parenleftig ρO(0)2 i/parenrightig/parenrightig −Tr/parenleftig O(0) iρ/parenrightig2 (327) ≤Tr/parenleftig O(0)2 i/parenrightig + 2 Tr/parenleftig ρO(0)2 i/parenrightig (328) ≤3 Tr/parenleftig O(0)2 i/parenrightig , (329) where in the second step we used that I−ρis positive semi-definite, and therefore Tr/parenleftig ρO(0)2 i/parenrightig ≤ Tr/parenleftig O(0)2 i/parenrightig . Finally, we conclude the proof by noting that Tr/parenleftig O(0)2 i/parenrightig ≤Tr/parenleftbig O2 i/parenrightbig . 42 Using the previous bound on the variance, we have that a number of copies N=O/parenleftbigg ε−2log (2M/δ) max i∈[m]/bracketleftbig Tr/parenleftbig O2 i/parenrightbig/bracketrightbig/parenrightbigg (330) suffices to estimate, for each i∈[m],Tr(Oiρ)up to precision εand with success probability at least 1−δ. Therefore, if we want to estimate observables with bounded 2-norm, we have that the protocol is sample-efficient. However, it is important to emphasize that sampling efficiency does not imply that the protocol described is computationally efficient; in fact, to guarantee it, we must also be able to compute the unbiased estimator in Eq. (302) in a time-efficient manner. This is known for observables that have a particular structure such as stabilizer states (i.e., states constructed by the application of a Clifford circuit to a computational basis state); in this case, one can take advantage of the fact that computing the overlap between two stabilizer states can be done efficiently in classical computational time [72]. See [1] for more details and also for checking the interesting case in which the uniform distribution over the unitary set formed by the tensor product of single-qubit Clifford gates is considered, which is better known as “random Pauli basis”. In this case, it turns out that the shadow protocol is both sample and time efficient for local observables like Paulis supported on a constant number of qubits. To explore additional examples of unitary distributions, one may refer for example to [96], which discusses the utilization of the fermionic Gaussian unitaries distribution, or to [97, 98], where unitary distributions generated by local

Question 748 multiple-choice

Group extensions and nilpotency are central themes in the study of abelian $p$-groups, where structural properties such as divisibility, exponent, and subgroup composition critically influence the behavior of extensions. Understanding when every extension of two abelian $p$-groups is nilpotent requires deep insight into infinite group theory and automorphism actions. Which of the following statements accurately describes the necessary and sufficient condition for every extension of an abelian group $A$ by the cyclic group of order 2 to be nilpotent, for the case $p=2$? 1) $A$ must be torsion-free and divisible. 2) $A$ must be a 2-group of finite exponent. 3) $A$ must be a 2-group of infinite exponent and non-divisible. 4) $A$ must have a homocyclic basic subgroup of infinite rank. 5) $A$ must be finite and simple. 6) $A$ must be a direct sum of copies of the cyclic group of order 4. 7) $A$ must be countable and have infinite exponent.

✓ Correct Answer:

The correct answer is 2) $A$ must be a 2-group of finite exponent..

📚 Reference Text:

Can. J. Math., Vol. XXXVIII, No. 5, 1986, pp. 1025-1052 NILPOTENT EXTENSIONS OF ABELIAN/^-GROUPS JOSEPH BUCKLEY AND JAMES WIEGOLD This paper arose out of an attempt to solve the following problem due to Suprunenko [5, Problem 2.77]. For which pairs of abelian groups A, B is every extension of A by B nilpotent? We obtain complete answers when A and B are /^-groups and (a) A has finite exponent or (b) B is divisible or (c) A has infinite exponent, is countable and B is non-divisible. The structure of a basic subgroup of A plays a central role in cases (b) and (c). 1. Introduction. At the outset we must say that the problem is too difficult to solve in complete generality. If G/A = 2?, then the nilpotency of G depends solely on properties of the associated homomorphism 0.B —» Aut A. Thus for instance if A is torsion-free and B finite, G is nilpotent if and only if the extension is a central one, and we would need detailed information on finite subgroups of the group Aut A. Such information is scanty apart from isolated results about free abelian groups of low rank. One could cite the examples of J. de Groot [2] of torsion free A for which Aut A is of order 2. Similar comments apply when A is a /?-group and B a g-group for distinct primes p, q\ nilpotent extensions again have to be central. Thus we have focused attention on the case where A and B are both /7-groups for the same prime p. Here, and generally, extensions constructed using non-nilpotent wreath products provide us with limits for possible general statements. Thus if A has a direct factor isomorphic to a restricted direct power *-<'> of a non-trivial group K, where |/| is no less than \B\ and B is infinite, then there is a non-nilpotent extension of A by B. Other insights come from looking at wreath products of the form C m wr C' „. For p = 2 everything is easy, due to the existence of the inverter automorphism; it is pretty obvious (Theorem 3.6) that every extension of an abelian group A by C 2, the cyclic group of order 2, is nilpotent if and only if A is a 2-group of finite exponent. Indeed for general p, the case where A has finite exponent is straightforward. When A has infinite exponent, everything is more complex and we are led to consider the three cases where B is divisible, finite, and infinite but not divisible respectively. In each case the conditions obtained for nilpotency are certain finite-rank conditions on the divisible part of A and on the homocyclic components of a basic subgroup of A. In all cases our results extend to general nilpotent Received May 27, 1983. 1025 https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press 1026 J. BUCKLEY AND J. WIEGOLD #, and we are able to give complete answers to the Suprunenko problem in the following cases:

Question 749 multiple-choice

In topological group theory, pseudocompact topologies provide a framework for studying groups where every continuous real-valued function is bounded. The concept of admissible cardinality is key to understanding which groups can support such topologies, especially in the context of infinite cardinals and set-theoretic assumptions. For infinite cardinals α and y, which condition is equivalent to the existence of a group of cardinality y admitting a pseudocompact group topology of weight α under ZFC + SCH? 1) y ≤ α and y ≥ m(α) 2) m(α) ≤ y ≤ α 3) m(α) < y < 2^α 4) y = 2^α 5) m(α) < y = α 6) m(α) < y ≤ 2^α 7) y < m(α) or y > 2^α

✓ Correct Answer:

The correct answer is 3) m(α) < y < 2^α.

📚 Reference Text:

pseudocompact group topology of weight a; appropriating a term from [14] we say that a cardinal y is admissible if some group of cardinality y admits a pseudocompact group topology. (According to this use of the term, the positive finite cardinals are admissible. Though we have little use for the finite cardinals in this context, this convention simplifies the Statement of some of our theorems and is otherwise harmless.) Using different notation and terminology, the infinite admissible cardinals were characterized in the axiom System ZFC + SCH (in terms of their arithmetic properties) by van Douwen [17]; see [8] for a more f ll account, and see [7] for proofs in succinct language of several of the results of [17]. In any event it is clear that for infinite cardinals α and y the following two conditions are equivalent: m (a) < y < 2a; some group G with | G | = y satisfies G e & (a). It should be noted explicitly however that not every group G such that | G \ is admissible admits a pseudocompact group topology. Isolated examples are given in [7], and the existence of large f amilies of such groups G will become apparent in Theorems 3.17 and 3.19 below. When a shift of emphasis is appropriate, we write ae.i/(y) in place of m(a) < y < T. § 3. Finding some pseudocompactifiable groups 3.1. Lemma. Lei α > ω and let F be a finite Abelian group with \F\ > 1. Then Bereitgestellt von | Technische Informationsbibliothek Hannover Angemeldet Heruntergeladen am | 27.03.18 14:50 328 W. W. Comfort, D. Remus Proof. It is enough to treat the case F = Z (pr) with p € P, 0 < r < ω; for if the result is known for such groups then from the isomorphism F « 0?= { Z (p r*) = [~] "= x Z (p · ') will follow i=l 2* 2« i=l 2* The case F= Z(/?r) is not difficult to prove; see Hewitt and ROSS [24] (Lemma 4.5). D 3.2. Theorem. Lei α «rf 7 6e infinite cardinals and let F be afinite Abelian group with \F\ > 1. IfF" contains a Gd-dense subset of cardinality y then Fa contains a Gd-dense copy of 0y F. Proof. Let ψ : F* -> @20lFbQ an isomorphism guaranteed by 3.1, let Z) be a G^-dense subset of F* with \D\ = y, and set with πξ : φ2«^ "* ^ the canonical projection. From 7 > ω and \F\ 1. Then the following Statements are equivalent. (a) m (a) < y

Question 750 multiple-choice

Efficient algorithms for scaling and normalizing tensors are fundamental in quantum information theory, machine learning, and computational mathematics. These algorithms often rely on properties of parameterization maps, symmetry groups, and randomized numerical techniques. Which statement best describes why the scaling algorithm for tensors with Gaussian integer entries and rational spectra is considered efficient and practical for high-dimensional applications? 1) It always produces exact marginals matching the target spectra in finite steps. 2) It uses non-randomized methods to guarantee deterministic success on every input. 3) It only applies to tensors parameterized by identity maps without exploiting symmetry. 4) It requires exponentially many parameters relative to the tensor's number of entries. 5) It does not utilize any group actions or equivariant properties in its computation. 6) It identifies highest-weight vectors without relating to parabolic subgroups. 7) It operates in randomized polynomial time, leveraging efficient parameterizations and group symmetries to ensure with high probability that marginals closely approximate the target spectra.

✓ Correct Answer:

The correct answer is 7) It operates in randomized polynomial time, leveraging efficient parameterizations and group symmetries to ensure with high probability that marginals closely approximate the target spectra..

📚 Reference Text:

of degree degpq d. Thus, the constant Min Proposition 6.9 is given by M2dK. Example 6.4 (All tensors) .IfXPpTenpn0;n1;:::;ndqqis the space of all tensors of a given format, as in Problem 1.1, then we can simply choose as the identity map. Thus, degpq1, soM2K. Example6.5 (Matrixproductstates) .Forsimplicity,weonlydiscusstranslation-invariantmatrixproduct states(see, e.g., [ VMC08]forthegeneral deﬁnition). Givenafamilyof NN-matricestMjuj1;:::;n, deﬁne a corresponding tensor in Tenp1;n;:::;nqby XrtMjus1;j1q;:::;jpdq:trrMjp1qMjpdqs: Theclosure Xofthe setof alltensorsof thisform iscalledthe varietyof matrixproduct states withbond dimension N. It is clear that Xis aG-stable subvariety of PpTenp1;n;:::;nqq. Moreover, :PpCnNNq99KTenp1;n;:::;nq;rtMjusÞÑXrtMjus isdominant. Itfollowsthat isagoodparametrizationof Xandthat Xisirreducible. Since degpqd, the constant Min Proposition 6.9 is given by M2dK. Notethatweparametrizeatensorwith ndentriesbyonly N2nparameters–thisisthepowerofmatrix productstates. Notealso that is equivariant with respect to the natural GLpnq-action on PpCnNNq– so we can implement Algorithm 1 by working solely in the small parameter space. The following is the main result of this section. Theorem 6.6 (Tensor scaling for good parametrizations) .LetXPpTenpn0;n1;:::;ndqqbe aG- stable irreducible projective subvariety. Let :PpCpq99KXbe a good parametrization (in the sense of Deﬁnition 6.1) with Gaussian integer coeﬃcients of bitsize no more than b. Also, letpPPpn1;:::;ndq withrationalentriesofbitsizenomorethan bsuchthatppiq ni¡0foralli1;:::;d. Finally,let "¡0. Then, with probability at least 1/2, Algorithm 2 either correctly identiﬁes that pRpXq, or it outputs XPXand gPGsuch that the marginals of YgXare"-close top(in fact, satisfy Eq. (5)). We’llalsoseethat,justlikeAlgorithm1,Algorithm2canbemodiﬁedtogivethesameguarantees even if the rank of some diagpppiqqis not full. Theorem6.7. LetbeasinthestatementofTheorem6.6,andfurthersupposethatdimensionofthedomain ofispolypNqand on inputs of bit-complexity c,can be computed in polypN;cqtime. Then there is a randomized algorithm running in time polypN;1{"q, that takes as input XPTenpn0;n1;:::;ndqwith Gaussianintegerentries(speciﬁedasalistofrealandcomplexparts,eachencodedinbinary,withbitsize ¤b) andpPPpn1;:::;ndqwith rational entries (speciﬁed as a list of numerators and denominators, each encoded in binary, with bit size ¤b). The algorithm either correctly identiﬁes that pRpXq, or it outputs a scalinggPGandXPXsuchthatthemarginalsof gXare"-closetothetargetspectra p. HereNisthe total bit-size of the input, N2n0n1ndb2pn1ndqb. 44 Remark 6.8. Themaps encounteredinthethreeexamplesbeforeTheorem6.6areindeedcomputablein time polynomial in the input size; as a corollary, there are eﬃcient (in the sense of Theorem 1.7) algorithms to correctlydeclare pRpXqoroutputsascaling gPGandXPXsuchthatthemarginalsof gXare"-close tothetargetspectra pwithprobabilityatleast 1{2forX;GasinExample6.2,Example6.3,Example6.4, andExample6.5. NotethatthatnorandomnessisrequiredatallforExample6.2,sothemainalgorithmic resultof[ BGO17]isaspecialcaseofTheorem6.7. Example6.3isalreadycoveredbyTheorem1.7,whichis a special case of Theorem 6.7. Input:pPPpn1;:::;ndqwith rational entries (speciﬁed as a list of numerators and denominators, each encoded in binary, with bit size ¤b) such thatppiq ni¡0for alli1;:::;d. Output: Either the algorithm correctly identiﬁes that pRpXq, or it outputs XPXandgPG suchthatthemarginalsof Y:gXsatisfyEq.(5);inparticularthemarginalsare "-closetothe target spectra p. Algorithm: 1.Let`¡0such that`ppiqhas integer entries for all i1;:::;d. LetZpZp1q;:::;Zppqqbe a vector with entries chosen independently uniformly at random from t1;:::;Mu, where M:2 degpqKandK:p`dmaxd i1niqdmaxd i1n2 i. SetX:pZq. 2. Fori1;:::;d, if the marginal piq Xis singular then output pRpXqand return. Otherwise, set g:pIn1{kXk;In2;:::;Indq. 3.Fort1;:::;T :16kln 2 "2°d i0log2pniqbdeg plog2plog2Mq , repeat the following: •ComputeY:gXand, fori1;:::;d, the one-body marginals piq Yand the dis- tances"piq:kpiq Ydiagpppiq Òqktr. •Select an index iPt1;:::;dufor which"piqis largest. If "piq¤", outputgand return. •Compute the Cholesky decomposition piq YRpiqpRpiqq:, whereRpiqis an upper- triangular matrix. Update gpiqÐdiagpppiq ni;:::;ppiq 1q1{2pRpiqq1gpiq. 4. OutputpRpXq. Algorithm 2: Scaling algorithm for Theorem 6.6 6.3 Randomization step We have the following extension of Corollary 2.7 for the parabolic subgroup. This allows us to ﬁnd andelementonwhichsomehighestweightvectordoesnotvanish. Theproofisalmostidentical to that of Corollary 2.7, but instead of scaling

Question 751 multiple-choice

In finite group theory, particularly the study of p-groups of nilpotency class 2, the structure of centralizers and the actions of elements on subgroups are crucial for classification. Elements of order 4 and their centralizers play a significant role in determining group properties. Suppose G is a finite group of nilpotency class 2, containing a maximal normal abelian subgroup A such that every element outside A acts by inversion on A. What is the group-theoretic structure of G under these conditions? 1) G is a direct product of copies of C4 and C2 2) G is a nontrivial central extension of a cyclic group of order 8 3) G is a quaternion group of order 8 4) G is a dihedral group of order 8 5) G is a non-abelian group with abelian centralizers for elements of order 2 only 6) G is a semidirect product, G = A ⋊ C2, where C2 acts on A by inversion 7) G is a Frobenius group with abelian kernel

✓ Correct Answer:

The correct answer is 6) G is a semidirect product, G = A ⋊ C2, where C2 acts on A by inversion.

📚 Reference Text:

½c;d/C138¼z, then hbc;adiGQ8andhbc;aditGsince ½a;bc/C138¼zu,½d;bc/C138¼1,½b;ad/C138¼1,½c;ad/C138¼zu. This contradicts our assumption. (5) If ½b;d/C138¼zand ½c;d/C138¼uz, then ½ad;c/C138¼zand ðc/C1adÞ2¼1 so that cadis an involution in G/C0W, a contradiction. (6) If ½b;d/C138¼uand ½c;d/C138¼u, thenhc;diGQ8andhc;ditGsince ½a;c/C138¼u, ½b;c/C138¼1,½a;d/C138¼1,½b;d/C138¼u. This contradicts our assumption. (7) If ½b;d/C138¼uand ½c;d/C138¼z, then ½bc;ad/C138¼1,ðbcÞ2¼zu,ðadÞ2¼zu,s ot h a t bcad is an involution in G/C0W, a contradiction. (8) If ½b;d/C138¼uand ½c;d/C138¼uz, then ½ad;c/C138¼zand ðc/C1adÞ2¼1 so that cadis an involution in G/C0W, a contradiction. In the rest of the proof we assume that jFðGÞj>4. First suppose that Ghas an element vof order 4 such that CGðvÞis non-abelian. By Lemma 1.16, there is a quaternion subgroup Q¼ha;bicontained in CGðvÞsuch that QVhvi¼f1g. SetZðQÞ¼hzi. Let Abe a maximal abelian subgroup of Gcontaining hvi/C2haiGC4/C2C4. It follows that Ais a maximal normal abelian subgroup of G since A>ZðGÞandGis of class 2. Moreover, we have W1ðAÞ¼W1ðGÞcZðGÞand FðGÞcW1ðAÞ. Suppose there is xAAsuch that x2Bhv2;ziand sohx2;v2;ziGE8. By Lemma 1.17, bdoes not commute with xand sohb;xiGH2since x20b2¼z. But then hb;x;vi¼hb;xi/C2hviGH2/C2C4which contradicts Lemma 1.16. We have proved that jW 1ðAÞj ¼4, so that Ais of type ð4;4;2;...;2Þ,AQ¼Ahbi, CGðhv;aiÞ¼A,FðAQÞ¼hv2;ziGE4and ðAQÞ0¼hzi. Since jFðGÞj>4, there is an element uAG/C0ðAQÞsuch that u2Bhv2;ziand therefore hu2;v2;ziGE8. Suppose that ucommutes with an element xof order 4 in hv;ai. We may set hv;ai¼hx;yifor a suitable element yof order 4 in hv;ai. The fact that CGðhx;yiÞ¼Aimplies that ½u;y/C13801, so that hu;yiGH2because u20y2. But W1ðhu;yiÞ¼hu2;y2iand so x2Bhu2;y2i(because hu2;x2;y2iGE8). We have hu;y;xi¼hu;yi/C2hxiGH2/C2C4, contrary to Lemma 1.16. We have proved that udoes not commute with any element of order 4 in hv;ai. Suppose that udoes not invert an element xof order 4 in hv;ai. Thenhu;xiGH2implies that xu¼xu2.I fy is any element of order 4 in hv;aiwith y20x2andyu0y/C01, then yu¼yu2. ButOn minimal non-abelian subgroups in ﬁnite p-groups 301 then ðxyÞu¼ðxu2Þðyu2Þ¼xy, a contradiction. Hence yu¼y/C01and noting that ðxyÞ2¼x2y20x2, we also get ðxyÞu¼ðxyÞ/C01. But we get then xu¼ð ðxyÞy/C01Þu¼ðxyÞuðyuÞ/C01¼ðxyÞ/C01y¼x/C01y/C01y¼x/C01; a contradiction. We have proved that uacts by inversion on hv;ai.I fucommutes with any element sAQ/C0hai¼Q/C0A,t h e n s2¼zand hu;v;si¼hu;vi/C2hsiGH2/C2C4; contrary to Lemma 1.16. Hence udoes not commute with any element of Q/C0hai. Suppose that uinverts an element wAQ/C0haiso that ½u;w/C138¼z. But then ½u;aw/C138¼½u;a/C138½u;w/C138¼zz¼1, a contradiction. Hence udoes not invert any element in Q/C0hai. Since hu;biGhu;abiGH2, the above results imply ½u;b/C138¼½u;ab/C138¼u2. But then u2¼½u;ab/C138¼½u;a/C138½u;b/C138¼zu2which gives z¼1, a ﬁnal contradiction. We have proved that the centralizer of each element of order 4 is abelian. Let again Abe any maximal normal abelian subgroup of G, so that we have W1ðGÞ¼W1ðAÞ¼ZðGÞsince CAðxÞ¼W1ðAÞfor each xAG/C0A. Also W1ðAÞ

Question 752 multiple-choice

Quantum computing leverages group-theoretic structures and superpositions to solve complex problems exponentially faster than classical algorithms. The shifted subset problem is a generalization of the abelian hidden subgroup problem, with applications in coding theory, cryptography, and combinatorics. Which statement best describes the exponential advantage provided by quantum algorithms for the shifted subset problem within the Boolean cube {0, 1}ⁿ? 1) Quantum algorithms can identify specific subsets, such as Hamming spheres, in polynomial time, whereas classical algorithms are believed to require exponential time. 2) Quantum algorithms for the shifted subset problem are only efficient for non-abelian groups and do not outperform classical methods in the Boolean cube. 3) Classical algorithms can efficiently solve the shifted subset problem using brute-force search in polynomial time. 4) Quantum algorithms rely exclusively on Grover’s search and do not exploit group structure for speedup. 5) The shifted subset problem does not admit any exponential quantum-classical separation for any known class of subsets. 6) Quantum algorithms for shifted subsets are limited to factoring integers and do not apply to Hamming spheres. 7) Classical algorithms generally outperform quantum algorithms for subset identification in group-theoretic problems.

✓ Correct Answer:

The correct answer is 1) Quantum algorithms can identify specific subsets, such as Hamming spheres, in polynomial time, whereas classical algorithms are believed to require exponential time..

📚 Reference Text:

Title: Quantum algorithms for shifted subset problems Year: 2008 Paper ID: b3e8076c484da8406352867cf6551bcb08d52432 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b3e8076c484da8406352867cf6551bcb08d52432 Abstract: We consider a recently proposed generalisation of the abelian hidden subgroup problem:the shifted subset problem. The problem is to determine a subset S of some abeliangroup, given access to quantum states of the form |S + x〉, for some unknown shift x.We give quantum algorithms to find Hamming spheres and other subsets of the booleancube {0, 1}n. The algorithms have time complexity polynomial in n and give rise toexponential separations from classical computation.

Question 753 multiple-choice

Quantum algorithms for computational group theory often exploit the structure of groups and their subgroups using techniques such as the quantum Fourier transform and specialized combinatorial procedures. These methods are particularly relevant for problems like the Hidden Subgroup Modulo Center (HSMC) in certain black-box group settings. Which combination of algorithmic components is essential for efficiently solving the Hidden Subgroup Modulo Center (HSMC) problem in semi-elementary black-box groups with unique encoding? 1) Classical subgroup membership testing and state purification 2) Grover search and abelian group decomposition 3) Quantum walk algorithms and non-abelian Fourier analysis 4) Quantum Fourier transform and efficient zero-sum subsequence finding 5) Measurement in the computational basis and group element sorting 6) Classical discrete logarithm computation and oracle-based state preparation 7) Randomized sampling and post-selection on coset representatives

✓ Correct Answer:

The correct answer is 4) Quantum Fourier transform and efficient zero-sum subsequence finding.

📚 Reference Text:

However, we can compute a state with essentially the same effect using several copies of the subgroup state and by applying an algorithm that ﬁnds zero sum subsequences of sufﬁcientlylong sequences of elements of L. Assume that we have a procedure that, for some S=S(p,n), given an element y =(y1,..., yS)∈LScomputes a non-empty subset J(y)of{1,..., S}such that/summationtext j∈J(y)yj=0. Then, for a sequence |a1H/angbracketright...|aSH/angbracketrightwe compute ﬁrst |L|−S/2/summationdisplay y∈LS|y/angbracketright|Py1(a1H)/angbracketright...|PyS(aSH)/angbracketright by applying the Fourier method outlined above component-wise. We next com- pute1√|G|/summationtext x∈G|x/angbracketrightin a fresh register and multiply by x−1the jth component of 123 Zero sum subsequences and hidden subgroups Page 13 of 23 14 |Py1a1H/angbracketright...|PySaSH/angbracketrightifj∈J(y).L e tχy:{1,..., S}→{ 0,1}denote the charac- teristic function of J(y). Then the state we obtained is1√|G|/summationtext x∈G|x/angbracketright|ψ(x)/angbracketright, where |ψ(x)/angbracketright=| L|−S/2/summationdisplay y∈LS|y/angbracketright|Py1a1Hx−χy(1)/angbracketright...|PySaSHx−χy(S)/angbracketright. Consider the term of |ψ(x)/angbracketrightcorresponding to any y.A s J(y)is non-empty, we have that for x1,x2not in the same left coset of LH, the appropriate terms of |ψ(xi)/angbracketright are orthogonal. As |y/angbracketrightalso appears in the corresponding term, we have that |ψ(xi)/angbracketright are also orthogonal. On the other hand, if x1,x2are in the same left coset of H, then these states are equal term by term. Finally, for x∈L,b y( 2), the term for yonly gets a phase change by/producttext j∈J(y)ω−(yj,x)=ω/summationtext j∈J(y)(yj,x)=ω0=1 by the choice of J(y). It follows that if x1and x2are in the same left coset of LH,|φ(x1)/angbracketright=|φ(x2/angbracketright. Thus our state is a puriﬁcation of /Xi1G,LH. As this holds for any ﬁxed S-tuple of left cosets ofH, by linearity we also obtain a puriﬁcation of /Xi1G,LHif we apply the procedure to copies of a puriﬁcation of /Xi1G,H. We obtained the following. Lemma 2 Assume that we have an exact quantum procedure (e.g., a deterministic poly- nomial time algorithm) that, given any sequence y 1,..., ySof S=S(p,n)elements ofZn p, in time T (p,n)≥S(p,n)ﬁnds a non-empty subset of {1,..., S}such that/summationtext j∈Jyj=0. Then we have an exact quantum procedure using n quantum Fourier transforms modulo p that converts S (p,n)copies of (a puriﬁcation) of /Xi1G,Hto a copy of (a puriﬁcation of) /Xi1G/L,HL/Lwhere L is subgroup of the center of G isomorphic toZn pin time T (p,n)poly(log|G|). In words, provided that we can efﬁciently solve the zero sum subsequence problem in the subgroup L∼=Zn pcontained in the center of G, we can convert copies of the subgroup state for HinGto a subgroup state for the subgroup HL/L(that is, “ H modulo L”) in the factor group G/L. Using the lemma in iteration and applying the exact abelian hidden subgroup algo- rithm of [ 16], we can derive the following. Proposition 2 Let G be a semi-elementary black-box group with unique encoding of order pn. Assume that the quantum Fourier transform modulo p and its inverse can be implemented by an efﬁcient exact algorithm and that, like in Lemma 2, we have an exact method to ﬁnd zero sum subsequences of sequences of S (p,n)elements of Zn pin time T(p,n)≥S(p,n). Then the problem HSMC can be solved by an exact quantum algo- rithm that uses poly(T(p,n)O(c)/lscript)elementary operations, poly(T(p,n)O(c)log|G|) applications

Question 754 multiple-choice

Poisson-Lie groups and groupoids are advanced structures in differential geometry that combine group-like symmetries with Poisson brackets, playing a crucial role in integrable systems and quantum group theory. Their classification often involves the behavior of certain commutator expressions and compatibility conditions for group multiplication. Which condition characterizes a triangular Poisson-Lie group, distinguishing it from the quasitriangular case? 1) The commutator expression [ρ12, ρ13] + [ρ13, ρ23] + [ρ12, ρ23] equals zero. 2) The bivector field Πρ is always proportional to a symmetric tensor T. 3) The bivector field Πρ does not arise from skew-symmetric elements in Λ^2g. 4) The group multiplication fails to be a Poisson map. 5) The commutator expression [ρ12, ρ13] + [ρ13, ρ23] + [ρ12, ρ23] is always proportional to a non-zero symmetric tensor. 6) The group is required to be abelian. 7) The coisotropic submanifolds must coincide with symplectic leaves.

✓ Correct Answer:

The correct answer is 1) The commutator expression [ρ12, ρ13] + [ρ13, ρ23] + [ρ12, ρ23] equals zero..

📚 Reference Text:

{f,g}=df⊗dg(Π). Recall that ( G,{,}) is called a Poisson-Lie group if the multiplication map m:G×G→Gis a Poisson map. Letρ∈Λ2gand consider the following bivector ﬁeld: Πρ=Rρ−Lρ, whereRρ(resp.Lρ) is the left-invariant (resp. right-invariant) bivector ﬁeld satisfying ( Rρ)|e=ρ(resp. (Lρ)|e=ρ); in other words, Rρ,Lρstand for the translates of ρby right and left shifts respectively. Proposition 5.1 (Drinfeld). The bivector Πρdeﬁnes a Poisson-Lie group structure on Gif and only if [ρ12,ρ13]+[ρ13,ρ23]+[ρ12,ρ23]∈(Λ3g)g. When this is the case, Gis called a coboundary Poisson-Lie group. Two cases are of special interest: 1. Theexists T∈(S2g)gsuchthat[ ρ12,ρ13]+[ρ13,ρ23]+[ρ12,ρ23] =1 4[T12,T23]. This implies that r=ρ+1 2Tsatisﬁes the classical Yang-Baxter equation, and Π ρ=Rr−Lr. In this case Gis called a quasitriangular Poisson-Lie group. 2. We have [ ρ12,ρ13]+ [ρ13,ρ23] +[ρ12,ρ23] = 0. In this case, Gis called a triangular Poisson-Lie group. 5.2. Poisson-Lie groupoids. It turns out that, in order to generalize this correspondence to the dynamical case, groups must be replaced by groupoids. Recall that a groupoid is a (small) category where all morphisms are is omor- phisms. It is equivalent to the following data: two sets XandP(the set of mor- phisms, or the groupoid itself, and the set of objects, or the base , respectively), two surjective maps s,t:X→P(the source and target maps), an injective mapE:P→X(the identity morphisms), a map m:{(a,b)∈X×X|t(a) = s(b)} →X(m(a,b) =b◦a, the composition of morphisms), and an involution 12 i:X→Xsuch that s(i(x)) =t(x), t(i(x)) =s(x), m(i(x),x) = Id s(x)and m(x,i(x)) = Id t(x)for allx∈X, satisfying some obvious axioms. One can visualize elements of Xas arrows s(a)a→t(a). Note that when |P|= 1, the notion of a groupoid coincides with the notion of a group. A Lie groupoid is a groupoid with a smooth structure (in particular, th e sets of objects and the sets of morphisms are smooth manifolds and the structure maps are smooth, see [M]). NowwewouldliketogeneralizethenotionofaPoisson-Liegrouptogro upoids. The usual deﬁnition does not generalize directly since if Xis a Lie groupoid and a Poisson manifold then the set of points ( a,b)∈X2for which the multiplica- tion is deﬁned is not necessarily a Poisson submanifold, so we cannot r equire that the multiplication map be Poisson. But this diﬃculty can be bypass ed using the following observation: Proposition 5.2. LetX,Ybe two Poisson manifolds and let f:X→Ybe a smooth map. Consider the graph Γf={(x,f(x))} ⊂X×Y, whereYis the manifold Y, with the opposite Poisson structure {,}Y=−{,}Y. Thenfis a Poisson map if and only if Γfis a coisotropic submanifold of X×Y, i.e if and only if for any z∈Γf,(TzΓf)⊥⊂T∗ z(X×Y)is an isotropic subspace with respect to the Poisson form ΠonTz(X×Y)∗. This gives rise to the following notion of a Poisson-Lie groupoid, ﬁrst in tro- duced by Weinstein [W]. Deﬁnition. ALie groupoid Xwith a Poissonstructureis called a Poisson-Lie groupoid if Γm⊂X×X×Xis a coisotropic submanifold. We now restrict ourselves to a particular class of Lie groupoids. Let Gbe a Lie group, let gbe its Lie algebra, h⊂ga subalgebra and Ha Lie subgroup ofGwith Lie algebra h. LetU⊂h∗be an open set. Consider the following groupoid: X=U×G×U,P=Uwiths(u1,g,u2) =u1,t(u1,g,u2) =u2. The composition m((u1,g,u2),(u3,g′,u4)) is deﬁned

Question 755 multiple-choice

Quantum Amplitude Estimation (QAE) is a key algorithm in quantum computing, enabling efficient estimation of probabilities and expectation values with applications in fields such as finance, chemistry, and machine learning. Recent advances have focused on making QAE more practical for near-term quantum devices by reducing reliance on complex subroutines. Which characteristic distinguishes the latest asymptotically optimal QAE algorithm designed for near-term quantum hardware from earlier approaches? 1) It increases the query complexity compared to classical Monte Carlo methods. 2) It depends heavily on the implementation of the Quantum Fourier Transform (QFT). 3) It achieves optimal query complexity only by increasing circuit depth significantly. 4) It reduces error rates by requiring entanglement between all qubits at every step. 5) It applies only to quantum computers with error correction capabilities. 6) It sacrifices numerical performance for theoretical optimality. 7) It eliminates the need for the Quantum Fourier Transform while matching the theoretical lower bound for query complexity.

✓ Correct Answer:

The correct answer is 7) It eliminates the need for the Quantum Fourier Transform while matching the theoretical lower bound for query complexity..

📚 Reference Text:

Title: Modified Iterative Quantum Amplitude Estimation is Asymptotically Optimal Year: 2022 Paper ID: c401c8ed57e2e31cb8f600cc9c4a05a9038a3f8a Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c401c8ed57e2e31cb8f600cc9c4a05a9038a3f8a Abstract: In this work, we provide the first QFT-free algorithm for Quantum Amplitude Estimation (QAE) that is asymptotically optimal while maintaining the leading numerical performance. QAE algorithms appear as a subroutine in many applications for quantum computers. The optimal query complexity achievable by a quantum algorithm for QAE is $O\left(\frac{1}{\epsilon}\log \frac{1}{\alpha}\right)$ queries, providing a speedup of a factor of $1/\epsilon$ over any other classical algorithm for the same problem. The original algorithm for QAE utilizes the quantum Fourier transform (QFT) which is expected to be a challenge for near-term quantum hardware. To solve this problem, there has been interest in designing a QAE algorithm that avoids using QFT. Recently, the iterative QAE algorithm (IQAE) was introduced by Grinko et al. with a near-optimal $O\left(\frac{1}{\epsilon}\log \left(\frac{1}{\alpha} \log \frac{1}{\epsilon}\right)\right)$ query complexity and small constant factors. In this work, we combine ideas from the preceding line of work to introduce a QFT-free QAE algorithm that achieves a query complexity upper bound of $\frac{62}{\epsilon}\ln\frac{6}{\alpha}$ which matches the optimal $O\left(\frac{1}{\epsilon}\log \frac{1}{\alpha}\right)$ query complexity. We supplement our analysis with numerical experiments comparing our performance with IQAE where we find that our modifications retain the high performance, and in some cases even improve the numerical results.

Question 756 multiple-choice

In algebraic geometry, the study of abelian varieties often involves valuating points, analyzing map bijectivity, and understanding conditions under which certain group-theoretic morphisms hold. The Riemann conditions play a fundamental role in characterizing the existence of complex torus structures associated with period matrices. Which statement accurately describes a key condition required for the bijectivity of the map φ from the quotient group Gg/Γ to the set Ak in the context of abelian varieties? 1) The map φ is bijective if the field is not algebraically closed. 2) Bijectivity holds only when ord. y is negative for all components. 3) Surjectivity of φ fails when matrix entries are diagonal. 4) Bijectivity is established by constructing preimages using algebraic closure and valuation properties. 5) The Riemann conditions are unnecessary for bijectivity in this setting. 6) The mapping φ is injective but never surjective. 7) Bijectivity requires that all monomials have identical exponents.

✓ Correct Answer:

The correct answer is 4) Bijectivity is established by constructing preimages using algebraic closure and valuation properties..

📚 Reference Text:

j/ > 0 equality occurs only when t = — 1. For ord. 7/ < 0, equality occurs only when t — 0. Proo/. (1) is clear. To prove (2) and (3) note that ί(2ί + a) > 0. Thus the results hold if ord. y = 0. If ord. y > 0, s > (2ί(2ί + α) + (4ί + or)) ord. 7/, while if ord. y < 0, s > (2ί(2ί + α) - (4ί + α)) |ord. y\. The calcu- lation is now straightforward. LEMMA II.7.3. Suppose y = (j/x, , yg) e Gα wife |ord. ^| < ^ord. S = {j: ord. ^ =^ 0}. Le£ α: {1,2, , g} -> {0,2} δe ίfeβ map that a~1(2) = iS. Tfeen ord. θa(y) = -2 2]^^ lord. ^|. Proo/. By (d) of § II.2, θa{y) = 2 M7 where ord. (δjT/7) = έ s^ = Σ (ί/2^ + a^) ord. g^ + (4t, + ^) ord. By Lemma Π.7.2, s^ > -2 |ord. ys\ for ieS. Thus ord. > ~2Σjes\wά- Vj\- Also if equality is to hold we must have ^ = 0 for j &S, tj = —1 when ord. ys > 0, and tj = 0 when ord. ^ < 0. So there is only one monomial for which equality holds, and the lemma follows. LEMMA II.7.4. Situation as in Lemma II.7.3. Suppose ord. y1 Φ 0. Define β3 e {0, ± 1,2} by setting βd = aά if j > 1, βλ = — 1 if ord. y1 > 0 and βι = l if ord. 2/i < 0. Then ord. 0/2/) = |ord. y1\ - 2 Σjes lord. ^|. Proof, Entirely similar to that of Lemma II.7.3. THEOREM II.7.5. Ak is bijectίve. Proof. Theorem II.3.2 shows that ψ is 1:1. To prove ontoness suppose PeAk. Let L be a complete algebraically closed extension of k. By Theorem Π.7.1 there is a y = (^, . - ,yg) e (L*)^ with ^(7/) = P, and we may assume |ord. yά\ < \ ord. qs. Suppose ord. yx Φ 0. Define a and β as in Lemmas II.7.2 and II.7.3. Then lord. yλ\ - ord. θβ(y) - ord. θa(y) = ord. (Z^(P)/Zβ(P)) . In particular there exists an xx e k* such that ord. x1 = ord. 2/1# Similarly choose Xj e it* so that ord. Xj = ord. ^ and let a? = (x19 - - >,xg). Then https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press ABELIAN VARIETIES 91 yx~ι e UgL, so φ(yx~ι) is a unit point. Since

Question 757 multiple-choice

Quantum algorithms often leverage group representation theory to efficiently solve problems involving symmetries, such as decomposing quantum states according to group representations. In systems involving symmetric groups, projectors and unitary operators play critical roles in identifying representation content and estimating computational complexity. Which statement correctly characterizes the role of maximal character values (χmax n,k) in the construction of unitary operators used for resolving representation projectors in symmetric group-based quantum algorithms? 1) They are used only to calculate the dimension of the Hilbert space associated with each representation. 2) They determine the order of the symmetric group relevant for the algorithm. 3) They serve exclusively as normalization constants for the projection operators. 4) They are used to label the irreducible representations via Young diagrams. 5) They specify which permutation the unitary operator should implement on the quantum state. 6) They correspond to the eigenvalues measured in quantum phase estimation for arbitrary operators. 7) They appear in the exponentials defining unitary operators whose eigenvalues distinguish projectors onto representation spaces labeled by Young diagrams.

✓ Correct Answer:

The correct answer is 7) They appear in the exponentials defining unitary operators whose eigenvalues distinguish projectors onto representation spaces labeled by Young diagrams..

📚 Reference Text:

our set of unitaries that will serve in the QPE of this system. For all n, we introduce the notation T(Sn) µwhich keeps its meaning of (2.1), but use the superscript Snas we are dealing with diﬀerent symmetric group in the present formalism. As usual µ will be restricted to [k,1n−k]. We also introduce χmax n,k= max R/turnstileleftn/hatwideχR(T(Sn) k) (7.4) and deﬁne the unitary operators U(1) m,k=e2πi χmax m,kT(Sm) k⊗1 U(2) n,k= 1⊗e2πi χmax n,kT(Sn) k U(3) m+n,k=e2πi χmax m+n,kT(Sm+n) k(7.5) There operators satisfy the eigenvalue equations U(1) m,kPR R1,R2=e2πi χmax n,k/hatwideχR1(T(Sm) k) PR R1,R2, U(2) n,kPR R1,R2=e2πi χmax n,k/hatwideχR2(T(Sn) k) PR R1,R2, U(3) m+n,kPR R1,R2=e2πi χmax m+n,k/hatwideχR(T(Sn+m) k) PR R1,R2(7.6) for anyk. This will play an important role in the QPE formalism. We need to use the values of k1∈{2,3,···,k∗(m)} k2∈{2,3,···,k∗(n)} k3∈{2,3,···,k∗(m+n)} (7.7) to detect the Young diagrams R1,R2andR, respectively. The problem of detecting LR-projectors is formulated as follows: “Given the set of unitaries U(i) /lscripti,ki,U(i) /lscripti,ki,...,U(i) /lscripti,ki,i= 1,2,3,/lscript1=m,/lscript2=n, /lscript3=m+n,ki∈[ [2,k∗(/lscripti)] ], and state PR R1,R2with unknown Ri/turnstileleftni,i= 1,2, R/turnstileleftm+n, determine which PR R1,R2we have.” – 29 – JHEP05(2023)191To obtain the query and gate complexities it suﬃces to use the same reasoning as previously. The diﬀerence in this setting is that they becomes function of the pair (m,n). The number of bits required to store the diﬀerent eigenvalues of the operators are listed below: Bits required to store eigenvalues of T(Sm) k1∼t1∼k1logm Bits required to store eigenvalues of T(Sn) k2∼t2∼k2logn Bits required to store eigenvalues of T(Sn+m) k3∼t3∼k3log(n+m)(7.8) We add up the complexities for the measurements of in each sector and ﬁnd its estimate: k∗(m)/summationdisplay k1=2k1log(m)∼k∗(m)2log(m),k∗(n)/summationdisplay k2=2k2log(n)∼k∗(n)2log(n) k∗(n+m)/summationdisplay k3=2k3log(m+n)∼k∗(m+n)2log(m+n) (7.9) Ifk∗(p)

Question 758 multiple-choice

Accurately modeling halogen bonding and the σ-hole phenomenon is essential for reliable predictions in molecular simulations, particularly regarding the anisotropic charge distributions of halogen atoms. Advances in force field parameterization have led to more realistic computational representations and improved agreement with experimental data. Which force field approach is most likely to yield generalizable and accurate bulk density predictions across diverse halogenated compounds, despite not being specifically trained on benchmark compounds? 1) OPLS force field with no explicit σ-hole representation 2) GAFF force field tuned for aromatic systems 3) CGenFF force field heavily parameterized for training compounds 4) Classical force field lacking anisotropic charge modeling 5) QMD-FF with quantum mechanics-derived parameters and explicit anisotropy 6) Empirical Lennard-Jones model for halogen interactions 7) Standard fixed-charge force field without σ-hole adjustments

✓ Correct Answer:

The correct answer is 5) QMD-FF with quantum mechanics-derived parameters and explicit anisotropy.

📚 Reference Text:

C-Br bond, the EPS arou nd the halogen goes from -0.3 a.u. near the center of the bond to -0.1 on the polar s ite of theσ-hole, correctly reproducing the electrostatic imbalance between the VdW sph ere’s side and pole. The OPLS description shows an inverted trend, with a more negati ve EPS on top of the Br atom. Inthecaseof I1,ontheotherhand,boththeOPLSandQMD-FFdescriptionsyie ld a more negative potential in the σ-hole with respect to the surrounding region, however the strongly negative value of ≃-0.3 a.u. for OPLS is reduced to ≃0.2 a.u. by QMD- FF, and its value is almost constant throughout all the Iodin e’s surface, as compared to a steeper slope in OPLS. From these examples, it is evident that the information about theσ-hole is present in the IPES sampled at the QM level, and that t he QMD-FF parameterization route is capable of getting close to the ex pected interaction proﬁle, by adjusting simultaneously the parameters of the halogen and its neighboring atoms. It is also clear, however, that the inclusion of an ad hoccenter, like a ESH, would be essential for a proper treatment of Iodine and Bromine’s anisotropy. 3.3.3 Validation against experiment Since the ultimate goal of any FF is to deliver accurate predi ction of the macroscopic properties, the performances in MD simulations of the twelv e QMD-FFs, obtained by joining the parameterized intra- and inter-molecular part s, were eventually tested against available experimental data, as well as compared to those ac hieved with popular trans- ferable FFs, reported in References [17,59]. For all compounds, extended MD simulations wereperformedintheNPTensemble, onsystemsofonethousand molecules, at1atmand at several temperatures. Unless otherwise stated, to make co nsistent comparisons with literature results, all MD parameters were set as suggested by Caleman and coworkers.17 Further details concerning the MD runs can be found in the Sup porting Information. The ﬁnal values of the computed thermodynamic properties fo r compounds C1-I2are reported in Tables 7-11. By looking at the last row of Table 7, where the standard deviation of the computed bulk densities with respect to the experiment is displayed, it appears as the overall performance of the QMD-FF (98 kg/m3) is comparable to the one delivered by OPLS (94 kg/m3) and remarkably better than the one given by GAFF (254 26 kg/m3). Within the chosen set of benchmark molecules, CGenFF yiel ds instead a better standard deviation (47 kg/m3), but it is worth recalling that many of these compounds were included in the CGenFF’s21training set, and their parameters tuned to reproduce experimentaldensityandvaporizationenthalpy. Morespec iﬁcally,theQMD-FFyieldsthe bulkdensitiesclosesttotheexperimentfor C1,C2,N1,N2andI2, whilethepercentage error oscillates from a minimum value of less than 1% ( C2) to a maximum one of ∼9 % (I1). As already pointed out, also the performances of the transf erable FFs appear to depend on the chosen target: for instance GAFF outperforms OP LS forC2, while the opposite is true for I2. The good agreement with the experiment found for QMD-FF and its performance versus transferable FFs is more than encour aging, especially considering both

Question 759 multiple-choice

The hidden subgroup problem is a foundational challenge in quantum computing, with solutions often relying on advanced group theory, linear algebra, and quantum measurement techniques. The "pretty good measurement" (PGM) approach is particularly significant for tackling nonabelian cases. Which statement accurately describes the role of the matrix sum problem in the implementation of the pretty good measurement (PGM) for the hidden subgroup problem in nonabelian groups? 1) It enables the reduction of the problem to an abelian group, allowing standard quantum algorithms to be used. 2) It provides the necessary algebraic framework to represent hidden subgroup states and is crucial for realizing the PGM measurement. 3) It constructs the automorphism group required for labeling cosets in the quotient group. 4) It identifies the normal subgroups within the original group to simplify the subgroup structure. 5) It transforms subgroup generators into vector space elements over Zp for efficient computation. 6) It partitions the group into conjugacy classes for improved measurement fidelity. 7) It determines the order of the subgroup H2 by solving characteristic equations.

✓ Correct Answer:

The correct answer is 2) It provides the necessary algebraic framework to represent hidden subgroup states and is crucial for realizing the PGM measurement..

📚 Reference Text:

ﬁnd H1/n⋊ttriangeqlleftG, then we know that H=H1, and we are done. Otherwise, we learn that H1/triangleleftequalG, and we proceed to factor out H1. The (left) cosets of H1inGcan be represented as follows: for g= (a,b)∈G, gH1={(a,b)(h,0) :h∈A1}={(a+ϕb(h),b) :h∈A1}={(a+h,b) :h∈A1}.(4) Thus the cosets can be labeled by b∈Zpanda∈A2:=A/A1. Now we work in the group G2:=G/H1∼=A2⋊ϕ2Zpwhose elements are these cosets. Note that G2inherits its deﬁning automorphism ϕ2from the original automorphism ϕ. Sincefis constant on cosets of H1≤H, we can consider it as a function f:G2→Swith the hidden subgroup H2:=H/H1. IfH1=HthenH2is trivial. If H1/ne}ati≀nslash=H, then we must have H=/an}bracketle{tH1,(a,1)/an}bracketri}htfor somea∈A, which can be seen as follows. Since pis prime, there must be some (a,1)∈H; but for any additional ( a′,1)∈H, we have (a′,1)(a,1)−1= (a′−a,0)∈H1, and hence (a′,1)∈ /an}bracketle{tH1,(a,1)/an}bracketri}ht. Also, note that /an}bracketle{t(a,1)/an}bracketri}ht∩H1=/an}bracketle{t(Φ(p)(a),0)/an}bracketri}ht. Thus, by the second isomorphism theorem,H2=H/H1∼=/an}bracketle{t(a,1)/an}bracketri}ht//an}bracketle{t(Φ(p)(a),0)/an}bracketri}ht, which is a cyclic group of order pgenerated by ( d,1) for somed∈A2. From now on we assume that the hidden subgroup is H=/an}bracketle{t(d,1)/an}bracketri}ht={(Φ(b)(d),b) :b∈Zp}for somed∈A. Note in particular that since |H|=p, we have Φ(p)(d) = 0. This reduction alone is enough to solve the hspover some semidirect product groups. For example, the hspover the wreath product groups Zn 2≀Z2(for which an eﬃcient quantum algorithm 1In some cases, the results of [31] may not be required; for exa mple, in the case A=Zr pconsidered in Section 6, H1can be viewed as a subspace, and it is suﬃcient to check whethe r this subspace is invariant under the action of the matrix µthat deﬁnes the automorphism ϕ. 4 was given in [28]) and Zpr⋊ϕZpwithϕ(a) = (pr−1+1)a(for which an eﬃcient quantum algorithm was given in [17]) are both reduced to instances of the abelia nhsp. However, in general, we are left with a nonabelian hsp, which we attempt to solve using the pretty good measurement . 4 The pretty good measurement approach In this section, we present the pretty good measurement appr oach to the hidden subgroup problem overG=A⋊Zp. We begin in Section 4.1 by describing the hidden subgroup st ates and expressing them in terms of a certain algebraic problem called the matri x sum problem. Then, in Section 4.2, we describe the pretty good measurement for distinguishing these states. In Section 4.3, we give an expression for the success probability of the measuremen t (as well as general upper and lower bounds), and in Section 4.4, we prove that the measurement is optimal. Finally, in Section 4.5, we explain how the measurement can be implemented by solving th e matrix sum problem. 4.1 Hidden subgroup states and the matrix sum problem According to (2), the hidden subgroup states are uniform mix tures of uniform superpositions over the left cosets of HinG. A complete set of left coset representatives of H=/an}bracketle{t(d,1)/an}bracketri}htinG=A⋊Zp is given by L={(ℓ,0) :ℓ∈A}, and we have the coset states |ψℓ,d/an}bracketri}ht:=1√p/summationdisplay b∈Zp|(ℓ,0)(Φ(b)(d),b)/an}bracketri}ht=1√p/summationdisplay b∈Zp|ℓ+Φ(b)(d),b/an}bracketri}ht. (5) The hidden subgroup state is ρd=1 |A|/summationdisplay ℓ∈A|ψℓ,d/an}bracketri}ht/an}bracketle{tψℓ,d|. (6) Fourier transforming the ﬁrst register (over A) gives ˜ρd=1 |A|/summationdisplay x∈A|˜ψx,d/an}bracketri}ht/an}bracketle{t˜ψx,d|where|˜ψx,d/an}bracketri}ht:=1√p/summationdisplay b∈Zpχx(Φ(b)(d))|x,b/an}bracketri}ht (7) and whereχx:A→Cforx∈Adenotes the xth group character of A,

Question 760 multiple-choice

In quantum algorithms addressing the hidden subgroup problem for semidirect product groups, efficient solution of certain polynomial systems over finite fields is essential. The algebraic structure of matrices, such as Jordan blocks, directly impacts algorithm design and computational complexity. When solving the matrix sum problem for a group of the form Z_p^r ⋊ Z_p, what is the typical number of solutions when the number of variables k equals the number of equations r? 1) O(1) 2) O(p^r) 3) O(r) 4) O(p^{k-r}) 5) O(k^r) 6) O(p) 7) O(r^k)

✓ Correct Answer:

The correct answer is 1) O(1).

📚 Reference Text:

the diagonal and ﬁrst superdiagonal, and that the elements on the ﬁrst sup erdiagonal are either 0 or 1. Since µp=I, the diagonal elements of this µmust all be equal to 1. Thus the various nonisomorphic groups of the form Zr p⋊Zpcorrespond to the partitions of r, where the partitions describe the sizes of the Jordan blocks. For simplicity, we consider the case of a single Jordan block of sizer; the extension to other cases will be clear. In other words, we consider the r×rmatrix µ= 1 1 0 ···0 0 1 1...... 0 0......0 .........1 1 0···0 0 1 . (46) Then thebth matrix sum M(b):=I+µ+µ2+···+µb−1is given by M(b)= /parenleftbigb 1/parenrightbig /parenleftbigb 2/parenrightbig /parenleftbigb 3/parenrightbig ···/parenleftbigb r/parenrightbig 0/parenleftbigb 1/parenrightbig /parenleftbigb 2/parenrightbig...... 0 0....../parenleftbigb 3/parenrightbig ........./parenleftbigb 1/parenrightbig /parenleftbigb 2/parenrightbig 0···0 0/parenleftbigb 1/parenrightbig modp. (47) Note thatM(0)=M(p)= 0. Ifµconsists of several Jordan blocks, then (47) can be applied i n each block. The matrix sum problem is thus the following: given uniforml y randomx∈(Zr p)kandw∈Zr p, ﬁndb∈Zk psuch that/summationtextk j=1M(bj)xj=w. This is a set of rpolynomial equations over Zpink variables. For example, in the case of a single Jordan block, we have k/summationdisplay j=1/bracketleftbigg/parenleftbiggbj 1/parenrightbigg xi,j+···+/parenleftbiggbj i/parenrightbigg x1,j/bracketrightbigg =wi (48) for all 1≤i≤r, where the ith equation is of degree i. Ifk < r, so that there are fewer unknowns than equations, we expect t hat the matrix sum problem will typically have no solutions. On the other hand, ifk > r, we expect that the ideal generated by (48) will typically have dimension k−r, and there will be O(pk−r) solutions. By choosingk=r, we can ensure that there are typically O(1) solutions, so that the pgmsucceeds, yet the solutions of the matrix sum problem are few in number a nd thus relatively easy to ﬁnd (and to quantum sample). Thisintuitioncanbeformalizedbycalculatingthemeanand varianceofthenumberofsolutions. Using such an argument, we ﬁnd the following: 13 Lemma 5. For the matrix sum problem of any group Zr p⋊Zpwithk=r,Pr(ηx w= 1or2)≥1 4. Proof.See Appendix B. By Lemma 2, this shows that Pr(success) ≥1/16, even if we only consider the cases in which the number of solutions is at most 2. Thus, the pretty good mea surement succeeds in identifying the hidden subgroup. Toeﬃciently implementthe pgm, wequantumsamplefromsolutionsofthematrixsumproblem, which can bedone by computing a list of all solutions. A list o f solutions can be found by ﬁrstusing Buchberger’s algorithm to compute a Gr¨ obner basis for the i deal and then using elimination theory [5]. In general, upper bounding the complexity of Buchberge r’s algorithm is a diﬃcult problem, and it is known that the algorithm may be very ineﬃcient in ter ms of the number of variables, the number of equations, and the degree of those equations. Howe ver, since we consider rconstant, such ineﬃciency is not an issue. The running time in terms of p, the size of the ﬁeld, enters only as an overall poly(log p) factor, accounting for the cost of performing arithmetic o perations in a ﬁnite ﬁeld (see for example [6]). Thus, for

Question 761 multiple-choice

Random walks on algebraic structures such as finite groups and quantum groups are important in probability, representation theory, and quantum information. Analyzing how quickly such random walks converge to uniformity requires advanced mathematical tools. Which property of quantum groups most directly enables the adaptation of the Diaconis-Shahshahani Upper Bound Lemma, originally formulated for classical groups, to the quantum case? 1) The existence of a commutative multiplication operation in all quantum groups 2) The presence of Haar measures identical to those on classical groups 3) The definition of quantum groups solely as sets equipped with a binary operation 4) The lack of representation theory in quantum groups 5) The absence of non-commutative geometry in quantum group theory 6) The structural similarities between the representation theory of quantum groups and classical groups 7) The restriction of random walks to abelian group structures in quantum settings

✓ Correct Answer:

The correct answer is 6) The structural similarities between the representation theory of quantum groups and classical groups.

📚 Reference Text:

Title: Random Walks on Finite Quantum Groups: Diaconis-Shahshahani Theory for Quantum Groups Year: 2017 Paper ID: ee52935d51150e042273f8ede148064e7ee3cabe Source: semantic-scholar URL: https://www.semanticscholar.org/paper/ee52935d51150e042273f8ede148064e7ee3cabe Abstract: The concept of a random walk on a finite group converging to random - and a way of measuring the distance to random after $k$ transitions - is generalised from the classical case to the case of random walks on finite quantum groups. A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The representation theory of quantum groups is very well understood and is remarkably similar to the representation theory of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for quantum groups. The Quantum Diaconis--Shahshahani Upper Bound Lemma is used to study the convergence of ergodic random walks on classical groups $\mathbb{Z}_n$, $\mathbb{Z}_2^n$, the dual group $\widehat{S_n}$ as well as the `truly' quantum groups of Kac & Paljutkin, and Sekine.

Question 762 multiple-choice

In mathematical physics and Lie algebra theory, Poisson brackets are modified in the presence of constraints using methods such as the Dirac bracket, especially when analyzing reductions via embedded subalgebras like sl₂. The structure of the reduced Poisson algebra and its surviving subalgebras provide insight into conserved quantities and symmetry. When reducing a Poisson algebra via the Dirac bracket corresponding to an embedded sl₂-subalgebra in a Lie algebra g, which property ensures that the series expansion for the operator R used in the reduction process converges? 1) The operator R is nilpotent for all embeddings. 2) The adjoint action adₜ⁺ always has a trivial kernel. 3) The centralizer C(i) consists solely of abelian subalgebras. 4) The Kirillov bracket is linear for every reduction. 5) The adjoint action with w commutes with all elements of g. 6) The lowering operator L possesses a degree-lowering property, guaranteeing eventual cancellation in the recursion. 7) The gradients defined by the pairing between gl_w and gh_w are always orthogonal.

✓ Correct Answer:

The correct answer is 6) The lowering operator L possesses a degree-lowering property, guaranteeing eventual cancellation in the recursion..

📚 Reference Text:

(perturbatively) written as R=∞/summationdisplay k=0Rkǫk(3.51) (we shall have to justify this later). Consider now the zerot h order part of the equation (3.50). It reads adt+R0= 0 (3.52) This means that R0≡F(/vector z) which is an arbitrary element of ker( adt+). The ﬁrst order equation is equal to adt+R1= Π([F,w]) (3.53) Obviously this equation is solved by R1=−L◦adwF (3.54) Proceeding with higher orders we ﬁnd Rk+1(/vector z;y) =−L◦adw(Rk(/vector z;y)) (3.55) 22 which means that R(/vector z;y) =1 1 +ǫL◦adwF(/vector z) (3.56) There are no convergence problems with this series since the operatorLlowers the degree by one which means that after 2 psteps it must cancel. We therefore ﬁnd /summationdisplay j,j′;µµ′z(µ) j{Jj,−j (µ),Jj′,−j′ (µ′)}∗(y)t(µ′) j′,−j′=−ady/parenleftbigg1 1 +Lad wF(/vector z)/parenrightbigg (3.57) Let againy=t++w∈gfix(i.e.w∈glw) andQ∈C∞(gfix). We then deﬁne gradyQ∈Ker(adt+)≡ghwby /parenleftig x,gradyQ/parenrightig ≡d dǫQ(y+ǫx)|ǫ=0for allx∈glw (3.58) Note that this uniquely deﬁnes gradyQbecause, as we saw before, glwandghware non- degenerately paired. It is now easy to derive a general (and c oordinate free) formula for the (Dirac) Poisson structure on C∞(gfix) induced by the Kirillov Poisson structure onC∞(g). ForQ1,Q2∈C∞(gfix) it reads {Q1,Q2}∗(y) =/parenleftbigg y,[gradyQ1,1 1 +L◦adwgradyQ2]/parenrightbigg (3.59) For the so called trivial embedding ( t0=t±= 0) the map Lis equal to the zero map and the above formula reduces to the ordinary Kirillov brack et as it should (because in that case gfix≡g). From now on denote the Kirillov Poisson algebra associated t o a (semi) simple Lie algebragbyK(g) and the Poisson algebra C∞(gfix) with Poisson structure (3.59) by W(i) whereiis again the sl2embedding in question. In general W(i) is a non-linear Poisson algebra as we shall see when we consi der examples. However it can happen that W(i) contains a subalgebra that is isomorphic to a Kirillov algebra. The next theorem deals with this. Theorem 4 The algebra W(i)has a Poisson subalgebra which is isomorphic to /circlemultiplydisplay j∈1 2NK(slnj)/circlemultiplydisplay (q−1)K(u(1)) (3.60) (where again qis the number of diﬀerent jvalues in the decomposition of the funda- mental representation). Proof: Remember that the centralizer of the sl2subalgebra i(sl2) is given by C(i) =/circleplusdisplay j∈1 2Nslnj/circleplusdisplay (q−1)u(1) (3.61) 23 and therefore the Kirillov algebra of C(i) is equal to (3.60). Since all elements of the centralizer are lowest (and highest) weight vectors w.r.t. i(sl2) the function Jais not constrained if ta∈C(i), i.e. all the elements Jaassociated to the centralizer survive the reduction. It is not obvious however that they still form the algebra (3.60) w.r.t. the Dirac bracket. This is what we have to show. The part of the equation /summationdisplay j,j′;µ,µ′z(µ) j{Jj,−j (µ),Jj′,−j′ (µ′)}∗(y)t(µ′) j′,−j′= [R(/vector z;y),y] (3.62) that determines the Poisson relations between the currents associated to the centralizer is/summationdisplay µ,µ′z(µ) 0{J0,0 (µ),J0,0 (µ′)}∗(y)t(µ′) 0,0= [R(/vector z;y),y](0)(3.63) The right hand side of this equation reads in more detail [R,y](0)= [R(0) lw,y(0)] + [R(0) 0,y(0)] +...+ [R(p),y(−p)] + [R(−1),t+] (3.64) Note thatR(0) lwandy(0)are both elements of C(i) which means that the ﬁrst term in the right hand side is also an element of C(i). We will now show that R(0) 0,R(1),...,R(p) andR(−1)do not depend on {z(µ)

Question 763 multiple-choice

Shor's algorithm is a quantum computational method for factoring large composite numbers efficiently, utilizing quantum parallelism and specific transformations to reveal periodicity in modular exponentiation. Its effectiveness relies on operations such as entanglement, quantum Fourier Transform (QFT), and the measurement of quantum states. Which step in Shor's algorithm directly utilizes the Quantum Fourier Transform (QFT) to extract the period of a modular exponential function, enabling factorization of a large number N? 1) Performing a unitary operation on an entangled quantum register containing superposed exponents 2) Initializing all quantum bits in the computational basis to zero 3) Measuring the modulus squared of the wavefunction after collapse 4) Generating classical random seeds for modular exponentiation 5) Applying greatest common divisor (gcd) computations to candidate factors 6) Using linear phase factors generated from impulse train translation 7) Collapsing the wavefunction to the peak probability outcome without further transformation

✓ Correct Answer:

The correct answer is 1) Performing a unitary operation on an entangled quantum register containing superposed exponents.

📚 Reference Text:

on -axis, the unknown translation , d, of the original impulse train (of periodicity r) having been transformed to a linear phase factor. S ince the probability of collapse of the wavefunction is proportional to the modulus squared of its values , this will not a ffect measurements of the impulse train periodicity 1/r. The high probability of collapse at the location of the peak of the wavefunction will allow the frequency of the impulse train to be determined, and hence the reciprocal value r, thus in turn allowing a factor of N to be found . This process can perhaps be made clearer by an explicit, simple numerical example. The factorizati on of the number 15 in to 3 an 5 is often used as example as 15 is the smallest number with two prime factors13. We take, for example, a seed value x = 7 and calculate 7a mod 15 to yield: 1,7,4,13,1,7,4,13…. We see this results in a periodic function of period (order) equal to 4. We divide by 2 and raise the seed to this power i.e. 74/2 to give 49 (note that this only works when the order is even; if it is odd we start again with a new trial seed value). Adding and subtracting 1 gives 48 and 50. The gcd of 15 and 48 is 3 and the gcd of 50 is 5, giving us 3 and 5 as the factors of 15. Thus if we can determine the order of the function generated by the modular exponentiation , we can factorize N. The Shor algorithm for determining the periodicity r using this example is summarised in graphical form in Figure 1 below. For large N calculating the modular exponentiation has to be repeated more than N 2 times which is not feasible on a conventional computer. However, if all values can be computed in parallel via a quantum computation, the modular exponentiation can be carried out in O(n3) for a n-bit number N. The period can then be determined from this using the QFT and from this the factors of N. However, to achieve this in O(poly log n) time rather than the exponential time required for a classical implementation requires the enta nglement of two quantum registers and a subsequent unitary operation, i..e. a QFT, on the partially collapsed state of one of the registers4. However, s ince the modular exponentiation requires O( n3) and the QFT ( n2) quantum gates for implementation, the implementation difficulties of the quantum algorithm are, as has been noted above , very technically demanding. Figure 1 (a). Diagrammatic summary of periodicity determination with the Shor algorithm . A sample is shown of the periodic function resulting from the factorization of 15 with seed x = 7, resulting in a periodic function of order 4. Register sample shown before a measurement. Figure 1(b). Suppose a measurement is made on the output r egister and the wavefunction collap ses, for example, to the value 13. The

Question 764 multiple-choice

Quantum computing relies on efficient circuit design to scale algorithms to large numbers of qubits, with circuit depth being a significant factor influencing performance and error rates. Innovations such as constant-depth quantum fan-out gates and mid-circuit feedforward are increasingly important for practical implementations. Which approach offers a scaling advantage for quantum circuits with more than 25 output qubits when classical feedforward is included? 1) Using error correction codes without mid-circuit measurements 2) Increasing the number of superconducting qubits per processor 3) Implementing classical post-processing after circuit execution 4) Employing only unitary gate sequences for fan-out operations 5) Utilizing constant-depth quantum fan-out gates with real-time feedforward 6) Applying quantum state tomography before each gate operation 7) Reducing circuit depth by removing all measurements

✓ Correct Answer:

The correct answer is 5) Utilizing constant-depth quantum fan-out gates with real-time feedforward.

📚 Reference Text:

Title: Constant-depth fan-out with real-time feedforward on a superconducting quantum processor Year: 2024 Paper ID: 82649d6fbcf888e485616c656b25fc9600cdc245 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/82649d6fbcf888e485616c656b25fc9600cdc245 Abstract: When using unitary gate sequences, the growth in depth of many quantum circuits with output size poses significant obstacles to practical quantum computation. The quantum fan-out operation, which reduces the circuit depth of quantum algorithms such as the quantum Fourier transform and Shor's algorithm, is an example that can be realized in constant depth independent of the output size. Here, we demonstrate a quantum fan-out gate with real-time feedforward on up to four output qubits using a superconducting quantum processor. By performing quantum state tomography on the output states, we benchmark our gate with input states spanning the entire Bloch sphere. We decompose the output-state error into a set of independently characterized error contributions. We extrapolate our constant-depth circuit to offer a scaling advantage compared to the unitary fan-out sequence beyond 25 output qubits with feedforward control, or beyond 17 output qubits if the classical feedforward latency is negligible. Our work highlights the potential of mid-circuit measurements combined with real-time conditional operations to improve the efficiency of complex quantum algorithms.

Question 765 multiple-choice

Quantum information theory leverages mathematical properties of graphs and compact Lie groups to establish bounds on mixing times, spectral gaps, and capacities of quantum channels. These concepts are fundamental in analyzing the efficiency and reliability of quantum communication protocols. Which property directly determines the rate at which a quantum process converges to equilibrium, and is foundational for establishing upper bounds on mixing and decoherence times? 1) The cb-norm of channel operators 2) The transference principle between classical and quantum settings 3) The explicit form of the heat kernel on 1D tori 4) The structure of resource theories in quantum information 5) The spectral gap of the Laplacian or generator 6) The universal bound conjecture for the CLSI inverse constant 7) The double counting property of geodesic paths

✓ Correct Answer:

The correct answer is 5) The spectral gap of the Laplacian or generator.

📚 Reference Text:

/divides.alt0/divides.alt0/summation.disp (f( cd)−f())2 =n1 /divides.alt0/divides.alt0(/summation.disp /summation.disp (c;d)∈E′/summation.disp (a;b)∈E;(c;d)∈)(f( cd)−f())2: Since, we may take geodesic, no (cd)is counted double. Thus forE′={(j;j+1)/divides.alt01≤j≤n}andE={(a;b)/divides.alt0a≠ b}, we deduce that EE(f)≤n3EE′(f): Thanks to the normalization factor1 nforLRTis implies that LSI(LNN)−1≤n2LSI(LRT)−1≤2n2lnn: Note that our estimate also implies that the spectral gap (LNN)≥1 2n2. Thanks to [24], we deduce t1;2(1/slash.lefte)LNN≤(1+1 4ln lnn!)2n2lnn ≤2n2(1+ln2n 2): By symmetry this implies tLNN 1;∞(1/slash.lefte2)≤4n2(1+ln(n))2: Using the spectral gap this yields tLNN 1;∞(")≤4n2((1+lnn)2−ln"): Note that we have an automatic cb-norm estimate in this case, and hence transference, allows us to estimate the decoherence time of the tensor swaps. Remark IV .2.The correct estimate MLSI(LRT)∼1 for the modiﬁed (not complete) logarithmic Sobolev inequality has only recently been found [30]. The standard inductive procedure appears not directly applicable for LNN, though. However, our proof shows thatLSI−1(LNN)≤cn2(1+ln2n), and hence trivially we ﬁnd a bound MLSI . We conjecture that for every 14 graph, we always have CLSI−1(LE)≤cln ln/divides.alt0V/divides.alt0LSI−1(LE): This would yield a bound of order cn2(1+lnn)3for the inverse of the CLSI constant. At any rate a better estimate of order cn2would be highly desirable. E. Compact Lie groups Here, we recall some well-known estimates for the heat semigroup deﬁned on various compact Lie groups. a)1-dimensional torus: The following estimate was derived in example 1 of Section 3 of [1]: /parallel.alt1h/uni21A6kt(gh−1)−1/parallel.alt12≤/radical.alt2 2+/radical.alt1 /slash.left2te−t: b)n-dimensional torus ( n>1)::The logarithmic Sobolev constant associated to the heat semigroup on the n dimensional torus Tnis known to achieve the bound 1/slash.leftc(SHeat)≤(SHeat)=1. In Theorem 5.3 of [1], the following upper bound on its kernel (and in fact on the kernel of any uniformly elliptic generator) was found: sup g∈Tn/parallel.alt1h/uni21A6kt(gh−1)−1/parallel.alt12 (47) ≤exp/parenleft.alt3−t+1 2log/parenleft.alt31 2nlogn/parenright.alt3+6/parenright.alt3: c) Matrix Lie groups:: In [1], precise estimates on the kernel of diffusion semigroups on various Riemannian manifolds were obtained starting from a curvature dimension inequality CD(;). Applying this to the curvature dimension inequalities satisﬁed by semisimple Lie groups [31], Saloff-Coste derived the following straightforward corollary (stated here as a theorem for sake of completeness): Theorem IV .3. Let(G;g)be a real connected semi-simple compact Lie group of dimension nendowed with the Riemannian metric induced by its Killing form. Then, the heat diffusion satisﬁes sup g∈G/parallel.alt1h/uni21A6kt(gh−1)−1/parallel.alt12 (48) ≤exp/parenleft.alt31+()/bracketleft.alt3−t+16 n+2 log/parenleft.alt31+1 2nlogn 4/parenright.alt3/bracketright.alt3/parenright.alt3; where()is the spectral gap of (G;g). In particular, for">0andtn=2(1+")logn, the above bound converges to 0. Moreover, the following bounds the logarithmic Sobolev and Poincaré constants hold: ()≥n 8(n−1)≥1 8; (49) c()≤4(n−1) n≤4: (50) V. C APACITY BOUNDS ,RESOURCE THEORIES AND ENTROPIC INEQUALITIES Here we will discuss how to apply the estimates provided in Section III-B to obtain upper bounds onthe (two-way) private, quantum, entanglement-assisted classical capacity and classical capacity of a transferred semigroup (Tt)t≥0converging to its associated conditional expectation Efix≡ENfix. Roughly speaking, the capacity is the ultimate rate of transmission of a certain resource through a quantum channel such that in the limit of inﬁnitely many uses of the channel, the success probability of the transmission of this resource converges to 1after encoding and decoding operations. We refer to [32, Chapter 8] for a precise deﬁnition of the

Question 766 multiple-choice

Quantum algorithms in computational number theory often utilize lattice invariants and specialized encodings to solve algebraic problems efficiently. Understanding how parameter choices affect quantum resource requirements is key to implementing practical algorithms. Which of the following best describes the dependence of the space complexity (qubit count) for a CHSP-based quantum algorithm that computes unit groups in number fields, in terms of its key parameters? 1) Exponential in the error tolerance η and linear in the lattice dimension m 2) Polynomial in the field discriminant D and independent of the unit rank m 3) Logarithmic in the unit rank m and independent of the encoding cost N_f 4) Linear in the Lipschitz constant of the oracle function f and independent of λ*_1 5) Exponential in the determinant of the lattice L and independent of the Lipschitz constant 6) Quadratic in the regulator R and logarithmic in the field degree n 7) Polynomial in the lattice dimension m and logarithmic in λ*_1, the Lipschitz constant of f, the determinant of L, the error tolerance τ, and the encoding cost N_f

✓ Correct Answer:

The correct answer is 7) Polynomial in the lattice dimension m and logarithmic in λ*_1, the Lipschitz constant of f, the determinant of L, the error tolerance τ, and the encoding cost N_f.

📚 Reference Text:

of L∗⊲Step 2 - Classical 7: Output the rows of (B−1)twhereBhas rows ˜u′ i. ⊲Step 3 - Classical Theorem 33 (Space complexity of the CHSP solver, Th 1 of [19]) We set8 η= 1/k2. LetNfbe the amount of qubits necessary to encode f. We ﬁxτthe error expected on basis’ vectors. Then, Algorithm 3 is correct an d requires a number of qubits: Nqubits=O(m3·log(m))+O(m3·log(Lip(f)))+O(m2·log(det(L))) +O/parenleftBig m·logLip(f) λ∗ 1/parenrightBig +O/parenleftBig m·log1 λ∗ 1·τ/parenrightBig +O(Nf).(2) The complexity of the algorithm depends on λ∗ 1which has the advantage that it is an invariant of the lattice. However, λ∗ 1is diﬃcult to lower bound and in some applications one has bounds on the coeﬃcients of BL, the matrix of some basis ofL. Lemma 34 2−3m/⌊ard⌊lBt L/⌊ard⌊l ≤1 λ∗ 1≤ /⌊ard⌊lBL/⌊ard⌊l. Proof. The ﬁrst inequality is [19, Corollary 6 in Appendix B.1]. The second inequality is direct. Let v∈L∗such that /⌊ard⌊lv/⌊ard⌊l2=λ∗ 1. By Nota- tion 23 we have /⌊ard⌊lBLv/⌊ard⌊l1≤ /⌊ard⌊lBL/⌊ard⌊l/⌊ard⌊lv/⌊ard⌊l∞. Sincev∈L∗,v/n⌉}ationslash= 0,BLv∈Znand then /⌊ard⌊lBLv/⌊ard⌊l1≥1. Also,/⌊ard⌊lv/⌊ard⌊l2≥ /⌊ard⌊lv/⌊ard⌊l∞, so 1≤ /⌊ard⌊lBL/⌊ard⌊lλ∗ 1, which proves the second inequality. 8The complexity of the CHSP solver depends on log1/ηand not on ηitself. Hence the contribution of ηis hidden in the ˜Onotation. 10 Razvan Barbulescu and Adrien Poulalion 3.2 Reduction of the computation of the unit group to CHSP In this subsection, we present the method of [22] to reduce th e computation of the unit group to the CHSP. For completeness, we reproduce th e description of [10, Sec ]. Let G:=Rn1+n2×(Z/2Z)n1×(R/Z)n2, and the mapping ϕ: (u1,...,u n1+n2,µ1,...,µ n,θ1,...,θ n2) /ma√sto→((−1)µ1eu1,...,(−1)µn1eun1,e2iπθ1eun1+1,...,e2iπθn2eun1+n2) Letσ1,σn1and respectively σn1+1,...,σ n1+n2be the real and the complex embeddings of K. We consider the Cartesian ring E=σ1(OK)×···×σn1+n2(OK). AnE-ideal is a sub- Z-latticeΛofEwhich is such that ∀x∈E,xΛ⊂Λ. We deﬁne g:G→ {E-ideals} x/ma√sto→g(x)E. Deﬁnition 35 (Ex. 5.3 of [22]) SetH=⊗i∈NC|i/an}⌊ra⌋k⌉tri}ht,HQ=⊗Q i=0C|i/an}⌊ra⌋k⌉tri}htandπQ: H → H Qthe projection on the ﬁrst Qqubits. We deﬁne the straddle encodings of parameterν,strν:R→ H,strm,ν:Rm→ Hmandf:{lattices of Rm} → Hm as follows: –|strν(x)/an}⌊ra⌋k⌉tri}ht= cos(π 2t)|k/an}⌊ra⌋k⌉tri}ht+sin(π 2t)|k+1/an}⌊ra⌋k⌉tri}ht, wherek=⌊x/ν⌋,t=x/ν−k; –|strm,ν(x1,...,x m)/an}⌊ra⌋k⌉tri}ht=⊗m i=1|strν(xi)/an}⌊ra⌋k⌉tri}ht; –|strlattice,m,ν(L)/an}⌊ra⌋k⌉tri}ht=γ−1/2/summationtext x∈Le−π/bardblx/bardbl2/s|strm,ν(x)/an}⌊ra⌋k⌉tri}ht withγ=/summationtext x∈Le−2π/bardblx/bardbl2/s2. Note that computing strlattice,m,νup to an error τrequiresO(mνlogτ)qubits. In Appendix A we propose an alternative function fwhich can be used for totally realKand doesn’t use the straddle encoding. Finally, one deﬁned fas follows: f:Gg− → {E-ideals}|str/an}bracketri}ht− −− → { quantum states }=Hm Q. (3) The following result is the conjunction of several results f rom [22]. Theorem 36 (Theorems 5.5, D.4 and B.3 of [22]) LetKbe a number ﬁeld of degreen, discriminant D, unit rank mand regulator R. LetL:= LogO∗ K which is the hidden period of the function fof Equation (3). Sets= 3·22n√ nDandν= 1/(4n(s√n)2n). Thenf5=⊗5fis an(r,a,ε)- oracle with ε= 243/1024,Lip(f) =a=√πns 4ν+1andr=s√nn−12ν√m, where cis an explicitly computable constant. In particular, ε<1/4,r<λ1(L)/6and log2Lip(f) =O/parenleftbig m2+mlogD/parenrightbig . The special case of cyclotomic ﬁelds in quantum algorithms f or unit groups 11 Proof. Note ﬁrst that if K=Qor is quadratic imaginary then m= 0and there is nothing to be computed. Also note that n/2−1≤m≤n−1so we can interchange O(m)andO(n)in the asymptotic complexity. By [22, Th 5.5] fis an(r,∞,ε)-oracle hiding the lattice L= logO∗ Kwith r= (s√n)n−12ν√nandε′=

Question 767 multiple-choice

Quantum algorithms often combine classical computation with quantum circuits and are analyzed using specific complexity metrics. The principle of deferred measurement plays a key role in simplifying quantum algorithm design. Which principle allows all measurements in a quantum computation to be postponed until the final step, enabling the chaining of quantum subroutines? 1) No-Cloning Theorem 2) Quantum Superposition Principle 3) Deferred Measurement Principle 4) Quantum Error Correction 5) Unitarity of Evolution 6) Ancilla Initialization 7) Entanglement Swapping

✓ Correct Answer:

The correct answer is 3) Deferred Measurement Principle.

📚 Reference Text:

every x, a measurement on the last mqubits ofU|x/an}⌊rack⌉⊔ri}h⊔|0ℓ/an}⌊rack⌉⊔ri}h⊔|0m/an}⌊rack⌉⊔ri}h⊔outputsf(x) with probability at least α. The exact location of the qubits that we measure for the output actually does not matter, since we can also apply SWA P gates (implementable by elementary gates) to swap them to the mlast positions. The extraℓqubits that are not part of the input/output are called ancilla qubits (or work space). Quantum Query Model. We use the standard form of the quantum query model: given a unitary O, we say that a circuit computes fwithoracle access toOif by augmenting the model with the unitary O, we can construct a circuit computing f. The number of queriesonOis the number of unitary Oin the circuit. Ifweﬁnd an eﬃcient algorithmfora problemin querycomplexit yandwe are given an explicit circuit realizing the black-box transformation of the oracle O, we will have an eﬃcient algorithm for an explicit computational proble m. Quantum Algorithms. We say that a quantum algorithm computes a function F:{0,1}∗→ {0,1}∗with probability αif there is a classical algorithm Awith quantum evaluation that outputs F(w) with probability αon inputw. By quan- tum evaluation we mean that the algorithm can, any number of times d uring the computation, build a quantum circuit and evaluate it, that is meas ure the stateU|0/an}⌊rack⌉⊔ri}h⊔whereUis the unitary implemented by the circuit. For a family of algorithms parameterised by n, thetime complexity T(n) is theclassicaltime complexityof Aplusthetime complexityofthecircuits,i.e.the number of gates. The classical space complexity S(n) is the space complexity of A(ignoring quantum evaluations). The quantum space complexity Q(n) is the maximal space complexity of all circuits, i.e. the maximum number of qu bits used. In the natural way, we say that a quantum algorithm has ora cle access toOif it produces circuits with oracle access to O. The query complexity of the algorithm is the sum of the query complexity of the circuits. Now t hat the quantum model of computation is properly deﬁned, we can express the fact that every reversible classical computation can be implemented by a quan tum com- puter, although at a non-negligible cost. Here reversible classical c omputation means that there is a one-to-one mapping between the inputs and t he outputs and that a transform exists to return from the output to the inpu t. Theorem 1 ([Ben89, LS90]). Given any ε >0and any classical compu- tation with running time Tand space complexity S, there exists an equivalent reversible classical computation with running time O(T1+ε/Sε)and space com- plexityO(S(1+ln(T/S))). Corollary 1. Given any ε >0and any classical computation with running timeTand space complexity S, there exists an equivalent quantum circuit of sizeO(T1+ε/Sε)usingO(S(1+ln(T/S)))qubits. In principle, it is always possible to turn a classical computation into a q uan- tum one (Corollary 1) and combine all quantum algorithms into one qua ntum 7 circuit by postponing all measurements until the very end of the co mputation, using the so-called principle of deferred measurement [NC11]. We will use this fact implicitly in the rest of this work and just assume that we can

Question 768 multiple-choice

Exceptional groups of Lie type, such as G2, F4, E6, E7, E8, and their twisted analogues, are central objects in finite group theory and algebraic geometry. Determining the structure and classification of their maximal subgroups, especially local maximal subgroups associated with elementary abelian subgroups, is a critical aspect of understanding their overall group structure. Which of the following statements is true regarding the methods used to classify local maximal subgroups of finite exceptional groups of Lie type? 1) The classification is only possible by applying the classification theorem of finite simple groups. 2) The results rely on the explicit determination of Sylow p-subgroups for all exceptional groups. 3) The analysis ignores twisted versions such as 2G2, 2F4, and 2E6. 4) The classification is restricted to classical groups and does not include exceptional types. 5) The classification of local maximal subgroups is achieved without relying on the classification of finite simple groups. 6) Only non-local maximal subgroups are considered in the classification process. 7) The methods require knowledge of sporadic groups for all cases.

✓ Correct Answer:

The correct answer is 5) The classification of local maximal subgroups is achieved without relying on the classification of finite simple groups..

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THE LOCAL MAXIMAL SUBGROUPS OF EXCEPTIONAL GROUPS OF LIE TYPE, FINITE AND ALGEBRAIC ARJEH M. COHEN, MARTIN W. LIEBECK, JAN SAXL, and GARY M. SEITZ (Received 18 May 1990-Revised 24 November 1990] Introduction Following the classification of finite simple groups, one of the major problems in finite group theory is the determination of the maximal subgroups of the almost simple groups-that is, of the groups X such that L ~ X ~ Aut L for some non-abelian finite simple group L. In the investigation of the maximal subgroups M of X, the analysis is often divided into two parts: the local case, in which M = Nx(E) for some elementary abelian subgroup E of X; and the non-local case, in which the socle of Mis a direct product of non-abelian simple groups. In this paper we determine the local maximal subgroups of the finite exceptional groups of Lie type in the families G2, F:i, £6, £7, £8, 2G2, 2F'.i and 2£6. (The maximal subgroups of the other families of exceptional groups, 2B2 and 3 D4, can be found in [29, 18].) It is a consequence of our main result, Theorem 1 (stated in § 1), together with the results of [23] and work on the other simple groups discussed in [21, I-III], that the local maximal subgroups of the almost simple groups are all explicitly known, apart from the 2-locals of the sporadic groups BM and M. Theorem 1 is used in the proof of [22, Theorem 2], where the study of maximal subgroups of finite exceptional groups of Lie type is reduced to the case of almost simple subgroups. We remark that the 'maximal local subgroups' of X form a larger class than the 'local maximal subgroups', and we make no attempt to determine the former class. For most of our proof of Theorem 1 we work in the simple algebraic group G corresponding to the finite exceptional group L. Our methods also yield the determination of the local maximal subgroups of simple algebraic groups of exceptional type over algebraically closed fields. The results are stated in Theorem 2 (the positive-dimensional subgroups) and Theorem 3 (the zero dimensional subgroups). Theorem 2 is used in [22, Theorem l], which determines all positive-dimensional maximal subgroups of G. One of the local subgroups occurring in the conclusion of Theorem 1 is a subgroup 53 .SL3(5) of E8(pu) (see Table 1). This turns out to be non-maximal when p = 2, because it lies in a subgroup L4(5) of £8(4) in that case. This embedding LiS) < £8(4), which may be of independent interest, is exhibited in§ 5. Finally, we remark that our proofs are all independent of the classification of finite simple groups. This work was supported by the Science and Engineering Research Council of Great Britain, through Visiting Research Fellowships for the first and fourth authors. The fourth author also acknowledges the support of an NSF grant. 1991 Mathematics Subject Classification: 20E28, 20006, 20G40. Proc. London Math. Soc. (3) 64 (1992)

Question 769 multiple-choice

Quantum machine learning leverages quantum computing principles to process and analyze quantum data, with model design and symmetry considerations playing critical roles in learning tasks. One advanced approach involves encoding symmetry information directly into quantum models to enhance performance and generalization. Which characterization best describes the role of group-equivariant quantum machine learning (GQML) in the context of learning from quantum data? 1) It restricts models to process only classical data embedded as quantum states. 2) It guarantees that all quantum models achieve zero training error regardless of design. 3) It eliminates the need for classical optimization in quantum machine learning workflows. 4) It ensures quantum channels are always unparameterized and fixed for all data types. 5) It removes the necessity to measure quantum states after processing. 6) It encodes symmetry properties of quantum data into the model, enabling improved efficiency and generalization. 7) It mandates the use of reinforcement learning exclusively for quantum state classification.

✓ Correct Answer:

The correct answer is 6) It encodes symmetry properties of quantum data into the model, enabling improved efficiency and generalization..

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are studying the same symme- try group and the same representation), and QML-relevant examples appear across several ﬁgures. Finally, the end of the article is capped oﬀ by a sort of ﬁeld guide with boxes containing important information and facts about several common symmetry groups. II. QUANTUM MACHINE LEARNING In this work we focus on the supervised QML problem of classifying labeled quantum data. However, we remark that the tools presented here can broadly be used in more general QML scenarios such as unsupervised learning, re- inforcement learning, and generative modeling. For our purposes, we assume that one is given repeated access to a dataset of the form S=fi;yigN i=1, whereiaren-qubit states belonging to a data domain Rin ad-dimensional Hilbert spaceH(withd= 2n), whileyiare real-valued labels in some label domain Y. We further assume that the data instances in Sare drawn i.i.d. from a distribution deﬁned overRY, such that the label yiassociated to the stateiis assigned according to some (unknown and potentially probabilistic) function f:R!Y. That is, f(i) =yi. We note that we will make no particular as- sumption regarding to how the data states iare created. That is,icould be the result of embedding classical data into quantum states (QML for classical data [9]), or theycould be obtained from some physical quantum mechanical process (QML for quantum data [55]). The goal is to train a model h:R!Y, withbeing trainableparameters, toproducelabelsthatmatchthoseof fwith high probability over the training set (small train- ing error), but also over new and previously unseen cases (small generalization error). QML models come in many forms and ﬂavors, leveraging both the power of quantum hardware (e.g., to compute some classically intractable ex- pectation value over the input data) as well as that of clas- sical computers (e.g., use an optimizer to train the param- eters, or analyze the quantum measurement outcomes with some classical neural network). In any case, the suc- cess of the QML model hinges on several factors, but per- haps the most important one is how the model is deﬁned, i.e., what are the inductive biases encoded in the model. As previously mentioned, the goal of GQML is to embed information about the symmetries of the data in Sinto the model. Here we will not concern ourselves with how to actually create QML models encoding these symmetries (we refer the reader to Refs. [25–31] for that), but rather we will present the tools to theoretically understand and handle the symmetries themselves. Throughout this work we will present several QML tasks and their respective symmetries. For the sake of simplicity, we will consider the case when the parameterized model his simply given by taking kcopies of the input states from the datasetS, sending them through a parameterized channel (usually called a quantum neural network ), and making a measurement at the output. That is, we will focus on QML models of the form h(i) = Tr W( k i)Mi ; (1) 4 whereW:B(H k)!B(H k0)is a trainable parameter- ized quantum channel (usually a unitary channel), and Mi

Question 770 multiple-choice

Quantum algorithms for the hidden subgroup problem (HSP) in Abelian groups are foundational in quantum computing, with significant implications for cryptography and computational complexity. Resource-efficient implementations are particularly important for near-term quantum devices. Which approach enables solving the hidden subgroup problem in finitely generated Abelian groups with a single control qubit and polynomially many operator applications, while preserving high probability of success? 1) Using multiple entangled control qubits and no classical post-processing 2) Applying uncontrolled unitary operations to all qubits without measurement 3) Implementing quantum error correction on each step of the algorithm 4) Deploying non-Abelian group generators for operator application 5) Using only classical computation after initial quantum state preparation 6) Repeatedly applying controlled rotations without resetting qubits 7) Employing semi-classical controls with qubit resetting, adaptive rotations, and classical post-processing

✓ Correct Answer:

The correct answer is 7) Employing semi-classical controls with qubit resetting, adaptive rotations, and classical post-processing.

📚 Reference Text:

fact do all the quantum controlled mult iplications with a single control qubit provided we can execute the ‘semi-cla ssical’ controls which allow us to reset a qubit to |0/an}brack⌉tri}ht+|1/an}brack⌉tri}htand perform a rotation dependent upon the previous measurements (the rotations could in fact be imple mented at any time after resetting the qubit and before applying the ﬁnal Hadam ard transform and measuring it; they could also be omitted provided we repeat e ach step a few extra times and do some additional classical post-processi ng as done in [Ki]). Alternatively, the control qubits could be a sequence of ﬂyi ng qubits which are measured (or prepared) in a way dependent upon the outcomes o f the previous measurements of control qubits. For the more general hidden subgroup problem in Abelian grou ps we would have a sequence of applications of Uf(xej)controlled by one qubit, which is measured, then reset to a superposition of |0/an}brack⌉tri}htand|1/an}brack⌉tri}htplus some rotation that is dependent upon the previous measurements. In summary: The hidden subgroup Kof a ﬁnitely generated Abelian group Ggenerated by e1,e2, . . .ek, corresponding to a function ffromGto a ﬁnite set X, can be found with probability close to 1by ‘semi-classical’ methods with only onecon- trol bit (or a sequence of ﬂying qubits) and polynomial in napplications of the operators |x/an}brack⌉tri}ht |f(y)/an}brack⌉tri}ht → |x/an}brack⌉tri}ht |f(y+xej)/an}brack⌉tri}htforj= 1,2, . . ., k , where nis the index ofKinG. Acknowledgments Many thanks to Peter Høyer for helping prepare this paper, to Mark Ettinger and Richard Hughes for helpful discussions and hospitality in Los Alamos, to BRICS (Basic Research in Computer Science, Centre of the Dan ish National Research Foundation), and to Wolfson College. This work was supported in part by CESG, the European TMR Rese arch Network ERP-4061PL95-1412, Hewlett-Packard, The Royal So ciety London, and the U.S. National Science Foundation under Grant No. PHY 94-07194. Part of this work was done at the 1997 Elsag-Bailey – I.S.I. Founda tion workshop on quantum computation, at NIS-8 division of the Los Alamos Nat ional Labora- tory, and at the BRICS 1998 workshop on Algorithms in Quantum Information Processing. 13 References [BBCDMSSSW] Barenco, A., Bennett, C.H., Cleve, R, DiVincen zo, D.P., Mar- golus, N., Shor, P., Sleater, T., Smolin, J., Weinfurter, H. :Phys. Rev. A 52, (1995) 3457. [BL] Boneh, D., and Lipton, R.J.: Quantum cryptanalysis of h idden linear functions (Extended abstract). Lecture Notes on Computer S cience 963 (1995) 424–437 [CEMM] Cleve, R., Ekert, E., Macchiavello, C., and Mosca, M. : Quantum Algorithms Revisited, Proc. Roy. Soc. Lond. A, 454, (1998) 339-354. [De] Deutsch, D. : Quantum Theory, the Church-Turing princi ple and the uni- versal quantum computer. Proc. Roy. Soc. Lond. A, 400, (1985) 97-117. [EM] Ekert, A., Mosca, M.: (note in preparation, 1998). [GN] Griﬃths, R.B. and Niu, C.-S.: Semi-classical Fourier T ransform for Quan- tum Computation, Phys. Rev. Lett. 76(1996) 3228-3231. [Gr] Grigoriev, D. Y.: Testing the shift-equivalence of pol ynomials by determin- istic, probabilistic and quantum machines. Theoretical Co mputer Science, 180(1997)

Question 771 multiple-choice

Quantum algorithms for hidden subgroup problems in abelian groups play a crucial role in computational number theory and cryptography. Advancements in these algorithms often involve generalizing methods to more complex algebraic structures and analyzing their time complexity. Which feature specifically distinguishes the "wandering" or "vintage Z_Q" generalizations of Shor’s algorithm for free abelian groups of finite rank? 1) They exclusively use non-abelian group structures to solve factoring problems. 2) They replace quantum Fourier transforms with classical algorithms for subgroup detection. 3) They deterministically select maximal cyclic subgroups without randomization. 4) They randomly select a cyclic direct summand and generate random characters of a finite group probe. 5) They rely solely on polynomial-time classical post-processing for hidden subgroup identification. 6) They avoid using any transversal sets in their algorithmic steps. 7) They are only applicable to groups of rank one and do not extend to higher ranks.

✓ Correct Answer:

The correct answer is 4) They randomly select a cyclic direct summand and generate random characters of a finite group probe..

📚 Reference Text:

Title: Quantum Hidden Subgroup Problems: A Mathematical Perspective Year: 2002 Paper ID: d82d40daa6f4483dd4ec4f65d16190aaff664fef Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d82d40daa6f4483dd4ec4f65d16190aaff664fef Abstract: The ultimate objective of this paper is to create a stepping stone to the development of new quantum algorithms. The strategy chosen is to begin by focusing on the class of abelian quantum hidden subgroup algorithms, i.e., the class of abelian algorithms of the Shor/Simon genre. Our strategy is to make this class of algorithms as mathematically transparent as possible. By the phrase "mathematically transparent" we mean to expose, to bring to the surface, and to make explicit the concealed mathematical structures that are inherently and fundamentally a part of such algorithms. In so doing, we create symbolic abelian quantum hidden subgroup algorithms that are analogous to the those symbolic algorithms found within such software packages as Axiom, Cayley, Maple, Mathematica, and Magma. As a spin-off of this effort, we create three different generalizations of Shor's quantum factoring algorithm to free abelian groups of finite rank. We refer to these algorithms as wandering (or vintage Z_Q) Shor algorithms. They are essentially quantum algorithms on free abelian groups A of finite rank n which, with each iteration, first select a random cyclic direct summand Z of the group A and then apply one iteration of the standard Shor algorithm to produce a random character of the "approximating" finite group A=Z_Q, called the group probe. These characters are then in turn used to find either the order P of a maximal cyclic subgroup Z_P of the hidden quotient group H_phi, or the entire hidden quotient group H_phi. An integral part of these wandering quantum algorithms is the selection of a very special random transversal, which we refer to as a Shor transversal. The algorithmic time complexity of the first of these wandering Shor algorithms is found to be O(n^2(lgQ)^3(lglgQ)^(n+1)).

Question 772 multiple-choice

Quantum algorithms for group-theoretic problems often exploit symmetries and group actions to achieve efficiency, but the computational complexity can vary dramatically depending on the algebraic structure involved. The Hidden Quadratic Polynomial Problem (HQPP) and its relationship to the Hidden Subgroup Problem (HSP) are central in understanding these complexities. Which of the following statements accurately characterizes the tractability of the Hidden Quadratic Polynomial Problem (HQPP) in relation to quantum polynomial-time algorithms and the structure of the underlying finite field? 1) HQPP is efficiently solvable on quantum computers for all prime fields regardless of size. 2) HQPP becomes intractable only when the field characteristic is divisible by 2. 3) HQPP over any finite field is classically polynomial-time solvable. 4) HQPP over fields of fixed odd characteristic is as hard as factoring integers. 5) HQPP is always equivalent to the discrete logarithm problem in every group setting. 6) HQPP over finite fields of constant characteristic is solvable in quantum polynomial time, but over prime fields it is as hard as the dihedral Hidden Subgroup Problem. 7) HQPP over fields of characteristic zero is always easier than over fields of positive characteristic.

✓ Correct Answer:

The correct answer is 6) HQPP over finite fields of constant characteristic is solvable in quantum polynomial time, but over prime fields it is as hard as the dihedral Hidden Subgroup Problem..

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(a)It has polynomial query complexity. (b)I tc a nb es o l v e di nq u a n t u mp o l y n o m i a lt i m ew h e n qis prime and |H|= Ω(q/polylog(q)). (c)I tc a nb es o l v e di nq u a n t u mp o l y n o m i a lt i m ew h e n qis the power of a ﬁxed prime. The case of Aﬀ q({±1}), whenqis an odd prime power, is particularly interesting. Ifqis itself an odd prime, then Aﬀ q({±1}) is just the familiar dihedral group Dq.I t turnsout thatthe HSSPover Aﬀ q({±1})forthe Frobeniuscomplementsisessentially the same problem as the HQPPoverFq. HIDDEN SYMMETRY SUBGROUP PROBLEMS 1997 Proposition 4.6. The following problems are polynomially equivalent: 1.HQPP(Fq). 2.HSSP(Aﬀq({±1}),Fq,◦,FC). 3.HSP(Aﬀq({±1}),FC). Proof. The ﬁrst two problems are equivalent as we claim that every f:Fq→S, as oracle input for HQPP(Fq) hides the polynomial Puif and only if as oracle input forHSSP(Aﬀq({±1}),Fq,◦,FC)i th i d e st h eF r o b e n i u sc o m p l e m e n t Hu. To see this, observe that the level sets of Puare of the form {x+u,−x+u},w h i c ha r ee x a c t l y the orbits of Hu. Therefore, we have the following equivalences: fhidespu⇐⇒πf={{x+u,−x+u}:x∈Fq} ⇐⇒π∗ f=Hu ⇐⇒fhidesHuby symmetries . The reduction from the second problem to the third one is provided by Theorem 4.4. Note thatwe can constructa base determinist icallyby choosingtwodiﬀerent elements of order two. For a reduction in the reverse direction, consider a function fon Aﬀq({±1}) which hides the subgroup Hu={(0,1),(2u,−1)}. Then all the collisions taken by fon elements of Aﬀ q({±1})a r ef(2u−b,−1) =f(b,1) forb∈Fq.W e deﬁne a new function f◦onFqasf◦(b)=m i n ( f(b,1),f(b,−1)). Examining the possible collisions gives that for b/negationslash=b/prime∈Fq,w eh a v e f◦(b)=f◦(b/prime) if and only if b/prime=2u−b=( 2u,−1)◦b. Together with Corollary 4.5 (c) the statements of this proposition imply the fol- lowing result. Theorem 4.7. HQPP (Fq)is solvable in quantum polynomial time over constant characteristic ﬁelds. We observe that in contrast to the constant characteristic case, the HQPPap- pears to be diﬃcult over prime ﬁelds Fp,a si ti se q u i v a l e n tt ot h e HSPin the dihedral groupDp∼=Aﬀp({±1}). Note that in [17] van Dam, Hallgren, and Ip gave a polynomial time solution to a problem which can be considered as a version of HSSP(Aﬀq(H),Fq,◦,FC), where the function hiding the complement is promised to be a shifted multiplicative characterχ:F ∗ q→C∗. This strong promise (in our oracle model we can only check for equality of the output values) makes the problem eﬃciently solvable even in the case H={±1},w h e r et h e HSSPwith general hiding function appears to be diﬃcult. We also remark that strong bases in the Frobenius group Aﬀ q(H) with|H|= (q−1)/2 play an important role (under the name factoring sets ) in certain algorithms for

Question 773 multiple-choice

Quantum algorithms on superconducting quantum hardware often struggle with coherent errors that can degrade performance and reliability. Advanced error mitigation techniques are essential for improving the accuracy and efficiency of variational algorithms like VQE used in quantum chemistry. Which statement best describes how hidden inverse (HI) gate replacements contribute to error mitigation in quantum circuits operating on NISQ devices? 1) HI gate replacements invert not only the gate operation but also certain correlated noise processes, thereby canceling systematic coherent errors and improving algorithmic outcomes. 2) HI gate replacements randomize the circuit structure to average out incoherent noise, similar to randomized compiling, but do not target coherent errors. 3) HI gate replacements replace all gates with their adjoints, which reduces decoherence but has minimal effect on coherent error sources. 4) HI gate replacements use additional ancilla qubits to detect and correct errors in real time during circuit execution. 5) HI gate replacements increase gate fidelity by optimizing pulse shapes, leading to indirect improvements in overall algorithm performance. 6) HI gate replacements are only effective when paired with error-correcting codes that require significant hardware overhead. 7) HI gate replacements primarily suppress measurement errors rather than mitigate gate-level noise processes.

✓ Correct Answer:

The correct answer is 1) HI gate replacements invert not only the gate operation but also certain correlated noise processes, thereby canceling systematic coherent errors and improving algorithmic outcomes..

📚 Reference Text:

Title: Quantum error mitigation by hidden inverses protocol in superconducting quantum devices Year: 2022 Paper ID: e3d89bdbb5fa3f4275fd92fde56f646f26c0dcd5 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/e3d89bdbb5fa3f4275fd92fde56f646f26c0dcd5 Abstract: We present a method to improve the convergence of variational algorithms based on hidden inverses (HIs) to mitigate coherent errors. In the context of error mitigation, this means replacing the hardware implementation of certain Hermitian gates with their inverses. Doing so results in noise cancellation and a more resilient quantum circuit. This approach improves performance in a variety of two-qubit error models where the noise operator also inverts with the gate inversion. We apply the mitigation scheme on superconducting quantum processors running the variational quantum eigensolver (VQE) algorithm to find the H 2 ground-state energy. When implemented on superconducting hardware we find that the mitigation scheme effectively reduces the energy fluctuations in the parameter learning path in VQE, reducing the number of iterations for a converged value. We also provide a detailed numerical simulation of VQE performance under different noise models and explore how HIs and randomized compiling affect the underlying loss landscape of the learning problem. These simulations help explain our experimental hardware outcomes, helping to connect lower-level gate performance to application-specific behavior in contrast to metrics like fidelity which often do not provide an intuitive insight into observed high level performance.

Question 774 multiple-choice

Quantum computing has revolutionized the study of group-theoretic problems, enabling efficient solutions for tasks that are classically difficult, such as subgroup identification and order finding in black-box groups. These advances are especially important in computational group theory, cryptography, and algorithmic algebra. Which of the following is a key advantage of the presented quantum algorithm for finding generators of hidden normal subgroups in solvable black-box and permutation groups? 1) It requires the group elements to have unique encodings. 2) It can only be applied when efficient quantum Fourier transform construction is available. 3) It is limited to Abelian groups and cannot handle non-Abelian cases. 4) It cannot be used for tasks related to quotient groups like G/N. 5) It depends on classical algorithms for order finding. 6) It efficiently finds generators without requiring efficient quantum Fourier transform construction. 7) It is restricted to groups where explicit multiplication in the secondary encoding is possible.

✓ Correct Answer:

The correct answer is 6) It efficiently finds generators without requiring efficient quantum Fourier transform construction..

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black-box group with unique encoding. Then, each of the ta sks listed in Corollary 5 can be solved in time polynomial in the in put size+ν(G)by a quantum algo- rithm.. Proof.Let usprove that task (b)inCorollary 5can besolved eﬃciently by aquantumalgorithm . In fact, wecan reducethetest toan instanceof theAbelianhidd ensubgroupproblemas follows. First, wecomputetheordersof theunderlyingelements (see [21] fo r example). Let theordersof h1,...,h r andgbes1,...,srands, respectively. Then for a tuple ( α1,...,α r,α) fromZs1×···×Zsr×Zs, set φ(α1,...,α r,α) =hα1 1···hαrrg−α. Clearly φis a homomorphism from Zs1×···×Zsr×ZsintoG, therefore this is an instance of the Abelian hidden subgroup problem, and its kernel can be found in polynomial time by a quantum algorithm. The kernel contai ns an element the last coordinate of which is relatively prime to sif and only if gis representable as a product of powers of hi’s. Also, from such an element an expression for gin the desired form can be constructed eﬃciently. This result generalizes the order ﬁnding algorithm of Watro us (Theorem 2 in [27]) for solvable groups. Also note that, even if Gis solvable, the way how quantum algorithms are used here is slightly diﬀerent from that of Watrous. 4.2 Hidden normal subgroup Assume now that Gis a black-box group with an encoding which is not necessaril y unique, and N is a normal subgroup of Ggiven as a hidden subgroup via the function f.We use the encoding of G for that of G/N. The function fgives us a secondary encoding for the elements of G/N. Although we do not have a machinery to multiply elements in the seconda ry encoding, Shor’s order-ﬁnding algorithm and even the treatment of the constructive member ship test outlined above are still applicable. 6 Theorem 7. Assume that Gis a black-box group with not necessarily unique encoding. S uppose thatNis a normal subgroup given as a hidden subgroup of G. Then all the tasks listed in Corollary 5 forG/Ncan be solved by quantum algorithms in time polynomial in the input size +ν(G/N). Proof.The proof is similar to the one of Theorem 6, where φ(α1,...,α r,α) =f(hα1 1···hαrrg−α) is taken. Letusnowturnbacktotheoriginalhiddensubgroupproblem. Weareabletosolveitcompletely when the hidden subgroup is normal. Note that Hallgren Russe ll and Ta-Shma [14] have already given a solution for that case under the condition that one ca n eﬃciently construct the quantum Fourier transform on G. The algorithm presented here does not require such a hypoth esis. Theorem 8. Assume that Gis a black-box group with not necessarily unique encoding. S uppose thatNis a normal subgroup given as a hidden subgroup of G. Then generators for Ncan be found by a quantum algorithm in time polynomial in the input size +ν(G/N). In particular, we can ﬁnd hidden normal subgroups of solvable black-box groups and pe rmutation groups in polynomial time. Proof.We use the presentation of G/Nobtained by the algorithm of Theorem 7 to ﬁnd generators forN. LetTbe the generating set from the presentation. If Tgenerates Gthen it is easy to ﬁnd generators for N. LetR0denote the set of elements obtained by

Question 775 multiple-choice

Quantum computing leverages principles of quantum mechanics to design algorithms that can outperform classical counterparts in certain tasks. Foundational quantum algorithms like Shor's and Grover's have demonstrated significant advantages by utilizing concepts such as entanglement, superposition, and group theory. Which mathematical concept is central to the exponential speedup achieved by Shor's quantum factoring algorithm? 1) The finite Fourier transform 2) Boolean logic gates 3) Classical error correction codes 4) Probabilistic Turing machines 5) Differential equations modeling 6) Hash functions 7) Neural network backpropagation

✓ Correct Answer:

The correct answer is 1) The finite Fourier transform.

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cuits, and in a 1992 paper with Richard Jozsa [27] described an algorithm that illustrates the power of quantum computation. Other algorithms demonstrating the power of quantum parallelism and "entanglement" in the solution of theoretical computational problems soon followed, and a 1994 result of Simon [67] inspired Shor's work. The purpose of this book is to make these quantum algorithms acces sible to the mathematically literate reader who is not familiar with the subject and who may have forgotten, or has never seen, some of the di verse tools and notation of the trade. The challenge has been to provide enough background to make the context and details of the algorithms understandable, but without lapsing into encyclopedic detail. The focus in this exposition is exclusively on the "software" side of the subject. While there is intense experimental work on the "hardware" side of quantum computation, progress has been much slower. For ex ample, the problems involved in creating stable microphysical devices for "quantum bits" have motivated experiments that are at the boundary of currently feasible technology. Indeed, there continues to be some question about the ultimate feasibility of a quantum computer complex enough to implement Shor's algorithm for even modestly large N. We begin in Chapter 1 by very briefly motivating the need for quantum mechanics and then developing and illustrating some of the mathemati cal framework of finite-dimensional quantum statics. There are no formal prerequisites, but the reader is assumed to be familiar with undergradu ate linear algebra. Key definitions and theorems from linear algebra are introduced in the context of the static model and then rewritten in Dirac notation, which is used unabashedly thereafter. References are given for the reader who wishes a more complete presentation of the history of Preface ix quantum mechanics and of the physical motivation for the mathematical model. In Chapter 2 the mathematical framework is further expanded to ac commodate multi-particle models and to illustrate the motivation for some of the notation. Wiring diagrams and quantum logic gates are in troduced, and some representative quantum subroutines are presented. In particular, a simple "teleportation" circuit is examined in detail. Chapter 3 illustrates the development of quantum algorithms from the explicit to the abstract. We begin with the Deutsch-Jozsa algorithm and describe what proves to be the typical structure of most quantum algo rithms. Grover's search algorithm differs somewhat from the others, and we examine it in a simple case, revealing the structure that applies in general. In a similar fashion we work through Shor's algorithm in the simple case of N = 15 and identify the techniques and the difficulties that occur in the more general context. A key step in Shor's factoring algorithm is the finite Fourier transform, and we illustrate its implemen tation as well. This leads to a discussion of the role of eigenvalues and eigenvectors, and the chapter concludes with a group theoretic model of quantum algorithms. Since the implementation of even the simplest logic gate is experimen tally challenging,

Question 776 multiple-choice

Quantum cryptography explores the security of cryptosystems against adversaries equipped with quantum computers, especially focusing on how new quantum algorithms threaten or bypass traditional cryptographic schemes. Code-based cryptosystems, such as McEliece and Niederreiter, are leading candidates for post-quantum security due to their resilience against known quantum attacks. Which cryptographic feature distinguishes the security of McEliece-type code-based cryptosystems against quantum Fourier sampling attacks compared to RSA and El Gamal? 1) Reliance on the computational difficulty of factoring large integers 2) Use of abelian group structures in encryption algorithms 3) Dependence on the hardness of discrete logarithms 4) Construction based on permutations of elliptic curve points 5) Vulnerability to quantum algorithms exploiting group commutativity 6) Security reduction to hard instances of the Hidden Subgroup Problem in non-abelian groups associated with linear codes 7) Utilization of small public keys to minimize attack surfaces

✓ Correct Answer:

The correct answer is 6) Security reduction to hard instances of the Hidden Subgroup Problem in non-abelian groups associated with linear codes.

📚 Reference Text:

McEliece and Niederreiter C ryptosystems That Resist Quantum Fourier Sampling Attacks Hang Dinh1, Cristopher Moore2,⋆, and Alexander Russell3,⋆⋆ 1Indiana University South Bend hdinh@cs.iusb.edu 2University of New Mexico, and Santa Fe Institute moore@cs.unm.edu 3University of Connecticut acr@cse.uconn.edu Abstract. Quantum computers can break the RSA, El Gamal, and elliptic curve public-key cryptosystems, as they can efﬁciently factor integers and extract dis- crete logarithms. This motivates the development of post-quantum cryptosystems: classical cryptosystems that can be implemented with today’s computers, that will remain secure even in the presence of quantum attacks. In this article we show that the McEliece cryptosystem over rational Goppa codes and the Niederreiter cryptosystem over classical Goppa codes resist pre- cisely the attacks to which the RSA and El Gamal cryptosystems are vulnerable— namely, those based on generating and measuring coset states. This eliminates the approach of strong Fourier sampling on which almost all known exponential speedups by quantum algorithms are based. Speciﬁcally, we show that the natu- ral case of the Hidden Subgroup Problem to which McEliece-type cryptosystems reduce cannot be solved by strong Fourier sampling, or by any measurement of a coset state. To do this, we extend recent negative results on quantum algorithms for Graph Isomorphism to subgroups of the automorphism groups of linear codes. This gives the ﬁrst rigorous results on the security of the McEliece-type cryp- tosystems in the face of quantum adversaries, strengthening their candidacy for post-quantum cryptography. We also strengthen some results of Kempe, Pyber, and Shalev on the Hidden Subgroup Problem in Sn. 1 Introduction If and when quantum computers are built, common public-key cryptosystems such as RSA, El Gamal, and elliptic curve cryptography will no longer be secure. Given that fact, the susceptibility or resistance of other well-studied public-key cryptosystems to quantum attacks is of fundamental interest. We present evidence for the strength of McEliece-type cryptosystems against quantum attacks, de monstrating that the quantum Fourier sampling attacks that cripple RSA and El Gamal do not apply to the McEliece or Niederreiter cryptosystems as long as the underlying code satisﬁes certain algebraic ⋆This work was supported by the NSF under grants CCF-0829931, 0835735, and 0829917 and by the DTO under contract W911NF-04-R-0009. ⋆⋆This work was supported by the NSF under grants 1117427 and 0835735 and by the DTO under contract W911NF-04-R-0009. P. Rogaway (Ed.): CRYPTO 2011, LNCS 6841, pp. 761–779, 2011. © International Association for Cryptologic Research 2011 762 H. Dinh, C. Moore, and A. Russell properties. While there are known classical att acks on these systems for the case of ratio- nal Goppa codes, our results also apply to the Niederreiter cryptosystem with classical Goppa codes, which to our knowledge is still believed to be classically secure. While our results do not rule out other quantum (or classical) attacks, they do demonstrate se- curity precisely against the types of quantum algorithms that have proven so powerful for number-theoretic problems. We also strengthen some results of Kempe et al. [9] on subgroups of Snreconstructible by Fourier sampling. McEliece-type cryptosystems. The McEliece

Question 777 multiple-choice

Quantum algorithms have demonstrated significant advantages over classical methods when solving hidden subgroup problems in certain non-abelian groups, such as q-hedral and dihedral groups. The efficiency and practicality of these quantum solutions often depend on group-theoretic properties and the choice of measurement basis in quantum Fourier sampling. In the context of reconstructing hidden subgroups of a q-hedral group $ Z_q \ltimes Z_p $ using quantum algorithms, which statement best explains why measuring in an adapted basis is crucial for distinguishing conjugate subgroups efficiently? 1) An adapted basis maximizes entanglement between quantum registers, reducing computational errors. 2) An adapted basis allows subgroup elements to be mapped directly onto irreducible representations of the full group. 3) Measuring in an adapted basis ensures polynomial-time classical algorithms can reconstruct subgroups. 4) Random basis measurements always provide uniform probability distributions, making them reliable for all subgroup types. 5) Measuring in an adapted basis aligns with the subgroup structure, making distinguishing features apparent, whereas a random basis requires exponentially more samples to differentiate conjugate subgroups. 6) An adapted basis eliminates the need for oracle functions in the hidden subgroup problem. 7) Measuring in an adapted basis is only necessary for abelian groups, as non-abelian groups do not benefit from basis adaptation.

✓ Correct Answer:

The correct answer is 5) Measuring in an adapted basis aligns with the subgroup structure, making distinguishing features apparent, whereas a random basis requires exponentially more samples to differentiate conjugate subgroups..

📚 Reference Text:

its conjugacy class, and we can information-theoretically reconstruct which conjugate it is using, Theorem 3or Theorem 4. Theorem 5. Subgroups of the q-hedral groups Z q/multicloseleftZp, including the aﬃne groups Ap, are information-theoretically reconstructible. Proof. As in Theorem 2, we can (fully) reconstruct normal subgroups, and so it suﬃces to consider nonnormal subgroups H. As discussed above, if we can determine |H|=q/prime, then we know that it is one of the conjugates of Hq/prime, and we can reconstruct it information-theoretically using Theorem 3 or Theorem 4. Now let the oracle be f:Zq/multicloseleftZ p→S, and let pα1 1...pαk kbe the prime factor- ization of q, in which case k≤/summationtext iαi=O(logq). For each i∈{1,...,k }and each α∈{0,...,α i}, we will determine if pα i||H|, and taking the largest such αfor each igives the prime factorization of |H|. To do this, for each i∈[k] and 1 ≤α≤αi, let Υα i:Zq/multicloseleftZ p→Zq/pα ibe the homomorphism given by Υα i:(a, b)/mapsto→apα i. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE POWER OF STRONG FOURIER SAMPLING 951 Then let Aαi i= ker Υα i={γ∈Zq/multicloseleftZ p|γpαi i=1}, where 1denotes the identity element of Zq/multicloseleftZ p.Aαi iis the subgroup of Zq/multicloseleftZ p consisting of all elements whose orders are a multiple of pα i. Now consider the function f/prime:Zq/multicloseleftZ p→S×Zq/pα i given by f/prime(γ)=(f(γ),Υα i(γ)). Observe that f/primeis constant (and distinct) on the left cosets of H∩Aα iand, furthermore, the subgroup H∩Aα ihas order pαif and only if pαdivides |H|. We may then determine ifH∩Aα ihas order pαby assuming that it does, reconstructing Hwith Theorem 4 using f/primeas the oracle, and checking the result against the original oracle f. This allows us to determine the prime factorization of |H|as desired, and the theorem follows. As in the dihedral case [6], we know of no polynomial-time algorithm which can reconstruct the most likely bfrom these queries. However, Kuperberg [18] gives a quantum algorithm for the HSP in the dihedral group, and more generally the hidden shift problem, that runs in subexponential (eO(log1/2p)) time. Since we can reduce the HSP on Zq/multicloseleftZ pto a hidden shift problem by focusing on two cosets of Zp, this algorithm applies to the q-hedral groups as well. 5. Random vs. adapted bases. In Theorems 3 and 5, we measured the high- dimensional representation ρin a speciﬁc basis which is adapted to the subgroup structure of Apand the q-hedral groups. In contrast, we show in this section that if we measure ρin arandom basis instead, then for all but the largest values of qwe need an exponential number of measurements in order to information-theoreticallydistinguish conjugate subgroups from each other. Theorem 6. Letpbe prime and let qbe a divisor of p−1for which q
0.L e t Pb(v)be the probability that we observe a basis vector vin the Fourier basis if the hidden subgroup is Hb q. If we measure ρin a random basis, then for any twob, b/prime, with high probability the /lscript1distance between these probability distributions is exponentially small.

Question 778 multiple-choice

Quantum computing explores algorithms for complex group-theoretic problems, including the Dihedral Hidden Subgroup Problem (DHSP), which connects with classical computational problems like subset sum. Understanding the computational hardness and measurement limitations is crucial for evaluating quantum algorithm capabilities in this domain. Which statement most accurately characterizes the role of the subset sum problem over the group ℤ/ℤ_N in relation to the Dihedral Hidden Subgroup Problem (DHSP) and quantum algorithm limitations? 1) It is efficiently solvable for all group sizes, enabling practical quantum algorithms for DHSP. 2) Its hardness is limited to classical computation; quantum algorithms always solve it efficiently. 3) It is only relevant for abelian groups and does not affect non-abelian DHSP cases. 4) Efficient quantum sampling guarantees polynomial-time solutions for all instances. 5) Its NP-completeness over ℤ/ℤ_N means that efficient solutions would yield efficient algorithms for DHSP, but quantum measurements are generally bounded by exponentially small success probabilities for large instances. 6) Legal subset sum instances can always be resolved by elimination observables in the dihedral group. 7) The subset sum problem's complexity is independent of quantum measurement strategies for hidden subgroup identification.

✓ Correct Answer:

The correct answer is 5) Its NP-completeness over ℤ/ℤ_N means that efficient solutions would yield efficient algorithms for DHSP, but quantum measurements are generally bounded by exponentially small success probabilities for large instances..

📚 Reference Text:

∕nπak Nnπak N1 21c o s 21cos .2 (4.5) For the second row, it is (() ) ()−∕ = ∕nπak Nnπak N1 21c o s 21sin .2 (4.6) We are now in the situation of the study by Ettinger and Høyer [29 ] and can apply the post-processing algorithm described (which is exponential in time) to determine awith high probability, for large N. 4.3 Relation to the subset sum problem Given /DoubleCapZ/DoubleCapZ () ( )=∈ ∕xxx N,… ,mm1 and/DoubleCapZ/DoubleCapZ∈∕rN , the problem of ﬁnding {}∈b 0, 1msuch that ⋅=bx r is called the subset sum problem over /DoubleCapZ/DoubleCapZ∕N. The vector bcorresponds to specifying a subset of the xx,…,m 1 that sum to r. Denote by {{ } }=∈ ⋅ =Sb b x r 0, 1 :rx m the set of subset sums for ()xr,. If such abexists, then ()xr,is called a legal instance. In the decision version of the subset sum problem, the problem is to determine whether a given ()xr,is a legal instance. In the study by Regev [12 ], it is shown that the ability to e ﬃciently ﬁnd an element ∈bSrxfor a large fraction of legal instances gives an e ﬃcient algorithm to solve DHSP. Furthermore, the study by Bacon et al. [24] shows that the ability to quantum sample from Srxallows us to e ﬃciently implement an optimal measure- ment to determine afrommDCP samples. The subset sum problem over /DoubleCapZis known to be an NP-complete problem. Since one can reduce the subset sum problem over /DoubleCapZto the subset sum problem over /DoubleCapZ/DoubleCapZ∕N, by choosing a large enough modulus N, it follows that the subset sum problem over /DoubleCapZ/DoubleCapZ∕N is also NP-complete. 4.4 Optimal measurements It is shown in the study by Ettinger and Høyer [33 ] that e ﬃcient elimination observables do not exist for the dihedral group. Further results can be found in the study by Bacon et al. [24]. In particular, let =∕νmNlog2 be the density de ﬁned in the study by Bacon et al. [24 ]. It is shown in [24 , Theorem 2] that if >+∕ν N 14 l o g2, the probability of determining ausing the optimal measurement on mDCP samples is ≥∕18. Furthermore, for any Nandm, the probability of determining aisThe dihedral hidden subgroup problem  11 ()≤∕=−N22 ,mν N 1l o g2 (4.7) which is exponentially small in Nlog2for any ﬁxed<ν1, and gives a trivial upper bound when ≥ν1. More general results on optimal measurements to distinguish conjugate hidden subgroups in certain groups can be found in the study by Moore and Russell [30 ]. In the study by Bacon et al. [24 ], the success probability of the optimal measurement is determined as: /DoubleCapZ/DoubleCapZ /DoubleCapZ/DoubleCapZ()∑∑ ⎟ ⎜ =⎛ ⎝⎞ ⎠+ ∈∕ ∈ ∕pNη1 2,mN mm xNr Nrx , 12 m where ∣∣≔ηSrx rx. Remark 4.8. For example, let =m 2,=N 2m, and=ν1. Computer calculations show that ≈p 0.6665mN, . On the other hand, we saw in Remark 3.10 that we can achieve a success probability of ≈0.9892 for=m

Question 779 multiple-choice

Quantum Fourier transform circuits are essential components in quantum algorithms, and their efficient implementation relies on specialized quantum gates and photonic technologies. The scaling of quantum discrete Fourier transform (DQFT) circuits compared to classical algorithms is a key factor in their computational advantage. In an optical quantum circuit using quantum dot-cavity systems and CRk gates, what is the number of gates required to implement a t-qubit discrete quantum Fourier transform (DQFT)? 1) t(t+1)/2 2) 2^t 3) t^2 4) t(t-1)/2 5) Θ(t2^t) 6) t+1 7) 2t

✓ Correct Answer:

The correct answer is 1) t(t+1)/2.

📚 Reference Text:

after the input state, |ϕ 〉in, passes through the CRk gate (QD1 and QD2 gates) in Fig. 3, the final state, |ϕ〉fin, can be given by xR Rx LR xR Lx eL L (14)i inCRkgate fin1 B1 A1 2B1 A1 3B1 A1 42/ 2 B1 A1kϕϕ=⇒== =++ +.π Consequently, the final state, |ϕ 〉fin, which is applied to the CRk gate using two QD-cavity systems, is identical to the state, | φ〉fin, in Eq. 4, which is applied to the CRk operation (UCRk: swapping paths and operation of rotation k). Now, using a simple example in Fig. 4, we show the implementation of the three-qubit DQFT scheme by directly utilizing the proposed CRk gates via the QD-cavity systems. As shown in Fig. 4, we can realize the the- oretical circuit of the three-qubit DQFT to three-photon DQFT scheme employing CRk gates (QD-cavity sys- tems) and HWPs by arranging our CRk gates in the manner shown in Fig. 3. Thus, for the scalability of the multi-photon DQFT scheme (for the implementation of multi-qubit DQFT), the structure of the three-photon 6 Scientific RepoRtS | (2019) 9:12440 | https://doi.org/10.1038/s41598-019-48695-z www.nature.com/scientificreports www.nature.com/scientificreports/DQFT scheme can be generalized (or expanded) to realize the multi-photon DQFT scheme by the composition of the CRk (k = 2, …, t) gates and HWPs. The structure of our DQFT scheme, which consists of CRk gates using QDs, is based on the theoretical circuit of DQFT in Fig. 1. When our scheme is expended to multi-photon DQFT scheme, the method of expansion is identical with the procedure of the theoretical circuit of multi-qubit DQFT algorithm, as described in Fig. 4. Thus, we can calculate the complexity2 about how many gates our DQFT scheme is utilizing on t photons, as follows: The total of t gates (a HWP and t − 1 CRk gates) is used on the first photon. And the total of t − 1 gates (a HWP and t − 2 CRk gates) is used on the second photon. Finally, the tth photon only utilizes one gate (a HWP). In our DQFT scheme, we can confirm that the number of gates for DQFT is as t (t + 1)/2 = t + (t − 1) + ⋅⋅⋅ + 2 + 1. Consequently, our DQFT scheme provides a Θ (t2) algorithm for the operation of DQFT on t photons. Furthermore, compared with the classical best algorithm as fast Fourier transform (FFT)2, which provides a Θ(t2t) algorithm for the discrete Fourier transform, our DQFT scheme on quantum computer is more efficient than FFT, which requires exponentially more operations, on classical computer. So far, we have proposed the use of the CRk gate, using the QD-cavity systems, to feasibly implement the CRk (controlled-rotation k) operation for the optical DQFT scheme with high efficiency and reliable performance. In the optical DQFT, our CRk gate with two QD-cavity systems (QD1 and QD2 gates) is the essential element with the stable storage of quantum information and the actual interaction between the photon and QD. Thus,

Question 780 multiple-choice

Quantum computing relies on efficiently constructing circuits from a finite set of gates, with practical error bounds and overhead, and often encodes logical qubits using advanced mathematical structures such as representation theory and Young tableaux. Understanding the theoretical foundations that enable accurate circuit approximation is crucial for scalable quantum computation. Which statement accurately describes the guarantee provided by the Solovay-Kitaev theorem for quantum circuit synthesis? 1) Any two-qubit gate can be implemented with zero error using only a finite universal gate set. 2) Any unitary operation in U(d) can be approximated to within any desired precision ε by a sequence of gates from a finite universal set, with the number of gates and classical computation time scaling polylogarithmically in 1/ε. 3) The total error in a quantum circuit is always less than the error in any individual gate. 4) Only Clifford gates can be efficiently synthesized using finite universal sets. 5) The operator norm of a gate increases when tensoring with identities on additional qubits. 6) Logical qubits cannot be encoded using paths in Young graphs or representation theory. 7) Quantum computational problem classes are generally sensitive to the gate count overhead introduced by circuit approximation.

✓ Correct Answer:

The correct answer is 2) Any unitary operation in U(d) can be approximated to within any desired precision ε by a sequence of gates from a finite universal set, with the number of gates and classical computation time scaling polylogarithmically in 1/ε..

📚 Reference Text:

Then, for every pair of normalized pure states |φ/an}bracketri}ht,|ψ/an}bracketri}ht ∈Cd, we have /vextendsingle/vextendsingle|/an}bracketle{tφ|U|ψ/an}bracketri}ht|2− |/an}bracketle{tφ|U′|ψ/an}bracketri}ht|2/vextendsingle/vextendsingle≤ǫ. The next theorem assures that universal sets of unitaries su ﬃce to eﬃciently ǫ-approximate an arbitrary quantum circuit. Theorem 7.3 (Solovay-Kitaev) LetG ⊂U(d)be a ﬁnite universal set of unitaries on a Hilbert space of constant dimension dwhich are closed under inverses, i.e. G−1=G. Then any unitary U∈U(d)can beǫ-approximated by a sequence of O/parenleftbig poly log(1/ǫ)/parenrightbig gates in G. Moreover, such a sequence can be found by a classical compute r inO/parenleftbig poly log(1/ǫ)/parenrightbig time. Fixing any ﬁnite, universal set of two-qubit unitaries G2⊂U(4), let Gn⊂U(2n) be the set of gates obtained by letting G2act on all pairs of neighboring qubits. Given a length m circuitUmUm−1···U1consisting of gates from Gnand any ﬁxeddesired degree of accuracy ǫ, the Solovay-Kitaev Theorem further implies that for each i, there is an ǫ/mapproximation U′ itoUiwhich is obtained by a sequence of O/parenleftbig poly log(m)/parenrightbig gates in G, and also that such a sequence can be found in O/parenleftbig poly log(m)/parenrightbig time (since ǫdoes not grow with m, it does not ﬁgure into the complexity estimate). Observe that for any un itariesUandU′, the operator norm is stable, meaning that ||U−U′||∞=||U⊗1d−U′⊗1d||∞for every ﬁnite dimension d. Therefore, the same approximations hold for the embeddings of the local gates into U(2n). By a telescoping sum and the triangle inequality, we have ||Um···U2U1−U′ m···U′ 2U′ 1||∞≤m/summationdisplay i=1||Ui−U′ i||∞. (7.2) Therefore, there is an ǫ-approximation to the desired circuit which uses O/parenleftbig mpoly log(m)/parenrightbig gates from Gn. So we see that working with ﬁnite universal sets of gates is j ust as good as working with all possible two-qubit unitaries, provided th at one can live with bounded errors and polylogarithmically more gates. In Section 8, we deﬁne c lasses of quantum computational problems which are insensitive to these limitations. Throughout this section, we ﬁx an ℓ≥5,ℓ/ne}ationslash= 6 and for each diagram λ, we abbreviate πλ≡π(2,ℓ) λandVλ=V(2,ℓ) λ. We begin by describing how to encode a single logical qubit i nto Pawel Wocjan and Jon Yard 25 1 /d38/d38/d77/d77/d77/d77/d77/d77/d77 ∅1/d47/d472/d57/d57/d115/d115/d115/d115/d115/d115/d115 1/d39/d39/d79/d79/d79/d79/d79/d792/d47/d47 2/d54/d54/d110/d110/d110/d110/d110/d110 Fig. 7.1. Qubit path basis for W1. The upper path corresponds to the basis state |t0/angbracketright, the lower path to |t1/angbracketright. V. This is a two-dimensional representation of B4with orthonormal basis |t0/an}bracketri}ht ≡/vextendsingle/vextendsingle/vextendsingle13 24/angbracketrightBig |t1/an}bracketri}ht ≡/vextendsingle/vextendsingle/vextendsingle12 34/angbracketrightBig . These basis states may also be viewed as paths in the Young gra ph (see Figure 7.1). Let t andt′be (2,ℓ)-tableaux of respective shapes [ j,j] and [k,k]. Recall the convention mentioned before Lemma 5.1 where we write tt′for the tableau of shape [ j+k,j+k] obtained by adding 2jto each box in t′then placing it to the right of t. Central to our proof is that tt′will always be a (2 ,ℓ)-tableaux. Recall that this allows us to deﬁne a natural emb edding of vector spacesV[j,j]⊗V[k,k]֒→V[j+k,j+k]via the mapping |t/an}bracketri}ht|t′/an}bracketri}ht /ma√sto→ |tt′/an}bracketri}ht. We encode nqubits into a subspaceW(n)of the irreducible representation V[2n,2n]ofB4nwhich is spanned by the 2n computational basis states/braceleftbig |tx1tx2···txn/an}bracketri}ht: (x1,x2,...,x n)∈ {0,1}n/bracerightbig .By our embedding the

Question 781 multiple-choice

In quantum field theory, representations of symmetry groups play a crucial role in describing particle behavior, and mathematical structures like cocycles and projective representations can encode physical effects such as anomalies. The treatment of fermions often involves sophisticated constructions in infinite-dimensional spaces. Which property of the infinite-dimensional spin representation allows for the identification of Schwinger terms and quantum anomalies in the context of fermion quantization? 1) Its inherently projective nature, which introduces a cocycle that manifests as anomalies in the group action 2) The ability to diagonalize all symmetry generators simultaneously 3) The restriction to finite-dimensional subspaces for computation 4) Its invariance under arbitrary gauge transformations 5) The use of normal ordering of operators to eliminate vacuum contributions 6) The existence of a unitary equivalence for all choices of complex structures 7) The commutativity of permutation operators acting on the Fock space

✓ Correct Answer:

The correct answer is 1) Its inherently projective nature, which introduces a cocycle that manifests as anomalies in the group action.

📚 Reference Text:

Title: Quantum electrodynamics in external fields from the spin representation Year: 1994 Paper ID: c12d9383111d95a87b3dc8c6c49ac6442eca8edb Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c12d9383111d95a87b3dc8c6c49ac6442eca8edb Abstract: Systematic use of the infinite‐dimensional spin representation simplifies and rigorizes several questions in quantum field theory. This representation permutes ‘‘Gaussian’’ elements in the fermion Fock space, and is necessarily projective: its cocycle at the group level is computed, and Schwinger terms and anomalies from infinitesimal versions of this cocycle are obtained. Quantization, in this framework, depends on the choice of the ‘‘right’’ complex structure on the space of solutions of the Dirac equation. We show how the spin representation allows one to compute exactly the S‐matrix for fermions in an external field; the cocycle yields a causality condition needed to determine the phase.

Question 782 multiple-choice

Efficient algorithms for solving banded circulant linear systems are significant in computational physics and engineering, especially as quantum and quantum-inspired classical methods become more prominent. Understanding the scaling of quantum resources and the underlying mathematical structures is crucial for advancing these computational techniques. Which approach enables quantum resource requirements to scale linearly with the bandwidth parameter K for solving banded circulant linear systems, as opposed to previous methods with exponential scaling? 1) Employing standard conjugate gradient methods on circulant matrices 2) Utilizing variational quantum algorithms without exploiting matrix structure 3) Applying naive matrix inversion on classical hardware 4) Using classical Fast Fourier Transform without quantum subroutines 5) Implementing quantum algorithms based solely on HHL for general sparse matrices 6) Decomposing banded circulant matrices into cyclic permutations and combining quantum states via convex optimization 7) Relying on randomized sampling without leveraging cyclic matrix structure

✓ Correct Answer:

The correct answer is 6) Decomposing banded circulant matrices into cyclic permutations and combining quantum states via convex optimization.

📚 Reference Text:

Hybrid quantum-classical and quantum-inspired classical algorithms for solving banded circulant linear systems Po-Wei Huang,∗Xiufan Li,†Kelvin Koor,‡and Patrick Rebentrost§ Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 (Dated: September 21, 2023) Solving linear systems is of great importance in numerous fields. In particular, circulant systems are especially valuable for efficiently finding numerical solutions to physics-related differential equa- tions. Current quantum algorithms like HHL or variational methods are either resource-intensive or may fail to find a solution. We present an efficient algorithm based on convex optimization of combinations of quantum states to solve for banded circulant linear systems whose non-zero terms are within distance Kof the main diagonal. By decomposing banded circulant matrices into cyclic permutations, our approach produces approximate solutions to such systems with a combination of quantum states linear to K, significantly improving over previous convergence guarantees, which require quantum states exponential to K. We propose a hybrid quantum-classical algorithm using the Hadamard test and the quantum Fourier transform as subroutines and show its PromiseBQP - hardness. Additionally, we introduce a quantum-inspired algorithm with similar performance given sample and query access. We validate our methods with classical simulations and actual IBM quan- tum computer implementation, showcasing their applicability for solving physical problems such as heat transfer. I. INTRODUCTION Quantum algorithms have been discussed in finding approximate solutions to linear systems of equations. Harrow et al. [1] famously gave an efficient quantum algorithm to prepare solutions to sparse, well-conditioned linear systems. However, the HHL algorithm consumes a large amount of quantum resources and may not be useful before fault- tolerant quantum computers are well established. Additionally, variational methods [2, 3] such as the variational quantum linear solver (VQLS) [4] are shown to be executable on noisy intermediate-scale quantum (NISQ) devices [5] but suffers from the barren plateau problem [6] as well as a lack of performance guarantees. To address these problems, Huang et al. [7] proposed a near-term solution that utilizes classical combinations of quantum states (CQS) and an Ansatz tree to provide a theoretical guarantee of finding the approximate solution with a limited amount of quantum resources. On the other hand, quantum-inspired classical algorithms that utilizes sample and query access [8] to achieve approximate solutions whose performance can potentially match quantum algorithms have also been discussed for finding approximate solutions for linear systems [9–13]. Solving circulant linear systems is a fundamental problem in numerous fields such as modeling physical systems, signal processing, and image reconstruction. Owing to their special properties, state-of-the-art classical algorithms often rely on the fast Fourier transform algorithm (FFT) to diagonalize such matrices [14] and obtain the solution of N-dimensional linear systems using O(NlogN) operations [15], providing a significant speedup over exact solutions of linear systems such as Gaussian elimination, which may take up to O(N3) operations. Further, it may also be faster than approximate solutions for ill-conditioned matrices, such as the conjugate gradient method which takes up to O(Nκ) operations [16], where κis the condition number of the linear system.

Question 783 multiple-choice

Quantum computing algorithms increasingly leverage specialized transforms to exploit the structure of physical systems, particularly those involving continuous variables and Gaussian distributions. The discrete Hermite transform is a powerful tool enabling efficient quantum simulations and learning tasks. Which feature distinguishes the discrete Hermite transform as a "Gaussian analogue" to the quantum Fourier transform in quantum algorithms? 1) It maps computational basis states to amplitudes proportional to Hermite functions, which are intrinsically linked to Gaussian distributions. 2) It directly encodes periodic functions into quantum states without incorporating Gaussian structures. 3) It operates exclusively on binary input spaces, mimicking classical logic gates. 4) It requires polynomial-time scaling with input size and error, limiting efficiency for large systems. 5) It is primarily designed for simulating spin systems rather than continuous variables. 6) It transforms only the phase information of quantum states, ignoring amplitude dynamics. 7) It is restricted to applications in cryptographic key generation and does not support property testing or learning.

✓ Correct Answer:

The correct answer is 1) It maps computational basis states to amplitudes proportional to Hermite functions, which are intrinsically linked to Gaussian distributions..

📚 Reference Text:

Title: Efficient Quantum Hermite Transform Year: 2025 Paper ID: c64064bcba843f3b8e163ca226eab491d8875f2e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c64064bcba843f3b8e163ca226eab491d8875f2e Abstract: We present a new primitive for quantum algorithms that implements a discrete Hermite transform efficiently, in time that depends logarithmically in both the dimension and the inverse of the allowable error. This transform, which maps basis states to states whose amplitudes are proportional to the Hermite functions, can be interpreted as the Gaussian analogue of the Fourier transform. Our algorithm is based on a method to exponentially fast forward the evolution of the quantum harmonic oscillator, which significantly improves over prior art. We apply this Hermite transform to give examples of provable quantum query advantage in property testing and learning. In particular, we show how to efficiently test the property of being close to a low- degree in the Hermite basis when inputs are sampled from the Gaussian distribution, and how to solve a Gaussian analogue of the Goldreich-Levin learning task efficiently. We also comment on other potential uses of this transform to simulating time dynamics of quantum systems in the continuum.

Question 784 multiple-choice

Swap algebras are central in studying permutation symmetries and their algebraic representations, especially in quantum information theory and combinatorial algebra. Gröbner bases are used to analyze the polynomial relations among swap variables, with their complexity and degree growing rapidly as the number of indices increases. Which of the following statements correctly describes the behavior of Gröbner bases for swap algebras as the number of indices $ n $ increases? 1) The size and maximal degree of Gröbner bases remain constant regardless of $ n $. 2) Gröbner bases for swap algebras always consist only of quadratic polynomials for any $ n $. 3) The number of elements in Gröbner bases decreases with increasing $ n $. 4) Gröbner bases are independent of the combinatorial structure of swap matrices. 5) The maximal degree of polynomials in Gröbner bases decreases as $ n $ grows. 6) Both the number of elements and the maximal degree of polynomials in Gröbner bases increase rapidly as $ n $ becomes larger. 7) Gröbner bases cannot be computed for irreducible representations of the symmetric group.

✓ Correct Answer:

The correct answer is 6) Both the number of elements and the maximal degree of polynomials in Gröbner bases increase rapidly as $ n $ becomes larger..

📚 Reference Text:

−1 2Swap15Swap26Swap48+1 2Swap15Swap26Swap47+1 2Swap15Swap26Swap38−1 2Swap15Swap26Swap37 +1 2Swap15Swap24Swap68−1 2Swap15Swap24Swap67−1 2Swap15Swap24Swap38+1 2Swap15Swap24Swap37 −1 2Swap15Swap23Swap68+1 2Swap15Swap23Swap67+1 2Swap15Swap23Swap48−1 2Swap15Swap23Swap47 +1 2Swap14Swap57Swap68−1 2Swap14Swap56Swap78−1 2Swap14Swap38Swap57+1 2Swap14Swap38Swap56 −1 2Swap14Swap37Swap68+1 2Swap14Swap37Swap58+1 2Swap14Swap36Swap78−1 2Swap14Swap36Swap58 −1 2Swap14Swap27Swap58+1 2Swap14Swap27Swap56+1 2Swap14Swap27Swap38−1 2Swap14Swap27Swap36 +1 2Swap14Swap26Swap58−1 2Swap14Swap26Swap57−1 2Swap14Swap26Swap38+1 2Swap14Swap26Swap37 +1 2Swap14Swap25Swap78−1 2Swap14Swap25Swap68−1 2Swap14Swap25Swap37+1 2Swap14Swap25Swap36 −1 2Swap14Swap23Swap78+1 2Swap14Swap23Swap68+1 2Swap14Swap23Swap57−1 2Swap14Swap23Swap56 +1 2Swap13Swap58Swap67−1 2Swap13Swap57Swap68−1 2Swap13Swap48Swap67+1 2Swap13Swap48Swap57 +1 2Swap13Swap47Swap68−1 2Swap13Swap47Swap58−1 2Swap13Swap26Swap58+1 2Swap13Swap26Swap57 +1 2Swap13Swap26Swap48−1 2Swap13Swap26Swap47+1 2Swap13Swap25Swap68−1 2Swap13Swap25Swap67 −1 2Swap13Swap25Swap48+1 2Swap13Swap25Swap47−1 2Swap13Swap24Swap68+1 2Swap13Swap24Swap67 +1 2Swap13Swap24Swap58−1 2Swap13Swap24Swap57−1 2Swap12Swap58Swap67+1 2Swap12Swap56Swap78 +1 2Swap12Swap47Swap58−1 2Swap12Swap47Swap56−1 2Swap12Swap45Swap78+1 2Swap12Swap45Swap67 +1 2Swap12Swap38Swap67−1 2Swap12Swap38Swap56−1 2Swap12Swap38Swap47+1 2Swap12Swap38Swap45 Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 81 −1 2Swap12Swap36Swap78+1 2Swap12Swap36Swap58+1 2Swap12Swap36Swap47−1 2Swap12Swap36Swap45 +1 2Swap12Swap34Swap78−1 2Swap12Swap34Swap67−1 2Swap12Swap34Swap58+1 2Swap12Swap34Swap56. To form a basis B4for the linear space spanned by products of at most four swap matrices, one takes B3and adds (Types 7) For each 7-tuple i

Question 785 multiple-choice

High-dimensional qudit systems extend beyond traditional qubit-based quantum computing by allowing quantum units to exist in more than two states. This creates new opportunities and challenges for quantum information processing, error correction, and algorithm design. Which of the following is a unique advantage of qudit systems compared to qubit systems in quantum computing? 1) They eliminate the need for error correction protocols altogether. 2) They can achieve higher information density per quantum particle, potentially enabling more compact circuits and efficient algorithms. 3) They guarantee universal fault-tolerance with simpler gates. 4) They restrict entanglement to only two particles at a time. 5) They require fewer references to group theory and higher mathematics. 6) They make benchmarking easier due to fewer parameters. 7) They prevent implementation in photonic or ion-trap platforms.

✓ Correct Answer:

The correct answer is 2) They can achieve higher information density per quantum particle, potentially enabling more compact circuits and efficient algorithms..

📚 Reference Text:

article introduces the basics of the high-dimensionalqudit systems and provides details about qudit gates, quditalgorithms and implementations on various physical systems.The article serves as a summary of recent developments of quditquantum computing and an introduction for newcomers to theﬁeld of qudit quantum computing. Furthermore we show the advantages and the potential for qudit systems to outperformqubit counterparts. Of course these advantages can come withchallenges such as possibly harder-to-implement universal gates,benchmarking [ 80, 94, 117 ]; characterization of qudit gate [ 68, 136] and error correction connected with the complexity of the Clifford hierarchy for qudits [ 157]. Compared to qubit systems, qudit systems currently have received less attention in both theoretical and experimentalstudies. However, qudit quantum computing is becomingincreasingly important as many topics and problems in thisﬁeld are ripe for exploration. Extending from qubits to qudits uses in some mathematical challenges, with these mathematicalproblems elegant and perhaps giving new insights into quantumcomputing in their own right. Connections between quantumresources such as entanglement, quantum algorithms and their improvements, scaling up qudit systems both to higher dimension and to more particles, benchmarking and errorcorrection, and the bridging between qudits and continuous-variable quantum computing [ 67] are examples of the fantastic research directions in this ﬁeld of high-dimensional quantum computing. AUTHOR CONTRIBUTIONS All authors discussed the relevant materials to be added and allparticipated in writing the article. ACKNOWLEDGMENTS We would like to acknowledge the ﬁnancial support by the National Science Foundation under award number 1839191-ECCS. BCS appreciates ﬁnancial support from NSERC and from the Alberta government. Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504 20Wang et al. Qudits and High-Dimensional Quantum Computing REFERENCES 1. Abrams DS, Lloyd S. Quantum algorithm providing exponential speed increase forﬁnding eigenvalues and eigenvectors. Phys Rev Lett (1999) 83:5162 –5. doi:10.1103/PhysRevLett.83.5162 2. Adcock MRA, Høyer P, Sanders BC. Quantum computation with coherent spin states and the close Hadamard problem. Quant Inf Process (2016) 15:1361 –86. doi:10.1007/s11128-015-1229-0 3. Aharonov D, van Dam W, Kempe J, Landau Z, Lloyd S, Regev O. Adiabatic quantum computation is equivalent to standard quantum computation. SIAM J Comput (2007) 37:166–94. doi:10.1137/S0097539705447323 4. Alber G, Delgado A, Gisin N, Jex I. Ef ﬁcient bipartite quantum state puri ﬁcation in arbitrary dimensional Hilbert spaces. J Phys Math Gen (2001) 34:8821 –33. doi:10.1088/0305-4470/34/42/307 5. Amin MHS, Dickson NG, Smith P. Adiabatic quantum optimization with qudits. Quant Inf Process (2013) 12:1819 –29. doi:10.1007/s11128-012-0480-x 6. Anwar H, Campbell ET, Browne DE. Qutrit magic state distillation. New J Phys (2012) 14:063006. doi:10.1088/1367-2630/14/6/063006 7. Aromí G, Aguilà D, Gamez P, Luis F, Roubeau O. Design of magnetic coordination complexes for quantum computing. Chem Soc Rev (2012) 41: 537–46. doi:10.1039/c1cs15115k 8. Aspuru-Guzik A, Dutoi AD, Love PJ, Head-Gordon M. Simulated quantum computation of molecular energies. Science (2005) 309:1704 –7. doi:10.1126/ science.1113479 9. Babazadeh A, Erhard M, Wang F, Malik M, Nouroozi R, Krenn M, et al. High- dimensional single-photon quantum gates: concepts and experiments. Phys Rev Lett(2017) 119:180510. doi:10.1103/PhysRevLett.119.180510 10. Barends R, Kelly J, Megrant A, Veitia A,

Question 786 multiple-choice

In representation theory, branching rules determine how representations of a group decompose when restricted to a subgroup, often using combinatorial objects such as tableaux. Quantum symmetric pairs and their q-analogues generalize classical symmetric pairs, introducing new structures relevant to quantum groups and mathematical physics. Which of the following statements correctly characterizes the combinatorial algorithm relating semistandard tableaux and symplectic tableaux in the context of the branching rule from GL_{2n}(ℂ) to Sp_{2n}(ℂ)? 1) It constructs a surjection from symplectic tableaux to semistandard tableaux of varying shapes. 2) It establishes an injection from symplectic tableaux to a subset of semistandard tableaux with fixed shape. 3) It defines a bijection between symplectic tableaux and semistandard tableaux of identical shapes. 4) It maps semistandard tableaux of fixed shape to a single symplectic tableau of that shape. 5) It creates a bijection between semistandard tableaux of fixed shape and a union of sets of symplectic tableaux of various shapes. 6) It partitions symplectic tableaux into subsets indexed by semistandard tableaux of arbitrary shapes. 7) It associates each symplectic tableau with multiple semistandard tableaux through a non-invertible mapping.

✓ Correct Answer:

The correct answer is 5) It creates a bijection between semistandard tableaux of fixed shape and a union of sets of symplectic tableaux of various shapes..

📚 Reference Text:

Title: Symplectic tableaux and quantum symmetric pairs Year: 2023 Paper ID: c2829fd7579df3865392a6b2402e77a41b0183cb Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c2829fd7579df3865392a6b2402e77a41b0183cb Abstract: We provide a new branching rule from the general linear group \mathrm{GL}_{2n}(\mathbb{C}) to the symplectic group \mathrm{Sp}_{2n}(\mathbb{C}) by establishing a simple algorithm which gives rise to a bijection from the set of semistandard tableaux of a fixed shape to a disjoint union of several copies of sets of symplectic tableaux of various shapes. The algorithm arises from representation theory of a quantum symmetric pair of type A\mathrm{II}_{2n-1} , which is a q -analogue of the classical symmetric pair (\mathfrak{gl}_{2n}(\mathbb{C}), \mathfrak{sp}_{2n}(\mathbb{C})) .

Question 787 multiple-choice

In machine learning, ensuring that models perform well across different subgroups of a population is critical for fairness and robustness, especially in applications with significant societal impact. Multi-group agnostic learning addresses the challenge of unknown or hidden subgroup stratification. Which approach is most effective for minimizing conditional risks across all identifiable and latent subgroups without disproportionately increasing computational burden? 1) Using ensemble methods to average predictions across multiple models 2) Employing adversarial training to simulate subgroup variations 3) Implementing simple and near-optimal algorithms tailored to multi-group agnostic learning 4) Relying solely on overall population risk minimization 5) Applying unsupervised clustering followed by standard model training 6) Increasing model complexity with deeper neural networks 7) Ignoring underrepresented subgroups during the evaluation phase

✓ Correct Answer:

The correct answer is 3) Implementing simple and near-optimal algorithms tailored to multi-group agnostic learning.

📚 Reference Text:

Title: Simple and near-optimal algorithms for hidden stratification and multi-group learning Year: 2021 Paper ID: c8847713b5beb84bd152d59735100df5a0ac6c36 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c8847713b5beb84bd152d59735100df5a0ac6c36 Abstract: Multi-group agnostic learning is a formal learning criterion that is concerned with the conditional risks of predictors within subgroups of a population. The criterion addresses recent practical concerns such as subgroup fairness and hidden stratification. This paper studies the structure of solutions to the multi-group learning problem, and provides simple and near-optimal algorithms for the learning problem.

Question 788 multiple-choice

Quantum error correction codes are essential for protecting quantum information against noise and imperfections, and the structure of their codewords—including normalization methods and symmetry properties—affects both theoretical performance and practical robustness. Understanding how these design choices influence infidelity and code behavior is critical for developing codes suited to various physical systems. In the context of extending quantum error-correcting codes to rotor systems, which feature distinguishes the rotor code's construction from its qudit counterpart and is crucial for preserving code symmetry and encoding structure? 1) Encoding based on discrete Fourier transforms using only rational phase multiples 2) The use of primitive roots of unity with phases approaching irrational values as the dimension increases 3) Implementation of error correction solely via permutation operators 4) Reliance on continuous measurement of angular momentum eigenstates 5) Covariance exclusively with respect to continuous U(1) symmetry 6) Normalization using sharp cutoff boundaries in Hilbert space 7) Codeword construction without enforcing any symmetry constraints

✓ Correct Answer:

The correct answer is 2) The use of primitive roots of unity with phases approaching irrational values as the dimension increases.

📚 Reference Text:

[this relation can be seen from Eq. (12)]. While in both normalized versions of the above code, using either the sharp or the smooth cutoff, we are careful to ensure that all code words are close to each other, it turns out that codewords with a sharp cutoff are in a regime where the infidelity is more sensitive to differences than the smooth cutoff. This sensitivity is because those code words haveincompatible supports. More precisely, for any state ρ, the infidelityﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1−F 2ðρ;ρþεXÞp , for a small perturbation ρ→ρþεX, can grow like the square root of εifρþ εXhas overlap outside of the support of ρ, while it grows linearly in εin well-behaved cases. The sharp cutoff belongs to the former regime, while in the case of the smooth cutoff the infidelity is better behaved.CONTINUOUS SYMMETRIES AND APPROXIMATE QUANTUM … PHYS. REV . X 10,041018 (2020) 041018-13 B. Five-rotor perfect code Here, we provide a rotor extension of the five-qubit perfect code [27,49] that can be tiled to construct holo- graphic codes [7]. While qudit [50] and oscillator [51] extensions have been considered, a rotor extension is not asstraightforward, because one has to take care of preservingthe phases in the code states as needed to error correcterasures. Our rotor code is the limit of a sequence of quditcode words whose constituent phases approach multiples ofanirrational number. This same trick has been used to obtain an irrational magnetic flux via a sequence of rational fluxes in the two-dimensional electron gas problem [52]as well as rotor limits of other Hamiltonians [46]. This limit is meant to be an idealization, since there is not enoughstorage space to measure an irrational number to infiniteprecision. Let the dimension Dof each of the five physical subsystems be finite for the qudit ½½5;1;3/C138/C138 ZDcode and infinite for the rotor ½½5;1;3/C138/C138Zcode. The general form of the unnormalized encoding for both codes is jxi→X j;k;l;m;n ∈ZDTðDÞ jklmnx jj; k; l; m; n i: ð43Þ We introduce the rotor code as a limiting case of the qudit code, obtaining a concise expression for the qudit perfect tensor TðDÞin the process. 1. Qudit version Consider first the known finite- Dcase, for which [53] TðDÞ jklmnx ¼δðDÞ x;jþkþlþmþnωjkþklþlmþmnþnj; ð44Þ where δðDÞ a;b¼1ifa¼bmodulo Dandωis a primitive Dth root of unity. Notice how the above expression makes the cyclic permutation symmetry naturally manifest. The deltafunction encodes the state label xinto the sum of the physical qudit variables, with the key difference from thesharply cutoff ½½3;1;2/C138/C138 Zcode being that the sum is modulo D. This property makes this code exactly error correcting andnotcovariant with respect to a Uð1Þsymmetry. Instead, this code is covariant with respect to a ZDsymmetry generated by Z⊗5, where Z¼P k∈ZDωkjkihkjis the qudit Pauli matrix. 2. Smooth cutoff To take the qudit-to-rotor limit, pick ω¼expð2πiL=D Þ with incommensurate integers L; D→∞such that L=D approaches a positive irrational number Φ. The indices in Eq.(43) now range over Z: Tð∞Þ jklmnx ¼δx;jþkþlþmþne2πiΦðjkþklþlmþmnþnjÞ;ð45Þand δis the usual Kronecker delta function. The final ingredient is to normalize the

Question 789 multiple-choice

Quantum simulation of topological systems often involves the use of engineered Hamiltonians and advanced algorithms to study phenomena such as band gaps and robust phases. Environmental noise, including both white and colored types, plays a key role in determining the fidelity and reliability of these simulations. In simulations of flat-band Haldane models using advanced quantum algorithms, which type of noise primarily leads to a slower decrease in band gap and a more gradual increase in bandwidth at a given noise strength? 1) Thermal noise with Markovian dynamics 2) Shot noise with zero correlation time 3) Amplitude noise with random telegraph statistics 4) Colored noise with finite correlation time 5) White noise with instantaneous correlations 6) Phase noise with non-Gaussian fluctuations 7) DC noise with infinite correlation time

✓ Correct Answer:

The correct answer is 4) Colored noise with finite correlation time.

📚 Reference Text:

Fig. 7, we compare the e ects of di erent noises on the energy gap and bandwidth. For a given , the colored noise has less e ect on the energy bands, in comparison with the white noise. This is seen from the slower dropping (increasing) of the band gap (width) in the presence of colored noise, as increases. 9 (b) (c) (a) FIG. 6. QQFT based quantum simulation of the ﬂat-band Haldane model. (a) Upper (solid) and lower (dash-dotted) energy bands for di erent strength of noise. E+andEdenote the eigenenergies in the upper and lower bands, respectively. (b) Noise strength ( ) dependence of the band gap Gand the band width W, which are averaged over 102random noise conﬁgurations. In (a) and (b), we set the Hamiltonian parameters M=0 and ==2. (c) Phase diagram determined by the Bott index B(colormap) and the Chern number C(black dash-dotted lines) at the noise strength =3102. We divide the Brillouin zone into a 16 16 grid. The magnitude of dis ﬁxed tojdj=2kHz, and the evolution time is T=1=(2) ms, as accessible to cold atom experiments. ( (a) (b) FIG. 7. Comparison of the e ects of di erent noises on the en- ergy bands of a ﬂat-band Haldane model. Panels (a) and (b) dis- play the band gap and band width, respectively, as a function of noise strength, which are averaged over 200 random noise conﬁg- urations. Lines of di erent types are for the white noise (solid) and exponential-correlation noise with c=300 (dash-dotted), respec- tively. In the simulation, we consider a 16 16 lattice. The Hamilto- nian parameters are chosen to M=4 and ==2. The magnitude ofdis ﬁxed tojdj=2kHz, and the evolution time is T=1=(2) ms, as accessible to cold atom experiments. For both the Poincar ´e crystal and ﬂat-band Haldane model, the colored noise has less e ect on our QQFT-based simula- tions, in comparison with the white noise. A rough explana- tion in the limiting case is given below. If we consider the dc-noise limit ( c! 1 ),sthen becomes a s-independent constant. The dc-noise modiﬁes both the QQFT ( ˆV) and its Hermitian adjoint ( ˆVy) by replacing the Hamiltonian ˆHpby ˆHp(1+s). Ifsis a constant, then such a replacement keeps ˆVˆVy=1. Because the evolution operator is ˆU=ˆVeiTˆHDˆVy, the modiﬁcation to ˆVhas no inﬂuence on the spectrum of ˆU.This explains why a dc-noise has less e ect on the energy bands than a white noise. V . CONCLUSION AND OUTLOOK We propose a QQFT based Hamiltonian engineering scheme, which allows ﬂexible quantum simulations of transla- tionally invariant Hamiltonian models. The Hamiltonian con- trol sequence to implement QQFT is analytically constructed, considering cold atoms conﬁned in a programmable opti- cal potential. We demonstrate its capability by investigating quantum simulations on the Poincar ´e spacetime crystal and topological ﬂatband, both of which are extremely challeng- ing to realize with conventional approaches for the required delicate long range couplings. Its robustness against imper- fections potentially existent in optical lattice experiments has been conﬁrmed

Question 790 multiple-choice

Lattice-based cryptography forms the backbone of many post-quantum cryptographic schemes due to its conjectured resistance to quantum attacks. Estimating the computational resources required for attacks on such schemes is critical for assessing their security and guiding parameter selection. Which parameter, when increased, most directly causes the exponential growth in the computational cost of BKZ lattice reduction attacks against cryptosystems such as Kyber and Saber? 1) Sample size (m) 2) FFT modulus (p) 3) Secret distribution entropy 4) Dimension for enumeration (k_enum) 5) Number of cryptographic schemes considered 6) FFT dimension (k_fft) 7) Block size (β)

✓ Correct Answer:

The correct answer is 7) Block size (β).

📚 Reference Text:

sage: %time results = runall() sage: save(results, "../data/estimates.sobj") sage: print(results_table(results)) """ fromsage.all import sqrt, log, exp, e, pi, RR, ZZ fromestimator.estimator.cost import Cost fromestimator.estimator.lwe_parameters import LWEParameters fromestimator.estimator.reduction import delta as deltaf fromestimator.estimator.reduction import RC, ReductionCost fromestimator.estimator.conf import red_cost_model as red_cost_model_default fromestimator.estimator.util import local_minimum, early_abort_range fromestimator.estimator.io import Logging fromestimator.estimator.schemes import ( Kyber512, Kyber768, Kyber1024, LightSaber, Saber, 50 FireSaber, ) fromestimator.estimator.schemes import TFHE630, TFHE1024 classChaLoy21(ReductionCost ): __name__ = "ChaLoy21" short_vectors = ReductionCost._short_vectors_sieve def__call__(self, beta, d, B=None): """ :param beta: Block size ≥2. :param d: Lattice dimension. :param B: Bit-size of entries. """ return ZZ(2) ** RR(0.2570 * beta) classMATZOV: """ """ C_prog = 1.0 / (1 - 2.0 ** (-0.292)) # p.37 C_mul = 32**2 # p.37 C_add = 5 * 32 # guessing based on C_mul @classmethod defT_fftf(cls, k, p): """ The time complexity of the FFT in dimension ‘k‘ with modulus ‘ p‘. :param k: Dimension :param p: Modulus ≥2 """ return cls.C_mul * k * p ** (k + 1) # Theorem 7.6, p.38 @classmethod defT_tablef(cls, D): """ Time complexity of updating the table in each iteration. :param D: Number of nonzero entries """ return 4 * cls.C_add * D # Theorem 7.6, p.39 @classmethod defNf(cls, params, m, beta_bkz, beta_sieve, k_enum, k_fft, p) : """ Required number of samples to distinguish with advantage. :param params: LWE parameters :param m: :param beta_bkz: Block size used for BKZ reduction :param beta_sieve: Block size used for sampling :param k_enum: Guessing dimension :param k_fft: FFT dimension :param p: FFT modulus """ mu = 0.5 k_lat = params.n - k_fft - k_enum # p.15 # p.39 lsigma_s = ( params.Xe.stddev ** (m / (m + k_lat)) 51 * (params.Xs.stddev * params.q) ** (k_lat / (m + k_lat)) * sqrt(4 / 3.0) * sqrt(beta_sieve / 2 / pi / e) * deltaf(beta_bkz) ** (m + k_lat - beta_sieve) ) # p.29, we’re ignoring O() N = ( exp(4 * (lsigma_s * pi / params.q) ** 2) * exp(k_fft / 3.0 * (params.Xs.stddev * pi / p) ** 2) * (k_enum * cls.Hf(params.Xs) + k_fft * log(p) + log(1 / mu)) ) return RR(N) @staticmethod defHf(Xs): return RR(1 / 2 + log(sqrt(2 * pi) * Xs.stddev)) / log(2.0) @classmethod defcost( cls, beta, params, m=None, p=2, k_enum=0, k_fft=0, beta_sieve=None, red_cost_model=red_cost_model_default , ): """ Theorem 7.6 """ ifmisNone: m = params.n k_lat = params.n - k_fft - k_enum # p.15 # We assume here that β_sieve ≈β N = cls.Nf( params, m, beta, beta_sieve ifbeta_sieve elsebeta, k_enum, k_fft, p, ) rho, T_sample, _, beta_sieve = red_cost_model.short_vect ors( beta, N=N, d=k_lat + m, sieve_dim=beta_sieve ) H = cls.Hf(params.Xs) T_guess = ( ((2 / sqrt(e)) ** k_enum) * (2 ** (k_enum * H)) * (cls.T_fftf(k_fft, p) + cls.T_tablef(N)) ) cost = Cost(rop=T_sample + T_guess, problem=params) cost["red"] = T_sample cost["guess" ] = T_guess cost["beta"] = beta cost["p"] = p cost["zeta"] = k_enum cost["t"] = k_fft 52 cost["beta_" ] = beta_sieve cost["N"] = N cost["m"] = m cost.register_impermanent( {"β’": False, "ζ": False, "t": False}, rop=True, p=False, N=False ) return cost def__call__( self,

Question 791 multiple-choice

Representation theory often examines how modules over algebras decompose under operations like tensor product, paying special attention to the behavior of almost-split sequences and the structure of radicals. In the context of finite-dimensional Hopf algebras and group algebras, properties such as projectiveness and the appearance of the trivial module play key roles in classification. In the study of finite-dimensional Hopf algebras with involutive antipodes, under what condition is the trivial module guaranteed to be a direct summand in the tensor product M ⊗ N for indecomposable modules M and N? 1) If the field characteristic divides the order of the Hopf algebra 2) If both M and N are simple modules 3) If the tensor product M ⊗ N is projective 4) If M is injective and N is simple 5) If N is isomorphic to the radical of M 6) If M and N are both semisimple 7) If N is projective

✓ Correct Answer:

The correct answer is 7) If N is projective.

📚 Reference Text:

αί)M translated to this context sends 7i,i®7i,i to Notice that the canonical generator of the radical of ΛfίjM + 1 is #%,! ® 7i,i + 7o,o ® 7i,i+1 because 70,o ® 7i,ί is the chosen generator oΐNitU+ί. Consequently the monomorphism of α0,0 ® NitU is the standard injection multiplied by Theorem 5.5. Let N^u be the finite dimensional Jjf-module from the vertex Sj to the vertex su of A™. Let α0,0 be the trivial almost split sequence of the module Λ/o,o Then NiίU ® α0,0 = qu α0,o ® ^«.« Proof. For u > i, recall that xitU = ~ ~^~ q and zί>M = - ^f/+1(w - i - l)β. 1 -^"~ί 1 1Consequently = —, = = -, — and their quotient value Xi,u ~ 1 (« - 0« ^i,« ~ ^l ^ (w - 0« is ^rM. The case i = u (simple modules) follows from the fact that St (x) α0,o = #M and αofo®Si = -ταίfί. The following is a consequence of Proposition 5.4, compare with [2]. Proposition 5.6. a) Let N and M be indecomposable finite dimensional ^f-modules, and let αM be the almost-split sequence ending at M. Then N (x) αM and αM (x) N are direct sums of almost-split sequences. b) // q is an nth root of unity of order d, let N and M be Jjf(q)/Id or kZn(q)/Id indecomposable modules. Then N (x) αM and αM (x) AT are direct sums of almost-split sequences and projective-injectives factors. Proof. Up to a non-zero scalar we have αM = M ® α0,o> where α0>0 is the almost- split sequence of the trivial module. Hence N ® αM = AT ® M ® α0,o; the result follows from the decomposition of Theorem 4.1 and the fact that JΓ® α0,o is almost-split. In case b) the eventual projective summands of N ® M provide the split factors. In [6, 2, 5] it is proved that for a modular fc-group algebra (or a finite dimen- sional fc-Hopf algebra with antipode of order 2), the trivial module is a direct summand of M ® N if and only if M ^ TV* and (p, dimfc AT) = 1, where Af and Λf are indecomposable modules and p is the characteristic of the field k. This result allowed Benson and Carlson [6] to give a proof of the semisimplicity of the Green 476 C. Cibils algebra of representations when the Hopf algebra is of finite representation type: they prove that there are no nilpotent elements in this commutative algebra. The next result shows that for the fc(g)-Hopf algebra $? the condition concern- ing the characteristic of the field is dropped; for q an nih root of unity of order d and for indecomposable J^(q) or kZn(q)/Id -modules this condition is replaced by requiring N not to be projective. Notice that both algebras have antipode of order 2d, with d > 1. Moreover the analog of the Green algebra of representations of Sweedler's 4-dimensional example has nilpotent elements.

Question 792 multiple-choice

Quantum Fourier sampling is a fundamental tool in quantum computing for solving group-theoretic problems, such as the hidden subgroup problem. Representation theory and probability distributions play crucial roles in understanding the limitations of these quantum algorithms. Which of the following statements best describes a scenario where strong Fourier sampling over a finite group G fails to efficiently distinguish a subgroup H from the trivial subgroup? 1) The probability distribution resulting from sampling is sharply peaked at a single representation. 2) The distribution resulting from strong Fourier sampling on H is nearly uniform over the basis. 3) The normalized character of every irrep on H is maximized. 4) The subgroup H is normal and has large index in G. 5) The dimension of every irrep is equal to one. 6) The total variation distance between distributions for H and the trivial subgroup is large. 7) The group G is Abelian and all subgroups are conjugate.

✓ Correct Answer:

The correct answer is 2) The distribution resulting from strong Fourier sampling on H is nearly uniform over the basis..

📚 Reference Text:

Tr/parenleftBig Πρ {1}/parenrightBig=/bardblb/bardbl2 dρ=1 dρ. 3.2 Distinguishability by QFS We ﬁx a ﬁnite group Gand consider quantum Fourier sampling over Gin the basis given by{Bρ}. For a subgroup H

Question 793 multiple-choice

In representation theory and computational complexity, Kronecker coefficients and their associated polytopes play a central role in understanding the structure and multiplicities of tensor products of symmetric group representations. The computational properties of the decision problems involving these polytopes have profound implications for algorithmic approaches in mathematics and theoretical computer science. What is the significance of the result that the KronPolytope decision problem lies in the complexity class NP ∩ coNP? 1) It proves that KronPolytope can be solved in logarithmic space. 2) It guarantees KronPolytope has a polynomial-time algorithm for all instances. 3) It shows KronPolytope is NP-complete under standard reductions. 4) It indicates both positive and negative instances of KronPolytope can be efficiently verified with appropriate certificates. 5) It implies KronPolytope is undecidable for large Young diagrams. 6) It establishes KronPolytope is always solvable by randomized algorithms. 7) It means KronPolytope is equivalent to the moment polytope membership problem for all Lie groups.

✓ Correct Answer:

The correct answer is 4) It indicates both positive and negative instances of KronPolytope can be efficiently verified with appropriate certificates..

📚 Reference Text:

(michael.walter@stanford.edu). 1arXiv:1511.03675v2 [cs.CC] 24 Jun 2017 2 P. BÜRGISSER, M. CHRISTANDL. K.D. MULMULEY, M. WALTER M= (Cm) 3by tensor products, (UA;UB;UC)7!UA UB UC. Then any triple of Young diagrams = (A;B;C)can be understood as a highest weight of G. The multiplicity g(A;B;C)of the corresponding irreducible representation of Gin the k-th symmetric power Symk(M)is known as the Kronecker coeﬃcient , where we may assume that all three diagrams have the same number of boxes k. The corresponding moment polytopes are known as the Kronecker polytopes , parameterized by m, and they are given by Kron(m) =(A;B;C) k:A;B;C`k;g(A;B;C)>0 R3m: We remark that Kron (m)is known to be a convex polytope of dimension 3(m1)[41]. We consider the problem KronPolytope of deciding whether (A;B;C)=k2 Kron (m), given as input a triple of Young diagrams A;B;C`k(each speciﬁed by its row lengths encoded in binary), where mdenotes the maximum among the heights of the three Young diagrams. Equivalently, KronPolytope is the problem of deciding whether there exists some positive integer l>0such that the stretched Kronecker coeﬃcient g(lA;lB;lC)>0. Our main result in the case of the Kronecker polytopes then is the following theorem: Theorem 1.The problem KronPolytope is in NP\coNP. That is, there exists polynomially-sized certiﬁcates such that both membership and non-membership can be veriﬁed in polynomial time. We discuss the implications of Theorem 1 in section 1.1 below. We now consider the general case. Let Gdenote a compact, connected Lie group andMa unitary representation of G. For each integer k>0, we denote by mG;M;k () the multiplicity of the irreducible G-representation with highest weight inSymk(M), thek-th symmetric power of M. Then the pairs (k;)for whichmG;M;k ()>0form a ﬁnitely generated semigroup and so the following set is a rational convex polytope, called the moment polytope associated with the G-representation M: G(M) = k:mG;M;k ()>0 G ZRit Here, Gdenotes the weight lattice and tthe dual of the Lie algebra of a maximal torusTofG. We shall assume that the moment polytope is of maximal dimension, i.e., that dimG(M) =dimT, the rank of the Lie group (equivalently, that generic points in the projective space P(M)have discrete G-stabilizer). We are interested in the problem MomentPolytope of deciding whether =k2 G(M), given as input a compact, connected Lie group G, a ﬁnite-dimensional unitary representation Mwith moment polytope of maximal dimension, a highest weight, and a positive integer k. Equivalently, MomentPolytope is the problem of deciding whether there exists some positive integer l>0such that the stretched multiplicity mG;M;lk (l)>0. We discuss the precise encoding of the input in section 3.1 below. Roughly speaking, the group is speciﬁed in terms of Dynkin diagrams and the representation in terms of its highest weights, given by its coeﬃcients in binary with respect to a basis of fundamental weights, together with the total dimension of the representation Min unary. The latter is a natural requirement, as it allows the algorithms to run in polynomial time in the dimension of the representation, which can be exponential in the speciﬁcation of the highest weights alone. In the

Question 794 multiple-choice

In quantum information theory, measurements play a critical role in inferring properties of quantum systems, with certain measurement schemes allowing for the complete reconstruction of quantum states. The mathematical formalism underlying these schemes often involves group theory, operator measures, and concepts from frame theory. Which measurement approach provides a set of experimental outcomes from which the expectation value of any operator can be estimated, links positive-operator valued measures to unitary irreducible representations of groups on the Hilbert space, and utilizes frame theory for constructive data processing? 1) Projective measurements associated with commuting observables 2) Measurement of position and momentum using Heisenberg's uncertainty principle 3) Von Neumann measurements of mutually exclusive states 4) Quantum measurements restricted to diagonalizable operators 5) Measurements of phase-space observables using classical statistical methods 6) Quantum state estimation via spectral decomposition 7) Informationally complete measurements realized through group-represented POVMs and frame theory

✓ Correct Answer:

The correct answer is 7) Informationally complete measurements realized through group-represented POVMs and frame theory.

📚 Reference Text:

arXiv:quant-ph/0310013v2 6 Oct 2003Informationally complete measurements and groups representation G. M. D’Ariano, P. Perinotti, and M. F. Sacchi § Unit` a INFM and Dipartimento di Fisica “A. Volta”, Universit` a di Pavia, via A. Bassi 6, I-27100 Pavia, Italy Abstract. Informationally complete measurements on a quantum system allow t o estimate the expectation value of any arbitrary operator by just averaging functions of the experimental outcomes. We show that such kind of measureme nts can be achieved through positive-operator valued measures (POVM’s) related to u nitary irreducible representations of a group on the Hilbert space of the system. Wit h the help of frame theory we provide a constructive way to evaluate the data-proce ssing function for arbitrary operators. PACS numbers: 03.65.Ta, 03.65.Wj 1. Introduction The aim of any measurement is to retrieve information on the state o f a physical system. In classical mechanics, measuring the location on the phas e space provides a complete information on the system. On the other hand, in quantu m mechanics there are inﬁnitely many elementary measurements—corresponding to diﬀ erent observables— that provide only partial information, whereas “complementary” in formations could be achieved only with mutually exclusive experiments where non-commut ing observables should be perfectly measured. The problem then arises on how to achieve a kind of quantum measure ment that is “complete” [1, 2], in the sense that it can be used to infer informatio n on all possible (also exclusive) observables. The main idea is to perform a generalize d “unsharp” measurement, described by a so-called POVM (positive-operator v alued measure), from which a speciﬁc type of information—i.e. a particular ensemble averag e of a given operator—isretrievedbyjustchangingthedata-processingofit sexperimentaloutcomes. Informationally complete measurements are relevant for foundat ions of quantum mechanics as a kind of “standard” for a purely probabilistic descript ion [3]. Moreover, the existence of such measurements with minimal number of outcom es is crucial for the quantum version of the de Finetti theorem [4]. The most popular example of informationally complete measurement is given by the coherent-state POVM for a single-mode of the radiation ﬁeld, w hose probability §To whom correspondence should be addressed (msacchi@unipv.it) Informationally complete measurements and groups represe ntation 2 distribution is the so-called Q-function (or Husimi function) [5, 6, 7]. Another example, though of completely diﬀerent kind, is the case of quantum tomogra phy [8], in which one measures an observable randomly selected from an information ally complete set—a ”quorum”. Investigations on informationally complete measurements have bee n extensively carried out in the framework of ”phase-space observables”: the monographs [9, 10, 11, 12, 13, 14, 15] review diﬀerent aspects of these developments. P hase-space observables are very useful in various ﬁelds of quantum physics, including quant um communication and information theory. They also lead to substantial advancemen t on some relevant conceptual issues, such as the problem of jointly measuring non-c ommuting observables, ortheproblemoftheclassicallimitforquantummeasurements. How ever, theproblemof classifying allpossibleinformationallycompletemeasurements, alsoin viewoffeasibility, was never investigated in generality, and only elementary physical s ystems have been considered: the

Question 795 multiple-choice

Quantum algorithms for the hidden subgroup problem (HSP) play a crucial role in tackling computational challenges in non-abelian groups such as the symmetric group. The extent to which joint measurements on coset states can efficiently extract information about hidden subgroups determines the feasibility of quantum speedups for problems like graph isomorphism. What is the minimum number of coset states that must be measured jointly in the symmetric group Sn to potentially enable a polynomial-time quantum algorithm for the hidden subgroup problem? 1) O(1) 2) Ω(n log n) 3) O(log n) 4) O(√n) 5) O(n) 6) O(n^2) 7) O(n!)

✓ Correct Answer:

The correct answer is 2) Ω(n log n).

📚 Reference Text:

of measurements oper- ating on more than one coset state (i.e., k >1) are known that give a polynomial amount of information about the hidden subgroup. Kuperberg [Kup03] gave a measure ment for the dihedral group operating on 2O(√ log|G|)coset states that also takes 2O(√ log|G|)time to implement. Bacon et al. [BCD05] gave an ef- ﬁciently implementable measurement for the Heisenberg gro up operating on two coset states, and similar efﬁcient measurements for some other groups operating on ac onstant number of coset states. The case of the symmetric group Snhas been much harder to understand. First it was shown that so me restrictedmeasurementsrelatedtotheabeliancasecannot solvetheproblem [HRT03]. Nextthenon-abelian 2 aspects ofthegroupwereattacked byGrigniet al.[GSVV04]w hoshowedthat forhidden subgroups in Sn, measuringtheFouriertransform ofasinglecosetstateusin grandomchoicesofbasesfortherepresentations ofSngives exponentially little information. They left open the question whether a clever choice of basis for each representation space can indeed give enough inform ation about the hidden subgroup. Recently, a breakthrough has been made byMoore, Russell and Schulman [M RS05] whoanswered this question in the negative for k= 1by showing that any measurement on asingle coset state of Sngives exponentially little information, i.e., any algorithm for the HSP in Snthat measures one coset state at a time requires at least exp(Ω(n))coset states. Subsequently, Moore and Russell [MR05] exten ded this result byshowing that any algorithm that jointly measures two coset states at a time re quires at least exp(Ω(√n/logn))coset states. However, their techniques fail for algorithms that jointly measure three or more coset states at a time, and they left the k≥3case open. In this paper, we show that no quantum measurement on k=o(nlogn)coset states can extract poly- nomial amount of information about the hidden subgroup in Sn. Thus, any algorithm operating on coset states that solves the hidden subgroup problem in Snin polynomial time has to make joint measurements onk= Ω(nlogn)coset states, matching the information theoretic upper bou nd. Our results apply to the hidden subgroups arising out of the reduction from isomorph ism of rigid graphs, and rules out any efﬁcient quantum algorithm that tries to solve graph isomorphism via the standard reduction to the HSP in Snusing measurements that act jointly onless than nlogncoset states at atime. Our lower bound holds for a more general setting: Given a grou pG, suppose we want to decide if the hidden subgroup is a conjugate of an a priori known order two s ubgroupH, or the identity subgroup. We show a lower bound on the total number of coset states require d by any algorithm that jointly measures at mostkstates at a time and that distinguishes between the above two cases. Our main theorem uses only properties of Gthat can be read off from the values of the characters at the tw o elements of H. We also prove atransfer lemma that allows us to transfer lower bounds proved for subgroups and quotient groups to larger groups. Using our main theorem and the transfer lem ma, we show lower bounds on the order of entangled measurements required to efﬁciently

Question 796 multiple-choice

Nearrings are algebraic structures that generalize rings by relaxing certain distributive and commutativity requirements, and local nearrings are those in which non-invertible elements form a subgroup of the additive group. The existence of local nearring structures depends intricately on the properties of the underlying group, such as nilpotency class and order. For primes greater than 3, how many non-abelian $p$-groups of nilpotency class 3 and order $p^4$ admit a local nearring structure with identity? 1) All four such groups 2) Three of the four groups 3) Two of the four groups 4) None of the four groups 5) Exactly those with abelian commutator subgroup 6) Only one of the four groups 7) Exactly those with exponent $p^2$

✓ Correct Answer:

The correct answer is 6) Only one of the four groups.

📚 Reference Text:

Title: Groups of nilpotency class $3$ of order $p^4$ as additive groups of local nearrings Year: 2025 Paper ID: 94f241433168ba986f192802a4d7a51a32f2de28 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/94f241433168ba986f192802a4d7a51a32f2de28 Abstract: A nearring is a generalization of the notion of a ring in the sense that the addition needs not to be commutative and only one distributive law is assumed. A nearing with identity is called local if the set of all non-invertible elements forms a subgroup of its additive group. However, it is not true that any finite group is an additive group of a nearring with identity. Therefore the determination of the non-abelian finite $p$-groups, which are an additive groups of local nearrings, is an open problem. We consider groups of nilpotency class $3$ of order $p^4$, which are the additive groups of local nearrings. It is shown that for $p>3$ there exists a local nearring on one of four such groups. Using the well-known system of computer algebra GAP, we construct examples of local nearrings with the specified properties.

Question 797 multiple-choice

Quantum Hidden Subgroup Problems (HSPs) are foundational in quantum computing, especially for algorithms like Shor’s, which rely on group-theoretic techniques to achieve computational speedups. One important challenge is handling infinite or extremely large groups in practical quantum algorithms. Which statement most accurately describes how the "pushing" technique enables quantum algorithms to solve HSPs for infinite groups such as the integers? 1) It replaces the original group with a non-abelian subgroup to increase algorithmic efficiency. 2) It uses a classical algorithm to directly simulate the infinite group on quantum hardware. 3) It approximates the infinite group by embedding it into a larger finite group with additional structure. 4) It partitions the original group into cosets based on the order of its subgroups. 5) It reduces the problem to a smaller group by restricting the original group's elements to a finite subset. 6) It constructs an injective homomorphism from the infinite group to a finite group for easier computation. 7) It substitutes the infinite group with a finite quotient group via an epimorphism and a transversal, allowing quantum algorithms to operate on the manageable group and map results back to the original.

✓ Correct Answer:

The correct answer is 7) It substitutes the infinite group with a finite quotient group via an epimorphism and a transversal, allowing quantum algorithms to operate on the manageable group and map results back to the original..

📚 Reference Text:

ij/parenrightBig/angbracketrightBig /radicalbigg/angbracketleftBig Φ/parenleftBig γ† ij/parenrightBig |Φ/parenleftBig γ† ij/parenrightBig/angbracketrightBig∈H/hatwideG⊗HS Step 5. Outputγijand stop. 4.Pushing HSPs for the generic QHS algorithm QRand For certain hidden subgroup problems (HSPs) ϕ:G−→S, the corresponding generic QHS algorithm QRand either is not physically implementable or is too ex- pensive to implement physically. For example, the HSP ϕis usually not physically implementable if the ambient group is inﬁnite (e.g., Gis the inﬁnite cyclic group Z), and is too expensive to implement if the ambient group is too large (e .g.,G is the symmetric group S10100). In this case, there is a standard generic way of ”tweaking” the HSP to get around this problem, which we will call pushing . Deﬁnition 2. Letϕ:G−→Sbe a map from a group Gto a setS. A map /tildewideϕ:/tildewideG−→Sfrom a group /tildewideGto the setSis said to be a pushofϕ, written /tildewideϕ=Push(ϕ), provided there exists an epimorphism µ:G−→/tildewideGfromGonto/tildewideG, and a transversal τ:/tildewideG−→Gofµsuch that /tildewideϕ=ϕ◦τ. IS GROVER’S ALGORITHM A QUANTUM HIDDEN SUBGROUP ALGORITHM ? 5 If the epimorphism µand the transversal τare chosen in an appropriate way, then execution of the generic QHS subroutine with input /tildewideϕ=Push(ϕ) , i.e., execution of QRand(/tildewideϕ) , will with high probability produce an irreducible representation /tildewideγof the group /tildewideG which is suﬃciently close to an irreducible representation γof the group G. If this is the case, then there is a polynomial time classical algorithm which up on input /tildewideγ produces the representation γ. Obviously, much more can be said about pushing. But unfortunately that would take us far aﬁeld from the objectives of this paper. For more information on pushing, we refer the reader to [24]. 5.Shor’s algorithm Shor’s factoring algorithm is a classic example of a QHS algorithm creat ed from the push of an HSP. LetNbe the integer to be factored. Let Zdenote the additive group of integers, andZ× Ndenote the monoid of integers under multiplication modulo N(i.e., the ring of integers modulo Nignoring addition.) Shor’s algorithm is a QHS algorithm that solves the following HSP ϕ:Z−→ Z× N m/ma√sto−→ammodN with unknown hidden subgroup structure given by the following comm utative dia- gram Zϕ−→Z× N νց ր ι Z/PZ, whereais an integer relatively prime to N, wherePis the hidden integer period of the mapϕ:Z−→Z× N, wherePZis the additive subgroup all integer multiples of P(i.e., the hidden subgroup), where ν:Z−→Z/PZis the natural epimorpism of of the integers onto the quotient group Z/PZ(i.e., the hidden epimorphism), and whereι:Z/PZ−→Z× Nis the hidden monomorphism. An obstacle to creating a physically implementable algorithm for this HS P is that the domain Zofϕis inﬁnite. As observed by Shor, a way to work around this diﬃculty is to push the HSP. In particular, as illustrated by the following commutative diagram Zϕ−→ Z× N µցտτրϕ=Push(ϕ) =ϕ◦τ ZQ, 6 SAMUEL J. LOMONACO, JR. AND LOUIS H. KAUFFMAN a push/tildewideϕ=Push(ϕ) is constructed by selecting the epimorphism µ:Z−→ZQof Zonto the ﬁnite cyclic group ZQof orderQ, where the integer Qis the unique power of 2 such that N2≤Q<2N2, and choosing the transversal3 τ:ZQ−→Z mmodQ/ma√sto−→m, where 0≤m

Question 798 multiple-choice

In quantum sensing and measurement, superconducting qubits such as transmons are used to detect external magnetic flux with high sensitivity. The Ramsey sequence, involving superpositions and coherence times, is central to optimizing measurement protocols and sensitivity limits. Which factor most directly determines the optimal delay time τ* for achieving maximum flux sensitivity in a quantum measurement protocol using a transmon qubit? 1) The frequency of the applied microwave control pulses 2) The temperature of the cryogenic environment 3) The capacitance between the transmon electrodes 4) The strength of the external magnetic field 5) The duration of the initial π/2 pulse 6) The coherence properties and dephasing time of the qubit 7) The resistance of the SQUID loop

✓ Correct Answer:

The correct answer is 6) The coherence properties and dephasing time of the qubit.

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frequency ωdof the π/2 pulses. The Ramsey sequence drives the transmon from its ground state into a coherent superpositionof ground and excited states with relative amplitudes determinedby the phase ϕ. The theoretical probability to ﬁnd the transmon in theﬁrst excited state is given by Pτ;ΔωðΦÞ ½/C138 ¼ 1 2þ1 2exp/C0τ=2T1 ðÞ γðτÞcosΔωðΦÞτ ½/C138 (1) and depends both on the delay time τand on the magnetic ﬂuxΦ through the frequency mismatch Δω(Φ). The decay function γ(τ) accounts for qubit dephasing, typically due to charge or ﬂux noise. By design, the transmon arti ﬁcial atom is rather insensitive to background charge ﬂuctuations. On the other hand, intrinsic 1/ f magnetic- ﬂux noise couples to the SQUID loop and is known to be a relevant source for dephasing in ﬂux qubits;21–23in addition, other decoherence mechanisms can be present, see below fordetails. The dephasing process can be described through anexternal classical noise source, see Methods. The particular shapeofγ(τ) depends on the noise spectral density at low frequencies. “White ”noise with a constant power density results in an exponential decay function γ(τ)=exp(−Γ wnτ), while 1/ f-noise produces a Gaussian decay γ(τ)=exp[−(Γ1/fτ)2]. We ﬁt our experimental curves P(τ,Δω(Φ)) by Eq. ( 1) using both an exponential and a Gaussian decay, see SI 2. For our sample witha relaxation time T 1of about 260 ns, we cannot distinguish between these two ﬁts, neither in the ‘sweet spot ’nor in the bias point. Fitting the Ramsey oscillation at different ﬂuxes one ﬁnds Γ/C01 wn/C251250 ns and Γ/C01 1=f/C25780 ns at the ‘sweet spot ’. At the bias point, these pure dephasing times reduce to 520 and 420 ns, respectively. The decay rates ΓwnandΓ1/fin the bias point then can be translated into equivalent white and 1/ fﬂux noises and we ﬁnd the spectral densities Swn=(5.9 × 10−8Φ0)2/Hz and S1/f(f)= (1.9 × 10−5Φ0)2/f[Hz], respectively (see Methods). The function γ(τ) determines the optimal delay time τwhere the sensitivity of the probability P(τ,Δω) to the changes in Δωand hence to a ﬂux is the highest. In the standard (classical) measurement approach, a minimal delay τ¼τ0/C28T2sets the frequency range Δω (Φ)∈[0,π/τ0] where the phase ϕand hence P(τ,Δω)c a nb e unambiguously resolved. This de ﬁnes the range ΔΦ=π(τ0dω01/ dΦ)−1where the magnetic ﬂux can be resolved with a precision scaling given by the standard quantum limit (see Methods), δΦ½/C138class¼dω01ðΦÞ dΦ/C12/C12/C12/C12/C12/C12/C12/C12/C011 τ0ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ t=Trepp ¼Aclassﬃﬃtp ; (2) where tis the total measurement or sensing time of the experiment and Trepis the time duration of a single Ramsey measurement. A better ﬂux sensitivity can be attained at larger delays τ, where the probability P[τ,Δω(Φ)] is more sensitive to changes in Δω. We obtain the best sensitivity at τ=τ*deﬁned by the condition (2 T1)−1−[lnγ(τ)]′=τ−1(see Methods), δΦ½/C138quant¼dω01ðΦÞ dΦ/C12/C12/C12/C12/C12/C12/C12/C12/C01e τ/C3ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ t=Trepp ¼Aquantﬃﬃtp : (3) The amplitudes Aclassand Aquant in Eqs. ( 2) and ( 3) quantify the magnetic ﬂux sensitivities. Measuring at the optimal delay τ=τ* improves the ﬂux resolution by a factor Aclass/Aquant=τ*/(eτ0), which depends on the qubit ’s coherence time, the latter serving as the quantum resource in

Question 799 multiple-choice

In computational quantum chemistry and materials science, constructing localized orbitals from Kohn-Sham orbitals is important for efficient simulations and improved physical insight. Various algorithms exist to achieve localization, with SCDM (Selected Columns of the Density Matrix) being notable for its simplicity and robustness. Which statement accurately describes a key advantage of the SCDM method for constructing localized orbitals in insulating systems? 1) It requires a non-convex optimization procedure to achieve localization. 2) It is only applicable to systems with zero electron density. 3) It relies on an initial guess for the localized orbitals. 4) It cannot be implemented using QRCP factorization algorithms. 5) It avoids non-convex optimization and does not require an initial guess for localization. 6) Its computational cost always scales linearly with the number of electrons. 7) It is primarily designed for topological insulators with non-zero Chern numbers.

✓ Correct Answer:

The correct answer is 5) It avoids non-convex optimization and does not require an initial guess for localization..

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of Mathematics, University of California, Berkeley, Berkeley, CA 94720 and Com- putational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720. Email: linlin@math.berkeley.edu zDepartment of Mathematics and Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA 94305. Email: lexing@math.stanford.edu 1arXiv:1604.06830v2 [math.NA] 20 Aug 2017 after truncating entries with relative magnitude smaller than a prescribed threshold. Here,Nis the number of grid points in the discrete real space representation of each Kohn-Sham orbital, and neis the number of orbitals. In the absence of spin degeneracy, neis also the number of electrons in the system. For a general matrix it may not be possible to construct such a Qand obtain with the desired structure. However, when represents a collection of Kohn-Sham orbitals of an insulating system, such localized orbitals generally exist. An impor- tant exception are topological insulators with non-vanishing Chern numbers [20, 7] |here we restrict our discussion to systems where localized functions are known to exist. Their construction can be justi ed physically by the \nearsightedness" principle for electronic matter with a nite HOMO-LUMO gap [23, 36]. The nearsightedness principle can be more rigorously stated as the single particle density matrix being exponentially localized along the o -diagonal direction in its real space representa- tion [4, 23, 6, 9, 8, 33, 25]. The recently developed selected columns of the density matrix (SCDM) procedure [10] provides a simple, accurate, and robust way of constructing localized orbitals. Unlike many existing methods [15, 30, 14, 34, 32], the SCDM method requires no initial guess and does not involve a non-convex optimization procedure. The core ideas behind it also readily extend to systems with Brillouin zone sampling [11]. The SCDM procedure constructs localized orbitals directly from a column selection procedure implicitly applied to the density matrix. Hence, the locality of the basis is a direct consequence of the locality of the density matrix. The SCDM method can be eciently performed via a single column pivoted QR (QRCP) factorization. Since ecient implementation of the QRCP is available both in serial and parallel computational environments through the LAPACK [1] and ScaLAPACK [5] libraries, respectively, the SCDM method can be readily adopted by electronic structure software packages. From a numerical perspective, the computational cost of a QRCP factorization scales asO(Nn2 e). The basic form of QRCP [17] is not able to take full advantage of level 3 BLAS operations. Hence for matrices of the same size, the QRCP factorization can still be relatively expensive compared to level 3 BLAS operations such as general matrix-matrix multiplication (GEMM). The computational cost of the single QRCP is not necessarily an issue when the SCDM procedure is used as a post-processing tool, but is a potential concern when the SCDM procedure needs to be performed repeatedly. This, for instance, could occur in geometry optimization and molecular dynamics calculations with hybrid exchange-correlation functionals [37, 19], where a localized representation of the Kohn-Sham invariant subspace needs to be constructed in each step to reduce the large computational cost associated with the Fock

Question 800 multiple-choice

Quantum Fourier transform (QFT) is a central operation in quantum computing, enabling efficient algorithms for problems like factoring and period finding. Its implementation on physical quantum computers demonstrates both the mathematical structure and experimental feasibility of quantum logic operations. Which statement most accurately describes the role of the quantum Fourier transform (QFT) within Shor’s factoring algorithm? 1) The QFT transforms quantum states to reveal periodicity, enabling extraction of the order that underlies efficient integer factorization. 2) The QFT provides error correction for entangled qubit states prior to measurement. 3) The QFT is used to create entanglement between all qubits in the quantum register. 4) The QFT acts as a universal gate for constructing any quantum algorithm. 5) The QFT amplifies the amplitude of the correct factors, ensuring deterministic output. 6) The QFT initializes the quantum register in the necessary input state for the algorithm. 7) The QFT measures the final quantum state, collapsing it to the factors of the integer.

✓ Correct Answer:

The correct answer is 1) The QFT transforms quantum states to reveal periodicity, enabling extraction of the order that underlies efficient integer factorization..

📚 Reference Text:

arXiv:quant-ph/9906059v1 16 Jun 1999Implementation of the Quantum Fourier Transform Yaakov S. Weinstein∗, Seth Lloyd∗♯, David G. Cory † ∗d’Arbeloﬀ Laboratory for Information Systems and Technolo gy Department of Mechanical Engineering, M.I.T., Cambridge, MA 02139 †Department of Nuclear Engineering, M.I.T Cambridge, MA 021 39 ♯Author to whom correspondence should be addressed November 26, 2024 Abstract The quantum Fourier transform has been implemented on a thre e bit nuclear magnetic resonance (NMR) quantum computer, providing a ﬁrst step towards the re alization of Shor’s factoring and other quantum algorithms. Implementation of the QFT is pres ented with ﬁdelity measures, and state tomography. Experimentally realizing the QFT is a cle ar demonstration of NMR’s ability to control quantum systems. PACS numbers 03.67.-a, 03.67.Lx, 02.70.-c, 89.70.+c Quantum computers are devices that process information in a way that preserves quantum co- herence. Unlike a classical bit, a quantum bit, or ‘qubit,’ can be in a superposition of 0 and 1 at once. This nonclassical feature of quan- tum information allows quantum computers to perform some computations faster than classical computers. For example, quantum computers, if constructed, could factor large numbers more rapidly [1], search data basis more quickly [2], and simulate quantum systems more eﬃciently [3] than is possible using current classical algo- rithms [4] [5] [6] [7] [8] [9] [10] [11]. A key subroutine of algorithms for factor-ing and simulation [12] [13] [14] is the quan- tum Fourier transform (QFT) [15] [16] [17]. In essence the QFT takes a ‘position’ state |x/angbracketrightto the corresponding ‘momentum’ state |p/angbracketrightand is deﬁned as follows: QFT q|x/angbracketright →1√qq−1/summationdisplay p=0e2πiap/q|p/angbracketright. (1) Where qis the dimension of the systems Hilbert space. In general the QFT qtransforms the input am- plitudes as, QFT q/summationdisplay xf(x)|x/angbracketright →/summationdisplay p˜f(p)|p/angbracketright. (2) 1 Where the coeﬃcients ˜f(p) are ˜f(p) =1√q/summationdisplay ae2πiap/qf(x). (3) For example, the two qubit QFT corresponds to the unitary operator, QFT 4=1 2 1 1 1 1 1i−1−i 1−1 1 −1 1−i−1i .(4) This operator shows the QFT separating the in- put states by 0 degrees in the ﬁrst row and col- umn, and then by 90 degrees, 180 degrees and 270 degrees, multiples ofπ 2. Equation (4) shows that the QFT has eﬀects similar to that of the classical Fourier transform. In particular, if f(a) is periodic with period r, then˜f(c) will exhibit a spike at c=r. This is the key to Shor’s algorithm which allows a quantum computer to factor very large numbers in poly- nomial time. The classical Fourier transform re- veals the periodicity in functions, the QFT re- veals periodicity of wavefunctions. As formulated by Coppersmith, the QFT can be constructed from two basic unitary opera- tions, Aj, operating on the jth qubit Aj=1√ 2/parenleftigg 1 1 1−1/parenrightigg (5) andBjkoperating on the jth and kth qubits Bjk= 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 eiθjk , (6) where θjk=π/2k−j.To implement the QFT, these gates, Bj,j+1Bj,j+2...Bj,L−1Aj (7) are implemented on the lead bit, j=L−1. Re- peating the above sequence of gates

Question 801 multiple-choice

Representation theory provides the mathematical foundation for analyzing symmetries in groups and Lie algebras, with profound applications in quantum information and quantum machine learning. Key results like Schur’s lemma and concepts such as commutants and Haar integration are central to understanding how linear maps and operators interact with symmetry. Which of the following statements correctly describes the constraint imposed by Schur’s lemma on equivariant linear maps between irreducible complex representations? 1) They must be orthogonal projections onto invariant subspaces. 2) They are always invertible linear transformations. 3) They can be any linear map commuting with the group action. 4) They are either zero or, if the representations are equivalent, scalar multiples of the identity. 5) They correspond to block-diagonal matrices with arbitrary entries in each block. 6) They must be unitary transformations preserving inner product. 7) They implement simultaneous diagonalization of commuting operators.

✓ Correct Answer:

The correct answer is 4) They are either zero or, if the representations are equivalent, scalar multiples of the identity..

📚 Reference Text:

tasks. Recall that we mentioned earlier that it is common abuse of lan- guage to call the vector space Va representation of Gorg without explicitly describing the homomorphism Rorr. Theorem 6 (Schur’s lemma) .LetVandWbe irreducible complex representations of a group Gor Lie algebra g, and let :V!Wbe an equivariant map. Then either = 0, or is an isomorphism and V=W. If :V!Wis an isomorphism, it must be that =1 for some2C. Schur’s lemma plays a central role throughout represen- tationtheoryanditsapplications. A priori, therearemany linear maps between vector spaces VandW, and the list of suspects for equivariant maps could be intimidatingly large. But Schur’s lemma tells us that the equivariance 15This averaging procedure is called “Weyl’s unitary trick”, hence the name for this theorem.condition, which can be interpreted as “the group action commutes with ”, i.e. (gv) =g (v)for allg2G, tightly constrains the space of equivariant maps: for in- equivalent irreducibles, only the 0 map is equivariant, and for equivalent irreducibles, the space of equivariant maps is the 1 dimensional space spanned by the identity map 1. Further, since Theorem 4 guarantees every unitary rep- resentation can be decomposed into irreducibles, Schur’s lemma in turn tightly constrains the space of equivariant maps between anyunitary representations. This is lever- aged heavily in e.g. the commutant structure Theorem 7. VII. SOME REPRESENTATION THEORY-RICH CONSTRUCTIONS IN QML Representation theory is all about block-diagonalization into irreps. When faced with symmetric tasks in QML, a powerful strategy is to understand how the task “acts” upon irreps. Keep this core idea in mind as we review some commonly encountered objects and operations within quantum computing. As a bird’s-eye-view of where we are headed, we begin with the key tool of Haar integration, which aﬀords us a version of integration that plays well with group structure. Just as Riemann integration grants access to a variety of useful averaging procedures, Haar in- tegration yields “group averaging” procedures, which will lead us naturally to the twirling operator. We then pivot to describing the commutant, the set of all operators com- muting with a representation. This is intimately tied to block-diagonalization and can be thought of as the natural extension of a fundamental statement in quantum mechan- ics: “twooperatorsaresimultaneouslydiagonalizableifand only if they commute”. This leads to the Schur-Weyl dual- ity, presented via example, which captures the interplay of a representation and its commutant. We wrap up this sec- tion with a brief discussion of some highlight applications of this theory within GQML. A. Haar integration Haar integration is ubiquitous in quantum science, be- cause it is in some sense the most “natural” probability measure to place on unitary groups. Depending on the level of generality, there are several well-exposited ways of deﬁning the Haar measure on a compact group G.16But 16In Hall [51], this is accomplished for matrix Lie groups via right- invariant diﬀerential forms furnished by the associated Lie algebra. 26 we will take a more pragmatic approach and just explain its key features. Firstly, the Haar measure dof a

Question 802 multiple-choice

In abstract algebra, group theory explores the structure and properties of mathematical groups, which are essential for understanding symmetry in physics, chemistry, and engineering. The process of generating groups from subgroups is fundamental to analyzing their composition and function. Which statement accurately characterizes the generation of finite non-abelian simple groups using Sylow subgroups? 1) They can be generated by a single Sylow subgroup corresponding to any prime dividing their order. 2) They require the use of only abelian Sylow subgroups for generation. 3) Generation is only possible with normal subgroups and not Sylow subgroups. 4) Two Sylow subgroups of the same prime are necessary for full generation. 5) They can be generated by two Sylow subgroups corresponding to distinct primes. 6) Only maximal intravariant subgroups are sufficient for their generation. 7) Generation necessarily involves the intersection of all Sylow subgroups of the group.

✓ Correct Answer:

The correct answer is 5) They can be generated by two Sylow subgroups corresponding to distinct primes..

📚 Reference Text:

Title: A Study About One Generation of Finite Simple Groups and Finite Groups: دراسة حول أحد توليدات الزمر البسيطة المنتهية والزمر المنتهية Year: 2021 Paper ID: 0532f4a311feb8b0b9de215898913c96c6af40e8 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0532f4a311feb8b0b9de215898913c96c6af40e8 Abstract: The group theory and its classifications are of great importance in many engineering, physical and chemical fields, especially those related to the concept of symmetry. In this paper, we study the problem of how a finite group can be generated by some subgroups. In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow P1- and P2-subgroups, where P1- and P2- are two different primes. We also, show that for a given different prime numbers P and q, any finite group can be generated by a Sylow P- subgroup and an intravariant q- subgroup. The paper consists of an introduction and two fundamental sections. In one section we study the problem of generating simple finite groups. In another section, we mention the fundamental results of the paper, that connected with generating the finite group from some subgroups.

Question 803 multiple-choice

In quantum gravity, the geometry of spacetime can be modeled using discrete building blocks such as curved tetrahedra, whose quantum states are described using advanced algebraic structures. Understanding how these structures incorporate curvature and the cosmological constant is crucial for modern approaches to quantizing gravity. Which mathematical framework generalizes the symmetry group su(2) to incorporate the effects of curvature associated with a non-zero cosmological constant in quantum gravity models of curved tetrahedra? 1) SL(2,C) representation theory 2) The classical group SO(3) 3) The braid group algebra 4) Affine Lie algebras 5) Quantum group Uq(su(2)) 6) Heisenberg algebra 7) The modular group PSL(2,Z)

✓ Correct Answer:

The correct answer is 5) Quantum group Uq(su(2)).

📚 Reference Text:

Title: Quantum curved tetrahedron, quantum group intertwiner space, and coherent states Year: 2024 Paper ID: f36b38fe261abfcc5ab4f7bbd84370e269af690b Source: semantic-scholar URL: https://www.semanticscholar.org/paper/f36b38fe261abfcc5ab4f7bbd84370e269af690b Abstract: In this paper, we construct the phase space of a constantly curved tetrahedron with fixed triangle areas in terms of a pair of Darboux coordinates called the length and twist coordinates, which are in analogy to the Fenchel–Nielsen coordinates for flat connections, and their quantization. The curvature is identified to the value of the cosmological constant, either positive or negative. The physical Hilbert space is given by the Uq(su(2)) intertwiner space. We show that the quantum trace of quantum monodromies, defining the quantum length operators, form a fusion algebra and describe their representation theory. We also construct the coherent states in the physical Hilbert space labeled by the length and twist coordinates. These coherent states describe quantum curved tetrahedra and peak at points of the tetrahedron phase space. This work is closely related to 3+1 dimensional loop quantum gravity with a non-vanishing cosmological constant. The coherent states constructed herein serve as good candidates for the application to the spinfoam model with a cosmological constant.

Question 804 multiple-choice

In the theory of division rings, the structure of subnormal subgroups in the multiplicative group is closely related to deep questions about non-commutative algebra and group theory. When considering subnormal subgroups that contain non-abelian solvable subgroups, certain properties of the center or the subgroup itself can force the existence of more complex subgroups. Which of the following conditions ensures that a subnormal subgroup of the multiplicative group of a division ring, containing a non-abelian solvable subgroup, must also contain a non-abelian free subgroup? 1) The subgroup is cyclic. 2) The subgroup is algebraic over the center, or the center is uncountable. 3) The subgroup is finitely generated and abelian. 4) The division ring is commutative. 5) The subgroup consists only of torsion elements. 6) The center is finite. 7) The division ring has characteristic two.

✓ Correct Answer:

The correct answer is 2) The subgroup is algebraic over the center, or the center is uncountable..

📚 Reference Text:

Title: On subnormal subgroups in division rings containing a non-abelian solvable subgroup Year: 2018 Paper ID: 503161fcb24efc5a0dc48967c1ecc3ecd768d287 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/503161fcb24efc5a0dc48967c1ecc3ecd768d287 Abstract: Let $D$ be a division ring with center $F$ and $N$ a subnormal subgroup of the multiplicative group $D^*$ of $D$. Assume that $N$ contains a non-abelian solvable subgroup. In this paper, we study the problem on the existence of non-abelian free subgroups in $N$. In particular, we show that if either $N$ is algebraic over $F$ or $F$ is uncountable, then $N$ contains a non-abelian free subgroup.

Question 805 multiple-choice

Quantum doubles of finite groups (D) play a central role in topological quantum computation and the study of link invariants, employing a rich algebraic and coalgebraic structure. Understanding the computational complexity of evaluating these invariants is essential for distinguishing the power of classical and quantum algorithms in low-dimensional topology. Which computational complexity class is associated with the exact evaluation of certain link invariants arising from finite image representations of the braid group, such as those derived from quantum doubles D? 1) BPP 2) #P-hard 3) NP 4) PSPACE 5) SBP 6) P 7) MA

✓ Correct Answer:

The correct answer is 2) #P-hard.

📚 Reference Text:

give quantum algorithms to additively approximate link invariants arising from irreps of D(G). We also give a classical randomized algorithm for ﬂuxon irreps. Then in Section 5, we show that additive approximations of these link invaria nts are BPP-hard, multiplicative approximations are SBP-hard, and exact evaluations are # P-hard. In Section 6we address the question of simulation of anyonic quantum computation. In Section 7, we describe quantum algorithms for the Clebsch-Gordan transform over D(G). Finally, in Section 8we present our conclusions and some open problems. Related work on ﬁnite image representations of the braid gro up. Finite image representations have been considered in recent papers. In [ 31], there is a discussion of two paradigms, one involving dens e images of the braid group and the other involving ﬁnite image s, along with conjectures on the complexity of link invariants. In [ 32], Rowell and Wang show how to “localize” certain ﬁnite image representations of the braid group. In [ 19] (see also [ 18]), Hastings, Nayak and Wang show that link invariants comin g from certain ﬁnite image representations of the braid group disc ussed in [ 32] are # Phard to evaluate exactly (and to evaluate a suﬃciently good multiplicative approxim ation). In [ 13], the image of the Ising anyon representation has been studied. 2 Quantum doubles of ﬁnite groups 2.1 Quantum doubles as Hopf algebras We recall the notion of Hopf algebra; more complete descript ions can be found in accounts by Kassel [ 21] and Majid [ 24]. AC-vector space Ais aHopf algebra if it possesses consistent algebra and coalgebra structure augmented with an antipode map that yields a natural notion of “inversion. ” To be more pr ecise, a Hopf algebra possesses the following structure: Algebra structure A multiplication map µ:A⊗A→Aand a unit map η:C→Awhich satisfy (Associativity) µ(µ⊗1)(a⊗b⊗c) =µ(1⊗µ)(a⊗b⊗c) for alla,b,c∈A, and (Unit)µ(a⊗1) =µ(1⊗a) =a, for alla∈A, where 1is the unique element of Afor whichη(c) =c1 forc∈C. Coalgebra structure A comultiplication rule ∆ : A→A⊗Aand a counit ǫ:A→Cwhich satisfy (Coassociativity) (1⊗∆)∆ = (∆⊗1)∆, and (Counit) (1⊗ǫ)∆ = (ǫ⊗1)∆ = 1. 3 Coherence and an antipode The algebra and coalgebra structure are related by the follo wing two axioms: (Coherence) The comultiplication map ∆ and the counit ǫare algebra homomorphisms (where A⊗A is given the natural tensor product algebra structure) and η(ǫ(a)) =ǫ(a)1. (Antipode) Anantipode mapS:A→Asatisfying µ(S⊗1)∆(a) =µ(1⊗S)∆(a) =ǫ(a)1. We will denote µ(a⊗b) asab. It follows from the axioms above that the antipode map S:H→His an antihomomorphism: S(a)S(b) =S(ba). (2.1) Furthermore, when His ﬁnite-dimensional (the only case we shall consider), Sis invertible as a linear operator. One perspective on the role played by the coalgebra and antip ode structure aﬀorded by a Hopf algebra is that it provides ring structure to the family of represent ations of the underlying algebra. Speciﬁcally, the comultiplication operator provides the structure of a r epresentation to the tensor product of two rep- resentations XandYofAby deﬁning the action of a∈Ato be that of ∆( a). The coherence axiom ∆(a)∆(b) =

Question 806 multiple-choice

In photonic quantum computing, dual-rail qubits are used to represent logical states, and entanglement generation often relies on interferometric measurements and optical elements. The creation and analysis of multipartite entangled states such as GHZ states require precise control over photon modes and measurement patterns. Which detection outcome pattern uniquely contributes to the output of an n-GHZ state analyzer employing Hadamard beamsplitters and post-selection, under the constraint that no adjacent modes are both 1 or both 0? 1) All zeros in all modes 2) Alternating (1,1,0,0,..) pattern 3) Patterns with exactly one photon per mode 4) Patterns where half of the modes are 1 and half are 0, in any order 5) Any pattern with at least one adjacent pair of 1's 6) Patterns where every mode is 1 except the last 7) The two patterns (1,0)n and (0,1)n

✓ Correct Answer:

The correct answer is 7) The two patterns (1,0)n and (0,1)n.

📚 Reference Text:

the generalized Xmea- surement) heralds success, and further that the four input modes do in fact correspond to a pair of dual-rail qubits. Similar to above, the terms affecting a successful Type I fusion have the form (with modes appropriately permuted) c(L) 10c(R) 10 ϕ(L) 10E ϕ(R) 10E |1,0,1,0⟩+c(L) 01c(R) 01 ϕ(L) 01E ϕ(R) 01E |0,1,0,1⟩,where we again assume that the internal states of the input photons are pure states. If the relevant input photons are all ideal (mutually indistinguishable), then the output state is (up to normalization) c(L) 10c(R) 10 ϕ(L) 10E ϕ(R) 10E |1,0⟩ ±c(L) 01c(R) 01 ϕ(L) 01E ϕ(R) 01E |0,1⟩, (9) 8 Figure 5: Then-GHZ state analyzer: Type II fusion is the special case with n= 2. We apply a Hadamard beamsplitter for each p∈ P, then post-select for patterns m= (m0, . . . , m 2n−1)with all m2i+1+m2i+2= 1(including m0+m2n−1= 1, due to the convention m2n=m0). This operation may be viewed as ngeneralized Xmeasurements between the modes of ndual-rail input states, where the n-GHZ state analyzer is considered successful if and only if all the generalized Xmeasurements are. where the sign is +1for measurement pattern (1,0)and−1for(0,1). If the photons are fully distinguishable, then for either measurement outcome, the output state is the even mixture of the two unentangled states c(L) 10c(R) 10 ϕ(L) 10E ϕ(R) 10E |1,0⟩, c(L) 01c(R) 01 ϕ(L) 01E ϕ(R) 01E |0,1,0,1⟩. (10) We now apply Proposition 1 to the study of the generalized GHZ state analyzer in Figure 5. The n-GHZ analyzer involves Hadamard beamsplitters on the mode pairs P={(1,2),(3,4), . . . , (2n−3,2n−2),(2n−1,0)}. (Note that the Hadamard is not symmetric between the two modes: our convention is that the second mode in the pair receives the nontrivial phase. In particular, note that with this convention it is mode 0,not2n−1, that receives the phase.) This is followed with post-selection as described in the figure. We begin with the following lemma, which is immediate from the definition of the n-GHZ state analyzer. We use the notation ⃗ m= (m0, . . . , m 2n−1),|⃗ m⟩=|m0, . . . , m 2n−1⟩. For convenience, we use the convention m2n=m0(to match with the mode pairs Pabove). We will make frequent reference to the 2n-tuples (1,0)n= (1,0,1,0, . . . , 1,0),(0,1)n= (0,1,0,1, . . . , 0,1). In the following lemma, the internal states of the input photons are suppressed from the notation. These states do not affect the content or the proof. Lemma 2. Suppose the final 2nmodes of the state |ϕ⟩are given as input to the n-GHZ state analyzer, and that the analyzer heralds success. Given this successful heralding, only terms of the following form may affect the output state: X ⃗ m∈Sc⃗ m|ϕ⃗ m⟩|⃗ m⟩, (11) where S={⃗ m:c⃗ m̸= 0, m2i+1+m2i+2= 1for all i}. If, for all ⃗ m∈ S and all 0≤i < n , we have (m2i, m2i+1)̸∈ {(1,1),(0,0)}, then only the two terms ⃗ m= (1,0)n,(0,1)nmay affect the measurement (and only if both have

Question 807 multiple-choice

Quantum algorithms for group-theoretic problems often rely on efficiently extracting subgroup information from quantum states associated with group symmetries. The choice of measurement basis in group representations can dramatically affect both the efficiency and the feasibility of reconstructing hidden subgroups or related structures. In the context of quantum algorithms addressing the hidden conjugate problem for nonabelian groups such as affine and q-hedral groups, which measurement strategy is crucial for achieving full reconstructibility of hidden conjugates with polynomially many measurements? 1) Measuring group representations in the computational basis 2) Applying random basis measurements to group representations 3) Performing measurements after treating the group as abelian 4) Measuring irreducible representations using the weak standard method 5) Using direct product decompositions for subgroup identification 6) Measuring in a basis unrelated to the group's subgroup structure 7) Measuring group representations in an adapted basis aligned with the group and its subgroups

✓ Correct Answer:

The correct answer is 7) Measuring group representations in an adapted basis aligned with the group and its subgroups.

📚 Reference Text:

“preferred” bases, those which give the matrices ρ(g) unusually structured or sparse form. In particular, Moore, Rockmore, and Russell [20]showed that so-called adapted bases yield highly eﬃcient algorithms for the quantum Fourier transform. 1.4. Contributions of this paper. As stated above, [13] and [10] leave an important open question, namely, whether there are cases where the strong standardmethod, with the proper choice of basis, oﬀers an advantage over a simple abeliantransform or the weak standard method. We settle this question in the aﬃrmative.Our results deal primarily with the q-hedral groups, i.e., semidirect products of the form Z q/multicloseleftZ p, where q|(p−1), and in particular the aﬃne groups Ap∼=Z∗ p/multicloseleftZ p. We begin in section 3 by focusing on full reconstructibility. We deﬁne the hidden conjugate problem (HCP) as follows: given a group G, a nonnormal subgroup H, and a function which is promised to be constant on the cosets of some conjugate Hb=bHb−1 ofH(and distinct on distinct cosets), determine the subgroup Hbby ﬁnding an element c∈Gso that Hc=Hb. We adopt the above classiﬁcation (fully, information- theoretically, quantum information-theoretically) for this problem in the natural way.Then we show that given a subgroup of suﬃciently small (but still exponentially large)index, hidden conjugates in A pare fully reconstructible (Theorem 1). This almost immediately implies that, for prime q=(p−1)/polylog( p), subgroups of the q-hedral groups Zq/multicloseleftZpare fully reconstructible (Theorem 2). Section 4 concerns itself with information-theoretic reconstructibility. We gener- alize the results of Ettinger and Høyer on the dihedral group and show that hiddenconjugates of any subgroup are information-theoretically reconstructible in the aﬃnegroups and, more generally, the q-hedral groups for all q(Theorem 3). We then show that we can identify the order, and thus the conjugacy class, of a hidden subgroup,and this implies that all subgroups of the aﬃne and q-hedral groups are information- theoretically reconstructible (Theorem 5). The results of sections 3 and 4 rely crucially on measuring the high-dimensional representations of the aﬃne and q-hedral groups in a well-chosen basis, namely an adapted basis that respects the group’s subgroup structure. We show in section 5 that we lose information-theoretic reconstructibility if we measure in a random basis instead. Speciﬁcally, we need an exponential number of measurements to distinguishconjugates of small subgroups of A p. This establishes for the ﬁrst time that the strong standard method is indeed stronger than measuring in a random basis: some basesprovide much more information about the hidden subgroup than others. For some nonabelian groups, the HSP can be solved with a “forgetful” approach, where we erase the group’s nonabelian structure and perform an abelian Fourier trans-form instead. In section 6 we show that this is not the case for the aﬃne groups.Speciﬁcally, if we treat A pas a direct product rather than a semidirect one, its con- jugate subgroups become indistinguishable. As an application, in section 7 we consider hidden shift problems. In the setting we consider, one must reconstruct a “hidden shift” s∈Zpfrom an oracle fs(x)= f(x−s), where fis any function that is constant on the

Question 808 multiple-choice

Quantum computing leverages principles of quantum mechanics to solve certain computational problems more efficiently than classical computers. One central challenge in this field is the Hidden Subgroup Problem (HSP), which has significant implications for both algorithm design and cryptography. Which of the following statements accurately describes the current state of quantum algorithms for the Hidden Subgroup Problem in nonabelian groups as of 2004? 1) Efficient quantum algorithms exist for all nonabelian instances of HSP. 2) Only classical algorithms have been proposed for the nonabelian HSP. 3) Quantum algorithms for nonabelian HSP are known to be inefficient for all group types. 4) The nonabelian HSP has been fully solved for all finite groups. 5) Progress has been made for specific nonabelian groups such as the dihedral group, but no general efficient solution is known. 6) Nonabelian HSP is provably unsolvable by quantum computers. 7) Nonabelian HSP algorithms are less efficient than their classical counterparts for cryptographic applications.

✓ Correct Answer:

The correct answer is 5) Progress has been made for specific nonabelian groups such as the dihedral group, but no general efficient solution is known..

📚 Reference Text:

Title: THE HIDDEN SUBGROUP PROBLEM - REVIEW AND OPEN PROBLEMS Year: 2004 Paper ID: 6716b47c1631d36772357c4e7855aac74c9a86b1 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/6716b47c1631d36772357c4e7855aac74c9a86b1 Abstract: An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the literature, and nota- tion is unified, making it easier to absorb the background necessary to begin research on the Hidden Subgroup Problem. Proofs are provided which give very concrete algorithms and bounds for the finite abelian case with little out- side references, and future directions are provided for the nonabelian case. This summary is current as of October 2004.

Question 809 multiple-choice

In cryptographic protocol analysis, security reductions are often used to relate the adversary's ability to break a scheme to solving a known hard problem, such as the root problem in a group of hidden order. Techniques like rewinding, probability bounds, and case analysis play important roles in these proofs. In a security reduction for a commitment scheme based on the root problem in a group G of rough order, which of the following best explains how the adversary's non-negligible probability of breaking the scheme is leveraged to solve the root problem? 1) By directly inverting the commitment function without interaction and using the known group order. 2) By using parallel repetition to amplify the adversary's failure probability and then brute-forcing all possible roots. 3) By extracting the solution from the transcript of a single execution without using rewinding or commitments. 4) By interacting with the adversary, employing rewinding to obtain multiple protocol transcripts, and using algebraic case analysis to construct a root solution when certain statistical events occur. 5) By reducing the group order to a smooth value and applying the Chinese Remainder Theorem to reconstruct the root. 6) By increasing the security parameter k until the adversary's probability of success becomes negligible. 7) By simulating the adversary's environment entirely offline and applying generic group algorithms to solve the root problem.

✓ Correct Answer:

The correct answer is 4) By interacting with the adversary, employing rewinding to obtain multiple protocol transcripts, and using algebraic case analysis to construct a root solution when certain statistical events occur..

📚 Reference Text:

LetR,(P,V),κ,Mandp()be as deﬁned above. Given any prover P ∗, there exists an algorithm A(P∗)that solves the root problem deﬁned by G(1k) with probabilityAdvκ,M,p (P∗,k) 9l(k)ifk≥6, and runs in time 448·tP∗(k)/κ(k)where tP∗(k)denotes the running time of P∗(1/l(k)is an lower bound on the proba- bility that a random element in GhasC(k)-rough order ). Proof.The algorithm claimed does the following: receive G,has an input. Set g=hαfor randomα∈[0..22B+k]. We sendg,hto the adversary, call Rewind and hope that we get get a set of bad values. However, we will only allow Rewind to do the protocol with the prover at most 448 /κ(k) times. If Rewindruns longer than this, we abort it and stop. If we obtained a set of bad values, we attempt to compute a root of has described below. 1Notethatthisisnotcompletelytrivial,as P∗isprobabilistic:althoughitsaverage success probability is /epsilon1view,P∗, it may not be equally successful for all choices of randomcoins.Itisessentialtogettheclaimedexpectedtimethat /epsilon1view,P∗>4/C(k), andnotjust >1/C(k) 136 IvanDamg˚ ardandEiichiroFujisaki It is immediately clear that this algorithm has the claimed running time. We now look at the success probability. We will assume that hhasC(k)-rough order.Sincethishappenswithprobabilityofatleast1 /l(k),itisenoughtoshow that the success probability under this assumption is at least the bound claimed timesl(k). Note that the distribution of G,h,gthatP∗receives here is exactly the same as in the real commitment scheme. Hence the probability of producing a view for whichMfails, is exactly Advκ,M,p(P∗,k). Note also that given any view view whereMfails, it must be the case that the values produced by Rewindare bad with probability of at least 1/2. If this was not the case, then Mcould expect to ﬁnd a way to open cafter calling Rewindtwice, which takes expected time 112/zepsilon

Question 810 multiple-choice

Quantum algorithms for the hidden subgroup problem (HSP) are central to advances in computational group theory and cryptography, especially for complex group structures such as affine groups over finite fields. Efficient solution of HSP in non-normal subgroups represents significant progress in quantum algorithm design. Which property enables the described quantum algorithm to efficiently solve the hidden subgroup problem for maximal cyclic non-normal subgroups of the affine group over a finite field of order $q$? 1) It requires the subgroup to be normal in the affine group. 2) The algorithm only works for subgroups isomorphic to the additive group of the field. 3) The success probability is fixed at exactly $0.5$ for all field sizes. 4) Its computational complexity grows polylogarithmically with the field size $q$ and the inverse of the error tolerance $\varepsilon$. 5) The affine group must have order $q^2$ for the algorithm to be efficient. 6) The algorithm is limited to subgroups generated by involutions. 7) It depends on classical brute-force search for subgroup identification.

✓ Correct Answer:

The correct answer is 4) Its computational complexity grows polylogarithmically with the field size $q$ and the inverse of the error tolerance $\varepsilon$..

📚 Reference Text:

Title: A quantum polylog algorithm for non-normal maximal cyclic hidden subgroups in the affine group of a finite field Year: 2013 Paper ID: a8ffb1b542e338428490fca84a7c4287e1a21b86 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a8ffb1b542e338428490fca84a7c4287e1a21b86 Abstract: We give an algorithm to solve the quantum hidden subgroup problem for maximal cyclic non-normal subgroups of the affine group of a finite field (if the field has order $q$ then the group has order $q(q-1)$) with probability $1-\varepsilon$ with (polylog) complexity $O(\log(q)^{R}\log(\varepsilon)^{2})$ where $R<\infty.$

Question 811 multiple-choice

Quantum algorithms often require precise estimation of eigenvalues for unitaries and Hamiltonians, with applications in quantum simulation and quantum walks. There is a fundamental distinction between obtaining n-bit digital estimates and achieving additive-error approximations of eigenvalues using quantum circuits. Which of the following best explains why perfect n-bit eigenvalue estimation using standard quantum phase estimation is fundamentally impossible, even though additive-error estimates are achievable? 1) Quantum circuits cannot maintain coherence during the measurement process required for n-bit estimation. 2) The eigenvalues of quantum Hamiltonians cannot be represented in a digital register due to their irrational nature. 3) Quantum phase estimation always introduces decoherence that limits digital precision. 4) Unitaries and Hamiltonians cannot be expressed in a form suitable for digital encoding in quantum circuits. 5) The mapping from eigenvalue to n-bit estimate is discontinuous, while quantum amplitudes vary continuously, so quantum circuits cannot exactly implement the discontinuous step required for perfect n-bit estimation. 6) Quantum algorithms are limited to only additive-error estimates due to physical noise in quantum hardware. 7) Perfect n-bit estimation would violate the no-cloning theorem in quantum mechanics.

✓ Correct Answer:

The correct answer is 5) The mapping from eigenvalue to n-bit estimate is discontinuous, while quantum amplitudes vary continuously, so quantum circuits cannot exactly implement the discontinuous step required for perfect n-bit estimation..

📚 Reference Text:

analysis of uncomputation (Lemmas 3,4). These will be used several times in this paper. Finally, we give a precise description and analysis of ‘textbook’ phase estimation (Proposition 5). Thus, this section clearly deﬁnes the problem and the previous state-of-the-art which we improve. Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 4 Our paper presents algorithms for phase, energy, and amplitude estimation. Amplitude estimation follows as a relativity simple corollary of energy estimation so we will only be talking about the prior two until Section 4. To talk about both phases and energies at once, we standardize input unitaries and Hamiltonians into the following form: U=/summationdisplay je2πiλj|ψj/angbracketright/angbracketleftψj| H=/summationdisplay jλj|ψj/angbracketright/angbracketleftψj| (6) We refer to the{λj}as the ‘eigenvalues’, and assume they live in the range [0,1). While any unitary can be put into this form, the form places the constraint 0/precedesequalH≺Ionto the Hamiltonian. Our goal is to compute |λj/angbracketrightan ‘n-bit estimate of λj’. In this paper, we take this to be ann-qubit register containing a binary encoding of the number ﬂoor (2nλj). Since λj∈[0,1)we are guaranteed that ﬂoor (2nλj)is an integer∈{0,...,2n−1}and thus has ann-bit encoding. Consider phase estimation using a quantum circuit composed of elementary unitaries and controlled- e2πiλj. Following an argument related to the polynomial method, we see that the resulting state must be of the form: /summationdisplay x∈{0,1}n/summationdisplay yαx,y(e2πiλj)|x/angbracketright|garbagex,y/angbracketright (7) whereαx,y(e2πiλj)is some polynomial of e2πiλj. We would like|x/angbracketrightto encode ﬂoor (2nλj), meaning that αx,y(e2πiλj) = 1ifx=ﬂoor(2nλj)andαx,y(e2πiλj) = 0otherwise. This is impossible: αx,y(e2πiλj)is a continuous function of λj, but the desired amplitude indicating x=ﬂoor(2nλj)is discontinuous. For energy estimation a similar argument applies, just that the amplitudes are of the form αx,y(λj). This argument even holds in the approximate case when we demand that αx,y≤δor≥1−δfor some small δ- the discontinuity is present regardless. To some extent, this issue stems from the awkwardness of the notion of an ‘ n-bit estimate’ since it requires rounding or ﬂooring, a discontinuous operation, when only con- tinuous manipulation of amplitudes is possible. A more comfortable notion is that of an additive-error estimate, used by [AA16, KP16] in their estimation algorithms. However, the primary application of our algorithms is a situation where this simpliﬁ- cation is not possible: Szegedy walks based on Hamiltonian eigenspaces [YA11, Lemi&19, JKKA20, WT21]. The original quantum Metropolis algorithm [Temme&09] implements a random walk over Hamiltonian eigenspaces, where the superposition is measured at every step and it is demonstrated that an additive error estimate is suﬃcient. But in order to harness a quadratic speedup due to quantum walks [Sze04], the measurement of energies must be made completely coherent, which is not achieved by an additive-error estimate unless the error is smaller than the gap between any eigenvalues. Therefore, we retain the notion of an ‘ n-bit estimate’ in this paper. We could always fall back to an estimate with additive error ε/2by computing n=ceil(log2(ε−1))bits of accuracy. However, the above argument still applies for additive-error estimation: the amplitude of any given estimate |ˆλ/angbracketrightis a continuous function of the eigenvalue λj. This means

Question 812 multiple-choice

In quantum chemistry, the accurate description of electronic state interactions, especially near conical intersections, is crucial for predicting molecular dynamics and nonadiabatic transitions. Advanced quantum algorithms seek to naturally provide electronic state representations that simplify these descriptions. Which feature distinguishes the state-average orbital-optimized variational quantum eigensolver (SA-OO-VQE) in its treatment of electronic state representations relevant to conical intersections? 1) It requires explicit transformation procedures to generate diabatic states. 2) It enforces strict adherence to the Born-Oppenheimer approximation. 3) It yields quasi-diabatic representations naturally through least-transformed block-diagonalization. 4) It limits analysis to single-state electronic problems. 5) It omits the use of orbital optimization in calculations. 6) It uses only residual descriptors to quantify diabaticity. 7) It is unsuitable for studying vibronic nonadiabatic couplings.

✓ Correct Answer:

The correct answer is 3) It yields quasi-diabatic representations naturally through least-transformed block-diagonalization..

📚 Reference Text:

Title: Transformation-Free Generation of a Quasi-Diabatic Representation from the State-Average Orbital-Optimized Variational Quantum Eigensolver. Year: 2025 Paper ID: 7082d1f5357aaaf8a8a17e130e1dfdf28c72254d Source: semantic-scholar URL: https://www.semanticscholar.org/paper/7082d1f5357aaaf8a8a17e130e1dfdf28c72254d Abstract: In the present work, we examine how the recent quantum-computing algorithm known as the state-average orbital-optimized variational quantum eigensolver (SA-OO-VQE), viewed within the context of quantum chemistry as a type of multiconfiguration self-consistent field (MCSCF) electronic-structure approach, exhibits a propensity to produce an ab initio quasi-diabatic representation "for free" if considered as a least-transformed block-diagonalization procedure, as alluded to in our previous work (Yalouz, S. et al. J. Chem. Theory Comput. 2022, 18, 776-794) and thoroughly assessed herein. To this end, we introduce intrinsic and residual descriptors of diabaticity and re-explore the definition and linear-algebra properties, as well as their consequences on the vibronic nonadiabatic couplings, of an optimal diabatic representation within this context and how much one may deviate from it. Such considerations are illustrated numerically in the prototypical case of formaldimine, which presents a well-known conical intersection between its ground and first-excited singlet electronic states.

Question 813 multiple-choice

In topological group theory and algebra, the relationship between group topology properties and algebraic structure is crucial for embedding and extension results. Foundational set theory axioms can play a decisive role in what constructions are possible. Which of the following properties is essential for the extension of Hausdorff ω-bounded group topologies from a subgroup to the whole abelian group in a way that preserves separability, but fails if only the Axiom of Determinacy is assumed? 1) The group is torsion-free. 2) The subgroup is open in the whole group. 3) The group is locally compact. 4) The Axiom of Choice holds. 5) The group is finite. 6) The topology is discrete. 7) The subgroup has finite index.

✓ Correct Answer:

The correct answer is 4) The Axiom of Choice holds..

📚 Reference Text:

Title: Each second countable abelian group is a subgroup of a second countable divisible group Year: 2008 Paper ID: 8bed07992a627f988a9b00ce1d72945a992dbedb Source: semantic-scholar URL: https://www.semanticscholar.org/paper/8bed07992a627f988a9b00ce1d72945a992dbedb Abstract: It is shown that each pseudonorm | · |H defined on a subgroup H of an abelian group G can be extended to a pseudonorm | · |G on G such that the densities of the pseudometrizable topological groups (H, | · |H) and (G, | · |G) coincide. We derive from this that any Hausdorff ω-bounded group topology on H can be extended to a Hausdorff ω-bounded group topology on G. In its turn this result implies that each separable metrizable abelian group H is a subgroup of a separable metrizable divisible group G. This result essentially relies on the Axiom of Choice and is not true under the Axiom of Determinacy (which contradicts to the Axiom of Choice but implies the Countable Axiom of Choice). This paper was motivated by the following question having its origin in functional analysis (see [PZ], [BRZ]): Is it true that every metrizable separable abelian topological group with no torsion is a subgroup of a metrizable separable divisible abelian group with no torsion? From now on all groups considered in the paper are commutative. We recall that a group G is divisible (resp. has no torsion) if for any element a ∈ G and a positive integer n the equation nx = a has a solution x ∈ G (resp. does not have two distinct solutions in G). According to the Baer Theorem [F, 21.1] each divisible group G is injective in the sense that each homomorphism h : B → G defined on a subgroup B of a group A can be extended to a homomorphism h̄ : A → G. A classical result of the theory of infinite abelian groups [F, 24.1] asserts that each group (with no torsion) is a subgroup of a divisible group (with no torsion). This result allows us to reduce the above question to the following one: Can every separable group topology on a subgroup H of a group G be extended to a separable group topology on G? Note that without the separability requirement this problem is trivial: just announce H to be an open subgroup of G and take the neighborhood base at the origin of H for a neighborhood base at the origin in the group G. However if the quotient group G/H is uncountable such an extension leads to an unseparable topology on G. So, another less direct approach should be developed. A classical result in the theory of topological groups asserts that each group topology is generated by a family of continuous pseudonorms, see [Tk, §2]. This observation allows us to reduce the problem of extending group topologies to the problem of extending pseudonorms. As usual, under a (continuous) pseudonorm of a (topological) group G we understand a (continuous) non-negative function | · | : G → [0,∞) such that |0| = 0 and |x −

Question 814 multiple-choice

Quantum comparators and modular arithmetic circuits are essential components in quantum algorithms, particularly on noisy intermediate-scale quantum (NISQ) computers where resource efficiency is critical. Limiting the number of ancillary qubits can make these algorithms more practical for current quantum hardware. Which design feature makes certain quantum comparators and modular arithmetic algorithms especially suitable for NISQ computers? 1) Employing multi-qubit entanglement exclusively for output encoding 2) Using classical bits to perform all comparison operations 3) Requiring a large number of ancilla qubits for temporary storage 4) Avoiding the use of the quantum Fourier transform subroutine 5) Limiting the requirement to at most one ancillary qubit 6) Restricting input to only classical integers 7) Implementing modular arithmetic solely via measurement operations

✓ Correct Answer:

The correct answer is 5) Limiting the requirement to at most one ancillary qubit.

📚 Reference Text:

Title: An improved QFT-based quantum comparator and extended modular arithmetic using one ancilla qubit Year: 2023 Paper ID: e74836ae06bb05cf9aa7aedd9843357a3494bf2c Source: semantic-scholar URL: https://www.semanticscholar.org/paper/e74836ae06bb05cf9aa7aedd9843357a3494bf2c Abstract: Quantum comparators and modular arithmetic are fundamental in many quantum algorithms. Current research mainly focuses on operations between two quantum states. However, various applications, such as integer factorization, optimization, and financial risk analysis, commonly require one of the inputs to be classical. It requires many ancillary qubits, especially when subsequent computations are involved. In this paper, we propose a quantum–classical comparator based on the quantum Fourier transform. Then we extend it to compare two quantum integers and modular arithmetic. Proposed operators only require up to one ancilla qubit, which is optimal for qubit resources. We analyze limitations in the current modular addition circuit and develop it to process arbitrary quantum states in the entire n-qubit space. The proposed algorithms reduce computing resources and make them valuable for noisy intermediate-scale quantum computers.

Question 815 multiple-choice

In the representation theory of semisimple Lie algebras, Verma modules and intertwining operators play a central role, especially in the study of weight decompositions and tensor product structures. Fusion matrices encode how representations combine and are essential in understanding categorical and quantum symmetries. Which of the following statements is TRUE regarding the properties of fusion matrices $ J_{WV}(\lambda) $ associated with tensor products of representations? 1) They are always symmetric matrices for any $ \lambda $. 2) They have entries that are polynomial functions of $ \lambda $. 3) Their invertibility is guaranteed by being upper triangular. 4) They satisfy a standard 2-cocycle condition independent of weights. 5) They are necessarily diagonal for irreducible modules. 6) Their entries depend irrationally on the weight parameter $ \lambda $. 7) They are strictly lower triangular matrices, rational functions of $ \lambda $, and satisfy a dynamical 2-cocycle condition.

✓ Correct Answer:

The correct answer is 7) They are strictly lower triangular matrices, rational functions of $ \lambda $, and satisfy a dynamical 2-cocycle condition..

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the induced module Mλ= Indg h⊕n+Cλ. Notice that Mλis a free U(n−)-module and can be identiﬁed with U(n−) as a linear space by the map U(n−)∼→Mλ,u/mapsto→u.xλ. Deﬁne a partial order on h∗by putting µ < νif there exist α1,...αr∈∆+, r >0, such that ν=µ+α1+...+αr. LetMλ=/circleplustext µ≤λMλ[µ] denote the decomposition of Mλinto weight subspaces. The following proposition is standard. Proposition 2.1. The module Mλis irreducible for generic values of λ. Deﬁne also the dual Verma module M∗ λto be the graded dual vector space/circleplustext µMλ[µ]∗equipped with the following g-action: (a.u)(v) =−u(a.v)∀a∈g, u∈M∗ λ, v∈Mλ. Letx∗ λbe the lowest weight vector of M∗ λsatisfying /a\}bracketle{txλ,x∗ λ/a\}bracketri}ht= 1. Now let Vbe a ﬁnite-dimensional g-module. Let V=/circleplustext ν∈h∗V[ν] be its decompositionintoweightsubspaces. Let λ,µ∈h∗andlet usconsider g-module intertwining operators Φ :Mλ→Mµ⊗V. If Φ is such an intertwining operator, deﬁne its ”expectation value” by /a\}bracketle{tΦ/a\}bracketri}ht=/a\}bracketle{tΦ.xλ,x∗ µ/a\}bracketri}ht ∈V[λ−µ]. Remark. This deﬁnition is similar to the notion of expectation value in quan- tum ﬁeld theory. Proposition 2.2. LetMµbe irreducible. Then the map Homg(Mλ,Mµ⊗V)→V[λ−µ],Φ/mapsto→ /a\}bracketle{tΦ/a\}bracketri}ht is an isomorphism. 3 Proof.By Frobenius reciprocity, we have Homg(Mλ,Mµ⊗V) = Hom h⊕n+(Cλ,Mµ⊗V) = Hom h⊕n+(Cλ⊗M∗ µ,V). Moreover, since Mµis irreducible, we have M∗ µ= Indh⊕n+ hC−µas anh⊕n+- module. In particular, Homh⊕n+(Cλ⊗M∗ µ,V) = Hom h(Cλ⊗C−µ,V) =V[λ−µ]. /squaresolid This proposition can be reformulated as follows: for any v∈V[λ−µ] there exists a unique intertwining operator Φv λ:Mλ→Mµ⊗Vsuch that Φv λ(xλ)∈xµ⊗v+/circleplusdisplay ν<µMµ[ν]⊗V. Notice that Φv λ(for ﬁxed v) is deﬁned only for generic values of λ. Identifying the Verma modules MλandMµwithU(n+), we can view Φv λas a linear map U(n+)→U(n+)⊗V. It is easy to see that the coeﬃcients of this map (in any basis) are rational functions of λ. We would now like to consider the ”algebra” of such intertwining opera tors. Let us denote by wt( u)∈h∗the weight of any homogeneous vector uin ag- module. Let V,Wbe two ﬁnite-dimensional g-modules, and let v∈V, w∈W be two homogeneous vectors. Let λ∈h∗and consider the composition Φw,v λ:MλΦv λ−→Mλ−wt(v)⊗VΦw λ−wt(v)−→Mλ−wt(v)−wt(w)⊗W⊗V. (Here and below we abuse notations and write Φ instead of Φ ⊗1). Then Φw,v λ∈Homg(Mλ,Mλ−wt(v)−wt(w)⊗W⊗V). Hence by Proposition 1.2, for genericλthere exists a unique element u∈V⊗W[wt(v) + wt(w)] such that Φu λ= Φw,v λ. It is clear that the assignment ( v,w)/mapsto→uis bilinear, and deﬁnes anh-linear map JWV(λ) :W⊗V→W⊗V, w⊗v/mapsto→ /a\}bracketle{tΦw,v λ/a\}bracketri}ht Deﬁnition. We call the operator JWV(λ) thefusion matrix ofVandW. We will now list some fundamental properties of fusion matrices. Firs t let us introduce an important piece of notation to be used throughout th is text. If A1,...Arare semisimple h-modules and F(λ) :A1⊗...⊗Ar→A1⊗...⊗Ar is a linear operator depending on λ∈h∗then, for any homogeneous a1,...ar we set F(λ−h(i))(a1⊗...⊗ar) :=F(λ−wt(ai))(a1⊗...⊗ar). Proposition 2.3. LetV,Wbe ﬁnite-dimensional g-modules. Then 4 1.JWV(λ)is a rational function of λ. 2.JWV(λ)is strictly lower triangular, i.e. J= 1+Nwhere N(W[ν]⊗V[µ])⊂/circleplusdisplay τ<ν,µ<σW[τ]⊗V[σ]. In particular, JWV(λ)is invertible. 3. LetU,V,W be ﬁnite-dimensional g-modules. Then the fusion matrices satisfy the following dynamical 2-cocycle condition: JU⊗W,V(λ)(JUW(λ−h(3))⊗1) =JU,W⊗V(λ)(1⊗JWV(λ)). onU⊗V⊗W. Proof.Statements 1. and 2. follow from the deﬁnitions and from the fact t hat the intertwining operators Φv λare rational functions of λ. To

Question 816 multiple-choice

In quantum information theory and operator algebras, conditioned noncommutative Lp norms and N-bimodule maps play a central role in quantifying properties of quantum channels and states. These mathematical frameworks generalize classical concepts like entropy and norm bounds to noncommutative settings. Which statement accurately describes the relationship between N-bimodule maps and conditioned noncommutative Lp norms when considering parameters $ p \leq q $? 1) The norm of an N-bimodule map depends on the choice of the parameter $ s $. 2) For N-bimodule maps and $ p \leq q $, the induced norms are equivalent and independent of the parameter $ s $. 3) Conditioned noncommutative Lp norms cannot be related to Hölder's inequality when $ p \leq q $. 4) N-bimodule maps only preserve left multiplication by elements of $ N $, not right multiplication. 5) The cb-norm of an N-bimodule map always increases as $ s $ increases, for any values of $ p $ and $ q $. 6) Induced norms on N-bimodule maps depend on matrix dimensions in all cases. 7) Conditioned noncommutative Lp norms reduce to zero when $ N = M $ and $ p \leq q $.

✓ Correct Answer:

The correct answer is 2) For N-bimodule maps and $ p \leq q $, the induced norms are equivalent and independent of the parameter $ s $..

📚 Reference Text:

a subalgebra N⊂Mwe deﬁne the conditioned Lq p(N⊂M)norm [19], [20] (see also [16], [21]) via /parallel.alt1x/parallel.alt1Lq p(N⊂M) (17) ∶=/uni23A7/uni23AA/uni23AA/uni23A8/uni23AA/uni23AA/uni23A9infx=ayb/parallel.alt1a/parallel.alt1L2r( )/parallel.alt1y/parallel.alt1Lq( )/parallel.alt1b/parallel.alt1L2r( )p≤q; sup/parallel.alt1a/parallel.alt1L2r( )/parallel.alt1b/parallel.alt1L2r( )≤1/parallel.alt1axb/parallel.alt1Lq( )p≥q: Here1 r=/divides.alt21 q−1 p/divides.alt2anda;bare elements in L2r(N). For N=M, we just ﬁnd another description of Lp(M), i.e.Lq p(N⊂M)=Lp(M). Note that for a selfadjoint elementx, we may assume a=a∗in (17). By Hölder’s inequality,Lq p(N⊂M)⊂Lp(M). In the particular case whenM=Mk(N)is the algebra of kbykmatrices with coefﬁcients in N, the spaces Lq p(N⊂M)≡Sk p(Lq(N)) coincide with Pisier’s vector-valued Lpspaces [19]. We will also be concerned with norms of linear maps between these Lpspaces. A map T∶Lp(M)→Lq(M) is called aN-bimodule map if, for any a;b∈Nand any x∈M: T(axb)=aT(x)b: For instance, when N=Nfix is the ﬁxed point subalgebra of a selfadjoint quantum Markov semigroup (Tt)t≥0acting onM, the mapsTtareN-bimodule maps with respect to N. ForN-bimodule maps and p≤q, the following was proved in Lemma 3.12 of [6], generalizing an earlier statement for vector valued Lpnorms (see Lemma 1.7 of [19]): for any s≥1: /parallel.alt1T∶Lp ∞(N⊂M)→Lq ∞(N⊂M)/parallel.alt1 (18) =/parallel.alt1T∶Lp s(N⊂M)→Lq s(N⊂M)/parallel.alt1: We refer to [19], [20] for motivation and further properties. We will also use the completely bounded version of these norms: /parallel.alt1T∶Lp s(N⊂M)→Lq s(N⊂M)/parallel.alt1cb =sup m/parallel.alt1idm⊗T∶Lp s(Mm⊗N⊂Mm⊗M) →Lq s(Mm⊗N⊂Mm⊗M)/parallel.alt1: which also does not depend on sforN-bimodule maps, as we discuss in more detail in Section VI. Noncommutative Lpnorms are closely related to the sandwiched Rényi divergences introduced in [17], [18]: Forp∈(1;+∞), these are deﬁned for two quantum states;∈D(H)as: Dp(/parallel.alt1)= /uni23A7/uni23AA/uni23AA/uni23AA/uni23AA/uni23A8/uni23AA/uni23AA/uni23AA/uni23AA/uni23A9log/parenleft.alt4Tr/bracketleft.alt4/parenleft.alt41−p 2p1−p 2p/parenright.alt4p /bracketright.alt4/parenright.alt4 p−1ker()⊆ker()orp∈(0;1) +∞; otherwise, (19) and, forp=∞, we set D∞(/parallel.alt1)=/uni23A7/uni23AA/uni23AA/uni23A8/uni23AA/uni23AA/uni23A9log/parenleft.alt2/parallel.alt1−1 2−1 2/parallel.alt1∞/parenright.alt2ker()⊆ker() +∞; otherwise. (20)One can then show that setting x=−1 2−1 2and (x)= Tr(x)in 15, we have: Dp(/parallel.alt1)=p p−1log(/parallel.alt1x/parallel.alt1Lp( )): Moreover, the sandwiched Rényi conditional entropy Hp(A/divides.alt0B)introduced in [17], [18] can be seen as a special case of the conditional Lpnorms deﬁned in 17. They are deﬁned for a bipartite state ∈D(HA⊗HB) as: Hp(A/divides.alt0B)=−min B∈D(HB)Dp/parenleft.alt1AB/divides.alt0/divides.alt0idB(HA)⊗B/parenright.alt1: It is then easy to see that −p′log/parenleft.alt1/parallel.alt1/parallel.alt1Lp 1(B(HB)⊂B(HA⊗HB)/parenright.alt1=Hp(A/divides.alt0B): forp′the Hölder conjugate. Thus, we see that the conditional norms can also be interpreted as a generalization of conditional entropies in which we condition w.r.t. to states on a subalgebra, which naturally includes the case of a subsystem. For two algebras N⊂Mwe will denote this conditional Rényi entropy byHp(M/divides.alt0N). Moreover, a bound on the norms deﬁned on (18) can be interpreted as a strong data processing inequality for conditional Rényi entropies. Indeed, given that /parallel.alt1T∶Lp ∞(N⊂M)→Lq ∞(N⊂M)/parallel.alt1 =/parallel.alt1T∶Lp 1(N⊂M)→Lq 1(N⊂M)/parallel.alt1≤c; we have log/parenleft.alt1/parallel.alt1T()/parallel.alt1Lq 1(N⊂M)/parenright.alt1≤log(c)+log/parallel.alt1/parallel.alt1Lp 1(N⊂M) holds for all states . Normalizing the expressions appropriately, we see that this implies that: Hq(M/divides.alt0N)T()≥q′ p′Hp(M/divides.alt0N)−log(c); which is stronger than a data-processing inequality for c=1andq≥p. B. Norm transference In [21], the authors proved the following factorization property: given the representation ∶g/uni21A6 g(:)=u(g)(:)u(g)∗of a ﬁnite or compact Lie group Gon the algebra B(H)of linear operators on a ﬁnite dimensional Hilbert space H, and for any t≥0, deﬁne the co-representation ∶B(H)→L∞(G;B(H)),x/uni21A6(g/uni21A6 g−1(x)). Then we may transfer properties of completely positive maps on L∞(G)to completely positive maps on B(H). Indeed, for every positive function konG, we deﬁne k()∶=/integral.dispk(g)u(g)∗u(g)d(g): (21) Hereis

Question 817 multiple-choice

Quantum speed limits define how quickly quantum systems can evolve between states, a crucial concept for designing rapid quantum gates and operations. Analytical solutions are rare for complex systems, making numerical methods essential for practical applications. In the context of optimizing quantum gate operations for multi-state quantum systems, which statement accurately describes the role of quantum optimal control theory (OCT)? 1) OCT provides exact analytical formulas for quantum speed limits in all quantum systems. 2) OCT relies solely on manual tuning of control fields without algorithmic optimization. 3) OCT is applicable only to single-qubit systems and cannot handle multi-qubit gates. 4) OCT determines quantum speed limits by calculating upper bounds analytically. 5) OCT focuses on maximizing energy expenditure to achieve desired quantum states. 6) OCT assesses performance by comparing initial and final Hamiltonians, ignoring target states. 7) OCT numerically minimizes a cost functional, often based on fidelity, to estimate quantum speed limits and derive time-optimal control fields for complex quantum gates.

✓ Correct Answer:

The correct answer is 7) OCT numerically minimizes a cost functional, often based on fidelity, to estimate quantum speed limits and derive time-optimal control fields for complex quantum gates..

📚 Reference Text:

any set{Ek(t)}of con- ceivable control ﬁelds—a requirement that typically limitsan analytical calculation of T QSLto simple systems [ 74]. We would like to stress that in some cases, it is nevertheless pos-sible to solve the entire control problem analytically, yieldingnot only T QSLbut also the time-optimal control ﬁelds {Ek(t)}. A popular method to achieve that is, for instance, the quantumbrachistochrone formalism [ 75], although its application has been mostly limited to simple systems [ 76–78]. Besides an analytical determination of T QSL, there are var- ious methods to approximate it . One prominent method isto calculate a lower bound T bound/lessorequalslantTQSLin order to get an estimate for TQSLitself. For the simplest case of a state-to-state control problem, lower bounds can be calculated analytically[24]. In contrast, in case of multiple pairs of initial and ﬁnal states, which describe the implementation of quantum gates,such lower bounds only exist for very simple systems [ 79]. In most cases, one needs to resort to numerical tools for esti-mating T QSL. In that context, quantum optimal control theory has proven to be very useful [ 80] as it not only estimates TQSL quite accurately but additionally yields the control ﬁelds that implement the desired dynamics, i.e., realizing the transitionfrom initial to target states, cf. Eq. ( 5). Since this is our method of choice, we introduce it in more detail in the following. B. Quantum optimal control theory Quantum optimal control theory (OCT) [ 81] is a toolbox providing analytical and numerical tools that allow to deriveoptimized control ﬁelds, which solve a given control problem,e.g., in shortest time or with minimal error. Mathematically,an optimal control problem is formulated by introducing thecost functional J[{ψ l},{Ek},T] =εT[{ψl(T)}]+/integraldisplayT 0Jt[{ψl(t)},{Ek(t)},t]dt,(6) where {ψl(t)}is a set of time-evolved states and {Ek(t)} a set of control ﬁelds to be optimized. The error measureε Tquantiﬁes the distance between the time-evolved states |ψl(T)/angbracketright=U(T,0;{Ek(t)})|ψin l/angbracketrightand the desired target states |ψtrgt l/angbracketrightat the protocol’s ﬁnal time T,c f .E q .( 5). The term Jtin Eq. ( 6) captures time-dependent running costs. In most cases, 023026-7 BASILEWITSCH, DLASKA, AND LECHNER PHYSICAL REVIEW RESEARCH 6, 023026 (2024) the error measure εTis the crucial ﬁgure of merit. In order to optimize for quantum gates, we use the error measure [ 82] εT[{ψl(T)}]=1−1 NtrgtNtrgt/summationdisplay l=1Re/braceleftbig/angbracketleftbig ψtrgt l/vextendsingle/vextendsingleψl(T)/angbracketrightbig/bracerightbig (7) and take |ψtrgt l/angbracketright=O|ψin l/angbracketrightas the desired target states for the target gate O.T h es e t {ψin l}runs over the Ntrgtlogical basis states affected by Owith Ntrgt=4,8,16 for two-, three- or four-body gates, respectively. Since the cost functional Jis formulated such that smaller values correspond to better solutions of the control problem,solving an optimal control problem becomes essentially aminimization task, i.e., to ﬁnd a set of control ﬁelds {E opt k(t)} that minimizes J, respectively εT. This is an optimization problem for which several numerical algorithms have beendeveloped [ 83–87]. Many of them are readily available in open-source software packages [ 88–92]. C. Optimization procedure and method In the context of determining TQSLnumerically via OCT, we search for the

Question 818 multiple-choice

In advanced algebra, constructions over fields of characteristic 2 often lead to distinctive group and ring structures, especially when considering modules, matrix algebras, and group presentations. Understanding how commutator subgroups interact with ideals and module generation is essential in this domain. In the described algebraic construction, what is the role of the subgroup G' within the group G, given that for every c in G', c² = 1? 1) G' is the center of G and consists of elements of infinite order. 2) G' is a cyclic subgroup of G of order greater than 2. 3) G' is a subgroup of G consisting exclusively of elements of order greater than 2. 4) G' is a normal subgroup of G where every element is a square in G. 5) G' is a subgroup of G isomorphic to G itself. 6) G' is the commutator subgroup of G and is an elementary abelian 2-group. 7) G' is a subgroup of G that contains only the identity element.

✓ Correct Answer:

The correct answer is 6) G' is the commutator subgroup of G and is an elementary abelian 2-group..

📚 Reference Text:

R k. Then, by (i), hx = 0 but, by (ii), v kx ^ 0 so vk & Ik as required. PROOF (of Lemma 1). Let AT be an arbitrary field of characteristic 2,kan arbitrary fixed positive integer. The algebra R k and the group H k were constructed in [3, Theorem 2'] in the following way. Let G be a group given by the presentation (7) G = {x ux2,...\ xf, [Xi,,x i2,xh], i, i u i2, h e N) and let G = G/ G. For each g e G put g = gG' e G. Note that for each ceG'we have c2 = 1 (because [Xj,Xj]2 = [xf,Xj] = 1 for every i,j e IN). Let T denote the ideal of the group algebra K(G) generated by all elements (8) )( ( )( ft] +1)- «i,&. ^3, ft e G. Denote S = K{G)/ T. For each / e ATG put / = (/ + T) e 5. Let M k be the left X"(G)-submodule of K(G) £ (g#G'), 1®1 and 1 (8) ([ft, ft] + 1) • • • (igTm-u hm\ + 1) (m ^k, g x,...g 2m e G). https://doi.org/10.1017/S1446788700002056 Published online by Cambridge University Press K(G) 0 0K(G)® KS K S 0(G)®j S K334 C. K. Gupta and A. N. Krasil'nikov [6] The algebra Rk is the algebra of matrices R* = which is the quotient algebra of the algebra fK(G) K(G)® KS modulo the ideal0 0 /°0 Vos0 0 0 0MA °0/sK The group Hk is the subgroup of the group of units of R k generated by the matrix C and all matrices g (g € G), where so that H t = BG is the semidirect product of B = sgp {Cg | g e G} with G = sgp {g \g € G], where (9) a = Now we are in position to prove that Ht satisfies the identity (5). Let hi,... , h6 € Ht, C\ = [h,,h2,h 3], C2 = [h4,h 5,h6]. Since Q, C 2 e B, they are products of elements of the form (9) so for some /,, gj e G o I / \o o i (entries denoted by * are not important for the argument). Therefore, /I 0 [C,,C 2]= 0 1 0 \0 0 1 https://doi.org/10.1017/S1446788700002056 Published online by Cambridge University Press [7] Solution of a problem of Plotkin and Vovsi 335 where Note that if fig'1 & G' for some i,j then gjf~x & G and On the other hand, if figj ' = c e G' for some i, j then g ;/; l = c~* = c € G' and ^-7,-So 7T1 ® M"1 -«/' ® ft/)"1 = 7~l ®c-j:l®c = o. Thus, P = 0 (mod M k) and [[hi, h 2, h3], [h4, h 5, h6]] = 1 for all h^ ... , h 6 e U k, that is Hi satisfies the identity (5). Let us check that H

Question 819 multiple-choice

In modern theoretical computer science and cryptography, discrete sums over natural numbers are frequently approximated by integrals to derive analytical bounds, particularly in lattice-based algorithms. Probabilistic guarantees involving high-probability events are crucial for ensuring the reliability of randomized algorithms. When comparing the discrete sum ∑_{i=a+1}^b √i to its continuous analogue ∫_a^b √x dx, which expression gives the exact value of the integral, commonly used to bound the sum in analytic estimates? 1) (b - a) / 2 2) b^2 - a^2 3) (2/3)(b^{3/2} - a^{3/2}) 4) (1/2)(b^{1/2} - a^{1/2}) 5) ln(b/a) 6) (b^{1/3} - a^{1/3}) 7) (b^2 + a^2)/2

✓ Correct Answer:

The correct answer is 3) (2/3)(b^{3/2} - a^{3/2}).

📚 Reference Text:

all a,b∈N, b/summationdisplay i=a+1√ i/greaterorequalslant/integraldisplayb a√xdx=2 3(b3/2−a3/2). Therefore, for any ℓ∈N, N(ℓ)/summationdisplay i=N(ℓ−1)+1√ i/greaterorequalslant2 3/parenleftig N(ℓ)3/2−N(ℓ−1)3/2/parenrightig . But check that N(ℓ−1)/lessorequalslantN(ℓ) so that /radicalbig N(ℓ)(N(ℓ)−N(ℓ−1))/lessorequalslantN(ℓ)3/2−N(ℓ−1)3/2/lessorequalslant3 2N(ℓ)/summationdisplay i=N(ℓ−1)+1√ i. It then easily follows that Gqc(DZn q,σ)/lessorequalslant3 21 ρσ(Zn)∞/summationdisplay ℓ=1ρσ(√ ℓ)N(ℓ)/summationdisplay i=N(ℓ−1)+1√ i=3 2Gqc(DZn,σ). This ﬁnishes the proof. ⊓ ⊔ 43 C Proofs for Section 5 C.1 Proof of Theorem 6 Below, we will establish the following claims: (1) For all ˜senum, the oracle ˆOinsideO(˜senum) is such that, for all ˜sﬀt, with probability at least 9 /10, ifFL(˜senum,˜sﬀt)>(1 + 2η)·CthenˆO(˜sﬀt) = 1 and ifFL(˜senum,˜sﬀt)/lessorequalslantCthenˆO(˜sﬀt) = 0. (2) For all ˜senum, with probability at least 9 /10, if there exists ˜sﬀtsuch that FL(˜senum,˜sﬀt)>(1+2η)·CthenO(˜senum) = 1. (3) For all ˜senum, with probability at least 9 /10, ifFL(˜senum,˜sﬀt)/lessorequalslantCfor all˜sﬀt thenO(˜senum) = 0. (4) With probability at least 9 /10,if the algorithmreturns ˜senum/n⌉}a⊔ionslash=⊥then there exists˜sﬀtsuch thatFL(˜senum,˜sﬀt)>C. Westartbyestablishingtheresultasfollowingfromtheclaimsandthe nestablish these claims below. Let xbe the output of the algorithm. By claim (4), if x/n⌉}a⊔ionslash=⊥ then, with probabilityat least9 /10,there exist ˜sﬀtsuch thatFL(˜senum,˜sﬀt)>C. Therefore,x∈SL C. Hence, this proves that x∈SL C∪{⊥}with probability at least 9/10. Now assume that SL (1+2η)C/n⌉}a⊔ionslash=∅and let˜senum∈SL (1+2η)C. Then by claim (2), with probability at least 9 /10,O(˜senum) = 1 so the algorithm will not return⊥, i.e.x/n⌉}a⊔ionslash=⊥. Proof of claim (1) . Fix˜senumand check that ˆO(˜sﬀt) returns 1 if and only if AO′ W((˜senum,˜sﬀt,1))>(1+η)·C D. NowOWis deﬁned in such a way that O′ W(j) =/parenleftig p q·yj,enum,/floorleftig p q·yj,ﬀt/ceilingrightig ,θ−p q·xT j·b/parenrightig whereyj,enumandyj,ﬀtare deﬁned as expected. Therefore, by Theorem 5, with probability at least 1 −δ, /vextendsingle/vextendsingle/vextendsingleAO′ W((˜senum,˜sﬀt,1))−fW((˜senum,˜sﬀt,1))/vextendsingle/vextendsingle/vextendsingle/lessorequalslantε. But one checks that /an}⌊rack⌉tl⌉{tOW(j),(˜senum,˜sﬀt,1)/an}⌊rack⌉tri}ht=/angbracketleftbigg/parenleftbiggp q·yj,enum,/floorleftbiggp q·yj,ﬀt/ceilingrightbigg ,θ−p q·xT j·b/parenrightbigg ,(˜senum,˜sﬀt,1)/angbracketrightbigg =p q·yT j,enum·˜senum+/floorleftbiggp q·yj,ﬀt/ceilingrightbiggT ·˜sﬀt+θ−p q·xT j·b. Therefore,fW((˜senum,˜sﬀt,1)) =1 D/summationdisplay jcos/parenleftbigg2π p/an}⌊rack⌉tl⌉{tOW(j),(˜senum,˜sﬀt,1)/an}⌊rack⌉tri}ht/parenrightbigg 44 =1 D/summationdisplay jcos/parenleftBigg 2π p/parenleftBigg p q·yT j,enum·˜senum+/floorleftbiggp q·yj,ﬀt/ceilingrightbiggT ·˜sﬀt+θ−p q·xT j·b/parenrightBigg +2π p·θ/parenrightBigg =1 Dℜ/parenleftBigg/summationdisplay jexp/parenleftBigg 2iπ p/parenleftBigg p q·yT j,enum·˜senum+/floorleftbiggp q·yj,ﬀt/ceilingrightbiggT ·˜sﬀt−p q·xT j·b/parenrightBigg +2iπ p·θ/parenrightBigg/parenrightBigg =1 Dℜ/parenleftBigg e2iπ pθ/summationdisplay jexp/parenleftBigg 2iπ p/parenleftBigg p q·yT j,enum·˜senum+/floorleftbiggp q·yj,ﬀt/ceilingrightbiggT ·˜sﬀt−p q·xT j·b/parenrightBigg/parenrightBigg/parenrightBigg =1 DFL(˜senum,˜sﬀt) sinceθwas computed so that ψ(˜sﬀt) =e−2iπ pθ. It follows that, with probability at least 1 −δ, /vextendsingle/vextendsingle/vextendsingle/vextendsingleAO′ W((˜senum,˜sﬀt,1))−1 DFL(˜senum,˜sﬀt)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantε=C D·η. Assume that this inequality holds. –IfFL(˜senum,˜sﬀt)>(1+2η)·CthenAO′ W((˜senum,˜sﬀt,1))>(1+2η)·C D−C D·η= (1+η)·C DsoˆO(˜sﬀt) = 1. –IfFL(˜senum,˜sﬀt)/lessorequalslantCthenAO′ W((˜senum,˜sﬀt,1))/lessorequalslantC D+C Dη= (1 +η)C Dso ˆO(˜sﬀt) = 0. Proof of claim (2) . Fix˜senum. If there exists ˜sﬀtsuch thatFL(˜senum,˜sﬀt)> (1 + 2η)·Cthen by claim (1), with probability at least 1 −δ,ˆO(˜sﬀt) = 1. It follows by Theorem 4 that the search will, with probability at least 9 /10, return i/n⌉}a⊔ionslash=⊥and therefore O(˜senum) will return 1. Proof of claim (3) . ForO(˜senum) to return 0, it is suﬃcient to have ˆO(˜sﬀt) = 0 for all ˜sﬀt. By claim (1), ˆO(˜sﬀt) = 0 with probability at least 1 −δwhen FL(˜senum,˜sﬀt)/lessorequalslantC. Hence, by Theorem 4, the search algorithm will return ⊥ with probability 9 /10 andO(˜senum) = 0. Note here that there is no need for a union bound because of Theorem 4. Proof of claim (4) . For the algorithm to return ˜senum, with probability 9 /10, wemust have O(˜senum) = 1.By claim(3), with probabilityat

Question 820 multiple-choice

In module theory, the concept of quasi-injective modules generalizes injective modules and plays a significant role in the study of abelian groups over their endomorphism rings. Understanding how properties like divisibility, direct sums, and fully invariant subgroups impact quasi-injectivity is fundamental in algebra. Which of the following statements is true regarding quasi-injectivity of abelian groups as modules over their endomorphism rings? 1) Every abelian group is quasi-injective over its endomorphism ring. 2) Only finite abelian groups can be quasi-injective over their endomorphism rings. 3) Quasi-injectivity of a direct sum of fully invariant subgroups requires only one summand to be quasi-injective. 4) A direct sum of fully invariant subgroups is quasi-injective over its endomorphism ring if and only if each summand is quasi-injective. 5) Divisible groups cannot be quasi-injective over their endomorphism rings. 6) Cyclic p-groups are never quasi-injective as modules over their endomorphism rings. 7) Quasi-injectivity is independent of the structure of fully invariant subgroups in abelian groups.

✓ Correct Answer:

The correct answer is 4) A direct sum of fully invariant subgroups is quasi-injective over its endomorphism ring if and only if each summand is quasi-injective..

📚 Reference Text:

Can. J. Math., Vol. XXIV, No. 4, 1972, pp. 617-621 ABELIAN GROUPS QUASI-INJECTIVE OVER THEIR ENDOMORPHISM RINGS GEORGE D. POOLE AND JAMES D. REID 1. Introduction. L. Fuchs has posed the problem of identifying those abelian groups that can serve as the additive structure of an injective module over some ring [1, p. 179], and in particular of identifying those abelian groups which are injective as modules over their endomorphism rings [1, p. 112]. Richman and Walker have recently answered the latter question, generalized in a non-trivial way [7], and have shown that the groups in question are of a rather restricted structure. In this paper we consider abelian groups which are quasi-injective over their endomorphism rings. We show that divisible groups are quasi-injective as are direct sums of cyclic ^-groups. Quasi-injectivity of certain direct sums (products) is characterized in terms of the summands (factors). In general it seems that the answer to the question of whether or not a group G is quasi- njective over its endomorphism ring E depends on how big Hom E(H, G) is, with H a fully invariant subgroup of G. All groups under discussion are abelian and our notation is standard. We write E(G) for the endomorphism ring of G when the dependence on G must be displayed; otherwise, we write simply E. We say that G is quasi- injective if G is a quasi-injective E-module when viewed as an E-module in the natural way. 2. Generalities. In this section we collect a few general facts, starting with PROPOSITION 2.1. Suppose H is a fully invariant subgroup of the quasi-injective group G and that restriction is a ring epimorphism of E{G) onto E(H). Then H is quasi-injective over E (H) . Proof. Let /: K -* H be an E(H)-map where K is fully invariant in H. Clearly K is an E(G) -submodule of H and / is an E(G)-map. By hypothesis on G, there is an a in the center of E(G) such that/ =

Question 821 multiple-choice

In quantum information theory, the no-cloning theorem establishes strict limits on when quantum states can be copied, but exceptions arise for sets of states with special properties. Understanding the mechanisms and requirements for cloning orthogonal quantum states is fundamental for quantum computing and cryptography. Which of the following conditions allows a quantum state to be cloned using only unitary operations? 1) The state is part of an unknown set of non-orthogonal states. 2) The state is a superposition of unknown basis states. 3) The state is chosen from a known set of mutually orthogonal states and the relevant parameter is known. 4) The state is entangled with an unknown environment. 5) The state belongs to a set of indistinguishable probability distributions. 6) The state is mixed with random noise. 7) The state is selected from a set of non-unitary evolutions.

✓ Correct Answer:

The correct answer is 3) The state is chosen from a known set of mutually orthogonal states and the relevant parameter is known..

📚 Reference Text:

a general unknown quantum state. However, if the states are chosen from a known set of mutually orthogonal states, it is well known that cloning is possible, as shown for completeness in the following proposition. Proposition 6.1. Let∣⟩∣ ⟩ ψψ ,…,aa m;1 ; be a set of mutually orthogonal states that depend on a parameter a. Suppose ∣⟩ ∣ ⟩=ψψai,for some index i (which is unknown ). If the value of a is known and we can encode a unitary operator Uasuch that ∣⟩∣⟩=Uψ iaai; ,then there is a unitary operation that copies ∣⟩ψ. Proof. First, note that we can copy any state ∣⟩iof the computational basis. Start with ∣⟩ ∣⟩ ∣⟩ ∣⟩ ∣⟩ ∣⟩=ii i0… 0 … 0 , n 0 where we have encoded the last two registers into nqubits, for nlarge enough. Applying a CNOT gate to the jth and()++jn 1th qubits∣⟩ ∣⟩i0j produces∣⟩ ∣⟩iijj for everyj. Hence, we can produce the state ∣⟩∣⟩ ∣⟩∣⟩ ∣ ⟩ ∣ ⟩ = ii ii i i…… . nn 00 The unitary operator Uahas the e ﬀect: ∣⟩∣ ⟩=Uψ i . aai; Starting with ∣⟩ ∣ ⟩ψ 0,ai; applyUato the ﬁrst register to obtain ∣⟩∣⟩i0.14 Imin Chen and David Sun Copy the state ∣⟩ito obtain ∣⟩∣⟩ii. Applying−Ua1to both pairs of registers gives ∣⟩ ∣⟩ψψ .ai ai;; □ Remark 6.2. Sinceais known, to encode Ua, we can use universality results (c.f. [35 , §4.5] or the Solovay –Kitaev Theorem [35, Appendix 3] for a fault-tolerant version). Later, we will need a slightly stronger version of Proposition 6.1 . Proposition 6.3. Let∣⟩∣ ⟩ ψψ ,…,aa m;1 ; be a set of mutually orthogonal states that depend on a parameter a, and assume that we can encode a unitary operator T such that ∣⟩ ∣ ⟩ ∣⟩ ∣ ⟩=Ta ψ a iai; . Suppose∣⟩ ∣⟩=ψψai;for some index i (which is unknown ).If we have the value of a in a register, then there is a unitary operation that copies ∣⟩ψ. Proof. Starting with ∣⟩ ∣ ⟩ ∣⟩ ∣⟩aψ 00 ,ai, applyTto obtain ∣⟩ ∣⟩ ∣⟩ ∣⟩ai 00 . Copy the states ∣⟩aand∣⟩ito obtain ∣⟩ ∣⟩ ∣⟩ ∣⟩aiai . Applying−T1to both pairs of registers gives ∣⟩ ∣ ⟩ ∣⟩ ∣ ⟩aψ aψ ,ai ai,, which we can permute to obtain ∣ ⟩ ∣⟩ ∣⟩ ∣ ⟩a ψψa .ai ai,, □ Proposition 6.4. If we can copy any given DCP sample (∣ ⟩ ∣ ⟩)=+−ψx y x1 2,aααa α ;(6.5) to produce a state of the form : (∣ ⟩ ∣ ⟩) (∣ ⟩ ∣ ⟩) ⊗= + ⊗ +−−ψψ xy x xy x1 21 2,aα aααa α αa α ;;(6.6) then we can determine the value of a from DCP samples for a . If a is known, then we can copy any given DCP sample for a using a unitary operation . Proof. Given samples of the form (6.6 ), we measure both registers, and with probability 1/2, we obtain ∣ ⟩ ∣⟩ ∣⟩ ∣ ⟩−−xy x y x x or .αa α

Question 822 multiple-choice

Quantum signal processing utilizes advanced transforms to optimize information representation and algorithmic efficiency on quantum hardware. Weighted fractional transforms and their quantum implementations are expanding the capabilities of quantum algorithms in various fields. Which quantum algorithm is essential for realizing quantum versions of weighted fractional transforms by facilitating the extraction of eigenvalues from unitary operators? 1) Grover's search algorithm 2) Quantum amplitude amplification 3) Quantum error correction 4) Quantum phase estimation 5) Quantum teleportation 6) Variational quantum eigensolver 7) Quantum annealing

✓ Correct Answer:

The correct answer is 4) Quantum phase estimation.

📚 Reference Text:

Title: Quantum Weighted Fractional-Order Transform Year: 2023 Paper ID: 812d3f85fc00534b61f938561e31178856dfe568 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/812d3f85fc00534b61f938561e31178856dfe568 Abstract: Quantum Fourier transform (QFT) transformation plays a very important role in the design of many quantum algorithms. Fractional Fourier transform (FRFT), as an extension of the Fourier transform, is particularly important due to the design of its quantum algorithm. In this paper, a new reformulation of the weighted fractional Fourier transform (WFRFT) is proposed in order to realize quantum FRFT; however, we found that this reformulation can be applied to other transformations, and therefore, this paper presents the weighted fractional Hartley transform (WFRHT). For the universality of application, we further propose a general weighted fractional-order transform (WFRT). When designing the quantum circuits, we realized the quantum WFRFT via QFT and quantum phase estimation (QPE). Moreover, after extending our design to the WFRHT, we were able to formulate the quantum WFRHT. Finally, in accordance with the research results, we designed the quantum circuit of the general WFRT, and subsequently proposed the quantum WFRT. The research in this paper has great value as a reference for the design and application of quantum algorithms.

Question 823 multiple-choice

Quantum algorithms for group theory often require efficient solutions to problems involving generators, coset translations, and encoding management, especially when working with black-box or solvable groups. Advanced procedures leverage specialized algorithms and error metrics to ensure robustness and computational efficiency. Which statement correctly describes the time complexity and error management of the ArbitraryAbelianTCS procedure for translating coset superpositions in arbitrary abelian groups? 1) It achieves polynomial quantum time complexity in the size of the group and guarantees zero error by post-selecting solutions. 2) The procedure operates in classical time exponential in the group order, tolerating errors only if the encoding length is minimal. 3) It employs a brute-force search taking exponential quantum time in the encoding length, with error bounded only by group structure. 4) Quantum time complexity is polynomial in group order and encoding length, with no dependence on error tolerance ε. 5) Quantum time complexity is exponential in O(√log|G|) and polynomial in encoding length and log(1/ε), implementing Translating Coset Superposition with error ε. 6) The algorithm fails for factor groups unless the encoding is incompatible, and its error depends solely on the choice of generators. 7) It requires classical subroutines for coset superposition and cannot guarantee fidelity within trace distance ε for large abelian groups.

✓ Correct Answer:

The correct answer is 5) Quantum time complexity is exponential in O(√log|G|) and polynomial in encoding length and log(1/ε), implementing Translating Coset Superposition with error ε..

📚 Reference Text:

Xdenotes a set of generators for G|φ1/angbracketright andua group element such that |u·φ1/angbracketright=|φ0/angbracketright. When no solution exists, we simply 2The sum is over all elements u∈Gand all lists Xconsisting of at most log |G|elements of G.I f there were no errors, αu,Xwould be zero for pairs ( u,X) which do not describe the coset translating φ0toφ1. Due to errors of the algorithm, some such coeﬃcients can be nonzero, although very small. 16 FRIEDL, IVANYOS, MAGNIEZ, SANTHA, AND SEN request the algorithm of Corollary 3.8 to set X=∅, and let ube any group element, instead of rejecting. Letvbe a ﬁxed group element such that |v·φ1/angbracketright=|φ0/angbracketright. Because of the choice of the parameters and by Corollary 3.8, the states |u/angbracketleftX/angbracketright/angbracketrightand|vG|φ1/angbracketright/angbracketrightare of trace distanceatmost ε. Thisimpliesthattheﬁnalstateofthealgorithmisoftracedistance at mostεfrom the following state without garbage: |φ0/angbracketright⊗t|φ1/angbracketright⊗t|vG|φ1/angbracketright/angbracketright. For an arbitrary abelian group G, we can modify procedure Elementary - AbelianTCS (G,α,t) by replacing the algorithm of Corollary 3.8 with an adapted version of Kuperberg’s subexponential method (see Theorem 7.1 of [31]) to solve Translating Coset . (We only need modiﬁcations to Kuperberg’s algorithm like the ones to TranslationFinding described in the proof of Corollary 3.8: We use the “conditionally swapped pairs of functions” trick presented in the proof of Corol-lary 3.6 and simulate the oracle with input quantum states.) Let us call the resulting procedure ArbitraryAbelianTCS (G,α,t). We obtain the following. Corollary 4.6. LetGbe a black-box abelian group with unique encoding of length/lscript.L e tαbe a group action of G,a n dl e t t=2 Ω(√ log|G|)be a positive inte- ger. Then ArbitraryAbelianTCS (G,α,t)implements Translating Coset Super- position (G,α,t)with error εin quantum time /lscriptO(1)2O(√ log|G|)log(1/ε). 4.1.2. Compatible encodings. We will apply recursion into factor groups of solvable groups. Therefore we need an eﬃcient procedure to design a unique encoding for these factor groups. Moreover, for the purpose of our algorithm we will require this encoding to be compatible with the originalencodingof the groupin the followingsense. Definition 4.7. LetGbe a black-box group with unique encoding encof length /lscript. LetNbe a normal subgroup of G. A unique encoding enc NforG/Niscompatible with encif 1.for every x∈G,t h e r ei s y∈xNsuch that encN(xN)=enc(y); 2.the partial unitary |enc(x)/angbracketright|0/angbracketright/mapsto→|enc(x)/angbracketright|encN(xN)/angbracketright,w h e r ex∈G,c a nb e implemented in quantum time poly(/lscript). Note that if Ghas encoding length /lscript, then a compatible encoding for G/Nalso has encoding length /lscript. From now on, we assume for simplicity that we have at our disposal a multiple r of|G|such that r=O (/lscript). This multiple is given or computed once for a group, and we keep the same value for all its subgroups. This assumption is reasonable since for solvable groups the cardinality of Gcan be computed in time poly( /lscript)[ 4 6 ] . In the following theorem, we assume for simplicity that we have at our disposal a zero-error quantum algorithm for computing the generalized discrete logarithm and for order ﬁnding. Theorem 4.8. LetGbe a black-box solvable group with unique encoding encof length/lscript.L e tNbe a normal subgroup of Gsuch that G/Nis abelian.

Question 824 multiple-choice

In the study of algebraic groups, particularly exceptional types such as $ F_4, E_6, E_7, E_8 $, the structure and classification of subgroups and their centralizers play a critical role in understanding group embeddings and maximality. Properties like the presence of tori, semisimplicity, and the types of factors in centralizers are key to subgroup analysis. Which of the following statements correctly describes the centralizer $ C_G(e) $ of a semisimple element $ e $ in a simple exceptional adjoint algebraic group $ G $ when $ C_G(e) $ is connected and contains a factor of type $ A_n $? 1) $ C_G(e) $ is always maximal in $ G $. 2) $ C_G(e) $ must have a non-trivial unipotent radical. 3) $ C_G(e) $ necessarily contains a maximal torus. 4) $ C_G(e) $ must contain a normal torus, contradicting the absence of maximal rank. 5) $ C_G(e) $ is never finite. 6) $ C_G(e) $ only consists of abelian elements. 7) $ C_G(e) $ is always parabolic.

✓ Correct Answer:

The correct answer is 4) $ C_G(e) $ must contain a normal torus, contradicting the absence of maximal rank..

📚 Reference Text:

17 completes the proof of Theorem 1, apart from the demonstration of the embedding L4(5) < E8( 4), which, as pointed out in the Introduction and in the proof of Lemma 2.10, is needed to show the non-maximality of the subgroup 53 .SL3(5) in Table 1 when p = 2. This embedding is exhibited in § 5. 3. Proof of Theorem 2 In this section we give a proof of Theorem 2. Thus let G be a simple exceptional adjoint algebraic group in characteristic/, and let S be a subgroup of Aut G such that D =(Sn G)0 is non-trivial, closed, and satisfies Conditions (1), (2) and (3) of Theorem 2. Suppose that D is not parabolic or of maximal rank, that is, D does not contain a maximal torus of G. By Condition (2), the group E = Q1(0r(Cc;(D))) is non-trivial. Clearly S normalizes E, so by Condition (3) we have D = C0(E)0 and N0(D) = Nc(E). Let Q=Ru(D). If Q:i=l or if r=l, then by [5,3.12], S normalizes a parabolic subgroup P containing N(Q) or N(E), forcing D = P; but then D contains a maximal torus, a contradiction. Thus Q = 1, and so Dis reductive. Moreover, if S normalizes a connected subgroup K * l,G then by (3), D = N(K)0, and hence N(K) contains no maximal torus. In particular Z(D)0 = 1, so Dis semisimple. As S normalizes DC(D), (3) gives C(D)0 ~ D, whence C(D) is finite. Since r * l, E consists of semisimple elements. We claim next that if 1 * E1 ~ E, then Cc(E 1) contains no non-trivial normal torus. For otherwise, if T

Question 825 multiple-choice

In quantum computing, the measurement of qubits using Pauli operators such as σX and σY enables the extraction of specific information about quantum states and operations. Understanding the relationship between measurement outcomes, norms, and error bounds is essential for analyzing quantum algorithms. When a qubit is measured with the Pauli operators σX and σY in the context of estimating ⟨ψ|U|ψ⟩ for a unitary U and state |ψ⟩, which statement about the expectation values and their connection to the real and imaginary parts of the inner product is correct? 1) Measuring σX yields the imaginary part of ⟨ψ|U|ψ⟩, while measuring σY yields the real part. 2) Both σX and σY measurements yield expectation values equal to the trace norm of U. 3) Measuring σX produces a random variable whose expectation value equals the norm squared of ⟨ψ|U|ψ⟩. 4) The expectation values of both σX and σY correspond to the sum of singular values of U. 5) Measuring σY yields the expectation value equal to the real part of ⟨ψ|U|ψ⟩. 6) Measuring σX yields an expectation value equal to the real part of ⟨ψ|U|ψ⟩, while measuring σY yields the expectation value equal to the imaginary part of ⟨ψ|U|ψ⟩. 7) The expectation values of σX and σY measurements are always zero for any unitary U and state |ψ⟩.

✓ Correct Answer:

The correct answer is 6) Measuring σX yields an expectation value equal to the real part of ⟨ψ|U|ψ⟩, while measuring σY yields the expectation value equal to the imaginary part of ⟨ψ|U|ψ⟩..

📚 Reference Text:

extra qubit is then equal to τC= Tr A|Ψ/an}bracketri}ht/an}bracketle{tΨ|CA =1 2TrA/parenleftbigg ψ ψU† Uψ UψU†/parenrightbigg =1 2/parenleftbigg 1 /an}bracketle{tψ|U†|ψ/an}bracketri}ht /an}bracketle{tψ|U|ψ/an}bracketri}ht 1/parenrightbigg =1 2/parenleftBig 12+σXRe/an}bracketle{tψ|U|ψ/an}bracketri}ht+σYIm/an}bracketle{tψ|U|ψ/an}bracketri}ht/parenrightBig . Since the ±1 eigenstates of σXare|±/an}bracketri}ht=1 2/parenleftbig |0/an}bracketri}ht ± |1/an}bracketri}ht/parenrightbig , measuring σXon the qubit yields a classical random variable X∈ {+1,−1}for which EX= Re/an}bracketle{tψ|U|ψ/an}bracketri}htas required. Similarly, by running the algorithm a second time, we can sample from Y, which satisﬁes EY= Im/an}bracketle{tψ|U|ψ/an}bracketri}ht. /square Proof of Lemma 7.2: By the triangle inequality, /vextendsingle/vextendsingle/an}bracketle{tφ|U|ψ/an}bracketri}ht/vextendsingle/vextendsingle2=/vextendsingle/vextendsingle/an}bracketle{tφ|U′|ψ/an}bracketri}ht+/an}bracketle{tφ|(U−U′)|ψ/an}bracketri}ht/vextendsingle/vextendsingle2≤/vextendsingle/vextendsingle/an}bracketle{tφ|U′|ψ/an}bracketri}ht/vextendsingle/vextendsingle2+/vextendsingle/vextendsingle/an}bracketle{tφ|(U−U′)|ψ/an}bracketri}ht/vextendsingle/vextendsingle2. The operator norm satisﬁes ||M||∞= max {|TrMX|:||X||1= 1}, where the trace norm ||X||1 of a square matrix Xis deﬁned to be the sum of its singular values. Thus, /vextendsingle/vextendsingle/an}bracketle{tφ|(U−U′)|ψ/an}bracketri}ht/vextendsingle/vextendsingle2=/vextendsingle/vextendsingleTr(U−U′)|ψ/an}bracketri}ht/an}bracketle{tφ|/vextendsingle/vextendsingle2≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleU−U′/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ∞≤ǫ2≤ǫ, Therefore,/vextendsingle/vextendsingle/an}bracketle{tφ|U|ψ/an}bracketri}ht/vextendsingle/vextendsingle2−/vextendsingle/vextendsingle/an}bracketle{tφ|U′|ψ/an}bracketri}ht/vextendsingle/vextendsingle2≤ǫ.A similar argument gives the same bound with U andU′swapped, proving the lemma. /square

Question 826 multiple-choice

In the study of code-based cryptography, mathematical tools from group representation theory are utilized to analyze the security of cryptosystems against both classical and quantum attacks. Key concepts include variance bounds, tensor product decomposition, and probability estimates related to group actions. Which mathematical technique is essential for bounding the variance of projections in cryptographic proofs involving group representations and is commonly applied using decompositions into irreducible components? 1) Fourier analysis of codewords 2) Reed-Solomon code interpolation 3) Syndrome decoding via linear algebra 4) Graph isomorphism testing 5) Lattice reduction algorithms 6) Error-correcting code concatenation 7) Character sums over irreducible group representations

✓ Correct Answer:

The correct answer is 7) Character sums over irreducible group representations.

📚 Reference Text:

of the symmetric groups. Invent. Math. 125(3), 451–485 (1996) 20. Ryan, J.A.: Excluding some weak keys in the McEliece cryptosystem. In: Proceedings of the 8th IEEE Africon, pp. 1–5 (2007) 21. Sendrier, N.: Finding the permutation betw een equivalent linear c odes: the suppor t splitting algorithm. IEEE Transactions on Information Theory 46(4), 1193–1203 (2000) 22. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing 26, 1484–1509 (1997) 23. Sidelnikov, V .M., Shestakov, S.O.: On insecurity of cryptosystems based on generalized Reed-Solomon codes. Discrete Mathematics and Applications 2(4), 439–444 (1992) 778 H. Dinh, C. Moore, and A. Russell 24. Simon, D.R.: On the power of quantum computation. SIAM J. Comput. 26(5), 1474–1483 (1997) 25. Stichtenoth, H.: On automorphisms of geometric Goppa codes. Journal of Algebra 130, 113– 121 (1990) 26. van Lint, J.H.: Introduction to coding theory, 2nd edn. Springer, Heidelberg (1992) Appendix A Proofs for the Main Theorem Proof of the Decoupling Lemma Proof (Proof of Lemma 4). Fix a vector b∈Vρ. To simplify notations, we shall write Πgas shorthand for Πρ Hg, and write gbforρ(g)b.F o ra n y g∈G,w eh a v e /bardblΠgb/bardbl2=/angbracketleftbig Πgb,Πgb/angbracketrightbig =/angbracketleftbig b,Πgb/angbracketrightbig =1 |H|/parenleftBigg /angbracketleftb,b/angbracketright+∑ h∈H\{1}/angbracketleftbig b,g−1hgb/angbracketrightbig/parenrightBigg . LetSg=∑h∈H\{1}/angbracketleftbig b,g−1hgb/angbracketrightbig .T h e n Varg/bracketleftbig /bardblΠgb/bardbl2/bracketrightbig =Varg[Sg] |H|2=Eg/bracketleftbig S2 g/bracketrightbig −Eg[Sg]2 |H|2. To bound the variance, we upper bound S2 gfor all g∈G.S i n c e Sgis real, applying Cauchy-Schwarz inequality, we have S2 g=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∑ h∈H\{1}/angbracketleftbig b,g−1hgb/angbracketrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ≤(|H|−1)/parenleftBigg ∑ h∈H\{1}/vextendsingle/vextendsingle/angbracketleftbig b,g−1hgb/angbracketrightbig/vextendsingle/vextendsingle2/parenrightBigg . As in Lemma 4.2 of [15], one can express the second moment of the inner product/angbracketleftbig b,g−1hgb/angbracketrightbig in terms of the projection of b⊗b∗into the irreducible constituents of the tensor product representation ρ⊗ρ∗. Speciﬁcally, for any h∈G,w eh a v e Eg/bracketleftbig |/angbracketleftbig b,g−1hgb/angbracketrightbig |2/bracketrightbig =∑ σ∈I(ρ⊗ρ∗)χσ(h) dσ/vextenddouble/vextenddouble/vextenddoubleΠρ⊗ρ∗ σ (b⊗b∗)/vextenddouble/vextenddouble/vextenddouble2 . It follows that Varg/bracketleftbig /bardblΠρ Hgb/bardbl2/bracketrightbig ≤|H|−1 |H|2∑ h∈H\{1}Eg/bracketleftBig/vextendsingle/vextendsingle/angbracketleftbig b,g−1hgb/angbracketrightbig/vextendsingle/vextendsingle2/bracketrightBig ≤Eh∈H\{1}/bracketleftBigg ∑ σ∈I(ρ⊗ρ∗)χσ(h) dσ/vextenddouble/vextenddouble/vextenddoubleΠρ⊗ρ∗ σ (b⊗b∗)/vextenddouble/vextenddouble/vextenddouble2/bracketrightBigg ≤∑ σ∈I(ρ⊗ρ∗) χσ(H)/vextenddouble/vextenddouble/vextenddoubleΠρ⊗ρ∗ σ (b⊗b∗)/vextenddouble/vextenddouble/vextenddouble2 . McEliece and Niederreiter Cryptosystems 779 Proof of the Main Theorem Proof (Proof of Theorem 1:). For any ρ∈L,s i n c e dρ≥D>d2 S,w em u s th a v e ρ/negationslash∈S. By Lemma 6, EH(ρ)≤4|H|2/parenleftbigg χ S(H)+Δd2 S D/parenrightbigg for all ρ∈L. Combining this with the fact that EH(ρ)≤4f o ra l l ρ/negationslash∈L, we obtain DH=Eρ[EH(ρ)]≤4|H|2/parenleftbigg χ S(H)+Δd2 S D/parenrightbigg +4Prρ[ρ/negationslash∈L]. To complete the proof, it remains to bound Pr ρ[ρ/negationslash∈L].S i n c eT r (Πρ H)≤dρ,w eh a v e P(ρ)=dρ|H| |G|Tr(Πρ H)≤d2 ρ|H| |G|. Since dρ

Question 827 multiple-choice

Quantum computing has revolutionized the study of computational problems, particularly those involving hidden structures in algebraic systems. A central problem is understanding when quantum algorithms offer significant speedups over classical ones in various algebraic frameworks. Which type of algebraic structure is known to permit a super-polynomial quantum speedup for the generalized Hidden Subgroup Problem, beyond what is possible in classical polynomial time? 1) Abelian groups 2) Non-abelian simple groups 3) 2-element Boolean rings 4) Certain powers of 2-element algebras 5) Finite fields of prime order 6) Commutative monoids 7) Cyclic groups of composite order

✓ Correct Answer:

The correct answer is 4) Certain powers of 2-element algebras.

📚 Reference Text:

Title: The Hidden Subgroup Problem for Universal Algebras Year: 2020 Paper ID: 3a4f27af8779fed5750d4719ef93dd5d4c813f90 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/3a4f27af8779fed5750d4719ef93dd5d4c813f90 Abstract: The Hidden Subgroup Problem (HSP) is a computational problem which includes as special cases integer factorization, the discrete logarithm problem, graph isomorphism, and the shortest vector problem. The celebrated polynomial-time quantum algorithms for factorization and the discrete logarithm are restricted versions of a generic polynomial-time quantum solution to the HSP for abelian groups, but despite focused research no polynomial-time solution for general groups has yet been found. We propose a generalization of the HSP to include arbitrary algebraic structures and analyze this new problem on powers of 2-element algebras. We prove a complete classification of every such power as quantum tractable (i.e. polynomial-time), classically tractable, quantum intractable, or classically intractable. In particular, we identify a class of algebras for which the generalized HSP exhibits super-polynomial speedup on a quantum computer compared to a classical one.

Question 828 multiple-choice

Quantum algorithms for hidden subgroup problems (HSP) play a foundational role in computational number theory and secure cryptographic systems. Challenges arise when extending these algorithms from discrete to continuous groups, especially in higher-dimensional settings. Which technique is specifically used to mitigate rounding errors when implementing quantum algorithms for hidden subgroup problems over continuous groups such as R^m? 1) Using classical enumeration of lattice points 2) Applying the shortest vector algorithm in discrete lattices 3) Encoding subgroup structures with unique oracle functions only 4) Restricting computations to finite Abelian groups 5) Representing each lattice point as a quantum superposition over a fine grid 6) Employing constant-degree number field reductions exclusively 7) Avoiding exponential transformations of lattices

✓ Correct Answer:

The correct answer is 5) Representing each lattice point as a quantum superposition over a fine grid.

📚 Reference Text:

The function gis periodic with pe- riodRand 1-1 within the period. However, exponential stretching and squeezing of lattices are not computationally trivial. Furthermore, the standard quantum algorithm for the hidden subgroup problem requires a unique represen- tation of the oracle function value (a representation up to an equivalence relation will not work). In [Hal07] these is- sues were addressed by using an intricate notion of “reduced ideals”. This method was extended to constant degree num- ber ﬁelds [Hal05, SV05], but it is diﬃcult to generalize this method to rings of higher degree. At a minimum, com- puting the required reduced ideals seems to require solving the shortest vector problem in ideal lattices of dimension n, and enumerating lattice points also seems necessary. Cryp- tosystems whose security relies on the hardness of solving problems in ideal lattices have been suggested for cryptog- raphy [PR07, LPR10]. Another problem is running the hid- den subgroup algorithm for the continuous group G=Rm, where rounding causes errors. Such errors are tolerable when mis ﬁxed, but worsen in higher dimensions. We propose a diﬀerent scheme, leading to a quantum re-duction from computing the unit group of a number ﬁeld of arbitrary degree nto solving an Abelian hidden subgroup problem over Rm, wherem=O(n). It involves several important ingredients. First, we represent a lattice by a re- duced basis (up to some precision). The exponential trans- formation is performed using repeated squaring of lattices. These lattices can be multiplied because they are also ide- als. Having obtained some basis of the lattice L=f(u), we construct a canonical quantum representation ofL, namely the Gaussian-weighted superposition of lattice points with a suﬃciently large dispersion. To ensure stability against rounding errors, each lattice point is represented by a su- perposition of nearby points in a ﬁne grid. (For example, in one dimension, such a superposition straddles two adjacent grid points.) The initial idea for handling this was using double Gaussian states as in [GKP01], which required a dif- ferent representation of points. In addition to showing how to classically compute approximate bases for the stretched lattices, we prove that the inner product of Gaussian lattice states has a hidden subgroup property. One byproduct of this work is a generalization of the HSP to uncountable topological groups such as R. Most expo- nential speedups by quantum algorithms either use or try to use the HSP [FIM+03, HMR+10]. In the HSP a func- tionf:G→Sis given on a group Gto some set S. For an unknown subgroup H⊆G, the function is constant on cosets ofHand distinct on diﬀerent cosets. The goal is to ﬁnd a set of generators for Hin time polynomial in the ap- propriate input size, e.g. log |G|. When Gis ﬁnite Abelian orZmthere is an eﬃcient quantum algorithm to solve the problem. Using the usual deﬁnition of the HSP for the group G=R does not work as can be seen by the following illustration. When the group is discrete the function can be evaluated on any group element. For example, it

Question 829 multiple-choice

In quantum computing, efficient implementation of quantum gates is crucial for reducing errors and circuit depth, especially on hardware with restricted qubit connectivity. Certain encoding schemes, such as LHZ encoding, offer novel methods for realizing universal gate sets and entangling operations between logical qubits. In the context of the LHZ encoding for quantum computation, which of the following statements correctly describes how a universal entangling gate between logical qubits is efficiently implemented? 1) A single Rz rotation on a physical parity qubit realizes a logical two-qubit entangling operation, such as a controlled phase gate, via stabilizer equivalence. 2) A sequence of SWAP gates between physically adjacent data qubits implements all entangling gates regardless of connectivity. 3) Entangling operations require applying Rx rotations exclusively to the data qubits corresponding to logical qubits. 4) Universal logical entangling gates are performed through simultaneous Hadamard operations on all parity qubits. 5) Only diagonal single-qubit gates contribute to entanglement in the LHZ scheme, eliminating the need for two-qubit gates. 6) Logical two-qubit entangling gates are implemented by global rotations on the entire physical qubit array. 7) Controlled phase gates are achieved by applying chains of CNOT gates between all pairs of logical qubits.

✓ Correct Answer:

The correct answer is 1) A single Rz rotation on a physical parity qubit realizes a logical two-qubit entangling operation, such as a controlled phase gate, via stabilizer equivalence..

📚 Reference Text:

operations on logical qubits can be constructed from the operators introduced inEqs. (6)–(7)using the decomposition U¼R zðαÞRxðβÞRzðγÞ: ð8Þ To obtain a universal gate set, an additional two-qubit entangling gate is necessary. In the LHZ encoding, a nativelogical two-qubit operation is obtained by performing asingle R zrotation on a physical parity qubit, RðijÞ zðαÞ¼exp/C18 −iα 2σðijÞ z/C19 ; ð9Þ which is stabilizer equivalent (i.e., the same up to the application of a constraint operator C¼σðiÞ zσðjÞ zσðijÞ z) to the operator exp/C18 −iα 2σðiÞ zσðjÞ z/C19 ð10Þ and thus effectively performs the two-body operation exp½−iðα=2Þ˜σðiÞ z˜σðjÞ z/C138on the logical qubits, as is obvious from Eq. (1). This operation can be transformed to a logical controlled phase gate CP ϕwith local Rzrotations only, as shown in the following. For the sake of simplicity, qubit indices are omitted. We start from an operation e−iα 2σz⊗σz¼diag ðe−iα 2;eiα 2;eiα 2;e−iα 2Þð 11Þ and the single-body Rzrotation, RzðθÞ¼e−iθ 2σz¼diag ðe−iθ 2;eiθ2Þ; ð12Þ and define UR≔½1⊗RzðβÞ/C138e−iα 2σz⊗σz½RzðγÞ⊗1/C138: ð13Þ Evaluating Eq. (13) for−α¼β¼γ¼ðϕ=2Þyields UR¼e−iϕ 4· diag ð1;1;1;eiϕÞ¼e−iϕ 4·C P ϕ; which corresponds, up to a global phase, to the controlled phase gate CP ϕ. Using the identities defined above, this can be implemented in our scheme with local operations on the physical parity qubits and data qubits as C˜Pði;jÞ ϕ¼RðiÞ z/C18ϕ 2/C19 RðijÞ z/C18 −ϕ 2/C19 RðjÞ z/C18ϕ 2/C19 : ð14Þ Here, iandjare the indices of the involved logical qubits, andRzindicates the rotation on the corresponding data orparity qubit. In particular, for ϕ¼π, we obtain the CZgate (controlled Z gate). This can in turn be transformed to a CNOT gate, by applying logical Hadamard gates before and after the CZgate on the desired target qubit. The operations ˜Rx,˜Rz, and C ˜Pϕform a universal gate set in the LHZ scheme, and we can not only build arbitrary quantum circuits by applying CNOT gates and controlling local fields (parameters only occur in single-qubit operations), but also exploit the comparably simple implementation of a con- trolled phase gate. The standard gate model requires two CNOT - and three single-qubit gates to implement a con- trolled phase gate, as depicted in Fig. 1. Platforms with limited connectivity typically need a large number of SWAP gates in addition. In contrast, a logical controlled phase gate in our approach only requires three parallel single- body rotations. The required resources for implementingcommon gates and gate sequences in our scheme are listed in Table I. The depth and the gate count for nondiagonal single-body operations result from the CNOT chains involved. The main advantage of the encoding clearly comes from the depth-1 implementation of diagonal gates. Note that while nondiagonal operations have a higher cost,their increased weight allows for intrinsic tolerance to bit-flip errors as an additional advantage. Following Eq. (8), any local unitary operation Uon a logical qubit ( i) can be performed in the LHZ scheme by performing it on the data qubit ( i) and surrounding it by the CNOT chains as in the logical ˜Rxrotation,TABLE I. The number of physical operations

Question 830 multiple-choice

Quantum algorithms often rely on efficient implementations of group-based transformations, such as the Quantum Fourier Transform (QFT), which is fundamental for tasks like factoring and phase estimation. Advances in generalizing these transformations can directly impact the performance of algorithms, especially when dealing with complex groups or problems. Which of the following statements most accurately describes a key benefit of providing explicit formulas for generalized QFT and Haar transformations in quantum algorithm design? 1) They enable classical algorithms to simulate quantum circuits without any efficiency loss. 2) They allow quantum circuits to be implemented directly with reduced gate complexity and circuit depth. 3) They eliminate the need for quantum error correction in most practical quantum algorithms. 4) They ensure that all non-Abelian hidden subgroup problems can be solved efficiently. 5) They guarantee optimal performance for quantum data compression regardless of input state. 6) They make Shor’s factoring algorithm obsolete for large integer factorization problems. 7) They allow signal processing tasks to be performed entirely with classical resources.

✓ Correct Answer:

The correct answer is 2) They allow quantum circuits to be implemented directly with reduced gate complexity and circuit depth..

📚 Reference Text:

Title: Generalization of Quantum Fourier Transformation Year: 2017 Paper ID: 7ace24a0c38c91ecb4e610beb37c647f8e57ed5e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/7ace24a0c38c91ecb4e610beb37c647f8e57ed5e Abstract: Quantum Fourier transformation is important in many quantum algorithms. In this paper, we generalize quantum Fourier transformation over the Abelian group $\mathbb{Z}_N$ from two different points to get more efficient unitary transformations. The obtained unitary transformations are given in concise and explicit formula which can be used directly. A relationship between the generalized quantum Fourier transformation and the dihedral hidden subgroup problem are discussed in this paper. This may lead a way to solve the dihedral hidden subgroup problem. The second goal of this paper is to give an explicit formula of quantum Haar transformation.

Question 831 multiple-choice

Quantum computing has demonstrated significant advantages over classical algorithms for certain computational problems, especially in group theory and number theory. The Hidden Subgroup Problem (HSP) is a foundational challenge that underlies many important quantum algorithms and remains a major focus of research, particularly for nonabelian groups. Which statement best describes the relationship between the quantum Fourier transform and efficient solutions to the Hidden Subgroup Problem for abelian groups? 1) The quantum Fourier transform is only applicable to nonabelian groups in solving HSP. 2) Efficient classical algorithms using the quantum Fourier transform can solve HSP for all groups. 3) The quantum Fourier transform is necessary but insufficient for solving HSP in abelian groups. 4) Efficient quantum algorithms for HSP in abelian groups rely on the quantum Fourier transform and Fourier sampling techniques. 5) The quantum Fourier transform has no relevance to the HSP in group theory. 6) Quantum Fourier transform-based algorithms have solved HSP efficiently only in nonabelian groups. 7) The quantum Fourier transform is primarily used for cryptographic key exchange rather than HSP.

✓ Correct Answer:

The correct answer is 4) Efficient quantum algorithms for HSP in abelian groups rely on the quantum Fourier transform and Fourier sampling techniques..

📚 Reference Text:

Sorbonne Paris-Cit´ e, Paris, France 75205 (frederic. magniez@univ-paris-diderot.fr). ¶C N R S ,L I A F A ,U n i v e r s i t ´ e Paris Diderot, Sorbonne Paris-Cit´ e, Paris, France 75205, and Cen- tre for Quantum Technologies, National University of Singapore, Singapore (miklos.santha@liafa.univ-paris-diderot.fr). /bardblSchool of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India 400005 (pgdsen@tcs.tifr.res.in). 1 2 FRIEDL, IVANYOS, MAGNIEZ, SANTHA, AND SEN While no classical algorithm can solve this problem with polynomial query com- plexityevenif Gisabelian, thebiggestsuccessofqua ntumcomputinguntilnowisthat it can be solved by a quantum algorithmeﬃciently for any abelian G. We will refer to this quantum algorithm as the standard algorithmfor Hidden Subgroup .T h em a i n tool for this solutionis Fourier sampling based on the (approximate)quantum Fourier transform for abelian groups which can be eﬃciently implemented quantumly [29]. Simon’s XOR-mask ﬁnding [42], Shor’s fac torization and discrete logarithm ﬁnding algorithms [41], and Kitaev’s algorithm [29] for the abelian stabilizer problem are all special cases of this general solution. Quantum algorithms of Hallgren [20, 21] andSchmidt and Vollmer [40] computing class groups and unit groups of number ﬁelds, including the solution of Pell’s equation, also follow along these lines. Finding an eﬃcient algorithm for Hidden Subgroup for nonabelian groups G is considered to be one of the most important challenges at present in quantum com- puting. Besides its intrinsic mathematical interest, the importance of this problem is enhanced by the fact that it contains as a special case the graph isomorphism problem. Unfortunately, although its query complexity is shown to be polynomial by Ettinger, Høyer, and Knill [14], nonabelian Hidden Subgroup seems to be much more diﬃcult than the abelian case. Although considerable eﬀort was spent on it in the last few years, only a small number of successes can be reported. They can be di- vided into twocategories. The standardabelianFourier samplingbasedalgorithmhasbeen extended to some nonabelian groups in [39, 22, 19, 16, 33, 12] using the quantum Fouriertransformoverthese (nonabelian) groups. Although eﬃcient quantum Fourier transform implementations are known for several nonabelian groups [8, 23, 37, 32], the power of the technique appears to be very limited. In a diﬀerent approach, Hid- den Subgroup was eﬃciently solved in the context of speciﬁc nonabelian black-box groups [5, 45] by [26] without using the Fourier transform on the group, and instead using Fourier transforms over abelian groups only. Similarly, only abelian Fourier transforms were used by [24, 6, 10, 27, 28] to solve the hidden subgroup problem insome speciﬁc kinds of nonabelian groups. See [11] for a more detailed review of hidden subgroup algorithms and related problems. In light of the apparent hardness of Hidden Subgroup in nonabelian groups, a natural line of research is to address subproblems of Hidden Subgroup which, in some groups, capture the main diﬃculty of the original problem. In a pioneeringpaper, Ettinger and Høyer [13], in the case of dihedral groups, implicitly considered another paradigmatic group problem, Hidden Translation .H e r ew ea r eg i

Question 832 multiple-choice

The hidden subgroup problem (HSP) is a central challenge in quantum computing, with applications to cryptography and computational group theory. Quantum algorithms for HSP exploit properties of group representations and the quantum Fourier transform to reveal hidden algebraic structures. In the quantum algorithm for solving the hidden subgroup problem over a finite group, which step is responsible for encoding subgroup structure into measurable quantum amplitudes using group theoretic information? 1) Initialization of the quantum registers into the trivial representation 2) Application of the quantum Fourier transform over the group 3) Entangling group elements with images using the unitary operator 4) Execution of classical post-processing after measurement 5) Preparation of the quantum system in a uniform superposition 6) Repeated application of the hidden subgroup oracle 7) Measurement of the quantum system in the computational basis

✓ Correct Answer:

The correct answer is 2) Application of the quantum Fourier transform over the group.

📚 Reference Text:

above diagram the hidden subgroup structure of the map ϕ:G−→S. Remark 1. The underlying intuition motivating this formal deﬁnition is as follows: Given a natural surjection (or epimorphism) ν:G−→G/Kϕ, an ”archvillain with malice of forethought” hides the algebraic structure of νby intentionally renaming all the elements of G/Kϕ, and ”tossing in for good measure” some extra elements to form a set Sand a map ϕ:G−→S. The hidden subgroup problem can be stated as follows: Problem 1 (Hidden Subgroup Problem (HSP) ).Given a map ϕ:G−→S with hidden subgroup structure, determine a hidden subgrou pKϕofG. An algo- rithm solving this problem is called a hidden subgroup algorithm . We will call a map with hidden subgroup structure a hidden subgroup problem (HSP). 1By saying that this diagram is commutative, we mean ϕ=ιϕ◦ν. This concept generalizes in an obvious way to more complicated diagrams. IS GROVER’S ALGORITHM A QUANTUM HIDDEN SUBGROUP ALGORITHM ? 3 The corresponding quantum form of this HSP is stated as follows: Problem 2 (Hidden Subgroup Problem: Quantum Version ).Let ϕ:G−→S be a map with hidden subgroup structure. Construct a quantum implementation of the mapϕas follows: LetHGandHSbe Hilbert spaces deﬁned respectively by the orthonormal ba ses {|g/an}bracketri}ht|g∈G}and{|s/an}bracketri}ht|s∈S}, and lets0=ϕ(1), where1denotes the identity of the ambient group G. Finally, letUϕbe a unitary transformation such that Uϕ:HG⊗HS−→ H G⊗HS |g/an}bracketri}ht|s0/an}bracketri}ht /ma√sto−→ | g/an}bracketri}ht|ϕ(g)/an}bracketri}ht, Determine the hidden subgroup Kϕwith bounded probability of error by making as few queries as possible of the blackbox Uϕ. A quantum algorithm solving this problem is called a quantum hidden subgroup (QHS) algorithm. 3.The generic QHS algorithm QRand Letϕ:G−→Sbe a map from a group Gto a setSwith hidden subgroup structure. We assume that all representations of Gare equivalent to unitary representations2. Let/hatwideGdenote a complete set of distinct irreducible unitary representations ofG. Using multiplicative notation for G, we let 1 denote the identity ofG, and lets0denote its image in S. Finally, let /hatwide1 denote the trivial representation of G. Remark 2. IfGis abelian, then /hatwideGbecomes the dual group of characters. The generic QHS algorithm is given below: Quantum Subroutine QRand(ϕ) Step 0. Initialization |ψ0/an}bracketri}ht=/vextendsingle/vextendsingle/vextendsingle/hatwide1/angbracketrightBig |s0/an}bracketri}ht∈H/hatwideG⊗HS Step 1. Application of the inverse Fourier transform F−1 GofGto the left register |ψ1/an}bracketri}ht=1/radicalbig |G|/summationdisplay g∈G|g/an}bracketri}ht|s0/an}bracketri}ht∈HG⊗HS, where|G|denotes the cardinality of the group G. 2This is true for all ﬁnite groups as well as a large class of inﬁ nite groups. 4 SAMUEL J. LOMONACO, JR. AND LOUIS H. KAUFFMAN Step 2. Application of the unitary transformation Uϕ |ψ2/an}bracketri}ht=1/radicalbig |G|/summationdisplay g∈G|g/an}bracketri}ht|ϕ(g)/an}bracketri}ht∈HG⊗HS Step 3. Application of the Fourier transform FGofGto the left register |ψ3/an}bracketri}ht=1 |G|/summationdisplay γ∈/hatwideG|γ|/summationdisplay g∈GTrace/parenleftBig γ(g)†|γ/an}bracketri}ht/parenrightBig |ϕ(g)/an}bracketri}ht=1 |G|/summationdisplay γ∈/hatwideG|γ|Trace/parenleftbig |γ/an}bracketri}ht/vextendsingle/vextendsingleΦ/parenleftbig γ†/parenrightbig/angbracketrightbig /parenrightbig ∈H/hatwideG⊗HS, where|γ|denotes the degree of the representation γ, whereγ†denotes the contragradient representation (i.e., γ†(g) =γ/parenleftbig g−1/parenrightbigT=γ(g)T), where Trace/parenleftbig γ†(g)|γ/an}bracketri}ht/parenrightbig =/summationtext|γ| i=1/summationtext|γ| j=1γ(g)ji|γij/an}bracketri}ht, andwhere/vextendsingle/vextendsingle/vextendsingleΦ/parenleftBig γ† ij/parenrightBig/angbracketrightBig =/summationtext g∈Gγji(g)|ϕ(g)/an}bracketri}ht. Step 4. Measurement of the left quantum register with respect to the ort honormal basis /braceleftBig |γij/an}bracketri}ht:γ∈/hatwideG, 1≤i,j≤|γ|/bracerightBig . Thus, with probability Probϕ(γij) =|γ|2/angbracketleftBig Φ/parenleftBig γ† ij/parenrightBig |Φ/parenleftBig γ† ij/parenrightBig/angbracketrightBig |G|2, γijis the measured result, and the quantum system ”collapses” to the state |ψ4/an}bracketri}ht=|γij/an}bracketri}ht/vextendsingle/vextendsingle/vextendsingleΦ/parenleftBig γ†

Question 833 multiple-choice

In photonic quantum information protocols, generating entangled states like the n-GHZ state requires careful handling of photon properties, particularly their internal states and mode assignments. Photon distinguishability can introduce errors that affect the success of entanglement generation and measurement. When photons in modes (0,1) have perfectly orthogonal internal states (Tr(η0η1) = 0), which outcome correctly describes the effect on the protocol after dual-rail projection and further processing? 1) All photons interfere, producing a pure n-GHZ state without the need for corrections. 2) The output state collapses into a single term representing indistinguishable photons in all modes. 3) The photons’ internal states become entangled with their spatial modes, leading to decoherence in the GHZ protocol. 4) Only mode 0 retains photons, while all other modes are emptied after projection. 5) The system produces a mixture of eight possible terms representing all photon placements. 6) The output state is a coherent superposition of all possible internal state assignments. 7) The output is an even mixture of four possible terms, each representing a different distribution of orthogonal internal states across the relevant modes.

✓ Correct Answer:

The correct answer is 7) The output is an even mixture of four possible terms, each representing a different distribution of orthogonal internal states across the relevant modes..

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multi-photon terms are dropped, and the operation may be viewed as taking the second half of each state in 1 2n/2(|1,0,1,0⟩ − |0,1,0,1⟩)⊗n, (A4) and projecting onto a known n-GHZ state. It is easy to see that this results in the output state being a n-GHZ state, up to Pauli corrections from the projection, which are accounted for in Step 5. b. Proof sketch for Theorem 4 We now consider the first-order error case studied in Theorem 4 above, in which there is a distinguishability error in the ith pair, modes (2i,2i+ 1) . Without loss of generality, we take i= 0, considering modes (0,1). We have internal states η0=|ψ0⟩⟨ψ0|, η1=|ψ1⟩⟨ψ1|, where η0is the ideal state and Tr(η0η1) = 0 . We have as input to modes (0,1)the state ρin which the non-ideal photon may be in either mode: thus, as in (16), ρis an even mixture of the two states a† 0(η0)a† 1(η1)|⃗0⟩ anda† 0(η1)a† 1(η0)|⃗0⟩. Calculating the image of each under Step 2 and simplifying, we obtain an even mixture of the states (a† 0(η0)a† 0(η1)−a† 1(η0)a† 1(η1))|⃗0⟩and(a† 0(η0)a† 1(η1)−a† 0(η1)a† 1(η0))|⃗0⟩. (A5) We then apply Step 3, giving a state on modes 0,1,2n,2n+1. Taking the dual-rail projection as discussed above Theorem 4, we drop all terms with 0or2photons in modes (0,1), giving a state ρ′that is an even mixture of the two states (a† 0(η0)a† 2n(η1)−a† 1(η0)a† 2n+1(η1))|⃗0⟩+ (a† 0(η1)a† 2n(η0)−a† 1(η1)a† 2n+1(η0))|⃗0⟩, (A6) (a† 0(η0)a† 2n+1(η1)−a† 1(η0)a† 2n(η1))|⃗0⟩+ (a† 1(η1)a† 2n(η0)−a† 0(η1)a† 2n+1(η0))|⃗0⟩. (A7) 21 (Note the dropped terms occur with equal weight in both cases, corresponding to terms in which an even number of photons travel from modes (0,1)to modes (2n,2n+1) via the 50:50 beamsplitters.) We consider (A6) in detail. This state has been expressed as a sum of two terms: in one, the photon with fully distinguishable internal state η1occupies modes (2n,2n+1); in the other, it is the photon with ideal internal state η0. Note that the later steps in Protocol 2 do not involve interactions between the mode pairs (0,1)and(2n,2n+ 1). Thus, since Tr(η0η1) = 0 , there can be no further interaction between the two terms of (A6). Reasoning similarly for (A7), we observe that it is equivalent to replace ρ′with the state ρ′′, an even mixture of the four terms (a† 0(η0)a† 2n(η1)−a† 1(η0)a† 2n+1(η1))|⃗0⟩= (|1,0,1,0⟩ − |0,1,0,1⟩)⊗η0⊗η1, (A8) (a† 0(η1)a† 2n(η0)−a† 1(η1)a† 2n+1(η0))|⃗0⟩= (|1,0,1,0⟩ − |0,1,0,1⟩)⊗η1⊗η0, (A9) (a† 0(η0)a† 2n+1(η1)−a† 1(η0)a† 2n(η1))|⃗0⟩= (|1,0,0,1⟩ − |0,1,1,0⟩)⊗η0⊗η1, (A10) (a† 1(η1)a† 2n(η0)−a† 0(η1)a† 2n+1(η0))|⃗0⟩=−(|1,0,0,1⟩ − |0,1,1,0⟩)⊗η1⊗η0. (A11) Here the right-hand size of each line expresses the state in terms of the occupation numbers and the internal states on modes (0,1,2n,2n+1); the second factor corresponds to the internal state in modes (0,1)and the third factor to the internal state in modes (2n,2n+ 1). To understand the state generated by Protocol 2 in this case, it suffices to consider each of the terms above and apply the n-GHZ state analyzer. Each of these are in a form that is easily studied using Theorem 1. Note

Question 834 multiple-choice

In quantum optimization, swap algebras and permutation structures are instrumental in formulating and simplifying semidefinite programming relaxations such as quantum max-cut. Gröbner bases and combinatorial properties of permutations are key tools in constructing efficient computational frameworks. Which of the following statements about 3-good permutations and their role in swap algebras is correct? 1) 3-good permutations are those that contain a decreasing subsequence of length three. 2) 3-good permutations cannot form a basis for the swap algebra due to algebraic dependencies. 3) The swap algebra is computationally simpler than the Pauli algebra when generating Gröbner bases. 4) Gröbner bases for swap algebras cannot be computed explicitly for small numbers of variables. 5) Veronese vectors for the non-commutative sum-of-squares hierarchy cannot be constructed from 3-good permutations. 6) The diagonal entries of matrices associated with swap algebra variables are always zero because they are involutions. 7) 3-good permutations form a basis for the swap algebra, enabling recursive expansion and efficient computation in quantum optimization hierarchies.

✓ Correct Answer:

The correct answer is 7) 3-good permutations form a basis for the swap algebra, enabling recursive expansion and efficient computation in quantum optimization hierarchies..

📚 Reference Text:

Since our variables sijare involutions, the diagonal of Md(L) in Eq. (5.5) is constantly 1 yielding a bound of 2 on each positive semidefinite Md(L). Remark 5.7. As pointed to us by Claudio Procesi, an alternative method for providing a polynomial time SDP hierarchy for the quantum max-cut can be based on the notion of good permutations [ Pro21 ]. Ford∈N, a permutation σ∈Snisd-bad if there are 1≤i1σ(i2)>···>σ(id). Otherwise it is d-good . Since each permutation is a product of transpositions, this notion can also be applied to certain elements of the swap algebra, e.g., to products of the sij. Procesi [ Pro21 , Theorem 8] shows that 3-good permutations form a basis of the swap algebra. Further, the proof of his theorem gives a recursive method for expressing an element of the swap algebra as a linear combination w.r.t. this basis. Hence providing an ordering for the set of 3-good permutations one can build increasing size Veronese vectors to be used in the ncSoS hierarchy. (Note that the transposition (i j) withj−i >1 is not 3-good, but an induction on j−itogether with Item (3) of Definition 3.1 can be used to express sijin terms of 3-good permutations.) It would be interesting to investigate this further and compare the performance of this method with the one presented here. 5.3 Gr ¨obner Bases for the Swap Algebra Ideally one would want to simplify Algorithm 2 to avoid expanding everything in terms of Paulis. This is indeed possible if one computes the Gr¨ obner basis for the Swap algebra. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 44 Unfortunately, the GB for the Swap algebra appears more involved that the one for the Paulis, and we were unable to identify its form in general. However, for small nit can be explicitly computed. Consider the lex order on pairs ( i,j), i.e., (i,j)<(k,ℓ) ifi

Question 835 multiple-choice

Quantum dynamical R-matrices play a central role in the theory of quantum groups and integrable systems, encoding the algebraic structure of quantum interactions. Their classification, gauge equivalence, and explicit construction provide deep insights into the symmetries of quantum dynamical systems, especially for the Lie algebra gl(n). In the classification of quantum dynamical R-matrices for gl(n), which property uniquely characterizes the "basic" rational or trigonometric solution among all possible gauge-equivalence classes? 1) It is associated to the empty subset of indices, yielding a trivial R-matrix. 2) It corresponds to a choice of q as a primitive root of unity. 3) It is defined exclusively by permuting the elementary matrices. 4) It involves only multiplicative 2-forms without any dependence on the subset X. 5) It quantizes only the non-reductive subalgebras of gl(n). 6) It is associated to the subset X = {1,.., n}, serving as the prototype to which all other solutions are gauge-equivalent. 7) It remains invariant under all possible gauge transformations, including those not preserving the quantum dynamical Yang-Baxter equation.

✓ Correct Answer:

The correct answer is 6) It is associated to the subset X = {1,.., n}, serving as the prototype to which all other solutions are gauge-equivalent..

📚 Reference Text:

a/ne}ationslash=bβab(λ)Eab⊗Eba(6.1) whereEijis the elementary matrix, and αab,βabare meromorphic functions h∗→C. So it is enough to look for solutions of this form. As in the classical case, we will give the classiﬁcation of solutions up to some group of transformations. Deﬁnition. Amultiplicative 2-form onVis a collection meromorphic func- tions{ϕab:h∗→C}n a,b=1satisfying ϕabϕba= 1 for all a,b. A multiplicative 2-form{ϕab(λ)}isclosedif for all a,b,c, ϕab(λ) ϕab(λ−ωc)ϕbc(λ) ϕbc(λ−ωa)ϕca(λ) ϕca(λ−ωb)= 1. Considerthe followingoperationsonmeromorphicweight-zeromaps R:h∗→ Endh(V⊗V) of the form (6.1): 1. R(λ)/mapsto→/summationdisplay aEaa⊗Eaa+/summationdisplay a/ne}ationslash=bϕab(λ)αab(λ)Eaa⊗Ebb+/summationdisplay a/ne}ationslash=bβab(λ)Eab⊗Eba, where{ϕab(λ)}is a closed multiplicative 2-form on V, 2.R(λ)/mapsto→R(λ−ν)whereνisapseudoconstant, i.e. ameromorphicfunction h∗→h∗such that ν(λ+ωi) =ν(λ) for alli(for example, a constant), 3.R(λ)/mapsto→(σ⊗σ)R(σ−1λ)(σ−1⊗σ−1) whereσ∈Snacts onVandh∗by permutation of coordinates. 15 Remark. Here we allow to perform transformation 2 only if the answer is meromorphic. Lemma 6.1. Tranformations 1−3preserve the set of quantum dynamical R- matrices. Two R-matrices which can be obtained one from the other by a seque nce of such transformations are said to be gauge-equivalent . 6.3. Classiﬁcation for q= 1.LetXbe a subset of {1,...n}and write X=X1∪...∪XkwhereXi={ai...bi}are disjoint intervals. Set RX(λ) =n/summationdisplay a,b=1Eaa⊗Ebb+k/summationdisplay l=1/summationdisplay a,b∈Xl a/negationslash=b1 λa−λb(Eaa⊗Ebb+Eba⊗Eab). Theorem 6.1 ([EV2]). LetX⊂ {1,...n}. Then RX(λ)is a quantum dy- namical R-matrix satisfying the Hecke condition with q= 1. Moreover, any dynamical R-matrix R:h∗→Endh(V⊗V)is gauge-equivalent to RX(λ)for a unique subset X⊂ {1,...n}. Remark. The function RX(λ/γ) is, up to a gauge transformation, a quanti- zation in the sense of Section 3 of the rational classical dynamical r -matrix (4.1) corresponding to the reductive subalgebra of gl(n) spanned by root subspaces gα,g−αforα∈X. Themostinterestingsolution RXcorrespondstothecasewhen X={1,...,n}. We will call it the basic rational solution of the QDYBE. 6.4. Classiﬁcation for q/\e}atio\slash= 1.Letε/\e}atio\slash∈2iπZand setq=eε. LetXbe a subset of {1,...n}and again write X=X1∪...∪XkwhereXi={ai...bi} are disjoint intervals. Set Rε X(λ) =/summationdisplay aEaa⊗Eaa+/summationdisplay a/ne}ationslash=bαab(λ)Eaa⊗Ebb+/summationdisplay a/ne}ationslash=bβab(λ)Eab⊗Eba, whereαab(λ) =q+βab(λ)andwhere βab(λ)isdeﬁnedasfollows: βab=q−1 qλa−λb−1 ifa,b∈Xlfor some 1 ≤l≤k,βab(λ) = 1−qotherwise if a > bandβab(λ) = 0 otherwise if a < b. Theorem 6.2 ([EV2]). LetX⊂ {1,...n}. Then Rε X(λ)is a quantum dy- namical R-matrix satisfying the Hecke condition with q=eε. Moreover, any dynamical R-matrix R:h∗→Endh(V⊗V)is gauge-equivalent to RX(λ)for a unique subset X⊂ {1,...n}. Remark. It can be checked that the Rε X(λ) yield (again up to gauge transfor- mations) quantizations of the trigonometric classical dynamical r- matrices with coupling constant εappearing in Theorem 4.2. Themostinterestingsolution Rε Xcorrespondstothecasewhen X={1,...,n}. We will call it the basic trigonometric solution of the QDYBE. 16 6.5. The fusion and exchange matrices for the vector represe ntation of classical and quantum gln.The above classiﬁcation can be applied to com- pute the fusion and exchange matrices for the vector represent ation. Namely, we have: Theorem 6.3 ([EV3]). 1. Letg=glnand letV=Cnbe the vector represen- tation. Then JVV(λ) = 1+/summationdisplay ab(λb−λa+a−b−1)(λb−λa+a−b+1) (λb−λa+a−b)2Eaa⊗Ebb. 2. LetV=Cnbe the representation of Uq(glN)which is the q-analog of the vector representation. Then JV V(λ) = 1+/summationdisplay ab(q2(λb−λa+a−b)−q−2)(q2(λb−λa+a−b)−q2) (q2(λb−λa+a−b)−1)2Eaa⊗Ebb. Proof.The proof relies on explicit computations and on the classiﬁcation of quantum dynamical R-matrices (Theorems 6.1 and 6.2). More precis ely, it is possibletocomputeexplicitlythe coeﬃcientsof

Question 836 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) often rely on representation theory and properties of group characters to establish computational complexity bounds. The nature of group structure, especially for non-abelian groups, dictates the necessity and extent of entanglement in quantum measurements. For which group does any efficient quantum algorithm solving the Hidden Subgroup Problem necessarily require joint measurements entangled across at least Ω(n log n) coset states, thereby ruling out product-state measurement strategies? 1) The cyclic group Zn 2) The dihedral group Dn 3) The alternating group An 4) The projective linear group PSL(2, Fq) 5) The symmetric group Sn and its wreath product Sn≀S2 6) The quaternion group Q8 7) The direct product group Z2 × Z2

✓ Correct Answer:

The correct answer is 5) The symmetric group Sn and its wreath product Sn≀S2.

📚 Reference Text:

of irreps of Gis|/hatwideG|=/parenleftbigp(n) 2/parenrightbig +2p(n)≤p(n)2,wherep(n)denotes the number of partitions of n. Inorder toapplyTheorem 2wechoose ε=n−αnforsomeconstant α>0tobedetermined later. Then Sε=/braceleftBig σ∈/hatwideG:|χσ(h)| dσ≥ε}={ϑλ,ϑ′ λ:dλ≤nαn/bracerightBig .Hence weobtain that /summationdisplay σ∈Sεdσ·|χσ(h)| ≤2/summationdisplay λ∈/hatwiderSn,dλ≤nαnd2 λ·dλ≤p(n)n2αn·nαn≤n3αneν√n. Herewehave estimated the partition number as p(n) =O(eν√n),whereν=π/radicalBig 2 3. Wealso compute that /summationdisplay σ∈Sεd2 σ≤2/summationdisplay λ∈/hatwiderSn,dλ≤nαnd4 λ≤p(n)n4αn≤n4αneν√n. Inorder toapply Theorem 2,wenow deﬁne α:= 1/4and obtain that δ1≤ε+/parenleftBigg/summationdisplay σ∈Sεdσ|χσ(h)|/parenrightBigg/parenleftBigg |/hatwideG| |G|/parenrightBigg1/2 ≤n−αn+n3αneν√n/parenleftbiggp(n)2 2(n!)2/parenrightbigg1/2 ≤n−1/4n+n3/4ne2ν√n √ 2n!=n−Ω(n), where wehave used the fact that n!≥(n/e)nfor largen. For theparameter δ2in Theorem 2 weobtain δ2= 2k(1+2kε)δ1/2 1+3kε+3k/summationtext σ∈Sεd2 σ |G| ≤2k/parenleftBig 1+2kn−1/4n/parenrightBig n−Ω(n)+3kn−1/4n+3knneν√n 2(n!)2= 2kn−Ω(n). 16 Hence, wehave proved the following corollary toTheorem 2: Corollary 14. Any algorithm operating on coset states that solves the hidd en subgroup problem in G= Sn≀S2in polynomial time has to make joint measurements on k= Ω(nlogn)coset states. The same is true for any algorithm that solves the hidden subgroup probl em inSnusing coset states. Also, any efﬁcient algorithm for isomorphism of two n-vertex graphs that uses the standard reduction to HSP in S2nand then uses coset states to solve the HSP needs to make measurements entangled across k= Ω(nlogn)coset states. Finally, we remark that if we apply Theorem 2 to all the full-s upport involutions in S2n, we only get a lower bound of k= Ω(n). This is because we use Roichman’s [Roi96] upper bound on the normalized characters of S2nin order to deﬁne Sǫ, as in [MRS05], and Roichman’s bound is always at least e−O(n). Since the involutive swaps form an exponentially small frac tion of all the full-support involutions, it is possible that an average hidden full-support involution ma y be distinguishable from the hidden identity subgroup by an O(n)-entangled POVM acting on nO(1)-coset states. However, no such POVM is known and the best upper bound for this problem continues to bethe k=O(nlogn)information-theoretic one. 4.2 The projective lineargroups PSL(2,Fq) The representation theory of the projective linear groups G= PSL(2,Fq)over any ﬁnite ﬁeld Fqis well- known. The following is a summary of the necessary results, f or more details we refer to Appendix B. We treat the cases qeven andqodd separately. In case qodd we have that |PSL(2,Fq)|=q(q2−1) 2. There is oneconjugacy class ofq(q±1) 2involutions (depending onwhether q≡1or3modulo4); lethdenote aﬁxed member of this conjugacy class. The degrees of the irreps are given by1,q,q±1, andq±1 2. The character values|χ(h)|can be upper bounded by 1,1,2, and1, respectively. There is a total number of |/hatwideG|=q+5 2 irreps. In order to apply Theorem 2,wechoose ε=2 q−1. Then Sε={σ∈/hatwideG:|χσ(h)| dσ≥ε}={11} contains only the trivial irrep. With this choice of thepara meterεwehave that /summationdisplay σ∈Sεdσ·|χσ(h)|= 1,/summationdisplay σ∈Sεd2 σ= 1,and/parenleftBigg |/hatwideG| |G|/parenrightBigg1/2 =/parenleftbigg(q+5)/2 q(q2−1)/2/parenrightbigg1/2 =O(q−1). Hence, wecan bound the parameter δ1used in Theorem 2asfollows: δ1≤ε+/parenleftBigg/summationdisplay σ∈Sεdσ|χσ(h)|/parenrightBigg/parenleftBigg |/hatwideG| |G|/parenrightBigg1/2 ≤2 q−1+1·O(q−1) =O(q−1). For theparameter δ2weobtain δ2= 2k(1+2kε)δ1/2 1+3kε+3k/summationtext σ∈Sεd2 σ |G| ≤2k/parenleftbigg 1+2k2 q−1/parenrightbigg O(q−1/2)+3k2 q−1+3k1 q(q2−1)/2≤2kO(q−1/2). 17 The caseq= 2n, where|PSL(2,F2n)|=|SL(2,F2n)|=q(q2−1), can be treated similarly. There we useε=1 q−1which implies that δ2≤2kO(q−1/2). Hence, using Theorem 2 we have shown the following result: Corollary 15. Letqbe a prime power. Then any algorithm operating on

Question 837 multiple-choice

Fault-tolerant quantum computing requires efficient decomposition of quantum gates into a set that can be reliably implemented on error-corrected hardware, such as the Clifford+T gate set. Minimization of two-qubit gates like CNOT is crucial for scalable, resource-efficient quantum algorithms. Which statement accurately describes a key computational complexity result regarding the minimization of T gate count in fault-tolerant quantum Fourier transform (QFT) implementations? 1) Finding the exact lower bound for T gate complexity in QFT is an NP-complete problem. 2) Minimizing the T gate count in QFT is always solvable in polynomial time. 3) The T gate count in QFT can be exactly minimized using only analytical methods. 4) NP-completeness applies only to minimizing the CNOT gate count, not the T gate count. 5) There is no known computational complexity classification for minimizing T gate count in QFT. 6) Exact T gate minimization in QFT is a trivial task for all circuit sizes. 7) The problem of minimizing T gate count in QFT is classified as BQP-complete.

✓ Correct Answer:

The correct answer is 1) Finding the exact lower bound for T gate complexity in QFT is an NP-complete problem..

📚 Reference Text:

Title: The exact lower bound of CNOT-complexity for fault-tolerant quantum Fourier transform Year: 2024 Paper ID: 5dc6c8d549ae685e111f5f9ddc59b8ab9be10cde Source: semantic-scholar URL: https://www.semanticscholar.org/paper/5dc6c8d549ae685e111f5f9ddc59b8ab9be10cde Abstract: The quantum Fourier transform (QFT) is a crucial subroutine in many quantum algorithms. In this paper, we study the exact lower bound problem of CNOT gate complexity for fault-tolerant QFT. First, we consider approximating the ancilla-free controlled-$R_k$ in the QFT logical circuit with a standard set of universal gates, aiming to minimize the number of T gates. Various single-qubit gates are generated in addition to CNOT gates when the controlled-$R_k$ is decomposed in different ways, we propose an algorithm that combines numerical and analytical methods to exactly compute the minimum T gate count for approximating any single-qubit gate with any given accuracy. Afterwards, we prove that the exact lower bound problem of T gate complexity for the QFT is NP-complete. Furthermore, we provide the transversal implementation of universal quantum gates and prove that it has the minimum number of CNOT gates and analyze the minimum CNOT count for transversally implementing the T gate. We then exactly compute the exact lower bound of CNOT gate complexity for fault-tolerant QFT with the current maximum fault-tolerant accuracy 10^{-2}. Our work can provide a reference for designing algorithms based on active defense in a quantum setting.

Question 838 multiple-choice

In three-dimensional Chern-Simons theory, quantum invariants of knot complements can be constructed using advanced techniques from topological quantum field theory, geometric quantization, and symplectic geometry. The interplay between classical invariants and their quantized counterparts is central to understanding the quantum topology of knots and 3-manifolds. Which concept provides the mathematical framework for interpreting the gluing of ideal tetrahedra in the construction of quantum invariants for knot complements in 3D Chern-Simons theory? 1) Homological mirror symmetry 2) Morse theory 3) Floer homology 4) Reidemeister torsion 5) Hodge decomposition 6) Symplectic reduction 7) Donaldson polynomials

✓ Correct Answer:

The correct answer is 6) Symplectic reduction.

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Title: Quantum Riemann Surfaces in Chern-Simons Theory Year: 2011 Paper ID: 8e2b58617a6bdbaa76b453f5ae66bdb6dfcfed5f Source: semantic-scholar URL: https://www.semanticscholar.org/paper/8e2b58617a6bdbaa76b453f5ae66bdb6dfcfed5f Abstract: We construct from first principles the operators $\hat A_M$ that annihilate the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups $SU(2)$, $SL(2,\mathbb{R})$, or $SL(2,\mathbb{C})$ on knot complements $M$. The operator $\hat A_M$ is a quantization of a knot complement's classical $A$-polynomial $A_M(\ell,m)$. The construction proceeds by decomposing three-manifolds into ideal tetrahedra, and invoking a new, more global understanding of gluing in topological quantum field theory to put them back together. We advocate in particular that, properly interpreted, 'gluing $=$ symplectic reduction.' We also arrive at a new finite-dimensional state integral model for computing the analytically continued 'holomorphic blocks' that compose any physical Chern-Simons partition function.

Question 839 multiple-choice

The quantum Fourier transform (QFT) is a foundational algorithm in quantum computing, enabling exponential speedups for problems such as factoring and discrete logarithms. Understanding the QFT's role in solving group-theoretic problems is key to appreciating quantum computational advantage. Which modification allows efficient quantum algorithms to solve hidden subgroup problems in any finitely-generated Abelian group, beyond cases where the hidden subgroup function is strictly distinct on different cosets? 1) Restricting quantum circuits to only powers of 2 2) Limiting the problem to cyclic subgroups of the integers 3) Relaxing the requirement that the hidden subgroup function be distinct on different cosets 4) Applying the quantum Fourier transform exclusively to continuous functions 5) Demanding that all subgroup functions be injective 6) Using only classical discrete Fourier transforms 7) Constraining solutions to complexity class MA

✓ Correct Answer:

The correct answer is 3) Relaxing the requirement that the hidden subgroup function be distinct on different cosets.

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Title: The quantum fourier transform and extensions of the abelian hidden subgroup problem Year: 2002 Paper ID: 1fa5fd82c096ddf9a4f6dfd419e070f30fb4b1c0 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/1fa5fd82c096ddf9a4f6dfd419e070f30fb4b1c0 Abstract: The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the Abelian hidden Subgroup problem, of which Shor's celebrated factoring and discrete log algorithms are a special case. We begin by addressing various computational issues surrounding the QFT and give improved parallel circuits for both the QFT over a power of 2 and the QFT over an arbitrary cyclic group. These circuits are based on new insight into the relationship between the discrete Fourier transform over different cyclic groups. We then exploit this insight to extend the class of hidden subgroup problems with efficient quantum solutions. First we relax the condition that the underlying hidden subgroup function be distinct on distinct cosets of the subgroup in question and show that this relaxation can be solved whenever G is a finitely-generated Abelian group. We then extend this reasoning to the hidden cyclic subgroup problem over the reals, showing how to efficiently generate the bits of the period of any sufficiently piecewise-continuous function on ℜ. Finally, we show that this problem of period-finding over ℜ, viewed as an oracle promise problem, is strictly harder than its integral counterpart. In particular, period-finding over ℜ lies outside the complexity class MA, a class which contains period-finding over the integers.

Question 840 multiple-choice

In three-dimensional supersymmetric abelian gauge theories, the study of non-perturbative sectors and the structure of perturbative expansions is highly sensitive to background geometry and deformation parameters. The interplay between localization, topological angles, and resurgence phenomena leads to novel mechanisms for extracting physical information. Which modification allows the Picard-Lefschetz decomposition to distinguish between topological sectors in 3D $$\mathcal{N}=2$$ supersymmetric gauge theories formulated on a squashed sphere? 1) Introducing a Fayet-Iliopoulos term for the gauge symmetry 2) Complexifying the squashing parameter to create an effective omega deformation 3) Adding a topological theta angle to the Lagrangian 4) Employing non-abelian gauge groups instead of abelian ones 5) Compactifying the theory on a torus instead of a sphere 6) Including higher-derivative terms in the action 7) Coupling the theory to additional chiral multiplets

✓ Correct Answer:

The correct answer is 2) Complexifying the squashing parameter to create an effective omega deformation.

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Title: Picard-Lefschetz decomposition and Cheshire Cat resurgence in 3D $$ \mathcal{N} $$ = 2 field theories Year: 2019 Paper ID: 8adb2054df46acd7ed5c617b8eed667e2487e1ab Source: semantic-scholar URL: https://www.semanticscholar.org/paper/8adb2054df46acd7ed5c617b8eed667e2487e1ab Abstract: Abstract We study three dimensional $$ \mathcal{N} $$ N = 2 supersymmetric abelian gauge theories with various matter contents living on a squashed sphere. In particular we focus on two problems: firstly we perform a Picard-Lefschetz decomposition of the localised path integral but, due to the absence of a topological theta angle in three dimensions, we find that steepest descent cycles do not permit us to distinguish between contributions to the path- integral coming from (would-be) different topological sectors, for example a vortex from a vortex/anti-vortex. The second problem we analyse is the truncation of all perturbative expansions. Although the partition function can be written as a transseries expansion of perturbative plus non-perturbative terms, due to the supersymmetric nature of the observable studied we have that each perturbative expansion around trivial and non-trivial saddles truncates suggesting that normal resurgence analysis cannot be directly applied. The first problem is solved by complexifying the squashing parameter, which can be thought of as introducing a chemical potential for the global U(1) rotation symmetry, or equivalently an omega deformation. This effectively introduces a hidden “topological angle” into the theory and the path integral can be now decomposed into a sum over different topological sectors via Picard-Lefschetz theory. The second problem is solved by deforming the matter content making manifest the Cheshire Cat resurgence structure of the supersymmetric theory, allowing us to reconstruct non-perturbative information from perturbative data even when these do truncate.

Question 841 multiple-choice

In molecular simulations, accurately capturing non-covalent interactions such as halogen bonding is crucial for modeling compounds containing halogens. Force field development often involves strategies to better represent the anisotropic electron distribution of halogen atoms. Which modification to classical force fields most directly targets the specific directional positive electrostatic potential known as the "sigma hole" on halogen atoms? 1) Increasing the van der Waals well depth (ϵ) for halogen atoms 2) Assigning larger partial negative charges to halogen atoms 3) Doubling the number of parameterization cycles during optimization 4) Including standard three-body interaction terms in the force field 5) Applying a global polarization function to all atoms 6) Introducing explicit auxiliary interaction sites with positive charge oriented opposite the halogen's covalent bond 7) Using the Picky algorithm to select more representative dimer configurations

✓ Correct Answer:

The correct answer is 6) Introducing explicit auxiliary interaction sites with positive charge oriented opposite the halogen's covalent bond.

📚 Reference Text:

was found for those Bromo-hydrocarbons, bearin g only one Bromine atom. Conversely, the largest deviations ( ∼2.3 kJ/mol) appeared for the two di-bromo alkanes, suggestingthatmorecomplexfunctionsshouldbea doptedinthiscasetoimprove the adherence to the QM IPES. Finally, I1andI2show an intermediate error ( ∼1.50 kJ/mol) between the aforementioned two groups. It could be h ypothesized that these errors are connected to the limited capability of the adopte d model functions to account for halogen bond. On this basis, ad hocmodiﬁcations of the standard FF functions ( e.g. the addition of auxiliary interacting sites like ESHs), coul d possibly solve the problem. Alternatively, the inclusion of many-body eﬀects through, for instance, polarization func- tions or even more complex algorithms to compute the FF could also be tempted, at least in the case of the rather small molecules here considered. Nonetheless, the inclusion of 22 ESHsand/or three-body contributions in a QMD-FF protocol is beyond the aims of this work, and it will be the object of future investigations. A deeper insight can be gained by monitoring the behavior of t he same key quantities along the parameterization cycles, as displayed in Figure 5 . In the bottom row the ratio between the bulk density achieved in the MC simulation of the monitored cycle and the ﬁnal one ( ρF) is displayed for all compounds. In almost all cases, conver gence 2 4 6 Cycle0.9511.05ρ/ρfinal C1 C2C3 C4 51015 ∆P (kJ/mol) 0.511.522.53σP (kJ/mol) 2 4 6 CycleB1 B2B3 B4 2 4 6 8 CycleN1 N2I1 I2 Figure 5: Selected properties monitored along the cycles of Picky parameterization, achieved for compounds C1-C4(left panels), B1-B4(center panels), N1-N2andI1-I2(right panels). From top to bottom: Picky standard deviation ( σP), convergence index (∆ P) and the ratio between the MC density at each cycle and the ﬁnal MC density. In the central p anels, the ∆ Pthreshold (0.5 kJ/mol) is displayed with a black dashed line. on density is reached after few cycles, being the changes aft er the ﬁfth cycle almost negligible. Conversely, as shown in the middle rows, the sha pe of the QMD-FF IPES converges more slowly, as it often requires two or more addit ional cycles to fall under the selected threshold. However, the largest ∆ Pvariations are made in most cases within the ﬁrst cycles, conﬁrming the eﬃciency of the Pickysampling algorithm to extract representative samples even with a relatively small number of dimers. The ﬁnal intermolecular parameters achieved through Pickyparameterization are 23 reported for compounds C1-C4in Table 6, and in Tables Tables I-K, for the rest of the benchmarked molecules. Here, the improved capability of the QMD-FF, with respect to Molecule prmC1C2H1H2Cl C1σ3.44 - 2.00 - 3.61 ǫ2.255 - 0.048 - 0.647 q-0.236 - 0.259 - -0.008 C2σ3.09 - 2.85 - 3.53 ǫ2.114 - 0.005 - 1.020 q-0.405 - 0.254 - -0.052 C3σ3.13 3.66 2.91 - 3.42 ǫ1.618 0.105 0.005 - 1.306 q-0.028 -0.221 0.098 - 0.026 C4σ3.55 3.25 2.83 2.77 3.52 ǫ0.598 1.659 0.005 0.0050 1.106 q0.005 -0.011 0.021 0.077 -0.209 Table 6: Pickyparameters for all

Question 842 multiple-choice

In algebraic number theory, the computation of the unit group of the ring of integers in a number field involves both classical and quantum algorithmic techniques, including lattice representations and hidden subgroup problems. Quantum algorithms can exploit unique encodings of mathematical structures to efficiently solve problems that are hard classically. In the context of quantum algorithms for computing the unit group of a number field, why is it necessary to map classical lattice representations to unique quantum states before applying the hidden subgroup problem (HSP) framework? 1) To decrease the computational complexity of classical algorithms for lattice reduction 2) To allow classical algorithms to distinguish non-isomorphic lattices efficiently 3) To increase the order of the unit group for cryptographic applications 4) To ensure the oracle function maps to multiple classical representations for redundancy 5) To provide a canonical and unique encoding suitable for quantum processing, avoiding ambiguity in lattice bases 6) To guarantee that the norm function on units remains multiplicative 7) To convert the unit group from a discrete to a continuous group for analysis

✓ Correct Answer:

The correct answer is 5) To provide a canonical and unique encoding suitable for quantum processing, avoiding ambiguity in lattice bases.

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matrix Gwith entriesGjk= tr(ω jωk). The discriminant D=D(O) de- pends only on the ring but not the basis. The extension degree,n, and the discriminant, D, constitute a natural set of parameters characterizing the “complexity” of the ring. Our algorithm for ﬁnding the group of units is polynomial inn+ log |D|. For various algorithmic tasks, e.g. the computation of the latticeeuO, the basis vectors of Omust be known with suﬃcient precision. We will use the fact that the embedding of elements of Kcan be found to any polynomial number of precision bits in polynomial time [Thi95]. 3. OVERVIEW OF THE ALGORITHM The group of units O∗consists of elements z∈ Osuch that N(z) =±1, and they are represented by conjugate vectors 295 of the form z=euv. Here u= (u(1),...,u(s+t))∈Rs+t satisﬁes the conditionP ju(j)= 0, and the components of v= (v(1),...,v(s+t))∈E=Rs×Ctare real or complex numbers of absolute value 1. Thus, the group of units O∗is contained in G=Rs+t−1×` (Z2)s×(R/Z)t´ . More speciﬁcally, O∗is the hidden subgroup in Gwhich corresponds to the following oracle: g:G→lattices inE: (u,v)/mapsto→euvO. (3.1) We give an eﬃcient classical realization of this function, where the output (i.e. a lattice L⊂E) is represented by some basis with a certain precision. Unfortunately, such a representation is not unique, and therefore gcannot be used as an oracle for a quantum HSP algorithm. To deal with this issue, we compose the function gwith another function: ˜f: lattices in E→quantum states : L/mapsto→ |˜f(L)/angbracketright, (3.2) where |˜f(L)/angbracketright is a uniquely deﬁned quantum superposition that encodes the lattice L. Thus, we obtain a usable quan- tum oracle f=˜f◦g:G→quantum states : (u, v)/mapsto→˛˛˜f` euvO´¸ . Note: By abuse of notation, we will later denote ˜fasf. For example, we will refer to the quantum state that encodes the latticeLas|f(L)/angbracketright. Finally, we reduce the HSP problem for Gto that for Rmand apply a general algorithm for ﬁnding the hidden subgroup in Rm. Thus, our algorithm for ﬁnding the group of units splits into three self-contained parts: •A classical algorithm for computing the function g: (u,v)/mapsto→veuO. Note that we cannot compute eube- cause it is an exponentially long number. Instead, we begin with representing uas 2lu0, where u0is suﬃ- ciently small, and compute the lattice eu0Odirectly. Then we apply the following procedure ltimes: given a basis {z1,...,zn}of the lattice Λ = ewO(for some wthat does not need to be known), we compute some basis of the lattice Λ2=e2wO. The repeated squaring yields a basis of the lattice euO; then we multiply it byv. The ideal squaring procedure is not trivial. We need to compute all products zjzkand ﬁnd some basis of the lattice they generate. This requires the detec- tion of linear dependencies with integer coeﬃcients as well as some way to prevent the vector lengths from growing. The algorithm for computing a basis for euO is new as far as we know. •A quantum procedure for the creation of the state |˜f(L)/angbracketright representing a lattice L⊂Rn. LetLbe given by some basis B. We ﬁrst make a Gaussian-weighted superposition of

Question 843 multiple-choice

In noncommutative algebra and optimization, certifying whether a polynomial is a sum of squares (SOS) often involves semidefinite programming (SDP) and ideal membership techniques. Gröbner bases and Veronese vectors play essential roles in efficiently representing and manipulating polynomials and ideals within these frameworks. Which statement correctly describes how SDP-based SOS membership testing for a noncommutative polynomial modulo an ideal I is typically reduced to a computational problem, assuming the ideal’s structure is known? 1) It requires direct computation of polynomial roots using symbolic algebra methods. 2) It is converted into a nonlinear optimization problem over the coefficients of the polynomial. 3) It reduces to solving a linear system for the entries of a positive semidefinite matrix, exploiting Gröbner bases or linear algebra for ideal membership. 4) It is addressed by integrating over the polynomial variables using numerical quadrature techniques. 5) It requires factoring the polynomial into irreducible noncommutative components. 6) It reduces to a combinatorial search over all possible monomial orderings in the polynomial ring. 7) It is solved by computing the discriminant of the polynomial modulo the ideal.

✓ Correct Answer:

The correct answer is 3) It reduces to solving a linear system for the entries of a positive semidefinite matrix, exploiting Gröbner bases or linear algebra for ideal membership..

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element sijin the polyno- mialhwith the corresponding swap matrix Swapij4, and we will investigate this particular situation more in the following section. For now we return to the case of a general ideal I. The Veroneses are column vectors, denoted Vd(n), which consist of degree dmonomials in then(n−1)/2 variables sij, i

Question 844 multiple-choice

Quantum algorithms often leverage block-encoding and polynomial transformations to efficiently manipulate and estimate properties of quantum systems, such as eigenvalues and amplitudes. These techniques are central to advancements in quantum phase, energy, and amplitude estimation, especially under hardware constraints like minimizing ancilla qubits. Which technique enables the amplification of eigenvalues in block-encoded matrices to approximate projector form (values close to 0 or 1) without requiring additional ancilla qubits? 1) Grover's search algorithm 2) Quantum amplitude amplification using ancilla qubits 3) Hamiltonian simulation via Trotter-Suzuki decomposition 4) Quantum error correction codes 5) Singular value transformation with amplifying polynomials 6) Quantum teleportation protocols 7) Classical randomized algorithms for eigenvalue estimation

✓ Correct Answer:

The correct answer is 5) Singular value transformation with amplifying polynomials.

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=M/summationdisplay k=⌈M/2⌉/parenleftBigg M k/parenrightBigg pk(1−p)M−k(4) Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 3 We can therefore apply such an ‘amplifying polynomial’ [Diak09] directly to the block- encoding using singular value transformation, which requires no new ancillae. In fact, using techniques from [LC17], we can construct a polynomial that performs the same task with much smaller degree. Now we have ampliﬁed the eigenvalues of the block-encoding to be close to 0 or 1, so we have a projector. To extract the 0/1-eigenvalue into a qubit we use the following novel technical tool: Theorem. Block-measurement. Say we have an approximate block-encoding of a pro- jector Π. Then there exists a quantum channel that approximately implements the map: |0/angbracketright⊗|ψ/angbracketright→| 1/angbracketright⊗Π|ψ/angbracketright+|0/angbracketright⊗(I−Π)|ψ/angbracketright (5) The channel is based on uncomputation [BBBV97], but the error analysis also features an interesting trick to deal with the case where the uncomputation fails. This theorem may ﬁnd applications elsewhere. Repeatingtheaboveprocedureforeachbitwhilecarefullyadjustingthephasesatevery step yields an algorithm that performs coherent phase estimation with no ancillae required. To estimate energies, we can construct a block-encoding with eigenvalues cos2(2n−k−1πλj) directly using the Jacobi-Anger expansion [GSLW18] rather than going through Hamilto- nian simulation, which boosts the performance further, although now the algorithm does require ancillae. Finally, to perform coherent amplitude estimation, we construct a block- encoding of a 1x1 matrix containing the amplitude to be estimated and then invoke energy estimation. A key research question of this paper is: Do these algorithms perform better than traditional phase estimation in practice? An asymptotic analysis is not suﬃcient to answer this question. Instead, we carefully bound the query complexity and failure probability of all algorithms involved using the diamond norm and carefully select constants to maximize the performance. Subsequently, we perform a numerical analysis of the query complexity. We ﬁnd that we improve the query complexity of phase estimation by a factor of about 13x, and the query complexity of energy estimation is improved by about 20x. Our amplitude estimation algorithm inherits the speedup of the energy estimation algorithm. This paper is structured as follows. In section 1 we carefully deﬁne the problem of coherent estimation and establish the notion of a rounding promise. Then we analyze the textbook method. In section 2 we present our novel algorithms for phase and energy estimation. In section 3 we discuss a numerical analysis of the query complexities of the algorithms. Then, in section 4 give a proof of the block-measurement theorem above and then show how to use energy estimation to perform non-destructive amplitude estimation in section 5. 1 Preliminaries In this section we formally deﬁne the estimation tasks we want to solve (Deﬁnition 2). This requires setting up the notion of a ‘rounding promise’ (Deﬁnition 1) which highlights an inherent complication with coherent estimation that is not present in the incoherent case. We argue that this complication is unavoidable, and should be taken into account when coherent estimation is performed in practice. Next, we set up some simple technical tools for dealing with the error

Question 845 multiple-choice

Quantum computing utilizes group-theoretic algorithms to solve problems such as the hidden subgroup problem (HSP), especially within semidirect product groups. Techniques like coset labeling, automorphism inheritance, and quantum measurements are critical for efficiently analyzing the structure of subgroups. In the recursive reduction of the hidden subgroup problem for groups of the form G = A ⋊ Z_p, which condition ensures that the subgroup H2 in the quotient group G2 is cyclic of order p and can be efficiently generated by a single element? 1) H1 is non-normal in G and A2 is nonabelian 2) H is the trivial subgroup and Z_p acts trivially on A 3) The automorphism ϕ2 is the identity on A2 4) H1 equals the center of G and A2 is simple 5) H is generated by all elements of Z_p 6) H1 contains all elements of A 7) H is generated by a single element of the form (a,1) for some a ∈ A

✓ Correct Answer:

The correct answer is 7) H is generated by a single element of the form (a,1) for some a ∈ A.

📚 Reference Text:

ﬁnd H1/n⋊ttriangeqlleftG, then we know that H=H1, and we are done. Otherwise, we learn that H1/triangleleftequalG, and we proceed to factor out H1. The (left) cosets of H1inGcan be represented as follows: for g= (a,b)∈G, gH1={(a,b)(h,0) :h∈A1}={(a+ϕb(h),b) :h∈A1}={(a+h,b) :h∈A1}.(4) Thus the cosets can be labeled by b∈Zpanda∈A2:=A/A1. Now we work in the group G2:=G/H1∼=A2⋊ϕ2Zpwhose elements are these cosets. Note that G2inherits its deﬁning automorphism ϕ2from the original automorphism ϕ. Sincefis constant on cosets of H1≤H, we can consider it as a function f:G2→Swith the hidden subgroup H2:=H/H1. IfH1=HthenH2is trivial. If H1/ne}ati≀nslash=H, then we must have H=/an}bracketle{tH1,(a,1)/an}bracketri}htfor somea∈A, which can be seen as follows. Since pis prime, there must be some (a,1)∈H; but for any additional ( a′,1)∈H, we have (a′,1)(a,1)−1= (a′−a,0)∈H1, and hence (a′,1)∈ /an}bracketle{tH1,(a,1)/an}bracketri}ht. Also, note that /an}bracketle{t(a,1)/an}bracketri}ht∩H1=/an}bracketle{t(Φ(p)(a),0)/an}bracketri}ht. Thus, by the second isomorphism theorem,H2=H/H1∼=/an}bracketle{t(a,1)/an}bracketri}ht//an}bracketle{t(Φ(p)(a),0)/an}bracketri}ht, which is a cyclic group of order pgenerated by ( d,1) for somed∈A2. From now on we assume that the hidden subgroup is H=/an}bracketle{t(d,1)/an}bracketri}ht={(Φ(b)(d),b) :b∈Zp}for somed∈A. Note in particular that since |H|=p, we have Φ(p)(d) = 0. This reduction alone is enough to solve the hspover some semidirect product groups. For example, the hspover the wreath product groups Zn 2≀Z2(for which an eﬃcient quantum algorithm 1In some cases, the results of [31] may not be required; for exa mple, in the case A=Zr pconsidered in Section 6, H1can be viewed as a subspace, and it is suﬃcient to check whethe r this subspace is invariant under the action of the matrix µthat deﬁnes the automorphism ϕ. 4 was given in [28]) and Zpr⋊ϕZpwithϕ(a) = (pr−1+1)a(for which an eﬃcient quantum algorithm was given in [17]) are both reduced to instances of the abelia nhsp. However, in general, we are left with a nonabelian hsp, which we attempt to solve using the pretty good measurement . 4 The pretty good measurement approach In this section, we present the pretty good measurement appr oach to the hidden subgroup problem overG=A⋊Zp. We begin in Section 4.1 by describing the hidden subgroup st ates and expressing them in terms of a certain algebraic problem called the matri x sum problem. Then, in Section 4.2, we describe the pretty good measurement for distinguishing these states. In Section 4.3, we give an expression for the success probability of the measuremen t (as well as general upper and lower bounds), and in Section 4.4, we prove that the measurement is optimal. Finally, in Section 4.5, we explain how the measurement can be implemented by solving th e matrix sum problem. 4.1 Hidden subgroup states and the matrix sum problem According to (2), the hidden subgroup states are uniform mix tures of uniform superpositions over the left cosets of HinG. A complete set of left coset representatives of H=/an}bracketle{t(d,1)/an}bracketri}htinG=A⋊Zp is given by L={(ℓ,0) :ℓ∈A}, and we have the coset states |ψℓ,d/an}bracketri}ht:=1√p/summationdisplay b∈Zp|(ℓ,0)(Φ(b)(d),b)/an}bracketri}ht=1√p/summationdisplay b∈Zp|ℓ+Φ(b)(d),b/an}bracketri}ht. (5) The hidden subgroup state is ρd=1 |A|/summationdisplay ℓ∈A|ψℓ,d/an}bracketri}ht/an}bracketle{tψℓ,d|. (6) Fourier transforming the ﬁrst register (over A) gives ˜ρd=1 |A|/summationdisplay x∈A|˜ψx,d/an}bracketri}ht/an}bracketle{t˜ψx,d|where|˜ψx,d/an}bracketri}ht:=1√p/summationdisplay b∈Zpχx(Φ(b)(d))|x,b/an}bracketri}ht (7) and whereχx:A→Cforx∈Adenotes the xth group character of A,

Question 846 multiple-choice

In quantum algorithms for abelian groups, understanding the structure of subgroups and their representation via quantum states and operators is crucial for efficient computation. The quantum Fourier transform (QFT) and group theory play essential roles in identifying hidden substructures. Which property of irreducible representations ensures that, after applying the quantum Fourier transform to a uniform superposition over a subgroup K of an abelian group G, only certain representation labels contribute to the outcome, enabling the identification of K? 1) The representations must be one-dimensional 2) The representations must have maximal character values on G 3) The representations must be trivial when restricted to K 4) The representations must correspond to conjugacy classes of G 5) The representations must be induced from proper subgroups of G 6) The representations must be regular on cosets of K 7) The representations must have nonzero trace on all elements of G

✓ Correct Answer:

The correct answer is 3) The representations must be trivial when restricted to K.

📚 Reference Text:

uniform superposition1√ |K|/summationtext k∈K|g0+k/an}bracketri}ht. Note that the state in eq. (15) may be written as a g0-shifted state: |g0+K/an}bracketri}ht=U(g0)|K/an}bracketri}ht (16) Our basic idea now is to introduce into Ha new basis {|χg/an}bracketri}ht:g∈G}of special states which are shift-invariant in the sense that U(g1)|χg2/an}bracketri}ht=eiφ(g1,g2)|χg2/an}bracketri}ht for allg1,g2 i.e. the |χg/an}bracketri}ht’s are the common eigenstates of all the shifting operations U(g). Note that theU(g)’s all commute (since the group is abelian) so such a basis of common eigenstates is guaranteed to exist. Then according to eq. (16) if we view |K/an}bracketri}htand|g0+K/an}bracketri}htin the new basis, they will contain the same pattern of labels determin ed by the subgroup Konly, and corresponding amplitudes will diﬀer only by phase factors. Thus the probability distribution of the outcomes of a measurement in the new basis will directl y provide information about the subgroup K. More precisely it may be shown [11] (and cf below) that this m easurement provides a uniform random sample from the so-called dual gro up ofKinG. The Fourier transform FonGis deﬁned to simply be the unitary transformation which rotates the shift invariant basis back to the standard basis : F |χg/an}bracketri}ht=|g/an}bracketri}ht for allg 9 Hence to read |ψ(g0)/an}bracketri}htin the new basis we just apply Fand read in the standard basis. To give an explicit construction of Fit suﬃces to give the states |χg/an}bracketri}htwritten as compo- nents in the standard basis. There is a standard way of calcul ating these components based on constructions from group representation theory. An intr oduction with further references is given in [10, 11] and here we will summarize the main points . If we write |χl/an}bracketri}ht=1/radicalbig|G|/summationdisplay gχl(g)|g/an}bracketri}htfor eachl∈G (17) then we can take the functions χl:G→ Cto be the |G|irreducible representations of the groupG. Then the basic theorems of group representation theory (cf for example [10]) guar- antee that the states |χl/an}bracketri}htare orthonormal and have the required shift invariant prope rty. Indeed shift invariance is a direct consequence of the basic deﬁning property of a represen- tation:χ(g1+g2) =χ(g1)χ(g2). For the group ZNthe irreducible representations are given byχk(j) = exp 2πijk/N forj,k∈ ZNand |χk/an}bracketri}ht=1√ NN−1/summationdisplay j=0e2πijk N|j/an}bracketri}ht leading to the Fourier transform formula given in eq. (4). Which labels lappear in F |g0+K/an}bracketri}ht? It suﬃces to consider F |K/an}bracketri}htand from eq. (17) we get directly F |K/an}bracketri}ht=1/radicalbig |G|/radicalbig |K|/summationdisplay l∈G /summationdisplay k∈Kχl(k) |l/an}bracketri}ht Now, for Abelian groups, the restriction of χlfromGtoKis an irreducible representation of Kand the orthogonality relations for irreducible represent ations give that/summationtext k∈Kχl(k) = 0 for allχl’s except the trivial representation deﬁned by χl(k) = 1 for all k∈K. In the latter case we have/summationtext k∈Kχl(k) =|K|. Hence F |K/an}bracketri}htis a uniform superposition of the |G|/|K|labels lsuch thatχlrestricts to the trivial representation on K. IfKhas a generator rthen the latter condition is equivalent to χl(r) = 1 as we saw in the example of factoring and discrete logarithms (where r= (y,1)). Thus we are able to uniformly sample from this set of labe ls, which distinguishes the possible K’s. This completes the quantum part of the algorithm

Question 847 multiple-choice

Transition metal complexes often exhibit complex electronic structures that present significant challenges for quantum chemical calculations, especially when strong electron correlation and multiple spin states are involved. Advances in computational methods have enabled more accurate treatment of these systems. Which computational approach has enabled the treatment of very large active spaces in multireference calculations for transition metal complexes, significantly expanding the range of accessible electronic states? 1) Hartree-Fock theory 2) Coupled cluster singles and doubles (CCSD) 3) Complete active space self-consistent field (CASSCF) 4) Density matrix renormalization group (DMRG) 5) Møller–Plesset perturbation theory (MP2) 6) Generalized gradient approximation (GGA) within DFT 7) Configuration interaction singles (CIS)

✓ Correct Answer:

The correct answer is 4) Density matrix renormalization group (DMRG).

📚 Reference Text:

Title: Multireference Approaches to Spin‐State Energetics of Transition Metal Complexes Utilizing the Density Matrix Renormalization Group Year: 2019 Paper ID: 49ee95f21d146d933ead67e29f548fc92531f6a2 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/49ee95f21d146d933ead67e29f548fc92531f6a2 Abstract: The accurate and reliable calculation of different electronic states in transition metal systems is a persistent challenge for theoretical chemistry. The widespread use of density functional theory for computing the relative energies of states with different spin in transition metal complexes not only has not discouraged, but often, owing to its limitations, has motivated the development and refinement of first‐principles wavefunction–based methods, including both single‐reference and multireference approaches. A significant boost for the latter has been the emergence of the density matrix renormalization group (DMRG) as a reliable tool in applied quantum chemistry. By enabling the use of larger active spaces than conventionally possible, DMRG has opened up areas of transition metal chemistry that were previously inaccessible to multireference approaches. The present perspective provides an overview of representative studies that make use of DMRG methods and discusses recent applications to spin‐state energetics of transition metal systems. These range from mononuclear to exchange‐coupled systems that define an important emerging field of DMRG applications. Major achievements are highlighted and potential pitfalls are identified with a view toward future methodological developments as well as extensions in the applicability of DMRG‐based approaches to problems of spin‐state energetics and exchange‐coupling in transition metal chemistry.

Question 848 multiple-choice

In homological algebra and mathematical physics, spectral sequences are utilized to compute cohomology groups of filtered complexes, with applications in the study of W-algebras and BRST cohomology. The behavior of spectral sequences can differ significantly between infinite-dimensional and finite-dimensional settings. When computing BRST cohomology for finite W-algebras, such as those associated with the principal sl2 embedding, which specific feature distinguishes their spectral sequence behavior compared to the infinite W-algebra case? 1) The spectral sequence collapses at the first term due to additional symmetries. 2) Cohomology computations always require the use of the universal coefficient theorem. 3) The reduced complex has no explicit generators or relations. 4) The presence of derivative terms ensures higher cohomology vanishes in finite cases. 5) BRST cohomology cannot be computed using spectral sequences for finite W-algebras. 6) The spectral sequence does not collapse at the second term, leading to a more intricate cohomological structure. 7) Infinite W-algebras lack filtration, making spectral sequences inapplicable.

✓ Correct Answer:

The correct answer is 6) The spectral sequence does not collapse at the second term, leading to a more intricate cohomological structure..

📚 Reference Text:

careful in deriving H∗(Ω;d) fromEp,q ∞. Finally, let us present another fact that will be useful late r. Lemma 3 SupposeAis a diﬀerential graded algebra, A=⊕n≥0An, with a diﬀerential of degree 1. Assume furthermore that Ahas a ﬁltration {0}=F0A⊂F1A⊂F2A⊂ · · · ⊂A, (5.13) such thatFpAFqA⊂Fp+qA, and thatdpreserves the ﬁltration, d(FpA)⊂FpA. If Hk(Fp+1A/FpA;d) = 0unlessk= 0, then we have the following isomorphism of vector spaces H0(A;d)≃ ⊕ p≥0H0(Fp+1A/FpA;d). (5.14) 41 Proof: One can assign a spectral sequence to such a ﬁltered gr aded algebra [29], whose ﬁrst term contains the cohomologies Hk(Fp+1A/FpA;d). If onlyH0/\e}atio\slash= 0, then the spectral sequence collapses at the ﬁrst term, and because th e ﬁltration is bounded from below ( {0}=F0A), one can collect the vector spaces that make up E∗,∗ ∞, to get the isomorphism (5.14). 5.4. The BRST Cohomology The computation of the BRST cohomology is simpliﬁed conside rably due to the introduction of the new set of generators ˆJa. The simpliﬁcation arises due to Theorem 3 If(Ω;d)denotes the BRST complex, with Ωgenerated by ˆJa,cαand bα, andd=d0+d1given by (5.8), then H∗(Ω;d)≃H∗(Ωred;d), where Ωredis the subalgebra of Ωgenerated by ˆJ¯αandcα. Proof: Apply the K¨ unneth theorem 2 to Ω red⊗(⊗αΩα), where Ω αis the algebra gener- ated by ˆJαandbα. Note that [ ˆJα,bα] = 0 and that the conditions of the K¨ unneth the- orem are satisﬁed. Therefore, H∗(Ω;d)≃H∗(Ωred;d)⊗(⊗αH∗(Ωα;d)). Now (Ω α;d) is essentially the same complex as the one we examined in the las t part of the previous sec- tion, and one easily proves that Hk(Ωα;d)≃Cδk,0. This shows H∗(Ω;d)≃H∗(Ωred;d) The reduced complex (Ω red;d) is described by the following set of relations: d0(ˆJ¯α) =fα¯α ¯γˆJ¯γcα, d0(cα) = −1 2fβγ αcβcγ, d1(ˆJ¯α) = −f¯αβ γχ(Jγ)cβ, d1(cα) = 0, [ˆJ¯α,ˆJ¯β] =f¯α¯β ¯γˆJ¯γ, [ˆJ¯α,cβ] = −f¯αγ βcγ, [cα,cβ] = 0. (5.15) Feigin and Frenkel [11] propose to use the spectral sequence for the double complex (Ω;d0;d1) to compute the BRST cohomology for the inﬁnite case, to obta in the stan- dardWalgebras. They claim that in that case, the spectral sequenc e collapses at the second term, and use this to identify the Walgebras as the centralizers of some vertex operators in a free ﬁeld algebra. Because ﬁnite Walgebras are the same as inﬁnite 42 Walgebras, with all dependence on the co-ordinates suppress ed (so that derivatives vanish), we would expect the same thing to happen for ﬁnite Walgebras. However, this turns out not to be the case here. Let us demonstrate what happens for the case of the principal sl2embedding in sl2,i.e.the embedding is given by the identity map. The reduced algeb ra Ω redis generated by ˆHof degree (0 ,0),ˆFof degree ( −1,1) andcof degree (0 ,1). The nontrivial relations between these generators are [ ˆH,ˆF] =−2Fand [ ˆH,c] =−2c. Furthermore, d1(ˆF) =d1(c) = 0,d1(ˆH) =−2c,d0(ˆH) =d0(c) = 0 andd0(ˆF) =ˆHc. To ﬁndE∗,∗ 1we computed0(ˆHaˆFbc) = 0 and d0(ˆHaˆFb) =bˆHa(ˆH+b−1)ˆFb−1c. (5.16) Therefore in H∗ d0the identity f(ˆH)ˆFbc=f(1−b)ˆFbcis valid, and E∗,∗ 1=C[ˆH]⊕C[ˆK]c. (5.17) Here we already disagree with [11], because they claim E1is only generated by

Question 849 multiple-choice

Quantum algorithms have transformed our understanding of certain group-theoretic problems, especially the Hidden Subgroup Problem (HSP), which has critical implications for the security of group-based cryptographic systems. The interplay between group structure and algorithmic complexity is central in assessing quantum-safe cryptographic platforms. Which statement most accurately explains why WalnutDSA, a post-quantum signature scheme based on braid groups and the Conjugacy Search Problem (CSP), is considered resistant to quantum attacks exploiting the Hidden Subgroup Problem (HSP)? 1) WalnutDSA relies on abelian group structures, for which quantum algorithms are inefficient. 2) There is no known reduction from CSP in braid groups to HSP, and braid groups lack non-trivial finite subgroups, making HSP-based quantum attacks inapplicable. 3) WalnutDSA's security is based on the equivalence of CSP to factoring, a problem already solved efficiently by quantum algorithms. 4) WalnutDSA uses infinite symmetric groups, which offer polynomial-time solutions to HSP using classical algorithms. 5) Grover’s algorithm directly solves the CSP in braid groups, compromising WalnutDSA’s security. 6) HSP in braid groups is efficiently solvable by quantum algorithms due to their subgroup structure. 7) The core problem in WalnutDSA is reducible to an NP-complete problem with many efficient quantum solutions.

✓ Correct Answer:

The correct answer is 2) There is no known reduction from CSP in braid groups to HSP, and braid groups lack non-trivial finite subgroups, making HSP-based quantum attacks inapplicable..

📚 Reference Text:

Ω(√ N)queries to solve the HSP. The classical cases in which HSP is easy are the cases in which Ghas only a polynomial number of subgroups, allowing brute-force for the function fon all subgroups. We provide a survey of results regarding the complexity of quantum algo- rithms for solving HSP in various group platforms. We also provide info rmation on the relationship between HSP and other computational problems . These re- sults provide insight into potential platforms for quantum safe cry ptography, when the underlying hard problem is reducible to HSP. 2 Group-based Cryptography Group-based cryptographycould be shown to be post-quantum if the underlying security problem is NP-complete or unsolvable; ﬁrstly, we need to an alyze the problem’s equivalence to HSP, then analyze the applicability of Grover ’s search problem. Cryptanalysis based on a reduction to solving HSP creates some ob- stacles as the groups under consideration below are mostly inﬁnite a nd do not have an eﬃcient algorithm for HSP. In the following cryptosystems a connection to HSP can assist in the analysis of security. For example in [2] a practical cryptanalysis of WalnutDSA was propos ed, a post-quantum cryptosystem using the conjugacy search prob lem (CSP) over braidgroupsthatwassubmittedtotheNISTcompetitionin2017 [3]. Ithasbeen argued since the braid group does not contain any non-trivial ﬁnite subgroups, there does not seem to be any viable way to connect CSP with HSP. It has been shown there is no reduction connection between the CSP and HSP, [ 4,5]. As for analysis via Grover’s algorithm [6], it has been mentioned that a major ity of the time for signature veriﬁcation in WalnutDSA is repeated E-Multiplicatio ns. There are alternative group-theoretic problems and classes of gr oups which havebeenproposedforpost-quantumcryptography.Forexam ple,theﬁrstproto- cryptosystembasedongroupswasproposedbyWagner-Magyar ikin[7]forwhich the wordchoiceproblemwashard.LateronFlores-Kahrobaei-Kob erdaproposed right-angled Artin groups for various other cryptographic proto cols [8], [9]. Eick and Kahrobaei proposed Polycyclic groups, using the Conjugacy S earch Prob- lem [10] for cryptography. Later on Gryak-Kahrobaei wrote a su rvey and pro- posed other group-theoretic problems for consideration using po lycyclic groups [11]. Kahrobaei-Koupparis [12] proposed a post-quantum digital sig nature using polycyclic groups. Kahrobaei-Khan proposed a public-key cryptos ystem using polycyclic groups [13]. Habeeb-Kahrobaei-Koupparis-Shpilrainpro posed the use of a semigroup of matrices with a semidirect product structure [14]. Thompson groups have been considered by Shpilrain-Ushakov base d on the Decomposition Search Problem [15]. Hyperbolic groups have been pro posed by Chatterji-Kahrobaei-Luusingpropertiesofsubgroupdistortion andtheGeodesic Length Problem [16]. Free metabelian groups have been proposed ba sed on the Subgroup Membership Search Problem by Shpilrain-Zapata [17]. Kah robaei- Shpilirain proposed Free nilpotent p-groups for a semidirect produc t public key cryptosystem[18].LineargroupswereproposedbyBaumslag-Fine -Xu[19].Grig- orchuk groups, have been proposed by [20]. Groups of matrices, for a Homomor- phic Encryption scheme were proposed by Grigoriev-Ponomarenko [21]. 3 Relation of HSP to Other Computational Problems Many computational problems are special cases of the HSP; an eﬃc ient algo- rithm for HSP over a certain group implies an eﬃcient algorithm for som e other computational problem. It is important to note that one method of

Question 850 multiple-choice

In quantum computing, the implementation of multi-qubit gates like ZZZ(γ) and ZZZZ(γ) requires precise control over system parameters to optimize speed and hardware efficiency. The quantum speed limit (QSL) defines the minimum time necessary to perform such gates, and control schemes can affect both QSL and gate fidelity. Which of the following statements correctly characterizes the quantum speed limit (QSL) for ZZZ(γ) and ZZZZ(γ) gates when γ > 0, considering phase-based control schemes in scalable quantum processors? 1) The QSL for these gates significantly increases as γ becomes larger than zero. 2) The QSL for ZZZ(γ) and ZZZZ(γ) is highly sensitive to the ratio Δ₁max/Ω₁max. 3) The QSL for these gates vanishes for all values of γ greater than zero. 4) The QSL is minimized only in adiabatic protocols compared to phase-based schemes. 5) The QSL decreases markedly for γ = π/2 due to the gates becoming non-entangling. 6) The QSL for both gates remains nearly constant for γ > 0 and is independent of Δ₁max/Ω₁max. 7) The QSL is always maximized when local operations are directly supported by the control scheme.

✓ Correct Answer:

The correct answer is 6) The QSL for both gates remains nearly constant for γ > 0 and is independent of Δ₁max/Ω₁max..

📚 Reference Text:

advantageous as it requires fewer hardware and controlresources. The fact that using ϕ ↑(t) as the only time-dependent con- trol ﬁeld sufﬁces for implementing ZZZ( γ) or ZZZZ( γ) originates from considering the gates’ phase-shifted versionsof Eq. ( 9). In detail, since the states |↓↓↓/angbracketright and|↓↓↓↓/angbracketright are the only states among the 8 or 16 basis states of ZZZ( γ)o r ZZZZ( γ) that technically require nonzero /Omega1 ↓(t) andϕ↓(t)i n order to be phase conﬁgurable, we simply avoid this require-ment by considering the gates’ phase-shifted version. For theremaining 7 or 15 basis states, which all have at least oneatom initially in the |↑/angbracketrightstate, the phase ϕ ↑(t) together with a constant /Omega1↑(t)=/Omega1maxis sufﬁcient to correctly adjust all phases. The “phase” conﬁguration thus represents a hardware-efﬁcient control scheme to realize fast ZZZ( γ) and ZZZZ( γ) gates. Our results furthermore reveal that the “sequential” con- ﬁguration, which was recently introduced in Ref. [ 65] and designed to implement high-ﬁdelity ZZZZ( γ) gates, seems not to be ideal when it comes to gate time as its conﬁguration-speciﬁc QSL is roughly twice as long as the QSLs for the otherconﬁgurations. However, it should be noted that a comparisonof the protocol from Ref. [ 65] with protocols at the QSL is not a fair comparison. On the one hand, the control scheme ofRef. [ 65] is based on adiabaticity—a regime that we are far away from in our numerical calculations. On the other hand,while the parameter γis a tunable variable in the adiabatic control scheme, the results in Figs. 3(d) and 3(e) are only valid for γ=π/4. If gates with a different γare required, one explicitly needs to optimize control ﬁelds for that purpose.While it is beyond the scope of this paper to examine whetherthere exists an analytical control scheme with conﬁgurable γ at the QSL, in the following, we nevertheless provide an anal-ysis of the QSLs for ZZZ( γ) and ZZZZ( γ) beyond γ=π/4. In detail, after having identiﬁed the “phase” conﬁguration asthe most reliable conﬁguration to determine a gate’s QSL, weprovide the QSLs for other γvalues in Fig. 4(b). We ﬁnd the QSLs to be almost constant for γ> 0 and only zero for γ=0, in which case ZZZ( γ) and ZZZZ( γ) coincide with the identity operation. Interestingly, we do not observe a decreasein the QSLs for γ=π/2 in which case the constraint gates are no longer entangling, cf. Fig. 4(a), and should therefore theoretically be implementable with local operations only. Wesuspect that we do not see a decrease of the QSLs sincewe do not consider any control ﬁelds for local operations inHamiltonian ( 1) and thus need to implement the local gates by means of the Rydberg levels. We moreover analyze the dependence of the QSLs on the ratio/Delta1 max//Omega1maxin Fig. 4(c). Surprisingly, we observe the QSLs for both ZZZ( γ) and ZZZZ( γ) to be independent on this ratio and ﬁnd /Delta1max=0 to be a viable option. In general, we observe that the QSLs for ZZZ( γ) and

Question 851 multiple-choice

Quantum computing leverages multi-level quantum systems known as qudits, such as qutrits, to offer greater computational power and flexibility beyond conventional two-level qubits. The quantum Fourier transform (QFT) is a pivotal operation for many quantum algorithms and requires specialized circuit decomposition techniques when implemented on hybrid qudit systems. Which method enables the implementation of generalized Hadamard and controlled-phase gates for hybrid qudit quantum Fourier transform circuits in Nuclear Magnetic Resonance (NMR) quantum emulators? 1) Optical pumping of spin states 2) Microwave-driven excitonic transitions 3) Superconducting Josephson junctions 4) Ion trap motional sidebands 5) Electron spin resonance pulses 6) Selective rotations controlling multi-level quantum states 7) Photonic path entanglement

✓ Correct Answer:

The correct answer is 6) Selective rotations controlling multi-level quantum states.

📚 Reference Text:

Title: Implementation of the quantum Fourier transform on a hybrid qubit–qutrit NMR quantum emulator Year: 2015 Paper ID: de086ca9a35dec0afc3632641d38be87236866de Source: semantic-scholar URL: https://www.semanticscholar.org/paper/de086ca9a35dec0afc3632641d38be87236866de Abstract: The quantum Fourier transform (QFT) is a key ingredient of several quantum algorithms and a qudit-specific implementation of the QFT is hence an important step toward the realization of qudit-based quantum computers. This work develops a circuit decomposition of the QFT for hybrid qudits based on generalized Hadamard and generalized controlled-phase gates, which can be implemented using selective rotations in NMR. We experimentally implement the hybrid qudit QFT on an NMR quantum emulator, which uses four qubits to emulate a single qutrit coupled to two qubits.

Question 852 multiple-choice

In quantum information science, the ability to fully reconstruct quantum states or estimate the expectation values of arbitrary operators is critical for applications such as quantum tomography and quantum computing. Advanced measurement schemes leverage mathematical frameworks and group theory to maximize informational efficiency. Which approach combines positive-operator valued measures (POVMs) with unitary irreducible group representations and frame theory to enable the estimation of any operator's expectation value from experimental data? 1) Projective measurements constructed from commuting observables 2) Classical probability distributions mapped onto Hilbert spaces 3) Quantum error correction codes using stabilizer formalism 4) Informationally complete measurements designed via group-theoretic POVMs and frame theory 5) Nonlinear optical measurements using entangled photon pairs 6) Decoherence-free subspaces engineered through dynamical decoupling 7) Ancilla-assisted measurement protocols with weak values

✓ Correct Answer:

The correct answer is 4) Informationally complete measurements designed via group-theoretic POVMs and frame theory.

📚 Reference Text:

Title: Informationally complete measurements and group representation Year: 2003 Paper ID: c4f38d0b54bd0978fe70b52d5ffc48103114c6c4 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c4f38d0b54bd0978fe70b52d5ffc48103114c6c4 Abstract: Informationally complete measurements on a quantum system allow one to estimate the expectation value of any arbitrary operator by just averaging functions of the experimental outcomes. We show that such kinds of measurement can be achieved through positive-operator valued measures (POVMs) related to unitary irreducible representations of a group on the Hilbert space of the system. With the help of frame theory we provide a constructive way to evaluate the data-processing function for arbitrary operators.

Question 853 multiple-choice

Quantum algorithms tackling hidden subgroup problems have significant implications for cryptography and computational complexity, especially in relation to group structures and quantum sample manipulation. Subexponential-time solutions have been developed through advanced sieving techniques that exploit quantum superposition and entanglement. Which algorithm achieves subexponential time, space, and query complexity for the Dihedral Hidden Subgroup Problem by combining quantum states with shared least significant bits? 1) Grover's search algorithm 2) Shor's factoring algorithm 3) Kuperberg's sieve algorithm 4) Quantum Fourier transform algorithm 5) Collimation sieve algorithm 6) Quantum walk algorithm 7) Simon's algorithm

✓ Correct Answer:

The correct answer is 3) Kuperberg's sieve algorithm.

📚 Reference Text:

equivalent to LWE up to polynomial loss in parameters [22 ]. Remark 3.35. Whenfis the normalized indicator function of {}0, 1, an extrapolated dihedral coset sample reverts to a dihedral coset sample. Also, the study by Brakerski et al. [22 ] imposes a stronger condition than (3.33). 4 Other approaches to DHSP and DCP 4.1 Subexponential algorithms Theﬁrst row of (3.17 ) can be encoded as: ( ∣ ⟩∣ ⟩ ∣ ⟩∣ ⟩) (∣ ⟩ ∣ ⟩)()∑∑ += ⊗ +− Nωk ω kNωω1 20111 201 . kNαk Naαk kNαk Nak Measuring the ﬁrst register yields the samples of the form: ∣⟩ ( ∣ ⟩ ∣ ⟩ )=+ ω Ψ1 201 , k Nak(4.1) wherekis known from the measurement. Let=N 2tfor simplicity and =⌈ − ⌉mt 1. The idea behind the subexponential algorithm in the study by Kuperberg [13 ] is to combine states of the form (4.1). In particular, we see that ∣ ⟩∣ ⟩ (∣ ⟩∣ ⟩ ∣ ⟩∣ ⟩)=+ +−ω ΨΨ1 2Ψ0 Ψ1 . pq p q Naq pq (4.2) Ifpandqhave the same mjleast signi ﬁcant bits, then ±pqstrictly increases the number of least signi ﬁcant bitspandqshare. With su ﬃciently many samples of the form Ψpthat havemjcommon least signi ﬁcant bits, it is shown in the study by Kuperberg [13 ] that combining the states as in (4.2) produces enough states with ()+mj 1common least signi ﬁcant bits. Thus, sieving from enough samples at the outset, we eventually produce states of the form: ∣⟩ ( )∣⟩=+ −−Ψ 01 1 ,a 2t1 which are su ﬃcient to determine the parity of a. It is shown in the study by Kuperberg [13 ] that the afore- mentioned method yields an algorithm that requires()2ON logtime, space, and queries. In the study by Regev [12], a modi ﬁed algorithm is given that requires( )2ON N log log logtime and ()N poly log space. An abstract description of this sieving process is given in previous studies [13, §9] and further improvements and general- izations can be found in the study by Kuperberg [15], in particular the so-called “collimation sieve. ” Remark 4.3. In the study by Kuperberg [13 ], it is shown that HSP for D2treduces to determining the parity of a.10 Imin Chen and David Sun 4.2 Query complexity In the study by Ettinger and Høyer [29 ], it is shown that a polynomial number of HSP samples is su ﬃcient to recoverHausing exponential time post-processing. A related result in the study by Ettinger and Høyer [32] using di ﬀerent methods shows that the HSP problem in a general ﬁnite group has polynomial quantum query complexity. Transposing ↔ij , and applying a Hadamard gate to the state in (3.17 ), gives the state () ( ) () ( )() ()⎛ ⎝−⎞ ⎠⎛ ⎝⎜⎞ ⎠⎟=⎛ ⎝⎜++ −−⎞ ⎠⎟− −− −−− −−ωω ωωωω ω ω ωω ω ω1 211 111 211 11.Nαk Naαk Naαk NαkNαk Nak Nαk Nak Nαk Nak Nαk Nak(4.4) The probability of observing the ﬁrst row is (() ) ()+∕ =

Question 854 multiple-choice

Quantum state preparation techniques are crucial for efficiently encoding classical data distributions in quantum algorithms. Methods for expanding limited quantum encodings to exploit the full Hilbert space impact scalability and computational power. Which approach enables the transition from a unary-encoded quantum state to a binary basis, allowing full Hilbert space utilization and richer superpositions in quantum algorithms? 1) Measurement-based quantum error correction 2) Variational quantum eigensolver optimization 3) Classical post-processing of measurement results 4) Application of a unitary transformation using CNOT and SWAP gates with a clean ancilla register 5) Implementation of Grover’s search algorithm 6) Use of amplitude damping noise channels 7) Repeated quantum teleportation protocols

✓ Correct Answer:

The correct answer is 4) Application of a unitary transformation using CNOT and SWAP gates with a clean ancilla register.

📚 Reference Text:

20th international conference on pattern recognition (IEEE, 2010) pp. 2366–2369. [32] Z.Wang, A.C.Bovik, H.R.Sheikh,andE.P.Simoncelli, Image quality assessment: from error visibility to struc- tural similarity, IEEE transactions on image processing 13, 600 (2004). [33] T. Acharya and P.-S. Tsai, Computational foundations of image interpolation algorithms., Ubiquity 2007, 4 (2007). [34] P. W. Shor, Polynomial-time algorithms for prime factor- ization and discrete logarithms on a quantum computer, SIAM review 41, 303 (1999). [35] S. Ramos-Calderer, https://github.com/qiboteam/ qft-resampling (2022). [36] Brought to my attention in a post by Craig Gidney in https://quantumcomputing.stackexchange.com /questions/5526/garbage-free-reversible- binary-to-unary-decoder-construction . 8 Appendix A: Enhancing a unary uploading We illustrate how the QFT resampling algorithm can enhance uploading techniques using the amplitude dis- tribution in the unary basis presented in Ref. [16]. This proposal requires only nearest-neighbor connectiv- ity and employs gates that are well suited for imple- mentation on near-term quantum devices. The distri- bution is uploaded in the unary basis where only one qubit is in state j1iwhile the others remain at j0i, ie. j10000i;j01000i;:::j00001i. This, however, reduces the Hilbert space available for computation and requires lin- ear depth with the number of amplitudes needed to up- load, limiting its usability. While this basis helps with gate application and control of the device, one would want to exploit the exponential Hilbert space that qubits support. In order to reclaim the lost Hilbert space using QFT interpolation we propose the following. Upload a prob- ability distribution into the unary basis over a 2nqubit register. Then, perform a unitary change of basis from unary to binary basis [36], as detailed in Alg. 1 and il- lustrated with a small example in Fig. 5 for n= 3. After the change of basis, the quantum state contains nqubits storingthesuperpositionand 2nncleanancillasatstate j0i. This allows the implementation of the QFT interpo- lation algorithm using the clean ancilla register to encode the high frequency components. Now the full extent of the available Hilbert space is used to encode the interpo- lated probability distribution. Shown in Fig. 6 are the simulation results of a 16qubit total QFT interpolation with unary uploading. u0 n0 u1 n1 u2 n2 u3 m0 u4 m1 u5 m2 u6 m3 u7 m4 Figure 5. Circuit example that encodes an 8to3basis change from the unary basis to the binary one. The uregister depicts the unary register where u0is the most signiﬁcant qubitinunaryrepresentation. Ontheleft-handsidethelabels forthebinaryqubits n, andtheancillaqubits m, aredepicted. The ﬁnal SWAP gates can be neglected if we keep track of the qubit order and perform classical reshuﬄing.Algorithm 1: Unary to binary encoding. 1Unary2Binary( n) 2c Circuit(2n) 3Ensure q= 0 4fori 0ton1do 5 qq 2ni1 6 c.add(CNOT( q; qq)) 7 forj 1toqq1do 8 c.add(CNOT( q+j; q)) 9 forj 1toqq1do 10 c.add(SWAP( q+j; q+j+qq).controlled_by( q)) 11 q q+qq 12c.add(X( 2n1)) 13return Binary basis in qubits f2n2igwithi nto1 Figure 6. Comparison of a Gaussian probability distribution uploadedusingtheunarybasisto 16qubits P4thatisenlarged intoP16using the QFT interpolation scheme to span the full 216Hilbert space. Appendix B: Trace-distance for pure

Question 855 multiple-choice

Quantum algorithms have revolutionized computational approaches to group-theoretic problems, particularly in the domains of cryptography and computational complexity. Problems such as the Hidden Subgroup and Hidden Translation are central to understanding quantum speed-ups for certain algebraic structures. For solvable black-box groups with unique encoding and constant exponent, which of the following statements best describes the quantum time complexity to solve the Hidden Translation problem with constant error? 1) It requires classical exponential time proportional to |G|. 2) It can be solved in quantum time proportional to O(|G|^2). 3) It is solvable in quantum time O(log|G|). 4) Quantum time complexity is O(|G|·log|G|). 5) It requires O((log|G|)^2) quantum time. 6) It can be solved in quantum time O((log|G|)^{O(loglog|G|)}). 7) Quantum algorithms offer no advantage for this problem.

✓ Correct Answer:

The correct answer is 6) It can be solved in quantum time O((log|G|)^{O(loglog|G|)})..

📚 Reference Text:

of constant exponent. Theorem 4.17. LetGbe a solvable black-box group with unique encoding of length/lscriptand of constant exponent. Then Hidden Translation (G)can be solved in quantum time /lscriptO(1)(log|G|)O(loglog |G|).F u r t h e r m o r e ,t h e Hidden Subgroup can be solved in quantum time /lscriptO(1)(log|G|)O(loglog |G|)in groups Gfor which G/primehas constant exponent. Finally, an application of Theorem 4.13 with e=1a n d s=|G|to the derived series of a solvable group gives the following. Theorem 4.18. LetGbe a solvable black-box group with unique encoding of length/lscript.T h e n Hidden Subgroup (G)andHidden Translation (G)can be solved with constant error in quantum time /lscriptO(1)(log|G|)O(√ log|G|·loglog|G|). Acknowledgments. We wish to thank Mark Ettinger and Peter Høyer for shar- ing their knowledge and ideas about Hidden Translation with us, and Martin R¨otteler for several useful discussions on Hidden Subgroup . We are also grateful to Cris Moore, Csaba Schneider, and Yong Zhang for helpful discussions regarding a previous version of this article, and for pointing out reference [17]. Part of the work of HIDDEN TRANSLATION AND TRANSLATING COSET 23 the last three authors was done while visiting MSRI, Berkeley, and part of the work of the second author was done during visits at the CQT in Singapore. REFERENCES [1]D. Aharonov ,Quantum computation: A review , in Annual Review of Computational Physics, Vol. VI, World Scientiﬁc, Singapore, 1998, pp. 1–77. [2]D. Aharonov, A. Kitaev, and N. Nisan ,Quantum circuits with mixed states , in Proceedings of the 30th ACM Symposium on Theory of Computing, 1998, pp. 20–30. [3]D. Aharonov and A. Ta-Shma ,Adiabatic quantum state generation and statistical zero knowl- edge, in Proceedings of the 35th ACM Symposium on Theory of Computing, 2003, pp. 20– 29. [4]L. Babai, G. Cooperman, L. Finkelstein, E. Luks, and A. Seress ,Fast Monte Carlo algorithms for permutation groups , J. Comput. System Sci., 50 (1995), pp. 296–308. [5]L. Babai and E. Szemer ´edi,On the complexity of matrix group problems I, in Proceed- ings of the 25th Annual IEEE Symposium on Foundations of Computer Science, 1984,pp. 229–240. [6]D. Bacon, A. Childs, and W. van Dam ,From optimal measurement to eﬃcient quantum algorithms for the hidden subgroup problem over semidirect product groups , in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 469– 478. [7]H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf ,Quantum ﬁngerprinting ,P h y s .R e v . Lett., 87 (2001), 167902. [8]R. Beals ,Quantum computation of Fourier transforms over symmetric groups , in Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 48–53. [9]K. Cheung and M. Mosca ,Decomposing ﬁnite abelian groups , Quantum Inf. Comput., 1 (2001), pp. 26–32. [10]D. P. Chi, J. S. Kim, and S. Lee ,Quantum algorithms for the hidden subgroup problem on some semi-direct product groups by reduction to abelian cases , Phys. Lett. A, 359 (2006), pp. 114–116. [11]A. M. Childs and W. van Dam ,Quantum algorithms for algebraic problems , Rev. Modern

Question 856 multiple-choice

In advanced combinatorics and algebra, partially well-ordered sets and finitely supported sequences play a key role in proofs involving orderings and termination. Constructing sets from products of simpler sets and defining injective mappings between structured spaces are common techniques to transfer and analyze order properties. Which statement accurately describes the property established by Lemma 13 regarding the injective mapping v from a set W to the sequence space Vq? 1) The mapping v is surjective and reverses the partial order on Vq. 2) The mapping v guarantees that every ordered pair in W maps to unordered pairs in Vq. 3) The mapping v preserves the non-injectivity of elements between W and Vq. 4) If v(w) <$ v(w') holds in Vq, then w <$ w' holds in W, meaning the order is preserved under v. 5) The mapping v ensures that every infinite sequence in W lacks an ordered subsequence. 6) The mapping v defines a bijection that disregards the partial order structure. 7) The mapping v only applies to sequences with infinitely many non-zero entries.

✓ Correct Answer:

The correct answer is 4) If v(w) <$ v(w') holds in Vq, then w <$ w' holds in W, meaning the order is preserved under v..

📚 Reference Text:

m. Then it is easy to check that m < m
)/ = £ £i'"(l)> where w(l) e ^#, m(l) < m'. Thus, (u>0)/ = 0 or (w)f = X> ft = n' t (1 Vq(S). Let w = (j u ... ,j q)m, m = /ni/n 2, where m, = ai'a^ •••+T,m 2 = ([a H,ai2] - 1) • • • (K_,, a, v] - 1) + T (/,<•••< i2/). Put https://doi.org/10.1017/S1446788700002056 Published online by Cambridge University Press [23] Solution of a problem of Plotkin and Vovsi 351 where s, = (|n,-|, sgn (n,), s3i), II, if i s = i for some s,l < s <2l; 0 otherwise. Clearly, v is injective. LEMMA 13. Let w,w' eW and wv <$ w'v. Then w <$ w'. PROOF. Let w = (j lt ... ,j q)m, w' = (/,',... ,j' q)m', m = mim 2, m' = m\m' 2, where m t, m' { are of the forms (29)-(31). Then wv = ((j\,... ,;

Question 857 multiple-choice

Quantum spin liquids (QSLs) are phases of matter in honeycomb-lattice spin systems characterized by strong quantum fluctuations and the absence of conventional magnetic order. Advanced numerical and theoretical techniques are used to probe their properties and connection to experimental observations in candidate materials. Which experimental signature is most directly connected to the presence of fractionalized excitations in a Kitaev-like quantum spin liquid on the honeycomb lattice? 1) Sharp spin-wave modes in the inelastic neutron scattering spectrum 2) A broad scattering continuum observed in dynamical structure factor measurements 3) Enhanced ferromagnetic order at low temperatures 4) Uniform magnetic susceptibility across all temperatures 5) An absence of two-particle excitations in the transfer matrix spectra 6) Distinct Bragg peaks in elastic neutron scattering experiments 7) Suppressed entanglement entropy in numerical simulations

✓ Correct Answer:

The correct answer is 2) A broad scattering continuum observed in dynamical structure factor measurements.

📚 Reference Text:

Title: Quantum spin liquid signatures in Kitaev-like frustrated magnets Year: 2017 Paper ID: 962ed9fe76c4f92c670c8d3cd53b5347270b7a22 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/962ed9fe76c4f92c670c8d3cd53b5347270b7a22 Abstract: Motivated by recent experiments on $\alpha$-RuCl$_3$, we investigate a possible quantum spin liquid ground state of the honeycomb-lattice spin model with bond-dependent interactions. We consider the $K-\Gamma$ model, where $K$ and $\Gamma$ represent the Kitaev and symmetric-anisotropic interactions between spin-1/2 moments on the honeycomb lattice. Using the infinite density matrix renormalization group (iDMRG), we provide compelling evidence for the existence of quantum spin liquid phases in an extended region of the phase diagram. In particular, we use transfer matrix spectra to show the evolution of two-particle excitations with well-defined two-dimensional dispersion, which is a strong signature of quantum spin liquid. These results are compared with predictions from Majorana mean-field theory and used to infer the quasiparticle excitation spectra. Further, we compute the dynamical structure factor using finite size cluster computations and show that the results resemble the scattering continuum seen in neutron scattering experiments on $\alpha$-RuCl$_3$. We discuss these results in light of recent and future experiments.

Question 858 multiple-choice

In lattice gauge theory and topologically ordered quantum systems, holonomy around loops and gauge invariance play critical roles in determining the structure of ground states and their entanglement properties. Understanding how group elements assigned to lattice edges relate to physical equivalence is essential for analyzing quantum codes and topological phases. In a quantum double model defined on a lattice, which condition ensures that two group basis states within a rectangular region A are physically equivalent under gauge transformations acting only on the interior vertices of A? 1) Both states must assign the same group element to every edge in A. 2) Both states must have identical trivial holonomy on the boundary but different interior edge assignments. 3) Both states must have trivial holonomy on all closed loops in A and identical group element assignments on the boundary of A. 4) Both states must differ only by a global group translation applied to all edges in the lattice. 5) Both states must have nontrivial holonomy but matching boundary conditions. 6) Both states must be related by a gauge transformation acting on boundary vertices only. 7) Both states must have the same group element assignments on the complement of A.

✓ Correct Answer:

The correct answer is 3) Both states must have trivial holonomy on all closed loops in A and identical group element assignments on the boundary of A..

📚 Reference Text:

a rectangular subregion A, contained in a simply connected region of the surface. (See Figure 1.) The rectangular assumption could be relaxed, at the cost of more complicated exposition. Assume that some state |ψ/angbracketrighton the entire lattice is invariant under all Av,Bpoperators whose support intersects with A. That is, we assume Av|ψ/angbracketright=|ψ/angbracketright Bp|ψ/angbracketright=|ψ/angbracketright(32) for allAv,Bpoperators such that v∈V(A)or∂p∩E(A)/negationslash=∅. One can think of such a state as “zero-energy density” on the region A, where the energy density is given by the quantum double Hamiltonian. We will show that all such states |ψ/angbracketrighthave the same reduced density matrix ρAon the region A, and we will explicitly construct ρA. Recall that the “group basis” refers to the basis of states on the lattice of the form |g/angbracketright ≡/circlemultiplytext e∈E|ge/angbracketright. For a such a basis state, it’s helpful to introduce the notion of the holonomy around a loop. Given a group basis state and an oriented closed loop on the lattice with ﬁxed base point, the “holonomy” of the loop is the product of group elements on the edges, composed right to left in loop order. While the choice of base point may generally aﬀect the holonomy, in this proof we will only be concerned with the notion of loops with “trivial holonomy,” i.e. whose holonomy is the identity element. In that case, the base point does not matter, because the property of trivial holonomy does not depend on base point. We discuss holonomies and their properties at greater length in Section §3.3. Write|ψ/angbracketrightin the group basis. Because it is invariant under the Bpoperators intersecting A, the only product states in this expansion will be the ones with trivial holonomy on all closed loops in A. Therefore we can write |ψ/angbracketrightas |ψ/angbracketright=/summationdisplay gAwith trivial holonomies on A|gA/angbracketrightA|φgA/angbracketright¯A, (33) where the sum is over all assignments gA= (ge)e∈E(A)of group elements to edges in A, such that all of the holonomies on Aare trivial. The states |φgA/angbracketright¯Aare some set of states on ¯Adepending on gA. Note that the states |gA/angbracketrightAare orthonormal, while the states |φgA/angbracketright¯Aare not normalized and not necessarily orthogonal. Next we will need the following result: For any two product states of group elements |gA/angbracketright and|g/prime A/angbracketrightwith trivial holonomies on A, and with the same group elements on the boundary ∂A, there exists a gauge transformation acting only on the vertices V(A)◦in the interior of Awhich transforms|gA/angbracketrightAto|g/prime A/angbracketrightA. That is, there is some gauge transformation Uintsupported on the interior ofAsuch that Uint|gA/angbracketrightA=|g/prime A/angbracketrightA. (34) To build such a gauge transformation Uintthat takes|gA/angbracketrightAto|g/prime A/angbracketrightAconsider the internal ver- tices ofA, ordered from left to right, then top to bottom. Start at the top left internal vetex v0, and choose the unique g0such thatAv0(g0)|gA/angbracketrightAmatches|g/prime A/angbracketrightAon the entire top left plaque- tte. Now move one vertex rightward, to internal vertex v1, and choose the unique g1such that Av1(g1)Av0(g0)|gA/angbracketrightAmatches|g/prime A/angbracketrightAon the top left two plaquettes. Continue in this manner until we have found a gauge transformation on the top row of internal vertices such that both states match elements

Question 859 multiple-choice

Graded rings of theta functions play a crucial role in algebraic geometry and representation theory, often involving intricate decompositions and generation properties. Understanding how basic components generate larger subspaces is essential for analyzing their algebraic structure. In the context of a graded ring of theta functions where R₂ is shown to generate R₂m for all m > 0, which method is primarily employed to prove this generation property? 1) Direct computation of all elements in R₂m 2) Construction using spectral sequences 3) Application of the Noether normalization lemma 4) Inductive argument based on parity decomposition 5) Use of Hilbert's syzygy theorem 6) Representation via cohomological techniques 7) Verification using explicit determinant calculations

✓ Correct Answer:

The correct answer is 4) Inductive argument based on parity decomposition.

📚 Reference Text:

The reduction of A Let R = Θo°° Rm be the graded ring of theta functions associated to the matrix (stfi3). If m is a positive integer, let Rmtβ denote the sub- space of Rm consisting of Laurent series with coefficients in Θ. The 2m- parity decomposition Rm = Θ(r) R£\ rόeZj2mZ, induces a decomposition Rm,Φ = ®(r) R%], where R<£. - R£> Π Rm,.. Let Rm = Rm>Φ/^Rm>0, R = eo°°5m. Then Rm is a direct sum of 1-dimensional subspaces R^ = R%]ΦI'JίR{Z]β over k. There is an obvious map Rmtβ-*li\Xt9 Xϊ1] given by 2] «5/jZ7—>Σ ^iX1- The kernel is evidently Jl Rmtβ9 so Rm identifies with a subspace of ϊclXiyXi1]. We now calculate what this subspace is. Rather than taking Tj to be elements of Z/2mZ we shall take rs to be integers with — m 0 and equality holds only when t3 — 0 or when tό — —1 and rs = m. Thus the reduc- tion Σ δ/Z1, of θ only involves monomials with \id\ < m. In particular the monomials X1 appearing in a generator of R^ are just those for which the following conditions hold: ij = Tj whenever \τό\ < m %. =z +m whenever τ3 = m . PROPOSITION II.1.1. R2 generates R2m for all m > 0. Proof. It suffices to show that RYRm — Rm+ι for all m > 1. If Rm+i = Θ(r> -Bm+i is the 2(m + l)-parity decomposition of Rm+1 it suffices to construct a non-zero element of RιRm of arbitrary 2(m + l)-parity (r) = (r1? . 9rg), — (m + 1) < r^ < (m + 1). We argue by induction on ΣlrΛ an(i define numbers Cj and d^ by: Cj = 0, d^ = r,- if Ir^l < m Cj = 1 if r^ = m, —m, m + 1 ^. = m — 1, 1 — m, m if ry = m, — m, m + 1 . Let θc generate R[c) and θd generate R^\ The monomials X1 appear- ing in θcθd are just those for which: I. — m or m — 2 i^ = —m or 2 — m i^ = +(m + 1) or ±(m — 1) In particular, a generator $r of RZ\ι occurs as a component of θcθd. By induction it will suffice to show that every other θs occurring in θcθd has Σ \SA < Σ \TJ\ Now Xs must appear in θcθd. So by the above, either sά — rj9 or \rό\ > m and sό = ±(m — 2) or ±(m — 1). If (s) ^ (r), we are in this latter case for at least one

Question 860 multiple-choice

Renormalization in quantum field theory (QFT) is closely linked to advanced structures in algebra and complex analysis, providing a rigorous mathematical framework for handling divergences in physical theories. Understanding these connections is essential for grasping modern theoretical physics. In the algebraic approach to renormalization, which operation is used to define the Lie bracket between one-particle irreducible Feynman graphs? 1) Tensor product of graphs 2) Decomposition into primitive elements 3) Contraction of internal edges 4) Addition of external legs 5) Integration over parameter space 6) Merging of vertices 7) Insertion of one graph into another

✓ Correct Answer:

The correct answer is 7) Insertion of one graph into another.

📚 Reference Text:

Title: Renormalization in Quantum Field Theory and the Riemann–Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem Year: 1999 Paper ID: e1b10e80013766521e82bc56babaab63c2265847 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/e1b10e80013766521e82bc56babaab63c2265847 Abstract: Abstract:This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann–Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra which is commutative as an algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of . We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop where C is a small circle of complex dimensions around the integer dimension D of space-time. Our main result is that the renormalized theory is just the evaluation at z=D of the holomorphic part γ+ of the Birkhoff decomposition of γ. We begin to analyse the group G and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. The analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title.

Question 861 multiple-choice

The Dihedral Hidden Subgroup Problem (DHSP) is a fundamental challenge in quantum computing with significant implications for cryptography and efficient algorithm design. Advances in DHSP algorithms can affect the feasibility of quantum attacks on certain cryptographic schemes. Which characteristic distinguishes the improved quantum algorithm for the Dihedral Hidden Subgroup Problem from Kuperberg’s original algorithm? 1) It achieves subexponential time complexity with only polynomial space requirements. 2) It solves DHSP in polynomial time using exponential space. 3) It requires classical post-processing after quantum computation. 4) It is limited to abelian groups rather than dihedral groups. 5) It increases the quantum query complexity compared to previous algorithms. 6) It provides exponential speedup for factoring large integers. 7) It relies exclusively on classical computational resources.

✓ Correct Answer:

The correct answer is 1) It achieves subexponential time complexity with only polynomial space requirements..

📚 Reference Text:

Title: A Subexponential Time Algorithm for the Dihedral Hidden Subgroup Problem with Polynomial Space Year: 2004 Paper ID: f4028939ba8a7a7b706f7b97d8ea451f2732465c Source: semantic-scholar URL: https://www.semanticscholar.org/paper/f4028939ba8a7a7b706f7b97d8ea451f2732465c Abstract: In a recent paper, Kuperberg described the first subexponential time algorithm for solving the dihedral hidden subgroup problem. The space requirement of his algorithm is super-polynomial. We describe a modified algorithm whose running time is still subexponential and whose space requirement is only polynomial.

Question 862 multiple-choice

In mathematical physics, the group SO(1, d+1) plays a pivotal role in understanding symmetries of both de Sitter spacetime and conformal field theories. Its representation theory is essential for describing quantum fields and their transformations under spacetime symmetries. Which property uniquely characterizes SO(1, d+1) as both the isometry group of de Sitter spacetime and the conformal group of flat d-dimensional space? 1) It preserves volume in all geometries. 2) It implements transformations that preserve causal structure and conformal angles. 3) It acts only on vector fields of integer spin. 4) It is compact and admits only finite-dimensional representations. 5) It is the symmetry group of Minkowski space for all d. 6) It is the maximal subgroup of SO(2, d). 7) It preserves the topology but not the geometry of spacetime.

✓ Correct Answer:

The correct answer is 2) It implements transformations that preserve causal structure and conformal angles..

📚 Reference Text:

Title: A note on the representations of SO(1, d + 1) Year: 2021 Paper ID: c3fb0c1c097d8f56855ed339f9053829f8138a34 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c3fb0c1c097d8f56855ed339f9053829f8138a34 Abstract: $\text{SO}(1, d+1)$ is the isometry group of $(d+1)$-dimensional de Sitter spacetime $\text{dS}_{d+1}$ and the conformal group of $\mathbb{R}^{d}$. This note gives a pedagogical introduction to the representation theory of $\text{SO}(1, d+1)$, from the perspective of de Sitter quantum field theory and using tools from conformal field theory. Topics include (1) the construction and classification of all unitary irreducible representations (UIRs) of $\text{SO}(1,2)$ and $\text{SL}(2,\mathbb R)$, (2) the construction and classification of all UIRs of $\text{SO}(1,d+1)$ that describe integer-spin fields in $\text{dS}_{d+1}$, (3) a physical framework for understanding these UIRs, (4) the definition and derivation of Harish-Chandra group characters of $\text{SO}(1,d+1)$, and (5) a comparison between UIRs of $\text{SO}(1, d+1)$ and $\text{SO}(2,d)$.

Question 863 multiple-choice

In analytic number theory and quantum computing, the behavior of additive character sums over multiplicative subgroups of finite fields is crucial for analyzing algorithms and understanding group structure. Asymptotic bounds on such sums depend on the order of the subgroup relative to the characteristic of the field. Which of the following gives the tightest asymptotic bound for the sum q⁻¹ ∑_{z=0}^{q-1} χₜ(aᶻ), where χₜ is a nontrivial additive character and a ∈ F∗ₚ has multiplicative order q, specifically when q ≥ p^{2/3}? 1) O(p^{3/4}) 2) O(p^{1/8}q^{5/8}) 3) O(p^{1/4}q^{3/8}) 4) O(q^{1/2}) 5) O(p^{2/3}) 6) O(q^{3/4}) 7) O(p^{1/2})

✓ Correct Answer:

The correct answer is 7) O(p^{1/2}).

📚 Reference Text:

of the paper we require bounds on additive characters taken over multiplicative subgroups of F∗ p. Such sums are discussed in detail in [17]. The speciﬁc bound we require is the following. Theorem 9. Letχtbe a nontrivial additive character of Fpanda∈F∗ pan element of multiplicative order q. Then q−1/summationdisplay z=0χt(az)=⎧ ⎪⎨ ⎪⎩O(p1/2) ifq≥p2/3, O(p1/4q3/8)ifp1/2≤q≤p2/3, O(p1/8q5/8)ifp1/3≤q≤p1/2. See [17, section 2] for a proof. Note that in the body of the paper, we use Zpto denote the additive group of integers modulo pandZ∗ pto denote the multiplicative group of integers modulo p. Acknowledgments. We are grateful to Wim van Dam, Julia Kempe, Greg Kuperberg, Frederic Magniez, Martin R¨ otteler, and Miklos Santha for helpful conver- sations and to Sally Milius and Tracy Conrad for their support. REFERENCES [1]D. Bacon, A. Childs, and W. van Dam ,From optimal measurement to eﬃcient quantum algorithms for the hidden subgroup problem over semidirect product groups , in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 469–478. [2]R. Beals ,Quantum computation of Fourier transforms over symmetric groups , in Proceedings of the 29th Annual ACM Symposium on the Theory of Computing, 1997, pp. 48–53. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 958 MOORE, ROCKMORE, RUSSELL, AND SCHULMAN [3]E. Bernstein and U. Vazirani ,Quantum complexity theory (preliminary abstract) , in Pro- ceedings of the 25th Annual ACM Symposium on the Theory of Computing, 1993, pp. 11–20. [4]W. van Dam, S. Hallgren, and L. Ip ,Quantum algorithms for some hidden shift problems , in Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms,2003, pp. 489–498. [5]I. B. Damg ˚ard,On the randomness of Legendre and Jacobi sequences , in Advances in Cryptology—CRYPTO ’88, Lecture Notes in Comput. Sci. 403, Springer-Verlag, Berlin,1990, pp. 163–172. [6]M. Ettinger and P. Høyer ,On quantum algorithms for noncommutative hidden subgroups , Adv. in Appl. Math., 25 (2000), pp. 239–251. [7]M. Ettinger, P. Høyer, and E. Knill ,The quantum query complexity of the hidden subgroup problem is polynomial , Inform. Process. Lett. 91 (2004), pp. 43–48. [8]K. Friedl, G. Ivanyos, F. Magniez, M. Santha, and P. Sen ,Hidden translation and orbit coset in quantum computing , in Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 2003, pp. 1–9. [9]W. Fulton and J. Harris ,Representation Theory: A First Course , Grad. Texts in Math. 129, Springer-Verlag, London, 1991. [10]M. Grigni, L. J. Schulman, M. Vazirani, and U. Vazirani ,Quantum mechanical algorithms for the nonabelian hidden subgroup problem , in Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, 2001, pp. 68–74. [11]L. Hales and S. Hallgren ,Quantum Fourier sampling simpliﬁed , in Proceedings of the 31st Annual ACM Symposium on Theory of Computing, 1999, pp. 330–338. [12]L. Hales and S. Hallgren ,An improved quantum Fourier transform algorithm and applica- tions, in Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, 2000, pp. 515–525. [13]S. Hallgren, A. Russell, and A. Ta-Shma ,Normal subgroup reconstruction and quantum computation using group

Question 864 multiple-choice

Quantum algorithms often exploit group-theoretic structures, with the Quantum Fourier Transform (QFT) and representation theory playing a central role in problems like the Hidden Subgroup Problem (HSP). Measurements involving positive operator-valued measures (POVMs) and projections onto invariant subspaces are crucial for extracting information from quantum states associated with group actions. For a finite group G, a subgroup H ≤ G, and an irreducible representation ρ of G, which formula gives the rank of the projection operator onto vectors fixed by every ρ(h) for h in H? 1) rρ = ∑_{h∈H} χρ(h) 2) rρ = dρ × |H| 3) rρ = (1/|G|) ∑_{g∈G} χρ(g) 4) rρ = χρ(1) 5) rρ = |H| × χρ 6) rρ = (1/dρ) ∑_{h∈H} χρ(h) 7) rρ = (1/|H|) ∑_{h∈H} χρ(h)

✓ Correct Answer:

The correct answer is 7) rρ = (1/|H|) ∑_{h∈H} χρ(h).

📚 Reference Text:

3]) that QFTGisaunitary transformation in C|G|. Forasubgroup H≤Gandρ∈/hatwideG,deﬁneρ(H) :=1 |H|/summationtext h∈Hρ(h). ItfollowsfromSchur’slemma(see e.g.[Ser77,Chapter2,Proposition4])that ρ(H)isanorthogonal projectiontothesubspaceof Vρconsisting of vectors that are point-wise ﬁxed by every ρ(h),h∈H. Deﬁnerρ(H) :=rank(ρ(H)); thenrρ(H) = 1 |H|/summationtext h∈Hχρ(h),whereχρdenotesthecharacter of ρ. Noticethat rρ(H) =rρ∗(H). Foranysubset S≤G we deﬁne |S/a\}bracketri}ht:=1√ |S|/summationtext s∈S|s/a\}bracketri}htto be the uniform superposition over the elements of S. Thestandard methodof attacking theHSPin Gusing coset states [GSVV04]starts byformingtheuniform su perposition 1√ |G|/summationtext g∈G|g/a\}bracketri}ht|0/a\}bracketri}ht. It then queries fto get the superposition1√ |G|/summationtext g∈G|g/a\}bracketri}ht|f(g)/a\}bracketri}ht. Ignoring the second register thereduced stateontheﬁrstregister becomesthed ensity matrix σG H=1 |G|/summationtext g∈G|gH/a\}bracketri}ht/a\}bracketle{tgH|, thatis thereduced stateisauniform mixture overallleft coset sta tes ofHinG. Itcanbeeasily seenthat applying QFTGtoσG Hgivesusthedensitymatrix|H| |G|/circleplustext ρ∈/hatwideG/circleplustextdρ i=1|ρ,i/a\}bracketri}ht/a\}bracketle{tρ,i|⊗ρ∗(H),whereρ∗(H)operatesonthe space of column indices of ρ. When measuring this state, we obtain an irrep ρwith probabilitydρ|H|rρ(H) |G|. Conditioned on measuring ρwe obtain a uniform distribution 1/dρon the row indices. The reduced state on the space of column indices after having observed an irrep ρand a row index iis then given by the state 5 ρ∗(H)/rρ(H),andabasictaskfor ahidden subgroup ﬁndingalgorithm isho wtoextract information about Hfrom it. If the the hidden subgroup is the trivial subgroup {1}, the probability of measuring ρis given by the so-called Plancherel distribution P(ρ) :=d2 ρ |G|. This distribution will be useful to us later on in the proof o f the maintheorem. POVMsarethemost general waytoobtain classical informati on from quantum states [NC00]. Theele- mentsof aPOVM MinCnare positive operators Ei≥0whichhave tosatisfy the completeness condition/summationtext iEi=11n. If the state of the quantum system is given by the density mat rixσ, then the probability pito observe outcome labeled iis given bythe Bornrule pi= Tr(σEi). Thefollowing observation iscrucial for the HSP case: since the states σG Hare simultaneously block diagonal in the Fourier basis for a nyH≤G, the elements of any POVM Moperating on these states can without loss of generality be a ssumed to have thesameblock structure. Fromthis itisclear that anymeasu rement toidentify Hwithout lossofgenerality ﬁrst applies the quantum Fourier transform QFTGtoσG H, measures the name ρof an irrep, the index iof a row,and then measures thereduced state onthe column space o fρusing aPOVM MρinCdρ. ThisPOVM Mρmaydepend on ρbut is independent of i. Furthermore, Mρcan be assumed to be a frame, i.e., a collection Bρ:={(ab,b)}, whereb∈Cdρwith /bardblb/bardbl= 1and0≤ab≤1such that/summationtext b∈Bab|b/a\}bracketri}ht/a\}bracketle{tb|=11dρi.e. a frame is a POVM with rank one elements. Orthonormal bases are special cases of frames in which ab= 1for allb∈ Bρ. We can assume that the POVM on the column space is a frame because any POVM can be reﬁn ed to a frame such that for any quantum state, the probabilities according to the original POVM are certain sums, independent of the state measured, of probabilities according tothe frame. If the the hidden subgroup is the trivial subgroup {1}, after observing an irrep ρand a row index i, the reducedstateonthespaceofcolumnindicesof ρisthetotallymixedstate11dρ dρ. Theprobability ofobserving a vectorbin frameBρis given by the so-called natural distribution on Bρdeﬁned by N(b|ρ) :=d2 ρ |G|. This distribution will be useful tous later on inthe proof of the m aintheorem. TheabovedescriptionwasforsingleregisterquantumFouri ersampling. Fouriersamplingon kregisters can be deﬁned analogously.

Question 865 multiple-choice

Distributed quantum computing leverages multiple quantum processors connected by classical communication and is challenged by errors arising from both quantum operations and network delays. Dynamic quantum circuits, which employ adaptive measurements and qubit resets, offer promising strategies for reducing certain error sources in distributed quantum algorithms. In distributed quantum phase estimation (DQPE) using dynamic quantum circuits, which factor most directly contributes to depolarization error when implementing the Quantum Fourier Transform across networked nodes? 1) The number of ancilla qubits used in the phase estimation 2) The fidelity of single-qubit gate operations 3) The measurement error rate of the quantum hardware 4) The entanglement between qubits on different nodes 5) The execution time of two-qubit gates within a single node 6) The initial state preparation error of computational qubits 7) The classical communication time relative to the qubit coherence time

✓ Correct Answer:

The correct answer is 7) The classical communication time relative to the qubit coherence time.

📚 Reference Text:

40, 41, 42]. 4.1 Dynamic quantum circuits Dynamic quantum circuits involve using mid-circuit measurements, qubit reset, reuse, and single-qubit conditional quantum gates [43]. Here, we explore using dynamic quantum circuits instead of distributed quantum gates in the DQC [44]. 6 One of the examples of dynamic quantum circuits is when we have a quantum Fourier transform followed by measurement [44]. Let’s consider a quantum circuit for the QFT on three qubits in Fig. 3, followed by measurement of the qubits. q0 H• P(−π/2)• q1 • H P(−π/4)• P(−π/2) q2 • • H Figure 3: Quantum Fourier transform on three qubits, where P(θ) refers to the phase gate Fig. 4 shows the corresponding dynamic quantum circuit. q0 H q1 P(−π/2) H q2 P(−π/4) P(−π/2) H Figure 4: Dynamic quantum Fourier transform on three qubits Note that the dynamic QFT does not suffer from two-qubit gate errors; it suffers from depolarization errors ϵd= 1−e−t/T 2[45], where t is the classical communication time, and T2is the coherence time of the qubit. For T2= 50µs, QPUs separated by 10 m, speed of light in fiber c= 2×108m/s, we get ϵd≈ 10−3. We can now illustrate the application of distributed dynamic QFT to the quantum phase estimation. 4.2 Distributed dynamic quantum phase estimation The problem of quantum phase estimation (QPE) is to determine the eigenvalue of the unitary gate that acts as U|ψ⟩=e2iπy|ψ⟩, where |ψ⟩is the eigenstate of U, andyis a fraction corresponding to the eigenvalue. The quantum circuit used to implement QPE is shown in Fig. 5. We can derive the fraction yfrom the out- come of the measurement. For instance, if we obtain the measurement outcome y0y1y2, where each yi∈0,1, then the fraction yis calculated as ( y0y1y2)10/23. To illustrate, let’s examine the implementation of QPE for the unitary gate U=RX(π)⊗RY(π) acting on two qubits. One of the eigenvalues for this op- erator is −1 (y= 1/2), corresponding to a measurement output of ”100”. We prepare the eigenstate |ψ⟩, create the distributed QPE (DQPE) for the follow- ing qubit distributions: 1: [0, 1, 2], 2: [3, 4], and apply the dynamic QFT to 7 |0⟩H • H |0⟩H • P(−π/2)H |0⟩H • P(−π/4) P(−π/2)H |ψ⟩ U22U21U20 Figure 5: Distributed dynamic quantum phase estimation, where the qubits are in different nodes. The controlled unitary gates are applied using DQCS, followed by dynamic QFT the distributed quantum circuit. The probability of the measurement outcome ”100” is shown in Fig. 6, where we consider the locally controlled unitary gates and quantum Fourier transform for the QPE (red) (Fig. 3). For the DQPE (blue), we consider distributed two-qubit quantum gates and dynamic QFT without two-qubit quantum gates for the quantum Fourier transform (Fig. 4). The probability of measuring ”100” is shown in Fig. 6. For the DQPE (blue), we consider distributed two-qubit quantum gates and dynamic QFT without two-qubit quantum gates for the quantum Fourier transform (Fig. 4). We con- sider the gate error ϵg, depolarization error ϵd, and the measurement error ϵm. In Fig. 6, we

Question 866 multiple-choice

In cryptographic protocols, proofs of knowledge are used to ensure that a prover can convince a verifier that they possess a secret without revealing it. Security definitions often rely on the concepts of relation generators, extractors, and parameters such as knowledge error and runtime bounds. Which of the following most accurately characterizes the role of the knowledge error function κ(k) in a proof of knowledge framework? 1) It measures the efficiency of the extractor machine in finding witnesses. 2) It specifies the maximum runtime allowed for the verifier during protocol execution. 3) It quantifies the completeness property of the protocol for honest provers. 4) It determines the set of valid witness-commitment pairs for a given relation R. 5) It reflects the probability that a cheating prover can make the verifier accept without knowing a witness, as a function of the security parameter. 6) It indicates the probability that the extractor fails to output a witness, regardless of protocol parameters. 7) It bounds the number of possible commitments generated by the relation generator.

✓ Correct Answer:

The correct answer is 5) It reflects the probability that a cheating prover can make the verifier accept without knowing a witness, as a function of the security parameter..

📚 Reference Text:

some time bound P∗can only win with some bounded (small) probability. 132 IvanDamg˚ ardandEiichiroFujisaki For the above deﬁnition we need to consider a relation generator , algorithm R, that takes 1kas an input and produces as an output a description of a binary relationR.Bythiswemeanastringcontaininginformationissuﬃcienttosample eﬃciently random pairs ( c,w)∈Rand to test membership in Reﬃciently. We will useRto denote both this description and the relation itself. For instance, Rmight be the key generator for a commitment scheme, and we can think of the public key pkas deﬁning a relation consisting of pairs ( c,(s,r)) for which verifypk(c,s,r)=accept. A prover in our setting is a machine Pwho getsRas an input, outputs a stringcand ﬁnally conducts the interactive proof with a veriﬁer VusingR,c as common input. For convenience, we want to be able to refer to P’s strategy when executing the proof as a separate machine. Therefore, from Pwe deﬁne a machinePviewwhich starts in the state Pis in after having seen view view and having produced c.Pviewthen conducts the protocol with VfollowingP’s algorithm. The view viewcontains all inputs, messages exchanged and random coins so in particular cis determined by view. Note that the distribution of view istakenovertherandomcoinsofboth PandR.W eletzepsilonκ(k), if the expected running time ofMusingP∗ viewas oracle, is greater thanp(k) /epsilon1view,P∗−κ(k). This deﬁnition is motivated by the fact that the standard knowledge soundness requirement from [6] says that the extractor must run in expected time at most the bound in the deﬁnition. Note that in any situation where P∗has produced c having seen view, it is well deﬁned whether Mfails or not. Intuitively, one may think of this as saying that if Mdoes not fail in a given situation, then P∗really “mustknow”awitnessfor c,inordertomake Vacceptwithaprobabilitybetter thanκ(k). Deﬁnition 2. LetRbe a probabilistic polynomial time relation generator, and let a protocol (P,V), a knowledge extractor M, polynomial p()and knowledge error function κ()be given. Consider the following experiment with input k: R:=R(1k),c:=P∗(R)(this deﬁnes view view). We deﬁne the advantage of P∗,Advκ,M,p(P∗,k)as the probability that Mfails on the view generated by this experiment. This probability is taken over the random coins of R,P∗. Finally, for a relation R, we let, as usual, LR={c|∃w:(c,w)∈R}.W ea r e now ready to deﬁne computationally convincing proofs of knowledge: AStatistically-HidingIntegerCommitmentScheme 133 Deﬁnition 3. LetRbe a probabilistic polynomial time relation generator. We say that(P,V)is a computationally convincing proof of knowledge for R, with knowledge error κ(), failure probability ν()and time bound t(), if the following hold: Knowledge Completeness. The

Question 867 multiple-choice

Quantum computing has shown promise for integer factorization, especially through algorithms like Shor's and innovative hardware approaches. Recent experimental efforts have sought to maximize limited quantum resources and explore alternative computational paradigms. Which of the following statements accurately describes the largest number factored using Shor's algorithm in a quantum experiment to date? 1) The largest numbers factored are 51 and 85, using 8 qubits. 2) The largest number factored is 21, using an iterative protocol. 3) The largest number factored is 2048, requiring millions of qubits. 4) The largest number factored is 15, using a single quantum register. 5) The largest number factored is 51, using universal quantum gates. 6) The largest number factored is 100, using quantum adiabatic computing. 7) The largest number factored is 30, using nuclear magnetic resonance techniques.

✓ Correct Answer:

The correct answer is 1) The largest numbers factored are 51 and 85, using 8 qubits..

📚 Reference Text:

yet the experiments cannot be applied to a larger number. Martín-López et al. [21] re-utilized qubits to factor 21 with Shor’s algorithm by adopting an iterative protocol. Geller et al. [ 22] employed Fermat numbers and eight qubits to factor 51 and 85, which are the largest numbers to be factored by Shor’s algorithm so far. According to Gidney [ 23], there should be 2 k+1 qubits to factor k-bit integers. From the perspective of universal quantum computation, there is still a long way to go before it could be practical. Shor’s algorithm [ 3] transforms the problem of factoring a given number Ninto an equivalent problem: Given a random positive integer a, where a

Question 868 multiple-choice

In topological quantum systems, the mathematical framework of fusion categories, such as Tambara-Yamagami categories and unitary modular tensor categories (UMTCs), plays a key role in describing anyons and symmetry defects. These structures utilize group theory, bicharacters, and diagrammatic calculus to encode fusion and braiding rules for quantum particles. In the Z₃-based UMTC with a Z₂ symmetry that interchanges ω and ω*, which mathematical object ensures the consistency of fusion and associativity in the corresponding Tambara-Yamagami fusion category? 1) The Frobenius algebra of Z₃ 2) A nondegenerate symmetric bicharacter and a square root of 3 3) The quantum dimension of ω 4) The modular S-matrix for Z₃ anyons 5) The central charge of the associated conformal field theory 6) The R-symbols of the Z₂-crossed category 7) The group cohomology class of Z₃

✓ Correct Answer:

The correct answer is 2) A nondegenerate symmetric bicharacter and a square root of 3.

📚 Reference Text:

in the next section tha t braiding of the symmetry defects leads to non-abelian statistics. 2.3.2.Z3anyons with charge conjugation duality. The UMTC Z3has three anyon types, labeled by 1 ,ω, and ω∗, with fusion rules given by the group multiplication in Z3. The category has a Z2symmetry given by interchanging ωandω∗, which lifts to a categorical Z2-action and hence admits a Z2-crossed extension which we write (Z3)× Z2={1,ω,ω∗}⊕{ X1}. As a fusion category, (Z3)× Z2is a Tambara-Yamagami (TY) category, whose algebraic data is known and relatively easy to describe [23]. Brieﬂy, g iven a ﬁnite group Aa TY category has simple objects A⊔{m}and fusion rules a⊗b=ab,a⊗m=m,and m⊗m=/circleplusdisplay a∈Aa. A choice of F-symbols for TY categories with this fusion are determined b y a choice of nondegenerate symmetric bicharacter χ:A×A→C×and a choice of square root of |A|. One can check that in order to correspond to a unitary Z2-crossed BFC the F-symbols are determined by a choice of a primitive cube root of unity. F or calculations in the next section we choose χto take the value ξ=e2πi/3on the non-identity diagonal elements of Z3×Z3and¯ξon the off-diagonal elements. ForA=Z3the fusion is given by ⊗ 1 ω ω∗X1 1 1 ω ω∗X1 ω ω ω∗1 X1 ω∗ω∗1 ω X1 X1X1X1X11⊕ω⊕ω∗ Formulas for Z2-crossed R-symbols were found for general Znanyons in [2]. 2.4. The G-crossed graphical calculus. Before we can apply UGxBFCs for computing we need to introduce the graphical calculus associated to th eir skeletonizations. The graph- ical calculus for defects generalizes the graphical calcul us for UMTCs in the sense that whenever the topological charges involved in a diagram are a ll anyons, we recover the graphical calculus of a UMTC. We refer the reader to [2, 8] for a review of the diagram- matic calculus for anyons. As with anyons, the consistency e quations that ensure the vari- ous categorical structures are compatible with one another are encoded in the topological properties of the diagrams. Here we point out only the parts o f the G-crossed graphical calculus that differ from the usual one for MTCs. Diagrams are read from bottom to top and interpreted as quant um mechanical processes between anyons and symmetry defects. For simplicity, in wha t follows we assume our the- ory is multiplicity-free and suppress the vertex labels. Re call the notationhagis shorthand forρh(ag), and ¯ g=g−1. 8 COLLEEN DELANEY AND ZHENGHAN WANG Trivalent vertices encode admissible splitting/fusion pr ocesses between anyons and de- fects, implicitly encoding the N-matrix entries Nagbg cgh. cghbh ag The F-symbols have the same diagrammatic deﬁnition as for anyons , except now the topological charges carry group element labels. ag bh ck egh dghk=∑ fhk/bracketleftig Fagbhck dghk/bracketrightig fhk,eghag bh ck fhk dghk As we saw above, the algebraic data of a GxBFC differs in three main ways: the R- symbols are extended to become the matrix entries of a G-braiding, and new symbols U andηarise. The G-crossed R-symbols are written in the same way as in a UMTC, although the y

Question 869 multiple-choice

Quantum algorithms are increasingly used to solve structured linear systems, such as those involving circulant matrices, by leveraging both quantum and classical computational techniques. Understanding the relationship between key parameters like condition number and truncation threshold is crucial for algorithmic efficiency and accuracy. For amplitude encoded input states in circulant linear system solvers employing hybrid quantum-classical algorithms, how does the minimum truncation threshold T scale with respect to the condition number κC? 1) T ∝ κC^2 2) T ∝ log κC 3) T ∝ κC^(2/3) 4) T ∝ κC^(1/2) 5) T ∈ o(κC) 6) T ∈ O(κC log κC) 7) T ∝ κC^0

✓ Correct Answer:

The correct answer is 3) T ∝ κC^(2/3).

📚 Reference Text:

logarithmic scale. Testing the performance of the hybrid algorithm on real quantum devices, we execute our hybrid algorithm on ibm-perth hardware. Due to the high consumption of quantum resource of the QFT subroutine on current quantum devices, we consider a smaller size of N= 23and set bas|0⟩⊗3. We set the shot number as 106. We find that on small systems, our algorithms can still produce results close to the values obtained by the matrix multiplication with the loss error not exceeding 0.05, as shown in Figure 5b. Lastly, we compare the condition number with the truncation threshold under different input states |b⟩. Setting the system size of N= 210, we evaluate on three different types of b, the zero state |010⟩, the GHZ state |010⟩+|110⟩/√ 2 and the amplitude encoded stateP1023 k=0k|k⟩/qP1023 k=0k2. We tune the parameter the parameter ξto adjust the condition number κC. Fixing a loss threshold υ= 10−2, we record the minimum truncation threshold Tthat achieves an MSE loss less than υas shown in Figure 5c. We note that from the slope of the log-log plot, we can obtain the exponent of κCwhen the truncation threshold Tis modeled as a polynomial function of κC. We note that for the zero and GHZ state, the exponent seems to converge to a small value, suggesting the T∈o(κC). Further, for the amplitude encoded state, we see that the slope is approximately2 3, indicating that T∝κ2/3 c. We note that our results all verify that the upper bound T∈O(κClogκC) provided by Proposition 4, however, we have not found a vector b that saturates this upper bound, hence providing no empirical suggestions of the tightness of the proposition. VIII. CONCLUSION This work explores an efficient strategy to solve circulant linear systems. Two different approaches are given—a hybrid quantum-classical algorithm and a sample and query access based quantum-inspired classical algorithm. As banded circulant systems can be expanded as a matrix polynomial of the cyclic permutation matrix Q, we adapt and modify the CQS approach proposed by Huang et al. [7] to convert the linear systems problem into a classical convex optimization problem with lower dimensions. In replacement of iteratively growing Ansatz trees using heuristical gradient expansion strategies, we utilize the properties of circulant systems and only explore the direct sum of two Krylov subspaces KT(Q,b) and KT(Q−1,b) to obtain our solution. The circulant matrix provides various properties that can provide convenience for solving such systems. For the hybrid algorithm, we make use of the efficient implementation of the powers of the cyclic permutation matrix Qdue to its eigendecomposition by QFT, as well as the availability of a tensor product decomposition of the diagonalized matrix Λ. For the quantum-inspired algorithm, we make use of the fact the Qevaluates to a simple shift in query indexes when applied to vector b. Such properties allow our algorithm to operate under lower time complexities compared to algorithms that solve generalized linear systems of equations. Further, we show the relevancy of the hybrid algorithm by discussing the PromiseBQP

Question 870 multiple-choice

Quantum simulations with ultracold atoms in optical lattices enable exploration of exotic phases of matter and spacetime symmetries. One such system, the Poincaré crystal, incorporates both discrete spacetime translations and Lorentz transformations in its symmetry group. In the context of simulating a Poincaré crystal using cold atoms, which specific property must the system's dispersion relation possess to ensure invariance under discrete Lorentz transformations? 1) It must be linear in momentum for all energy values. 2) It must depend solely on the lattice site index modulo the lattice size. 3) It must allow arbitrary long-range tunneling without symmetry constraints. 4) It must be symmetric under time reversal for every momentum state. 5) It must scale with the tunneling strength J in a non-monotonic manner. 6) It must be Lorentz-invariant under the discrete group of spacetime transformations. 7) It must exhibit periodic revivals only in the presence of Gaussian white noise.

✓ Correct Answer:

The correct answer is 6) It must be Lorentz-invariant under the discrete group of spacetime transformations..

📚 Reference Text:

1D QQFT, tak- ing a few hundred ms, if we consider Li atoms in an optical lattice of size L=32 with a laser wavelength 1064 nm and re- coil energy ER=~=225:12kHz, using a tunneling strength J=0:01ER. This time scale is accessible to the cold atom experiments. k 2=(La)01234567891011121314151617181920212223242526272829303132 Ek 2=(LT)061215243030915122702712189303241521606211824339182127 TABLE I. A Lorentz-invariant dispersion relation. III. QUANTUM SIMULATION OF POINCAR ´E CRYSTAL A. Hamiltonian engineering of Poincar ´e crystal In order to benchmark our QQFT-based quantum simula- tion protocol, we ﬁrst apply this scheme to quantum engi- neering of Poincar ´e crystal, which is deﬁned by the pres- ence of discrete spacetime translation and Lorentz symme- tries48,49. This quantum spacetime crystal has a robust col- lapse and revival dynamics, i.e., crystallization in spacetime, as opposed to the typical ballistic expansion for the standard quantum wave evolution. Its single-particle propagator forms a Lorentz-invariant spacetime pattern. The crystallization in spacetime is related to the discrete Lorentz symmetry, unlike the Floquet time crystal whose robustness relies on the lo- calization and the topological -mode50–54. However, the di-rect experimental realization of Poincar ´e crystal is extremely challenging for its requirement on delicate long-range cou- plings49. In detail, the Poincar ´e crystal is a lattice model whose sym- metry group is the discrete Poincar ´e group, which consists of discrete spatial translations, temporal translations and Lorentz transformations. The ratio of spatial to temporal translation periods is a=T=c=p 21, where 2 is an integer and cis the invariant speed in the Lorentz transformation (e.g. the speed of light). andcdetermine an elementary Lorentz transformation, which in the 1 +1-dimensional spacetime is p 21=c cp 21 ! : (26) For convenience, we choose the units of time and length to beTanda, respectively. In these new units, the elementary 6 (a) (b) (c) FIG. 3. QQFT based quantum engineering of Poincar ´e crystal. We simulate quantum walk of a single particle on the Poincar ´e crystal. The particle is initialized at n=0. The color indexes jGnsj2, the probability of ﬁnding the particle at the lattice site nat the time sT. Here, we consider di erent strength of the Gaussian white noise: (a) =0, (b)=5103, and (c)=2102. The system size is chosen to be L=33. Lorentz matrix becomes an integer matrix, reading L= 1 21 ! : (27) Lgenerates a cyclic group ˜L=fLnjn2Zg, dubbed the dis- crete Lorentz group, while the discrete Poincar ´e group is the direct product of ˜Land spacetime translations. The quantum theory of Poincar ´e crystal is exactly the unitary representation of the discrete Poincar ´e group. We consider a Poincar ´e crystal with periodic boundary con- dition, whose Hamiltonian in momentum space reads ˆHPC=X kEkˆcy kˆck: (28) The Lorentz symmetry requires that the dispersion relation must be invariant under L. This is a strong restriction. The method of ﬁnding Lorentz-invariant dispersion relations was discussed in Ref. [49]. An example exists as =2 and L=33 (the Lorentz symmetry requires L, i.e., the lattice size, to take some speciﬁc integers). The corresponding dispersion rela- tion is

Question 871 multiple-choice

In quantum computing, the quantum Fourier transform (QFT) is composed of quantum gates that facilitate efficient computation in algorithms such as Shor's. The entanglement properties and tensor network representations of QFT operators are crucial for understanding their classical simulability. Which property most directly enables the core QFT operator (Qn) to be efficiently represented as a matrix product operator (MPO) with constant bond dimension, even as the number of qubits increases? 1) Exponential decay of its Schmidt coefficients beyond the first 2) Uniform distribution of its Schmidt coefficients 3) Maximal operator entanglement across all partitions 4) Implementation solely with Hadamard gates 5) Linear growth of bond dimension with system size 6) Absence of controlled phase gates in its circuit 7) Bit-reversal operation preceding its application

✓ Correct Answer:

The correct answer is 1) Exponential decay of its Schmidt coefficients beyond the first.

📚 Reference Text:

H=1√ 2/parenleftbigg11 1−1/parenrightbigg P(θ)=/parenleftbigg10 0eiθ/parenrightbiggRnQn FIG. 1. A 4-qubit example of the QFT circuit, i.e., Fnwith n= 4, consisting of two parts, Qnand the bit-reversal operation Rn. Qncan be decomposed into Hadamard gates and controlled phase gates, with deﬁnitions of each given at the bottom. operator Schmidt decomposition in the context of unitary operators. Deﬁnition 1.— The operator Schmidt decomposition of a unitary operator Uacting on a bipartite space A⊗B with dimension dim (A⊗B)=dim(A)dim(B)=Nis a decomposition of the form U=√ Nχ−1/summationdisplay k=0σkAk⊗Bk,( 4 ) where {Ak}and{Bk}are operators acting on Aand B, respectively, each satisfying Tr (A† kAk/prime)=δk,k/primeand Tr(B† kBk/prime)=δk,k/prime, with δk,k/primebeing the Kronecker delta. The constant χis the Schmidt rank and is at most min(dim(A)2, dim(B)2). The set of non-negative numbers {σk}are called “Schmidt coeﬃcients,” which in our case are organized in descending order and are l2normalized, i.e.,σ0≥σ1≥··· and/summationtextσ2 k=1. Equivalently, the operator Schmidt decomposition can be interpreted as the Schmidt decomposition of the vectorized operator |U/angbracketright/angbracketright. The resulting state has norm/radicalbig Tr(U†U)=√ Nand thus there is an extra√ Nscalar in Eq.(4). The Schmidt coeﬃcients represent the operator’s expressibility as an MPO, which is shown in more detail in Sec. III. Additionally, it is also a rough estimation of the operator’s entanglement generation on a state, which is discussed in Sec. II Cand Appendix E. The exact operator Schmidt decomposition of Fnwas calculated in Refs. [ 7,8], where both papers showed that the Schmidt coeﬃcients are uniform. This suggests that Fnis maximally entangled and its MPO representation requires a bond dimension growing exponentially withthe number of qubits. Through simulations, however, we noticed the core part of F n, the operator Qn, has Schmidt coeﬃcients decaying exponentially quickly. This was pre- viously and independently reported in Ref. [ 9], where theauthors gave numerical evidence that the Schmidt coeﬃ- cients decay very fast and the operator can be compressed to a constant-bond-dimension MPO with exponentially small error. However, no analytic analysis on the upper bounds of the Schmidt coeﬃcients was given, and the underlying reasons for this low-entanglement property were not studied. In this paper, we ﬁll this gap by proving the following theorem. Theorem 1.— (Exponential decay of the QFT’s Schmidt coeﬃcients). Consider the quantum Fourier transform Qn deﬁned in Eq. (3)with the order of qubits at the output reversed. We partition the Hilbert space into Awith qubits 1t o jandBwith qubits j+1t o n, so that Qnhas the operator Schmidt decomposition Qn=√ 2nmin(2j,2n−j)−1/summationdisplay k=0σk n,jAk n,j⊗Bk n,j.( 5 ) It then follows that as n→∞ ,f o r k≥2,σk n,jhas upper bound ˜σk j=lim n→∞σk n,j≤1√ kexp/parenleftbigg −2k+1 2log/parenleftbigg4k+4 eπ/parenrightbigg/parenrightbigg . (6) Additionally, the following identity holds: k0/summationdisplay k=0/parenleftBig σk n+1,j/parenrightBig2 ≤k0/summationdisplay k=0/parenleftBig σk n,j/parenrightBig2 (7) for any n,j,a n d k0. We prove this theorem in Sec. II B. Roughly, in words, Eq.(6)says as the qubit number goes to inﬁnity, each Schmidt coeﬃcient is still bounded by a quantity decay- ing (more than) exponentially fast. Note this bound is also independent of j, the partition of the system. Equation

Question 872 multiple-choice

Quantum algorithms for the hidden subgroup problem (HSP) often require sophisticated measurement strategies, and their complexity is closely related to classical computational problems such as subset sum. The interplay between quantum sampling and classical hardness has significant implications for algorithmic efficiency and group-theoretic cryptography. Which statement best describes the resource requirements for determining the least significant bit of the hidden subgroup in the dihedral hidden subgroup problem using quantum measurements? 1) It requires as many hidden subgroup states as computing the full hidden subgroup answer. 2) It can be accomplished with substantially fewer quantum states than solving the entire problem. 3) It is efficiently solvable using classical algorithms alone. 4) It only involves measuring a single qubit in the computational basis. 5) It bypasses the need for quantum sampling from subset sum solutions. 6) It can be reduced to a trivial problem for non-abelian groups. 7) It is independent of the density parameter in subset sum quantum sampling.

✓ Correct Answer:

The correct answer is 1) It requires as many hidden subgroup states as computing the full hidden subgroup answer..

📚 Reference Text:

determining just the least signiﬁcant bit of the answer requires the same number of hidden subgroup states. Finally, we considered the implementation of the measure- ment by a quantum circuit and found that it is closely related to the subset sum problem. We considered the special (but well-motivated) case in which the measurement ﬁrst determines the block x, and then performs the opti- malpovm within that block. For a given number of copies of the hidden 20 DAVE BACON, ANDREW M. CHILDS, AND WIM VAN DAM subgroup state, we showed that this measurement can be implemented eﬃ- ciently if and only if one can quantum sample from subset sum solutions at the corresponding density. Many open questions remain. First, given Kuperberg’s subexponential time algorithm for the dhsp usingk= 2O(√logN)copies of the hidden sub- group state, as well as the Flaxman-Przydatek algorithm for ﬁnding a subset sum solution at the corresponding density, it seems promising to look for an implementation of the optimal measurement by quantum sampling from subset sum solutions at this density. As an intermediate step, it would be interesting simply to ﬁnd an algorithm for producing a subset sum solution uniformly at random. Of course, implementing the optimal measurement at the Kuperberg den- sity would not yield an improvement over previous algorithms, so it would be more interesting to ﬁnd an implementation of the optimal measurement at still lower density. If one pursues the natural strategy of ﬁrst measuring the blockx, then our results show that this approach is at least as hard as solving the subset sum problem, in which case one could simply apply Regev’s Theorem 5. However, as discussed above, one could consider imple- menting the optimal measurement without ﬁrst measuring x, which might give an improved algorithm for the dhsp without yielding an algorithm for subset sum. Finally, it would interesting to consider optimal measurements for other non-abelian hidden subgroup problems. Can such measurements be imple- mented eﬃciently in any of the cases where eﬃcient algorithms are already known? Or more ambitiously, can any new quantum speedups be found in this way? Presumably the subset sum problem has some analog for other groups, and such a problem might be interesting in its own right. acknowledgments We thank Carlos Mochon and Frank Verstraete for helpful discussions of Theorem 1, and Abie Flaxman for correspondence regarding the algorithm in [16] and an earlier version thereof. DB acknowledges support from NSF grant No. EIA-0086038 and the Santa Fe Institute. AMC received support from the National Science Foun- dation under Grant No. EIA-0086038, and was also supported in part by the Fannie and John Hertz Foundation, by the Department of Energy under cooperative research agreement DE-FC02-94ER40818, and by the National Security Agency and Advanced Research and Development Activity under Army Research Oﬃce contract DAAD19-01-1-0656. WvD was supported in part by the Cambridge-MIT Institute, the Department of Energy un- der cooperative research agreement DE-FC02-94ER40818, and the Disrup- tive Technology Oﬃce (DTO) under Army Research Oﬃce (ARO) contract number W911NF-04-R-0009.

Question 873 multiple-choice

Quantum state tomography is a fundamental technique used to reconstruct quantum states and benchmark quantum processors. Accurate estimation of state fidelity and modeling of errors such as decoherence are crucial for evaluating quantum hardware performance. Which method most efficiently reduces the number of projective measurements required for quantum state estimation from an exponential to a linear scaling with the number of qubits, while still estimating the pure state most closely aligned with observed reduced density matrices? 1) Using the 3n projective measurement approach instead of traditional 4^n tomography 2) Applying a depolarizing channel to the experimental state before measurement 3) Maximizing fidelity between experimental and theoretical states without measurement reduction 4) Employing ancillary qubits for universal unitary corrections 5) Measuring only in the Z basis for all qubits 6) Averaging measurement counts across all Pauli bases 7) Performing full quantum process tomography for each input state

✓ Correct Answer:

The correct answer is 1) Using the 3n projective measurement approach instead of traditional 4^n tomography.

📚 Reference Text:

state j01iin ibmq santiago ,measuring a single qubit. Basis 0 1 IX 4103 4089 IY 7923 269 IZ 2230 1866 XI 162 3934 Y I 2274 1822 ZI 4115 4077 TABLEII.Experimental results for the basis input state j01iin ibmq santiago ,measuring twoqubits. Basis 00 01 10 11 XX 202 120 3981 3889 XY 231 182 7266 513 XZ 158 147 4272 3615 Y X 2299 2166 1874 1853 Y Y 4350 101 3587 154 Y Z 2433 2016 2023 1720 ZX 2213 2052 2039 1888 ZY 4078 111 3875 128 ZZ 2204 1996 2167 1825 Rev.Mex.Fis. 68031402 4 C.ALLENDE, A. FONSECA DE OLIVEIRA, AND E. BUKSMAN TABLEIII.Optimalpure state jÁiforibmq santiago . jÁi00jÁi10jÁi11 0.565 0.559 0.586 0.450+ 0.031i -0.480 - 0.006i -0.026 - 0.453i 0.485 - 0.010i 0.534 + 0.021i -0.498 0.488 - 0.063i -0.414 + 0.015 0.015 + 0.450i FIGURE3.Unitarycorrection Ucfor anyinput state. 4.1. Experimental resultswith the isotropicindex PerformingtheQSTonthisdataweobtainedtheexperimen- tal reconstructed density matrix for ibmqsantiago, which wasdeemed too largeto ﬁt this article. Thetheoreticaloutputstatecorrespondingtoinput j01iis jÃi01=j¡ij +yi,where j+yi=j0i+ij1i. Forthisstate, the pure state jÁi01with the maximum alignment with ½exp isfound: jÁi01=2 6640:524 0:003 + 0 :484i ¡0:519 + 0 :064i ¡0:057¡0:463i3 775: (15) Finally the unitary rotation matrix U01is calculated which maps the state jÁ01ito the output theoretical state jÃi01. Similarly, the pure states jÁiijobtained for the other inputbasis states are presented in the TableIII. This procedure was performed for the four resulting states, obtaining the four unitary transformations. Using two ancillary qubits, a unitary transformation Ucis deﬁned that controls which element of the unitary set should be applied, so that the model works for any input state. This model is shownin Fig. 3. In order to model the decoherence, a total Depolarizing Chanel is applied on the theoretical state jÃi, and the result- ingstate ½Dch=!0I 4+ (1 + !0)jÃihÃj; (16) iscomparedwiththetransformationoftheexperimentalstate under U:½0=U½expUy. Although the decoherence error generated by the computer does not have to ﬁt with a cho- sen Depolarizing Channel, this simple error model does not changethealignmentofthestatesandcanestimatetheeffect ofdecoherence. FIGURE4.Fidelities Ec. ( 17) for the method using the isotropic index. F1inwhite bar, F2in gray bar,and F3inblack bar. TABLEIV.Fidelities, Eq. ( 17) of the second method, using 4n projections,for ibmq santiago . Input state F1 F2 F3 j00i 0.87940 0.88928 0.99213 j01i 0.90030 0.90551 0.99236 j10i 0.87870 0.88957 0.99553 j11i 0.88217 0.89308 0.99589 Finally in Table IV and as shown in Fig. 4, the results are compared. This comparison is done by calculating the ﬁdelity in each step F1; F2; F3deﬁnedas: F1=Fid(½exp;jÃi); F2=Fid(½0;jÃi); F3=Fid(½0; ½Dch); (17) where jÃiis the expected theoretical state and ½0= U½expUy. 4.2. Experimental resultsUsing only 3nprojections Incontrastwiththepreviousmethod,nowwelookforanes- timate of the state jÁi, closer to the state ½expwithout the explicit representation of the density matrix, using only 3n projective measurements. Those projections are chosen so that the unitary matrix brings the state ½expcloser to the par- ticular basis state, and as far as possible from the other three basisstates. Theideaistolookforthepurestatethatismost aligned with each of the reduced one qubit matrices. With this purpose we measured the projections X; Y;

Question 874 multiple-choice

In quantum information theory, understanding how random quantum states behave under measurement is crucial for analyzing high-dimensional systems and cryptographic protocols. Powerful mathematical tools allow us to bound the probability that certain observables deviate significantly from their mean values. When analyzing the probability that the expectation value of a Hermitian operator on a Haar-random quantum state deviates from its average by more than ε, which mathematical result most strongly suppresses the probability of large deviations as the dimension increases? 1) Jensen's inequality 2) The Central Limit Theorem 3) Chebyshev’s inequality using the variance 4) Markov’s inequality applied to moments 5) Levy’s lemma, utilizing the Lipschitz constant and dimension 6) The law of large numbers for quantum observables 7) The Schmidt decomposition theorem

✓ Correct Answer:

The correct answer is 5) Levy’s lemma, utilizing the Lipschitz constant and dimension.

📚 Reference Text:

threshold ε>0decays exponentially with the number of qubits n: Prob |ψ⟩∼µH(|⟨ψ|P|ψ⟩|≥ε) = Prob |ψ⟩∼µH/parenleftig |⟨ψ|P|ψ⟩|2≥ε2/parenrightig ≤1 (2n+ 1)ε2. (260) The same holds if the expected value is taken with respect to a state 2-design. Another concentration inequality that can be particularly useful when analyzing the averages over Haar-random states is Levy’s lemma. Lemma 53 (Levy’s lemma [85]) .Consider the set S2d−1:={v∈Cd:∥v∥2= 1}. Letf:S2d−1→ Rbe a function satisfying the Lipschitz condition |f(v)−f(w)|≤L∥v−w∥2. For allε≥0, we have the probability bound: Prob |ϕ⟩∼µH/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(ϕ)−E |ψ⟩∼µH[f(ψ)]/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥ε/bracketrightbigg ≤2 exp/parenleftbigg −2dε2 9π3L2/parenrightbigg . (261) Using Levy’s lemma, we can improve the dependence on the number of qubits nto a double exponential form compared to what we obtained by a simple application of Markov’s inequality in Eq.(260) , as shown in the next example. However, this double exponential form comes at the cost of requiring the state to be drawn from the Haar measure rather than a state 2-design. Example 54. LetO∈Herm/parenleftbig Cd/parenrightbig be a Hermitian operator. For all ε≥0, we have: Prob |ψ⟩∼µH/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle⟨ψ|O|ψ⟩−Tr(O) d/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥ε/parenrightbigg ≤2 exp/parenleftigg −dε2 18π3∥O∥2 ∞/parenrightigg . (262) In particular, if Ois a Pauli string P∈{I,X,Y,Z}⊗n\{I⊗n}, we have: Prob |ψ⟩∼µH(|⟨ψ|P|ψ⟩|≥ε)≤2 exp/parenleftbigg −2nε2 18π3/parenrightbigg . (263) Proof. To apply Levy’s lemma, we consider the function f(ψ) =⟨ψ|O|ψ⟩and compute its expected value and Lipschitz constant. First, we observe that E |ψ⟩∼µH[f(ψ)] = E |ψ⟩∼µH[⟨ψ|O|ψ⟩] = Tr/bracketleftbigg OE |ψ⟩∼µH|ψ⟩⟨ψ|/bracketrightbigg =Tr(O) d. (264) Next, we determine the Lipschitz constant. We have |f(v)−f(w)|=|Tr [(|u⟩⟨u|−|v⟩⟨v|)O]|≤∥O∥∞∥|u⟩⟨u|−|v⟩⟨v|∥1 (265) 35 where we used the matrix H¨ older inequality. We then observe that: ∥|u⟩⟨u|−|v⟩⟨v|∥1= 2/radicalig 1−|⟨u|v⟩|2. (266) This can be seen by considering Q:=|u⟩⟨u|−|v⟩⟨v|. The Hermitian matrix Qhas a rank of at most 2, which means it can have at most two non-zero eigenvalues denoted as λ1andλ2. Since the trace of Qis zero, we have λ2=−λ1. Additionally, we know that Tr/parenleftbig Q2/parenrightbig =λ2 1+λ2 2= 2λ2 1, and Tr/parenleftbig Q2/parenrightbig = 2(1−|⟨u|v⟩|2). Therefore, we can conclude that λ1=/radicalig 1−|⟨u|v⟩|2. The 1-norm of Qis given by∥Q∥1=|λ1|+|λ2|, which simplifies to ∥Q∥1= 2/radicalig 1−|⟨u|v⟩|2. Moreover, we have the following inequality: 2/radicalig 1−|⟨u|v⟩|2≤2/radicalbig 2−2|⟨u|v⟩|≤2/radicalbig 2−2 Re(⟨u|v⟩) = 2∥u−v∥2. (267) This implies that ∥|u⟩⟨u|−|v⟩⟨v|∥1≤2∥u−v∥2. (268) Hence, we have|f(v)−f(w)|≤2∥O∥∞∥u−v∥2. By applying Levy’s lemma with the Lipschitz constant of fbeing 2∥O∥∞, we can conclude. Combining Markov’s inequality with higher moments provides a useful trick for computing bounds. By using Prob (X≥ε)≤Prob/parenleftbig Xk≥εk/parenrightbig ≤E[Xk] εk, we can derive bounds based on the k-th moment of a random variable. This approach is particularly useful in situations involving k-unitary designs, where we can often compute moments only up to a given order. Another useful inequality when averaging over the full Haar measure is the following. If α>0, then Prob [X≥ε]≤Prob [exp(αX)≥exp(αε)]≤exp(−αε)E[exp(αX)]. To illustrate an applica- tion of this, let us consider an example provided also in the notes of Richard Kueng [45]. Example 55. Consider a n-qubit state|ϕ⟩∈Cdwithd= 2n. If we randomly pick a state |ψ⟩ from the Haar measure, the probability that the overlap between |ψ⟩and|ϕ⟩is larger than ε >0 decays double exponentially with the number of qubits n: Prob |ψ⟩∼µH/bracketleftig |⟨ψ|ϕ⟩|2≥ε/bracketrightig ≤2

Question 875 multiple-choice

In the study of Hopf algebras and rigid braided tensor categories, structures such as trace forms, ribbon elements, and balancing play crucial roles in quantum algebra and topology. These features enable the construction of invariants and encode symmetries of the underlying algebraic systems. Which statement accurately describes the condition required for balancing to exist in the double D of a Hopf algebra? 1) Balancing exists if the antipode S is involutive, i.e., S² = id. 2) Balancing exists precisely when the canonical trace µ_D is degenerate. 3) Balancing occurs if every group-like element in A is central. 4) Balancing is possible only when the ribbon element v equals the identity. 5) Balancing exists if the group-like element ω satisfies ω² = 1. 6) Balancing exists when one can find square roots of certain group-like elements in A⁰ and A, specifically k = √α (√a)⁻¹. 7) Balancing is guaranteed if the category is modular and semisimple.

✓ Correct Answer:

The correct answer is 6) Balancing exists when one can find square roots of certain group-like elements in A⁰ and A, specifically k = √α (√a)⁻¹..

📚 Reference Text:

ces of Proposition 6 for doubles. It is easy to see that anelement of a Hopf algebra w∈DwithS2(y) =wyw−1provides us with an isomorphism C(D)/tildewidest−→Co(D) :λ/ma√sto→λ⊳w. HereCo(D) denotes as in [Dr1] the space of traces on D. Given the two canonical elements in (2.19) we wish to compute the respective traces for µD. To this end deﬁne the following linear forms on D(A): χ:D(A).−1−−→Ao⊗A<,>−→k ˆχ:D(A).−1−−→Ao⊗A1⊗S−1−−−→Ao⊗A<,>−−→k (2.45) The forms on D(A) and the canonical elements are now related as follows. Proposition 7 χ=µD⊳u , ˆχ=ω−1µD⊳ˆu , (2.46) and both χandˆχare nondegenerate traces onD(A). Proof :From previous considerations it is clear that µD⊳uandµD⊳ˆuare traces. The rest of the proof are straightforward computations: (µD⊳u)(λ.y) = (µD⊳u)(y.λ) =/summationdisplay iµD(S(fi).ei·y.λ) =/summationdisplay iµD(S2(λ)·S(fi).ei·y) =/summationdisplay i(S2(λ)S(fi))(x)µ(S(y)S(ei)) 17 =/summationdisplay ifi(x ↽ S2(λ))µ(S(y)ei) =µ/parenleftBig S(y)(x ↽ S2(λ))/parenrightBig =µ/parenleftBig S(y)βl(S2(λ))/parenrightBig =βrβl(S2(λ))(S(y)) by(2.30)=S(λ)(S(y)) =λ(y) =χ(λ.y). Similarly, (µD⊳ˆu)(λ.y) =/summationdisplay iµD(fi·λ.y·ei) =/summationdisplay ifi·λ(x)µ(S(yei)) =µ/parenleftBig S(y(λ ⇀ x))/parenrightBig by(2.28)=ωµ(ay(λ ⇀ x)) =ωµ(ayβr(λ)) =ωβrβr(λ)(ay) by(2.31)=ωS(a⊲λ)(ay) =ωS(λ)(y) =ωˆχ(λ.y). Nondegeneracy of χand ˆχfollow directly from nondegeneracy of µD. ✷ 4.) Balancing in Doubles : In a rigid BTC any object Xis isomorphic to its double conjugate X∨∨. Yet the only isomorphism that is a priori canonical is between Xand X∨∨∨∨. Thus in addition to the usual axioms deﬁning a BTC one often require s the existence of a⊗−natural isomorphism of the functor X→X∨∨to the identity, which squares to the canonical one from X→X∨∨∨∨to the identity. For the representation category of a quasitriangular Hopf algebra Dthis is equivalent to the existence of a group like element kwith: k∈G(D), g=k2,andS2(y) =kyk−1. (2.47) It is clear that a balancing does not have exist since often gis not a square in G(D). If it does it is unique up to multiplication with central, group like elements of o rder two, i.e., elements in Σ( D) :=2G(D)∩Z(D). Equivalently, we can consider the corresponding element v:=u·k−1= ˆu−1·k. Inspecting (2.21) it is easily veriﬁed that vdeﬁnes a balancing elements iﬀ v∈Z(D), S(v) =v,andM=v⊗v∆(v−1) (2.48) From these conditions ǫ(v) = 1 and v2=zfollow. This point of view has been introduced in [RT0] where vis called a ribbon element . In their context the eigenvalue of vin an irreducible representation yields the framing anomalies of colored link . For a double D(A) the existence of a balancing can be phrased as a property of the moduli ofA. Proposition 8 18 1.kis a balancing of D(A)if and only if k=√α.(√a)−1(2.49) where√α∈G(Ao),√a∈G(A)square to αandarespectively and S2=ad∗(√α)◦ad(√a−1)onA. (2.50) 2. To a given balancing we associate the number νdeﬁned by ν−1=χ(k) =√α(√a). (2.51) IfD(A)admits a balancing νis a root of unity, ν4=ωand ν2does not depend on the choice of balancing. Proof :From Lemma 1 and (2.47) we infer that khas to be a product of group likes of the special subalgebras. By deﬁnition ofdoubles (2.50) isthe sameas S2(y) =kyk−1∀y∈A. The inverse adjoint of (2.50) yields the same equation on Aoand thereby (2.47). For part 2.)we remark that two balancings kandk′are related by k′=k.RwhereR=ρ.r∈ Σ(D(A))∼=Σ(Ao)⊕Σ(A). Then χ(k′)·χ(k)−1=√α(r)ρ(√a)ρ(r) which is of order two since ρandrare. ✷ In particular, the last statement implies that

Question 876 multiple-choice

Quantum computation leverages principles such as superposition and entanglement to solve problems more efficiently than classical computers. Algorithms in quantum computing often showcase dramatic speedups for specific computational tasks. Which quantum algorithm is specifically known for providing an exponential speedup in factoring large integers, thereby posing a threat to classical cryptographic systems? 1) Grover's algorithm 2) Deutsch-Jozsa algorithm 3) Quantum walk algorithm 4) Simon's algorithm 5) HHL (Harrow-Hassidim-Lloyd) algorithm 6) Hamiltonian simulation algorithm 7) Shor's algorithm

✓ Correct Answer:

The correct answer is 7) Shor's algorithm.

📚 Reference Text:

Title: Quantum Computing: Lecture Notes Year: 2015 Paper ID: f02ebadcd13a9c62bd02e9c7dad2687c637e9675 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/f02ebadcd13a9c62bd02e9c7dad2687c637e9675 Abstract: This is a set of lecture notes suitable for a Master's course on quantum computation and information from the perspective of theoretical computer science. The first version was written in 2011, with many extensions and improvements in subsequent years. The first 10 chapters cover the circuit model and the main quantum algorithms (Deutsch-Jozsa, Simon, Shor, Hidden Subgroup Problem, Grover, quantum walks, Hamiltonian simulation and HHL). They are followed by 2 chapters about complexity, 4 chapters about distributed ("Alice and Bob") settings, and a final chapter about quantum error correction. Appendices A and B give a brief introduction to the required linear algebra and some other mathematical and computer science background. All chapters come with exercises, with some hints provided in Appendix C.

Question 877 multiple-choice

In representation theory, Lie groups and Lie algebras play a central role in understanding symmetries and their mathematical structure. The adjoint representation is fundamental in both mathematics and physics, especially for semisimple Lie groups and their associated algebras. Which of the following statements accurately describes the adjoint representation of a Lie group G and its Lie algebra g? 1) It maps group elements to scalars via the trace function. 2) It sends group elements to permutation matrices acting on the underlying set. 3) It represents group elements as diagonal matrices acting on a vector space of the same dimension. 4) It is defined only for abelian groups and assigns each element a phase factor. 5) It acts by projecting elements of the Lie algebra onto the Cartan subalgebra. 6) It is the representation induced by the determinant function on the group. 7) It maps elements of the Lie algebra g to linear transformations given by the Lie bracket, ad_X = [X, Y], and for the group, Ad_g = gXg⁻¹.

✓ Correct Answer:

The correct answer is 7) It maps elements of the Lie algebra g to linear transformations given by the Lie bracket, ad_X = [X, Y], and for the group, Ad_g = gXg⁻¹..

📚 Reference Text:

of Vinto invariant subspaces: observe that for all H2su(2),H 1+1 Hmaps an- tisymmetric vectors to antisymmetric vectors and maps symmetric vectors to symmetric vectors, so V=Sym2(C2)Alt2(C2)as Lie algebra represen- tations. practically useful. From a theoretical stand-point there is an “upward direction” where we can take simple group rep- resentations and build more complex ones. More interest- ingly, however, is the “downward direction” where we can try to understand very complex representations of a group in terms of simpler ones. From a practical perspective, we will see that change of representations are extremely useful in certain types of quantum neural networks [63, 64]. 1. Constructing new representations, changing representation Note that in what follows we will construct new rep- resentations from two given representations R1;R2. The generalization to R1;:::;Rmis straightforward. We start by noting that up to this point we have cor- responding deﬁnitions at the level of the Lie group Gand the Lie algebra g. Really, we should have Theorem 2 and Corollary 1 at the front of our mind for each and every one of these: the deﬁnition at the group level induces the algebradeﬁnition, andthedeﬁnitionatthealgebralevelin- duces the group deﬁnition. It is straightforward (albeit notpedagogically wise) to show this equivalence using these theorems, often using some version of the Taylor series ex- pansion computational trick (Eq. (17)). First, let us recall that every Lie group Ghas a particu- larly important representation associated to it.11 Deﬁnition 8. LetGbe a Lie group with associated Lie algebra g. Theadjoint representation Ad:G!GL(g)is the mapg7!Adgdeﬁned by Adg(X) =gXg1;for allX2g: Using Theorem 2, this induces the adjoint representation ofg, ad:g!L(g)is the map X7!adX= [X;]given by adX(Y) = [X;Y ];for allY2g: Since quantum science primarily works with complex representations, it is common to write, as we do in this article, the adjoint representation as V=gwhile meaning the complexiﬁed Lie algebra V=g C. Next, we let us highlight the remarkable fact that that given two representations of a group, their direct sum is also a representation. Deﬁnition 9. LetR1;R2be representations of a Lie group Gacting on vector space V1;V2. Thedirect sum R1R2 is a representation of Gacting onV1V2deﬁned by (R1R2(g)) (v1;v2) = (R1(g)v1;R2(g)v2);for allg2G: Likewise, if r1;r2are representations of a Lie algebra gon V1;V2, thedirect sumr1r2is a representation of gacting onV1V2deﬁned by (r1r2(X)) (v1;v2) = (r1(X)v1;r2(X)v2);for allX2g: It can be useful to think of the direct sum in block diag- onal form : the representation R1R2acting onV1V2 can be written as (R1R2)(g) = R1(g) 0 0R2(g) ;for allg2G: Of course, a similar form holds for r1r2. Notice that a direct sum decomposition appeared in both of our key examples in Examples 4, 5, 7, 8. Block diagonalization will be of the utmost importance throughout this paper. We ﬁnally note that one can also take the direct sum of the 11Liegroupstructureishighlydeterminedbythisrepresentation, and it plays a central role in the classiﬁcation of so-called semisimple Lie groups. 21 same representation, i.e., R1R1, in which case we say thatR1has multiplicity of two, and we write (R1R1)(g) =112 R1(g);for allg2G: For instance,

Question 878 multiple-choice

Quantum algorithms for algebraic problems often depend on structural properties of the underlying algebra, such as the existence of special term operations. The Hidden Kernel Problem (HKP) generalizes the Hidden Subgroup Problem and its computational complexity varies with these properties. Which structural property of a finite algebra is directly linked to the existence of efficient quantum algorithms for solving HKP, and is possessed by all groups? 1) Having a Maltsev term 2) Having a cube term 3) Being idempotent 4) Being a member of a congruence distributive variety 5) Satisfying the absorption property 6) Having a near unanimity term 7) Being commutative

✓ Correct Answer:

The correct answer is 2) Having a cube term.

📚 Reference Text:

4.2, Theorem 5.1, and Theorem 5.2provides a completeclassiﬁcation ofthe quan- tumandclassicalalgorithmiccomplexityofHKP (Bn),where Bis a 2-element algebra. This classiﬁcation is summarized in the theorembelow andin Figure 5. Theorem6.1. LetBbea2-element algebra. 1. IfMPT∞ 0/√re⌋edesequalB,MPT∞ 1/√re⌋edesequalB, orDM /√re⌋edesequalB, then thereexistbothclassicalandquantumpolynomial-time algorithmsfor HKP(Bn). 2. IfAP /√re⌋edesequalB/√re⌋edesequal A, then aquantum polynomial-timeal- gorithm solving HKP(Bn)exists. Furthermore, no clas- sical polynomial-timealgorithm for HKP(Bn)exists. 3. IfB/√re⌋edesequal/logicalandtext.1,B/√re⌋edesequal/logicalortext.1, orB/√re⌋edesequal Uthen no quantum or clas- sical polynomial-timealgorithm for HKP(Bn)exists. Thisclassiﬁcationcanbeseenasabroadextensionofthe results of Simon [ 19], which are included in item 2 of the theoremwhenB=Z2. There are many open questions surrounding the Hidden KernelProblem,afewofwhichwedetailbelow.Theorems 6.1 LICS’20,July8–11,2020,Saarbrücken,Germany Ma/t_thew Moore and TaylorWalenczyk and5.1statethatbothclassicallyandusingaquantumalgo- rithm,HKP (A)iseitherin PorEXPwhenAisthepowerof a 2-elementalgebra. Question. Forarbitrary ﬁxed ﬁnite algebras A, precisely which complexity classes (quantumor classical) are param- eterized by HKP (A)? Turningthisproblemaround,wecaninsteadaskforalge- braic properties which enforce the existence of an eﬃcient quantumsolution to HKP (An). Question. Consider HKP (An), whereAis an arbitrary ﬁ- nite algebra. What structural conditions on Aensure that HKP(An)always admits a polynomial-time quantum solu- tion? In an algebraA, the term operation t(x1,...,xn)is said to be acube term if for every i∈ [n]there is a choice of u1,...,un∈ {x,/y.alt}withui=/y.altsuch that the equation t(u1,...,un)=xholds inA. The clone APhas a cube term given by x+/y.alt+zandDMhas a cube term given bymaj(x,/y.alt,z).Moregenerally,everygrouphasacubeterm given byx/y.alt−1z. Thestudyofcubetermsoriginatedwithalgebraicapproach to theConstraint Satisfaction Problem (CSP) in [2], and the existenceofacubetermfor Aisassociatedwithsomequite strong regularity conditions on the structure of powers of A. IfAhas a cube term, then subalgebrasof Anhave a gen- erating set whichisboundedby apolynomialin n.Thisap- pliesin particulartocongruencesof An.ItfollowsthatifA has a cube term, then counting arguments similar to those in theproofsofLemmas 5.5and5.7will notbesuﬃcient to ruleouttheexistenceofaquantumalgorithm.Thisleadsus to conjecturethe following. Conjecture. IfAhasacubeterm, then HKP(An)has an ef- ﬁcient quantum solution. Asmentionedabove,everygrouphasacubeterm.Aneﬃ- cientquantumsolutiontotheaboveconjecturewouldthere- fore restrict to an eﬃcient quantum solution to the hidden normalsubgroup problem. References [1] Dave Bacon, AndrewM Childs,and Wimvan Dam. 2005. From opti- malmeasurementtoeﬃcientquantumalgorithmsforthehiddensub- group problem over semidirect product groups. In 46th Annual IEEE Symposium onFoundationsofComputerScience(FOCS’05) . IEEE,469– 478. [2] JoelBerman,PawełIdziak,PetarMarković,RalphMcKenzie,Ma tthew Valeriote, and Ross Willard. 2010. Varieties with few subalge- bras of powers. Trans. Amer. Math. Soc. 362, 3 (2010), 1445–1473. h/t_tps://doi.org/10.1090/S0002-9947-09-04874-0[3] Aaron Denney,CristopherMoore,andAlexanderRussell.2010. Find- ing conjugate stabilizer subgroups in PSL (2; q) and related groups. Quantum Information& Computation 10,3 (2010),282–291. [4] Paul Adrien Maurice Dirac. 1939. A new notation for quantum me- chanics. In Mathematical Proceedings of the Cambridge Philosophical Society, Vol.35.Cambridge UniversityPress,416–418. [5] Dmitry Gavinsky. 2004. Quantum solution to the hidden subgroup problem for poly-near-Hamiltonian groups. Quantum Information & Computation 4,3(2004),229–235. [6] Sean Hallgren. 2007. Polynomial-time quantum algorithms for P ell’s equationandtheprincipalidealproblem. JournaloftheACM(JACM) 54,1(2007),4. [7] Gábor Ivanyos, FrédéricMagniez, and Miklos Santha. 2003. Eﬃcient quantum algorithms for some instances of the non-abelian hidden subgroup problem. International Journal of Foundations of Computer Science14,05(2003),723–739. [8] Gábor Ivanyos, Luc Sanselme, and Miklos Santha. 2007. An eﬃcient quantum algorithm for the hiddensubgroup problem in extraspecia l groups. In Annual Symposium onTheoretical Aspects ofComputerSci- ence.Springer,586–597. [9] Gábor Ivanyos, Luc Sanselme, and Miklos Santha. 2008. An eﬃcient quantum algorithmforthehiddensubgroup probleminnil-2groups. InLatin American Symposium on Theoretical Informatics . Springer, 759–771. [10] Bjarni Jónsson. 1967. Algebras whose congruence lattices are distributive. Math. Scand. 21

Question 879 multiple-choice

In quantum information theory and high-dimensional probability, the statistical distinguishability of quantum states is often analyzed using the trace norm (1-norm) between probability distributions derived from random orthonormal vectors. Concentration inequalities and probabilistic bounds are crucial for understanding the behavior of these distributions in large dimensions. Suppose S and T are probability distributions obtained by measuring quantum mixed states formed from two disjoint sets of random orthonormal vectors in complex d-dimensional space, with set sizes p and q and background size r. What is the asymptotic lower bound for the 1-norm difference between S and T with high probability in terms of p, q, and r? 1) O((p+q)/d) 2) Ω(√p/(p+r) + √q/(q+r)) 3) Θ((pq)/(d^2)) 4) O(1/d) 5) Θ((p+q)/r) 6) Ω((p−q)/d) 7) O(√(p+q)/d)

✓ Correct Answer:

The correct answer is 2) Ω(√p/(p+r) + √q/(q+r)).

📚 Reference Text:

thefollowing holds: Deﬁne α=dδ2−2log(p+q+r). Supposeα= Ω(1). Then, with probability at least 1−exp(−Ω(α))over the choice of W, /bardblS−T/bardbl1= Ω/parenleftbigg√p p+r+√q q+r/parenrightbigg . Proof: Generateaset W′={a′1,...,a′p}∪{b′1,...,b′q}∪{c′1,...,c′r}ofp+q+rrandom independent unit vectors in Cdasdescribed inSection 3.2. Let α′i j,β′i j,γ′i j,j={1,...,d}denote theGaussians used to generate therandom unit vectors a′i,b′i,c′irespectively. Then, a′i j=α′i j/summationtextd l=1|α′i l|2, b′i j=β′i j/summationtextd l=1|β′i l|2, c′i j=γ′i j/summationtextd l=1|γ′i l|2. ByFact3,withprobability atleast 1−2exp(−dδ2)over thechoice oftheGaussians, thenormalisation factor in the denominator of a given vector in W′is/radicalbig (1±ǫ)dwhereǫ=O(δ). LetE0be the event that the normalisation factors in the denominators of all vector s inW′are/radicalbig (1±ǫ)d. By the union bound on probabilities, E0occurs with probability at least 1−2exp(−α)over the choice of theGaussians. Sinceα= Ω(1),δ >10√ d. By Fact 2, for any w′i,w′j∈W′,i/ne}ationslash=j,|/an}bracketle{tw′i|w′j/an}bracketri}ht| ≤δ+10√ d<2δ with probability at least 1−2exp(−dδ2)over the choice of the Gaussians. Let E1denote the event that |/an}bracketle{tw′i|w′j/an}bracketri}ht|<2δfor allw′i,w′j∈W′,i/ne}ationslash=j. By the union bound on probabilities, E1occurs with probability at least 1−2exp(−α)over thechoice of the Gaussians. Using Fact 3 we see that for any ﬁxed coordinate j, with constant probability at least θover the choice of the Gaussians, each of the following three events occurs: p/summationdisplay i=1|α′i j|2>p+√p,q/summationdisplay i=1|β′i j|2r+1. Since these are independent events, all three of them hold at coordinatejsimultaneously with constant probabilityatleast θ3. Callsuchacoordinate jgood. LetE2denotetheeventthatmorethandθ3 2coordinates jaregood. ByFact5, E2occurs withprobability at least 1−exp(dθ3/8)over thechoice of the Gaussians. Now suppose that all three events E0,E1,E2occur. We Gram-Schmidt orthonormalise W′to get the random orthonormal set W={a1,...,ap} ∪ {b1,...,bq} ∪ {c1,...,cr}. LetS′,T′be the analogous probability vectors deﬁned withrespect to W′instead ofW. FromProposition 3,wesee that /bardbl|w/an}bracketri}ht/an}bracketle{tw| −|w′/an}bracketri}ht/an}bracketle{tw′|/bardbltr<20·δ·(p+q+r) =O((p+q+r)−1) for corresponding vectors w∈W,w′∈W′. Deﬁnedensity matrices σ=1 p+r/parenleftBiggp/summationdisplay i=1|ai/an}bracketri}ht/an}bracketle{tai|+r/summationdisplay i=1|ci/an}bracketri}ht/an}bracketle{tci|/parenrightBigg , σ′=1 p+r/parenleftBiggp/summationdisplay i=1|a′i/an}bracketri}ht/an}bracketle{ta′i|+r/summationdisplay i=1|c′i/an}bracketri}ht/an}bracketle{tc′i|/parenrightBigg , τ=1 q+r/parenleftBiggq/summationdisplay i=1|bi/an}bracketri}ht/an}bracketle{tbi|+r/summationdisplay i=1|ci/an}bracketri}ht/an}bracketle{tci|/parenrightBigg , τ′=1 q+r/parenleftBiggq/summationdisplay i=1|b′i/an}bracketri}ht/an}bracketle{tb′i|+r/summationdisplay i=1|c′i/an}bracketri}ht/an}bracketle{tc′i|/parenrightBigg . Then,S,S′,T,T′are the probability distributions got by measuring the stat esσ,σ′,τ,τ′in the standard basis ofCd. By triangle inequality, /bardblσ−σ′/bardbltr=O((p+q+r)−1)and/bardblτ−τ′/bardbltr=O((p+q+r)−1). Hence,/bardblS−S′/bardbl1=O((p+q+r)−1)and/bardblT−T′/bardbl1=O((p+q+r)−1). 16 For agoodcoordinatej |S′ j−T′ j|>p (p+r)(1+ǫ)d−q (q+r)(1−ǫ)d+r+1 (1+ǫ)d/parenleftbigg1 p+r−1 q+r/parenrightbigg +√p (p+r)(1+ǫ)d+√q (q+r)(1−ǫ)d =−2qǫ (q+r)(1−ǫ2)d+q−p (p+r)(q+r)(1+ǫ)d+√p (p+r)(1+ǫ)d+√q (q+r)(1−ǫ)d >−O/parenleftbigg1 (p+q+r)1/2(q+r)(1−O((p+q+r)−3))d/parenrightbigg +Ω/parenleftbigg1 d/parenleftbigg√p p+r+√q q+r/parenrightbigg/parenrightbigg = Ω/parenleftbigg1 d/parenleftbigg√p p+r+√q q+r/parenrightbigg/parenrightbigg . The ﬁrst, third and fourth steps above follow from the fact th atǫ=O(δ) =O((p+q+r)−3/2)andp≤q without loss of generality. Now, /bardblS′−T′/bardbl1≥/summationdisplay j:jgood|S′ j−T′ j|>dθ3 2·Ω/parenleftbigg1 d/parenleftbigg√p p+r+√q q+r/parenrightbigg/parenrightbigg = Ω/parenleftbigg√p p+r+√q q+r/parenrightbigg . Finally, /bardblS−T/bardbl1≥ /bardblS′−T′/bardbl1−/bardblS−S′/bardbl1−/bardblT−T′/bardbl1 = Ω/parenleftbigg√p p+r+√q q+r/parenrightbigg −2·O/parenleftbigg1 p+q+r/parenrightbigg = Ω/parenleftbigg√p p+r+√q q+r/parenrightbigg . Theconﬁdence bound is Pr[E0∧E1∧E2]>1−4exp(−α)−exp(dθ3/8) = 1−exp(−Ω(α)), sinceδ=O(1). Thiscompletes the proof of the lemma. Wecan prove thefollowing lemma inasimilar fashion asLemma 2. Lemma3 LetW={a1,...,ap}be a random orthonormal set of pvectors in Cd. Letai jdenote thejth coordinate ofvector ai. Deﬁnethed-dimensional probability vector Sasfollows:Sj=1 p/summationtextp i=1|ai j|2. Then there existsδ=θ(p−1)such that the following holds: Deﬁne α=dδ2−2logp. Supposeα= Ω(1). LetU denote the uniform probability distribution on {1,...,d}. Then, with probability at least 1−exp(−Ω(α)) over the choice of V,/bardblS−U/bardbl1= Ω(p−1/2). Proof:(Sketch)Generateaset W′={a′1,...,a′p}ofprandomindependentunitvectorsin Cdasdescribed in Section 3.2. Let α′i j,j={1,...,d}denote the Gaussians used to generate the random unit vector s a′i. Then,a′i j=α′i j/summationtextd l=1|α′i l|2. We Gram-Schmidt orthonormalise W′to get the random orthonormal set W={a1,...,ap}. LetE0be the event that the normalisation factors in the denominat

Question 880 multiple-choice

Quantum walks are a powerful computational paradigm leveraging quantum superposition and entanglement to explore complex spaces efficiently. The matrix representations of shift and control operators are essential for implementing quantum walk algorithms on quantum circuits. Which of the following statements correctly describes the use of the exchange matrix J in representing and manipulating quantum walk shift operators within a multi-qubit framework? 1) J is used to generate arbitrary unitary rotations on position qubits. 2) J enables direct encoding of velocity in the binary representation of position. 3) J is responsible for sparse matrix multiplication in quantum walk propagation. 4) J is employed to diagonalize Toeplitz matrices in the velocity-position space. 5) J reverses the order of basis states and, when conjugating a Toeplitz matrix, yields its transpose, facilitating efficient operator decomposition. 6) J implements modulo addition for position updates during the quantum walk. 7) J serves as a control register for conditional quantum gates in the walk circuit.

✓ Correct Answer:

The correct answer is 5) J reverses the order of basis states and, when conjugating a Toeplitz matrix, yields its transpose, facilitating efficient operator decomposition..

📚 Reference Text:

or one qubit ﬂipXgate), which as a matrix in our coordinate representation is the N×Nmatrix X= 0 0 0 ···1 1............ 0......0 0 ......1 0 0 0···0 1 0 . (3) We may check that X|x/an}⌊ra⌋ketri}ht=|x+1/an}⌊ra⌋ketri}ht, where|x/an}⌊ra⌋ketri}htis as in Eq. (2). The matrix for propagation σ on the joint velocity-position space is a controlled shift of position |x/an}⌊ra⌋ketri}htby velocity |v/an}⌊ra⌋ketri}ht, with |v/an}⌊ra⌋ketri}ht=|+1/an}⌊ra⌋ketri}htaﬀecting the right shift Xand|v/an}⌊ra⌋ketri}ht=|−1/an}⌊ra⌋ketri}htaﬀecting the left shift X⊤. Its matrix is σ=/bracketleftbigg X 0 0 X⊤/bracketrightbigg . (4) where0is the matrix of all zeros. Figure 1 shows the circuit for this description of QW with ˆSfollowed by σ. 4 EFFICIENT AND SCALABLE QUANTUM WALK ALGORITHMS |v/angbracketright |x1/angbracketright |xn/angbracketrightS X,XT Figure 1. A quantum pseudo-circuit for QW. The control |v/an}⌊ra⌋ketri}htselectsXorX⊤ 2.1.Implementing thepropagation σ.Thepropagation σinEq.(4)hasadecomposition that we can implement using a single right shift Xfrom Eq. (3), as we describe next. Let us ﬁrst state the deﬁnition and a property of Toeplitz matrices, wh ich we then apply toward our decomposition. Recall that a Toeplitz matrix is a square matrix wit h each descending diagonal from left to right a constant, i.e., a N×NmatrixAis Toeplitz if A= a0a−1a−2···a−(N−1) a1a0a−1...... a2a1......a−2 ............a−1 aN−1···a2a1a0 . for some set of numbers {a−(N−1),...,a 0,...,a N−1}. So if we number the rows and columns ofAby{0,...,N−1}, thenAi,j=ai−jfor all 0≤i,j≤N−1. Also recall the deﬁnition of anexchange matrix (see Section 1 .2.11 of [9]). An exchange matrix is a square matrix with all its entries 0 except those on the anti-diagonal which are all 1 . Denote the N×N exchange matrix by J, J= 0···0 0 1 0......1 0 ...0...0... 0 1......0 1 0 0 ···0 . The transpose of a Toeplitz matrix can be obtained as follows (Sectio n 4.7 of [9]). A⊤=JAJ. As the shift matrix Xin Eq. (3) is Toeplitz, by the above property, we can express σin Eq. (4) as σ=/bracketleftbigg X 0 0JXJ/bracketrightbigg =/bracketleftbigg IN0 0J/bracketrightbigg/bracketleftbigg X 0 0 X/bracketrightbigg/bracketleftbigg IN0 0J/bracketrightbigg , (5) whereJis theN×Nexchange matrix and INis theN×Nidentity matrix. Let us denote the ﬁrst (and the last) matrix on the right side product decomposit ion byCv(J), Cv(J) =/bracketleftbigg IN0 0J/bracketrightbigg . (6) EFFICIENT AND SCALABLE QUANTUM WALK ALGORITHMS 5 Thenσin Eq. (5) may be written σ=Cv(J)(Iv⊗X)Cv(J), (7) whereIvis the identity operator on the velocity space. 2.1.1.Specializing to qubits. Let us assume further that the lattice is of size N= 2nfor some ﬁniten, natural for multi-qubit systems. The binary expansion of the pos itionx x=n−1/summationdisplay j=02jxj, (8) withxj∈ {0,1},j∈ {0,...,n−1}, gives usa bijective map x↔(xn−1...x0). This identiﬁes the bases of position space and of nqubits:|x/an}⌊ra⌋ketri}ht ↔ |xn−1...x0/an}⌊ra⌋ketri}ht. We also identify the velocity basis states with the usual qubit states as |+1/an}⌊ra⌋ketri}ht ↔ |0/an}⌊ra⌋ketri}ht, and|−1/an}⌊ra⌋ketri}ht ↔ |1/an}⌊ra⌋ketri}ht. Combined, the velocity-position space is thus identiﬁed with n+1 qubits, H ↔Cn+1 |+1,x/an}⌊ra⌋ketri}ht /ma√sto→ |0,xn−1...x0/an}⌊ra⌋ketri}ht |−1,x/an}⌊ra⌋ketri}ht /ma√sto→ |1,xn−1...x0/an}⌊ra⌋ketri}ht For the rest of this section when we refer to any of the vector spa ces, we interpret that as the corresponding multi-qubit space. Let us turn to the operators. We observe that the exchange mat rixJ, in

Question 881 multiple-choice

Quantum algebraic structures such as H-bialgebroids and Hopf algebroids play a central role in understanding the relationship between quantum groups and their classical counterparts, especially through deformation and representation theory. The conditions placed on algebra generators often determine whether their properties align with classical expectations. Which condition, when imposed alongside the Quantum Dynamical Yang-Baxter Equation (QDYBE) with the quantum parameter q not a nontrivial root of unity, ensures that the polynomial spaces in the H-bialgebroid $\bar{A}_R$ are free modules over $M_T \otimes M_T$ with ranks matching the classical matrix algebra case? 1) The cocycle condition 2) The involutive property 3) The braid relation 4) The unitarity condition 5) The reflection equation 6) The fusion procedure 7) The Hecke condition

✓ Correct Answer:

The correct answer is 7) The Hecke condition.

📚 Reference Text:

even if Rsatisﬁes the QDYBE, the H-bialgebroid ¯ARmay not be completely satisfactory. In particular, one may ask the following question: does¯ARdeﬁne a ”good quantum matrix algebra”? More precisely, does the Hilbert series of ¯ARequal to that of the function ring on the matrix algebra, i.e. (1−t)−dim(V)2? In general, the answer is no, even if the quantum dynamical Yang-Baxter equation is satisﬁed. In fact, here is the place where the Hecke condition comes handy. N amely, we have the following proposition, which is a generalization of a well kno wn proposition in the theory of quantum groups (due to Faddeev, Res hetikhin, Takhtajan). Proposition 7.3 ([EV2]). Suppose that Rsatisﬁes the QDYBE and the Hecke condition with qnot equal to a nontrivial root of 1. Then the space ¯Am Rof polynomials of degree min generators Lαβin¯ARis a free MT⊗MT-module, and the ranks of these modules are given by /summationdisplay m≥0rk(¯Am R)tm= (1−t)−n2. Remark. The proof of this proposition, like the proof of its nondynamical analog, is based on the fact that under the assumptions of the Pro position, ¯Am Ris a representation of the Hecke algebra. This justiﬁes the name ”H ecke condition”. 7.4. The H-Hopf algebroid AR.Suppose now that Rsatisﬁes the QDYBE and the Hecke condition where qis not a nontrivial root of 1. In this case, it turns out that, analogously to the nondynamical case, a suitable lo calization ARof¯ARis actually a Hopf algebroid. Namely, deﬁne ARby adjoining to ¯AR a new element L−1, with the relation LL−1=L−1L= 1. It is easy to see that the structure of an H-bialgebroid on ¯ARnaturally extends to AR, and it can be shown that ARadmits a unique antipode Ssuch that S(L) =L−1. This antipode equips ARwith a structure of an H-Hopf algebroid. This H- Hopf algebroid is a quantization of the group GLnin the same sense as the H-bialgebroid ¯ARis a quantization of the matrix algebra Matn. 7.5. Quasiclassical limit. In conclusion of the section, we would like to explain why the H-Hopf algebroid ARconsidered here (for the basic rational or trigonometric solution Rof the QDYBE) should be regarded as a quantization of the Poisson groupoid Xrcorresponding to the basic rational, respectively trigonometric, solution rof the CDYBE (for the deﬁnition of Xr, see Section 5). 22 To see this, consider the H-Hopf algebroid ARwithT=h∗for some ﬁnite dimensional abelian Lie algebra h, andMTreplaced with the ring of regular functions on some open subset Uofh∗. Introduce a formal parameter γ(like in Section 3), and make a change of variable λ→λ/γin the deﬁning relations for AR. It is easy to see that the resulting algebra Aγ RoverC[[γ]] is a deformationof a commutative algebra. The above result about the Hilbert series imp lies that this deformation is ﬂat, so the quotient algebra A0 R:=Aγ R/(γ) obtains a Poisson structure. Let Xbe the spectrum of A0 R; it is an algebraic Poisson manifold. It is not diﬃcult to show that the the moment maps, coproduct, cou nit, and antipode of ARdeﬁne maps s,t,m,E,i (see Section 5) for X,

Question 882 multiple-choice

Quantum algorithms such as Grover's and Shor's have revolutionized computational possibilities by leveraging group structures and the hidden subgroup problem (HSP) framework. Understanding their connections and limitations is central to advancing quantum algorithm theory. Which statement accurately reflects the relationship between Grover's algorithm, the hidden subgroup problem (HSP), and the standard non-abelian quantum hidden subgroup (QHS) algorithm? 1) Grover's algorithm is an efficient abelian HSP algorithm for cyclic groups. 2) The standard non-abelian QHS algorithm solves the HSP for all symmetric groups, including those relevant to Grover's algorithm. 3) Grover's algorithm is asymptotically faster than any possible HSP-inspired quantum algorithm for unstructured search. 4) Shor's algorithm is primarily used to solve search problems on non-abelian groups. 5) The HSP framework is not applicable to Grover's or Shor's algorithms. 6) Grover's algorithm can be interpreted as solving a non-abelian HSP on the symmetric group, but the standard non-abelian QHS algorithm cannot solve this specific HSP. 7) Any quantum algorithm based on HSP for unstructured search is guaranteed to outperform Grover's algorithm in asymptotic complexity.

✓ Correct Answer:

The correct answer is 6) Grover's algorithm can be interpreted as solving a non-abelian HSP on the symmetric group, but the standard non-abelian QHS algorithm cannot solve this specific HSP..

📚 Reference Text:

arXiv:quant-ph/0603140v1 15 Mar 2006IS GROVER’S ALGORITHM A QUANTUM HIDDEN SUBGROUP ALGORITHM ? SAMUEL J. LOMONACO, JR. AND LOUIS H. KAUFFMAN Abstract. The arguments given in this paper suggest that Grover’s and Shor’s algorithms are more closely related than one might at ﬁrst expect. Speciﬁcally, we show that Grover’s algorithm can be viewed a s a quantum al- gorithm which solves a non-abelian hidden subgroup problem (HSP). But we then go on to show that the standard non-abelian quantum hidd en subgroup (QHS) algorithm can not ﬁnd a solution to this particular HSP . This leaves open the question as to whether or not there is som e mod- iﬁcation of the standard non-abelian QHS algorithm which is equivalent to Grover’s algorithm. Contents 1. Introduction 1 2. Deﬁnition of the hidden subgroup problem (HSP) and hidden subgr oup algorithms 2 3. The generic QHS algorithm QRand 3 4. Pushing HSPs for the generic QHS algorithm QRand 4 5. Shor’s algorithm 5 6. Description of Grover’s algorithm 6 7. The symmetry hidden within Grover’s algorithm 7 8. A comparison of Grover’s and Shor’s algorithms 9 9. However 10 10. Conclusions and Open Questions 11 References 12 1.Introduction Is Grover’s algorithm a quantum hidden subgroup (QHS) algorithm ? We do not completely answer this question. Instead, we show that G rover’s algorithm is a QHS algorithm in the sense that it can be rephrased as a q uantum algorithm which solves a non-abelian hidden subgroup problem (HSP) o n the sym- metric group SN. But we then go on to show that the standard non-abelian QHS algorithm cannot solve the Grover HSP. This leaves unanswered an intriguing question: Date: March 12, 2006. 1991Mathematics Subject Classiﬁcation. [2000]Primary 81P68; Secondary 81P99. 1 2 SAMUEL J. LOMONACO, JR. AND LOUIS H. KAUFFMAN Question. Is there an extension or modiﬁcation of the standard non-abe lian QHS on the symmetric group SNwhich solves the non-abelian HSP associated with Grover’s algorithm? It should be mentioned that, because of a result of Zalka [31], such a n algorithm, if it exists, could not be asymptotically faster than Grover’s algorith m. We hope that the results found in this paper will lead to a better unde rstanding of quantum algorithms. 2.Definition of the hidden subgroup problem (HSP) and hidden subgroup algorithms What is a hidden subgroup problem ? What is a hidden subgroup algorith m ? Deﬁnition 1. A mapϕ:G−→Sfrom a group Ginto a setSis said to have hidden subgroup structure if there exists a subgroup KϕofG, called a hidden subgroup , and an injection ιϕ:G/Kϕ−→S, called a hidden injection , such that the diagram Gϕ−→S νց ր ιϕ G/Kϕ is commutative1, whereG/Kϕdenotes the collection of right cosets of KϕinG, and whereν:G−→G/Kϕis the natural surjection of GontoG/Kϕ. We refer to the group Gas theambient group and to the set Sas thetarget set . If Kϕis a normal subgroup of G, thenHϕ=G/Kϕis a group, called the hidden quotient group , andν:G−→G/Kϕis an epimorphism, called the hidden epimorphism . We will call the

Question 883 multiple-choice

Quantum hypothesis testing involves distinguishing between different quantum states using measurements, with techniques from matrix analysis and representation theory playing a critical role. Concepts such as pinching, relative entropy, and permutation-invariant measurements are central to understanding the limits of quantum state discrimination. Which statement accurately reflects the relationship guaranteed by Horn's Lemma in the context of quantum information theory? 1) Every doubly stochastic matrix can be expressed as the entrywise square modulus of a unitary matrix's coefficients. 2) Every positive semidefinite matrix can be decomposed into tensor products of maximally mixed states. 3) Every permutation-invariant operator can be written as a convex combination of orthogonal projections. 4) The spectrum of any quantum state can be uniquely determined by its pinching in a random basis. 5) Every quantum measurement yields a doubly stochastic matrix when applied to a pure state. 6) Every irreducible representation of the symmetric group corresponds to a unique quantum state. 7) Every empirical distribution in quantum hypothesis testing is invariant under unitary conjugation.

✓ Correct Answer:

The correct answer is 1) Every doubly stochastic matrix can be expressed as the entrywise square modulus of a unitary matrix's coefficients..

📚 Reference Text:

for the matrix coeﬃcients of Dit holdsDij=|uij|2for alli,j∈[d]. This can be done, thanks to Horn’s Lemma [32]. Deﬁne ρ:=/summationtext i¯λi|ei/an}bracketri}ht/an}bracketle{tei|, then/an}bracketle{tei,UρU†ei/an}bracketri}ht=f(i) for all i∈[d], and this means that the pinching of ρto the basis {U†ei}d i=1satisﬁes our requirements. Now take any state ρwith spectrum rρand pinching ˜ rρin some (not exactly speciﬁed) basis. If ρ=1 d /BD there is nothing to prove, since every pinching then satisﬁes ˜ rρ=rρand obviously choosing D=Idis consistent with elementary fact that Knrρ,nrρ>0 for every nsuch that n/d∈N. Thus,ρ/ne}ationslash=1 d /BD. Choosing σ=1 d /BDit follows that D(ρ/bardblσ)>0 and using our projections Pnwe see from equation (57) that there is N∈Nsuch that for all n≥Nthere is a projection Pf,λ/ne}ationslash= 0 that has the property /bardbl¯f−˜rρ/bardbl ≤n−1 4and/bardbl¯λ−rρ/bardbl ≤n−1 4, proving our claim. 11 Proof of Theorem 4. We will ﬁrst show the statements in the case that S′is ﬁnite, and then apply them to the case of an inﬁnite set S′. The issues arising in the latter setting are mainly due to ’roundness’ of the set of states. It could possibly happen that S′contains parts of the boundary of the set of S(Cd), and it can then (thinking of the Bloch sphere representation is helpf ul here) be impossible to include S′ into the convex hull of some ﬁnite set that lies entirely within the set o f states itself. Robustiﬁcation, Part I - Finite S.Let a state ρˆsnbe given, and let ˆ snhavetype N:S→N(meaning that the short-hand N(·) =N(·|ˆsn) is valid). Let p∈P(S) be deﬁned by p(s) :=1 nN(s). It then holds (see [20], the proof of Lemma 2.3) that p⊗n(TN)≥(2n)−|S|. Since our measurement is permutation-invariant, it follows tr{( /BD−Pn)(/summationdisplay s∈Sp(s)ρs)⊗n}= tr{( /BD−Pn)/summationdisplay sn∈Snp⊗n(sn)ρsn} (73) ≥tr{( /BD−Pn)/summationdisplay sn∈TNp⊗n(sn)ρsn} (74) = tr{( /BD−Pn)/summationdisplay sn∈TNp⊗n(sn)ρˆsn} (75) =p⊗n(TN)tr{( /BD−Pn)ρˆsn} (76) ≥(2n)−|S|tr{( /BD−Pn)ρˆsn}, (77) implying, via equation (57) and our choice of Pn, that tr{Pnρˆsn}= 1−tr{( /BD−Pn)ρˆsn} (78) ≥1−(2n)|S|tr{( /BD−Pn)(/summationdisplay s∈Sp(s)ρs)⊗n} (79) ≥1−(2n)|S|2−n(αε2+2d2 nlog(2n))(80) ≥1−2−n(αε2+2d2+|S| nlog(2n)). (81) By linearity of the trace, this immediately implies that we can distinguish the sequence (conv({ρsn}sn∈Sn}))n∈Nof sets from the sequence ( σ⊗n)n∈N. We still have to show that we have the right error exponent, but th is follows immediately from the Sanov case by noting that for arbitrary sn∈Snand the type N(·) :=N(·|sn) we have by Lemma 2.3 in [20] tr{Pnρsn} ≤(2n)|S|tr{Pn(/summationdisplay s∈S¯N(s)ρs)⊗n}. (82) Proof of the converse part for ﬁnite |S|.Let tr{Aρsn} ≥1−νfor allsn∈Sn. Like before, we can without loss of generality assume that Ais permutation-invariant. Then again by Lemma 5 in [28] we get A≤(2n)d2/summationdisplay f,λ,jcf,λ,jPf,λ,j, (83) wherePf:=/summationtext λPf,λ,Pλ:=/summationtext fPf,λandPfPλ=PλPfholdsandtheindex jreferstoanydecomposition Vf,λ=⊕λ,jVf,λ,jintoirreducible components. Obviously, Pf,λ,jthen denote the orthogonal projections onto the Vf,λ,j. Let us pick p∈P(S) and an sn∈Snhaving type abbreviated by N(·) =N(·|sn). We may additionally assume that Nsatisﬁes /bardbl¯N−p/bardbl ≤δn(whereδn= 2|S|/n). Then assume that a λ∈Yd,nsatisﬁes /bardbl¯λ−spec(¯ρ)/bardbl> δ >0 for some δand the state ¯ ρ:=/summationtext sp(s)ρs. For such λit follows from [20], Lemma 12 2.3, that tr{Pλρsn}=1 |TN|/summationdisplay sn∈TNtr{Pλρsn} (84) ≤(2n)|S|¯N⊗n(TN)/summationdisplay sn∈TNtr{Pλρsn} (85) ≤(2n)|S|/summationdisplay sn∈Sn¯N⊗n(sn)tr{Pλρsn} (86) = (2n)S|tr{Pλ(/summationdisplay s¯N(s)ρs)⊗n} (87) ≤(2n)|S|(2n)d22−nD(¯λ/bardblspec(/summationtext s¯N(s)ρs))(88) ≤(2n)|S|+d22−nα/bardbl¯λ−spec(/summationtext s¯N(s)ρs)/bardbl2(89) ≤(2n)|S|+d22−nα(/bardbl¯λ−spec(¯ρ)/bardbl−/bardblspec(¯ρ)−spec(/summationtext s¯N(s)ρs)/bardbl)2(90) ≤(2n)|S|+d22−nα(/bardbl¯λ−spec(¯ρ)/bardbl−δn)2(91)

Question 884 multiple-choice

Quantum Fourier Transform (QFT) and its approximate variant (AQFT) are essential operations in quantum computing, particularly for algorithms involving phase estimation and frequency analysis. Understanding the trade-offs between fidelity and circuit complexity is crucial for scalable quantum device design. In an n-qubit AQFT circuit, what is the primary consequence when the maximum phase error Δmax introduced by omitting certain phase gates exceeds π/2? 1) The output amplitudes become uniformly distributed regardless of input state. 2) The AQFT circuit performs a classical Fourier transform instead of a quantum one. 3) The overall circuit depth is reduced without affecting output fidelity. 4) The probability of successful error correction increases. 5) The transformation's computational speed is maximized at the expense of all accuracy. 6) The qubit register fails to encode more than two distinct states. 7) The output states lose coherent phase structure, resulting in scrambled phase information and reduced algorithmic correctness.

✓ Correct Answer:

The correct answer is 7) The output states lose coherent phase structure, resulting in scrambled phase information and reduced algorithmic correctness..

📚 Reference Text:

K. Mølmer, and J. Dalibard, J. Opt. Soc. Am. B 10, 524 (1993). [15] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). [16] R. B. Griﬃths and C. S. Niu, Semiclassical Fourier Trans form for Quantum Computation, Carnegie Mellon University preprint (1995), quant–ph/9511007 (unpublished). [17] U. Feige, P. Raghavan, D. Peleg, and U. Upfal, SIAM J. Com put.23, 1001 (1994). 12 a3a2a1a0 A θ32A θ31θ21A θ30θ20θ10A ✈✈ ✈✈ ✈ ✈ c3 c2 c1 c0 Figure 1: QFT network operating on a four–bit register. The p hases θjkthat appear in the operations Bjkare related to the “distance” of the qubits ( j−k) and are given byθjk=π/2j−k. The network should be read form the left to the right: ﬁrst th e gateAis eﬀected on the qubit a3, then B(φ32) ona2anda3, and so on. a3a2a1a0 A θ32A θ21A θ10A ✈✈✈ c3 c2 c1 c0 Figure 2: AQFT network operating on a four–bit register when m= 2. Figure 3: (a) Function f(a) in Eq. (12). The parameters are l= 8,r= 10 and the number of bits in the calculation is L= 9 so that numbers up to 2L−1 = 511 can be encoded. (b) |˜f(c)|, obtained from f(a) by a QFT. The inset shows ¯ c3= 155 which is the closest integer to 3 ·512/10 = 153 .6. (See the Appendix for more details). Figure 4: Diﬀerent orders of approximation in the AQFT perfo rmed on a state |Ψ/an}bracketri}ht for which f(a) =δ9, amod 10 . Figure 5: As Fig. 4, but showing the phase of the amplitudes, i .e., arg( ˜f(c)). Figure 6: The quality factor Qas a function of mfor selected values of δ. The register sizes are (a) L= 9 and (b) L= 16. Statistical errors are too small to be represented on the graph. 13 Figure 7: The quality factor Qas a function of the register size Lfor QFT, with varying levels of decoherence, from δ= 0.1 (top line) to δ= 0.5 (bottom line). Statistical errors are too small to be represented on the gra ph. Figure 8: (a) Argand diagram corresponding to the sum of the p hases that appear in the expression of f(¯c) for ¯cclose to one of the values n2L/r. Prob(¯ c) is the norm of the vector resulting from the sum of each vector in the diag ram. (b) Prob(¯ c) is bounded by the worst case situation in which we have taken θmaxinstead of θ¯c, in this case the phases lie on an interval [0 ,π] on the Argand diagram and a closed form expression can be found. Figure 9: (a) Argand diagram for the AQFT. Vectors are rotate d by an angle ∆(jr,c). (b) To obtain a closed form for a bound for Prob A(¯c) we consider the worst case in which half of the phases pick up a factor ∆ max. Figure 10: In the case that the order mof the AQFT is such that ∆ max> π/2, the individual phases can get scrambled in such a way

Question 885 multiple-choice

In molecular simulations, force fields are used to model the interactions between atoms, with parameters such as Lennard-Jones σ and ǫ, and atomic charges, playing crucial roles in determining physical properties and chemical behavior. The treatment of halogen atoms, especially heavier ones, is particularly significant in accurately capturing phenomena like halogen bonding. Which statement best describes the difference in charge assignment to bromine atoms between quantum mechanics-derived force fields (QMD-FF) and empirical force fields such as OPLS, and its impact on the predicted electrostatic potential surface (EPS)? 1) QMD-FF assigns a highly negative charge to bromine, creating a stronger negative EPS than OPLS. 2) OPLS and QMD-FF both assign nearly neutral charges to bromine, resulting in similar EPS distributions. 3) OPLS assigns a positive charge to bromine, while QMD-FF assigns a negative charge, reversing the EPS polarity. 4) QMD-FF assigns nearly zero charge to bromine, whereas OPLS assigns a more negative charge, leading to differing EPS distributions. 5) Both force fields assign highly positive charges to bromine, which enhances halogen bonding in simulations. 6) QMD-FF and OPLS assign identical charges to bromine, but differ only in Lennard-Jones parameters. 7) OPLS assigns a charge of zero to bromine, while QMD-FF assigns a charge close to –1 atomic unit, exaggerating the EPS.

✓ Correct Answer:

The correct answer is 4) QMD-FF assigns nearly zero charge to bromine, whereas OPLS assigns a more negative charge, leading to differing EPS distributions..

📚 Reference Text:

investigated chloro-hydrocarbons: σ(˚A),ǫ(kJ/mol) and q(a.u.). transferable FFs, to preserve chemical identity even at ato mic level emerges even more clearly than for the intramolecular parameters. To not over whelm the reader with undue information, the comparison with literature parameter set s will be discussed only with respect to OPLS11parameters, since it covers the largest number of the molecu les here investigated, while GAFF and CGenFF data were available17,59only for few compounds. For instance, the QMD-FF LJ’s σvalues for the carbon atoms reported in Table 6, show very diﬀerent values, going from 3.09 ˚A, found for dichloromethane to 3.55 ˚A, obtained for the methyl atom in chloroethane. This range is even more exte nded (3.68 ˚A, for molecule I1) if one takes into account all the investigated compounds. O n the contrary, empirical FFs (see for instance OPLS, in Tables F-H), to preserve transferability, tend to adopt the sameσvalue for all (alkane) carbons, with a consequent loss of pre cision in the IPES. On the same foot, the strength of the LJ interaction for carbon a toms is constrained to 0.28 kJ/mol, whereas in QMD-FF, ǫmay vary from a minimum of 0.1 kJ/mol to a maximum value of 2.3 kJ/mol, only considering the small C1-C4set. Similar conclusions can be drawn by extending such comparative analysis to all compoun ds and all intermolecular parameters, reported in detail in the Supporting Informati on. 24 1 AC1 - OPLS 1 AC1 - QMD-FF 1 AC4-OPLS 1 AC4- QMD-FF 1 A B -OPLS 1 A -QMD-FF 1 AI1 -OPLS 1 AI1 - QMD-FF Figure 6: Visual 2D comparison (minimap) of the OPLS11and QMD-FF descriptions of molecules C1, C4,B2,I1. Each molecule is displayed on the molecular plane containi ng the X-C-C or X-C-H triplet (X = halogen). For each atom, the assigned charge is reported ; LJ parameters are displayed through white (repulsive) circles with radius = σ, and grey (attractive) rings, whose opacity is proportiona l toǫ. The electrostatic potential (EPS) is plotted along the inte rsection between the molecular plane and the VdW spheres of radii σ, with colors varying from dark blue (0.7 a.u.) to dark red (-0 .4 a.u.). The same values of ESP are also plotted, projected onto the xaxis, below each minimap. Some of the parameterized QMD-FFs are visualized in Figure 6 (2D diagrams), along with their OPLS counterparts. As concerns the treatment of th e halogen atoms, B2and I1provide interesting insights about the aforementioned σ-hole problem (which is known to be more evident in the heaviest halogens). By looking at th e Bromine atom in B2, it may be worth noticing that the charge assigned by QMD-FF is close to zero (-0.010 a.u.), while it is more negative in OPLS (-0.110 a.u.). Indee d, the electrostatic potential (EPS, red and blue curves), which is expected to be negative o n the nucleophilic side of the atom’s VdW surface, is now determined by the strongly nega tive charge assigned to 25 the carbon atom: as seen from above the

Question 886 multiple-choice

Quantum computing leverages the principles of superposition and entanglement to perform computations using quantum bits (qubits) and, in advanced architectures, multivalued quantum digits (qudits) such as qutrits. Efficient implementation of algorithms like the Quantum Fourier Transform (QFT) often requires specialized gates that match the underlying quantum system. Which quantum gate is the direct ternary generalization of the binary Hadamard gate, used specifically to create uniform superpositions across qutrit states using complex third roots of unity? 1) Toffoli gate 2) Fredkin gate 3) Controlled-Z gate 4) Pauli-X gate 5) Chrestenson gate 6) SWAP gate 7) Phase gate

✓ Correct Answer:

The correct answer is 5) Chrestenson gate.

📚 Reference Text:

to follow. 3.2.1 Transforming 2nElements—Qubit Description The speedup in deriving fast quantum algorithms is due to decomposable realization. The 2nquantities by nqubits are represented as y¼y0þ2y1þ22y2þ/C1/C1/C1 2n/C01yn/C01and the decomposable states are: jyi¼jy0ijy1i/C1/C1/C1jyn/C01i. An applica- tion of QFT is then: 1ﬃﬃﬃﬃﬃ 2npX y2f0;1gne/C02/C25i 2nxyjyi¼1ﬃﬃﬃﬃﬃ 2npX y2f0;1gne/C02/C25i 2nxPn l¼1yl2ljy0y1/C1/C1/C1yn/C01i:The key step, analogous to the FFT, is the decomposition X y2f0;1gne/C02/C25i 2nxyjyi¼X y02f0;1gn/C01e/C02/C25i 2nx2y0jy00i þX y02f0;1gn/C01e/C02/C25i 2nxð2y0þ1Þjy01i by the least significant bit. Then, QFT is (up to a multiplicative constant) reduced to decomposable states: X y02f0;1gn/C01e/C02/C25i 2nx2y0jy0ij0iþe/C02/C25ix 2nj1i/C16/C17 : By using the same construction recursively, QFT reduces to: ðj0iþe/C0/C25ix 20j1iÞð j0iþe/C0/C25ix 21j1i Þ/C1/C1/C1ð j 0iþe/C0/C25ix 2n/C01j1iÞ: ð1Þ By dividing xwith 2l, the expression 0:xl/C01xl/C02...x0 denotes the fraction obtained from the least significant l qubits of xas 0:xl/C01xl/C02/C1/C1/C1x0¼2/C01xl/C01þ2/C02xl/C02þ/C1/C1/C1þ 2/C0lx0 and the expansion is rewritten into: j0iþe/C02/C25i0:x0j1i/C0/C1 j0iþe/C02/C25i0:x1x0j1i/C0/C1 /C1/C1/C1 j0iþe/C02/C25i0:xn/C01xn/C02/C1/C1/C1x0j1i/C0/C1 : This representation is amenable to the efficient quantum circuit implementations. Each term, multiplied with the appropriate constant, is the unitary transform, built using the Walsh-Hadamard and the controlled rotation gates. Theleftmost product term is, by definition, equal to the functionperformed by the Walsh-Hadamard gate j0iþe /C02/C25i0:x0j1i/C0/C1 ¼j0iþe/C0/C25ix 0j1i/C0/C1 ¼/C0 j0iþð /C0 1Þx0j1i/C1 : The remaining terms are realized following the construction in [5]. For the lthterm, we first apply the W2gate, leading to 1ﬃﬃ 2pðj0iþð /C0 1Þxl/C01j1iÞ. For each k 2Þcome quite naturally in many cases [13], there exist a few explicitconstructions of q-valued gates. In developing the repre- sentation of q-valued states, we find useful the general- ization of the Walsh-Hadamard gate, which we call the Chrestenson gate after the Chrestenson Transform [11]. Example 4. For the case q¼3, the complex third root of unity is: a¼e /C02/C25i 3¼cos/C02 3/C25i/C18/C19 þisin/C02 3/C25i/C18/C19 ¼/C0 0:5/C0i/C30:866: The Chrestenson gate performs the mapping given by: CH¼1ﬃﬃﬃ 3p11 1 1aa2 1a2a2 43 5: The adjoint of CH is given as: 11 1 1aa2 1a2a2 43 5y ¼11 1 1a/C3a2ðÞ/C3 1a2ðÞ/C3a/C32 43 5¼11 1

Question 887 multiple-choice

In computational group theory and quantum computing, efficient subgroup representations and algorithmic manipulation are critical for solving problems related to group structure and membership. Matrix normal forms are widely used to encode and analyze subgroups of finite abelian groups and their lattices. Which of the following properties uniquely distinguishes the Hermite normal form (HNF) of an integer matrix when representing a subgroup of $ \mathbb{Z}_m^k $? 1) It diagonalizes the matrix, with entries reflecting invariant factors of the subgroup. 2) It relies on non-unique encoding, requiring equality-testing oracles for subgroup membership. 3) It provides a matrix with orthogonal columns corresponding to subgroup generators. 4) It yields a unique canonical basis for the lattice, encoding the subgroup structure. 5) It performs matrix reduction over finite fields, not over the integers. 6) It creates a block upper-triangular matrix regardless of subgroup properties. 7) It always produces a matrix where all entries are either 0 or 1.

✓ Correct Answer:

The correct answer is 4) It yields a unique canonical basis for the lattice, encoding the subgroup structure..

📚 Reference Text:

factors, that is, a sequence of subgroupsf1g=G0

Question 888 multiple-choice

Gauge symmetries play a fundamental role in quantum field theory, particularly in the classification of physical states and the structure of vacuum configurations. The distinction between global and local gauge symmetries leads to profound implications for quantization, topological invariants, and mechanisms of symmetry breaking. In non-abelian gauge theories, which of the following statements correctly describes the significance of a non-trivial residual local gauge group after reduction, especially with respect to vacuum structure and physical states? 1) It allows all gauge bosons to remain massless due to unbroken symmetries. 2) It eliminates the possibility of topological invariants arising from the gauge group. 3) It prevents the quantization of fields by introducing additional redundancies. 4) It ensures that spontaneous symmetry breaking always leads to Goldstone bosons. 5) It restricts physical states to only those invariant under global transformations. 6) It enables the existence of gauge-invariant topological invariants that classify vacuum structure and permits constructing physical states via the vacuum representation of the local field algebra. 7) It requires the residual group to be reduced to the identity for deterministic field evolution.

✓ Correct Answer:

The correct answer is 6) It enables the existence of gauge-invariant topological invariants that classify vacuum structure and permits constructing physical states via the vacuum representation of the local field algebra..

📚 Reference Text:

Title: Why Gauges? Gauge symmetries for the classification of the physical states Year: 2024 Paper ID: 9912c7d2e472bcb215593b428e564216624c2135 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/9912c7d2e472bcb215593b428e564216624c2135 Abstract: This note focuses the problem of motivating the use of gauge symmetries (being the identity on the observables) from general principles, beyond their practical success, starting from global gauge symmetries and then by emphasizing the substantially different role of local gauge symmetries. In the latter case, a deterministic time evolution of the local field algebra, necessary for field quantization, requires a reduction of the full local gauge group $\cal{G}$ to a residual local subgroup $\cal{G}_r$ satisfying suitable conditions. A non-trivial residual local gauge group allows for a description/construction of the physical states by using the vacuum representation of a local field algebra, otherwise precluded if $\cal{G}$ is reduced to the identity. Moreover, in the non-abelian case the non-trivial topology of the a residual $\cal{G}_r$ defines the (gauge invariant) topological invariants which classify the vacuum structure with important physical effects; furthermore, it provides a general mechanism of spontaneous symmetry breaking without Goldstone bosons.

Question 889 multiple-choice

In the study of quantum Markov semigroups and their relation to classical Markov processes, spectral gap estimates and norm bounds play a crucial role in quantifying convergence to equilibrium. Understanding how quantum and classical mixing times relate provides insight into decoherence and thermalization in open quantum systems. Which statement best characterizes the relationship between classical mixing time and quantum decoherence time for quantum Markov semigroups constructed from classical processes? 1) Quantum decoherence time is always strictly less than the classical mixing time. 2) The classical mixing time cannot be used to estimate the quantum decoherence time. 3) Quantum decoherence time is independent of the classical mixing time for all processes. 4) Classical mixing time is always exactly equal to the quantum decoherence time. 5) Quantum decoherence time is strictly greater than classical mixing time for hypoelliptic systems. 6) For transferred quantum Markov semigroups, quantum decoherence time is bounded above by the classical mixing time. 7) Quantum decoherence time and classical mixing time are unrelated due to differing normalization conventions.

✓ Correct Answer:

The correct answer is 6) For transferred quantum Markov semigroups, quantum decoherence time is bounded above by the classical mixing time..

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and min(L)≥min(L)be their respective spectral gaps. Then for all t≥0 /parallel.alt1Tt−Efix∶L1(B(H))→L1(B(H))/parallel.alt1 ≤/parallel.alt1Tt−Efix∶L1(B(H))→L1(B(H))/parallel.alt1cb ≤/parallel.alt1St−EG∶L1(G)→L1(G)/parallel.alt1(29) and for alls;t≥0 /parallel.alt1Tt+s−Efix∶L1(B(H))→L2 1(Nfix⊂B(H))/parallel.alt1 ≤e−min(L)s/parallel.alt1kt−1/parallel.alt12 (30) ≤e−min(L)s/parallel.alt1kt−1/parallel.alt12: (31) Proof. Equation (29) is a direct consequence of the deﬁnition of the cb norm and of Lemma III.3. In order to prove Equations (30) and (31), we just note that we haveTtEfix=EfixTt. This implies Tt+s−Efix=(Ts−Efix)(Tt−Efix): and(Ts−Efix)Tt=Tt+s−Efix. Therefore, using Theorem III.2, /parallel.alt1Tt+s−Efix∶L1(B(H))→L2 1(Nfix⊂B(H))/parallel.alt1 ≤/parallel.alt1Tt−Efix∶L1(B(H))→L2 1(Nfix⊂B(H))/parallel.alt1 ×/parallel.alt1Ts−Efix∶L2 1(Nfix⊂B(H))→L2 1(Nfix⊂B(H))/parallel.alt1 ≤/parallel.alt1St−EG∶L1(G)→L1(G)/parallel.alt1 ×/parallel.alt1Ts−Efix∶L2(B(H))→L2(B(H))/parallel.alt1: The result follows by the deﬁnition of the spectral gap as well as Equation (28). Note that the norms in the statement above are just the trace and diamond distance between the channels. For Hörmander systems, the following kernel estimates go back to the seminal work of Stein and Rothshield [7], see also [23]. Theorem III.5. LetV={V1;:::;Vm}be a Hörmander system such that Kiterated commutators span a Lie algebra of dimension d. ThenLVhas a spectral gap and there exists a constant CV>0such that, for all 00also hold in general. We shall discuss several examples of such estimates in Section VI. From a quantum information perspective, such bounds can be interpreted as saying that the maximum output min entropy of these semigroups is bounded by the logarithmof the R.H.S. of (32). That is, they quantify how fast the semigroup "spreads out" all over the group. IV. E XAMPLES Here, we illustrate the method developed in the previous sections by listing examples of known QMS, that can be seen as transferred from classical semigroups. We focus on the decoherence-time of the QMS, a notion that generalizes the mixing-time to non-primitive evolutions. We recall that for a general QMS (not necessarily selfadjoint), the decoherence time is deﬁned for any">0as tdeco(") (33) ∶=inf{t≥0;/parallel.alt1T† t()−E† fix()/parallel.alt11≤"∀∈D(H)}: In all the examples below however, the QMS is selfadjoint. We also recall the deﬁnition of the mixing time of a classical primitive Markov process (St)t≥0 tmix(")=inf{t≥0/divides.alt0/parallel.alt1St(f)−EG(f)/parallel.alt1L1(G) ≤"∀f≥0;EG[f]=1}:(34) Remark the difference in the normalization of the norms in both deﬁnitions. In the quantum case, density matrices are normalized with respect to the unnormalized trace whereas in the classical case, we look at the evolution of states normalized with respect to the probability distribution G. Then, Equation (29) in Corollary III.4 implies that for a transferred QMS (Tt)t≥0with associated classical semigroup (St)t≥0, we have for any ">0: tdeco(")≤tmix("): (35) This shows that the decoherence time of a QMS is controlled by the mixing time of any classical Markov semigroup from which it can be transferred. In the examples below, the classical mixing time is estimated thanks to known results on functional inequalities that we mainly introduce in Section VI and Section C, apart for the last example in Section IV-D which is directly computed in Section C. A. The depolarizing QMS Perhaps the simplest QMS that one can think of is the depolarizing semigroup on B(Cn): Ldep()=−ICn n; Tdep t()=e−t+(1−e−t)ICn n: (36) This QMS can be seen to be transferred from the uniform walk

Question 890 multiple-choice

In computational group theory and quantum computing, efficient algorithms for problems in group structures often rely on decomposing complex groups into simpler components. Nilpotent groups, and their lower central series, play a critical role in such decompositions, especially for quantum algorithms targeting group-theoretic problems. For a finite nilpotent group whose lower central series yields factor groups that are elementary abelian, which property allows quantum algorithms designed for abelian groups to be effectively applied to these factor groups? 1) The factor groups are always cyclic. 2) The factor groups are always non-abelian. 3) The factor groups are vector spaces over a finite field. 4) The factor groups contain only the identity element. 5) The factor groups are isomorphic to symmetric groups. 6) The factor groups have a nilpotency class greater than the original group. 7) The factor groups are direct products of simple groups.

✓ Correct Answer:

The correct answer is 3) The factor groups are vector spaces over a finite field..

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of the group oracle, calls to the oracle computing the puriﬁcation of the subgroup state; and the inverses of these. (The elements of G are assumed to beuniquely encoded by strings of length /lscript.) Proof We compute the lower central series G=G 0>G1> ... > Gc={ 1} using the method presented in Subsection 2.2.A s G/G/primeis elementary abelian, so are the factors Gi−1/Gi(i=1,..., c). This is because the factor groups Gi−1/Giare homomorphic images of tensor powers (as Z-modules) of the G/G/prime, see Theorem 5.2.5 123 14 Page 14 of 23 M. Imran, G. Ivanyos of [27]. Also, isomorphisms of Gi−1/Giwith Znipcan be efﬁciently computed using the method of [ 16]. Iteration of Lemma 2gives a procedure to convert/producttextc i=2S(ni)copies of a puriﬁcation of /Xi1G,Hto a copy of a puriﬁcation of /Xi1G/G/prime,HG/prime/G/prime. The composition of instances of the original subgroup state creating procedure (the calls to the oracle) with the conversion gives a procedure for creating a puriﬁcation of /Xi1G/G/prime,HG/prime/G/prime.W e can use this as the oracle input for the exact hidden subgroup algorithm of [ 16]i n Zn1p. Fori=1,..., c,w eh a v e S(p,ni)≤S(p,n)and T(p,ni)≤T(p,n)because Znip can be embedded in Zn pas a subgroup. /intersectionsq/unionsq As an example, consider the group Gof unitriangular 4 by 4 matrices modulo 3. We can visualize Gas⎛ ⎜⎜⎝1∗∗∗ 1∗∗ 1∗ 1⎞ ⎟⎟⎠Here and below in the example, a matrix with asterisks denotes the set of matrices obtained by substituting the asterisks by integers modulo 3, independently at each place. Assume that His the subgroup⎛ ⎜⎜⎝1∗ 1 1 1⎞ ⎟⎟⎠. Gis a semi-elementary matrix group. The lower central series of GisG0=G, G1=⎛ ⎜⎜⎝1∗∗ 1∗ 1 1⎞ ⎟⎟⎠,G2=⎛ ⎜⎜⎝1∗ 1 1 1⎞ ⎟⎟⎠, and G3={I}, thus the nilpotency class is 3. In the ﬁrst round, we convert copies subgroup states for Hto those for HG 2/G2=⎛ ⎜⎜⎝1∗∗ 1 1 1⎞ ⎟⎟⎠/slashbig⎛ ⎜⎜⎝1∗ 1 1 1⎞ ⎟⎟⎠and next to HG 1/G1=⎛ ⎜⎜⎝1∗∗∗ 1∗ 1 1⎞ ⎟⎟⎠/slashbig⎛ ⎜⎜⎝1∗∗ 1∗ 1 1⎞ ⎟⎟⎠in the elementary abelian group G/G1. There we can solve the HSMC for Gby using the exact abelian hidden subgroup algorithm from [ 16]i nG/G1. In the non-exact setting, essentially the same proof gives the following. Proposition 3 Let G be a semi-elementary black-box group with unique encod- ing of order pn. Assume that there exists a quantum (or a randomized) algorithm that ﬁnds zero sum subsequences of sequences of S (p,n)elements of Zpnin time T (p,n)≥S(p,n)with high probability. Then the problem HSMC can be solved by a quantum algorithm that uses poly(T(p,n)O(c)/lscript)elementary operations, poly(T(p,n)O(c)log|G|)applications of the group oracle and calls to the oracle computing the puriﬁcation of the subgroup state. 5 Zero sum subsequences in Zn p In this section, we assume that our input is a sequence of vectors from Zn p.W ea l s o assume that pis an odd prime as for p=2 a zero sum subsequence can be obtained from n+1 vectors in the form of a zero linear combination. As subsequences can be 123 Zero sum subsequences and hidden subgroups

Question 891 multiple-choice

In computational algebra, the structure and properties of vector spaces over finite fields are fundamental for algorithms in coding theory and cryptography. Key aspects include spanning sets, unique solutions to linear systems, and applications of determinants such as the Vandermonde determinant. Which statement accurately describes the role of the Vandermonde determinant in proving that sets of vectors constructed from powers of elements in $\mathbb{Z}_p^n$ span the space of polynomials up to degree $p-1$? 1) It demonstrates that any collection of vectors is necessarily linearly dependent in $\mathbb{Z}_p^n$. 2) It ensures that only polynomials of degree less than $p-1$ can form a basis over $\mathbb{Z}_p$. 3) It is used exclusively for verifying the invertibility of matrices over real numbers, not finite fields. 4) It shows that all vectors in $\mathbb{Z}_p^n$ are orthogonal under standard inner product. 5) It guarantees linear independence of vectors constructed from powers of distinct elements, allowing these vectors to span the relevant space. 6) It implies that the sum of all powers of elements in $\mathbb{Z}_p$ is always zero. 7) It provides a probabilistic bound on the likelihood of randomly sampled vectors spanning the space.

✓ Correct Answer:

The correct answer is 5) It guarantees linear independence of vectors constructed from powers of distinct elements, allowing these vectors to span the relevant space..

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the y(p−1)for the previously sampled y’s. This immediately implies that if our sample size is of the order of the dimension of Z(p−1) p[x1,...,x n], the span of y(p−1)for the samples yisZ(p−1) p[x1,...,x n] with high probability. In that case, the linear equations y(p−1)·U= 1 have exactly one solution, which is u∗. From this unique solution one can easily recover a vector vsuch that v=aufor some 0

Question 892 multiple-choice

In quantum algorithms for hidden subgroup problems, coset states and quantum Fourier transforms play a central role in distinguishing group-theoretic structures using measurement protocols. The effectiveness of these protocols is often analyzed using concepts from representation theory and probability. Which parameter is central to bounding the average total variation distance between measurement outcomes associated with coset states of conjugate subgroups and the trivial subgroup in certain groups relevant to graph isomorphism problems? 1) The order of the group G 2) The dimension of the subgroup H 3) The number of coset states k 4) The quantity δ2, which depends on k and the pair (G, H) 5) The eigenvalues of the quantum Fourier transform 6) The maximum character value of the irreducible representations 7) The trace of the frame B used in measurement

✓ Correct Answer:

The correct answer is 4) The quantity δ2, which depends on k and the pair (G, H).

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Here one starts off with kindependent copies of the coset state σG H, i.e., with the state(σG H)⊗k∼=σGk Hkand applies QFT⊗k Gto it. HereGk,Hkdenote thek-fold direct product of G,H respectively. Notethat since/hatwiderGk∼=/hatwideG⊗k,wehavethat QFTGk= QFT⊗k G. Wecanexpress anirrep ρofGk asρ=⊗k i=1ρi,ρi∈/hatwideG;observe that Vρ=⊗k i=1Vρi. Weadopt theconvention that multiregister vectors and representations are denoted in boldface type. After applyi ngQFT⊗k G, we measure the name ρof an irrep ofGk, i.e, irrepsρ1,...,ρkofG. After that, we measure a row index of ρi.e., row indices of ρ1,...,ρk, and then measure the resulting reduced state in the column sp ace ofρusing a frame BofVρ. The frame B used depends onthe observed ρbut not onthe observed rowindices. Noticethat only the appl ication of the frameBmaybeanentangled measurement, theapplication of QFTGkandmeasurement of ρtogether with arow index of ρare single register operations. 3 The maintheorem LetGbe a group and h∈Gbe an involution, that is, H:={1,h}is an order two subgroup of G. We letHg:=gHg−1denote the conjugate of Hbyg∈G. Letkbe a positive integer. Fix a POVM Mon C[G]⊗k∼=C[Gk]. LetMHg,M{1}denote the classical probability distributions obtained b y measuring the statesσ⊗k Hg,σ⊗k {1}respectively according to M. We will show that the average total variation distance 6 betweenMHgandM{1}over conjugates Hg,g∈Gis at most 2ktimes a quantity that depends purely on the pair(G,H). In the next section, we will show that this quantity is expon entially small for many pairs (G,H)of interest, including when G=Sn≀S2andHis generated by an involutive swap, i.e., the case relevant toisomorphism of rigid graphs. Theorem 2 (Main theorem). LetGbe a ﬁnite group and H:={1,h}be an order two subgroup of G. Letk≥1be an integer. Fix a POVM MonC[G]⊗kand letMHg,M{1}denote the classical probability distributions obtained by measuring the states σ⊗k Hg,σ⊗k {1}respectively according to M. Forε >0, deﬁne the set Sε:=/braceleftbigg τ∈/hatwideG:|χτ(h)| dτ≥ε/bracerightbigg . Suppose that 2kε<1holds. Deﬁne δ1:=ε+1 |G|·/parenleftBigg/summationdisplay τ∈Sεdτ|χτ(h)|/parenrightBigg · /summationdisplay ν∈/hatwideGdν ≤ε+/parenleftBigg/summationdisplay τ∈Sεdτ|χτ(h)|/parenrightBigg ·/parenleftBigg |/hatwideG| |G|/parenrightBigg1/2 , and δ2:= 2k(1+2kε)δ1/2 1+3kε+3k |G|·/summationdisplay τ∈Sεd2 τ. Then Eg[/vextenddouble/vextenddoubleMHg−M{1}/vextenddouble/vextenddouble 1]≤δ2, where theexpectation is taken over the uniform distributio n ong∈G. By ak-entangled POVM Fontcoset states, we mean that Fconsists of a sequence of POVM’s (Mi)i∈[t′], where each Mioperates on a fresh set of at most k-coset states and t′≤t. The number of coset states operated upon by Fis at mostt. The outcome of Fis a sequence of length t′corresponding to the outcomes of Mi. The choice of Mimay depend on the observed outcomes of M1,...,Mi−1. If required, further classical postprocessing may be done on t he outcome of F. We now prove the following corollary of Theorem 2. Corollary 3. SupposeFis ak-entangled POVMon tcoset states. Then for at least a fraction of 1−√tδ2 conjugate subgroups Hg,g∈G,/vextenddouble/vextenddoubleFHg−F{1}/vextenddouble/vextenddouble 1≤/radicalbig tδ2. Proof.Using Theorem 2 and triangle inequality, it iseasy tosee tha tEg[/vextenddouble/vextenddoubleFHg−F{1}/vextenddouble/vextenddouble 1]≤tδ2. Applying Markov’s inequality to the expectation over g∈Gﬁnishes the proof. The remainder of the section is devoted to proving Theorem 2. We ﬁrst give some notation that will be useful for the proofs of various lemmas. Our notation and set up is inspired to a large extent by the notation in[MR05]. Asarguedintheprevioussection, wecanassumewithoutloss ofgenerality that

Question 893 multiple-choice

Shor's algorithm is a quantum method for integer factorization, utilizing both quantum and classical computational techniques. Quaternions are a mathematical structure that generalizes complex numbers and is essential in representing spatial rotations. Which key mathematical property did Hamilton sacrifice to generalize complex numbers into quaternions, enabling the construction of a four-dimensional normed division algebra? 1) Associativity 2) Existence of multiplicative inverses 3) Commutativity 4) Distributivity 5) Closure under addition 6) Existence of an identity element 7) Linearity

✓ Correct Answer:

The correct answer is 3) Commutativity.

📚 Reference Text:

of a. Ifj=85 is obtained, the continued fraction expansion of85 512is used to determine the period r. We have 85 512=1 6+1 42+1 2(3.41) and the set of convergents is1 6;42 253;85 512 : (3.42) Since r

Question 894 multiple-choice

Quantum phase estimation is a fundamental procedure in quantum computing, essential for tasks such as Hamiltonian simulation and quantum chemistry. Recent algorithmic improvements emphasize maintaining quantum coherence and optimizing accuracy with minimal resource overhead. When using coherent iterative estimators for quantum phase estimation, how does increasing the number of estimation bits from n to n+r affect the gap parameter α and what is the relationship between r and the target accuracy α? 1) It increases α linearly with r, and r is proportional to α. 2) It leaves α unchanged, and r depends on the square root of α. 3) It doubles α for each additional bit, and r is inversely proportional to α. 4) It reduces α logarithmically, with r given by the formula r = ⎡log₂(1/2α)⎤. 5) It increases the number of gaps but widens them, with r independent of α. 6) It decreases accuracy, and r grows exponentially with decreasing α. 7) It eliminates all gaps, and r is always equal to n.

✓ Correct Answer:

The correct answer is 4) It reduces α logarithmically, with r given by the formula r = ⎡log₂(1/2α)⎤..

📚 Reference Text:

perform median ampliﬁcation with gap parameter η0we are guaranteed that values of λ(x) jthat fall into a rounding gap will at least be rounded to an adjacent bin. Wecanusethisfacttoachievean ∼α−1dependence. Recallthat αisthefractionofthe range [0,1)whereλjare not allowed to appear due to the rounding promise, independent ofn. If we perform phase estimation with n+ 1rather than n, then the width of each disallowed region is cut in half from α2−ntoα2−n−1- but we also double the number of disallowed regions. However, if we simply ignore the ﬁnal bit, then, because we are amplifying to η0, any values of λjthat fall into one of the newly introduced regions will simply be rounded. Thus, we cut the gaps in half without introducing any new gaps, so αis cut in half. We will make this argument more formal in a moment. So, in order to achieve a particular α, we observe from Figure 1 that if we amplify to η0then the gaps have width1 2·2−n. If we estimate an additional rbits, then the gaps have width1 2·2−n−r. Solving1 2·2−n−r≤α2−nforr, we obtain r:=/ceilingleftbigg log2/parenleftbigg1 2α/parenrightbigg/ceilingrightbigg . (67) We brieﬂy make the nexplicit inλ(x) jby writingλ(n,x) j. After running the protocol at the beginning of the proof with n+rbits, we obtain, for /vector x∈{0,...,2n−1}M, the state: /summationdisplay /vector x|median (/vector x)/angbracketright⊗M/circlemultiplydisplay l=12r−1/summationdisplay t=0β(λ(n+r,2rxl+t) j )|2rkl+t/angbracketright⊗|ψj/angbracketright (68) =2n−1/summationdisplay x=0|x/angbracketright⊗/summationdisplay /vector xwhere median (/vector x)=kxM/circlemultiplydisplay l=12r−1/summationdisplay t=0β(λ(n+r,2rxl+t) j )|2rxl+t/angbracketright⊗|ψj/angbracketright (69) =2n−1/summationdisplay x=0√pj,xeiϕj,x|x/angbracketright⊗|garj,x/angbracketright⊗|ψj/angbracketright (70) where in the last step we have deﬁned the normalized state |garx/angbracketright, the probability pj,x, and the phase ϕj,xvia: √pj,xeiϕj,x|garj,x/angbracketright:=/summationdisplay /vector xwhere median (/vector x)=xM/circlemultiplydisplay l=12r−1/summationdisplay t=0β(λ(n+r,2rxl+t) j )|2rxl+t/angbracketright (71) Now if we can demonstrate that pj,x≥1−δmedwhenx=ﬂoor(2nλj)then the same Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 15 argument as in (60-65) holds and we are done. Observe that: pj,x:=/summationdisplay /vector xwhere median (/vector x)=xM/productdisplay l=12r−1/summationdisplay t=0/vextendsingle/vextendsingle/vextendsingleβ(λ(n+r,2rxl+t) j )/vextendsingle/vextendsingle/vextendsingle2(72) Sayx=ﬂoor(2nλj). Then, consider t:=ﬂoor(2n+rλj)−x2r, which is an integer ∈{0,...,2r−1}. Then, looking at Figure 1, we see that: |β(λ(n+r,2rx+t) j )|2+|β(λ(n+r,2rx+t+1) j )|2≥γ(λ(n+r,2rx+t) j ) +γ(λ(n+r,2rx+t) j−1)≥1 2+η0 (73) Therefore the probability that the ﬁrst nbits arexis/summationtext2r−1 t=0/vextendsingle/vextendsingle/vextendsingleβ(λ(n+r,2rx+t) j )/vextendsingle/vextendsingle/vextendsingle2≥ 1/2 +η0, which is all that was required to perform the median ampliﬁcation argument above. So we can conclude that pj,k≥1−δmed, so the modiﬁed algorithm retains the same accuracy. 2 Coherent Iterative Estimation In this section we present the novel algorithms for phase estimation and energy estimation. Asdescribedintheintroduction, bothofthesefeatureastrongsimilarityto‘iterativephase estimation’ [Kit95], where the bits of the estimate are obtained one at a time. Unlike iterative phase estimation however, the state is never measured and the entire process is coherent. We therefore name these algorithms as ‘coherent iterative estimators’. Another similarity that these new algorithms share with the original iterative phase estimationisthattheless signiﬁcantbitsaretakenintoaccountwhenobtainingthecurrent bit. This greatly reduces the amount of ampliﬁcation required for the later bits, so the runtime is vastly dominated by the estimation of the least signiﬁcant bit. We will go into more detail on this later. To permit discussion of coherent iterative estimation of phases and energies in a uniﬁed manner, we ﬁt this

Question 895 multiple-choice

Non-perturbative techniques in quantum field theory are essential for studying strongly correlated systems where traditional perturbative methods are insufficient. Spectral representations and dimensional regularisation play important roles in maintaining theoretical consistency in these analyses. Which of the following approaches preserves space-time, chiral, and gauge symmetries while providing a non-perturbative framework for calculating real-time correlation functions in strongly interacting quantum field theories? 1) Spectral renormalisation with dimensional regularisation 2) Lattice discretisation without symmetry constraints 3) Bare perturbative expansion using Feynman diagrams 4) Mean-field approximation in imaginary time formalism 5) Hard cutoff regularisation applied to Green’s functions 6) Wick rotation combined with ultraviolet momentum integration 7) Boltzmann transport equation for classical fields

✓ Correct Answer:

The correct answer is 1) Spectral renormalisation with dimensional regularisation.

📚 Reference Text:

Title: Spectral functions in the $\phi^4$-theory from the spectral DSE Year: 2020 Paper ID: e1e95927964a17037d01de2b5d49e96b3ce7067b Source: semantic-scholar URL: https://www.semanticscholar.org/paper/e1e95927964a17037d01de2b5d49e96b3ce7067b Abstract: We develop a non-perturbative functional framework for computing real-time correlation functions in strongly correlated systems. The framework is based on the spectral representation of correlation functions and dimensional regularisation. Therefore, the non-perturbative spectral renormalisation setup here respects all symmetries of the theories at hand. In particular this includes space-time symmetries as well as internal symmetries such as chiral symmetry, and gauge symmetries. Spectral renormalisation can be applied within general functional approaches such as the functional renormalisation group, Dyson-Schwinger equations, and two- or $n$-particle irreducible approaches. As an application we compute the full, non-perturbative, spectral function of the scalar field in the $\phi^4$-theory in $2+1$ dimensions from spectral Dyson-Schwinger equations. We also compute the $s$-channel spectral function of the full $\phi^4$-vertex in this theory.

Question 896 multiple-choice

Cryptographic key exchange protocols often rely on mathematical group structures, with non-commutative groups being explored for enhanced post-quantum security. Minimal non-abelian metacyclic groups, particularly of the Millera Moreno type, offer unique algebraic challenges relevant to computational hardness. Which feature of metacyclic groups of the Millera Moreno type makes them especially promising for constructing secure key exchange protocols resistant to both classical and quantum attacks? 1) Their order is always a prime number 2) They are isomorphic to all finite cyclic groups 3) Every element commutes with every other element 4) The subgroup structure is trivial 5) The discrete logarithm problem is easy in these groups 6) The conjugacy problem is computationally intractable 7) They contain no non-trivial subgroups

✓ Correct Answer:

The correct answer is 6) The conjugacy problem is computationally intractable.

📚 Reference Text:

Title: Some Approach to Key Exchange Protocol Based on Non-commutative Groups Year: 2020 Paper ID: 6b72899d7368ede9753c0ea9fd9d57d4e026a027 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/6b72899d7368ede9753c0ea9fd9d57d4e026a027 Abstract: We consider non-commutative generalization of CDH problem [1,2] on base of metacyclic group G of type Millera Moreno (minimal non-abelian group). We show that conjugacy problem in this group are intractable. The algorithm of generating (desinging) of common key in non-commutative group with 2 mutually commuting subgroups are constructed by us.

Question 897 multiple-choice

Quantum algorithms often rely on optimal measurements to extract information from quantum states, especially in group-theoretic problems such as the hidden subgroup problem. The pretty good measurement (PGM) is a central concept in distinguishing between quantum states associated with different properties of hidden parameters. In the context of the dihedral hidden subgroup problem, which property makes the constructed measurement operators E+ and E- optimal for determining the least significant bit (parity) of the hidden shift d? 1) They minimize the trace distance between all possible quantum states. 2) They rely solely on classical post-processing of measurement outcomes. 3) They equalize the success probability for all values of d by symmetrizing over group elements. 4) They utilize adaptive measurement strategies based on previous outcomes. 5) They satisfy the positivity and completeness conditions required by a general theorem for optimal quantum measurements. 6) They only distinguish between subsets with zero group elements. 7) They are constructed using entanglement between multiple quantum registers.

✓ Correct Answer:

The correct answer is 5) They satisfy the positivity and completeness conditions required by a general theorem for optimal quantum measurements..

📚 Reference Text:

BACON, ANDREW M. CHILDS, AND WIM VAN DAM To establish this result, we proceed as before: we ﬁrst identify the opti- mal measurement, then derive an expression for its success probability, and ﬁnally place bounds on this expression. Our goal is to determine the least signiﬁcant bit of d, i.e., whether dis even or odd. In other words, we would like to distinguish the two density matrices (45) ρ(k) ±:=2 N/summationdisplay deven,oddρ⊗k d. Sinceρ(k) ++ρ(k) −=2 NG(withGgiven in (21)), the pgm for these states has the two povm operators E±:=N 2G−1/2ρ(k) ±G−1/2(46) =/summationdisplay deven,oddEd. (47) Now for simplicity we assume Nis even such that we have the identities /summationdisplay devenωd(p−q)=/braceleftBigg N 2p=qorp=q+N/2 0 otherwise(48) and /summationdisplay doddωd(p−q)=  N 2p=q −N 2p=q+N/2 0 otherwise(49) which can then be used to obtain ρ(k) ±=1 (2N)k/summationdisplay x∈Zk N/summationdisplay r∈ZN(ηx r|Sx r,x/angbracketright/angbracketleftSx r,x| ±/radicalBig ηxrηx r+N/2|Sx r,x/angbracketright/angbracketleftSx r+N/2,x|)(50) E±=1 2/summationdisplay x∈Zk N/summationdisplay r∈ZN(|Sx r,x/angbracketright/angbracketleftSx r,x| ± |Sx r,x/angbracketright/angbracketleftSx r+N/2,x|).(51) We claim that this pgm is the optimal measurement for determining the least signiﬁcant bit of the shift. To see this, check the conditions of Theo- rem 1. We have /summationdisplay i∈ZNρ⊗k iEi=ρ(k) +E++ρ(k) −E− (52) =1 (2N)k/summationdisplay x∈Zk N/summationdisplay r∈ZN/parenleftBig ηx r+/radicalBig ηxrηx r+N/2/parenrightBig |Sx r,x/angbracketright/angbracketleftSx r,x| (53) OPTIMAL MEASUREMENTS FOR THE DIHEDRAL HSP 15 since the cross terms cancel. A similar calculation veriﬁes (16). Then (54)/summationdisplay i∈ZNρ⊗k iEi−ρ(k) ±= 1 (2N)k/summationdisplay x∈Zk N/summationdisplay r∈ZN/radicalBig ηxrηx r+N/2(|Sx r,x/angbracketright/angbracketleftSx r,x| ∓ |Sx r,x/angbracketright/angbracketleftSx r+N/2,x|), which is clearly a positive matrix, verifying (17). The success probability of this optimal measurement is independent of whetherdis even or odd, and is given by ˜p:= trE+ρ(k) + (55) =1 2(2N)k/bracketleftBig/summationdisplay x∈Zk N/summationdisplay r∈ZNηx rtr(|Sx r,x/angbracketright/angbracketleftSx r,x|+|Sx r,x/angbracketright/angbracketleftSx r+N/2,x|) (56) +/summationdisplay x∈Zk N/summationdisplay r∈ZN/radicalBig ηxrηx r+N/2tr(|Sx r,x/angbracketright/angbracketleftSx r,x|+|Sx r,x/angbracketright/angbracketleftSx r+N/2,x|)/bracketrightBig =1 2/bracketleftBig 1 +1 (2N)k/parenleftBig/summationdisplay x∈Zk N/summationdisplay r∈ZN/radicalBig ηxrηx r+N/2/parenrightBig/bracketrightBig (57) where we have used the fact that/summationtext x∈Zk N/summationtext r∈ZNηx r= (2N)k. With these expressions in hand, we are ready to prove Theorem 4: Proof. We bound the expression (57) for the success probability of the op- timal measurement. Because the η’s are integers we have for all r /summationdisplay x∈Zk N/radicalBig ηxrηx r+N/2≤/summationdisplay x∈Zk Nηx rηx r+N/2. (58) For theN−2 casesr/negationslash= 0,N/2 it holds that /summationdisplay x∈Zk Nηx rηx r+N/2= (2k−1)(2k−2)Nk−2, (59) while for the remaining r= 0 orr=N/2 we have /summationdisplay x∈Zk Nηx 0ηx N/2= (2k−1)(2k−2)Nk−2+ (2k−1)Nk−1. (60) In both equalities the (2k−1)(2k−2)Nk−2term was obtained by considering a non-empty subset S(of which there are 2k−1) and a distinct non-empty subsetT(of which there are 2k−2) and two elements i,jsuch that either i∈S−Tandj∈T, ori∈T−Sandj∈S(such a choice is always possible becauseSandTare non-empty and distinct). Then for any values of the remainingk−2 elements, there is exactly one choice for elements iandj such that the elements in Ssum torand the elements in Tsum tor+N/2. For the case r= 0,N/2 we have to add the possibility that Sis the empty 16 DAVE BACON, ANDREW M. CHILDS, AND WIM VAN DAM subset and Tis a non-empty subset with element isuch that for any

Question 898 multiple-choice

In representation theory and quantum algorithms, Kronecker coefficients play a crucial role in understanding the tensor products of symmetric group representations and their computational complexity. The algebraic structure K(n), built from the symmetric group Sn, is deeply connected to combinatorial objects and operator algebras. Which statement accurately describes the relationship between the dimension of K(n) and Kronecker coefficients for the symmetric group Sn? 1) The dimension of K(n) equals the product of the orders of all irreducible representations of Sn. 2) The dimension of K(n) is the sum of all Kronecker coefficients for pairs of irreducible representations of Sn. 3) The dimension of K(n) is given by the sum over squared Kronecker coefficients indexed by triples of irreducible representations of Sn. 4) The dimension of K(n) is always equal to n! for any n. 5) The dimension of K(n) corresponds to the total number of conjugacy classes in Sn. 6) The dimension of K(n) is determined exclusively by the Clebsch-Gordan coefficients for Sn. 7) The dimension of K(n) is unrelated to ribbon graph enumeration or representation theory.

✓ Correct Answer:

The correct answer is 3) The dimension of K(n) is given by the sum over squared Kronecker coefficients indexed by triples of irreducible representations of Sn..

📚 Reference Text:

[ 32,64]. These algorithms are based on regarding K(n)as Hilbert spaces. The dimensions of these Hilbert spaces grow as n!at largen(see [65] for all orders asymptotic formulae for these dimensions). An important motivation for this paper has been to take advantage of known exponential improvements due to quantum algorithms [ 58] to formulae eﬃcient algorithms based onK(n)for Kronecker coeﬃcients, since an important element of the interest in Kronecker coeﬃcients comes from computational complexity theory [ 66–69]. Our perspective is that, while the intersection of representation theory in mathematics and computational complexity in computer science is a fascinating interface with many interesting outcomes, the mathematical question of combinatorial constructibility of representation theory quantities is sometimes fruitfully kept distinct from the complexity theoretic questions associated with the precise choice of computational task based on Kronecker coeﬃcients and the eﬃciency of algorithms for the chosen task. Informally speaking we would argue that construction problems, such as making hay and manufacturing needles, are distinct from complexity questions, such as the diﬃculty of ﬁnding a needle in a haystack. We review the basics ofK(n)from [31] and apply QPE to projector detection task in K(n). The projector is labelled by a triple of Young diagrams with non-vanishing coeﬃcient. – 24 – JHEP05(2023)191Consider C(Sn)⊗CC(Sn), simply denoted C(Sn)⊗C(Sn). Introduce the tensor product elementsσ1⊗σ2and sum over all their diagonal conjugates as σ1⊗σ2→/summationdisplay γ∈Snγσ1γ−1⊗γσ2γ−1(6.1) Now, consider the C-vector subspaceK(n)⊂C(Sn)⊗C(Sn)spanned by all/summationtext γ∈Snγσ1γ−1⊗ γσ2γ−1,σ1andσ2∈Sn: K(n) = SpanC/braceleftBig/summationdisplay γ∈Snγσ1γ−1⊗γσ2γ−1, σ1,σ2∈Sn/bracerightBig (6.2) The dimension of K(n)maps to the number of ribbon graphs with nedges, and equivalently to a sum over triples R1,R2,R3of irreducible representations (irreps) of Sn[31]: |Rib(n)|= Dim(K(n)) =/summationdisplay R1,R2,R3/turnstileleftnC(R1,R2,R3)2(6.3) The Kronecker coeﬃcient C(R1,R2,R3)is a non-negative-integer that yields the number of timesR3appears in the tensor product decomposition R1⊗R2. The Kronecker coeﬃcient is expressed in terms of characters as C(R1,R2,R3) =1 n!/summationdisplay σ∈SnχR1(σ)χR2(σ)χR3(σ) (6.4) In an analogous way of section 3, there are operators that multiplicatively generate the centre ofK(n). At anyn≥2, we deﬁne elements in C(Sn)⊗C(Sn) T(1) k=Tk⊗1 =/summationdisplay σ∈Ckσ⊗1, T(2) k= 1⊗Tk=/summationdisplay σ∈Ck1⊗σ, T(3) k=/summationdisplay σ∈Ckσ⊗σ. (6.5) The sum of products of the T(i) k’s,k= 1,...,n, generates the centre Z(K(n))ofK(n). In fact, one does not need the entire set k= 1,...,nto generate the centre, only a fewer number of them is enough k= 1,...,k∗(n)≤n. K(n)has a matrix basis QR1,R2,R3τ1,τ2=dR1dR2 (n!)2C(R1,R2,R3)/summationdisplay τ=1/summationdisplay σ1,σ2∈Sn/summationdisplay i1,i2,i3,j1,j2 CR1,R2;R3,τ1 i1,i2;i3CR1,R2;R3,τ2 j1,j2;i3DR1 i1j1(σ1)DR2 i2j2(σ2)σ1⊗σ2 (6.6) obeying QR1,R2,R3τ1,τ2QR/prime 1,R/prime 2,R/prime 3 τ/prime 1,τ/prime 2=δR1,R/prime 1δR2,R/prime 2δR3,R/prime 3δτ2,τ/prime 1QR1,R2,R3 τ1,τ/prime 2(6.7) DR ij(σ)are the matrix elements of the linear operator DR(σ)in an orthonormal basis for the irrepR. The index τruns over an orthonormal basis for the multiplicity space of dimension – 25 – JHEP05(2023)191equals to the Kronecker coeﬃcient C(R1,R2,R3).κR1,R2,R3is a normalization factor. CR1,R2;R3,τ1 i1,i2;i3are Clebsch-Gordan coeﬃcients of the representations of Sn(see the appendices of [31] for the properties needed to prove that this expression gives a Wedderburn-Artin basis forK(n)). The centreZ(K(n))is spanned by PR1,R2,R3=/summationdisplay τQR1,R2,R3τ,τ This is equal to an expression in terms of characters ˜PR1,R2,R3= ∆(PR3)(PR1⊗PR2) =1 (n!)3dR1dR2dR3/summationdisplay σ1,σ2,σ3∈SnχR1(σ1)χR2(σ2)χR3(σ3)σ3σ1⊗σ3σ2(6.8) The equality PR1,R2,R3=˜PR1,R2,R3can be shown using the properties: PR/prime

Question 899 multiple-choice

Computational chemistry strategies are essential for simulating vibrational spectra of molecules in complex environments, balancing accuracy with computational efficiency. Hybrid quantum mechanical and molecular mechanical approaches have been developed to address limitations in traditional high-level methods. Which computational strategy enables accurate simulation of solvent-induced vibrational frequency shifts while significantly reducing computational cost compared to fully high-level quantum calculations? 1) Applying only classical molecular mechanics for all frequency calculations 2) Using vibrational configuration interaction (VCI) exclusively along the entire molecular dynamics trajectory 3) Relying solely on electrostatics-based spectroscopic mapping methods 4) Implementing path integral molecular dynamics for every spectral evaluation 5) Combining high-level quantum calculations for reference frequencies with low-level QM/MM methods for frequency shifts along the trajectory 6) Neglecting nuclear quantum effects and solvent interactions in all calculations 7) Modeling all vibrational couplings as strictly harmonic oscillators

✓ Correct Answer:

The correct answer is 5) Combining high-level quantum calculations for reference frequencies with low-level QM/MM methods for frequency shifts along the trajectory.

📚 Reference Text:

method. Intramolecular (and intermolecular) mode-coupling can also be treated by using vibrational configuration interaction (VCI) as implemented in the MULTIMODE program, but the computational costs are prohibitive for computation of time-dependent vibrational frequencies. A linear scaling strategy is clearly desired. To further reduce computational costs, we examined a hybrid, or composite strategy that can yield accurate peak position and line shape. In this scheme, the reference-state vibrational frequency (eq 1) is determined with a highly accurate method only once, but the vibrational frequency shifts at each instantaneous configuration along the MD trajectory are evaluated using the QVP method with a lower-level QM/MM potential. In the HCl–water cluster test cases, we compared the spectral result determined by using CCSD(T)-F12b on the entire system with that from the hybrid strategy with CCSD(T)-F12b reference frequency and M06-2X frequency shift. The agreement is very good, especially considering the tremendous cost-saving. Importantly, the inclusion of nuclear quantum effects, either using the semiclassical (eq 5) or cumulant expansion (eq 6) algorithm, has a large effect on the computed spectra and is found to be in better agreement with experiment than that obtained by Fourier transform of the classical dipole autocorrelation function. We found that anharmonicity in the H–Cl stretch is significant and has a large solvent effect in the relaxation process exhibited in the computed 2D-IR spectra. To illustrate the general applicability, solvent effects on the carbonyl stretch of acetone in water were determined using QVP1 with a dual-level approach using the CCSD(T) results as the reference state coupled with combined QM(AM1)/MM simulations to account for solute–solvent interactions. The computed solvent-induced spectral shifts are in accord with experimental results. The present QVP method complements other computational approaches, including path integral-based molecular dynamics, and represents a major improvement over the electrostatics-based spectroscopic mapping procedures. Supplementary Material Refer to Web version on PubMed Central for supplementary material. Acknowledgments We thank Martin A. Suhm and Michal Fárník for providing the raw experimental data. This research has been supported by grants from the National Natural Science Foundation of China (Grant Nos. 21273094 and 21533003), Program for New Century Excellent Talents in University and the National Institutes of Health (Grant No. GM46376). References 1. Skinner JL, Pieniazek PA, Gruenbaum SM. Vibrational Spectroscopy of Water at Interfaces. Acc. Chem. Res. 2012; 45:93–100. [PubMed: 22032305] Xue et al. Page 12 J Chem Theory Comput . Author manuscript; available in PMC 2018 February 01. Author Manuscript Author Manuscript Author Manuscript Author Manuscript 2. Fried SD, Boxer SG. Measuring Electric Fields and Noncovalent Interactions Using the Vibrational Stark Effect. Acc. Chem. Res. 2015; 48:998–1006. [PubMed: 25799082] 3. Fried SD, Bagchi S, Boxer SG. Extreme electric fields power catalysis in the active site of ketosteroid isomerase. Science. 2014; 346:1510–1514. [PubMed: 25525245] 4. Kim H, Cho M. Infrared Probes for Studying the Structure and Dynamics of Biomolecules. Chem. Rev. 2013; 113:5817–5847. [PubMed: 23679868] 5. Reppert M, Tokmakoff A. Electrostatic frequency shifts in amide I vibrational spectra: Direct parameterization against experiment. J. Chem. Phys. 2013; 138:134116. [PubMed: 23574217]

Question 900 multiple-choice

Quantum operations on mixed states can be complicated when transformations depend on classical parameters such as space or time, especially in experimental settings like nuclear magnetic resonance (NMR). Understanding how density matrices evolve under these conditions is crucial for accurate modeling and error mitigation in quantum information processing. When the applied quantum transformation and the ensemble’s probability distribution both depend on a classical parameter, such as spatial position, which statement correctly describes the evolution of the final density matrix? 1) The final density matrix is always obtained by applying the transformation to the initial state, then averaging over measurement outcomes. 2) The final density matrix generally cannot be obtained by simply averaging the transformation over the classical parameter applied to the initial density matrix. 3) There always exists a linear superoperator mapping any initial density matrix to the correct final density matrix in this scenario. 4) The evolution is guaranteed to remain completely positive for any choice of transformation and probability distribution. 5) The initial mixed state must be replaced by a pure state to correctly model the system’s evolution. 6) Incoherent errors are never caused by spatial variations of experimental parameters. 7) Non-linear transformations are always preferable in practical quantum information processing.

✓ Correct Answer:

The correct answer is 2) The final density matrix generally cannot be obtained by simply averaging the transformation over the classical parameter applied to the initial density matrix..

📚 Reference Text:

via weak interactions with a Markovian environment, it is not diﬃcult to construct situation s in which not only complete positivity, but even the distinction between a state and its transformations breaks down [24, 25, 26, 27]. Such situations can only arise when the initial st ate of the system is not pure, but can be represented by a probability distribution ove r an ensemble of pure states with density matrix ρini. Suppose the ensemble’s probability distribution depends on some classical parameter c(usually space or time) and that ρinialso represents an average overc(see below for a concrete example). Then if the applied transforma tionKlikewise depends on c, so that the transformation is correlated with the states in the ensemble, then the ﬁnal density matrix ρﬁnwill generally not be equal to the result of applying the average ofKoverctoρini. Indeed there will usually be no superoperator which maps every po s- sibleρinito the correct ρﬁn. In principle one could deﬁne a non-linear transformation that produces this mapping, but this is not very useful in practice becau se the large amount of 7 information needed to deﬁne a general non-linear transformation is neither readily available nor easy to work with. Incoherent errors are precisely those which depend upon some cla ssical parameter that labels the members of an ensemble, and so generate a correlation be tween the states in the ensemble and this parameter. In the NMR experiments described be low, the main source of incoherent errors is the spatial inhomogeneity in the RF (radio-f requency) ﬁeld over the sample volume. This may be expressed by writing the Hamiltonian for th e interaction of the spins with the RF ﬁeld in the form HRF(t;/vector r) =α(t)γB1(/vector r)e−i∆ωtσz/2σxei∆ωtσz/2, (11) whereα(t) is the envelope of the ﬁeld, γis the gyromagnetic ratio of the spins, B1(/vector r) is the magnitude of the ﬁeld at the point /vector r, ∆ωis the diﬀerence between the frequency of the RF ﬁeld and the resonance frequency of the spins in the static magn etic ﬁeld B0, and the σ’s are the usual Pauli matrices. Thus, the unitary operation U= exp(/integraltext dtHRF) depends upon the position /vector rin the sample as well as time, i.e. U=U(t;/vector r), as does the resulting spin state ρ(t;/vector r) =U(t;/vector r)ρ(0;/vector r)U†(t;/vector r). The equilibrium state of the spins in the static ﬁeldB0is independent of /vector rover the sample volume, so we may take ρ(0;/vector r) =ρ(0) to be constant. The density operator that is observed is the integral o ver the sample volume V, i.e. ρ(t) =/integraldisplay Vd/vector r ρ(t;/vector r) =/integraldisplay Vd/vector r U(t;/vector r)ρ(0)U†(t;/vector r)≡S(t)ρ(0). (12) whereS(t) denotes the net superoperator of the actual quantum operat ion implemented. We can express this in matrix form as [13] col(S(t)ρ(0)) =/parenleftBig/integraldisplay Vd/vector rU(t;/vector r)⊗U(t;/vector r)/parenrightBig col(ρ(0)), (13) whereUis the complex conjugate of Uand the “ col” operator maps N×Nmatrices to (N2)-dimensional column vectors, as described above. Describing the evolution of an ensemble by integrating over its spatia l degrees of freedom works well for

Question 901 multiple-choice

Family symmetries and Grand Unified String Theories (GUSTs) are explored as frameworks to understand why quarks and leptons have their observed mass spectra and mixing patterns. These models often involve embedding the Standard Model within larger gauge groups and addressing the challenges of symmetry breaking using advanced group-theoretic methods. Which gauge group construction allows the incorporation of an SU(3)H family symmetry within a rank 16 Grand Unified String Theory model derived from four-dimensional heterotic superstring theory with free fermions? 1) SU(4) × U(1) × (SU(2) × U(1))H 2) SO(14) × (SU(2) × U(1))H ⊂ SO(18) 3) SU(5) × SU(3)H ⊂ SO(10) 4) SO(10) × (SU(3) × U(1))H ⊂ SO(16) 5) SU(6) × (SU(2) × U(1))H 6) E(7) × SU(2)H ⊂ E(8) 7) SU(5) × SU(2) × U(1)H

✓ Correct Answer:

The correct answer is 4) SO(10) × (SU(3) × U(1))H ⊂ SO(16).

📚 Reference Text:

Title: Non-Abelian Gauge Family Symmetry in Rank 8 and 16 Grand Unified String Theories Year: 1995 Paper ID: 4aecde5e64a9a493638658ab2963e1d923c0478d Source: semantic-scholar URL: https://www.semanticscholar.org/paper/4aecde5e64a9a493638658ab2963e1d923c0478d Abstract: The one of the main points of the investigations in high energy physics is to study the next chain: a law of the quark and lepton mass spectra → the puzzles of the quark and lepton family mixing → a possible new family dynamics. The new family symmetry dynamics might be connected to the existence of some exotic gauge or matter fields or something yet. For this it will better to study the possibilities of the appearence this gauge symmetry in the framework of the Grand Unified String Theories. In the framework of four dimensional heterotic superstring with free fermions we investigate the rank eight Grand Unified String Theories (GUST) which contain the SU(3)H-gauge family symmetry. We explicitly construct GUST with gauge symmetry G = SU(5)×U(1)× (SU(3)×U(1))H and G = SO(10)× (SU(3)×U(1))H ⊂ SO(16) or E(6)× SU(3)H ⊂ E(8) in free complex fermion formulation. As the GUSTs originating from Kac-Moody algebras (KMA) contain only low-dimensional representations it is usually difficult to break the gauge symmetry. We solve this problem taking for the observable gauge symmetry the diagonal subgroup G of rank 16 group G×G ⊂ SO(16)×SO(16) or (E(6)× SU(3)H) ⊂ E(8)× E(8). We discuss the possible fermion matter and Higgs sectors in these models. In these GUST there has to exist ”superweak” light chiral matter(mfH < MW ). The understanding of quark and lepton mass spectra and family mixing leave a possibility for the existence of an unusually low mass breaking scale of the SU(3)H family gauge symmetry (some TeV).

Question 902 multiple-choice

Quantum algorithms often leverage group-theoretic structures to solve problems efficiently, especially when dealing with the Hidden Subgroup Problem. Understanding the properties of nilpotent groups, p-groups, and their subgroup structures is critical in this domain. Which property uniquely characterizes nil-2 p-groups of exponent p, making them particularly suitable for efficient quantum algorithms in subgroup identification? 1) Their commutator subgroup is always trivial. 2) All elements have infinite order. 3) Both the group’s quotient by its commutator subgroup and the commutator subgroup itself are elementary abelian p-groups. 4) They cannot be broken down into abelian components through a normal series. 5) Their lower central series does not terminate at the trivial subgroup. 6) The center of the group is a non-abelian subgroup. 7) The commutator operation fails to be bilinear.

✓ Correct Answer:

The correct answer is 3) Both the group’s quotient by its commutator subgroup and the commutator subgroup itself are elementary abelian p-groups..

📚 Reference Text:

fact whose proof is immediate from Lemma 1 in [11] recast s the existence of the standard algorithm for the abelian HSPin the context of hiding sets. Fact 1. LetGbe a ﬁnite abelian group. If there exists an eﬃcient quantum p rocedure which hides the subgroup HofGthen there is an eﬃcient quantum algorithm for ﬁnding H. 2.2 Nilpotent groups LetGbe a ﬁnite group. For two elements g1andg2ofG, we usually denote their product by g1g2. If we conceive group multiplication from the right as a group action of Gon itself, we will use the notation g1·g2 forg1g2. We write H≤GwhenHis a subgroup of G, andH < Gwhen it is a proper subgroup. Normal subgroups and proper normal subgroups will be denoted respect ively byH/triangleleftequalGandH⊳G. For a subset X ofG, let/a\}bracketle{tX/a\}bracketri}htbe the subgroup generated by X. Thenormalizer ofXinGisNG(X) ={g∈G:gX=Xg}. For an integer n, we denote by Znthe group of integers modulo n, and for a prime number p, we denote by Z∗ pthe multiplicative group of integers relatively prime with p. The commutator [ x,y] of elements xandyisx−1y−1xy. For two subgroups XandYofG, let [X,Y] be/a\}bracketle{t{[x,y] :x∈X,y∈Y}/a\}bracketri}ht. The derived subgroup G′ofGis deﬁned as [ G,G], and its center Z(G) as{z∈G:gz=zgfor allg∈G}. Thelower central series ofGis the series of subgroups G= 3 A1/trianglerightequalA2/trianglerightequalA3...,whereAi+1= [Ai,G] for every i >1. Theupper central series ofGis the series of subgroups {1}=Z0/triangleleftequalZ1/triangleleftequalZ2...,whereZi+1={x∈G: [x,g]∈Zifor allg∈G}for every i >0. Clearly A2=G′andZ1=Z(G). The group Gisnilpotent if there is a natural number nsuch that An+1={1}. If nis the smallest integer such that An+1={1}thenGisnilpotent of class n. It is a well known fact that Gis nilpotent of class nif and only if Zn=Gin the upper central series. Nilpotent groups of class 1 are simply the nontrivial abelian groups. A nilpotent group of class at mos tnis called a nil-ngroup. A detailed treatment of nilpotent groups can be found for example in Hall [7]. Let us just recall here that nilpotent groups are solvable, and that every p-group is nilpotent, where a p-group is a ﬁnite group whose order is a power of some prime number p. 2.3 Nil- 2p-groups of exponent p It is clear from the deﬁnition of nilpotent groups that Gis a nil-2 group if G′≤Z(G).It is easy to see that this property implies that the commutator is a bilinear function in the f ollowing sense. Fact 2. LetGbe a nil-2group. Then for every g1,g2,g3,g4inG, [g1g2,g3g4] = [g1,g2][g1,g3][g2,g3][g2,g4]. The quantum part of our algorithm will deal only with special nilpotent groups of class 2, which are also p-groups and are of exponent p. The structure of these special groups is well known, and is expre ssed in the following simple fact. Fact 3. LetGbe a p-group of exponent pand of nilpotency class 2. Then there exist positive integers m andd, group elements x1,...,x m∈Gandz1,...,z d∈G′such that 1.G/G′∼=Zm pandG′∼=Zd p, 2. for every g∈Gthere exists a unique (e1,...,e m,f1,...fd)∈Zm+d psuch that g=xe1 1...xem mzf1 1...zfd d, 3.G=/a\}bracketle{tx1,...,x m/a\}bracketri}htandG′=/a\}bracketle{tz1,...,z d/a\}bracketri}ht. We will say that a nil-2 p-groupGof exponent phas parameters ( m,d) ifG/G′∼=Zm pandG′∼=Zd

Question 903 multiple-choice

In group theory, permutation groups model the rearrangement of elements and are fundamental to understanding symmetry and structure. The concept of "nudgability" examines whether certain subgroup properties remain invariant under inversion of permutations. Which subgroup is guaranteed to be nudgable, based on the symmetry properties of permutation groups? 1) Any abelian subgroup of a permutation group 2) The full symmetric group on n elements 3) Any standard dihedral group 4) All alternating groups for every n 5) All subgroups of the permutation group on 3 elements 6) The other half of alternating groups that are not non-nudgable 7) Every subgroup of the permutation group on 4 elements

✓ Correct Answer:

The correct answer is 6) The other half of alternating groups that are not non-nudgable.

📚 Reference Text:

Title: Non-nudgable subgroups of permutations Year: 2017 Paper ID: 3ba7d409a3c6f8d88c4d795dc4968224d5fd4420 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/3ba7d409a3c6f8d88c4d795dc4968224d5fd4420 Abstract: Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is called non-nudgable if the cardinality of this set always remains the same when replacing the initial permutation with its inverse. It is called nudgable otherwise. We show that all full permutation groups, standard dihedral groups, half of the alternating groups, and any abelian subgroup are non-nudgable. In the right probabilistic sense, it is thus quite likely that a randomly generated subgroup is non-nudgable. However, the other half of the alternating groups are nudgable. We also construct a smallest possible nudgable group, a 6-element subgroup of the permutation group on 4 elements.

Question 904 multiple-choice

Quantum process tomography (QPT) is a key technique in experimental quantum computing, allowing researchers to characterize the performance of quantum gates implemented on physical qubits. The quantum Fourier transform (QFT) is a fundamental operation used in several quantum algorithms and is realized through sequences of basic quantum gates. Which of the following gate types is essential for implementing the quantum Fourier transform on qubits, specifically to encode phase relationships between qubits? 1) Toffoli gates 2) NOT gates 3) Pauli-X gates 4) Conditional phase gates 5) Measurement gates 6) Identity gates 7) CNOT gates

✓ Correct Answer:

The correct answer is 4) Conditional phase gates.

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on the model used here were also used in designing the strongly modu lating control ﬁelds by which both the desired unitary operation as well as all the unitary op erations needed for the QPT procedure were implemented [10]. This was done by minimizing the diﬀ erence between the desired qubit rotation operator and the quantum operation ob tained by simulating the eﬀect of the control ﬁelds on the spins in the molecule used for the e xperiments. As a result, any incorrectness in the model directly aﬀects the reliability of the e xperiments, but in ways that, by deﬁnition, diﬀer from the simulations. This can suggest way s to improve the model, after which further simulations will pinpoint the remaining experiment al errors. The operation on which QPT was performed is the quantum Fourier tr ansform (QFT). The QFT constitutes a key subroutine in several quantum algorithm s [16, 17] as well as in certain methods for simulating quantum dynamics on a quantum com puter [18, 19]. In algorithms such as Shor’s factoring algorithm, the QFT is used to ext ract periodic features of wave functions, while in simulations of quantum dynamics it is used to move between the position and momentum representations. The QFT is deﬁned as follow s: UQFT|x/an}bracketri}ht=1√ NN−1/summationdisplay x′=0e2πixx′/N|x′/an}bracketri}ht. (1) 3 The QFT has been expressed [20, 21] as a sequence of one-qubit Ha damard gates Hj, which rotate the jth qubit from a computational basis state to an equal superpositio n of compu- tational basis states, and two-qubit conditional phase gates Bjk(θ), which rotate the phase of qubitkbyθif qubitjis in the state |1/an}bracketri}ht. In this notation, the complete gate sequence of the three-qubit QFT is (reading from right-to-left) Swap13H3B23/parenleftbigπ 2/parenrightbig B13/parenleftbigπ 4/parenrightbig H2B12/parenleftbigπ 2/parenrightbig H1, (2) whereSwapjkis a swap gate between qubits jandk(see Fig. 1). This gate sequence has been implemented via NMR to demonstrate the ability of the QFT to ext ract periodicity [8], and as part of the quantum baker’s map [22]. The remainder of this paper is organized as follows. Section II gives a n overview of the experimental and computational procedures used for QPT, toge ther with the metrics by which the results were compared to those of the theoretical mode l. Section II describes incoherent errors and their eﬀects on QPT, in particular the appar ent lack of complete posi- tivity of the results, while section III describes the experimental s ystem and procedures used to implement QPT in detail. Section IV presents a complete description of the system-plus- apparatus model by which the results of QPT were interpreted. Th is is followed in section V by an overview of the experimental results, and in section VI by a d etailed comparison of the results with the model’s predictions. Finally, section VII conta ins an analysis of the discrepancies between the experimental and simulated results, wit h the goal of determining their probable origins. The paper concludes with brief discussion of t he implications of our ﬁndings for the

Question 905 multiple-choice

Topological quantum models often utilize group theory and algebraic structures to describe phases and excitations in condensed matter systems. Representation theory plays a crucial role in decomposing Hilbert spaces and analyzing quantum symmetries. In the context of quantum double models based on finite groups, how is the Hilbert space L² most precisely decomposed in terms of irreducible representations? 1) As a direct sum over conjugacy classes of G 2) As a tensor product of two copies of the regular representation 3) As a direct sum over character tables associated with G 4) As a direct product of G with its dual group 5) As a direct sum over all group elements indexed by their class 6) As a direct sum over all irreducible representations μ of G and their duals, i.e., L² = ⨁_μ H_μ ⊗ H_μ* 7) As a single irreducible representation corresponding to the identity element

✓ Correct Answer:

The correct answer is 6) As a direct sum over all irreducible representations μ of G and their duals, i.e., L² = ⨁_μ H_μ ⊗ H_μ*.

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demonstrated that an analogous result cannot hold for excited states. Namely, contrary to intuitions one might have from typical gauge theory models, Wilson loop operators do not form a complete set of commuting observables. Finally, we have also used our detailed analysis of the groundstatestoanalyzethetopologicalentanglemententropy, inparticulardemonstratingthatthe algebraic deﬁnition of entanglement entropy yields the correct topological entanglement entropy, correcting claims made previously in the literature. As was mentioned in the introduction, Kitaev’s models can be generalized from ﬁnite groups to HopfC∗-algebras; the latter reduce to the former when the Hopf-algebra is taken to be the group algebra C[G]associated to G. It is interesting to ask to what extent the techniques we have used can be adapted to the Hopf-algebra case. Given the equivalence between these generalized Kitaev models and the Levin-Wen string net models, a successful generalization would therefore consti- tute a proof that the Levin-Wen models have TQO as well. We leave this question for future study. Acknowledgments. SC would like to thank Zhenghan Wang for bringing the problem studied in this paper to his attention. We would like to thank Matthew Hastings, Patrick Hayden, Alexei Kitaev, and Xiao-Liang Qi for their helpful discussions. BR gratefully acknowledges NSF grant PHY 1720397. DD would like to thank God for all of His provisions and is supported by a National Defense Science and Engineering Graduate Fellowship. XH is supported by a Stanford Graduate Fellowship. SC acknowledges support from Simons Foundation, Virginia Tech, and Purdue University. A The “irrep basis” and the non-Abelian Fourier transform We restrict our attention to ﬁnite groups, for the purpose of the quantum double model, but analogous results hold for compact groups. Consider the Hilbert space L2(G) =span{|g/angbracketright:g∈G}. The groupGacts onL2(G)in at least two natural and distinct ways: left multiplication by g, i.e. asLg|h/angbracketright≡|gh/angbracketright, and right multiplication by g−1, i.e. asRg|h/angbracketright≡|hg−1/angbracketright. These actions commute, and therefore we can simultaneously decompose the Hilbert space into irreducible representations of both actions. In particular, it turns out the Hilbert space decomposes as L2(G) =/circleplusdisplay µ∈ˆGHµ⊗Hµ∗ (80) where the sum is over all inequivalent irreducible representations (“irreps”) µ, with the set of irreps denoted by ˆG. The Hilbert space of an irrep µis denoted by Hµ, andµ∗denotes the dual representation. In the above decomposition, action of Gby left multiplication acts non-trivially only on the left factors Hµ, whereas the action of Gby right multiplication (of g−1) acts non- trivially only on the right factors Hµ∗. The above decomposition is not necessarily obvious, but it is a basic fact in the representation theory of ﬁnite groups. Accepted in Quantum2020-09-17, click title to verify. Published under CC-BY 4.0. 21 Compatible with the above decomposition, we can deﬁne the orthonormal basis |µ;a,b/angbracketrightof L2(G), whereµruns over all irreps, where a= 1,...,dµis a label of some basis of irrep µwith dimensiondµ, and where b= 1,...,dµis a label of the corresponding dual basis of µ∗, noting dµ=dµ∗. This is called the “irrep basis.” A foundational fact in the representation theory of ﬁnite groups yields the following transfor- mation

Question 906 multiple-choice

In large-scale electronic structure calculations, efficient generation of localized orbitals is essential for reducing computational cost and enabling simulations of complex systems. Algorithms that balance speed, accuracy, numerical stability, and scalability are especially valuable in this domain. Which statement best characterizes the impact of combining randomized localization methods with refinement steps in producing localized orbitals for electronic structure calculations? 1) The combination results in orbitals with significantly worse sparsity compared to traditional approaches. 2) The randomized method with refinement leads to poorly conditioned matrices, making computations unstable. 3) The refined randomized approach produces orbitals that remain substantially less localized than those from established algorithms. 4) The refined randomized strategy achieves orbital locality and sparsity nearly equivalent to established methods, while greatly improving computational efficiency and numerical stability. 5) The approach eliminates the need for orthogonal transformations in the localization process. 6) The combined method is slower than traditional algorithms but offers improved accuracy. 7) The algorithm requires extremely high condition numbers for numerical stability.

✓ Correct Answer:

The correct answer is 4) The refined randomized strategy achieves orbital locality and sparsity nearly equivalent to established methods, while greatly improving computational efficiency and numerical stability..

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well. However, the output is clearly not as good as that produced by the original SCDM algorithm. However, once the re nement algorithm is applied we see that, while not identical, the locality of the localized orbitals basically matches that of the localized orbitals generated by Algorithm 1. This is further illustrated in Figure 4.4, which shows isosurfaces for a localized orbital generated by each of the three methods. Once again the columns of the density matrix we ultimately select are very well conditioned, in fact the condition number is less than two. Table 4.1 illustrates the computational cost of our new algorithms as compared to the original version. The re nement algorithm is over nine times faster than the original algorithm, though it cannot be perform without rst approximately localizing the orbitals. Hence, we also provide the time of getting the orthogonal transform out of Algorithm 4, which is the sum of the preceding three lines in the table. Even taking the whole pipeline into account we see a speed up of about a factor of close to seven. 4.3. Water molecules. We now consider a three dimensional system consisting of 256 water molecules (part of the atomic con guration shown in Figure 4.5). In this example,N= 7;381;125 andne= 1024. Figure 4.6 shows histograms of the fraction of non-zero entries after truncation for the randomized method, the re nement procedure applied to the output of the randomized method, and the original SCDM algorithm when applied to the 256 water molecule system. As before, the randomized method actually serves to localize the orbitals rather well. However, there is still visible di erence between the result of the randomized algorithm and the original SCDM algorithm. Application of the re nement algorithm achieves a set of localized orbitals that broadly match the quality 15 2.2 2.4 2.6 2.8 33.2 3.4 3.6 3.8 44.2 percent non-zero051015202530354045number of orbitalsRandomized localization 2.2 2.4 2.6 2.8 33.2 3.4 3.6 3.8 44.2 percent non-zero051015202530354045number of orbitalsRefined localization 2.2 2.4 2.6 2.8 33.2 3.4 3.6 3.8 44.2 percent non-zero051015202530354045number of orbitalsOriginal SCDM Fig. 4.3: Histogram of localized orbitals for the alkane chain example computed by three di erent algorithms based on fraction of non-zero entries after truncation. (top) output of the randomized algorithm, (middle) output of the re nement algorithm ap- plied to the output of the randomized algorithm, and (bottom) output of the original SCDM algorithm. of the ones computed by the original SCDM algorithm. Similar to the previous two examplesP:;Cis very well conditioned, its condition number is once again less than two. Table 4.2 illustrates the computational cost of our new algorithms as compared to Algorithm 1. As before, the randomized algorithm for computing is very fast and in this case much faster than the matrix-matrix multiplication required to construct the localized orbitals themselves. This makes the algorithm particularly attractive in practice when is given and, as in many electronic structure codes, the application ofQto may be e ectively parallelized. Furthermore, the complete procedure for getting

Question 907 multiple-choice

Quantum algorithms often rely on the ability to distinguish quantum states efficiently, especially in the context of the Hidden Subgroup Problem (HSP). The structure of group representations and the choice of measurement basis are fundamental to the success of these algorithms. In the context of applying random Fourier sampling techniques to the Hidden Subgroup Problem, which property of the underlying group is most crucial for ensuring that a single random POVM can efficiently distinguish coset states by at least a constant times their Frobenius distance? 1) The group’s irreducible representations have polynomially bounded rank over the hidden subgroup. 2) The group is non-Abelian with exponentially many irreducible representations. 3) The group’s coset states are always orthogonal. 4) The group forms a non-primitive permutation group. 5) The group has a trivial center. 6) The group’s representation theory is entirely reducible. 7) The group is simple and does not admit Gel’fand pair structure.

✓ Correct Answer:

The correct answer is 1) The group’s irreducible representations have polynomially bounded rank over the hidden subgroup..

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Title: Random Measurement Bases, Quantum State Distinction and Applications to the Hidden Subgroup Problem Year: 2005 Paper ID: 0f7e18b4fa28dd43c36a2bb5a3c413311a44c671 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0f7e18b4fa28dd43c36a2bb5a3c413311a44c671 Abstract: We show that measuring any two low rank quantum states in a random orthonormal basis gives, with high probability, two probability distributions having total variation distance at least a universal constant times the Frobenius distance between the two states. This implies that for any finite ensemble of quantum states there is a single POVM that distinguishes between every pair of states from the ensemble by at least a constant times their Frobenius distance; in fact, with high probability a random POVM, under a suitable definition of randomness, suffices. There are examples of ensembles with constant pairwise trace distance where a single POVM cannot distinguish pairs of states by much better than their Frobenius distance, including the important ensemble of coset states of hidden subgroups of the symmetric group (Moore et al., Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005).We next consider the random Fourier method for the hidden subgroup problem (HSP) which consists of Fourier sampling the coset state of the hidden subgroup using random orthonormal bases for the group representations. In cases where every representation of the group has polynomially bounded rank when averaged over the hidden subgroup, the random Fourier method gives a POVM for the HSP operating on one coset state at a time and using totally a polynomial number of coset states. In particular, we get such POVMs whenever the group and the hidden subgroup form a Gel’fand pair, e.g., Abelian, dihedral and Heisenberg groups. This gives a positive counterpart to earlier negative results about random Fourier sampling when the above rank is exponentially large (Grigni et al., Combinatorica 24(1):137–154, 2004), which happens for example in the HSP in the symmetric group.

Question 908 multiple-choice

In particle cosmology, asymmetric dark matter (ADM) scenarios explore how dark matter relic densities can arise from particle-antiparticle asymmetries, often involving hidden sectors and non-standard cosmological histories. The production and evolution of such asymmetries are tracked through detailed particle interactions and model parameters. Which condition would most directly suppress transfer reactions between hidden sector dark matter states, thereby impacting the final asymmetric dark matter relic density? 1) Increasing the hidden sector gauge coupling constant g_x 2) Lowering the mass of the hidden sector vector boson m_x 3) Raising the neutralino mass above 10 GeV 4) Decreasing the asymmetry parameter 𝜖 to zero 5) Allowing moduli to decay only into Standard Model particles 6) Raising the reheating temperature significantly 7) Setting m_x1 greater than the sum of m_𝜒 and m_𝜒'

✓ Correct Answer:

The correct answer is 7) Setting m_x1 greater than the sum of m_𝜒 and m_𝜒'.

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MSSM neutralino will also continue to decay to the hidden sector through gauge kinetic mixing, now with additional decay modes 0 1! 1;2. As before, the net 0 1lifetime is expected to be short relative to the cosmological timescales of interest. B. Moduli Reheating and Asymmetric Dark Matter The and states will both act as ADM if they are created in the moduli reheating process slightly more often than their antiparticles. The production of the asymmetry can be accommodated within a set of Boltzmann equations similar to Eq. (8) as follows: dn dt+ 3Hn = (1 +=2)N ' m''hvi n n (neq )2 (38) hvitrans(n2 2n2 ) dn dt+ 3Hn= (1 +=2)N' m''hvi nn(neq )2 (39) hvitrans(2n2 n2 ); 20 with a similar set of equations for the anti-DM and , but with!. Here,N and Nare the mean number of and produced per modulus decay. This includes particles created directly in moduli decays, rescattering, and from the cascade decays of other states. The thermally-averaged cross sections hvi ;describe the and annihilation, while hvitransin each equation corresponds to the transfer reaction $ mediated by U(1)x gaugino exchange with = 2 (m =m)2K2(m =T)=K2(m=T). Asymmetry generation in this scenario is parametrized by the constant . It could arise directly from moduli decays or from the interactions of intermediate moduli decay products along the lines of one of the mechanisms of Refs. [118{124]. Indeed, this theory can be viewed as a simpli ed realization of the supersymmetric hylogenesis model studied in Ref. [125]. Relative to that work, we undertake a more detailed investigation of the relic density resulting from di erent choices for the moduli parameters, and we do not attempt to link the DM asymmetry to the baryon asymmetry. The annihilation cross section hvi is dominated by the !XX channel to hidden vector bosons and is given by hvi =1 16g4 x m2 1m2 x m2 3=2 1m2 x 2m2 2 (40) '(1:51024cm2=s)gx 0:0541 GeV m 2 : (41) The scalar annihilation rate is similar. For the transfer reaction, we have hvitransg4 x 8s 1m2 m2 3X k=1(A2 kB2 k)m x k m2 x k+m2 m2 2 ; (42) whereAk=Z 11Pk3andBk=Z12Pk3withPk3the HS gaugino content of x kandZijis a unitary matrix that diagonalizes the scalar mass matrix of Eq. (36). Note that the transfer reaction can be suppressed relative to annihilation for m x 1>m +m. C. Relic Densities and Constraints To investigate the relic densities of and in this theory following moduli reheating and the corresponding constraints upon them, we set all the dimensionful hidden parameters to be xed ratios of the U(1)xgaugino soft mass Mx= 4g2 xm3=2=(4)2: mAx= 100= 50Y= 100mx= 250b1=2 Y= 250meY=Mx: (43) With these choices, the mass spectrum for gx= 0:1 andm3=2= 200 TeV is m = 1 GeV; m = 0:97 GeV; m x 1= 5:1 GeV; mx= 0:51 GeV; mhx 1= 0:5 GeV: 21 m3/2 [100 T e V] mΨ [Ge V] κΩadm/Ωcdm forgx= 0.1 and/epsilon1= 10−4CMB Excl.Ωadm/Ωcdm DM

Question 909 multiple-choice

In quantum information theory and representation theory, understanding the relationship between unitary group actions, tensor products, and operator algebras is crucial for analyzing symmetries and invariant subspaces. Schur-Weyl duality and moment operators play a central role in classifying operator commutants in such settings. Which statement most accurately characterizes the role of permutation operators in the commutant of k-fold tensor powers of the unitary group U(d)? 1) Permutation operators form an orthonormal basis for the commutant under the Hilbert-Schmidt inner product. 2) Permutation operators are invariant under all linear transformations on the tensor product space. 3) Permutation operators generate the algebra of all unitary operators in the tensor product space. 4) Permutation operators commute only with diagonal elements of U(d)⊗k but not with arbitrary unitaries. 5) Permutation operators span the commutant of U(d)⊗k and form a basis (but not an orthonormal basis) for it. 6) Permutation operators do not represent group-like properties when acting on tensor product spaces. 7) Permutation operators project any operator onto the symmetric subspace exclusively.

✓ Correct Answer:

The correct answer is 5) Permutation operators span the commutant of U(d)⊗k and form a basis (but not an orthonormal basis) for it..

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of the Haar measure. 3. SinceA∈Comm(U(d),k), we have: M(k) µH(A) =E U∼µH/bracketleftbig U⊗kAU†⊗k/bracketrightbig =E U∼µH/bracketleftbig AU⊗kU†⊗k/bracketrightbig =A. (17) Theorem 7 (Projector onto the commutant) .The moment operator M(k) µH(·) =E U∼µH/bracketleftbig U⊗k(·)U†⊗k/bracketrightbig is the orthogonal projector onto the commutant Comm := Comm(U( d),k)with respect to the Hilbert-Schmidt inner product. In particular, let P1,...,P dim(Comm) be an orthonormal basis of Comm and letO∈L((Cd)⊗k). Then, we have: M(k) µH(O) =dim(Comm)/summationdisplay i=1⟨Pi,O⟩HSPi. (18) Proof. Let us extend the orthonormal basis of the commutant with the orthonormal operators Pi fori∈{dim (Comm) + 1 ,..., dim (V)}, whereV:=L/parenleftbig (Cd)⊗k/parenrightbig . This extended basis forms an orthonormal basis for V. Therefore we have: M(k) µH(O) =dim(V)/summationdisplay i=1⟨Pi,M(k) µH(O)⟩HSPi (19) =dim(Comm)/summationdisplay i=1⟨M(k) µH(Pi),O⟩HSPi+dim(V)/summationdisplay i=dim(Comm)+1⟨Pi,M(k) µH(O)⟩HSPi (20) =dim(Comm)/summationdisplay i=1⟨M(k) µH(Pi),O⟩HSPi+ 0 (21) =dim(Comm)/summationdisplay i=1⟨Pi,O⟩HSPi, (22) where in the second line we used the fact that the moment operator is self-adjoint (Lemma 6.1), in the third line that M(k) µH(O)∈Comm (Lemma 6.2) and that Piwithi∈ {dim (Comm) + 1,..., dim (V)}are in its orthogonal complement, in the fourth line that Pi∈Comm for i∈ {1,..., dim (Comm)}and Lemma 6.3. 4 We have just shown that the moment operator is intimately related to the k-order commutant of the unitary group, i.e. all the matrices that commute with U⊗kfor any unitary U. A set of operations that surely commutes with U⊗kis if we exchange some tensor factors. For this purpose, we will now define the operators that implement such transformations, namely the permutation operators. Definition 8 (Permutation operators) .Givenπ∈Skan element of the symmetric group Sk, we define the permutation matrix Vd(π)to be the unitary matrix that satisfies: Vd(π)|ψ1⟩⊗···⊗|ψk⟩=/vextendsingle/vextendsingleψπ−1(1)/angbracketrightbig ⊗···⊗/vextendsingle/vextendsingleψπ−1(k)/angbracketrightbig , (23) for all|ψ1⟩,...,|ψk⟩∈Cd. Note that from this definition it follows that Vd(σ)Vd(π) =Vd(σπ)andVd(π−1) =V† d(π). Equivalently, we can write the permutation matrix as: Vd(π) =/summationdisplay i1,...,ik∈[d]k/vextendsingle/vextendsingleiπ−1(1),...,iπ−1(k)/angbracketrightbig/angbracketleftbig i1,···,ik/vextendsingle/vextendsingle. (24) Thus, we have the following property: Vd(π) (A1⊗···⊗Ak)V† d(π) =Aπ−1(1)⊗···⊗Aπ−1(k), (25) forA1,...,Ak∈L/parenleftbig Cd/parenrightbig . A crucial and much celebrated result of representation theory is now that the permutation operators characterize allpossible matrices in the commutant – this is the Schur-Weyl duality. Theorem 9 (Schur-Weyl duality [53]) .Thek-th order commutant of the unitary group is the span of the permutation operators associated to Sk: Comm(U(d),k) = span/parenleftig Vd(π) :π∈Sk/parenrightig . (26) We will omit the proof of this theorem here, except for the cases where k= 1andk= 2, which we will show later in this section. Interested readers can refer to Refs. [46, 53, 54] for a detailed exposition. However we can easily check that span/parenleftig Vd(π) :π∈Sk/parenrightig ⊆Comm(U(d),k). To see why this is true, consider an arbitrary permutation Vd(π)withπ∈SkandU∈U (d). We have: Vd(π)U⊗k|ψ1⟩⊗···⊗|ψk⟩=Vd(π)(U|ψ1⟩)⊗···⊗ (U|ψk⟩) (27) =U/vextendsingle/vextendsingleψπ−1(1)/angbracketrightbig ⊗···⊗U/vextendsingle/vextendsingleψπ−1(k)/angbracketrightbig (28) =U⊗kVd(π)|ψ1⟩⊗···⊗|ψk⟩, (29) for all|ψ1⟩,...,|ψk⟩∈Cd. Hence we have that/bracketleftbig Vd(π),U⊗k/bracketrightbig = 0for allπ∈Sk, from which the claim follows. The permutation matrices form a basis for the k-th order commutant of the unitary group, capturing its essential structure. However, it is important to note that they are not orthonormal with respect to the Hilbert-Schmidt inner product. Therefore, we cannot directly apply Theorem 7. Nevertheless, we have an alternative

Question 910 multiple-choice

In quantum information theory, averages over the Haar measure and representations of operators using tensor networks play a central role in analyzing random quantum processes and entanglement. The structure of moment operators and their commutants reveals invariant properties under unitary transformations. Which statement best characterizes the mathematical relationship between the rank of the moment operator M(k) and its commutant in the context of Haar measure on the unitary group? 1) The rank of M(k) is always equal to the dimension of the underlying Hilbert space. 2) The rank of M(k) with respect to Haar measure equals the dimension of its commutant. 3) The rank of M(k) is determined by the number of permutation operators in the symmetric group Sk. 4) The commutant of M(k) is trivial for k > 1. 5) The rank of M(k) depends solely on the dimension of the swap operator. 6) The moment operator M(k) is invertible for all k and dimensions. 7) The rank of M(k) is maximized when the Hilbert-Schmidt inner product vanishes.

✓ Correct Answer:

The correct answer is 2) The rank of M(k) with respect to Haar measure equals the dimension of its commutant..

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rank of M(k) µHis equal to the dimension of the commutant, i.e., r= dim (Comm). Consequently, we can conclude the proof by noting that for all O∈L/parenleftbig (Cd)⊗k/parenrightbig : M(k) µH(O) = vec−1/parenleftig vec(M(k) µH)|O⟩⟩/parenrightig = vec−1/parenleftiggr/summationdisplay i=1|Pi⟩⟩⟨⟨Pi|O⟩⟩/parenrightigg =r/summationdisplay i=1⟨Pi,O⟩HSPi. (118) To address the issue of the permutation matrices not being orthonormal, we can introduce a vectorized version of Eq. (35) that provides a concise representation of the moment operator. E U∼µH/bracketleftbig U⊗k⊗U∗⊗k/bracketrightbig =/summationdisplay π,σ∈SkWg(π−1σ,d)|Vd(π)⟩⟩⟨⟨Vd(σ)|. (119) Proof. Using Eq.(35), we have: E U∼µH/bracketleftbig U⊗k⊗U∗⊗k/bracketrightbig |O⟩⟩=E U∼µH|U⊗kOU†⊗k⟩⟩ (120) =/summationdisplay π,σ∈SkWg/parenleftbig π−1σ,d/parenrightbig |Vd(π)⟩⟩Tr/parenleftig V† d(σ)O/parenrightig (121) =/summationdisplay π,σ∈SkWg/parenleftbig π−1σ,d/parenrightbig |Vd(π)⟩⟩⟨⟨Vd(σ)|O⟩⟩. (122) Since this equation holds for all vectors |O⟩⟩in (Cd)⊗2k, we conclude the proof. 16 We can now write the moment operator equations for k= 1andk= 2presented in Example 13 in their vectorized form: E U∼µH[U⊗U∗] =1 d|I⟩⟩⟨⟨I|, (123) E U∼µH/bracketleftbig U⊗2⊗U∗⊗2/bracketrightbig =1 d2−1/bracketleftbigg |I⟩⟩⟨⟨I|−1 d|I⟩⟩⟨⟨F|−1 d|F⟩⟩⟨⟨I|+|F⟩⟩⟨⟨F|/bracketrightbigg . (124) Proof. For Equation (123), we have: E U∼µH[U⊗U∗]|O⟩⟩=E U∼µH|UOU†⟩⟩=Tr(O) d|I⟩⟩=1 d|I⟩⟩⟨⟨I|O⟩⟩, (125) where we used⟨⟨I|O⟩⟩= Tr(IO) = Tr(O). For Equation (124), we have: E U∼µH/bracketleftbig U⊗2⊗U∗⊗2/bracketrightbig |O⟩⟩=E U∼µH|U⊗2OU†⊗2⟩⟩ (126) =Tr(O)−d−1Tr(FO) d2−1|I⟩⟩+Tr(FO)−d−1Tr(O) d2−1|F⟩⟩ (127) =1 d2−1/bracketleftbigg |I⟩⟩⟨⟨I|O⟩⟩−1 d|I⟩⟩⟨⟨F|O⟩⟩−1 d|F⟩⟩⟨⟨I|O⟩⟩+|F⟩⟩⟨⟨F|O⟩⟩/bracketrightbigg ,(128) where we used that F†=F,⟨⟨I|O⟩⟩= Tr(O) and⟨⟨F|O⟩⟩= Tr(FO). Equations like the ones presented above can be effectively visualized and manipulated using Tensor Network diagrams. 6 Tensor network diagrams Tensor Networks provide a graphical representation that simplifies the understanding and analysis of tensor operations [62]. In these diagrams, tensors are represented as boxes (or nodes), and tensor contractions are represented as connections between boxes. To illustrate the concept, we use specific diagrams for ket-states, bra-states, and matrices. A ket-state |ψ⟩∈Cdis represented with a box and a legas follows: |ψ⟩ . (129) Similarly, a bra-state ⟨ψ|is represented as: ⟨ψ| . (130) Matrices, such as A∈L/parenleftbig Cd/parenrightbig , are depicted like boxes with one leg in and one leg out as: ⟨ψ|AAB . (131) In particular, the identity matrix I∈L/parenleftbig Cd/parenrightbig is simply represented by a line: ⟨ψ|AAB . (132) 17 The trace Tr(A)of an operator A∈L/parenleftbig Cd/parenrightbig is denoted as: ⟨ψ|AABAA=AAAA AT=AT=AB . (133) GivenA,B∈L/parenleftbig Cd/parenrightbig , their product ABis represented by placing the box representing matrix A on the left side of the box representing matrix B. This can be illustrated as follows: ⟨ψ|AABAA=AAAA AT=AT=AB . (134) The transpose ATof the matrix Ais depicted by: ⟨ψ|AABAA=AAAA AT=AT=AB . (135) GivenA∈L/parenleftbig Cd/parenrightbig andB∈L/parenleftig Cd′/parenrightig , their tensor product A⊗Bis denoted by arranging the box representing matrix Bbelow the box representing matrix A: ⟨ψ|AABAA=AAAA AT=AT=AB . (136) A∈L/parenleftbig Cd/parenrightbig ⊗L/parenleftig Cd′/parenrightig is represented with a box that has two (different) legs on the left and two on the right: ⟨ψ|AABAA=AAAA AT=AT=AB . (137) The partial trace with respect to the second tensor space Tr2(A)is denoted as: ⟨ψ|AABAA=AAAA AT=AT=AB , (138) and similarly for the partial trace with respect to the first tensor space. 18 Particularly important is the non-normalized maximally entangled state |Ω⟩=|I⟩⟩, which is the vectorization of the identity and is denoted as: 𝕀===|𝕀⟩⟩=UU* . (139) Given an operator A∈L/parenleftbig Cd/parenrightbig , its vectorization |A⟩⟩=A⊗I|Ω⟩is represented

Question 911 multiple-choice

In the study of flavor symmetries in particle physics, group theory is used to constrain the possible forms of Yukawa couplings and mass matrices for fermions and Higgs fields. The invariance of these couplings under discrete group transformations has profound implications for particle mass hierarchies and mixing angles. When constructing invariant Yukawa couplings under a discrete symmetry group GF, which procedure ensures that invariance holds for all group elements? 1) Verifying invariance under the representations of the group's generators 2) Checking invariance only for the identity element of the group 3) Ensuring invariance for a randomly selected subset of group elements 4) Applying invariance conditions to the conjugacy classes exclusively 5) Using invariance constraints derived from the center of the group 6) Testing invariance after spontaneous symmetry breaking 7) Verifying invariance for each element individually without using generators

✓ Correct Answer:

The correct answer is 1) Verifying invariance under the representations of the group's generators.

📚 Reference Text:

3 invariant under the following set of transformations: L′ αL= (AL)α,βLβL, l′ βR= (Al)β,γlγR, (2) ν′ βR= (Aν)β,δνδR,Φ′ i= (AΦ)i,kΦk. In the above formulas AL,Aν,Alstand for 3-dimensional irreducible (in general diﬀer- ent) representations of GF. On the other hand, AΦdenotes the 2-dimensional irreducible representation of the same group. The dimensions of these repre sentations follow from the existence of 3families of fermions and 2Higgs doublets which are assumed in the model. The requirement of the invariance of Eq.(1) under the transforma tionsgiven inthe above- mentioned formulas leads to the equations restricting the form of Y ukawa couplings [5]: /summationdisplay i=1,2(Aφ)i,k(AL)† αγ(hl i)γδ(Al)δβ= (hl k)αβ, /summationdisplay i=1,2(Aφ)∗ i,k(AL)† αγ(hν i)γδ(Al)δβ= (hν k)αβ, (3) which after some indices manipulation take the form of 2 eigenproblem s to the eigenvalue 1: N1Γl= Γl,N2Γν= Γν, (4) where (Γl,ν)T=/bracketleftBig (hl,ν 1)11,...,(hl,ν 1)33,(hl,ν 2)11,...,(hl,ν 2)33/bracketrightBig (5) and N1= (AΦ)T⊗(AL)†⊗(Al)T, (6) N2= (AΦ)†⊗(AL)†⊗(Aν)T. (7) Thus in order to get the desired form of Yukawa couplings one should simply ﬁnd the eigensubspaces of matrices N1andN2corresponding to the eigenvalue 1. However, the grouphas got many elements, each of them has got it sown set of irreducible representations. Thus, should we solve Eq.(4) for each group’s ele ment separately? Luckily, the answer to this question is negative. It turns out that one has t o take only generators’ representations in order to obtain the Yukawa couplings which are in variant with respect to the whole group GF[5]. 4 Theorem 1. If the relation : /summationdisplay i=1,2(Aφ)i,k(AL)† αγ(hl i)γδ(Al)δβ= (hl k)αβ holds for the generators’ representations of some group GFthen it holds for the representa- tions of all group’s elements. Proof: Every element of the group Gcan be presented as a unique combination of its generators {A,B,C,..}: G=An1Bn2Cn3... wheren1,n2,n3,...are natural numbers. Therefore it is suﬃcient to prove that if the relations : /summationdisplay i=1,2(AΦ)ik(B)(AL)† αγ(B)(hl i)γδ(Al)δβ(B) = (hl k)αβ, /summationdisplay i=1,2(AΦ)ik(C)(AL)† αγ(C)(hl i)γδ(Al)δβ(C) = (hl k)αβ are fulﬁlled for the representations AΦ,AL,ARof some generators B,Cthen they are also valid for the representations AΦ,AL,ARof their product: /summationdisplay i=1,2(AΦ)ik(BC)(AL)† αγ(BC)(hl i)γδ(Al)δβ(BC) = (hl k)αβ. The above-mentioned statement holds indeed: /summationdisplay i=1,2(AΦ)ik(BC)(AL)†(BC)(hl i)(Al)(BC) = =/summationdisplay i,m=1,2(AΦ)ik(B)(AΦ)km(C)(AL)†(C)(AL)†(B)(hl i)(Al)(B)(Al)(C) = =/summationdisplay i,m=1,2(AΦ)km(C)(AL)†(C)(AΦ)ik(B)(AL)†(B)(hl i)(Al)(B)(Al)(C) = =/summationdisplay i,m=1,2(AΦ)km(C)(AL)†(C)(hl k)(Al)(C) = (hl m) (8) We arrive thereby to the conclusion that only group’s generators a re necessary to ﬁnd in- variant Yukawa couplings hl,ν 1,2. After solving Eq.(4) for each of the generators one will get some set of eigenspaces Wito the eigenvalue 1 (this eigenvalue is in general degenerate). 5 In order to get the appropriate solution, one has to ﬁnd the commo n eigenspace Uof these individual eigenspaces . Then, it is necessary to ﬁnd the basis vecto r ofUwhich constitutes the ultimate result. At this point, knowledge about the form of Yukawa matrices makes t he construction of mass matrices feasible: Ml α,β=1√ 2(v1hl 1+v2hl 2), Mν αβ=1√ 2(v1hν 1+v2hν 2). (9) Then, we are able to ﬁnd the UPMNSmixing matrix, which is composed of the matrices that diagonalize MlandMν: Vl† LMlVl R=diag(me,mµ,mτ), Vν† LMνVν R=diag(mνe,mνµ,mντ),

Question 912 multiple-choice

Quantum algorithms have dramatically changed the computational landscape for problems such as integer factorization, threatening cryptographic systems reliant on classical hardness assumptions. A key innovation in these algorithms is the use of quantum period-finding, which utilizes quantum superposition and the quantum Fourier transform to efficiently extract periodicity from functions. Which technique allows quantum algorithms to estimate the period of a function using only a small number of applications of a specially defined unitary operator, thereby providing exponential speed-ups over classical factoring methods? 1) Application of the quantum Fourier transform combined with a unitary operator Uf that shifts inputs and encodes function values 2) Repeated application of classical trial division enhanced by quantum random number generators 3) Simulating probabilistic index calculus within a quantum register 4) Direct measurement of function outputs in the computational basis without superposition 5) Use of deterministic elliptic curve methods accelerated by quantum entanglement 6) Employing heuristic number field sieve methods implemented on a quantum circuit 7) Applying the Grover search algorithm to locate factors in polynomial time

✓ Correct Answer:

The correct answer is 1) Application of the quantum Fourier transform combined with a unitary operator Uf that shifts inputs and encodes function values.

📚 Reference Text:

N. This uses onlyO(n3) or exp( clogn) elementary quantum operations, for c= 3 + o(1) (or c= 2 + o(1) if we use fast Fourier transform techniques). Other dete rministic factoring methods will factor NinO(√ N) or exp( cn) steps, where c= 1/2 + o(1). The best known rigorous probabilistic classical algor ithm (using index calculus methods) [LP] uses exp( c(nlogn)1/2) elementary classical operations, c= 1 + o(1). There is also an algorithm with a heuristic expected run ning time of exp( c(n1/3(logn)2/3) elementary classical operations (see [MOV] for an overview and references) for c= 1.902 + o(1). Thus, in terms of elementary operations, a quantum computer provides a drastic improvem ent over known classical methods to factor integers. 4.2 Finding the Period of a Function The above algorithm, as pointed out in [BL], can be applied to a more general setting. Replace the mapping from ttoatwith any function ffrom the integers to some ﬁnite set X. Deﬁne Uf(x)to be an operator that maps f(y) tof(y+x). This is a generalisation of Uaxexcept it does not matter how it is deﬁned on values not in the range of f, as long as it is unitary. Deﬁne Uf(x)to be an operator which maps |x/an}brack⌉tri}ht |f(y)/an}brack⌉tri}htto|x/an}brack⌉tri}htUf(x)|f(y)/an}brack⌉tri}ht=|x/an}brack⌉tri}ht |f(y+x)/an}brack⌉tri}ht. The following are eigenvectors of Uf(x): |Ψk/an}brack⌉tri}ht=r−1/summationdisplay t=0exp(−2πikt/r )|f(t)/an}brack⌉tri}ht, k= 0,1, . . ., r −1, (7) with respective eigenvalues exp(2 πikx/r ). As in (3), we can start with |0/an}brack⌉tri}ht |f(0)/an}brack⌉tri}ht=r−1/summationdisplay k=0|0/an}brack⌉tri}ht |Ψk/an}brack⌉tri}ht except with our new, more general, deﬁnition of |Ψk/an}brack⌉tri}ht. We apply F2nto the ﬁrst register to produce (4), and then apply Uf(x)to produce (5), followed by F−1 2nto get (6). Observing the ﬁrst register will give an estimate ofk/rfor an integer kchosen uniformly at random, and the same analysis as in the pr evious section applies to ﬁnd r. One important issue is how to compute Uf(x)only knowing how to compute f. Note that from (4) to (5) (using the modiﬁed deﬁnition of |Ψk/an}brack⌉tri}ht) we simply 7 go from 2n−1/summationdisplay x=0|x/an}brack⌉tri}ht |f(0)/an}brack⌉tri}ht=2n−1/summationdisplay x=0/parenleftBiggr−1/summationdisplay k=0|x/an}brack⌉tri}ht |Ψk/an}brack⌉tri}ht/parenrightBigg (8) to 2n−1/summationdisplay x=0|x/an}brack⌉tri}ht |f(x)/an}brack⌉tri}ht=2n−1/summationdisplay x=0|x/an}brack⌉tri}ht/parenleftBiggr−1/summationdisplay k=0exp(2πixk/r )|Ψk/an}brack⌉tri}ht/parenrightBigg (9) which could be accomplished by applying Uf, which we do have, to the starting state 2n−1/summationdisplay x=0|x/an}brack⌉tri}ht |0/an}brack⌉tri}ht. Thus even if we do not know how to explicitly compute the opera torsUf(x), any operator Ufwhich computes the function fwill give us the state (9). This state permits us to estimate an eigenvalue of Uf(x)which lets us ﬁnd the period of the function fwith just O(1) applications of the operator UfandO(n2) other elementary operations. The equality in (9) is the key t o the equivalence between the two approaches to these quantum algorithms. On t he left hand side is the original approach ([Si], [Sh], [BL]) which considers the target register in the standard computational basis. We can analyse the Fourie r transform of the preimages of these basis states, which is less easy when the F ourier transforms do not exactly correspond to the group G. On the right hand side of (9) we consider the target register

Question 913 multiple-choice

The intersection of computational complexity theory and low-dimensional topology reveals surprising connections between problems involving 3-manifolds and classical notions such as NP-completeness and #P-completeness. Group theory and topological quantum computation play key roles in understanding the hardness of decision and counting problems for topological spaces. Which statement correctly characterizes the computational complexity of counting homomorphisms from the fundamental group of a homology 3-sphere to a fixed finite non-abelian simple group? 1) It is in the class P for all such groups. 2) It is PSPACE-complete for abelian groups only. 3) It is NP-complete for any non-abelian group. 4) It is #P-complete for any fixed finite non-abelian simple group. 5) It is undecidable for infinite groups. 6) It is coNP-complete for solvable groups. 7) It is polynomial-time computable for cyclic groups.

✓ Correct Answer:

The correct answer is 4) It is #P-complete for any fixed finite non-abelian simple group..

📚 Reference Text:

Title: Computational complexity and 3-manifolds and zombies Year: 2017 Paper ID: bbaba44c7120966d3cb3f796cce5445c60cdbe50 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/bbaba44c7120966d3cb3f796cce5445c60cdbe50 Abstract: We show the problem of counting homomorphisms from the fundamental group of a homology $3$-sphere $M$ to a finite, non-abelian simple group $G$ is #P-complete, in the case that $G$ is fixed and $M$ is the computational input. Similarly, deciding if there is a non-trivial homomorphism is NP-complete. In both reductions, we can guarantee that every non-trivial homomorphism is a surjection. As a corollary, for any fixed integer $m \ge 5$, it is NP-complete to decide whether $M$ admits a connected $m$-sheeted covering. Our construction is inspired by universality results in topological quantum computation. Given a classical reversible circuit $C$, we construct $M$ so that evaluations of $C$ with certain initialization and finalization conditions correspond to homomorphisms $\pi_1(M) \to G$. An intermediate state of $C$ likewise corresponds to a homomorphism $\pi_1(\Sigma_g) \to G$, where $\Sigma_g$ is a pointed Heegaard surface of $M$ of genus $g$. We analyze the action on these homomorphisms by the pointed mapping class group $\text{MCG}_*(\Sigma_g)$ and its Torelli subgroup $\text{Tor}_*(\Sigma_g)$. By results of Dunfield-Thurston, the action of $\text{MCG}_*(\Sigma_g)$ is as large as possible when $g$ is sufficiently large; we can pass to the Torelli group using the congruence subgroup property of $\text{Sp}(2g,\mathbb{Z})$. Our results can be interpreted as a sharp classical universality property of an associated combinatorial $(2+1)$-dimensional TQFT.

Question 914 multiple-choice

Quantum algorithms for non-abelian groups rely on advanced representation theory and specialized data structures to efficiently compute Fourier transforms. One such approach uses graphical tools to organize basis vectors and manage complex embedding operations. In the context of quantum Fourier transforms over finite non-abelian groups, which structure is specifically used to efficiently index representation paths, simplify embedding operations, and implement twiddle factors by providing an adapted basis? 1) Cayley graph 2) Bratteli diagram 3) Character table 4) Young tableau 5) Schreier tree 6) Conjugacy class lattice 7) Gelfand-Tsetlin pattern

✓ Correct Answer:

The correct answer is 2) Bratteli diagram.

📚 Reference Text:

GiinGin whichglies. At the endofthe computation,we havea pair ofpathsinthe Bra tteli diagram, s=s1···smandt=t1···tm, which index the rows and columns of the representations ρofG. These paths begin empty and grow as we work our way up the tower; after having performedthe Fourier transform f orGi, the paths p=p1···piandq=q1···qiof length i indextherowsandcolumnsofrepresentations σofGi. With a compact encoding, one could store αin the same registers as sandt, at each step replacing a coset representative αiwith a pair of edges si,ti. However,our algorithmis simpler to describe if we double t he numberof qubits and store αands,tin separate registers. Padding out α,s, andtto length mwith zeroes, our computational basisconsistsofunitvectorsoftheform |α/an}bracketri}ht|s,t/an}bracketri}ht=/vextendsingle/vextendsingleαm···αi+10i/angbracketrightbig ⊗/vextendsingle/vextendsingles1···si0m−i,s1···si0m−i/angbracketrightbig . Keepinmindthebasis {|s,t/an}bracketri}ht},wheresandthavelength iandendinthesamerepresentation,isjustapermutationof our adapted Gel’fand-Tsetlinbasis {|σ,j,k/an}bracketri}ht}forˆGi, whereσrangesoverthe representationsof Giand 1≤j,k≤dσ indexitsrowsandcolumns. Therefore,wewillsometimesabu senotationbywriting ˆf(s,t)andˆf(σ)j,kfortheFourier transformover Giindexedin thesetwo differentways. EachstageofthealgorithmconsistsofcalculatingtheFour iertransformover Gi+1fromthatover Gi. Byinduction it sufﬁces to consider the last stage, where we go from H=Gm−1toG=Gm. Speciﬁcally, choose a transversal Tof HinGsuchthatevery g∈Gcanbewritten αhwhereα∈Tandh∈H. Foreach α∈T,deﬁneafunction fαonHas fα(h)=f(αh); thisis therestrictionof fto thecoset αH, shiftedinto H. AfterhavingperformedtheFouriertransformon H,ourstate will be ∑ α∈T|α/an}bracketri}ht ⊗∑ s,tof length m−1ˆfα(s,t)|s,t/an}bracketri}ht=∑ α∈T|α/an}bracketri}ht ⊗∑ (σ,j,k)∈ˆHˆfα(σ)j,k|σ,j,k/an}bracketri}ht. (4) Ourgoalistotransformthisstate intotheFourierbasisof G,namely |0/an}bracketri}ht ⊗∑ s,toflength mˆf(s,t)|s,t/an}bracketri}ht=|0/an}bracketri}ht ⊗∑ (ρ,j,k)∈ˆGˆf(ρ)j,k|ρ,j,k/an}bracketri}ht. (5) where|0/an}bracketri}htoccupiesthe registerthatheldthecoset representative αbefore. This transformation is greatly simpliﬁed by the following t wo observations, which are common to nearly every algorithmfortheFFT.First,asdescribedinEquation(2)ab ove,ˆfcanbewrittenasasumovercontributionsfrom f’s valuesoneachcoset αH,giving ˆf(ρ)=∑ α∈Tρ(α)·ˆfα(ρ). (6) Sincefαhassupportonlyin H, the matrix ˆfα(ρ)is a direct sum of sub-matricesof the form ˆfα(σ), summedoverthe σappearingin ρ. Inthequantumsettingweaccomplishthisviaan embedding operationwhichreversestherestriction toH, |σ/an}bracketri}ht→∑ ρ:σappears in ρ|HAσ,ρ|ρ/an}bracketri}ht (7) wherethis“scale factor”is Aσ,ρ=/radicalBigg |H| |G|dρ dσ. (Notethat ∑ρ|Aσ,ρ|2=1.) Thusthealgorithmconsistsof (i.)embeddingthe σinthe appropriate ρ,(ii.)applyingthe“twiddlefactor” ρ(α), and(iii.)summing over the cosets. However, in general, doing these th ings efﬁciently is no simple matter. First, a givenσmight appear in a given ρwith an arbitrary change of basis; the twiddle ρ(α)could be an arbitrary unitary 6 matrix of exponential size; and summing over an exponential number of cosets will take exponential time unless parallelizedinsomeway. It is here that the Bratteli diagram proves to be extremely he lpful. It allows us to implement the twiddle factors ρ(α)efﬁciently whencoupledwith a stronggeneratingset as disc ussed in Section 3.2 by providingan adaptedbasis. It simpliﬁes the embedding operation as well: ﬁrst note that ˆfα(s,t)is nonzero only when sandtend in the same representation σofGt, i.e. in the same vertex of the diagram. Moreover, recall tha t the Bratteli diagram indexes an adapted basis in which ρ|His block-diagonal with the σjas its blocks. This means that the σappear in the ρin an extremelysimpleway: namely,where sandtareextendedbyappendingthesameedge etoboth. Let adoptsome notation. Givena path sin the Bratteli diagramof length m−1orm, denotethe representationin which it ends by σ[s]orρ[s]respectively, and if s=s1···sm−1, denotes1···sm−1easse. We will index the edges of each vertex {1,...,k}where it has out-degree k. It will be convenientto carry out this embeddingonly if the register containingthecoset representativeiszero,andleaveothe rbasisvectorsin (T∪{0})⊗ˆHﬁxed. Then(7)becomes U:/braceleftbigg|0/an}bracketri}ht|s,t/an}bracketri}ht→|0/an}bracketri}ht∑eAσ[s],ρ[se]|se,te/an}bracketri}ht |α/an}bracketri}ht|s,t/an}bracketri}ht→|α/an}bracketri}ht|s,t/an}bracketri}htforallα∈T(8) wherethesumisoverall outgoingedges eofσ[s]=σ[s]. Note that we have not deﬁned Uon the entire space; in particular, since we are moving proba bilityfrom ˆHtoˆG, basis vectors |0/an}bracketri}ht|se,te/an}bracketri}ht∈(T∪{0})⊗ˆGcannot stay ﬁxed. As we will see below, it does not matter prec isely how U behaves on the

Question 915 multiple-choice

Quantum algorithms frequently utilize measurements in the Fourier basis to solve group-theoretic problems, such as the Hidden Subgroup Problem (HSP), which underpins tasks like factoring and graph isomorphism. Understanding entangled measurements across quantum registers is key to leveraging quantum parallelism in these algorithms. Which statement accurately describes the measurement strategy that enables quantum algorithms to address HSPs relevant to graph isomorphism and dihedral groups? 1) It applies a classical random sampling technique to independent quantum registers. 2) It measures each quantum register separately in the computational basis. 3) It performs an entangled measurement in the Fourier basis across $k = \log_2 |G|$ quantum registers, with each subset of registers providing useful information. 4) It uses only a single quantum register and applies the inverse Fourier transform. 5) It relies exclusively on unentangled measurements in the Pauli-X basis. 6) It collapses the quantum state using amplitude amplification without basis change. 7) It limits the measurement to subsets containing exactly two registers.

✓ Correct Answer:

The correct answer is 3) It performs an entangled measurement in the Fourier basis across $k = \log_2 |G|$ quantum registers, with each subset of registers providing useful information..

📚 Reference Text:

Title: Explicit Multiregister Measurements for Hidden Subgroup Problems Year: 2005 Paper ID: b4a31848cc239de91c0d4977ed517e58e1c25925 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b4a31848cc239de91c0d4977ed517e58e1c25925 Abstract: We present an explicit measurement in the Fourier basis that solves an important case of the Hidden Subgroup Problem, including the case to which Graph Isomorphism reduces. This entangled measurement uses $k=\log_2 |G|$ registers, and each of the $2^k$ subsets of the registers contributes some information. While this does not, in general, yield an efficient algorithm, it generalizes the relationship between Subset Sum and the HSP in the dihedral group, and sheds some light on how quantum algorithms for Graph Isomorphism might work.

Question 916 multiple-choice

Quantum Phase Estimation (QPE) is a fundamental quantum algorithm used to estimate the eigenvalues of unitary operators and is crucial for applications in quantum chemistry, factoring, and simulation. The algorithm's reliability is heavily influenced by noise processes, which can introduce errors during computation. Which noise channel is commonly used to benchmark error-correcting schemes due to its representation of complete qubit decoherence, where with probability p the qubit is replaced by a maximally mixed state? 1) Bit flip channel 2) Phase flip channel 3) Bit-phase flip channel 4) Amplitude damping channel 5) Depolarizing channel 6) Generalized amplitude phase channel 7) Collective dephasing channel

✓ Correct Answer:

The correct answer is 5) Depolarizing channel.

📚 Reference Text:

with an environment.11,12In this paper, we will study the impact of (1) bit flip, (2) phase flip, (3) bit-phase flip, and (4) depolarizing channels on QPE. These four noise processes are fundamental in the sense that other single-qubit unital noise processes can be described in terms of these processes. This paper is organized as follows: In Section 2, we provide a brief overview of the QPE algorithm, noise models, and incorporation of noise models in the QPE algorithm. The simulation results and discussion is presented in Section 3. The conclusions are presented in Section 4. ∗Corresponding author: muhammad.faryad@lums.edu.pkarXiv:2307.15675v3 [quant-ph] 3 Dec 2023 2. MODELING NOISE IN QPE The QPE algorithm finds the eigenvalue of a unitary operator Usuch that U|v⟩=e2πiθ|v⟩where 0 ≤θ≤1 and |v⟩is an eigenvector of U. The main idea of this algorithm is to provide an n-bit approximation of θin a single run. For the work presented in this paper, we took Uas a single qubit operator that has eigenvectors |0⟩and |1⟩with eigenvalues 1 and e2πiθ, respectively. This algorithm uses two registers. The first register is initialized to|0⟩⊗n, where nis the number of qubits to estimate θ. The second register starts in the state |v⟩and contains the number of qubits necessary to store |v⟩. The algorithm is performed in four steps: (1) creating superposition of all possible states and performing the controlled- U2joperation for j≥0, (2) applying the inverse quantum Fourier transform on the first register, (3) measurement in the computational basis to read out the state of the first register, and (4) classical post-processing to extract estimated θ. The circuit for quantum phase estimation is illustrated in Fig. (1) for the simplest unitary operator Uacting on a single qubit.1 ......... ···|0⟩⊗nH FT† |2nθ⟩H H |v⟩ U U2U2n−1|v⟩ Figure 1: The circuit for quantum phase estimation for estimating the eigenvalue of a unitary operator U. For the simulation results in this paper, we considered four noisy quantum channels to represent noise processes. The first is the bit flip channel that represents a noise process that flips the qubit with probability p and is given as E(ρ) = (1 −p)ρ+pXρX, where ρis the density matrix representing the state of a single qubit and Xis the Pauli Xgate. This channel applies the Xgate to the qubit with probability p. The second noise model we used in this work is the phase-flip channel that models the random flipping of the phase of the qubit state |1⟩. This noise model can be written as E(ρ) = (1 −p)ρ+pZρZ, where Zis the Pauli Zgate. The third noise model we considered is the combination of bit-flip and phase flip channel, called bit-phase flip channel defined as E(ρ) = (1 −p)ρ+pY ρY. Since Y=iXZ, this channel applies both the phase flip and bit flip with probability p. The fourth channel is called the depolarizing channel. Under this noise model, the environment interacts with the system in such a way that with the probability p, a qubit is depolarized, i.e., the state of the system is

Question 917 multiple-choice

Quantum algorithms often rely on the quantum Fourier transform (QFT) for problems involving periodicity, but practical implementations face challenges from noise and limited gate precision. The approximate quantum Fourier transform (AQFT) offers a trade-off between accuracy and resource requirements in these contexts. Which advantage does the approximate quantum Fourier transform (AQFT) offer when implemented on quantum hardware with limited coherence times and noisy environments? 1) It guarantees exact determination of all periodicities in quantum states. 2) It requires more precise quantum gates than the standard QFT. 3) It eliminates the need for quantum registers entirely. 4) It increases the overall gate count and circuit depth. 5) It introduces greater sensitivity to decoherence effects. 6) It reduces resource requirements and can improve robustness against decoherence by omitting low-impact gates. 7) It prevents the use of quantum superposition in algorithmic operations.

✓ Correct Answer:

The correct answer is 6) It reduces resource requirements and can improve robustness against decoherence by omitting low-impact gates..

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arXiv:quant-ph/9601018v1 21 Jan 1996Approximate Quantum Fourier Transform and Decoherence Adriano Barencoa, Artur Ekerta, Kalle-Antti Suominenband P¨ aivi T¨ orm¨ ac aClarendon Laboratory, Department of Physics, University o f Oxford, Parks Road, OX1 3PU Oxford, U.K. bTheoretical Physics Division, Department of Physics, Univ ersity of Helsinki, PL 9, 00014 Helsingin yliopisto, Finland cResearch Institute for Theoretical Physics, University of Helsinki, PL 9, 00014 Helsingin yliopisto, Finland November 26, 2024 Abstract We discuss the advantages of using the approximate quantum F ourier trans- form (AQFT) in algorithms which involve periodicity estima tions. We analyse quantum networks performing AQFT in the presence of decoher ence and show that extensive approximations can be made before the accura cy of AQFT (as compared with regular quantum Fourier transform) is compro mised. We show that for some computations an approximation may imply a bett er performance. PACS: 89.70.+c, 03.65.-w, 42.50.Lc Submitted to Phys. Rev. A. (Jan. 96) 1 Introduction In the course of history many ingenious mechanical, acousti c and optical devices have been invented for performing Fourier transforms [1] (inclu ding nature’s own such as the human ear). Most of them are now of merely historical inte rest since the arrival of the computer–based algorithm known as the fast Fourier tran sform (FFT) [2, 3] which eﬃciently computes the discrete Fourier transform. The FFT algorithm can also be phrased in terms of quantum dynamics, i.e., in terms of unita ry operations performed by a quantum computer on quantum registers. Indeed, all know n quantum algorithms employ the quantum version of Fourier transforms, either ex plicitly or indirectly. It is used for the periodicity estimation in the Shor algorithm [4] and its approximate version (the Hadamard transform) is commonly used to prepar e quantum registers in coherent superpositions of diﬀerent values. In this paper we analyse the performance of the quantum Fouri er transform (QFT) in the presence of decoherence. In particular we show t hat as far as the peri- 1 odicity estimation is concerned the approximate quantum Fo urier transform (AQFT) can yield better results than the full Fourier transform. In the following we use some terms which were originally adop ted from the classical theory of information and computer science and be came standard in the lore of quantum computation. More detailed descriptions can be f ound e.g. in Refs. [5, 6, 7] and in some recent reviews [8]. •Aqubitis a two–state quantum system; it has a chosen ‘computationa l basis’ {|0/an}bracketri}ht,|1/an}bracketri}ht}corresponding to the classical bit values 0 and 1. Boolean op erations which map sequences of 0’s and 1’s into another sequences of 0 ’s and 1’s are deﬁned with respect to this computational basis. A collecti on of Lqubits is called a register of size L. •Information is stored in the registers in binary form. For ex ample, number 6 is represented by a register in state |1/an}bracketri}ht⊗|1/an}bracketri}ht⊗|0/an}bracketri}ht. In more compact notation: |a/an}bracketri}ht stands for the direct product |aL−1/an}bracketri}ht ⊗ |aL−2/an}bracketri}ht...|a1/an}bracketri}ht ⊗ |a0/an}bracketri}htwhich represents a quantum register prepared with the value

Question 918 multiple-choice

Quantum computing operations rely on the manipulation of qubits using quantum gates and transforms, which differ fundamentally from classical logic gates due to their reversible and unitary nature. Efficient quantum algorithms often exploit the structure and properties of multi-qubit gates and quantum transforms to achieve computational speedups. Which statement accurately describes a key property of the Walsh-Hadamard gate used in quantum computing? 1) It is its own inverse, meaning applying it twice restores the original qubit state. 2) It can only operate on multi-qubit systems and cannot create superpositions. 3) Its matrix representation is always non-unitary for real-valued entries. 4) It is used exclusively for phase shift operations rather than amplitude changes. 5) It produces entanglement between two qubits when applied to a single qubit. 6) Its action cannot be represented as a 2x2 matrix over complex numbers. 7) It is irreversible, meaning the output cannot be mapped uniquely back to the input.

✓ Correct Answer:

The correct answer is 1) It is its own inverse, meaning applying it twice restores the original qubit state..

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quantum computing. 2.2 Binary Quantum Gates As with classical circuits, quantum operations can beperformed by networks of gates. Among a large number of possible quantum gates, we now review only the gates used in quantum Fourier transforms. For reversibility, each gate must have the same number of inputs and the outputs. First, single-input quantum gates are defined by 2/C22unitary matrices over complex numbers. Example 2. Consider the Walsh-Hadamard gate: W 2¼1ﬃﬃﬃ 2p11 1/C01/C20/C21 : Its application to state jx>is: W2jxi¼1ﬃﬃﬃ 2pðj0iþð /C0 1Þxj1iÞ; which is easily verified by applying it to states j0>and j1>. This gate is equal to its own inverse since ðW2Þ2¼I. Binary quantum gates with ninputs and noutputs are given by unitary matrices with 2n/C22nentries. Multiple- input, multiple output gates perform unitary operations over a tensor product of single quantum states. There are several common constructions of multiple qubit gates [15]. The controlled gate approach can be applied to any single- qubit gate G. The first (control) qubit is left unchanged, while the second qubit is affected by Gonly when the first qubit is j1>. Example 3. The phase shift gate performs multiplication: S/C11¼10 0ei/C11/C20/C21 : The controlled phase shift gate R/C11performs the phase shift if the value of the control qubit is 1. The effect of the circuit is given as: j00i!j 00i;j01i!j 01i;j10i!j 10i;j11i!ei/C11j11i: Other multi-qubit constructions include that of nindependent single-qubit operations. In that case, the function is given by the Kronecker product of the single- qubit matrices. 3Q UANTUM TRANSFORMS The Fourier Transform rep resents elements over an arbitrary Abelian (commutative) group by an expansion over orthogonal sets of basis vectors. We will first describe the traditional development of quantum transforms in the case of multivariate binary inputs, i.e., ðZ2Þn, as well as discrete Fourier transforms performed over binary qubits, in which case the group is Zn 2.ZILIC AND RADECKA: SCALING AND BETTER APPROXIMATING QUANTUM FOURIER TRANSFORM BY HIGHER RADICES 203 TABLE 1 Standard Quantum Operation Notation 3.1 Quantum Walsh-Hadamard Transform The first construction is given in terms of n-variable quantum functions over binary digits. Then, for vectors xandy, QFT equals the quantum Walsh-Hadamard Transform, W2njxi¼1ﬃﬃﬃﬃﬃ 2npX y2f0;1gnð/C01Þxyjyi; which we denote in shorthand as: jxi!1ﬃﬃﬃﬃﬃ 2npX y2f0;1gnð/C01Þxyjyi: The transformation of n-variable functions is performed by a Kronecker product of univariate transforms. Since the n-fold Kronecker product is equivalent to the parallel application ofnsingle-qubit functions, the overall transform is performed by only nWalsh-Hadamard gates, Fig. 1. The classical fast Walsh Transform requires Oðn2 nÞoperations. 3.2 Quantum Discrete Fourier Transform In the general case, Quantum Fourier Transforms defined over a set of tvalues are somewhat more complex to perform. For numbers xand y, the transform has the univariate form: jxi!1ﬃﬃ tpX y2f0;1;/C1/C1/C1;t/C01ge/C02/C25ixy tjyi: Unlike in previous cases, there is no apparent general way to represent the transform by means of the Kronecker product of the decomposable bases. Similarly to the developments in the traditional FFT, several factorizations are possible. We concentrate on the factorization that willallow us to develop the ternary transform in the section

Question 919 multiple-choice

Functional analysis and quantum probability often rely on concepts like operator semigroups, matrix-valued function spaces, and functional inequalities to study convergence and stability properties in complex systems. Hypercontractivity and logarithmic Sobolev inequalities play a central role in understanding mixing and regularization effects in both classical and quantum settings. Which statement best describes the significance of the equivalence between completely bounded hypercontractivity (cHCq) and complete logarithmic Sobolev inequality (cLSIq) for operator semigroups acting on matrix-valued function spaces? 1) It establishes that key regularization and entropy decay properties are intrinsically linked, allowing extension of convergence results from finite Markov chains to general measure spaces and quantum systems. 2) It shows that the generator of the semigroup must always be bounded and self-adjoint for mixing to occur in matrix-valued settings. 3) It provides a criterion for when the algebra of bounded measurable functions fails to be dense in Lp spaces. 4) It implies that the Dirichlet form cannot be defined for positive matrix-valued functions with spectra bounded away from zero. 5) It restricts the use of normalized traces in entropy computations to classical Markov processes only. 6) It demonstrates that hypercontractivity is irrelevant for convergence to equilibrium in infinite-dimensional systems. 7) It proves that Lq-entropy and the associated Dirichlet form are only applicable in commutative function algebras.

✓ Correct Answer:

The correct answer is 1) It establishes that key regularization and entropy decay properties are intrinsically linked, allowing extension of convergence results from finite Markov chains to general measure spaces and quantum systems..

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fact, it will be convenient to assume for technical reasons that the following hypothesis, already used in [66], [67], holds: Hypothesis B.7. There exists an algebra Aof bounded measurable functions, containing all the constants, dense in all the spaces Lp(),p≥1as well as in dom(L), that is stable under composition with multivariate smooth functions. We also assume that for any sequence {fn} ofAthat converges to a function finL2(), and every smooth bounded function ∶R→Rwith bounded derivatives, there exists a subsequence {(fnk)}of {(fn)}converging towards (f)inL1()and such thatL(fnk)converges to L(f)inL1(). For anym∈N, the algebra Mm(L∞()) coincides with the algebra L∞(E;Mm)of bounded measurable functions with values in Mm, with norm /parallel.alt1f/parallel.alt1L∞(Mm)deﬁned as supx∈E/parallel.alt1f(x)/parallel.alt1Mm. For sake of simplicity, we denote this norm by /parallel.alt1:/parallel.alt1∞. Next, for any f∈L∞(Mm), deﬁne the following trace on L∞(Mm): (f)∶=/integral.dispE1 mTrMm(f(x))(dx): We denote the completions of the Lpnorms associated to that state Lp(Mm),p≥1, and denote the norms associated to it by /parallel.alt1:/parallel.alt1Lp( ). To simplify the notations, we will get rid of the indices identifying the underlying spaces and introduce the normalized traces tr∶=1 mTr . The main difference to [64] in our setting arises from the possible unboundedness of the generator Lof the classical semigroup (St)t≥0, which will not be an issue as long as we carry out our differentiations in the algebras A(Mm)++∶={g=(gij);gij∈A∀ij∈{1;:::;m};g>0}of positive matrix valued functions with coefﬁcients in A, with spectrum uniformly bounded away from 0. Then, we deﬁne theLq-entropy and the Dirichlet form as follows: given elements f;g∈A(Mm)++, Entq(f)= (fqlogfq)− (fq)log (fq); Eq;idMm⊗L(f)∶= (fq−1(idMm⊗L)(f)): Next, we deﬁne the notions of completely bounded hypercontractivity and of complete logarithmic Sobolev inequality: Given q≥1, the semigroup (St)t≥0is said to4 - beq-completely hypercontractive if there exist c> 0;d≥1such that for all p≥qand allt≥c 2logp−1 q−1: /parallel.alt1St∶Lq(CIM⊂M)→Lp q(CIM⊂M)/parallel.alt1cb≤d1 q−1 p (cHCq(c;d)) - satisfy aq-complete logarithmic Sobolev inequality if there exist c≥0,d≥1such that for all m∈N, and allf∈A(Mm)++: Entq(f)≤cEq;idMm⊗L(f)+log(d)/parallel.alt1f/parallel.alt1q Lq(Mm): (cLSIq(c;d)) The equivalence between cLSIq(c;d)andcHCq(c;d) was proved in [64] in the case of a Markov chain deﬁned on a ﬁnite sample space (and even for quantum Markov semigroups in ﬁnite dimensions). Here, we extend this equivalence to the present abstract setting, which in particular incorporates the case of classical diffusions. Theorem B.8. Let(St)t≥0be a classical Markov semigroup deﬁned on the algebra L∞(E;F;)of bounded measurable functions on some measure space (E;F;), and assume that is an invariant measure of(St)t≥0for which (St)t≥0is reversible. Further assume the existence of a subalgebra Asatisfying Hypothesis B.7. Then, (i) If cHCq(c;d)holds, then cLSIq(c;d)holds. (ii) If cLSI 2(c;d)holds, then cHCq(c;d)holds for any q≥2. We now brieﬂy sketch a proof of Theorem B.8: As usual, the ﬁrst step towards establishing a Gross lemma is to provide a formula for the differential at p=qof p/uni21A6/parallel.alt1St(p)∶Lq ∞(CIE⊂L∞)→Lp ∞(CIE⊂L∞(E))/parallel.alt1cb =sup m/parallel.alt1idMm⊗St(p)∶Lq q(Mm⊂L∞(Mm))→ Lp q(Mm⊂L∞(Mm))/parallel.alt1; for some increasing, twice differentiable function t∶ [1;∞)→[0;∞). The proof of the differentiability follows closely the one of Lemma 9 of [64]: given 4These inequalities in particular imply the primitivity of the semigroup. One could easily extend these inequalities to non-primitive classical semigroups, which however play no role

Question 920 multiple-choice

In the study of complex semisimple Lie algebras, constructing explicit representations and understanding their geometric and combinatorial properties is fundamental for applications in mathematics and physics. Efficient algorithms for these computations are especially important when dealing with large representation spaces. Which approach enables polynomial-time computation of representation matrices for exceptional Lie algebras such as G₂, F₄, and E₈? 1) Utilizing Gelfand-Tsetlin patterns directly for exceptional types 2) Applying the Molev basis for all exceptional algebras 3) Embedding exceptional Lie algebras into general linear algebras glₐ(ℂ) and leveraging glₐ(ℂ) algorithms 4) Decomposing representations solely using negative roots 5) Restricting computations to classical Lie algebra types A, B, C, and D 6) Constructing moment polytopes without combinatorial bases 7) Using Lakshmibai monomial basis exclusively for exceptional types

✓ Correct Answer:

The correct answer is 3) Embedding exceptional Lie algebras into general linear algebras glₐ(ℂ) and leveraging glₐ(ℂ) algorithms.

📚 Reference Text:

the simple roots 2PS(G)can be computed in polynomial time. Indeed, if 2P(G)is not simple then e andf can computed from the above by using poly(hGi)many Lie brackets. We proceed case by case: Type A(gC=gln(C)): Let GT denote the set of Gelfand-Tsetlin patterns, which can be computed in time poly(hGi;dimM). Let GTthe corresponding Gelfand- Tsetlin basis; it is a basis of weight vectors. Using the explicit formulas given in [ 16,32] (cf. Theorem 2.3 in [ 32]), each matrix element of the corresponding representation matrices of e ,f , andh can be computed in polynomial time given the pair of Gelfand-Tsetlin patterns indexing the entry. It follows that these representation matrices can be computed in time poly(hGi;dimM). Types B, D (gC=son(C)): Using the corresponding Gelfand-Tsetlin basis [ 17], the proof proceeds as for type A. Type C (gC=spn(C)): In this case, we can use Molev’s basis [ 31] which has similar properties. Exceptional Types : Since there are only ﬁnitely many exceptional Lie groups, they can all be embedded into some ﬁxed GLa(C). Correspondingly, gCcan be embedded into gla(C); let us ﬁx such an embedding for each type. Now let O(gla(C))ldenote the degree-lpart of the coordinate ring of gla(C). Write=P ii!i, where the !iare the fundamental weights of g. Letd:=P idi; it is easy to see using Weyl’s dimension formula that d=O(dimM). It can be shown that Moccurs as a subrepresentation ofM0:=O(gla(C))lfor somel=O(d) =O(dimM)that can be computed explicitly from. The dimension of M0is equal to the number of monomials of degree lina2 variables, and therefore O(la2) =O(poly(dimM)), sinceais a constant. We now compute the representation matrices of the generators of gla(C)with respect to the basis of M0consisting of the usual degree- lmonomials in a2variables, which we will denote by B0. From this, we in turn obtain the representation matrices for thegeneratorsof gCwithrespecttothesamebasisbyusingtheexplicitembeddingﬁxed above. After this, we can compute a basis of highest weight vectors for Gby ﬁnding those weight vectors that are annihilated by the action of the e for 2PS(G). For each highest weight vector, we then compute the corresponding Lakshmibai monomial basis expressed in terms of the usual monomial basis by using Lemma 10. In this way we obtain a second basis BofM0. At last, we compute the inverse change of basis matrix and use it to express the representation matrices for the generators of gCwith respect to the basis B. By restricting these representation matrices to the Lakshmibai monomial basis BBcorresponding to a copy of M=MinM0, we ﬁnally obtain the desired representation matrices. All this can be done in time O(poly(dimM0)) =O(poly(dimM)). 14 P. BÜRGISSER, M. CHRISTANDL. K.D. MULMULEY, M. WALTER 3.3. Inequalities for moment polytopes. We now recall the description of the moment polytope G(M)from [41]. Let (M) =f'1;:::;'Dgdenote the multiset of weights of the representation M, whereD=dimM, and consider a corresponding decomposition of Minto one-dimensional weight spaces, M=LD i=1C i, where iis a weight vector of weight 'i. Recall that N(G)denotes the set of negative roots of G as deﬁned at the beginning of section 3.

Question 921 multiple-choice

Quantum computing architectures are evaluated on their ability to execute algorithms rapidly and reliably, balancing gate operation speed against error accumulation. The quantum speed limit (QSL) sets fundamental lower bounds on gate execution times for different hardware platforms. Which statement best characterizes the comparative quantum speed limits when implementing the quantum Fourier transform and QAOA algorithms on neutral atom versus superconducting qubit platforms under optimal control theory? 1) Neutral atom platforms exhibit significantly faster circuit QSLs than superconducting qubits for both algorithms. 2) Both neutral atom and superconducting qubit platforms show similar weighted circuit QSLs relative to system size and gate type. 3) Superconducting qubits outperform neutral atoms in QSL exclusively for multi-qubit gates but not for two-qubit gates. 4) The QSL for both algorithms is determined solely by the connectivity of the platform, not gate types or system size. 5) Parity mapping introduces a substantial QSL advantage for superconducting qubits over neutral atoms. 6) Neutral atom platforms have slower QSLs due to inherently longer error time scales. 7) Circuit depth is not influenced by the choice of quantum speed limit or hardware platform.

✓ Correct Answer:

The correct answer is 2) Both neutral atom and superconducting qubit platforms show similar weighted circuit QSLs relative to system size and gate type..

📚 Reference Text:

Title: Comparing planar quantum computing platforms at the quantum speed limit Year: 2023 Paper ID: 702c6bde1e0111ca5ab6299cdabe21962381980b Source: semantic-scholar URL: https://www.semanticscholar.org/paper/702c6bde1e0111ca5ab6299cdabe21962381980b Abstract: An important aspect that strongly impacts the experimental feasibility of quantum circuits is the ratio of gate times and typical error time scales. Algorithms with circuit depths that significantly exceed the error time scales will result in faulty quantum states and error correction is inevitable. We present a comparison of the theoretical minimal gate time, i.e., the quantum speed limit (QSL), for realistic two- and multi-qubit gate implementations in neutral atoms and superconducting qubits. Subsequent to finding the QSLs for individual gates by means of optimal control theory we use them to quantify the circuit QSL of the quantum Fourier transform and the quantum approximate optimization algorithm. In particular, we analyze these quantum algorithms in terms of circuit run times and gate counts both in the standard gate model and the parity mapping. We find that neutral atom and superconducting qubit platforms show comparable weighted circuit QSLs with respect to the system size.

Question 922 multiple-choice

Quantum computing has shown significant promise for solving certain computational problems by leveraging properties such as quantum superposition and entanglement. One area of active research is the application of quantum algorithms to combinatorial challenges, including those involving group-theoretic structures. Which statement best explains the quantum computational limitation encountered when tackling the graph isomorphism problem via the hidden subgroup problem framework? 1) Exponential quantum speedups are readily achievable for all group-based problems, regardless of group structure. 2) Single or pair-wise measurements on coset states in non-abelian groups yield enough information for efficient algorithms. 3) The use of quantum Markov chain sampling consistently provides exponential speedups for graph isomorphism. 4) Quantum algorithms for abelian hidden subgroup problems generally fail to outperform classical methods. 5) Efficient quantum algorithms for non-abelian hidden subgroup problems exist for every group class. 6) Coset states in abelian groups are less informative than those in non-abelian groups for hidden subgroup problems. 7) Extracting useful information for graph isomorphism via quantum hidden subgroup algorithms requires highly entangled measurements across at least Ω(n log n) coset states, which are experimentally challenging to implement.

✓ Correct Answer:

The correct answer is 7) Extracting useful information for graph isomorphism via quantum hidden subgroup algorithms requires highly entangled measurements across at least Ω(n log n) coset states, which are experimentally challenging to implement..

📚 Reference Text:

arXiv:quant-ph/0511148v1 15 Nov 2005Limitationsof QuantumCosetStatesforGraphIsomorphism Sean Hallgren,MartinR¨ otteler, and Pranab Sen NECLaboratories America, Inc. 4 IndependenceWay, Suite200 Princeton, NJ08540,U.S.A. {hallgren,mroetteler,pranab }@nec-labs.com Abstract It has been known for some time that graph isomorphism reduce s to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum co mputation are obtained by solving in- stances of the HSP. A common feature of the resulting algorit hms is the use of quantum coset states, whichencodethehiddensubgroup. Anopenquestionhasbeenh owharditistousethesestatestosolve graph isomorphism. It was recently shown by Moore, Russell, and Schulman [MRS05] that only an exponentially small amount of information is available fro m one, or a pair of coset states. A potential sourceof powerto exploitare entangledquantummeasuremen tsthat act jointly onmanystates at once. We show that entangled quantum measurementson at least Ω(nlogn)coset states are necessary to get useful information for the case of graph isomorphism, match ing an informationtheoretic upper bound. This may be viewed as a negative result because highly entang led measurements seem hard to imple- ment in general. Our main theorem is very general and also rul es out using joint measurementson few cosetstatesforsomeothergroups,suchas GL(n,Fpm)andGnwhereGisﬁniteandsatisﬁesasuitable property. 1 Introduction Almost all exponential speedups that have been achieved in q uantum computing are obtained by solving some instances of the Hidden Subgroup Problem (HSP). In part icular, the problems underlying Shor’s al- gorithms for factoring and discrete logarithm [Sho97], as w ell as Simon’s problem [Sim94], can be nat- urally generalized to the HSP: given a function f:G→Sfrom a group Gto a setSthat is constant on left cosets of some subgroup H≤Gand distinct on different cosets, ﬁnd a set of generators for H. Ideally, we would like to ﬁnd Hin time polynomial in the input size, i.e. log|G|. The abelian HSP [Kit95, BH97, ME98], i.e., when Gis an abelian group, lies at the heart of efﬁcient quantum al- gorithms for important number-theoretic problems like fac toring, discrete logarithm, Pell’s equation, unit group of anumber ﬁeldetc. [Sho97, Hal02,Hal05,SV05]. It has been known for some time that graph isomorphism reduce s to the HSP over the symmetric group [Bea97,EHK99a], anon-abelian group. Whilethenon-a belian HSPhasreceived muchattention asa result, efﬁcient algorithms are known only for some special classes of groups [IMS03, FIM+03, MRRS04, BCD05]. Ontheother hand, theHSPpresents asystematic wayt otryand approach thegraph isomorphism problem, and this approach is rooted in developing a deeper u nderstanding of how far techniques and tools that have worked in the abelian case can be applied. Tothe bes t of our knowledge, the only other approach to solve graph isomorphism on a quantum computer is by creati ng a uniform superposition of all graphs 1 isomorphic to a given graph. It has been proposed to create th is superposition via quantum sampling of Markov chains [AT03],however, very little isknown about th is. One of the key features of a quantum computer is that it can com pute functions in superposition. This fact alone does not lend itself to exponential speedups, for instance for unstructured search problems it merely leads to a

Question 923 multiple-choice

The Hidden Subgroup Problem (HSP) is a central topic in computational group theory and quantum computing, with query complexity bounds being crucial for understanding algorithmic efficiency. In classical deterministic algorithms, the structure of the group and subgroup directly affects the minimum and maximum number of queries required to solve HSP instances. What is the asymptotic lower bound for the decision tree depth required by any classical algorithm to identify the hidden subgroup in HSP+, given a finite Abelian group G and subgroup H, where there are at least |H| possible outputs? 1) Ω(|G|·|H|) 2) Ω(log|G|·log|H|) 3) Ω(log|H|·log(|G|·|H|)) 4) Ω(√(|G|·|H|)) 5) Ω(|H|·log|G|) 6) Ω(log|G|·|H|) 7) Ω(√(|G|)·log|H|)

✓ Correct Answer:

The correct answer is 3) Ω(log|H|·log(|G|·|H|)).

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It is worth noting that W1is non-decrease after each iteration. Since W1andW1×{0}are actually the same set, we essentially query all the elements not queried in W1in Step 3 each time. Thus, in the i-th iteration, the number of queries in Step 3 is ai−ai−1; also, the number of queries in Step 6 is at most ( ti+1)bi. Thus, the total number of queries is at most al+/summationtextl i=1(ti+1)bi. In the following, we ﬁrst prove |Hi|≤pki−ti i|Hi−1|. i) IfHi=Hi−1, then ti=kiand thus|Hi|=pki−ti i|Hi−1|. ii) IfHi/\⌉}atio\slash=Hi−1, then there exists hsuch thathi=pti iandHi=Hi−1+/a\}⌊ra⌋k⌉tl⌉{th/a\}⌊ra⌋k⌉tri}ht. By Lemma 2, we have |Hi|≤pki−ti i|Hi−1|. Hence,|Hi|≤pki−ti i|Hi−1|≤/producttexti m=1pkm−tmm. Since|Vi|=/producttexti m=1ptmm, we hace |Hi|·|Vi|=/producttexti m=1pkmm=|Gi|. Thus,|V|=|Vl|=|Gl| |Hl|=|G| |H|. LetS1={i|ri=ri−1,1≤i≤l}andS2={i|ri=ri−1+ 1}. Then |S1|=l−rand|S2|=r. Moreover, for i∈S1, we have ti≥1 and bi≤⌈/radicalbig |Vi|/max{1,r}⌉≤⌈/radicalbig Vi⌉≤/radicalbig Vi+1; fori∈S2, we have ti= 0. Similar to Eq. (1), we have/summationtexti m=1/radicalbig |Vm| ≤ 1 1−1/√ 2/radicalbig |Vi|. Sincet2 i≤2ti+2≤4·pti iforti≥1, we have ti≤2√ |Vi|√ |Vi−1|. So ti/radicalbig |Vi−1|≤2/radicalbig |Vi|. Since |V|=l/productdisplay i=1ptm m, we have l/summationdisplay i=1ti≤l/summationdisplay i=1tmlogpm= log|V|, which means/summationtextl i=1ti≤logp|V|. Moreover, since |V|=l/productdisplay i=1ptmm≥/productdisplay i∈S1ptmm≥/productdisplay i∈S1pm≥2l−r, 14 we have l−r≤log|V|. Thus, we have /summationdisplay i∈S1(ti+1)bi≤/summationdisplay i∈S1(ti+1)(/radicalbig |Vi−1|+1) =l/summationdisplay i=1(ti/radicalbig |Vi−1|+/radicalbig |Vi−1|)+l/summationdisplay i=1ti+1+l−r ≤l/summationdisplay i=1(2/radicalbig |Vi|+/radicalbig |Vi−1|)+2log|V| ≤l/summationdisplay i=13/radicalbig |Vi|+2log|V| ≤3 1−1/√ 2/radicalbig |V|+2log|V| =O(/radicalbig |V|), /summationdisplay i∈S2(ti+1)bi=/summationdisplay i∈S2bi ≤/summationdisplay i∈S2(/radicalBigg/radicalbig |V| max{1,ri−1}+1)

Question 924 multiple-choice

In computational quantum physics and signal processing, algorithms such as the Fast Fourier Transform (FFT) and Quantum Fourier Transform (QFT) are used for analyzing data, with tensor network representations like Matrix Product States (MPS) playing a key role in efficient simulations. The efficiency of these algorithms can depend heavily on the structure and compressibility of the input data. Which statement accurately describes the computational advantage of simulating the Quantum Fourier Transform (QFT) using Matrix Product States (MPS) for large, structured datasets? 1) QFT simulation is always exponentially faster than FFT, regardless of data structure or compressibility. 2) QFT simulation becomes slower than FFT as the bond dimension of the MPS decreases. 3) For highly compressible data with low MPS bond dimension and grid size exponent n greater than 18, QFT simulation can be orders of magnitude faster than FFT even accounting for MPS conversion time. 4) FFT outperforms QFT for any data that originates from quantum simulation. 5) QFT simulation efficiency is independent of whether the data is in MPS form. 6) The crossover point where QFT becomes faster than FFT does not depend on the grid size or data structure. 7) QFT simulation on classical hardware is feasible for all quantum states, including those used in Shor’s algorithm.

✓ Correct Answer:

The correct answer is 3) For highly compressible data with low MPS bond dimension and grid size exponent n greater than 18, QFT simulation can be orders of magnitude faster than FFT even accounting for MPS conversion time..

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approach gives no advantage for random data. Using structured data, we ﬁnd a crossover point around n=18, where the QFT approach is faster than the FFT approach, even when we account for the time needed to convert the data to MPS form. We consider three example functions, with more details about these functions given in Appendix K. The ﬁrst is a single cosine function making one period over the interval [0, 1]. This function com- presses exactly into an MPS of bond dimension χm=2. The next example is a sum of 20 cosine functions of vari- ous frequencies, combining to make a function resembling a sinc function centered in the interval [0, 1]. The last example is a single cosine function plus a sum of four functions of the form ae−b|x−c|for various values of a,b, and c, which results in the function having sharp cusps. For these last two examples, the MPS bond dimensions are around χm∼10, with the dimensions being somewhat more uniform for the cosine-plus-cusps example. The time of the QFT step is approximately linear in n, although it depends on the speciﬁc bond dimensions needed to repre- sent a function on a grid size of 2n, which can have a more complicated but mild dependence on n. Beyond n>18 the QFT approach becomes orders of magnitude faster than the FFT approach if one does not include the cost of constructing the data MPS. This is reasonable in certain contexts, such as if the function to be transformed is an MPS solution optimized over adiﬀerential equation represented by some tensor network [49–51] or the result of a quantum simulation with an MPS state representation [ 52–54]. Even when the construction of the data MPS is included, with data as compressible as in these simple cases one ﬁnds an improvement of about 1 order of magnitude in comparison with the FFT. Eﬃcient algorithms for compressing functions into an MPS are an ongoing area of research [ 45,55–57]. V. CONCLUSION We have shown that the maximum-entanglement prop- erty of the standard QFT comes mainly from the trivial bit-reversal operation, while the core part of QFT has Schmidt coeﬃcients decaying exponentially quickly. We have given rigorous bounds on the Schmidt coeﬃcients, demonstrating the exponential decay, and we have given a method to construct the constant-bond-dimension QFT MPO with an exponentially small error in O(n2)time using a tensor network representation of QFT. Thus, simulating the QFT on a classical computer with high precision can be done in O(n)time if a low-bond-dimension MPS rep- resentation of the initial state is given. In tests comparing the QFT and the FFT for Fourier transforming data in a purely classical context, we have found that when the data are compressible, the QFT approach has better scaling andcan be signiﬁcantly faster, even when the time needed to convert the data to an MPS is accounted for. We emphasize that this does not mean thew QFT is eﬃciently classically simulable in general (e.g., states in Shor’s algorithm), since

Question 925 multiple-choice

The Hidden Subgroup Problem (HSP) is a fundamental challenge in quantum computing with significant cryptographic implications, especially in the context of non-abelian groups. Recent research explores efficient quantum algorithms for solving the HSP in structured non-abelian groups, such as nilpotent groups, by leveraging their algebraic properties. Which technique enables the recursive reduction of the Hidden Subgroup Problem in nilpotent groups to an abelian case, thereby allowing the use of efficient quantum algorithms? 1) Applying Fourier analysis over the entire group without decomposition 2) Utilizing group cohomology to classify subgroup extensions 3) Transforming the problem through successive images in quotient groups along a central series 4) Employing randomized algorithms for subgroup membership testing 5) Mapping group elements to their character tables for direct diagonalization 6) Using quantum walks to traverse the subgroup lattice 7) Implementing non-deterministic algorithms for subgroup intersection problems

✓ Correct Answer:

The correct answer is 3) Transforming the problem through successive images in quotient groups along a central series.

📚 Reference Text:

Quantum Information Processing (2024) 23:14 https://doi.org/10.1007/s11128-023-04228-2 Zero sum subsequences and hidden subgroups Muhammad Imran1·Gábor Ivanyos2 Received: 17 April 2023 / Accepted: 17 December 2023 / Published online: 5 January 2024 © The Author(s) 2024 Abstract We propose a method for solving the hidden subgroup problem in nilpotent groups. The main idea is iteratively transforming the hidden subgroup to its images in the quotient groups by the members of a central series, eventually to its image in the commutativequotient of the original group, and then using an abelian hidden subgroup algorithm to determine this image. Knowing this image allows one to descend to a proper subgroup unless the hidden subgroup is the full group. The transformation relies on ﬁnding zerosum subsequences of sufﬁciently large sequences of vectors over ﬁnite prime ﬁelds. We present a new deterministic polynomial time algorithm for the latter problem in the case when the size of the ﬁeld is constant. The consequence is a polynomial timeexact quantum algorithm for the hidden subgroup problem in nilpotent groups having constant nilpotency class and whose order only have prime factors also bounded by a constant. Keywords Hidden subgroup problem ·Nilpotent group ·Zero sum subsequence · Exact quantum algorithm 1 Introduction The standard version of the hidden subgroup problem (HSP for short) is the following.Given a function fon the group G→{ 0,1} rwith the property that there is a subgroup Hsuch that f(x)=f(y)if and only if xand yare in the same left coset of H, ﬁnd the subgroup H. Perhaps Kitaev was the ﬁrst who observed that Shor’s factoring and B Muhammad Imran muh.imran716@gmail.com Gábor Ivanyos Gabor.Ivanyos@sztaki.hun-ren.hu 1Department of Algebra and Geometry, Budapest University of Technology and Economics, Egry József u. 1, Budapest 1111, Hungary 2Institute for Computer Science and Control, Hungarian Research Network, Kende u. 13-17, Budapest 1111, Hungary 123 14 Page 2 of 23 M. Imran, G. Ivanyos discrete logarithm algorithms can be generalized to solve the HSP in ﬁnite abelian groups (and also in certain inﬁnite commutative groups) in polynomial time. Much less is known about the complexity of the problem in non-commutative groups. Themost general result is due to Ettinger, Hoyer and Knill. They showed in [ 1] that the query complexity of the problem in ﬁnite not necessarily abelian groups is polynomial. Regarding the time complexity, Kuperberg’s subexponential time quantum algorithm[2] for the HSP in dihedral and very similar groups is perhaps the best known result. It has a remarkable extension by Alagic et al. [ 3] to a special HSP in a class including non-solvable groups. There are some classes of groups in which the HSP can be solvedin polynomial time. See the survey papers by Lomont [ 4] and by Wang [ 5] for early results of this kind. The paper [ 6] by Lomonaco and Kauffman proposes interesting derivatives and generalizations of the Shor-Kitaev algorithm. The paper [ 7] by Horan and Kahrobaei discusses cryptographic aspects of the HSP and reports also on more recent results. The hidden shift problem in abelian groups

Question 926 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) are central to advancing computational capabilities, especially for non-abelian groups such as Weyl-Heisenberg, which play key roles in quantum mechanics and error-correcting codes. Modern approaches leverage advanced representation theory and resource-efficient techniques to address these challenging problems. Which innovation in quantum algorithms for the HSP over Weyl-Heisenberg groups specifically enables the use of only two coset states per iteration, thereby improving resource efficiency compared to previous methods? 1) Employing commutative Fourier analysis of subgroup elements 2) Utilizing tensor powers of abelian group representations 3) Applying stabilizer code constructions directly to group elements 4) Exploiting high-dimensional irreducible representations without decomposition 5) Introducing a novel relabeling technique for irreducible representations to produce low-dimensional components 6) Implementing quantum error correction during coset state preparation 7) Using classical post-processing to simulate non-commutative measurements

✓ Correct Answer:

The correct answer is 5) Introducing a novel relabeling technique for irreducible representations to produce low-dimensional components.

📚 Reference Text:

Title: An Efficient Quantum Algorithm for the Hidden Subgroup Problem over Weyl-Heisenberg Groups Year: 2008 Paper ID: b9257e8a94113583ad31b87b3b0a6b09b36ba6fb Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b9257e8a94113583ad31b87b3b0a6b09b36ba6fb Abstract: Many exponential speedups that have been achieved in quantum computing are obtained via hidden subgroup problems (HSPs). We show that the HSP over Weyl-Heisenberg groups can be solved efficiently on a quantum computer. These groups are well-known in physics and play an important role in the theory of quantum error-correcting codes. Our algorithm is based on non-commutative Fourier analysis of coset states which are quantum states that arise from a given black-box function. We use Clebsch-Gordan decompositions to combine and reduce tensor products of irreducible representations. Furthermore, we use a new technique of changing labels of irreducible representations to obtain low-dimensional irreducible representations in the decomposition process. A feature of the presented algorithm is that in each iteration of the algorithm the quantum computer operates on two coset states simultaneously. This is an improvement over the previously best known quantum algorithm for these groups which required four coset states.

Question 927 multiple-choice

In algebraic cryptography, the structure of vector spaces over finite fields plays a crucial role in constructing secure primitives for post-quantum signature schemes. Understanding how elements within these algebras organize into groups enables efficient and secure parameter generation. In a 4-dimensional algebra over a finite field GF'(p) used for post-quantum signature schemes, which of the following statements is true regarding invertible vectors that are not scalar multiples of the identity? 1) Each invertible non-scalar vector uniquely belongs to one of three distinct types of commutative groups within the algebra. 2) Every invertible vector is a generator for multiple non-commutative subgroups. 3) All invertible vectors form a single large cyclic group. 4) Invertible vectors are distributed randomly across commutative and non-commutative groups. 5) Only scalar vectors can be part of commutative groups in this algebra. 6) The order and count of commutative groups cannot be determined for invertible vectors. 7) Invertible vectors may simultaneously belong to multiple distinct commutative groups.

✓ Correct Answer:

The correct answer is 1) Each invertible non-scalar vector uniquely belongs to one of three distinct types of commutative groups within the algebra..

📚 Reference Text:

Title: STRUCTURE OF A 4-DIMENSIONAL ALGEBRA AND GENERATING PARAMETERS OF THE HIDDEN DISCRETE LOGARITHM PROBLEM Year: 2022 Paper ID: 084bf010e964a9ce8d5586ae9ea31c1da91002a0 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/084bf010e964a9ce8d5586ae9ea31c1da91002a0 Abstract: Structure of a 4-dimensional algebra and generating parameters of the hidden discrete logarithm problem the field GF'(p) is studied in connection with using it as algebraic support of the hidden discrete logarithm problem that is an attractive primitive of post-quantum signature schemes. It is shown that each invertible 4-dimensional vector that is not a scalar vector is included in a unique commutative group representing a subset of algebraic elements. Three types of commutative groups are contained in the algebra and formulas for computing the order and the number of groups are derived for each type. The obtained results are used to develop algorithms for generating parameters of digital signature schemes based on computational difficulty of the hidden logarithm problem.

Question 928 multiple-choice

Quantum algorithms for group-theoretic problems often leverage the structure of non-abelian groups to solve hidden subgroup problems efficiently. Specialized variants such as the Hidden Subgroup Modulo Commutator (HSMC) have proven particularly useful for certain classes of groups in quantum computation. In quantum hidden subgroup algorithms, which property of semi-elementary p-groups enables HSMC strategies to reduce the problem to an efficiently solvable abelian case? 1) Their group order is always a power of two 2) The quotient G/G' is elementary abelian 3) They have only cyclic subgroups 4) Their center is trivial 5) Every subgroup is normal 6) All elements commute 7) Their commutator subgroup is abelian

✓ Correct Answer:

The correct answer is 2) The quotient G/G' is elementary abelian.

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suitable for such a reduction. In the standard version, the input is given by an oracle which is a unitary map computing |x/angbracketright|f(x)/angbracketrightfrom|x/angbracketright|0/angbracketright. Many important applications belong to the case when the group Gacts as a permutation group on a (large) set /Omega1, and we are interested in computing the stabilizer of an element ω∈/Omega1. In that case, the stabilizer is the subgroup hidden by the function f:G→/Omega1, where x/mapsto→xωthe action of xonω. In fact, Kitaev used this framework to generalize Shor’s algorithms. The usual hidden subgroup algorithms start with computing the superposition 1√|G|/summationtext x∈G|x/angbracketright|f(x)/angbracketrightusing the oracle, and most of them ignore the second register that holds the value of fand work with the coset superpositions |xH/angbracketright=1√|H|/summationtext y∈H|xy/angbracketright in the sequel, see e.g., [ 6]. These methods, as noted in [ 2], remain applicable in the context where the oracle is assumed to generate copies of a mixture of the coset super-positions. This holds in particular in the case of the exact abelian hidden subgroup algorithm of [ 16]. Speciﬁcally, we consider the state 1√|G|/summationtext x∈G|x/angbracketright|f(x)/angbracketrightas a puriﬁcation of the mixed state /Xi1G,H=1 |G:H|/summationtext x∈X|xH/angbracketright/angbracketleftxH|, where Xis any left transversal of H inG, in order to have a unitary oracle. The state /Xi1G,His referred to as a (hidden) subgroup state. We assume that our hidden subgroup His given by a unitary map (referred as oracle) that, on zero input, returns a copy of an arbitrary (though ﬁxed) puriﬁcation of the subgroup state /Xi1G,H. It will be convenient to introduce a subtask of the HSP , namely computing the hidden subgroup modulo the commutator subgroup of G, that is, the subgroup HG/prime where His the hidden subgroup. We use the shorthand HSMC for this problem. To illustrate the power of HSMC in nilpotent groups note that it naturally includes thecommutative case of the HSP and that having computed the subgroup HG /primeand it is a proper subgroup of G; then, we can descend to it to compute H, while if HG/prime=G then H=Gbecause in a nilpotent group every maximal subgroup contains the commutator. We give a high-level description of a strategy for solving the problem HSMC in a class of nilpotent groups. We call a group G semi-elementary ifGis a p-group for some prime psuch that G/G/primeis elementary abelian. The Heisenberg group modulo p, that is, the group of 3 ×3 upper triangular matrices having entries integers modulo pand 1s in the diagonal, is a semi-elementary p-group, as well as the groups of the unitriangular matrices of higher dimension groups modulo p, while the Heisenberg group modulo p2is a p-group which is not semi-elementary. In a semi-elementary group G, our strategy for computing the hidden subgroup modulo the commutator is based on iterating the following procedure. Assume that Lis an elementary abelian subgroup contained in the center of G. Then we create a copy of the subgroup state corresponding to HL/Lin the quotient group G/Lfrom sufﬁciently many copies of the subgroup state for HinG. We refer to this procedure (as well

Question 929 multiple-choice

Quantum information science often involves identifying entanglement structures within multipartite quantum systems, which is central to the verification and characterization of quantum resources. Advanced quantum algorithms leverage group-theoretic techniques and Fourier sampling to efficiently solve problems related to hidden symmetries. Which statement accurately describes the relationship between the hidden cut problem and the State Hidden Subgroup Problem (StateHSP) in the context of quantum algorithms for entanglement testing? 1) The hidden cut problem is equivalent to the classical Hidden Subgroup Problem for non-Abelian groups. 2) The hidden cut problem cannot be formulated using group actions or quantum state invariance. 3) The hidden cut problem requires non-polynomial time algorithms for efficient solution. 4) The hidden cut problem only applies to Haar-random quantum states and not arbitrary product states. 5) The hidden cut problem is solved using Grover's algorithm for database search. 6) The hidden cut problem is formulated as a StateHSP using an Abelian group action, allowing Fourier sampling techniques analogous to Simon’s algorithm for efficient identification of the hidden cut. 7) The hidden cut problem is unrelated to symmetry discovery and does not utilize group-theoretic quantum algorithms.

✓ Correct Answer:

The correct answer is 6) The hidden cut problem is formulated as a StateHSP using an Abelian group action, allowing Fourier sampling techniques analogous to Simon’s algorithm for efficient identification of the hidden cut..

📚 Reference Text:

Title: The State Hidden Subgroup Problem and an Efficient Algorithm for Locating Unentanglement Year: 2024 Paper ID: 6cfa931b369056d22068c199e95832be16fd7fdc Source: semantic-scholar URL: https://www.semanticscholar.org/paper/6cfa931b369056d22068c199e95832be16fd7fdc Abstract: We study a generalization of entanglement testing which we call the “hidden cut problem.” Taking as input copies of an n-qubit pure state which is product across an unknown bipartition, the goal is to learn precisely where the state is unentangled, i.e. to determine which of the exponentially many possible cuts separates the state. We give a polynomial-time quantum algorithm which can find the cut using O(n/є2) many copies of the state, which is optimal up to logarithmic factors. Our algorithm also generalizes to learn the entanglement structure of arbitrary product states. In the special case of Haar-random states, we further show that our algorithm requires circuits of only constant depth. To develop our algorithm, we introduce a state generalization of the hidden subgroup problem (StateHSP) which might be of independent interest, in which one is given a quantum state invariant under an unknown subgroup action, with the goal of learning the hidden symmetry subgroup. We show how the hidden cut problem can be formulated as a StateHSP with a carefully chosen Abelian group action. We then prove that Fourier sampling on the hidden cut state produces similar outcomes as a variant of the well-known Simon’s problem, allowing us to find the hidden cut efficiently. Therefore, our algorithm can be interpreted as an extension of Simon’s algorithm to entanglement testing. We discuss possible applications of StateHSP and hidden cut problems to cryptography and pseudorandomness.

Question 930 multiple-choice

Quantum metrology leverages the principles of quantum coherence to achieve measurement precision beyond classical limits. The scaling exponent α in the relationship between information gain and phase accumulation time is a key indicator of whether an algorithm operates in the classical or quantum regime. For a quantum metrological algorithm, which value of the scaling exponent α in ΔI ~ τϕ^α corresponds to the Heisenberg limit, indicating the ultimate quantum precision? 1) α = 0.3 2) α = 0.5 3) α = 0.62 4) α = 0.65 5) α = 1 6) α = 1.5 7) α = 2

✓ Correct Answer:

The correct answer is 5) α = 1.

📚 Reference Text:

and their scaling behavior at large sensing times coincides with the scaling behavior resulting from a regular passport function. The quantum algorithms suffer, however, fromthe same irregularity problem at larger sensing times, not shownin Fig. 3. Finally, we discuss how our metrological algorithms use the quantum resource of qubit coherence in order to acquireinformation about the measured ﬂux. We quantify the quantum coherence resource spent in a given measurement by the total phase accumulation time τ ϕ¼NP kτkmk, where mkis the number of calls for the Bayesian learning subroutine with delayτk. The amount of information ΔIacquired during the measure- ment is given by a decrease of the Shannon entropy ΔI=H(P0)− H(P), where P0(Φi) and P(Φi) are the initial and ﬁnal probability distributions and H(P)=/C0P iPΦiðÞlog2PΦiðÞðÞ . The scaling behavior ΔIτϕ/C0/C1 /τα ϕseparates the classical domain with 0 < α≤ 0.5 from the quantum domain with 0.5 < α≤1, where α=1 corresponds to the ultimate Heisenberg limit. Indeed, in the idealcase where no relaxation and decoherence phenomena arepresent, the quantum algorithms double the ﬂux resolution (squeeze the ﬂux distribution function into a twice narrower range) for each next step of the procedure. This means that theassociated Shannon entropy decreases by ln(2) and one learns one bit of information for each doubling of the Ramsey delay time. In contrast, the classical procedure with N/C291 repetitions results in a Gaussian probability distribution of the measured quantitywhere the precision scales as δΦ/1=ﬃﬃﬃ ﬃ Np . Hence the associated Shannon entropy scales as τ 0.5with an invested total phase accumulation time τ=Nτ0. We run each quantum algorithm 25 times for every ﬂux value and average over the obtained information gains and phase times. The resulting scaling dependence is shown in Fig. 4and demonstrates that both Kitaev and Fourier algorithms indeed belong to the quantum domainwith a scaling exponent within [0.624, 0.654] (95% con ﬁdence interval). The scaling exponent is still below the Heisenberg limit, which is a consequence of the ﬁnite dephasing time T 2: at large time delays τ~T2the visibility of the Ramsey interference pattern decreases, requiring more Ramsey measurements in order to learn the next bit of information.10-5123456 10-410-310-2 Fig. 4 Information (in bits) inferred by the Kitaev (circles) and Fourier ( ﬁlled diamonds) algorithms as a function of the total phase accumulation time. The Kitaev algorithm was run for ﬁve different tolerance level constants εat each step (indicated by color). The color of diamonds indicates the different starting time of the Fourieralgorithm. The dashed red and blue lines refer to the Heisenberg and shot-noise scaling laws with the corresponding scaling exponents 1 and 1/2. The thin solid lines show the numericalsimulation for the 6-step Kitaev algorithm with an idealized passportfunction given by Eq. ( 1) at different dephasing times T 2ranging from 10 μs (red line) to 340 ns (blue line). One can clearly see that at large dephasing times the Kitaev procedure approaches the Heisenberg limit, while at smaller T2the scaling exponent decreases to the standard quantum limit 0.5. The observed

Question 931 multiple-choice

Advances in quantum computing present significant challenges to the security of widely used cryptosystems, prompting the development of new cryptographic approaches. Understanding which mathematical problems underpin cryptographic security is essential for designing systems resistant to quantum attacks. Which mathematical problem currently forms the foundation of many post-quantum cryptosystems due to its resistance against all known efficient quantum algorithms? 1) Integer factorization 2) Discrete logarithm 3) Elliptic curve discrete logarithm 4) Dihedral hidden subgroup problem 5) Extrapolated dihedral coset problem 6) Order finding 7) Lattice problems

✓ Correct Answer:

The correct answer is 7) Lattice problems.

📚 Reference Text:

Title: Quantum algorithms for typical hard problems: a perspective of cryptanalysis Year: 2020 Paper ID: 244738b569cde48980ff4586f821bc55afe75e47 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/244738b569cde48980ff4586f821bc55afe75e47 Abstract: In typical well-known cryptosystem, the hardness of classical problems plays a fundamental role in ensuring its security. While, with the booming of quantum computation, some classical hard problems tend to be vulnerable when confronted with the already-known quantum attacks, as a result, it is necessary to develop the post-quantum cryptosystem to resist the quantum attacks. With the purpose to bridge the two disciplines, it is significant to summarize known quantum algorithms and their threats toward these cryptographic intractable problems from a perspective of cryptanalysis. In this paper, we discussed the designing methodology, algorithm framework and latest progress of the mathematic hard problems on which the typical cryptosystems depend, including integer factorization problem, discrete logarithmic problem and its variants, lattice problem, dihedral hidden subgroup problems and extrapolated dihedral coset problem. It illustrated the reason why some cryptosystems such as RSA and ECC are not resistant to quantum attacks, yet some of them like lattice cryptosystems remain intact facing quantum attacks.

Question 932 multiple-choice

Quantum error correction is essential for protecting quantum information from noise, but its compatibility with symmetries in quantum systems poses significant theoretical and practical challenges. In quantum metrology and holographic codes, the interplay between perfect error correction and global symmetries can affect both signal sensitivity and code performance. Which principle explains why finite-dimensional quantum error-correcting codes cannot support a continuous symmetry acting transversally while enabling the evolution needed for quantum sensing? 1) The Gottesman-Kitaev-Preskill encoding theorem 2) The Eastin-Knill theorem 3) The Shannon noiseless coding theorem 4) The Landauer limit 5) The Hamming bound 6) The Ryu-Takayanagi formula 7) The Lieb-Robinson bound

✓ Correct Answer:

The correct answer is 2) The Eastin-Knill theorem.

📚 Reference Text:

a sum of local charges, the Eastin-Knill theorem poses a challenge to the application of error-correcting techniques; namely,error correction prevents probe states from evolving, obscuring the signal. The infinite-dimensional counter- examples of Ref. [12] show that it is nonetheless possible to correct against local noise andadmit a charge that is a sum of noise operators, granted one has non-normalizable code words. The bounds and example codes of this papercharacterize the imperfect code performance when the codewords are normalizable. Our results suggest that one might improve sensitivity to the signal using normalizable probe states by sacrificingCONTINUOUS SYMMETRIES AND APPROXIMATE QUANTUM … PHYS. REV . X 10,041018 (2020) 041018-27 some error-correction fidelity (cf. also Ref. [24]). However, to fruitfully apply our results to quantum metrology, additional steps are needed; these are the subject offollow-up work in progress. First, since we are trying tomeasure an unknown parameter (and not necessarily to protect quantum information per se ), we should account for the fact that a code is required only to reconstruct a logicalstate that would yield a precise reading of said parameter.Second, our results are stated in terms of the worst-case entanglement fidelity, but for applications to metrology one would prefer different figures of merit, such as the precisionat which the probe can sense magnetic fields or the abilityof a quantum clock to tell time accurately. Finally, it would be desirable to consider noise models that are more relevant to quantum metrology, such as fluctuating backgroundmagnetic fields that induce dephasing errors. B´ eny’s characterization of approximate quantum error correction of algebras [105] provides a potential approach to address- ing these challenges, because one can precisely specifywhich observables need to be faithfully reproduced afteraction by the noise and a possible recovery operation. Approximate quantum error-correcting codes also arise naturally in many-body quantum systems [8,9]. We antici- pate that constraints on correlation functions of many-body quantum states can be derived from the covariance proper-ties of the corresponding codes. Finally, the interplay of symmetry and quantum error correction has a prominent role in the AdS/CFT holo-graphic correspondence. Although covariance with respect to a continuous symmetry is incompatible with perfect correctability of erasure of physical subsystems for anyfinite-dimensional quantum code, nevertheless we expectthat in the AdS/CFT code continuous time evolution of the boundary system corresponds to continuous time evolution of the encoded logical bulk system. Our results relieve thetension between these two observations, because near-perfect correctability can be achieved if either the number of physical subsystems or the dimension of each physical subsystem becomes very large. Both these provisos applyto the continuum limit of a regulated holographic boundaryconformal field theory, as the number of lattice sites per unit volume is very large in this limit, and the number of degrees of freedom per site is also very large if semi-classical gravity accurately describes the bulk geometry (the“large- Nlimit”). Recent results indicate that not just exact continuous symmetries, but also exact discretesymmetries, are incompatible with the quantum error-correction properties of the AdS/CFT code [15,16] .A n intriguing topic

Question 933 multiple-choice

Algebraic structures such as permutation centralizer algebras and Kronecker algebras are fundamental in mathematical physics and representation theory, often encoding deep combinatorial and symmetry properties relevant to quantum algorithms. Detecting certain projectors in these algebras is closely related to problems involving Young diagrams and combinatorial invariants. In the context of quantum detection tasks for projectors in Kronecker algebra K(n), which labeling property directly determines whether a projector is nontrivial and relates to the combinatorial complexity of tensor model correlators? 1) The dimension of the underlying vector space 2) The order of the symmetric group Sn used in the algebra 3) The eigenvalue spectrum of the associated central element 4) The number of permutations commuting with each element 5) Triples of Young diagrams with non-vanishing Kronecker coefficients 6) The trace of the corresponding representation operator 7) The number of cycles in the permutation representation

✓ Correct Answer:

The correct answer is 5) Triples of Young diagrams with non-vanishing Kronecker coefficients.

📚 Reference Text:

its connection to centres of symmetric group algebras presents a concrete problem of comparison of complexities for a detection task. The estimates we arrive at in section 4 involve Λ(n)as well as some intrinsic complexities of gravitational measurements which we leave for future investigation. A more complete treatment of his holographic comparison of complexities should be useful for other instances of holographic CFT-quantum-state/AdS- gravitational-geometry correspondences (see the recent reference for a detailed discussion involvingAdS 3spacetimes e.g. [ 28]). To get some additional perspective on the quan- tum detection of the projectors PR, we describe a randomized classical algorithm, of the kind used in [ 29], for the same task in section 5 and give the corresponding complex- ity estimates. – 2 – JHEP05(2023)191Section 6 considers the problem of detecting projectors in a Kronecker algebra K(n)[30– 32] which is a sub-algebra of C(Sn)⊗C(Sn). This algebra is an example of a Permutation Centralizer Algebra (PCA) as deﬁned in [ 30]. The algebraK(n)is isomorphic to a direct sum of matrix algebras labelled by triples of Young diagrams (R,S,T )withnboxes, where these triples have a non-vanishing Kronecker coeﬃcient C(R,S,T ). This is the multiplicity of the trivialrepresentationinthetensorproductof Snirreduciblerepresentations VSn R⊗VSn S⊗VSn T. The matrix algebras have dimension C(R,S,T )2. These algebras control the combinatorics and correlators of invariant observables and correlators of tensor models [ 31]. We imagine a scenario where Bob constructs the projector and sends it to Alice. Alice can use QPE to identify the triple of Young diagrams in time which is polynomial in n. If the triple (R,S,T )has vanishing Kronecker coeﬃcient, there is no projector for Bob to send to Alice. For this reason the projector detection problem falls short of being a quantum algorithm for determining whether or not a triple of Young diagrams has a non-vanishing Kronecker coeﬃcient. We outline another algorithm, involving communication from Bob to Alice of a simple combinatorial-basis state in K(n), which has an expansion in projectors associated with all triples (R,S,T )having non-vanishing Kronecker coeﬃcients. Determining whether or not a given triple of Young diagrams appears in this expansion would be a quantum algorithm which determines whether or not the Kronecker coeﬃcient for that triple is zero. We leave the problem of estimating the complexity of this algorithm for future research. Section 7 concerns a third example of projector detection, based on the algebra A(m,n) labelled by two integers m,nand deﬁned using equivalence classes of permutations in Sm+n moduloconjugationbypermutationsin Sm×Sn. Thesealgebrasariseinthestudyof2-matrix invariants relevant to ﬂuctuations of half-BPS giant gravitons in AdS/CFT. The projectors are labelled by a triple of Young diagrams such that the associated Littlewood-Richardson coeﬃcient is non-zero. The algebra was described in [ 30] along with its decomposition into matrix algebras following earlier work on orthogonal bases for CFT4 correlators in the 2-matrix sector [ 33–39]. A formulation of a dual classical computation in terms of strings/branes or gravity conﬁgurations is an interesting problem for the future. Additional future research directions are given at the end of most of the

Question 934 multiple-choice

Quantum computing utilizes specialized algorithms such as the discrete quantum Fourier transform (DQFT) to achieve exponential speedup for certain computational tasks. Solid-state quantum technologies, like quantum dots in optical cavities, offer promising platforms for implementing scalable quantum circuits. Which feature most directly enables the scalable construction of multi-qubit DQFT circuits using quantum dot-cavity systems? 1) Utilization of two-photon absorption for state initialization 2) Integration of quantum error correction codes at the hardware level 3) Use of time-bin encoding to improve coherence times 4) Modular composition of controlled-rotation (CRk) gates realized via photon–quantum dot interactions 5) Application of dynamic decoupling sequences to suppress decoherence 6) Embedding quantum dots in photonic crystal waveguides for enhanced emission 7) Implementation of frequency conversion techniques for photon routing

✓ Correct Answer:

The correct answer is 4) Modular composition of controlled-rotation (CRk) gates realized via photon–quantum dot interactions.

📚 Reference Text:

Title: Photonic scheme of discrete quantum Fourier transform for quantum algorithms via quantum dots Year: 2019 Paper ID: bc8ddc06d70134ba9cbd4166666d9b5bf98e7e43 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/bc8ddc06d70134ba9cbd4166666d9b5bf98e7e43 Abstract: We propose an optical scheme of discrete quantum Fourier transform (DQFT) via ancillary systems using quantum dots (QDs) confined in single-sided cavities (QD-cavity systems). In our DQFT scheme, the main component is a controlled-rotation k (CRk) gate, which utilizes the interactions between photons and QDs, consisting of two QD-cavity systems. Since the proposed CRk gate can be experimentally implemented with high efficiency and reliable performance, the scalability of multi-qubit DQFT scheme can also be realized through the simple composition of the proposed CRk gates via the QD-cavity systems. Subsequently, in order to demonstrate the performance of the CRk gate, we analyze the interaction between a photon and a QD-cavity system, and then indicate the condition to be efficient CRk gate with feasibility under vacuum noise and sideband leakage.

Question 935 multiple-choice

Quantum phase estimation is a fundamental technique in quantum algorithms for extracting eigenphases of unitary operators with high precision. The efficiency of this process is crucial for applications such as factoring and order finding. Which statement correctly describes the key requirement that enables efficient extraction of multiple bits of a phase parameter φ using the quantum phase estimation algorithm? 1) The ability to initialize the quantum system in a pure eigenstate of the unitary operator. 2) The possibility to perform simultaneous measurements on all control qubits without error. 3) The use of unitaries whose eigenvalues are irrational numbers to guarantee uniqueness. 4) The exponential suppression of error probability solely by increasing the measurement time. 5) The efficient implementation of both the unitary operation U and its powers U^(2^j) in polynomial time. 6) The existence of classical algorithms for exact calculation of eigenvalues before measurement. 7) The restriction to unitary operators with infinite order to avoid degeneracy.

✓ Correct Answer:

The correct answer is 5) The efficient implementation of both the unitary operation U and its powers U^(2^j) in polynomial time..

📚 Reference Text:

we can measure p0(i.e.φ) to precision δwith error probability ǫ. Note that for ﬁxed δthe error probability ǫdecreases exponentially with ti.e. t=O(log(1/ǫ) but the precision δ(for ﬁxedǫ) cannot be eﬃciently improved – for each extra bit of precision, δ→δ/2 , we require t→4tin (19) to maintain a constant level of ǫ. Hence by this direct method, the number of bits of precision can be improved only by a correspondingly exponential increase in computing eﬀort – O(4l) steps forlbits of precision. This is unacceptable. To get around this d iﬃculty 15 let us suppose that not only Uis eﬃciently computable (i.e. in poly( n) steps) but also that: Assumption :U(2j)can be computed in poly( j,n) steps (20) This assumption is valid in our application of Ubeing “multiplication by y”.U2j is then “multiplication by y2j” which can be implemented by a sequence of jre- peated squarings, starting with y. It will not, however, be valid for a general unitary transformation U. Now assuming (20) we can eﬃciently improve the precision δas follows i.e. obtain lbits ofp0with computing eﬀort poly( l). Note that |λ/an}bracketri}htis an eigenstate of U2jwith eigenvalue exp (2 πi[2jφmod 1]). To obtain lbits ofφwith error probability ≤ǫwe measure (as above) the values of 2jφmod 1 forj= 0,...,l −1, to a ﬁxed precision δ= 1/8 with error probability ≤ǫ/l. Now if we write φin binary then 2jφhas the point shifted jplaces to the right and “mod 1” removes the integer part. Thus knowing 2jφmod 1 to ±1 8gives the ﬁrst few bits of 2jφmod 1 i.e. bits jandj+ 1 ofφitself. Hence we get about lbits of precision of φ. The probability that all these bits are correct exceeds (1 −ǫ/l)l≥1−ǫ. This completes the eﬃcient approximation ofφunder the assumption (20) above. Generally (as in Kitaev’s factoring algorithm) we will not h ave available a pure eigenstate of Ubut instead some superposition/summationtextaλ|λ/an}bracketri}ht. If we apply PROC to this state withtcontrol bits we will obtain/summationtextaλ|ψλ/an}bracketri}ht...|ψλ/an}bracketri}ht|λ/an}bracketri}htso that a measurement of the control bits will yield one of the eigenvalues λwith probabilities |aλ|2. i.e. if we trace out the eigenstate register |λ/an}bracketri}htthetcontrol qubits are in a mixture of the repeated states |ψλ/an}bracketri}ht...|ψλ/an}bracketri}htwith probabilities |aλ|2. Note that we must apply PROCttimes before any measurement of the control qubits is made. Otherwise eac h successive measurement will provide information about a diﬀerent eige nvalue and ﬁnally we will only obtain information about the average value of the λ’s weighted by |aλ|2, rather than about some oneof theλ’s. In most cases the eigenvalues exp(2πiφ) will have rational values ofφ,φ=a/b. This is because the U’s of interest will have ﬁnite order i.e. Um=Ifor somemso thatφ=k/mfor somek. For example if Uis “multiplication by y” thenUr=Iso thatφmust have the form k/r(as noted previously). In this situation we can ﬁnd φ exactly, rather than just approximately, by choosing a suit ably high precision δ. The minimum separation between any two rational numbers with de nominators ris 1/r so we can get φ=k/rexactly by measuring it to precision 1 /2r>1/2Ni.e. 1+logN bits. Thus we

Question 936 multiple-choice

Quantum algorithms are increasingly investigated for tasks in image processing, aiming to leverage quantum speedup for operations like filtering and denoising. Achieving practical performance gains depends on constraints related to encoding and signal quality in quantum representations of classical images. Which condition must be satisfied for a quantum image filtering algorithm to provide substantial speedup over classical methods? 1) The image must be grayscale rather than colored. 2) The image must be efficiently encodable into quantum states with a sufficient signal-to-noise ratio. 3) The filtering operation must involve only linear transformations. 4) The image must be compressed using classical lossless algorithms prior to quantum processing. 5) The quantum algorithm must utilize entanglement between all qubits. 6) The algorithm must be limited to images with less than 256 pixels. 7) The image must have a uniform intensity distribution.

✓ Correct Answer:

The correct answer is 2) The image must be efficiently encodable into quantum states with a sufficient signal-to-noise ratio..

📚 Reference Text:

Title: Achieving quantum advantages for image filtering Year: 2023 Paper ID: 6502d56403ca31a31437b5f29d521e3784086a67 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/6502d56403ca31a31437b5f29d521e3784086a67 Abstract: Image processing is a fascinating field for exploring quantum algorithms. However, achieving quantum speedups turns out to be a significant challenge. In this work, we focus on image filtering to identify a class of images that can achieve a substantial speedup. We show that for images with efficient encoding and a lower bound on the signal-to-noise ratio, a quantum filtering algorithm can be constructed with a polynomial complexity in terms of the qubit number. Our algorithm combines the quantum Fourier transform with the amplitude amplification technique. To demonstrate the advantages of our approach, we apply it to three typical filtering problems. We highlight the importance of efficient encoding by illustrating that for images that cannot be efficiently encoded, the quantum advantage will diminish. Our work provides insights into the types of images that can achieve a substantial quantum speedup.

Question 937 multiple-choice

In the theory of finite groups of Lie type, understanding the existence and structure of subgroups often depends on properties like commutator relations and the splitting of group extensions. These considerations are essential when embedding classical groups within exceptional groups over fields of various characteristics. Which of the following statements accurately describes the condition under which the subgroup M is isomorphic to L₄(5) within a larger group G? 1) M is isomorphic to L₄(5) if and only if p = 5, due to the presence of six I-invariant subgroups of order 5. 2) M is isomorphic to L₄(5) only when the maximal torus H is not contained in R. 3) M is isomorphic to L₄(5) if the extension splits for all odd primes p. 4) M is isomorphic to L₄(5) precisely when p = 2, because (U±α) and (U±γ) commute, allowing the embedding of L₄(5) in E₈(4). 5) M is isomorphic to L₄(5) only if N₀'/H is trivial. 6) M is isomorphic to L₄(5) whenever R ≅ 5⁴ and Nₚ/V acts reducibly on R/V. 7) M is isomorphic to L₄(5) for all p ≠ 2,5 if the Curtis-Tits relations do not hold.

✓ Correct Answer:

The correct answer is 4) M is isomorphic to L₄(5) precisely when p = 2, because (U±α) and (U±γ) commute, allowing the embedding of L₄(5) in E₈(4)..

📚 Reference Text:

2. From our construc tion of E in Lemma 2.10, we thus see that Eis conjugate to E. Hence P = N0(E) = E.SLJ(S), and there is a unique complement D to E which contains T. Then jj contains precisely six I-invariant subgroups of order 5, and we have jj = ( u±IX' u±/3). That is, we have a new root subgroup u _a:' opposite to U"'. Set M = ( U±rx, U±f.l• U±r ). We shall show that if p = 2 then M = L4(5). To do this, it suffices to show that [ ( U±y)' u_(\'] = 1; for then [ ( u±IX >' ( U±y > l = 1 and an application of the Curtis-Tits relations [12, Theorem 1.4] implies that M is an image of L4(5). To this end, set v = Ua:+{3UO!+f3+r v = u/HyUIX+{3+y and R = vvu{:i = Uo:+f!.Ua:+f3+rUf.l Uf.l+r· We check that R == 54, and also Np(V)<"'l = R ( U±y) and N.P(V)<"'l = R ( U±"' ). We now consider N0(R). Working in C0(e) we see that Cc(V)0 is a maximal torus, say H, so Np(V):;;; N0(H). As Np(V)/V is irreducible on R/V and 53 does not divide IW(E8)I, we have R:;;; H. Hence N0(R) normalizes C0(R)0 =H. Consider the actions of ( U±IX) and ( U±r) on R. We claim that these groups commute modulo H. To see this, note that N0(R)' induces on R an irreducible subgroup of SLi5) of order dividing JW(E 8)j, and containing a 5-element centralized by SLi(5) (corresponding to the image of U"' x ( U±r):;;; P). It follows that N0(R)' /H == SL2(5) 0SLz(5), corresponding to a subgroup Q4(2) x Q4(2) or Qt(4) in Qt(2). The former subgroup comes from W(A4) x W(A4), which does not act faithfully and irreducibly on R. Hence No(R)'/H corresponds to Qt(4). Let t be the involution in T centralizing ( U ±a) and ( U ±r). Then ( tH) = Z(N(H)/H). Hence CN(R)'(t)=Q 1(02(H)).(SL 2(5)0SL2(5)), showing that (U±a:) and ( U ±r) commute modulo Q 1 ( Oz(H)). If p = 2 then 02(H) = 1, so ( U±a:) and ( U±r) commute, which implies that M = L4(5). Moreover, all the groups occurring in the proof exist in £8(4), so this embeds L4(5) in £8(4), as required. It remains to show that if p -=!= 2, 5 then G has no subgroup Li5). To do this, we shall demonstrate that the extension CN(R).(t) = 28.(SL2(5) o SL2(5)) does not split. This implies that ( U±a) and ( U±r) do not commute. Hence if G had a subgroup L4(5), we could have performed all the above calculations within this subgroup to obtain a contradiction. Thus suppose for a contradiction that the extension splits. We have CN(RJ.(t) = 29. (Alt5 x Alt5) with central involution t. This is a subgroup of Cc(t) = D8. Let N=28 be the image of_!he normal 29 in C0(t)/(t) =PQ16(K), where K = \FP. Let N be the preimage of Nin Q16(K),

Question 938 multiple-choice

Isogenies between supersingular elliptic curves and their algebraic representations play a fundamental role in both arithmetic geometry and modern cryptographic systems. The Deuring correspondence establishes a bridge between the theory of elliptic curves and quaternion algebras through the structure of ideals and orders. Which statement best describes the explicit relationship between a separable isogeny φ: E₀ → E₁ of supersingular elliptic curves and the associated ideal I in the endomorphism ring of E₀? 1) The ideal I consists of all endomorphisms that fix the kernel of φ pointwise. 2) The ideal I consists of all endomorphisms of E₀ that annihilate the kernel of φ. 3) The ideal I is generated by the image of φ acting on E₁. 4) The ideal I corresponds to automorphisms of E₁ that extend φ. 5) The ideal I is defined as the intersection of left and right orders of the endomorphism ring of E₁. 6) The ideal I is the set of all morphisms from E₀ to E₁. 7) The ideal I consists of endomorphisms of E₁ that commute with φ.

✓ Correct Answer:

The correct answer is 2) The ideal I consists of all endomorphisms of E₀ that annihilate the kernel of φ..

📚 Reference Text:

can deﬁne the Eichler order associated toIasOL(I)∩OR(I). In that case, it can be shown that OL(I)∩OR(I) =Z+I (see [FKL+20]). 2.3 The Deuring correspondence Fixasupersingularellipticcurve E0,andanorder O0≃End(E0).Thecurve/order correspondence allows one to associate to each outgoing isogeny ϕ:E0→E1 an integral left O0-ideal, and every such ideal arises in this way (see [ Koh96] for instance). Through this correspondence, the ring End( E1) is isomorphic to the right order of this ideal. This isogeny/idealcorrespondence is deﬁn ed in [Wat69], and in the separable case, it is explicitly given as follows. Deﬁnition 2.1. GivenIan integral left O0-ideal coprime to p, we deﬁne the I- torsionE0[I] ={P∈E0(Fp2) :α(P) = 0for allα∈I}. ToI, we associate the separable isogeny ϕIof kernelE0[I]. Conversely given a separable isogeny ϕ, the corresponding ideal is deﬁned as Iϕ={α∈ O0:α(P) = 0for allP∈ker(ϕ)}. WesummarizepropertiesoftheDeuringcorrespondenceinTable1, borrowed from [FKL+20]. Title Suppressed Due to Excessive Length 7 Supersingular j-invariants over Fp2Maximal orders in Bp,∞ j(E) (up to Galois conjugacy) O∼=End(E) (up to isomorphism) (E1,ϕ) withϕ:E→E1 Iϕintegral leftO-ideal and rightO1-ideal θ∈End(E0) Principal ideal Oθ deg(ϕ) n(Iϕ) Table 1. The Deuring correspondence, a summary [ FKL+20]. 2.4 Isogeny representation In this subsection,we lookatisogeniesthrougha morealgorithmicpr ism.Specif- ically, we consider the following question: what does it mean to “compu te” an isogeny? A natural answer is a rational map representation of the isogeny. Other representations are however possible, and in [ PL17, Sec 2.4] and [ Ler22a] it is argued that any such representation should allow eﬃcient evaluatio n at arbi- trary points (for a more complete study, look at [ Ler22b, chapter 4]). More formally, Leroux deﬁnes an isogeny representation as some data sφassociated to an isogeny φ:E→E′of degreeNsuch that there are two algorithms: one to “verify” and one to “evaluate” φ. The motivation to have a veriﬁcation algorithm is found in a cryptogra phic context wherean adversarymight tryto cheat byrevealingsomet hing that is not avalidisogenyrepresentation.But,inthemorecryptanalaticpoint ofviewofthis paper, we can assume that we work with a valid isogeny representat ion. This is why we take a relaxed deﬁnition of isogeny representation where we only require an evaluation algorithm (a veriﬁcation algorithm can probably be deriv ed from the evaluation algorithm anyway). Moreover, we assume that the r epresentation is “eﬃcient” meaning that is has polynomial size and the evaluation algo rithm is polynomial-time in the log of the degree and the prime. We give a detaile d version below. In our context, it is suﬃcient that the evaluation algo rithm gives evaluation ofpoints up to a (common) scalarwhich is why we qualify our isogeny representation as weak. Deﬁnition 2.2. A weak isogeny representation for the isogeny φ:E→E′of degreeN, is a datasφof sizeO(polylog(p+N))(associated to a unique isogeny φ), such that there exists an algorithm Ethat takessφand a point Pof the curve Eof orderdin input and computes λ(d)φ(P)inO(polylog(d+N+|P|))for any pointPofE, where|P|is the bitsize of the representation of P. The notion of isogeny representation is particularly relevant when t he degree Nis a big prime and the kernel points are deﬁned over an Fp-extension of big degree (this is exactly the

Question 939 multiple-choice

In fault-tolerant quantum computing, optimizing the number of T gates in quantum circuits is essential due to their high resource cost. The Approximate Quantum Fourier Transform (AQFT) circuit can be improved by employing efficient gate implementations and cancellation strategies. Which of the following formulas provides a straightforward estimation of the total T-count required for an improved AQFT circuit with n qubits and parameter b, enabling rapid comparison of resource requirements? 1) 4n(b+1) 2) n(b−2)+8 3) 16n(b−1) 4) 8(n−2)b 5) n+3b−4 6) 8b(n−1) 7) 8n(b−1)

✓ Correct Answer:

The correct answer is 7) 8n(b−1).

📚 Reference Text:

gates and three T gates (includinginverses) as shown in Fig. 5. We know from our construction above that each controlled-Z agate in the AQFT is implemented using 8 T gates (of which 4 are used to remove the control, and 4 toimplement the target via the adder). Therefore, instead of relyingon inducing the gradient operation through the adder, we implement controlled-P gates directly, according to Fig. 5. Next, we consider controlled-T gates. As per Fig. 1, we see that each controlled-T gate in the AQFT neighbors a controlled-P gatein the following layer of controlled-Z agates in the target qubit line. Since we implement controlled-P gates according to Fig. 5, we may obtain T-count savings via gate cancellation (TT†=Id)b y rewriting the controlled-T gate as the controlled-Z3∕4gatefollowed by the controlled-Z−1∕2, where the controlled-Z−1∕2gate is implemented according to Fig. 5, inducing T-count reduction by 2 on the ‘target ’of controlled-Z−1∕2and controlled-T gates, and by another 2 for each layer of controlled-Zagates by cancellations on the ‘control ’line, and the controlled-Z3∕4gate is implemented directly as per the top panel of Fig. 2, which costs 5 T gates. Altogether, the above implementation of the controlled-T and controlled-P gate pair requires 7( =5+3+3−2−2) T gates. This is in comparison to 16 T gates that would otherwise have been used by the implementation based on the adder. What remains to be investigated at this point is the modi ﬁcation that needs to be made to the gradient operation so as to induce a partial gradient operation, i.e., kjiψdþ1;bþ1/C12/C12 i7!e2πik=2bþ1kjiψdþ1;bþ1/C12/C12 i,w h e r e k<2b−d, d≤b, and ψdþ1;bþ1/C12/C12 iis the state ψbþ1/C12/C12 iwithout ﬁrstd+1 qubits, to implement the remaining Zagates in a layer. To obtain the partial gradient operation, we analyze how the gradient operation works. Firstly, we formally de ﬁne the state ψdþ1;bþ1/C12/C12 i:¼1ﬃﬃﬃﬃﬃﬃﬃ 2b/C0dpP2b/C0d/C01 j¼0e/C02πij=2bþ1jji. The application of ( b−d)- bit addition (see ref.31)t o kjiψdþ1;bþ1/C12/C12 iresults in two cases: k+j< 2b−dand k+j≥2b−d. In order for the partial gradient operation to work, we need k+j↦k+jmod2b−d. This may be achieved by applying Z1=2dgate to the most signi ﬁcant bit of the modular addition circuit. Since in our case d=2, this amounts to applying a T gate for each gradient operation. This means that the overall result of our optimization detailed in this section is by about 8( n− 2) T gates. Comparisons to prior work Our improved implementation of AQFT nwith n>b> 2 requires the qubit count of nq=n+3b−4, the CNOT-gate count of 7 :5n/C0 13þPn/C01 l¼3(16 min( b−2,l−2)−5)þPminðb;n/C01Þ b0¼3CCNOTðRUS b0Þ= pb0,a n dt h eT - c o u n to f7 n/C011þPn/C01 l¼3ð8m i n ðb/C02;l/C02Þþ1ÞþPminðb;n/C01Þ b0¼3CTðRUS b0Þ=pb0, where CgðRUS b0Þdenotes the count of the fault-tolerant gate gin the RUS circuit synthesizing z/C01=2b0 , and pb0denotes the success probability of the RUS circuit. As follows from our constructions, the T gate count can be fairly accurately approximated by the simple formula 8 n(b−1). This may be compared to the previous state of the art that uses a variant of[ref.30, Fig. 10] to implement

Question 940 multiple-choice

Quantum Amplitude Estimation (QAE) is a quantum algorithm that offers significant speedup over classical methods for estimating expected values and integrals, with practical implementation challenges arising from hardware constraints and gate decompositions. Efficient circuit synthesis is critical for running quantum algorithms on devices with limited qubit connectivity, such as Linear Nearest Neighbor (LNN) architectures. Which of the following best explains why decomposing controlled-Ry gates that share a target qubit is crucial for Quantum Amplitude Estimation circuits on LNN architectures? 1) It enables parallel execution of all gates, regardless of connectivity. 2) It reduces circuit depth and gate count, making the circuit feasible on hardware with restricted qubit connectivity. 3) It allows for the direct use of native three-qubit gates available on all quantum processors. 4) It guarantees error-free operation by eliminating the need for SWAP gates. 5) It automatically transforms the circuit into a classical reversible computation. 6) It permits the use of only Hadamard and CNOT gates without additional decomposition. 7) It ensures that only single-qubit gates are required for amplitude encoding.

✓ Correct Answer:

The correct answer is 2) It reduces circuit depth and gate count, making the circuit feasible on hardware with restricted qubit connectivity..

📚 Reference Text:

quantum hardware, specifically the IBM_Nairobi10 and Rigetti Aspen-1111 systems. We also illustrate the decom- position of controlled- Ry gates that share a target qubit using our proposed method. This particular circuit is often found in QAE, which is anticipated to supplant classical Monte Carlo integration methods24,25. By provid- ing these results, we aim to highlight the practicality and effectiveness of our approach in real-world quantum computing applications. The remainder of this paper is organized as follows: in the “ Background ” section, we provide a brief overview of quantum circuits, QFT, QPE, and QAE. The proposed approach section outlines our method for construct - ing LNN QFT circuits. In the results and discussion section, we present the outcomes of transpilation on IBM quantum computers, display the experimental results of QPE executions on quantum hardware, and illustrate how to convert a circuit of controlled- Ry gates sharing the target qubit into an LNN circuit using our proposed method. We also address the limitations of our study and suggest potential future research directions. Finally, we conclude the paper with a summary of our findings and their implications for the field of quantum computing. Background Quantum circuit. Quantum circuits consist of qubits and gates. Qubits store a quantum state, a vector in a Hilbert space, and each gate represents a unitary transformation on the Hilbert space. The matrix representa- tions of the gates used in this paper are as follows: Quantum fourier transform. QFT is a quantum version of the discrete Fourier transform. The definition of n-qubit QFT and its inverse are as follows:(1)H=1√ 2� 11 1−1� ,CNOT = 1000 0100 0001 0010 , Rn=� 10 0eiπ 2n−1� ,Rz(θ)=� e−iθ 20 0eiθ 2� ,Ry(θ)=� cos(θ/2)sin(θ/2) sin(θ/2)cos(θ/2)� , SWAP= 1000 0010 0100 0001 ,C(Rn)= 100 0 010 0 001 0 000 eiπ 2n−1 . 3 Vol.:(0123456789) Scientific Reports | (2023) 13:8638 | https://doi.org/10.1038/s41598-023-35625-3 www.nature.com/scientificreports/The conventional n-qubit QFT circuit requires n(n−1)/2 C(Rn) gates and n Hadamard ( H ) gates if qubit reordering is allowed8 (see Fig. 1). Synthesizing a C(Rn) gate demands two CNOT and three Rz gates26. There- fore, n(n−1) CNOT gates are required to construct an n-qubit QFT circuit. However, if the LNN architecture is required to implement the QFT, the number of CNOT gates is much larger than n(n−1)14–18. One of the most important uses of QFT is the QPE algorithm8,13. QPE is an algorithm for finding an eigen- vector of a unitary operator using QFT. Let a unitary operator, an eigenvalue of the unitary operator, and a cor - responding eigenstate be U , e2πiθ , and |u� , respectively. Then, QPE can find θ if the state |u� is prepared and the controlled- U operators are implemented. The canonical QPE is executed according to the following process: first, prepare the state /vextendsingle/vextendsingle0�⊗t|u� , where t is a positive integer related to the precision of QPE. Second, apply t Hadamard ( H ) gates to /vextendsingle/vextendsingle0�⊗t . Third, apply the controlled- Uj operator to the total state, where the controlled-

Question 941 multiple-choice

Quantum photonics leverages programmable devices such as Spatial Light Modulators (SLMs) and metalens arrays to physically implement quantum algorithms using single photons. Accurate signal detection and state manipulation are crucial for extracting computational results from interference patterns formed by photon superposition. Which experimental technique directly improves the reliability of single-photon measurements in quantum photonic setups by signaling the likely presence of a photon correlated with another detected event? 1) Amplitude modulation via iris adjustment 2) Heralding 3) Classical light calibration 4) Unitary matrix transformation using metalens arrays 5) Polarization conversion with Quarter Wave Plates 6) Gated detection with a fixed coincidence window 7) Pattern decoding using an interference library

✓ Correct Answer:

The correct answer is 2) Heralding.

📚 Reference Text:

Module (SPCM) (Excelitas-SPCM-800-14-FC), triggeringtheSinglePhotonAvalancheDiode(SPAD)camera(PiImag-ingSPAD512S)locatedattheendofthesignalarm.TheSPADcamera operates in gated mode with a 16-ns coincidencewindow. The heralding process only triggers imaging forthose photons with vertical (V) polarization that arrive attheSpatialLightModulator(SLM)(HoloeyePluto2.1)inthesignal arm. The SLM generates the input state and selec-tivelyexcitesthemetalens.Asmentionedearlier,tocontrol the amplitude from the phase-SLM, some light is diverted and blocked by an iris positioned at the focal plane of alens with f 1=40cm. A pair of orthogonal Quarter Wave Plates (QWP) is placed before and after the metasurfacetoconvert thelinearlypolarizedincident beamtocircularpolarization(CP)andviceversa.TheﬁrstQWP,withafastaxis oriented at the diagonal, converts V-polarized light toLCP. Upon interacting with the metasurface, the photonsdiffract into different output directions. The second QWP,with a fast axis oriented at the anti-diagonal, and an H-polarizerareusedtoconvertthepolarizationbacktolinearpolarizationthatisorthogonaltotheinputlightinthesignalarm.Consequently,interferencepatternscorrespondingtodifferent output directions are formed at the focal planeof the metalens array and captured by the SPAD camera. Extendedexposuretime(atotalof20,000frameswitheachframelasting0.3s)isutilizedfordataacquisitionduetotheextremelylowphotoncountfromthequantumsource,andthe use of the heralding technique is essential to increasethesignal-to-noiseratiofordetectioninthecaseofquantumlightsource. 2.3 Experimental characterization We ﬁrst examine the operations of the metasurface usingclassical light. In this case, we replace the part before theSLM inFigure 2(d) with an 810nm laser (OBIS LX 808nm 150mW Laser). The utilization of classical light for exci-tation effectively enhances the signal, resulting in shorteracquisitiontimes(5–10ms).Italsoleadstohigherresolutionin the interference patterns formed at the focal plane ofthe metalens array because the higher power can be dis-tributed among more pixels on the single-photon camera.Theseinvestigationsfacilitatethemorechallengingtaskofextractinginformationfromtheinterferencepatternsinthesubsequentstagewithaquantumlightsource. Figure 3 displays the simulation and experimental resultsobtainedusingaclassicallightsourcefortwoopera-tions:GS(left4columns)andQFT(right4columns).Foreach operation,wetestitusingfourdifferentinputstates,which arelabeledatthetopofeachcorrespondingresultcolumn.Letusbeginwiththeﬁrstcolumn,whichcorrespondstothe Figure 3:SimulationandexperimentalresultsusingaclassicallightsourcefortheGrover-Search(left)andQFT(right)operations.(a)SLMphase profileforeachinputstateforthetwooperations:fourinputstatesforeachoperationlistedatthetopofeachcolumn.Colorcode:redfor1,cyanfor −1,greenfor i,andpurplefor −i,shadedregions(phasegradient)areeffectivelyinactive.(b)Numericalsimulationofthefar-fieldinterference pattern.(c)Experimentalresultusingaclassicallightsource.Theinputandoutputstatesarelabeledatthetopandbottomofeachcolumn.Thelibraryfordecodingtheinterferencepatternintotheoutputstatefortherepresentativecasesofvalues1and0islistedinthebottominset, whereconstructiveinterference(1)exhibitsabrightcenternodeanddestructiveinterference(0)exhibitsadarkcenternode. 932—R.S.Tanuwijayaetal.:Metasurfaceforprogrammablequantumalgorithms input state (−1,1,1,1)for the GS algorithm. In the current case, the GS algorithm essentially picks the basis with 𝜋 phaseshift(auniquemarkedelementinthedatabase)andoutput the state with zero components except the corre-sponding basis with value 1: (1, 0, 0, 0), with ideally unit probability on the corresponding basis using the metalens arraytoimplementtheunitarymatrix U GS(asshowninthe matrixmultiplicationin Figure1(d) ).Theinputstateisini- tiallyencodedasthephaseproﬁleontheSLM( Figure3(a) ) using the four upper-left lenses in a row-major manner: a region marked as −1 has a constant SLM phase of π(cyan color),whilearegionmarkedas1hasaconstantSLMphase of 0 (red color). Only the corresponding metalenses are selectivelyexcitedtoimplementtheGSalgorithm,whiletheother regions display a phase gradient to deﬂect the light awayfromthemetalensarray.Inthefarﬁeld(practicallyat focal plane of the metalens), the interference patterns areformedattheoutputdirections,asshowninthenumericalsimulation(byusingangularspectrummethodtopropagate the complex ﬁeld to the focal plane of the metalens) in Figure 3(b) and the experimental results captured by the single-photon camera in Figure 3(c) . The interference pat- terns (intensity) for the two results closely resemble each otherandrepresenttheoutputstateintheupperleftregion,alsoindexedinarow-majormanner.Theintensityatdirec- tionk i, proportional to|||𝜓(out) i|||2 , can then be determined from the center of each interference pattern. It is worth notingthatonlytheupperleft4directionsexhibitnotable interferencepatterns(asseen Figure1(c) ).Toachievehigher accuracy,weextracttheintensityusingtheentireinterfer- ence pattern. Some representative cases from the library areshownatthebottomof Figure3,illustratingthediffer- ent possible patterns for value 0 (fully destructive, result-ing in dark centers) and 1 (fully constructive, resulting in a bright center). By comparing the interference patterns with the library, we ﬁnd that the output state agrees withthe expected result (1 ,0,0,0)for the GS algorithm. In the meantime, for the sake of convenience of our discussion, wediscretizetheoutputstatetobeeitherfullydestructiveor fully constructive interference. More details are giveninSupplementaryMaterials regardingthedecodingprocess forthefullspectrumofintensityvaluesotherthan0and1, whichwillbeusedinthenextstep. Forcompleteness,theresultsfortheotherthreediffer- entinputstateswith 𝜋phasechangeatthecorresponding inputbasisareshowninthesecondtofourthcolumns.Thesimulationandexperimentresultsexhibitgoodagreement, whilethedecodedoutputstatesareshownbelowtheexperi- mentalresults.TotestthequantumFouriertransform(QFT)using the same technique, we select the input states to be(1,1,1,1),(1 ,−i,−1,i),(1,−1,1,−1)and(1,i,−1,−i)inorder to detect the different “frequencies” in the state. The

Question 942 multiple-choice

Quantum process tomography is essential for evaluating the accuracy and reliability of quantum operations, especially in experimental quantum computing platforms like NMR quantum processors. Effective error analysis is critical for advancing quantum gate fidelity and the implementation of algorithms such as the quantum Fourier transform. Which type of error predominantly causes a lack of complete positivity in the superoperator obtained from quantum process tomography on a three-qubit NMR quantum processor? 1) Coherent errors due to systematic calibration issues 2) Decoherent errors arising from random noise 3) Incoherent errors resulting from deterministic but system-wide variations 4) Gate errors from faulty quantum measurements 5) Crosstalk errors between adjacent qubits 6) Leakage errors to non-computational states 7) State-preparation errors prior to gate application

✓ Correct Answer:

The correct answer is 3) Incoherent errors resulting from deterministic but system-wide variations.

📚 Reference Text:

Title: Quantum process tomography of the quantum Fourier transform. Year: 2004 Paper ID: 876bdca9c1e7f2e12e03ff69f8542a80ae1a981e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/876bdca9c1e7f2e12e03ff69f8542a80ae1a981e Abstract: The results of quantum process tomography on a three-qubit nuclear magnetic resonance quantum information processor are presented and shown to be consistent with a detailed model of the system-plus-apparatus used for the experiments. The quantum operation studied was the quantum Fourier transform, which is important in several quantum algorithms and poses a rigorous test for the precision of our recently developed strongly modulating control fields. The results were analyzed in an attempt to decompose the implementation errors into coherent (overall systematic), incoherent (microscopically deterministic), and decoherent (microscopically random) components. This analysis yielded a superoperator consisting of a unitary part that was strongly correlated with the theoretically expected unitary superoperator of the quantum Fourier transform, an overall attenuation consistent with decoherence, and a residual portion that was not completely positive-although complete positivity is required for any quantum operation. By comparison with the results of computer simulations, the lack of complete positivity was shown to be largely a consequence of the incoherent errors which occurred over the full quantum process tomography procedure. These simulations further showed that coherent, incoherent, and decoherent errors can often be identified by their distinctive effects on the spectrum of the overall superoperator. The gate fidelity of the experimentally determined superoperator was 0.64, while the correlation coefficient between experimentally determined superoperator and the simulated superoperator was 0.79; most of the discrepancies with the simulations could be explained by the cumulative effect of small errors in the single qubit gates.

Question 943 multiple-choice

In modern cryptography, zero-knowledge proofs are used to allow a prover to demonstrate knowledge of a secret related to group-based commitments without revealing the secret itself. Soundness and zero-knowledge properties are critical for ensuring both security and privacy in these protocols. Which scenario poses a particular vulnerability for soundness in commitment schemes using plural bases, requiring special analysis to ensure security? 1) When challenge differences are large and have only large prime factors 2) When challenge differences are small or divisible by small prime factors 3) When group order is easily computable and known to all parties 4) When commitments use a single group element as the base 5) When the group operation is commutative but not associative 6) When the protocol operates in cyclic groups of prime order 7) When the prover uses random challenges unrelated to the verifier's input

✓ Correct Answer:

The correct answer is 2) When challenge differences are small or divisible by small prime factors.

📚 Reference Text:

e tG, choose h∈GwithC(k)-rough order, set g=hαand output descr(G),g,h. Then deﬁne the relation Rmult={((c1,c2,c3),(x1,r1,b1,x2,r2,b2,x3,r3,b3))|ci,bi∈G, ci=µigxihri,µl i=1,i=1,2,3}.This leads to: Theorem 3. If the root assumption is (t/prime(k),zepsilon<(k))-secure, the above protocol is a computationally convincing proof of knowledge for Rwith knowledge error κ(k), time bound t(k)and failure probability ν(k), whereν(k)=9zepsilon<(k)l(k),t(k)< t/prime(k)/448andκ(k) = max(4/C(k),448t(k)/t/prime(k)).I f−T≤x1,x2,x3≤T(as they will be when the prover is honest), the protocol is honest veriﬁer statistical zero-knowledge. For the space limitation, we omit the proof, which can be easily derived from the proof of Theorem 2. 6 WhatIstheMajorDiﬀerencefromtheEarlierProof in[14]? For completeness, we brieﬂy indicate here what is mainly diﬀerent from theearlier work [14] in terms of the proof of soundness. As mentioned above, themain gap we ﬁll here does not appear in the proofs in related works [8,3,10,4].This is because the gap only appears in the proofs for protocols associated withcommitment using plural bases (i.e.,c=g shrsuch as in [14]). Theseprotocolssuggestedin[14]areverysimilartotheoneswesuggesthere, inparticulartheyhavethesame3-moveform,withachallenge efromtheveriﬁer AStatistically-HidingIntegerCommitmentScheme 139 asthesecondmessage.So[14]usesarewindingargumentaswedohere,toobtain correct answers from the prover to challenges e,e/prime. However, a problem occurs in thelastpartoftheproof,whichcorrespondstothelastcaseinouranalysis,thatis, Case 2: “( e−e /prime) dividesα(u−u/prime)+(v−v/prime)”. Translated into our notation, we have now (e−e/prime),u−u/prime, andv−v/primesuch thathα(u−u/prime)+(v−v/prime)=ce−e/prime.I fe−e/prime divides both of ( u−u/prime) and (v−v/prime), we are essentially done since we then have c=µg(u−u/prime)/(e−e/prime)h(v−v/prime)/(e−e/prime)whereµe−e/prime= 1. In the earlier work [14], it is claimed that the adversary can make Case 2 occur only with negligible probability unlesse−e/primedivides both of (u−u/prime)and(v−v/prime), because he doesn’t have enough information about α, whereg=hα. However, if e−e/primeis a small number, then this case may in fact happen with signiﬁcant probability, even without e−e/prime dividing both u−u/primeandv−v/prime. This problem was not taken into account in [14]. Later in the full paper version of [14], it was shown that when the knowledgeextractorrewinds P ∗andmakeshimoutput e,e/prime,itonlyhappenswithnegligible probability that e−e/primeissmall, see [15]. However, even if e−e/primeis large, there is still a problem: if e−e/primehas a small prime factor p, there may be a signiﬁcant probability that pdividesα(u−u/prime)+(v−v/prime) whereas it does not divide both u−u/primeandv−v/prime. The additional idea we provide here essentially ﬁlls this gap, and indeed seems necessary for this type of proof to go through. As for [8,3,10,4], what corresponds to Case 2 is the event “( e−e/prime) divides (u−u/prime)”; there is no gap to go to c=µg(u−u/prime)/(e−e/prime), whereµe−e/prime= 1. Hence, the related works above do not suﬀer from the problem we need to consider. 7 ApplyingtheSchemeinClassGroupsandBeyond Wedonotgiveanydetailedintroductiontoclassgroupshere–ormoreprecisely,class groups of quadratic number ﬁelds. It is enough to know, that each suchgroup is deﬁned by a single number, the discriminant ∆. Given this number, one can choose elements in the group and compute the group and inversionoperations. Finding the order of the class group (the class number) from ∆ appears to be a hard problem, and is at least as hard as factoring ∆(if∆is composite). Therefore, root extraction also appears to be a hard problem, andit seems reasonable to conjecture that if ∆is chosen randomly from a large set of values, then the class number will contain large and random prime factors,and will not have a very large factor consisting of only

Question 944 multiple-choice

In Lie algebra theory and mathematical physics, the construction of the Dirac bracket is crucial for Hamiltonian reduction and maintaining a consistent Poisson structure in constrained systems. Techniques from representation theory, such as sl2 embeddings and graded decomposition, play a fundamental role in this construction. Which mathematical property ensures the unique solvability of recursive equations for components of an element R in the graded decomposition of a Lie algebra, particularly when constructing the Dirac bracket in the presence of sl2 embeddings? 1) The commutativity of the Lie algebra 2) The existence of a central element in the algebra 3) The invariance of the algebra under conjugation 4) The bijectivity of the adjoint action on graded subspaces 5) The nilpotency of the raising operator 6) The degeneracy of the Cartan-Killing form 7) The compactness of the algebra

✓ Correct Answer:

The correct answer is 4) The bijectivity of the adjoint action on graded subspaces.

📚 Reference Text:

(3.38) are satisﬁed, then k/summationdisplay ij=1zi{Ji,Jj}∗(y)tj=k/summationdisplay i=1{QR(/vector z;y),Ji}∗(y)ti =k/summationdisplay i=1{QR(/vector z;y),Ji}(y)ti =dim(g)/summationdisplay i=1{QR(/vector z;y),Ji}(y)ti = [R(/vector z;y),y] (3.44) 20 where eqn.(3.38) was used in the ﬁrst step. This proves the th eorem . Note that conversely in order to show the existence of the Dir ac bracket on gfit is suﬃcient to prove that equations (3.37) and (3.38) of the abo ve theorem are solvable within R. We will now show that this is the case when gf≡gfixassociated to an arbitrarysl2embedding. Theorem 3 Let|i|be the total number of sl2multiplets in the branching of the adjoint representation of g. The equations t++ [R(/vector z;y),y]∈gfixfor all/vector z∈C|i|andy∈gfix QR(/vector z;y)|gfix=/summationdisplay j;µz(µ) jJj,−j (µ)+ (R(/vector z;y),t+)for allz∈C|i|andy∈gfix have a unique solution R∈ Rfor arbitrary embeddings. Proof: First solve the second equation. QR(/vector z;y)(y) =/parenleftig R(p)+...+R(−p),t++y(0)+...+y(−p)/parenrightig = (R(−1),t+) + (R(p),y(−p)) +...+ (R(0),y(0)) Now,y(−j)∈g(−j) lwwhich means that y(−j)=/summationtext µyj,−j (µ)t(µ) j,−j. AlsoR(j)=R(j) 0+R(j) hwwhere R(j) 0∈Im(adt+) andR(j) hw∈Ker(adt+). Let {tj,−j (µ)}µ⊂g(j)be such that ( t(µ) j,−j,tj′,−j′ (µ′)) = δj′ jδµ µ′(remember that g(j)andg(−j)are non-degenerately paired). Then take R(j) hw=/summationdisplay µz(µ) jtj,−j (µ) (3.45) Obviously the second equation of the theorem is then satisﬁe d. Let’s now solve the ﬁrst equation. For k>0 it reads [R,y](k)= 0. Writing this out gives [R(k−1) 0,t+] = [y(0),R(k)] +...+ [y(k−p),R(p)] (3.46) Remember that adt+:g(k−1) 0→g(k)is bijective which means that if we already know R(k),...,R(p)then we can uniquely solve this equation for R(k−1) 0(remember that for l≥0 we have already determined the quantities R(l) hw). The initial term of this sequence of equations is obviously R(−p)=R(−p) hw(i.e.R(−p) 0= 0). In this way we can completely determine the positive and zero grade piece of R(/vector z;y). For−k≥0 the situation is similar but slightly diﬀerent. What we now have to demand is that [ R,y](−k) 0= 0. This leads to the equation [t+,R(−k−1)] = [R(−k),y(0)]0+...+ [R(−k+p),y(−p)]0 (3.47) 21 where we used that [ t+,R(−k−1)] is already an element of g(−k) 0. Remember that adt+: g(−k−1)→g(−k) 0is bijective so there exists a unique R(−k−1)(z) such that the equation is solved, assuming we know R(−k),...,R(−k+p)already. In this way we can grade by grade determine the negative grade piece of the matrix R. It is also clear from the above construction that R∈ R. This proves the theorem . It is possible to derive a general formule for R(/vector z;y) using arguments similar to those used in [8]. Let again glw=/circleplusdisplay j∈1 2Ng(−j) lw= span( {t(µ) j,−j}j;µ (3.48) and let Π be the orthogonal (w.r.t. the Cartan-Killing form) projection onto Im( adt+). Obviously the map adt+: Im(adt−)→Im(adt+) is invertible. Denote the inverse of this map, extended by 0 to gbyL. As before what we want to do is solve the equation [R,y] =x∈glw (3.49) fory∈gfix. Noting that y=t++wwherew∈glwand applying Π this equation reduces to Π◦adt+R=ǫΠ([R,w]) (3.50) where we introduced a parameter ǫin the right hand side which we want to put to 1 later. Note that the left hand side is equal to adt+Rsince this is already an element of Im(adt+). Assume now that Rcan be

Question 945 multiple-choice

In the study of quantum algorithms for group-theoretic problems, representation theory plays a key role in enabling efficient computation over nonabelian groups such as affine and q-hedral groups. Understanding the structure and decomposition of these group representations is crucial for solving problems like the Hidden Subgroup Problem. Which of the following properties makes the (p−1)-dimensional representation of the affine group Ap particularly useful for quantum algorithms aimed at solving the Hidden Subgroup Problem for embedded q-hedral subgroups? 1) Its restriction to a q-hedral subgroup decomposes into q-dimensional representations that align with subgroup structure, allowing efficient quantum Fourier sampling. 2) It only contains one-dimensional irreducible components, making all computations abelian. 3) It is always reducible over any subgroup, providing trivial solutions to HSP. 4) It is induced from a trivial representation of the cyclic subgroup, which simplifies all quantum transformations. 5) Its character values remain constant across all elements, preventing any distinction between subgroups. 6) It lacks block-diagonal structure, making adapted bases inefficient in quantum computation. 7) It does not interact with normal subgroups, preventing any reconstructibility of hidden conjugates.

✓ Correct Answer:

The correct answer is 1) Its restriction to a q-hedral subgroup decomposes into q-dimensional representations that align with subgroup structure, allowing efficient quantum Fourier sampling..

📚 Reference Text:

HandHb, respectively, consisting of lines whose slope is a power of a. Construction of the representations of Aprequires that we ﬁx a generator γof Z∗ p. Deﬁne log : Z∗ p→Zp−1to be the isomorphism log γt=t. Let ωpdenote the pth root of unity e2πi/p. Then Aphasp−1 one-dimensional representations σs, namely the representations of Z∗ p∼=Zp−1given by σt((a, b)) = ωtloga p−1. In addition, Aphas one ( p−1)-dimensional representation ρgiven by (2) ρ((a, b))j,k=/braceleftbiggωbj p,k=ajmod p, 0 otherwise ,1≤j, k < p, Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE POWER OF STRONG FOURIER SAMPLING 945 where the indices iandjare elements of Z∗ p. See [24, section 8.2] for a more detailed discussion. Similarly, given a prime pand a divisor qofp−1, we consider the q-hedral groups , namely semidirect products Zq/multicloseleftZ p. These embed in Apin a natural way, namely, as the normal subgroups Nqdeﬁned above. The dihedral groups are the special case where q=2 . The representations of Zq/multicloseleftZ pinclude the qone-dimensional representations of Zqgiven by σ/lscript((at,b)) = ω/lscriptt qfor/lscript∈Zqand the ( p−1)/qdistinct q-dimensional representations ρkgiven by ρk((au,b))s,t=/braceleftbigg ωkasb p,t=s+umod q, 0 otherwise for each 0 ≤s ,tN q>{1}, then ρbecomes block-diagonal, with ( p−1)/qblocks of size q, and these blocks are exactly the representations ρkofNq. (See [20] for an introduction to adapted bases and their uses in quantum computation.) We will use this fact in sections 4 and 5below. The aﬃne and q-hedral groups are metacyclic groups, i.e., extensions of a cyclic group Z pby a cyclic group Zq. In [14], Høyer shows how to perform the nonabelian Fourier transform over such groups (up to an overall phase factor) with a polynomial,i.e., polylog( p), number of elementary quantum operations. 3. Full reconstructibility. In this section we show that conjugates of suﬃ- ciently large subgroups of the aﬃne groups are fully reconstructible in polynomialtime. For some values of pandq, this allows us to completely solve the HSP for the q-hedral group Z q/multicloseleftZ p. Theorem 1. Letpbe prime and let qbe a divisor of p−1for which (p−1)/q= polylog( p). Then the hidden conjugates Hb qofHqinApare fully reconstructible. Proof. First, consider the maximal nonnormal subgroup H=/angbracketleft(γ,0)/angbracketright, where γis a generator of Z∗ p. Carrying out Steps 1 through 3 of the Fourier sampling procedure outlined in the introduction results in a state ψ3over the group Gwhich is uniformly supported on a random left coset of the conjugate Hb. Using the procedure of [14], we now compute the quantum Fourier transform of this state over Apin the basis

Question 946 multiple-choice

Code-based cryptography is a promising approach for post-quantum security, leveraging the difficulty of decoding linear error-correcting codes. Quasi-cyclic codes and their applications in cryptosystems are particularly relevant for resisting quantum attacks. Which property of non-binary quasi-cyclic codes is crucial for ensuring that Niederreiter cryptosystems resist quantum attacks based on the hidden subgroup problem using quantum Fourier sampling? 1) Low minimum distance relative to code length 2) Use of binary alphabets in all code constructions 3) Reliance on number-theoretic hardness assumptions 4) Vulnerability to Shor's algorithm 5) Absence of cyclic structural properties 6) Satisfaction of specific criteria that prevent efficient quantum Fourier sampling from solving HSP 7) Requirement that codes are always linear over the field GF(2)

✓ Correct Answer:

The correct answer is 6) Satisfaction of specific criteria that prevent efficient quantum Fourier sampling from solving HSP.

📚 Reference Text:

Title: Niederreiter cryptosystems using quasi-cyclic codes that resist quantum Fourier sampling Year: 2019 Paper ID: 2193618928c53afce81a5f2f42b52f5a09667400 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/2193618928c53afce81a5f2f42b52f5a09667400 Abstract: McEliece and Niederreiter cryptosystems are robust and versatile cryptosystems. These cryptosystems work with many linear error-correcting codes. They are popular these days because they can be quantum-secure. In this paper, we study the Niederreiter cryptosystem using non-binary quasi-cyclic codes. We prove, if these quasi-cyclic codes satisfy certain conditions, the corresponding Niederreiter cryptosystem is resistant to the hidden subgroup problem using weak quantum Fourier sampling. Though our work uses the weak Fourier sampling, we argue that its conclusions should remain valid for the strong Fourier sampling as well.

Question 947 multiple-choice

Quantum field theory seeks to model particles and fields in a way that is consistent with both quantum mechanics and the symmetries of spacetime, notably those encoded by the Poincaré group. Mathematical constructions such as Fock spaces, Weyl operators, and induced representations play crucial roles in ensuring relativistic covariance and proper particle statistics. Which mathematical procedure is essential for constructing covariant quantum fields that respect mass and spin by utilizing representations induced from the stabilizer subgroups of the Lorentz group? 1) Induced representations from little groups to form Hilbert spaces and associated Fock spaces 2) Diagonalization of the Hamiltonian in coordinate space 3) Applying the Heisenberg uncertainty principle to momentum operators 4) Using gauge fixing to eliminate redundant degrees of freedom 5) Introducing path integrals over spacetime trajectories 6) Employing canonical quantization directly on classical field equations 7) Separating variables in the Schrödinger equation for free particles

✓ Correct Answer:

The correct answer is 1) Induced representations from little groups to form Hilbert spaces and associated Fock spaces.

📚 Reference Text:

Title: COVARIANT QUANTUM FIELDS VIA LORENTZ GROUP REPRESENTATION OF WEYL OPERATORS Year: 2019 Paper ID: c99b4cf7c9cfc7990baca8be45865bf87efb9e04 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c99b4cf7c9cfc7990baca8be45865bf87efb9e04 Abstract: The building blocks of Hudson-Parthasarathy quantum stochastic calculus start with Weyl operators on a symmetric Fock space. To realize a relativistically covariant version of the calculus we construct representations of Poincare group in terms of Weyl operators on suitably constructed, Bosonic or Fermionic based on the mass and spin of the fundamental particle, Fock spaces. We proceed by describing the orbits of homogeneous Lorentz group on R4 and build fiber bundle representations of Poincar\'e group induced from the stabilizer subgroups (little groups) and build the Boson Fock space of the Hilbert space formed from the sections of the bundle. Our Weyl operators are constructed on symmetric Fock space of this space and the corresponding annihilation, creation, and conservation operators are synthesized in the usual fashion in relativistic theories for space-like, time-like, and light-like fields. We achieve this by constructing transitive systems of imprimitivity (second-quantized SI), which are dynamical systems with trajectories dense in the configuration space, by induced representations. We provide the details of the field operators for the case of massive Bosons as the rest are similar in construction and indicate the ways to construct adapted processes paving way for building covariant quantum stochastic calculus.

Question 948 multiple-choice

Quantum computing leverages principles such as superposition and entanglement to tackle mathematical problems more efficiently than classical computers. The hidden subgroup problem (HSP) and its extensions are fundamental to the development of quantum algorithms for group-theoretic problems. Which of the following statements is true regarding the continuous hidden symmetry subgroup problem on $\mathbb{R}^n$ as addressed by quantum algorithms? 1) It cannot be reduced to the continuous hidden subgroup problem under any conditions. 2) An efficient quantum algorithm exists for the continuous hidden symmetry subgroup problem on $\mathbb{R}^n$. 3) Only classical algorithms are capable of solving the continuous hidden symmetry subgroup problem on $\mathbb{R}^n$. 4) Quantum algorithms require the group to be finite in order to address the continuous hidden symmetry subgroup problem. 5) The continuous hidden symmetry subgroup problem on $\mathbb{R}^n$ has been proven unsolvable by any quantum algorithm. 6) The continuous hidden symmetry subgroup problem on $\mathbb{R}^n$ is equivalent to integer factorization. 7) The continuous hidden symmetry subgroup problem on $\mathbb{R}^n$ can only be solved efficiently for abelian groups.

✓ Correct Answer:

The correct answer is 2) An efficient quantum algorithm exists for the continuous hidden symmetry subgroup problem on $\mathbb{R}^n$..

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Title: Quantum Algorithm for Solving the Continuous Hidden Symmetry Subgroup Problem Year: 2021 Paper ID: 9975f3ce31fb8c7e09e22a82f964e747d34b915e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/9975f3ce31fb8c7e09e22a82f964e747d34b915e Abstract: Quantum computers are expected to be able to outperform classical computers. In fact, some computational problems such as integer factorization can be solved on quantum computers substantially faster than classical computers. Interestingly, these problems can be cast in a framework of the hidden symmetry subgroup problem. However, only a few of quantum algorithms for efficiently solving this problem have been known, and the approaches used in all previous results can be applied to particular groups with specific group actions. In this paper, we introduce new technique for solving the hidden symmetry subgroup problem which can be applicable for any groups and any group actions with a certain condition. In addition, we define the continuous hidden symmetry subgroup problem on a group by employing a continuous oracle function, and prove that if the group is a metric space and the group action satisfies some condition, then the continuous hidden symmetry subgroup problem can be efficiently reduced to the continuous hidden subgroup problem. In particular, we show that there exists an efficient quantum algorithm to solve the continuous hidden symmetry subgroup problem on $\mathbb {R}^{n}$ , while it has not yet been shown that the original hidden symmetry subgroup problem on $\mathbb {R}^{n}$ can be efficiently solved by a quantum computer.

Question 949 multiple-choice

The Drinfeld double of a finite-dimensional Hopf algebra is a fundamental construction in quantum algebra, combining a Hopf algebra with its dual in a specific way. Understanding the structure and properties of this double is essential for applications in quantum groups and topological field theories. Which of the following statements correctly describes the relationship between group-like elements and one-dimensional representations in the Drinfeld double D of a finite-dimensional Hopf algebra A? 1) The group-like elements and one-dimensional representations of D can be constructed naturally from those of A and its dual Ao via a group isomorphism. 2) Only the group-like elements of Ao contribute to the structure of D; those of A are not involved. 3) In D, group-like elements exist independently and are unrelated to representations of A or Ao. 4) One-dimensional representations of D cannot be derived from those of A or Ao. 5) The group-like elements of D correspond solely to those of A, with no contribution from Ao. 6) There is no algebraic relationship between group-like elements and representations in D. 7) Each group-like element of D must be a trivial element originating from the base field.

✓ Correct Answer:

The correct answer is 1) The group-like elements and one-dimensional representations of D can be constructed naturally from those of A and its dual Ao via a group isomorphism..

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roots of unity. 8 Acknowledgements I thank D. Kazhdan for many, very useful discussions and S. Majid for discussion and help with the references. I thank H. Wenzl and the Department of Mathematics of UCSD, where part of this work was done, for kind hospitality. This work was supported in part by NSF grant DMS-9305715. 9 2. Mapping Class Group Action on Doubles 1.) Double Algebras and Balancing : Inthissection we recall somebasicfactsandnotions on Hopf algebras that we will use later. For a ﬁnite dimensional Hopf a lgebraAover a ﬁeldkwe denote byAothe dual Hopf algebra with opposite comultiplication. We shall always assume that Ahas a counit ǫand an invertible antipode S. The antipode of Ao is thus given by S−1∗. Forλ,µ∈Aoanda,b∈Awe have the following relations: < λµ,a > =< λ⊗µ,∆(a)> < ∆(λ),ab >=< λ,b⊗a > (2.11) ∆⊗id(R) =R13R23id⊗∆(R) =R13R12(2.12) < S∓1(l),a >=< l,S±1(a)> S⊗id(R) =id⊗S−1(R) =R−1(2.13) Here<,>:Ao⊗A→kistheusualcontractionand Risthecanonicalelement R∈A⊗Ao. Thus, if{ei}is a basis ofAand{fi}the respective dual basis of Aowe can write R=/summationtext iei⊗fi. A bicrossed product of the two algebras is a Hopf algebra Dwhich containsAandAoas sub-Hopf algebras such that .:A⊗Ao/tildewidest−→Dis an isomorphism, where.is the multiplication in D. Clearly, a bicrossed structure is uniquely determined by an isomorphism ⊲⊳:A⊗Ao/tildewidest−→Ao⊗Awhich by.⊲⊳=.deﬁnes an associative product, such that the coproduct on Ddeﬁned by the coproducts on AandAoextends to a homomorphism into D⊗2. In[Dr0]itis shown thatthere isprecisely onebicrossed product D(A), the double, such that R∆(y) = ∆′(y)Rfor ally∈D(A). (2.14) Here ∆′=τ◦∆ is the opposite comultiplication and τis the ﬂip τ(a⊗b) =b⊗a. The bicrossed structure is given explicitly by ⊲⊳:A⊗Aoτ− →Ao⊗A∆⊗∆′−−−→Ao⊗Ao⊗A⊗A1⊗S⊗1⊗1−−−−−→Ao⊗Ao⊗A⊗A 1⊗<,>⊗1−−−−−→Ao⊗A∆′⊗∆−−−→Ao⊗Ao⊗A⊗A1⊗<,>⊗1−−−−−→Ao⊗A (2.15) If we use the usual abbreviation ∆( a) =/summationtext ia′ i⊗a′ i=a′⊗a′′, ∆2(a) =a′⊗a′′⊗a′′′(2.15) can be summarized in the formula ⊲⊳(y⊗λ) =λ′′⊗y′′/an}bracketle{tλ′,y′/an}bracketri}ht/an}bracketle{tS(λ′′′),y′′′/an}bracketri}ht The inverse is given similarly by ⊲⊳−1:Ao⊗A∆′⊗∆−−−→Ao⊗Ao⊗A⊗A1⊗S⊗1⊗1−−−−−→Ao⊗Ao⊗A⊗A1⊗<,>⊗1−−−−−→Ao⊗A ∆⊗∆′−−−→Ao⊗Ao⊗A⊗A1⊗<,>⊗1−−−−−→Ao⊗Aτ− →A⊗Ao(2.16) 10 For a Hopf algebra D, withdim(D)<∞we denote by G(D) the ﬁnite group of group likeelements g, characterized by ∆( g) =g⊗g. Also we shall use the notation Ch(D)∼= G(D∗) for the group of one dimensional representations of D. For doubles we have the following easy fact: Lemma 1 For a Hopf algebra Athe multiplication map .:A⊗Ao/tildewidest−→D(A)yields a group isomorphism: G(A)⊕Ch(A)/tildewidest−→G(D(A)). Similarly the sum of restrictions yields: Ch(D(A))/tildewidest−→Ch(A)⊕G(A). Proof:Forb∈D(A) letbbe the corresponding element in End(A). For the coproduct this means ∆( b)(x⊗y) = ∆(b(yx))which for b∈G(A)has to equal b(x)⊗b(y). Inserting y= 1 and applying ǫ⊗1 we ﬁnd that ais of rank one and a=g.γwhereg=a(1) and γ=ǫ◦a. Inserting instead x= 1 and y= 1 (applying ǫ⊗ǫ) shows that g∈G(A) (γ∈Ch(A)). The adjoint action of Ch(A) on the double D(A) stabilizesAand, there, coincides with the coadjoint action, i.e.,we have γ.y.γ−1=γ ⇀ y ↽ γ−1for ally∈A. Since the coadjoint action on group likes in Ais trivial the images of G(A) andCh(A) centralize each other and the inclusion factors into the direct sum. Injectivity now follows from linear independence of group likes, see [Ab], and injectivity of .. ✷ Here we used the notation ⇀(↽) as in [Ab] for the left (right) action of D∗on a Hopf algebra D. Similarly, we use ( a⊲λ)(y) :=λ(ya)

Question 950 multiple-choice

Quantum algorithms have revolutionized the process of factoring large integers, especially those that are products of two primes—a critical challenge in cryptography. Shor’s algorithm exploits periodicity in modular exponentiation to efficiently identify nontrivial factors. When attempting to factor a large integer N using the quantum order-finding method, which condition must be satisfied by the order r of a random integer a modulo N to guarantee a nontrivial factorization, assuming a and N are coprime? 1) The order r must be even, and neither a^(r/2) − 1 nor a^(r/2) + 1 is a multiple of N. 2) The order r must be prime, and a^(r) ≡ 2 mod N. 3) The order r must divide φ, and N must be odd. 4) The order r must be odd, and a^(r) ≡ −1 mod N. 5) The order r must be a power of two, and a^(r/2) ≡ 1 mod N. 6) The order r must equal N − 1, and a^(r) ≡ 1 mod N. 7) The order r must be a divisor of N, and a^(r) ≡ 0 mod N.

✓ Correct Answer:

The correct answer is 1) The order r must be even, and neither a^(r/2) − 1 nor a^(r/2) + 1 is a multiple of N..

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we will have found a factor ofNand we are ﬁnished! However it is overwhelmingly likely that a randomly chosen awill be coprime to N(e.g. ifNis the product of two large primes). If ais coprime to N, then Euler’s theorem of number theory guarantees that ther e is a power of awhich has remainder 1 when divided by N. Letrbe the smallest such power: ar≡1 modN andris the least such power (11) (Ifais not coprime to Nthen no power of ahas remainder 1). ris called the order ofa moduloN. Next we show that the information of rcan provide a factor of N. Suppose that we have a method for determining r(c.f. later) and suppose further that r comes out to be an evennumber. Then we can rewrite eq. (11) as ar−1≡0 modNand factorise as a diﬀerence of squares: (ar/2−1)(ar/2+ 1)≡0 modN (12) Letα=ar/2−1 andβ=ar/2+ 1. ThenNexactly divides the product αβ. If neither αnor βis a multiple of NthenNmust divide partly into αand partly into β. Thus computing 5 the highest common factor of Nwithαandβ(again using Euclid’s algorithm) will generate a non-trivial factor of N. As an example take N= 15 and choose the coprime number a= 7. By computing the powers of 7 modulo 15 we ﬁnd that 74≡1 mod 15 i.e. the order of 7 modulo 15 is 4. Thus 15 must exactly divide the product (74/2−1)(74/2+ 1) = (48)(50). Computing the highest common factor of 15 with 50 and 48 gives 5 and 3 respectively, w hich are indeed nontrivial factors of 15. Our method will give a factor of Nprovided that rcomes out to be even and that neither of (ar/2±1) are exact multiples of N. To guarantee that these conditions occur often enough (for randomly chosen a’s) we have Theorem: Let Nbe odd and suppose that a

Question 951 multiple-choice

In the classification of subgroups within simple Lie-type and algebraic groups, the structure and automorphism actions on classical groups like D₄ and their interaction with symmetric and symplectic groups play a crucial role. Specific embeddings and centralizer properties determine the uniqueness and possible automorphism groups of certain subgroups. Which of the following statements correctly describes the action of the group S₃ on the D₄ subgroup in the context of Lie-type group automorphisms? 1) S₃ acts as field automorphisms on D₄, permuting its root subgroups cyclically. 2) S₃ acts as diagonal automorphisms on D₄ induced by a maximal torus of A₇. 3) S₃ acts as graph automorphisms on D₄, corresponding to the triality phenomenon unique to D₄. 4) S₃ acts trivially on all automorphisms of D₄ due to lack of outer automorphisms. 5) S₃ acts by conjugation through central elements of E₈ containing D₄. 6) S₃ acts as Frobenius automorphisms on D₄ in the finite group case. 7) S₃ embeds into the center of D₄ fixing all elements of order 2.

✓ Correct Answer:

The correct answer is 3) S₃ acts as graph automorphisms on D₄, corresponding to the triality phenomenon unique to D₄..

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Proposition 2.7 implies that (C(e) n C(y)0)' is of type E, or C4; as this subgroup lies in C(e )0 = A7, it must be of type C4• Hence, also by Proposition 2.7, C(e )0 n C(f)0 = D4, that is, Cc(E)0 = D4 (of ad joint type). Further, E#=e0nE andf, fe are conjugate in C0(e)=A 7(w2) (regarding the action off as the 'inverse-transpose' action, f and fe are conjugate by a suitable diagonal element of A7). Repeating this for each element of E#, we see that N0(E)/C 0(E) = S3. We claim that the group S3 here acts as graph automorphisms on the factor D = C0(E)0 = D4. For suppose to the contrary that there is a 3-element x centralizing D. Now D has two composition factors on the minimal 56-dimensional module for G, each of dimension 28 (each is the skew-square of a natural 8-dimensional module). It follows that x has precisely two distinct eigenvalues on the module, and C0(x) acts on the corresponding eigenspaces, each of which has dimension 28. But D is irreducible on each eigenspace, so by [16, 14.1] the only possibility for Cc;(x )0 is A7. This is absurd as A7 centralizes only 2-elements. Hence our claim is proved. To conclude this case (assuming E extraspecial), we consider the situation in the finite group Ga. We have proved that the G-class of E is uniquely determined, and Nc(E) =(Ex D4).S3. Applying [26, I, 2.7 and 2.8], we see that the Ga-classes of subgroups (E1)a for a-stable E 1 E £0 are in bijective correspon dence with the classes of elements in the coset S4a = (E.S3)a. The only such classes giving Klein 4-groups (E1)a are represented by a and ea, where e E £#. In the latter case, however, the corresponding Klein 4-group does not have an element of order 3 acting on it. Hence the G0-class of E is uniquely determined, and N0JE) =(Ex Inndiag(D 4(q)).S 3, as in Table 1. We may now assume that E is elementary abelian. Thus each involution in £# has centralizer of type A1D6• We view G as a subgroup of E8, centralizing a fundamental subgroup SL2, say 18 (in the notation of Lemma 2.14). We may then use the relations in Lemma 2.14, setting (z)=(e 8)=Z(G), and e=e1. Let Ce( e) = 11 Y, a commuting product of 11 =A 1 and Y = D6. Note that e l and e1 z are not conjugate in G, since otherwise Cc;(e) would not be connected, a contradiction. Suppose that there exists J E (En Y)\(e, z ). Choose a maximal torus T of 11 Y containing e and f Then 11 is contained in 11 T ~ Ct;(/) = Jlf P for some g E G. Since 11 is T-invariant, it follows that 11 ~ P and so [11, 11] = 1. Hence J E Jlf ~ C(J1) = Y. Thus J lies in a fundamental SL2 of Y, so E ~ Ca(e, /) = Ct;(e,

Question 952 multiple-choice

Quantum algorithms have revolutionized computational group theory, especially in solving problems related to subgroup structure and membership within finite groups. A key application is the efficient solution of the Hidden Subgroup Problem (HSP) for Abelian groups using quantum techniques. Which of the following is essential for efficiently solving the Abelian Hidden Subgroup Problem (HSP) using quantum algorithms? 1) Using classical enumeration of all subgroup elements 2) Employing a quantum oracle that maps elements from different cosets of the hidden subgroup to orthogonal quantum states 3) Applying only random sampling over group elements 4) Generating uniform superpositions over individual group generators 5) Constructing the Cayley graph of the group 6) Utilizing brute-force search for subgroup membership 7) Replacing the quantum Fourier transform with classical Fourier analysis

✓ Correct Answer:

The correct answer is 2) Employing a quantum oracle that maps elements from different cosets of the hidden subgroup to orthogonal quantum states.

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substituting the gene rators in T into the relators, and let N0stand for the normal closure (the smallest normal subgroupc ontaining) ofR0. ThenN=N0sinceN0≤NandG/N0=G/Nby deﬁnition of TandR0. Still some care has to be taken since it is possible that Tgenerates Gonly modulo N, that is it might generate a proper subgroup of G.Therefore some additional elements should be added to R0. LetSbe the generating set for G. Using the constructive membership test for G/N,we express the original generators from SmoduloNwith straight line programs in terms of the elements of T. For each element x∈Swe form the quotient y−1xwhereyis the element obtained by substituting the generators from Tinto the straight line program for xmoduloN. LetS0be the set of all the quotients formed this way. Note that TandS0generate together G. Then one can verify that the normal closure of R0∪S0inGisN. Thus,from R0andS0wecanﬁndgeneratorsfor Nintimepolynomialintheinput size+ ν(G/N) using the normal closure algorithm of [1]. We obtained the de sired result. 4.3 Unique encoding and solvable normal subgroup We conclude this section with some results obtained as combi nation of the ideas presented above with those of Watrous described in [27]. Assume that the enco ding of the elements of Gis unique and a normal solvable subgroup NofGis given by generators. We use the encoding of Gfor that of G/N. The identity test in G/Ncan be implemented by an eﬃcient quantum algorithm for testi ng membership in Ndue to Watrous (Theorem 2). We are also able to produce (sever al copies of) the uniform superposition |N/a\}bracketri}ht=1√ |N|/summationtext x∈N|x/a\}bracketri}hteﬃciently. For solvable subgroups N, we can again apply the result of Watrous (Theorem 2) to produce |N/a\}bracketri}htin polynomial time. We will now show that having suﬃciently many copies of |N/a\}bracketri}htat hand, we can use ideas of Watrous for computing orders of elements of G/Nand even for performing the constructive membership test in Abelian subgroups of G/N. Thus, we will have an eﬃcient quantum implementation of the Beals-Babai algorithms for G/N. We will ﬁrst state a lemma which says that we can eﬃciently so lve the HSP in an Abelian group if we have an appropriate quantum oracle. Lemma 9. LetAbe an Abelian group, and let Xbe a ﬁnite set. Let H≤A, and let f:A→CX (given by an oracle) such that: 7 1. For every g∈A,|f(g)/a\}bracketri}htis a unit vector, 2.fis constant on the left cosets of H,and maps elements from diﬀerent cosets into orthogonal states. Then there exists a polynomial time quantum algorithm for ﬁnd ing the hidden subgroup H. Proof.First we extend naturally ftoG/H: on a coset of H, it takes the value f(h) for an arbitrary member hof the coset. The algorithm is the standard quantum algorith m for the Abelian hidden subgroup problem. We repeat several times the following ste ps to ﬁnd a set of generators for H. •Prepare the initial superposition: |1G/a\}bracketri}ht|0m/a\}bracketri}ht. •Apply the Abelian quantum Fourier transform in Aon the ﬁrst register:/summationtext g∈A|g/a\}bracketri}ht|0m/a\}bracketri}ht. •Callf:/summationtext g∈A|g/a\}bracketri}ht|f(g)/a\}bracketri}ht. •Apply again the Fourier transform in A:/summationtext g∈A/H,h∈H⊥χh(g)|h/a\}bracketri}ht|f(g)/a\}bracketri}ht. •Observe the ﬁrst register. By hypothesis, the states |f(g)/a\}bracketri}htare orthogonal

Question 953 multiple-choice

Quantum Ordered Binary Decision Diagrams (OBDDs) are an important framework in quantum computing for efficiently representing and evaluating Boolean functions. Techniques such as fingerprinting and polynomial representations are often employed to optimize algorithms within restricted quantum computation models. Which quantum algorithmic technique enables optimal resource usage for testing if a Boolean matrix is a permutation matrix within the framework of quantum OBDDs? 1) Quantum Fourier Transform 2) Grover's Search Algorithm 3) Quantum Walks 4) Amplitude Amplification 5) Quantum Phase Estimation 6) Quantum Error Correction 7) Fingerprinting combined with characteristic polynomial representation

✓ Correct Answer:

The correct answer is 7) Fingerprinting combined with characteristic polynomial representation.

📚 Reference Text:

S.BarryCooper &Vincent Danos (eds.): FifthWorkshop on Developments inComputational Models—Computational Mode ls FromNature EPTCS9, 2009, pp. 1–11, doi:10.4204/EPTCS.9.1c/circlecopyrtF.Ablayev & A.Vasiliev Thiswork islicensed under the Creative Commons AttributionLicense.AlgorithmsforQuantumBranchingPrograms BasedonFingerprinting Farid Ablayev∗ Institutefor Informatics Kazan, Russian Federation fablayev@gmail.comAlexanderVasiliev† Institutefor Informatics Kazan, Russian Federation Alexander.Vasiliev@ksu.ru Inthepaperwedevelopamethodforconstructingquantumalg orithmsforcomputingBooleanfunc- tions by quantum ordered read-once branching programs (qua ntum OBDDs). Our method is based onﬁngerprintingtechniqueandrepresentationofBooleanf unctionsbytheircharacteristicpolynomi- als. We use circuit notationfor branchingprogramsfor desi red algorithmspresentation. For several known functions our approach provides optimal QOBDDs. Name ly we consider such functions as MODm,EQn,Palindrome n, andPERMn(testing whether given Boolean matrix is the Permutation Matrix). We also propose a generalization of our method and a pply it to the Boolean variant of the HiddenSubgroupProblem . 1 Introduction During the last two decades different types of quantum compu tation models based on Turing Machines, automata, and circuits have been considered. For some of the m different examples of functions were presentedforwhichquantummodelsappeartobemuchmore(ex ponentially) efﬁcientthantheirclassical counterparts. In this paper we consider a restricted model of computation k nown asOrdered Read-Once Quan- tum Branching Programs . In computer science this model is also known as Ordered Bina ry Decision Diagrams (OBDDs). The main reason for the investigation of r estricted models of quantum computers was proposed by Ambainis and Freivalds in 1998 [1]. Consider ing one-way quantum ﬁnite automata, they suggested that ﬁrst quantum-mechanical computers wou ld consist of a comparatively simple and fast quantum-mechanical part connected toaclassical comp uter. Twomodels of quantum branching programs were introduced by Ablayev, Gainutdinova, Karpinski [2] (leveled programs ), and by Nakanishi, Hamaguchi, Kashiwabara [11] ( non-leveled programs ). Later it wasshown bySauerhoff [12] that these twomodels are polyn omially equivalent. For this model we develop the ﬁngerprinting technique introduced in [6]. The basic ideas of this approach are due to Freivalds (e.g. see the book [10]). It was later successfully applied in the quantum automata setting by Ambainis and Freivalds in1998 [1] (later improve d in [5]). Subsequently, the same technique was adapted for the quantum branching programs by Ablayev, Gainutdinova and Karpinski in 2001 [2],and waslater generalized in [6]. Forourtechniqueweusethepresentation ofBooleanfunctio nsknownas characteristic polynomials . Our deﬁnition of the characteristic polynomial differs fro m that of [4],though it uses similar ideas. We display several known functions for which our method prov ides optimal QOBDDs. Namely, these functions are MODm,EQn,Palindrome n, andPERMn. ∗Workwas inpartsupported bythe RussianFoundation for Basi c Researchunder the grant 08-07-00449a †Workwas inpartsupported bythe RussianFoundation for Basi c Researchunder the grant 08-07-00449a 2 Algorithms for Quantum Branching Programs 2 Preliminaries We use the notation |i/an}⌊ra⌋k⌉tri}htfor the vector from Hd, which has a 1 on the i-th position and 0 elsewhere. Obviously, the set of vectors |1/an}⌊ra⌋k⌉tri}ht,...,|d/an}⌊ra⌋k⌉tri}htforms anorthonormal basis in Hd. Deﬁnition 1. AQuantum Branching Program Q over the Hilbert space Hdis deﬁned as Q=/an}⌊ra⌋k⌉tl⌉{tT,|ψ0/an}⌊ra⌋k⌉tri}ht,Maccept/an}⌊ra⌋k⌉tri}ht, where T is a sequence of l instructions: T j=/parenleftbig xij,Uj(0),Uj(1)/parenrightbig is determined by the variable x ijtested on the step j, andU j(0),Uj(1)areunitary transformations in Hd. Vectors|ψ/an}⌊ra⌋k⌉tri}ht∈Hdarecalledstates(statevectors) ofQ, |ψ0/an}⌊ra⌋k⌉tri}ht∈HdistheinitialstateofQ,andM accept –isaprojector ontheaccepting subspace Hd accept(i.e. itisadiagonal zero-one projection matrix,which determines

Question 954 multiple-choice

In the study of computational group theory, automata are used to model the structure and properties of subgroups within complex algebraic systems, such as direct products of free and abelian groups. Innovations like enriched automata facilitate algorithmic analysis of subgroup intersections and index calculations in these settings. Which concept enables a computable bijection between finitely generated subgroups of free-times-abelian groups and automata that encode both free and abelian components? 1) Schreier graphs with labeled edges 2) Coset enumeration trees 3) Standard finite state automata 4) Cayley graphs of abelian groups 5) Group presentations with relators 6) Enriched automata with abelian labels 7) Nielsen generators for subgroups

✓ Correct Answer:

The correct answer is 6) Enriched automata with abelian labels.

📚 Reference Text:

Title: Stallings automata for free-times-abelian groups: intersections and index Year: 2019 Paper ID: 40a07d189ee819cfbb061b6aca83fbb67357eb56 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/40a07d189ee819cfbb061b6aca83fbb67357eb56 Abstract: We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which - as it happens in the free group - is computable in the finitely generated case. This approach provides a neat geometric description of (even non finitely generated) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric solution to the subgroup intersection problem and the finite index problem, providing recursive bases and transversals respectively.

Question 955 multiple-choice

Advanced quantum chemistry methods are essential for accurately modeling atomic electronic structure and understanding phenomena such as electron correlation and phase transitions in solids. Sophisticated wavefunctions can describe both localized and delocalized electron states, which are crucial for studying material properties. Which of the following features most clearly distinguishes SOGI wavefunctions from Hartree-Fock methods in the context of atomic and solid-state calculations? 1) They are unable to model multi-electron interactions. 2) They require more extensive basis sets to achieve convergence. 3) They do not support spin function optimization. 4) They are limited to single-electron systems. 5) They rely exclusively on delocalized orbitals. 6) They cannot describe phase transitions in materials. 7) They intrinsically provide both localized and delocalized orbitals without requiring additional localization steps.

✓ Correct Answer:

The correct answer is 7) They intrinsically provide both localized and delocalized orbitals without requiring additional localization steps..

📚 Reference Text:

4>3x ' '~ ' " ' ' ' ' ' /:~,, ' ' ' : /,-~ .... ' ..... ' ' ,u., ........ ... ,,\._,...,_)) ' '.::::::.~ ......... ' 4>4x 8 ~ ----... .... .... ' ' .... ,-.... \ 8 I • \ I ,_) I I I I I cf>ty I / I I I 0 I ,....-, I I e \ \ ',_ .... ' , ____ ,. ... ~ 8 cf>2y ~ 0 8 ~ tP3y r-:.::-.... / I(,--·<',', / I \I e I ) I II \ ,'-:r;/"" / / \ ... / / / ',_, I / / / /.//~ 4>4y (c) 1079 .... .... Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp1080 R. C. LADNER AND W. A. GODDARD III TABLE I. SOGI and G1 wavefunctions for the ground state of Li (Z/2=0.217°±0.001 °). Function t 1SOGI 1G1 ls 3.0 1.14368 1.14370 4s 5.33 -0.10348 -0.10348 3s 5.40 -0.07808 -0.07810 3s 3.00 0.00542 0.00550 3s 1.347 0.01569 0.01549 3s 0.841 -0.00255 -0.00292 4s 0.732 0.03458 0.03446 3s 0.62 -0.03799 -0.03793 E -2.84409 -2.8427 (orbital energy) transitions are of theoretical interest and have been suggested by Mott17 to be relevant in the discussion of conduction and other properties of such solids as NiO. Since the SOGI wavefunctions dissociate correctly and can naturally lead directly to either localized or delocalized orbitals without requiring additional Wannier-like localizations as in the HF method, it would seem that the SOGI wavefunctions would form a suitable foundation on which to discuss ideas such as the Mott transition. III. RESULTS AND DISCUSSIONS Since the spatial SOGI equations are of exactly the same form as the G1 equations, we solve them in the same way. We expand the unknown orbitals in terms of a finite basis set (26) and solve for the coefficients iteratively. (The equa tions could also be solved numerically.) Existing three and four-electron G1 programs were adapted to do SOGI calculations by changing only the sections which calculate the D matrices. A. Three-Electron Doublet States First we will consider some three-electron doublet states. In this case there are two linearly independent spin states, j~'=2. The standard representation is given in Paper I (Appendix A) and the general transforma tions L to new spin functions is given by ( cos(:E:/2) L(:E:) = -sin(:E:/2) sin(:E:/2)) cos(:E:/2) (27) Thus in addition to optimizing the three orbitals cf>a, cp,, and cf>c, we must optimize the single parameter :E:, which 17 N. F. Mott, Proc. Phys. Soc. (London) A62, 416 (1949). 2SOG1 2G1 sSOGI sG1 0.60987 0.60984 0.02804 0.03189 0.12715 0.12709 -0.00228 -0.00208 0.07854 0.07853 0.00060 0.00069 0.27260 0.27246 0.02045 0.02124 -0.00398 -0.00362 0.28602 0.28571 -0.00004 0.00063 0.63560 0.63501 -0.01551 -0.01531 0.06863 0.06854 0.01738 0.01731 0.05573 0.05576 -2.46128 -2.4588 -0.19616 -0.19615 just corresponds to a rotation angle in the two-dimen sional space of spin functions. From (21) we have that (28) or expanding w (?Lcf>acf>bcf>ca.Ba =jet { cf>az,cf>c[ a,Ba ( 1 + i cos:E:+ iVJ sinS:)

Question 956 multiple-choice

In representation theory and quantum machine learning, symmetries are often analyzed by examining how groups and their associated Lie algebras act on vector spaces. Understanding the relationship between Lie groups and Lie algebras is essential for constructing neural network layers that respect such symmetries. Which approach most effectively reduces the challenge of constructing equivariant quantum neural network layers to a tractable problem when working with symmetry groups? 1) Translating the equivariance condition from the Lie group to its Lie algebra and verifying commutation with infinitesimal generators 2) Directly enumerating all possible unitary matrices compatible with the symmetry group 3) Optimizing neural network weights using gradient descent without regard for symmetry constraints 4) Decomposing the underlying vector space into orthogonal subspaces irrespective of group action 5) Applying only real-valued transformations and ignoring complex structure 6) Checking group invariance solely by evaluating random group elements 7) Designing layers to commute with a single fixed group element rather than the full representation

✓ Correct Answer:

The correct answer is 1) Translating the equivariance condition from the Lie group to its Lie algebra and verifying commutation with infinitesimal generators.

📚 Reference Text:

in agreement with Theo- rem 2, this representation similarly respects the de- composition of Vinto invariant subspaces: observe that for all H2su(2),[H;1] = 0trivially, and [H;su(2)]su(2)because su(2)isaLiealgebraand thus closed under commutators. So V=su(2)C1 as Lie algebra representations. physicists will say the Paulis, which are Hermitian, gen- erate su(2), whereas the equivalent statement for mathe- maticians is that su(2) =spanR(iX;iY;iZ ). There are two comments to be made here: •When we work on the Lie groups and Lie algebras themselves,thedistinctionbetweentheseconventions is crucial. We recommend using the “Taylor expan- sion trick” (Equation 17) from the beginning of this section to avoid getting confused. •In practice for quantum machine learning, we are almost always working on complex representations : since we are allowing complex scalars on the repre- sentation side, the representations of a Lie algebra g are the same as for the complexiﬁcation of that Lie algebra su(2) C, which just means we allow com- plex linear combinations of our basis. Here, the dis- tinctions between the physicist’s convention and the mathematician’s is no longer problematic. Do note that this comes at a price if we need other informa- tion beyond complex representations of the Lie alge- 19 bra. For instance, the Lie groups SU(2)andSL(2) are very diﬀerent (e.g. the former is compact while the latter is not); but their complexiﬁed Lie algebras are identical su(2) C=sl2 C. All the previous algebra theory to connect Lie groups to Lie algebras is nice and good. But how can we use this for QML? We have already stated that dealing with Lie groups can be cumbersome, whereas dealing with Lie algebras is more manageable. In particular we recall the mantra: When we have a problem with Lie group symme- try, pass to the Lie algebra, analyze it, and return to the Lie group.Lets put this to practice for the task of constructing an equivariant quantum neural network. Consider a QML model as in Eq. (1) of the form h(i) = Tr[W(i)Mi] = Tr W()iWy()Mi ;(18) whereW:B(H)!B(H)withW(i) =W()iWy()a parameterized unitary quantum neural network. Our goal here is to construct a Wrespecting the symmetries in the dataset. Recall from Eq. (3) that Wwill be equivariant if W()UgiUy gWy() =UgW()iWy()Uy g;(19) for allginG, and for all. That is, we need [W();Ug] = 0;8;8g2G: (20) Solving Eq. (20) means determining allthe unitaries W()that commute with allthe representations Ugof the elements of the symmetry group G. Quite the daunting task! However, we can use the trick of passing to the Lie algebra, solving there, and going back to the group. Ex- plicitly, let us consider the case where W()is composed of a single “layer”, which is a fancy way of saying that W() =eiH, for some Hermitian operator H, and for some trainable parameter 2R. In particular, since (20) must hold for all , it must hold for inﬁnitesimal parame- ters. We can again use the Taylor expansion trick as in Eq. (17) to expand around = 0. SoeiH=11+H+O(2) in Eq. (20) yields [W();Ug] =[H;Ug] +O(2); (21) which is

Question 957 multiple-choice

In quantum computing, mapping between computational bases and bases structured by group representation theory is essential for leveraging symmetries in quantum algorithms. Techniques such as the Schur transform employ concepts from symmetric and unitary group representations to efficiently decompose and manipulate quantum states. Which step in the conversion of a computational basis vector to a block diagonal basis labeled by |λ, i, j⟩ directly involves applying the Fourier transform over permutation modules, and what is its primary purpose in the algorithm? 1) Remapping entries of the computational basis vector to fit within the range [n], ensuring compatibility with the symmetric group action 2) Computing the type of the mapped vector and constructing the symmetric group element as a product of transpositions 3) Tracking the mapping (pe) between original and remapped entries for later reconstruction 4) Rewriting the basis vectors in terms of semistandard Young tableaux and Gelfand-Tsetlin patterns 5) Using the type T to perform the Fourier transform for permutation modules, thereby decomposing the space into irreducible components 6) Verifying the completeness of the basis states by checking all possibilities for type T and mapping pe 7) Bounding the overall runtime of the algorithm by a polynomial function in the key parameters

✓ Correct Answer:

The correct answer is 5) Using the type T to perform the Fourier transform for permutation modules, thereby decomposing the space into irreducible components.

📚 Reference Text:

the block diagonal basis |λ,i,j/angbracketright, whereλis the symmetric or unitary group irrep, ilabels the Young-Yamanouchi basis vector in the symmetric group and j labels the Gelfand-Tsetlin basis vector in the unitary group irrep. Proof. To prove the runtime claim, we describe the steps in more detail with an example and bound the run time. 16 1. Given a basis vector in the computational basis, ﬁrst we map the entries of the vector that are greater thannto entries inside nand keep track of the map. So a basis vector |e/angbracketright=|e1,...,en/angbracketrightbecomes |pe,˜e/angbracketright, where ˜eis a vector with entries in [ n] andpeis the map. This map need not be global and can be speciﬁc to the vector |e/angbracketright. This can be done in poly( n) steps. As a running example, let us considern= 5,d= 10 and take the vector |5,5,10,3,9/angbracketright. In the ﬁrst step, this gets mapped to |10→1,9→2/angbracketright⊗|5,5,1,3,2/angbracketright. This can be done in O(poly( n,logd)) steps. 2. Compute the type of the vector ˜ eand the symmetric group element (as a product of a set of transpo- sitions) needed to convert the standard basis vector i.e., where the entries appear in ascending order to ˜e. The basis vector |e/angbracketrightis converted to|pe,T,t/angbracketright, whereTis the type and tis the transversal element of the subgroup YTinSn. For the example, we would have the standard basis vector as |1,1,2,3,4/angbracketright, the type is (1 ,1,1,0,2) i.e., one 1, one 2, one 3, zero 4 and two 5s. The transversal as a product of transpositions would be (15)(52)(53). This takes poly( n) time. 3. Recall that the register with |t/angbracketrightcan be viewed as the induced representation of YTinSni.e., a permutation module. Conditioned on the type T, we can apply the Fourier transform for permutation modules to obtain the basis |pe,T,λ,i,j/angbracketright. This step takes poly( n,log/epsilon1−1) time. 4. Rewriting this as |λ,(pe,T,i),j/angbracketright, we can convert pe,T,iinto a SSYT by using the information in pe andTand rewriting the basis i, which consists of SSYT of shape λand contentT, to match the GT basis (we prove below that this is gives the GT basis). The claim of the dependence on /epsilon1follows from the fact the each of the steps in the algorithm can be done with O(polylog /epsilon1−1) elementary gates based on the results described in section 3.1 and 3.2. This shows that the overall runtime is O(poly(log d,n,log/epsilon1−1)). To prove the theorem, we now need to prove that the basis obtained in the last step (labeled by SSYTs) is exactly the GT basis. The (unnormalized) basis states can be written in the form (from Eq. 3.9) |λ,pe,T,i,j/angbracketright=/radicalBigg dλ |G|/|YT|/summationdisplay g[λ(gYT)]i,j|gYT/angbracketright, (4.1) whereGis the symmetric group Sn,YTis a Young subgroup of the form ( Sµ1×Sµ2×...Sµk) for some k≤d. This follows from the next lemma. Lemma 4. LetGbe the symmetric group SnandYTbe a Young subgroup corresponding to the type T. Consider the unnormalized basis states written as |λ,pe,T,i,j/angbracketright=/radicalBigg dλ |G|/|YT|/summationdisplay g[λ(gYT)]i,j|gYT/angbracketright, (4.2) where the sum over gis over the cosets of YTinG. WhenTruns over all possible types and peruns over all

Question 958 multiple-choice

Quantum groups, such as Uq(SL(2)), play a significant role in representation theory and integrable systems, introducing q-deformed algebraic structures and modified symmetry properties. The interplay between tensor products, singlet state projections, and intertwining operators is central to understanding their applications in mathematical physics. In the context of Uq(SL(2)) representation theory, which formula correctly encodes the q-deformed antisymmetrization for the singlet state constructed from the two-dimensional representation? 1) S = |+⟩|+⟩ − q|−⟩|−⟩ 2) S = q|+⟩|−⟩ + |−⟩|+⟩ 3) S = |+⟩|−⟩ − q|−⟩|+⟩ 4) S = |+⟩|−⟩ + q|−⟩|+⟩ 5) S = q|+⟩|+⟩ − |−⟩|−⟩ 6) S = |−⟩|+⟩ − q|+⟩|−⟩ 7) S = |+⟩|−⟩ − q²|−⟩|+⟩

✓ Correct Answer:

The correct answer is 3) S = |+⟩|−⟩ − q|−⟩|+⟩.

📚 Reference Text:

diﬀerent from V′⊗W′in the case of quantum groups. 5 Using this projection, if it is not occasionally orthogonal to the image ofE⊗E′, one can build a new intertwining operator Γ :ˆV⊗ˆV′E⊗E′−→V⊗W⊗W′⊗V′I⊗π⊗I−→V⊗V′, (8) which possesses the property Γ(g⊗g) = (g⊗g)Γ (9) for any group element gsuch that ∆(g) =g⊗g. (10) 3. It now remains to take a matrix element of (9) between four stat es, V′/an}bracketle{tk′|V/an}bracketle{tk|(g⊗g)Γ|n/an}bracketri}htˆV|n′/an}bracketri}htˆV′=V′/an}bracketle{tk′|V/an}bracketle{tk|Γ(g⊗g)|n/an}bracketri}htˆV|n′/an}bracketri}htˆV′ (11) and rewrite this identity in terms of generating functions (1). For illustrative purposes we give now an explicit example of this calculat ion for the case ofSLq(2). Formulas for SL(2) arise in the limit q= 1. 3 Explicit equations for SLq(2) 3.1 Bilinear identities To begin with, ﬁx the notations. We consider generators T+,T−andT0ofUq(SL(2)) with commutation relations qT0T±q−T0=q±1T±, [T+,T−] =q2T0−q−2T0 q−q−1,(12) and comultiplication 6 ∆(T±) =qT0⊗T±+T±⊗q−T0, ∆(qT0) =qT0⊗qT0.(13) Verma module Vλwith highest weight λ(not obligatory half-integer), consists of the elements |n/an}bracketri}htλ≡Tn −|0/an}bracketri}htλ, n≥0, (14) such that T−|n/an}bracketri}htλ=|n+1/an}bracketri}htλ, T0|n/an}bracketri}htλ= (λ−n)|n/an}bracketri}htλ, T+|n/an}bracketri}htλ≡bn(λ)|n−1/an}bracketri}htλ, bn(λ) = [n][2λ+1−n],[x]≡qx−q−x q−q−1, ||n||2 λ≡λ/an}bracketle{tn|n/an}bracketri}htλ=[n]! Γq(2λ+1) Γq(2λ+1−n)λ∈Z/2=[2λ]![n]! [2λ−n]!.(15) Now, (∆(T−))n=qnT0⊗Tn −+[n]T−q(n−1)T0⊗Tn−1 −q−T0+...+ +[n]Tn−1 −qT0⊗T−q−(n−1)T0+Tn −⊗q−nT0.(16) Let us manifestly derive equations (11) taking for Wan irreducible spin-1 2representation ofUq(SL(2)). Then ˆV=Vλ±1 2,V=Vλand the highest weights of ˆVinW⊗VorV⊗W are5: |0/an}bracketri}htλ+1 2=|+/an}bracketri}ht|0/an}bracketri}htλ,|+/an}bracketri}ht ≡ |0/an}bracketri}ht1 2, or|0/an}bracketri}htλ|+/an}bracketri}ht;(17) |0/an}bracketri}htλ−1 2=|+/an}bracketri}ht|1/an}bracketri}htλ−q(λ+1 2)[2λ]|−/an}bracketri}ht|0/an}bracketri}htλ,|−/an}bracketri}ht ≡ |1/an}bracketri}ht1 2, or/parenleftig q→q−1/parenrightig |1/an}bracketri}htλ|+/an}bracketri}ht−q−(λ+1 2)[2λ]|0/an}bracketri}htλ|−/an}bracketri}ht.(18) Entire Verma module is generated by the action of ∆( T−): 5Hereafter we omit the symbol of tensor product from the notatio ns of the states |+/an}bracketri}ht⊗|0/an}bracketri}htλetc. 7 |n/an}bracketri}htλ+1 2= (∆(T−))n|0/an}bracketri}htλ+1 2−→ qn/2/parenleftig |+/an}bracketri}ht|n/an}bracketri}htλ+q−(λ+1 2)[n]|−/an}bracketri}ht|n−1/an}bracketri}htλ/parenrightig , orq−n/2/parenleftig |n/an}bracketri}htλ|+/an}bracketri}ht+q(λ+1 2)[n]|n−1/an}bracketri}htλ|−/an}bracketri}ht/parenrightig ;(19) |n/an}bracketri}htλ−1 2= (∆(T−))n|0/an}bracketri}htλ−1 2−→ qn/2/parenleftig |+/an}bracketri}ht|n+1/an}bracketri}htλ+q(λ+1 2)[n−2λ]|−/an}bracketri}ht|n/an}bracketri}htλ/parenrightig , orq−n/2/parenleftig |n+1/an}bracketri}htλ|+/an}bracketri}ht+q−(λ+1 2)[n−2λ]|n/an}bracketri}htλ|−/an}bracketri}ht/parenrightig ;(20) Step 2 to be made in accordance with our general procedure is to pr oject the tensor product of two diﬀerent W’s onto singlet state S=|+/an}bracketri}ht|−/an}bracketri}ht−q|−/an}bracketri}ht|+/an}bracketri}ht:6 (A|+/an}bracketri}ht+B|−/an}bracketri}ht)⊗(|+/an}bracketri}htC+|−/an}bracketri}htD)−→AD−qBC. (21) With our choice of Wwe can now consider two diﬀerent cases: (A) bothˆV=Vλ−1 2andˆV′=Vλ′−1 2, or (B)ˆV=Vλ−1 2andˆV′=Vλ′+1 2: Case A: |n/an}bracketri}htλ−1 2|n′/an}bracketri}htλ′−1 2−→ −→qn′−n−1 2/parenleftig [n′−2λ′]qλ′|n+1/an}bracketri}htλ|n′/an}bracketri}htλ′−[n−2λ]q−λ|n/an}bracketri}htλ|n′+1/an}bracketri}htλ′/parenrightig .(22) Case B: |n/an}bracketri}htλ+1 2|n′/an}bracketri}htλ′−1 2−→ −→qn′−n−1 2/parenleftig [n′−2λ′]qλ′|n/an}bracketri}htλ|n′/an}bracketri}htλ′−[n]q+λ+1|n−1/an}bracketri}htλ|n′+1/an}bracketri}htλ′/parenrightig .(23) 6This state is a singlet of Uq(SL(2)). In the case of Uq(GL(2)) one should account for the U(1) non- invariance of S. This is the origin of the factor det qgat the r.h.s. of the ﬁnal equations (29), (34) and (36). 8 Now we proceed to the step 3. Consider any “group element”, i.e. an elementgfrom some extension of Uq(G), which possesses the property: ∆(g) =g⊗g, (24) and take matrix elements of the formula (9): λ′/an}bracketle{tk′|λ/an}bracketle{tk|(g⊗gΓ = Γg⊗g)|n/an}bracketri}htˆλ|n′/an}bracketri}htˆλ′. (25) The action of operator Γ can be represented as: Γ|n/an}bracketri}htˆλ|n′/an}bracketri}htˆλ′=/summationdisplay l,l′|l/an}bracketri}htλ|l′/an}bracketri}htλ′Γ(l,l′|n,n′), (26) and in these terms (25) turns into: /summationdisplay m,m′Γ(k,k′|m,m′)||k||2 λ||k′||2 λ′ ||m||2 ˆλ||m′||2 ˆλ′/an}bracketle{tm|g|n/an}bracketri}htˆλ/an}bracketle{tm′|g|n′/an}bracketri}htˆλ′= =/summationdisplay l,l′/an}bracketle{tk|g|l/an}bracketri}htλ/an}bracketle{tk′|g|l′/an}bracketri}htλ′Γ(l,l′|n,n′).(27) In order to rewrite this as a diﬀerence equation, we use our deﬁnitio n ofτ-function: τλ(t,¯t|g)≡ /an}bracketle{tλ|etT+ge¯tT−|λ/an}bracketri}ht=/summationdisplay m,n/an}bracketle{tm|g|n/an}bracketri}htλtm [m]!¯tn [n]!. (28) Then, one can write down the generating formula for the equation ( 27), using the manifest form (22)-(23) of matrix elements Γ( l,l′|n,n′): Case A: /radicalig M− ¯tM+ ¯t′/parenleftig qλ′D(0) ¯t¯t′D(2λ′) ¯t′−q−λ¯tD(2λ) ¯tD(0) ¯t′/parenrightig τλ(t,¯t|g)τλ′(t′,¯t′|g) = = [2λ][2λ′](detqg)/radicalig M− tM+ t′/parenleftig q−(λ+1 2)t′−q(λ′+1 2)t/parenrightig τλ−1

Question 959 multiple-choice

Quantum error correction is a vital area of quantum information science, enabling reliable quantum computation in the presence of noise and decoherence. Different families of quantum codes vary in their structure, efficiency, and ability to isolate types of errors. Which class of quantum error-correcting codes separates the correction of bit-flip errors from phase-flip errors, thereby simplifying their implementation in quantum circuits? 1) Shor's nine-qubit code 2) Stabilizer codes 3) CSS (Calderbank-Shor-Steane) codes 4) Five-qubit code 5) Abstract quantum error correction 6) Reversible codes 7) Quantum complexity codes

✓ Correct Answer:

The correct answer is 3) CSS (Calderbank-Shor-Steane) codes.

📚 Reference Text:

correction ........... . 4.3 Shor's nine-qubit error-correcting code 4.4 A seven-qubit quantum error-correcting code . 4.5 A five-qubit error-correcting code . . . 4.6 Stabilizers and the five-qubit code . . . 4.7 Theoretical aspects of stabilizer codes . 4.8 CSS codes . . . . . . . . . . . . . . . . 4.9 Abstract quantum error correction 4.10 Further aspects of quantum error-correcting codes Afterword References Index Contents 57 64 68 74 81 81 85 89 92 96 99 . 106 .113 .115 . 122 125 127 135 Preface In 1994 Peter Shor [65] published a factoring algorithm for a quantum computer that finds the prime factors of a composite integer N more efficiently than is possible with the known algorithms for a classical com puter. Since the difficulty of the factoring problem is crucial for the se curity of a public key encryption system, interest (and funding) in quan tum computing and quantum computation suddenly blossomed. Quan tum computing had arrived. The study of the role of quantum mechanics in the theory of computa tion seems to have begun in the early 1980s with the publications of Paul Benioff [6]' [7] who considered a quantum mechanical model of computers and the computation process. A related question was discussed shortly thereafter by Richard Feynman [35] who began from a different perspec tive by asking what kind of computer should be used to simulate physics. His analysis led him to the belief that with a suitable class of "quantum machines" one could imitate any quantum system. A separate line of research into the thermodynamics of computation in the 1970s and 1980s had led to the concept of "reversible programming." Motivated by mechanistic and biological models of computation, the idea was that the energy expenditures of computation could be reduced if no machine state had more than one logical precursor. (See [8] for a survey of the field at that time and associated references.) Since a quantum me chanical model of computation would involve unitary transformations, which are indeed invertible, there was the additional incentive to explore the advantages and limitations of computing devices based on quantum viii Preface mechanics. In particular it was soon noticed that the logic gates of re versible programming [74], for example, could also be modeled as specific unitary maps of collections of two-state quantum systems, as in [36] where a simple addition routine is described. Building on these ideas David Deutsch [25] developed the thesis that quantum theory and a "universal quantum computer" constitute the cor rect context for an extended Church-Turing principle, that is, that "ev ery finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means." He also analyzed the role of quantum parallelism, which we will see is a key el ement in the theory of quantum algorithms, and he commented on the role of quantum complexity theory. In [26] Deutsch expanded on the concept of quantum computational cir

Question 960 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) have profoundly impacted computational group theory and cryptography, especially when dealing with complex group structures such as semi-direct products. Efficient solutions to the HSP for non-abelian groups are an active area of research, leveraging advanced algebraic techniques. Which condition is crucial for the existence of an efficient quantum algorithm to solve the Hidden Subgroup Problem in groups of the form G = Z_N ⋊ Z_q^s? 1) N must be a power of 2 and q must be even. 2) N must be square-free and s must equal q. 3) Every prime dividing N must also divide s. 4) The group Z_q^s must be abelian and N must be odd. 5) The order of Z_N must equal the order of Z_q^s. 6) There must exist an integer t such that N divides q^t − 1. 7) N must have a prime factorization, q must be an odd prime, s a positive integer, and there must exist the smallest positive integer t with q^t = 1, along with specific divisibility conditions on the prime factors of N and q.

✓ Correct Answer:

The correct answer is 7) N must have a prime factorization, q must be an odd prime, s a positive integer, and there must exist the smallest positive integer t with q^t = 1, along with specific divisibility conditions on the prime factors of N and q..

📚 Reference Text:

the restriction of ftoB, which hides H2inB. The solution of the HSP in the groups AandBwith functions f1andf2determines generators for the subgroups H1 andH2, respectively. The group Ais abelian and the group Bwas addressed in [5,6] and recently generalized by [4]. Therefore, we obtain the following result: Theorem 3.1. LetNbe a positive integer with prime factorization pr1 1:::prnn,qan odd prime such that q6=piandsa positive integer. De ne the semi-direct product group G=ZNo Zqsfor an 2Z N. Lett2f1;:::;sgbe the smallest positive integer such that qt= 1. Let us assume that there exist a 1knsuch thatqtjpk1andq-pi1 for alli6=k. Then there exists an ecient quantum algorithm that solves the HSP in the semi-direct product groups ZNo Zqs. 4 Conclusion We have addressed the HSP on the semi-direct product groups G=ZNoZqswhereN is factorized as N=pr1 1:::prnnand given a 1ts, there exists a 1knsuch thatqt dividespk1q-pi1 for alli6=k. By employing an isomorphism between ZNoZqsandProceeding Series of the Brazilian Society of Applied and Computational Mathematics, Vol. 3, N. 2, 2015. DOI: 10.5540/03.2015.003.02.0045 020045-4 © 2015 SBMAC 5 the direct product of Zpro Zqswith cyclic groups we have shown that the HSP can be reduced to similar HSPs the solutions of which are already known. This provides a new ecient solution for the HSP on G. Acknowledgements The authors would like to tank the CNPq and Faperj for nancial support and the anonymous reviewers for their valuable comments and suggestions to improve the manuscript. References [1] R. Beals, Quantum computation of Fourier transforms over symmetric groups, Proc. 29th ACM Symp. on Theory of Computing, pages 48{53, New York, (1997). [2] D. P. Chi, J. S. Kim and S. Lee, Quantum algorithms for the hidden subgroup problem on some semi-direct product groups by reduction to Abelian cases, Physics Letters A, pages 114{116, (2006). [3] A. M. Childs and W. van Dam, Quantum algorithms for algebraic problems, Rev. Mod. Phys., 82(1): 1{52, (2010). [4] W. van Dam and S. Dey, Hidden Subgroup Quantum Algorithms for a Class of Semi-Direct Product Groups, Proc. of 9th Conference on the Theory of Quantum Computation, Communication and Cryptography, TQC14, pages 110{117, (2014). [5] D. N. Gon calves, R. Portugal, and C. M. M. Cosme, Solutions to the hidden sub- group problem on some metacylic groups, Proc. 4th Worshop on Theory of Quantum Computation, Communication and Cryptography, LNCS, Springer-Verlag, (2009). [6] D. N. Goncalves and R. Portugal, Solution to the Hidden Subgroup Problem for a Class of Noncommutative Groups, Quantum Physics, Abstract quant-ph/1104.1361, (2011). [7] C. Lomont, The Hidden Subgroup Problem - Review and Open Problems, Quantum Physics, Abstract quant-ph/0411037, (2004). [8] M. Mosca, Quantum algorithms, Encyclopedia of Complexity and Systems Science, pages 7088{7118, (2009). [9] O. Regev, Quantum Computation and Lattice Problems, SIAM Journal on Comput- ing, 33(3):738{760, (2004). [10] D. R. Simon, On the Power of Quantum Computation, Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pages 116{123, (1994).Proceeding Series of the Brazilian Society of Applied and Computational Mathematics, Vol. 3, N. 2, 2015. DOI: 10.5540/03.2015.003.02.0045 020045-5 © 2015

Question 961 multiple-choice

In quantum information theory, the Quantum Fourier Transform (QFT) and Schmidt decomposition are fundamental tools for analyzing entanglement and optimizing quantum algorithms. Understanding how truncation errors behave as quantum systems scale is crucial for ensuring the reliability of these methods. In the context of QFT-based quantum systems, which property ensures that the maximal truncation error from Schmidt decomposition remains uniform across different system sizes and partition positions as the system size becomes large? 1) The dependence of error bounds on the specific arrangement of basis vectors 2) The growth of Schmidt coefficients with increasing system size 3) The variation of truncation error with different phase arrangements 4) The dominance of off-diagonal elements in the operator's decomposition 5) The lack of normalization in the associated eigenvectors 6) The sensitivity of error bounds to the order of bits 7) The symmetry of the QFT under bit reversal and transposition, leading to identical Schmidt coefficients at cuts j and n−j

✓ Correct Answer:

The correct answer is 7) The symmetry of the QFT under bit reversal and transposition, leading to identical Schmidt coefficients at cuts j and n−j.

📚 Reference Text:

bits represented in the reverse order, i.e., Rj|uk n,j/angbracketright. This suggests the operator Schmidt decomposi- tion of /Omega1n,jis /Omega1n,j=√ 2nχ−1/summationdisplay k=0σk n,jdiag(Rj|uk n,j/angbracketright)⊗diag(|uk n,n−j/angbracketright), (J10) where diag (|v/angbracketright)denotes the diagonal matrix with diagonal elements generated from the vector |v/angbracketright. The error can be then calculated as follows: ˜emax=sup |φ/angbracketright/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble√ 2n/summationdisplay k≥χσk n,j(Ak n,j⊗Bk n,j)|φ/angbracketright/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 =2n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay k≥χσk n,jdiag(Rj|uk n,j/angbracketright)⊗diag(|uk n,n−j/angbracketright)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 ≤2n⎛ ⎝/summationdisplay k≥χσk n,jmax x(|ωx n,jvk,x n,j|)max x(|ωx n,n−jvk,x n,n−j|)⎞ ⎠2 ≤2n⎛ ⎝/summationdisplay k≥χ˜σk jmax x(|vk,x n,j|)max x(|vk,x n,n−j|)⎞ ⎠2 , (J11) where we can see the extra phases and reordering are also irrelevant to the error bounds. We now consider the behav- ior of the error in the limit n→∞ , where |vk n,j/angbracketrightapproaches the DPSSs |˜vk j/angbracketright, 2j−1/summationdisplay y=0sin(π(x−y)/2j) π(x−y)˜vk,y j=˜λk j˜vk,y j, (J12) while |vk n,n−j/angbracketrightapproaches some continuous functions by the principle of Riemann sum, /integraldisplay1/2j 0sin/parenleftbig 2jπ(˜x−˜y)/parenrightbig sin(π(˜x−˜y))fk j(˜y)d˜y=˜λk jfk j(˜x), (J13) where ˜xand˜yare the continuous representation of x/2n andy/2n,d˜yis 1/2nin the limit, and the continuous func- tion fk j(˜x)is√ 2nvk,x n,n−jin the limit. The extra√ 2ninfk j ensures it has a nonvanishing inner product with itself over the range [0, 1 /2j], i.e., /integraldisplay1/2j 0/parenleftBig fk j(˜x)/parenrightBig∗ fk j(˜x)d˜x =lim n→∞2n−j−1/summationdisplay x=0√ 2n/parenleftBig vk,x n,n−j/parenrightBig∗√ 2nvk,x n,n−j1 2n 040318-28 QUANTUM FOURIER TRANSFORM... PRX QUANTUM 4,040318 (2023) =lim n→∞2n−j−1/summationdisplay x=0/parenleftBig vk,x n,n−j/parenrightBig∗ vk,x n,n−j, =1, (J14) and thus fk jhas a well-deﬁned value at every point in the region. We saw in Appendix Bthat fk jbelong to a spe- cial kind of function known as DPSWFs, with coordinates shifted by W=1/2j+1. DPSWFs are known to have the same eigenvalues as their corresponding DPSSs [ 14], and this property is illustrated here as |˜vk j/angbracketrightand fk jarise from the same partition j. Therefore, as nincreases, the maxi- mum of |vk,x n,j|and√ 2n|vk,x n,n−j|will approach the maximum of their limiting cases, which will be dependent only on j: /braceleftBigg max x(|vk,x n,j|)→max x(|˜vk,x j|),√ 2nmax x(|vk,x n,n−j|)→max ˜x(|fk j(˜x)|).(J15) Therefore, the upper bound on the maximal error for truncation is independent of n, i.e., emax(Dχ n,j)≤⎛ ⎝/summationdisplay k≥χ˜σk jmax x(|˜vk,x j|)max ˜x(|fk j(˜x)|)⎞ ⎠2 . (J16) We now use induction to show that once the upper bound is independent of n, it is also independent of j.F i r s t ,w e note that Qnis symmetric under bit reversal followed by ...j=1n=2 n=3 n=4 n=5 ... j=2 j=3 j=4 ...n=6 j=5 FIG. 9. Diagrammatic illustration on how emax(Dχ n,j)are related for diﬀerent nand j. The horizontal lines denote that the error on bond jhas an upper bound independent of n,w h i c hi s evident from Eq. (J16) . The diagonal lines arise from the sym- metry of the QFT, i.e., the Schmidt coeﬃcients are identical atcuts jand n−j. Therefore, one can form a path between two bonds at any nandj; thus, the upper bound on e max(Dχ n,j)is also independent of j.transposition, i.e., (R† nQnRn)T=Qn, which means we can get exactly the same argument by replacing jwith n−j. This means ˜emaxonDχ n,jshares the same bound with

Question 962 multiple-choice

Quantum computation leverages fundamental differences from classical computation through the use of qubits, superposition, and unitary operations. Understanding the roles of various quantum operators and mathematical constructs is essential for building scalable quantum algorithms. Which mathematical operation allows the extension of single-qubit gates to multi-qubit systems, thereby constructing the computational basis for quantum circuits? 1) Direct sum 2) Cross product 3) Scalar multiplication 4) Hadamard product 5) Matrix inversion 6) Tensor (Kronecker) product 7) Outer difference

✓ Correct Answer:

The correct answer is 6) Tensor (Kronecker) product.

📚 Reference Text:

αjj2and βjj2. This is in stark contrast to classical digital computation whose operations must always exclusively evaluate to a value of either 0 or 1. Quantum computation allowsan algorithm to simultaneously visit bothlogical states ∣0iand∣1iof a single qubit. If nqubits (i.e. multiple qubits) are applied, then a quantum system, in principal, has the potential to simultaneously visit 2 nlogical states (again, with their associated probabilities). This exponential computational capacity is the source of quantum parallelism. However, there is a catch. Only when some observable is measured can we ascertain the current logical state of the system. Hence, quantum computersrequire large samples of measurements in order to build up the statistics necessaryto determine the outcome of any given algorithm. 2.1 Unitary operators The time evolution operator Uassociated with a quantum system must be unitary meaning that U †U¼I (7) where U†is the conjugate transpose of U. A major implication of this require- ment is that the forward time system evolution must (at least mathematically) be 2Real Perspective of Fourier Transforms and Current Developments in Superconductivity reversible. This requirement, in turn, constrains computations that are implemented by quantum operators to be reversible . Therefore, logical operations such as AND, OR, and XOR (exclusive-or) would not be a quantum mechanical possibility unless some additional input information were to be pre-served. This is because, in the absence of information about the input, measuringthe output of these operations is not enough to ascertain the values of the inputs.Hence, these boolean processes, by themselves, are not reversible. However, thereis a theory of reversible computation that can augment these logical operations sothat input information is recoverable. Furthermore, much thought has gone intophrasing reversible computation in the context of unitary operators. Given the discussion so far, it is appropriate to give a short list of standard single qubit operators: I¼10 01/C20/C21 ,H¼ 1ﬃﬃﬃ 2p11 1/C01/C20/C21 ,Rϕ¼10 0eiϕ/C20/C21 (8) X¼01 10/C20/C21 ,Y¼0/C0i i0/C20/C21 ,Z¼10 0/C01/C20/C21 (9) The reader can check that these are all unitary. As a simple example of how to apply such operators, consider the action of Xon the basis vector ∣0i X∣0i¼01 10/C20/C21 ∣0i¼01 10/C20/C211 0/C20/C21 ¼0 1/C20/C21 ¼∣1i (10) where ∣0iand∣1iare ‘swapped ’, indicating a form of logical inversion. His a Hadamard transform (i.e. a DFT for a sequence of length N=2). X,YandZare Pauli matrices. Rϕis a generalization of Z¼RπandI¼R0. While these are single quhit operators, the next sections discuss how they can be extended to the multiple qubit case. Amazingly, this set of quantum operators can be applied to devise some verypowerful quantum algorithms (e.g. QFT computation) [3, 10]. 2.2 Tensor product (Kronecker product) The Kronecker product of an m/C2nmatrix Awith a p/C2qmatrix Bis defined to be A⊗B¼a 11Ba 12B⋯a1nB a21Ba 22B⋯a2nB ⋮⋮ ⋮ ⋮ am1Ba m2B⋯amnB2 66666643 7777775: (11) Furthermore, assuming the dimensions are compatible for matrix multiplication, the following identity often proves useful A⊗B ðÞ C⊗D ðÞ ¼ ACðÞ⊗BDðÞ (12) for matrices A,B,C,D. The computational basis can be extended to any number of qubits using the tensor product. For example, if two

Question 963 multiple-choice

In quantum signal processing and time-frequency analysis, Discrete Prolate Spheroidal Sequences (DPSSs) and their periodic counterparts (PDPSSs) are important for optimal spectral concentration and energy localization. Understanding the spectral properties and eigenvalue bounds of the associated matrices is crucial for error analysis and algorithm design. Which statement most accurately describes the relationship between the eigenvalues of the finite periodic DPSS (PDPSS) matrix Tₙ,ⱼ and those of the infinite-size DPSS matrix ˜Tⱼ for all matrix sizes n ≥ 0, assuming the eigenvalues form a smoothly decaying curve? 1) For all k, λₖₙ,ⱼ is strictly greater than ˜λₖⱼ. 2) The largest eigenvalue of Tₙ,ⱼ increases as n increases and always exceeds the largest eigenvalue of ˜Tⱼ. 3) No upper bound on the nonleading eigenvalues of ˜Tⱼ applies to Tₙ,ⱼ for finite n. 4) The smallest eigenvalue of Tₙ,ⱼ is always larger than the smallest eigenvalue of ˜Tⱼ. 5) All eigenvalues of Tₙ,ⱼ are equal to those of ˜Tⱼ for any n. 6) Any upper bound on nonleading eigenvalues of ˜Tⱼ applies directly to those of Tₙ,ⱼ for all n ≥ 0, with λ₀ₙ,ⱼ ≥ ˜λ₀ⱼ and λ₁ₙ,ⱼ ≤ ˜λ₁ⱼ. 7) The eigenvalues of Tₙ,ⱼ are unrelated to those of ˜Tⱼ due to finite-size effects.

✓ Correct Answer:

The correct answer is 6) Any upper bound on nonleading eigenvalues of ˜Tⱼ applies directly to those of Tₙ,ⱼ for all n ≥ 0, with λ₀ₙ,ⱼ ≥ ˜λ₀ⱼ and λ₁ₙ,ⱼ ≤ ˜λ₁ⱼ..

📚 Reference Text:

PDPSSS In this appendix, we prove that in the context of the QFT, with a mild assumption, the nondominant eigenvalues of DPSSs are upper bounds to those of PDPSSs. Consider the PDPSS matrix arising from the QFT Tn,j[x,y]=1 2nsin(π(x−y)/2j) sin(π(x−y)/2n), (D1) where [ x,y] denotes the (x+1)th row and (y+1)th col- umn and x,y∈[0, 2j−1]. We argue that as nincreases, the ﬁrst (largest) eigenvalue of Tn,jwill decrease and all other eigenvalues will increase. Therefore, if we consider the DPSS matrix arising from an inﬁnite-sized QFT ˜Tj[x,y]=lim n→∞Tn,j[x,y]=sin(π(x−y)/2j) π(x−y), (D2) any upper bound on the nonleading eigenvalues ( ˜λk j)o f˜Tj can be directly applied to those of Tn,j(λk n,j) for all n≥0. The assumption we make is that both Tn,jand˜Tj have the set of eigenvalues for k≥2NW=1 forming a smoothly decaying curve, which is evident from numerical simulations and the early analysis of the spectral concen-tration problem [ 14,28]. Together with the majorization condition in Theorem 1, this implies that there exists a number k 0such that λk n,j≥˜λk jfork0 for all xandy; thus, by the Perron-Frobenius theorem, the ﬁrst eigenvector v0 n,jcan be chosen to have strictly positive components. This means that dλ0 n,j dn=/parenleftBig v0 n,j/parenrightBigTdTn,j dnv0 n,j≤0. (D5) Thus, as nincreases, the ﬁrst eigenvalue cannot increase, resulting in the inequality λ0 n,j≥˜λ0 j. We next prove λ1 n,j≤˜λ1 j. To do this, we use three facts. Firstly, we observe that Tn,jis a Toeplitz matrix, i.e., Tn,j[x,y]=Tn,j[x+1,y+1] for x,y∈[0, 2j−2], and it is symmetric. This suggests that all the eigenvectors are either symmetric or antisymmetric, depending on the par- ity: vk n,j[x]=⎧ ⎨ ⎩vk n,j[2j−1−x]i f k=0, 2, 4 ..., −vk n,j[2j−1−x]i f k=1, 3, 5 ...,(D6) where v[x] denotes the (x+1)th element of the vector v. Therefore, application of any symmetric Toeplitz matrix onv1 n,jmust also produce an antisymmetric vector. Sec- ondly, it is known that vk n,jhas exactly knodes, i.e., sign(vk n,j[x])=−sign(vk n,j[x+1])occurs ktimes among x∈[0, 2j−2], where sign (v[x])denotes the sign of the vector element v[x][28]. Therefore, we can choose v1 n,jto be non-negative in the ﬁrst half and nonpositive in the sec-ond half. Thirdly, we note that using Eq. (D4) , one can verify that for the range x,y∈[0, 2 j−1−1], dTn,j dn[x,2j−1−y]≤dTn,j dn[x,y]≤0. (D7) Combing all three facts, we can conclude that the ﬁrst half of (dTn,j/dn·v1 n,j)is non-negative, i.e., for 040318-18

Question 964 multiple-choice

In quantum algorithms for non-Abelian groups, the Clebsch-Gordan transform is used to decompose tensor products of irreducible representations, which is critical for problems like the hidden subgroup conjugacy problem (HSCP). The Heisenberg group over a finite field, Hp, provides a key example due to its rich subgroup and representation structure. Which aspect of solving the hidden subgroup conjugacy problem (HSCP) for the Heisenberg group Hp specifically requires finding an appropriate measurement basis for the multiplicity space after the Clebsch-Gordan transform? 1) To determine the group operation rules for Hp 2) To distinguish conjugacy classes of hidden subgroups using quantum measurements 3) To efficiently enumerate all normal subgroups of Hp 4) To compute the explicit formulas for one-dimensional irreducible representations 5) To implement the inverse of the Clebsch-Gordan transform within a quantum circuit 6) To discard the entire nontrivial action space regardless of measurement outcomes 7) To generate upper triangular matrices over the finite field Fp

✓ Correct Answer:

The correct answer is 2) To distinguish conjugacy classes of hidden subgroups using quantum measurements.

📚 Reference Text:

irrep labels, and perform a Clebsch-Gordan transform. 5. Measure the irrep label after the Clebsch-Gordan transform, µ. 6. ThrowawaythespacewheretheirrepactsnontriviallyintheClebs ch-Gordantransform, leavingonlythe spacearisingfromthe multiplicity ofthe µth irrep. Fromthis remaining state, a measurement should be performed which solves the hidden subgroup conjugacy problem, alongwiththe classicaldataresultingfromthe irrepmeasu rements, µ1,...,µ m andµ. The above protocol can be used without a loss of computational po wer for distinguishing hid- den subgroup conjugacy states. Notice that the ﬁnal step is a st ep which requires a judicious choice of measurement. Since the Clebsch-Gordan transform doe s not yield a canonical basis for the multiplicity space, this implies that all of the diﬃculty of distingu ishing HSCP states lies in determining a basis for this measurement. Further, getting int o this basis should be accomplished using an eﬃcient quantum circuit. At this point one might wonder whether it is every possible to make such a judicious choice. It is precisely the go al of this paper to show that such a choice is possible and can lead to eﬃcient quantum algorith ms for the HSCP. 6 The Heisenberg Group We now turn from our general discussion of the HSP and the HSCP fo r any ﬁnite group Gto a discussion where Gis a particular group, the Heisenberg group. In this section we collec t much of the relevant information about this group, its subgroups, and its irreps. We are interested in the Heisenberg group Hp=Zp⋊ Z2 pwherepis prime. This is the group of upper right triangular 3 ×3 matrices with multiplication and addition over the ﬁeld Fp. We denote elements of this group by a three-tuple of numbers, ( x,y,z), withx,y,z∈Zp. The multiplication rule for this group is then given by (x,y,z)(x′,y′,z′) = (x+x′,y+y′+xz′,z+z′). (51) The inverse of an element ( x,y,z) of the Heisenberg group is the element ( −x,−y+xz,−z). The representation theory and subgroup structure of the Heise nberg group is easily de- duced. We will use (up to a change in the way we write group elements) the notation from the paper of Radhakrishnan, R¨ otteler and Sen[41]. The Heisenber g group has p2diﬀerent one dimensional irreps and p−1 diﬀerent pdimensional irreps. The one dimensional irreps are given by χa,b((x,y,z)) =ωax+bz, (52) wherea,b∈Zpandω= exp/parenleftBig 2πi p/parenrightBig . Thepdimensional irreps are given by σk((x,y,z)) =ωky/summationdisplay r∈Zpωkzr|r+x/an}bracketri}ht/an}bracketle{tr|, (53) 18How a Clebsch-Gordan Transform Helps to Solve ... wherek∈Z∗ p. For future reference, the characters of the pdimensional irreps are χk((x,y,z)) =δx,0δz,0ωkyp. (54) The Heisenberggrouphassubgroupsoffourdiﬀerentorders, p3,p2,p, and1. InTable1we catalog these subgroups and describe their generators. The ﬁrs t two groups in Table 1 are the self-explanatory group itself and the trivial subgroup. The Nαgroups are normal subgroups ofHp,Nα⊳Hp. These groups are isomorphic to the Abelian group Zp×Zp. TheAi,j subgroups are Abelian subgroups which are not normal in Hp. These groups are subgroups of the appropriate Nisubgroups. In particular they are normal subgroups Aα,j⊳Nαand Aα,j/⋉⋊tsubseteqlNα′ifi/ne}ati≀nslash=i′. Finally, Cis the center of Hp. Table 1. Subgroups of the Heisenberg group, Hp. Subgroup Generators Label Range Order Hp(1,0,0),(0,1,0),(0,0,1) - p3 T (0,0,0)

Question 965 multiple-choice

Quantum circuits utilize gates such as CNOT and SWAP to manipulate multi-qubit states, with controlled unitaries playing a crucial role in complex operations like the Quantum Fourier Transform. Understanding the mathematical structure and decomposition of these gates is fundamental to quantum algorithm design. Which statement best describes the matrix form of the CNOT (Controlled-NOT) gate acting on two qubits, where the first qubit is the control and the second is the target? 1) The CNOT gate is represented by X⊗I, applying an X gate to both qubits simultaneously. 2) The matrix form of CNOT is I⊗X, flipping the target qubit regardless of the control state. 3) The CNOT gate is expressed as E1⊗I2 + E2⊗X, where E1 and E2 are projectors onto the control qubit states, and X acts on the target qubit. 4) The CNOT gate is equivalent to a SWAP operation between the two qubits. 5) The CNOT gate is represented by U⊗V, where U and V are arbitrary single-qubit unitaries. 6) The matrix form of CNOT is I2⊗E1 + X⊗E2, applying X to the control qubit based on the target qubit state. 7) The CNOT gate is a controlled-Z gate with an additional Hadamard transformation.

✓ Correct Answer:

The correct answer is 3) The CNOT gate is expressed as E1⊗I2 + E2⊗X, where E1 and E2 are projectors onto the control qubit states, and X acts on the target qubit..

📚 Reference Text:

the input to this circ uit is a rank-1 tensor x2=x1⊗x2, the output of the circuit is simply Ux1⊗x2, and a classical simulation of the circuit has the same cost a s its executing on a quantum computer. Notice that the 2 ×2 identity matrix does not require a quantum gate. In general, if x2is a two qubit state of maximal rank 2, cfr. (26), the circuit p erforms the computation y2=α1·(Ux(1) 1)⊗x(1) 2+α2·(Ux(2) 1)⊗x(2) 2, 10 which requires two products with Uto compute classically. It follows that if an n-qubit state has rank O(2n), the quantum eﬃciency of quantum computers can lead to exponenti al speedups in comparison to classical computers. This type of diagram can be easily generalized for unitary tr ansformations applied to state vectors encoded by several qubits. In the context of the QFT, a particularly int eresting unitary transformation is the one represented by the matrices of Lemma 5. As we saw in (27), the action on the ( j+ 1)st tensor factor x(i) j+1depends on the coeﬃcients of the ﬁrst tensor factor, E1x(i) 1andE2x(i) 1. If the ﬁrst tensor factor is equal to e1, then the identity matrix is applied to x(i) j+1. Ifx(i) 1is equal toe2, then the ( j+1)st factor changes to Rj+1x(i) j+1. Consequently, this type of transformation is called a controlled unitary with the ﬁrst qubit as the controlling qubit and the ( j+1)st qubit as the target qubit. In general, x(i) 1can be in a superposition of e1ande2and the result is the linear combination (27) which can increase the tensor rank as we dis cussed in Section 5.1. The diagrammatic notation for this controlled unitary is drawn on the left in the diagra m below. Note that a solid circle is drawn on the line representing the controlling qubit. It is connected to the q ubit to be transformed via a vertical line. Lemma 5 shows the equivalence between the controlled unitaries on t he left and the right for controlled- Rgates. q1• q1R3 q2 =q2 q3R3 q3• q4 q4 Another useful controlled unitary is the controlled -NOT or CNOT unitary. As a 2-qubit operator with the ﬁrst qubit being the control and the second qubit being the ta rget, it can be written as Xc=E1⊗I2+E2⊗X, whereXis the NOT operator X=/bracketleftbigg0 1 1 0/bracketrightbigg , that mapse1toe2and vice versa. The conventional diagram for a CNOT gate in qu antum computing is • • X= One particular use of the CNOT gate is to implement the SWAP op erator deﬁned as SWAP(ψ1⊗ψ2) =ψ2⊗ψ1. This is sometimes represented by q1×q2 q2×q1 in a quantum circuit. It can be easily veriﬁed that a SWAP gate can be decomposed as the product of three CNOTS with alternating controlling qubits. As a result, the CNOT implementation of a SWAP gate can be drawn as × •• ×= • Controlled unitaries can also be deﬁned to have multiple tar get qubits. The matrix E1⊗Ik+E2⊗Ukis a controlled- Ukgate with the ﬁrst qubit as a control and

Question 966 multiple-choice

Lattice algorithms play a critical role in computational number theory and cryptography, relying on mathematical structures such as bases, dual lattices, and canonical forms to solve problems efficiently. Various classical algorithms and norms are used to ensure both correctness and practical performance. Which canonical matrix form is specifically employed to compute the index of one lattice inside another, thereby aiding in the classification of sublattices? 1) Jordan Normal Form 2) Reduced Row Echelon Form 3) Cholesky Decomposition 4) Hermite Normal Form 5) Smith Normal Form 6) QR Decomposition 7) LU Decomposition

✓ Correct Answer:

The correct answer is 5) Smith Normal Form.

📚 Reference Text:

sampler, δλ∗ 1(L) = 1/2/⌊ar⌈⌊lBM/⌊ar⌈⌊l) 4:yi←BDD(/tildewideyi,M∗,1/2/⌊ar⌈⌊lBM/⌊ar⌈⌊l) 5:end for 6:UseBuchmann-Pohst algorithm on(y1,...,y t). Compute the Hermite normal form to ﬁnd a basis (y′ 1,...,y′ m)ofL∗from(y1,...,y k). Here the vectors have exact integer coordinates in a basis of M∗. ⊲Step 2 - Classical 7: Compute the Smith normal form (SNF) of (y′ 1,...,y′ m)to obtain the exact value of[L:M]. Output (B−1)t∈1 [L:M]Matm(Z), whereB∈Matm(Z)is the matrix a basis ofL∗written in a basis of M∗. ⊲Step 3 - Classical Notation 23 For any matrix B∈GLn(C)and anyα,β∈R+/uniontext{∞}the oper- ator norm is /⌊ard⌊lB/⌊ard⌊lα,β= max{/⌊ard⌊lMv/⌊ard⌊lα| /⌊ard⌊lv/⌊ard⌊lβ≤1}. Whenαandβare not speciﬁed7,/⌊ard⌊lB/⌊ard⌊ldenotes/⌊ard⌊lB/⌊ard⌊l∞,1: /⌊ard⌊lB/⌊ard⌊l=/⌊ard⌊lB/⌊ard⌊l∞,1= max j/summationdisplay i|Bi,j|. Whenα=βwe simply write /⌊ard⌊lB/⌊ard⌊lα. Note that /⌊ard⌊lB/⌊ard⌊l∞,1≥ /⌊ard⌊lB/⌊ard⌊l2= max i/radicalBigg/summationdisplay j|Bi,j|2≥ /⌊ard⌊lB/⌊ard⌊l∞,1/√m. In this article, the complexity of the algorithms depends on O(log/⌊ard⌊lB/⌊ard⌊l∞,1) = O(log/⌊ard⌊lB/⌊ard⌊l2+logm). Sincelog/⌊ard⌊lB/⌊ard⌊l=Ω(m), this justiﬁes that we drop the indices of the operator norm. Theorem 24 a) Algorithm 1 solves Problem 1 if δλ∗ 1(M)<1/(2/⌊ard⌊lBM/⌊ard⌊l). The number of qubits is O(Qm)withQgiven in Equation (1). We prepare the proof with two lemmas. Lemma 25 (Lemma 5in [19]) We notek=α(m+mlog2R+log2(detL)), for an absolute constant α>1. 7The norm operator used in the analysis of the Buchmann-Pohst algorithm [19, Ap- pendix B] is/⌊ar⌈⌊l·/⌊ar⌈⌊l∞,1. The special case of cyclotomic ﬁelds in quantum algorithms f or unit groups 7 Let/tildewidey1,/tildewidey2,...,/tildewideykbe the ﬁrst kvectors output by a dual basis sampler. For i= 1,2,...,k putyi= CVP(/tildewideyi,L). Then for any value of the absolute constant α>2we have Prob(y1,...,y kgenerateL)≥1−cm, wherec<1is an explicitly computable constant. From now on, we ﬁx such a k, and we use it in Algorithm 1. Algorithm 2 Babai’sBDD solver. Input: /tildewidey∈Rmsuch that d(/tildewidey,M∗)<1/(2/⌊ar⌈⌊lBM/⌊ar⌈⌊l) Output: CVP(/tildewidey,M∗), as a vector and as coordinates in a basis of M∗ compute /tildewidez:=Bt M/tildewidey; roundz= (z1,...,z n) := (⌊/tildewidez1⌉,...,⌊/tildewiderzn⌉) returny:=BM∗z∈M∗andz∈Zm. Let us putδsuch thatd(/tildewidey,M∗) =δλ∗ 1(M). The following result can be seen as a reformulation of a result of Babai. Lemma 26 ([2] Eq. (4.3)) Algorithm 2 solves BDD(/tildewidey,M∗,δλ∗ 1(M))in clas- sical polynomial time when δλ∗ 1(M)<1/(2/⌊ard⌊lBM/⌊ard⌊l2), whereBMis the matrix of a basis ofM. Proof. We claim that ⌊Bt M/tildewidey⌉=Bt My′wherey′= CVP(/tildewidey,M∗). Indeed /vextenddouble/vextenddoubleBt M/tildewidey−Bt My′/vextenddouble/vextenddouble ∞≤/vextenddouble/vextenddoubleBt M/tildewidey−Bt My′/vextenddouble/vextenddouble 2≤/⌊ar⌈⌊lBM/⌊ar⌈⌊l2·/vextenddouble/vextenddouble/tildewidey−y′/vextenddouble/vextenddouble 2=/⌊ar⌈⌊lBM/⌊ar⌈⌊l2δλ∗ 1(M)<1/2. Since⌊Bt M/tildewidey⌉=Bt My′, by Lemma 21, y=BM∗= ((BM)t)−1(Bt My′) =y′, so the algorithm is correct. Proof (Proof of Theorem 24.a)). We structure the proof in several steps. step 1. We apply Lemma 25 to the lattice L∗. Then, with probability 1−cm, the vectors y1,y2,...,y kgenerateL∗. step 2. Let us ﬁxi∈ {1,2,...,k}. By the deﬁnition of the dual sampler, with probability greater than 1−η,dist(/tildewideyi,L∗)<δλ∗ 1(M∗). Hence Lemma 26 applies and Line 4 is executed in classical polynomial time. Since,L∗⊂M∗andy′ i:= CVP(/tildewideyi,L∗)belongs to M∗. Sinced(/tildewideyi,L∗)< λ∗ 1(M)/2,y′ iis the closest vector of M∗to/tildewideyi, or equivalently y′ i=yi. We summarize by yi= CVP(/tildewidey,L). step 3. Algorithm 2 can output zin addition to y. This means that the vectors yi∈L∗are given by their integer coordinates with respect to a basi s ofM∗. In Line 6, one computes the HNF [16, Sec 2.4.2] by manipulatin g integer co- eﬃcients only. This is done in classical polynomial time wit hout loss of precision. step 4.

Question 967 multiple-choice

In spectral graph theory and quantum computing, group representation theory is leveraged to simplify the analysis of quantum Hamiltonians associated with graphs. Irreducible representations (irreps) of the symmetric group play a crucial role in decomposing the spectrum and computational complexity of these Hamiltonians. Which of the following correctly expresses the dimension of the [n−k, k] irreducible representation of the symmetric group S_n using combinatorial quantities? 1) n! / (k! × (n−k)!) 2) (n−2k+1) × n[k] / (k! × (n−k+1)) 3) n[k] / (n−k+1)! 4) (n−k+1) × C(n, k) / (n−2k+1) 5) (n−2k+1) × k! × C(n, k) 6) (n−2k+1) / (n−k+1) × n[k] / k! = (n−2k+1) / (n−k+1) × C(n, k) 7) n[k] × k! × C(n, k)

✓ Correct Answer:

The correct answer is 6) (n−2k+1) / (n−k+1) × n[k] / k! = (n−2k+1) / (n−k+1) × C(n, k).

📚 Reference Text:

annvertex graph with edge set E(G)and edge weights wij. Then define the QMC irrep Hamiltonian Hλ Gby Hλ G=ρλ /summationdisplay (i,j)∈E(G)2wij(I−(i j))  (2.23) We will frequently use the phrase “all two row irrep Hamiltonians of G” to refer to the set of irrep Hamiltonians {H[n−k,k] G: 1≤k≤⌊n/2⌋}. (2.24) Now we state a straightforward corollary of Theorem 2.7 which makes the significance of the two row irrep Hamiltonians clear. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 13 Corollary 2.9. The spectrum of the QMC Hamiltonian of Gis given by the union of the spectra of all two row irrep Hamiltonians of G. That is eigs(HG) =⌊n 2⌋/uniondisplay k=0eigs(H[n−k,k] G) (2.25) In particular, we have eigmax(HG) =⌊n 2⌋ max k=0/parenleftig eigmax(H[n−k,k] G)/parenrightig (2.26) Proof. Immediate from Theorem 2.7. Corollary 2.9 suggests an immediate technique for computing the max eigenvalue of a QMC Hamiltonian – rather than computing the maximum eigenvalue of the matrix HG directly we can instead compute the maximum eigenvalue of each of the H[n−k,k] Gmatri- ces. When combined with brute force computation this yields a modest computational advantage, since the dimension of each H[n−k,k] Gmatrix is smaller than that of HG.3In the next lemma we calculate these dimensions. It should be pointed out that the dimension of these irreps still scales exponentially with n, meaning this approach does not give a polynomial time algorithm for computing the max eigenvalue of HG. In Section 2.4.1 we give the results of some simple experiments regarding which irreps provide the maximizing eigenvalue for QMC Hamiltonians. We calculate the dimension of each irrep by computing the character (or trace) of the identity element in the irrep. When working with the symmetric group, we will use e∈Sndenote the identity and χ[n−k,k]:Sn→Cdenote the character of ρ[n−k,k], so χ[n−k,k](π) := Tr/parenleftig ρ[n−k,k](π)/parenrightig for allπ∈Sn. Then, in particular, χ[n−k,k](e) = Tr/parenleftig ρ[n−k,k](e)/parenrightig (2.27) is the dimension of the irrep ρ[n−k,k]ofSn. Lemma 2.10. χ[n−k,k](e) =n−2k+ 1 n−k+ 1n[k] k!=n−2k+ 1 n−k+ 1/parenleftigg n k/parenrightigg . (2.28) Here we used the falling factorial notation n[k], defined by n[k]:=n! (n−k)!=n(n−1)···(n−k+ 1), (2.29) andn[0]= 0[k]= 1by definition. 3This approach also requires computing the form of the H[n−k,k] G matrix in some basis. This can be done, for example, by explicitly computing matrix representations of transpositions in the [ n−k, k] irrep via their action on the Specht basis (see [ Jam06 ] for details), then summing these matrices to obtain H[n−k,k] G . While involved, all these operations have runtime polynomial in the dimension of the [ n−k, k] irrep. In practice, these computations can be also be performed using a convenient software package, for example [Fon20], which was the approach used in this paper. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 14 Proof. The dimension of the irrep of the partition [ n−k,k] is equal to the trace, or character, of the identity element in this irrep, that is, χ[n−k,k](e). We compute this quantity using the well-known hook length formula. Straightforward

Question 968 multiple-choice

Loop quantum gravity (LQG) is a background-independent approach to quantizing general relativity, where quantum states of space are represented by spin networks. Mathematical tools like recoupling theory and the Temperley-Lieb algebra play essential roles in the computation of quantum geometric properties. Which property of spin network states, particularly in trivalent expansions, is ensured by the introduction of a specific scalar product in loop quantum gravity, allowing clear probabilistic interpretation and symmetry of geometric operators? 1) They exhibit entanglement with all possible configurations. 2) Their eigenvalues are always degenerate. 3) They become invariant under diffeomorphisms. 4) They form an orthonormal basis of the quantum state space. 5) Their edges correspond exclusively to fermionic representations. 6) They guarantee non-locality of area operators. 7) Their nodes are restricted to four-valent intersections.

✓ Correct Answer:

The correct answer is 4) They form an orthonormal basis of the quantum state space..

📚 Reference Text:

Title: Geometry eigenvalues and the scalar product from recoupling theory in loop quantum gravity. Year: 1996 Paper ID: 8ac718ac78e89cdb06826cff351c7ec510abd7ef Source: semantic-scholar URL: https://www.semanticscholar.org/paper/8ac718ac78e89cdb06826cff351c7ec510abd7ef Abstract: We summarize the basics of the loop representation of quantum gravity and describe the main aspects of the formalism, including its latest developments, in a reorganized and consistent form. Recoupling theory, in its graphical tangle-theoretic Temperley-Lieb formulation, provides a powerful calculation tool in this context. We describe its application to the loop representation in detail. Using recoupling theory, we derive general expressions for the spectrum of the quantum area and the quantum volume operators. We compute several volume eigenvalues explicitly. We introduce a scalar product with respect to which area and volume are symmetric operators, and (the trivalent expansions of) the spin network states are orthonormal.

Question 969 multiple-choice

Moufang permutations are specialized group permutations related to biadditive mappings and play a role in the study of finite Moufang loops, particularly those constructed from abelian group structures. The classification of such permutations depends critically on the structure of the underlying group, especially its components of prime power order. For a finite abelian group X, which condition on its 2-primary component ensures the existence of Moufang permutations that are not automorphisms? 1) The 2-primary component is trivial. 2) The 2-primary component is cyclic of order four. 3) The 2-primary component is non-cyclic and of order greater than four. 4) The 2-primary component is of odd order. 5) The 2-primary component is a direct sum of two cyclic groups of order two. 6) The 2-primary component is a cyclic group of prime order. 7) The 2-primary component is isomorphic to the integers modulo 2.

✓ Correct Answer:

The correct answer is 3) The 2-primary component is non-cyclic and of order greater than four..

📚 Reference Text:

Title: Moufang permutations over small abelian groups Year: 2025 Paper ID: 69e1b3b6f03a449d724f5432416a8c06c48ebf42 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/69e1b3b6f03a449d724f5432416a8c06c48ebf42 Abstract: . Moufang permutations are certain permutations on an abelian group X that differ from an automorphism of X by a symmetric alternating biadditive mapping. It is known that every finite split abelian-by-cyclic 3-divisible Moufang loop is obtained from a Moufang per-mutation of the abelian normal subgroup. In this paper we investigate Moufang permutations for small abelian groups. We prove that a finite abelian group X possesses non-automorphic Moufang permutations if and only if the 2-primary component of X is of order more than four and is not cyclic. The automorphism group of X acts by conjugation on the set of Moufang permutations of X and the orbits of this action provide a partial answer to the corresponding isomorphism problem. We explicitly find all Moufang permutations for small abelian groups, including small elementary abelian 2-groups.

Question 970 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) leverage the mathematical structure of finite groups, representation theory, and quantum measurements to efficiently determine hidden subgroups. The quantum Fourier transform and associated measurement probabilities play a crucial role in extracting information from quantum states prepared during these algorithms. In the context of quantum algorithms for the Hidden Subgroup Problem over a finite group G, which formula gives the probability of observing an irreducible representation ρ upon measurement after applying the quantum Fourier transform, when the hidden subgroup H is trivial? 1) dρ / |G| 2) (1/|H|) Σh∈H χρ(h) 3) dρ |H| / |G| 4) rρ / dρ 5) dρ² / |G| 6) |G| / dρ² 7) |H| / dρ

✓ Correct Answer:

The correct answer is 5) dρ² / |G|.

📚 Reference Text:

3]) that QFTGisaunitary transformation in C|G|. Forasubgroup H≤Gandρ∈/hatwideG,deﬁneρ(H) :=1 |H|/summationtext h∈Hρ(h). ItfollowsfromSchur’slemma(see e.g.[Ser77,Chapter2,Proposition4])that ρ(H)isanorthogonal projectiontothesubspaceof Vρconsisting of vectors that are point-wise ﬁxed by every ρ(h),h∈H. Deﬁnerρ(H) :=rank(ρ(H)); thenrρ(H) = 1 |H|/summationtext h∈Hχρ(h),whereχρdenotesthecharacter of ρ. Noticethat rρ(H) =rρ∗(H). Foranysubset S≤G we deﬁne |S/a\}bracketri}ht:=1√ |S|/summationtext s∈S|s/a\}bracketri}htto be the uniform superposition over the elements of S. Thestandard methodof attacking theHSPin Gusing coset states [GSVV04]starts byformingtheuniform su perposition 1√ |G|/summationtext g∈G|g/a\}bracketri}ht|0/a\}bracketri}ht. It then queries fto get the superposition1√ |G|/summationtext g∈G|g/a\}bracketri}ht|f(g)/a\}bracketri}ht. Ignoring the second register thereduced stateontheﬁrstregister becomesthed ensity matrix σG H=1 |G|/summationtext g∈G|gH/a\}bracketri}ht/a\}bracketle{tgH|, thatis thereduced stateisauniform mixture overallleft coset sta tes ofHinG. Itcanbeeasily seenthat applying QFTGtoσG Hgivesusthedensitymatrix|H| |G|/circleplustext ρ∈/hatwideG/circleplustextdρ i=1|ρ,i/a\}bracketri}ht/a\}bracketle{tρ,i|⊗ρ∗(H),whereρ∗(H)operatesonthe space of column indices of ρ. When measuring this state, we obtain an irrep ρwith probabilitydρ|H|rρ(H) |G|. Conditioned on measuring ρwe obtain a uniform distribution 1/dρon the row indices. The reduced state on the space of column indices after having observed an irrep ρand a row index iis then given by the state 5 ρ∗(H)/rρ(H),andabasictaskfor ahidden subgroup ﬁndingalgorithm isho wtoextract information about Hfrom it. If the the hidden subgroup is the trivial subgroup {1}, the probability of measuring ρis given by the so-called Plancherel distribution P(ρ) :=d2 ρ |G|. This distribution will be useful to us later on in the proof o f the maintheorem. POVMsarethemost general waytoobtain classical informati on from quantum states [NC00]. Theele- mentsof aPOVM MinCnare positive operators Ei≥0whichhave tosatisfy the completeness condition/summationtext iEi=11n. If the state of the quantum system is given by the density mat rixσ, then the probability pito observe outcome labeled iis given bythe Bornrule pi= Tr(σEi). Thefollowing observation iscrucial for the HSP case: since the states σG Hare simultaneously block diagonal in the Fourier basis for a nyH≤G, the elements of any POVM Moperating on these states can without loss of generality be a ssumed to have thesameblock structure. Fromthis itisclear that anymeasu rement toidentify Hwithout lossofgenerality ﬁrst applies the quantum Fourier transform QFTGtoσG H, measures the name ρof an irrep, the index iof a row,and then measures thereduced state onthe column space o fρusing aPOVM MρinCdρ. ThisPOVM Mρmaydepend on ρbut is independent of i. Furthermore, Mρcan be assumed to be a frame, i.e., a collection Bρ:={(ab,b)}, whereb∈Cdρwith /bardblb/bardbl= 1and0≤ab≤1such that/summationtext b∈Bab|b/a\}bracketri}ht/a\}bracketle{tb|=11dρi.e. a frame is a POVM with rank one elements. Orthonormal bases are special cases of frames in which ab= 1for allb∈ Bρ. We can assume that the POVM on the column space is a frame because any POVM can be reﬁn ed to a frame such that for any quantum state, the probabilities according to the original POVM are certain sums, independent of the state measured, of probabilities according tothe frame. If the the hidden subgroup is the trivial subgroup {1}, after observing an irrep ρand a row index i, the reducedstateonthespaceofcolumnindicesof ρisthetotallymixedstate11dρ dρ. Theprobability ofobserving a vectorbin frameBρis given by the so-called natural distribution on Bρdeﬁned by N(b|ρ) :=d2 ρ |G|. This distribution will be useful tous later on inthe proof of the m aintheorem. TheabovedescriptionwasforsingleregisterquantumFouri ersampling. Fouriersamplingon kregisters can be deﬁned analogously.

Question 971 multiple-choice

Quantum metrology utilizes advanced algorithms and quantum resources to enhance measurement sensitivity, often beyond classical limits. Various magnetometer technologies differ in sensitivity, dynamical range, and suitability for small-scale applications. Which statement correctly describes both the scaling behavior and practical sensitivity achieved by quantum phase estimation algorithms using a single transmon qubit as a magnetometer? 1) They achieve the same scaling as classical techniques and are limited by thermal noise in small volumes. 2) Their sensitivity is highest at short sensing times and decreases with increased quantum coherence. 3) They achieve atomic spatial resolution and sensitivity in the nanotesla range without the need for quantum coherence. 4) Their scaling exponent remains at 1/2, matching the standard quantum limit with no improvement from quantum effects. 5) They exhibit scaling exponents above 1/2, surpass the classical shot-noise limit, and achieve flux sensitivity near 6×10⁻⁶ Φ₀ Hz⁻¹/² with dynamical range set by transmon coherence time. 6) Their performance is typically inferior to atomic magnetometers when volumes are reduced to the size of a transmon. 7) Their sensitivity and dynamical range are exclusively determined by step-dependent tolerances used in the Fourier algorithm.

✓ Correct Answer:

The correct answer is 5) They exhibit scaling exponents above 1/2, surpass the classical shot-noise limit, and achieve flux sensitivity near 6×10⁻⁶ Φ₀ Hz⁻¹/² with dynamical range set by transmon coherence time..

📚 Reference Text:

experimentalscaling behavior shows that both Fourier and Kitaev algorithm areindeed quantum with a scaling exponent above the standardquantum limit 1/2, see the dash-dotted cyan line connecting theKitaev (at 0.2% tolerance) dataFig. 3 Observed scaling behavior of the ﬂux resolution versus total sensing time for the three different metrological procedures, Kitaev(colored circles), Fourier (black diamonds) and standard (redcrosses). The Kitaev algortihm has been run with constant tolerancesϵ k=ϵfor each step k=1,…, 5 and for ﬁve different values of ϵas indicated by different colors. The Fourier algorithm has been performed with the step-dependent tolerances ϵk=0.182, 0.076, 0.039, 0.02, 0.01 for k=1,…, 5. We show the result of the Fourier algorithm only for the ﬁnal two steps, k=4(ﬁlled diamonds) and k =5 (empty diamonds), running the algorithm with four different starting delays, τ(s)=300, 320, 340, and 360 ns (all collapsed to the same data points). The phase estimation algorithms lag behind in precision at short times when compared to the standard procedure, but rapidly gain precision at longer times. The black solid linerepresents the scaling law for a numerical simulation of the standardprocedure with a regular passport function given by Eq. ( 1). The crossover to the red solid line is due to the irregularity of thepassport functionQuantum-enhanced magnetometry by phase estimation algorithms S Danilin et al. 6 npj Quantum Information (2018) 29 Published in partnership with The University of New South Wales DISCUSSION We have used a single transmon qubit as a magnetic- ﬂux sensor and have implemented two quantum metrological algorithms inorder to push the measurement sensitivity beyond the standardshot-noise limit. In our experiments, we utilize the coherentdynamics of the qubit as a quantum resource. We demonstrateexperimentally, on the same sensor sample, that suitably modi ﬁed Kitaev and Fourier algorithms both outperform the classical shot- noise-limited measurement procedure and approach the Heisen- berg limit. Both algorithms exhibit a similar asymptotic ﬂux- sensitivity A Φ~6×1 0−6Φ0Hz−1/2or magnetic- ﬁeld sensitivity AB ~ 20.7 pT Hz−1/2within a dynamical range ΔΦ =AΦ/C24417ﬃﬃﬃﬃﬃ ﬃ Hzp at a coherence time T2~260 ns of the qubit. Finally, we can compare the characteristics of our qubit sensor with other magnetometers. dc-SQUID sensors typically feature a 1μΦ0Hz−1/2sensitivity and a much larger dynamical range /C24106ﬃﬃﬃﬃﬃ ﬃ Hzp , see ref.27. However, conventional dc-SQUIDs are operated with a current bias close to critical, which limits theirsensitivity to 10 −8–10−6Φ0Hz−1/2, see refs.27–29, due to intrinsic thermal noise ﬂuctuations of excited quasi-particles. Atomic magnetometers30can approach a magnetic ﬁeld sensitivity ~0.1–1.0 fT Hz−1/2. However, these magnetometers measure the ﬁeld in a ﬁnite macroscopic volume ~1 cm3and their sensitivity translated to the (100 μm)3volume range of a transmon sensor reduces to 0.1 –1.0 pT Hz−1/2, with a dynamical range ~104–105 ﬃﬃﬃﬃﬃ ﬃ Hzp compatible with dc-SQUID sensors. NV centers in diamond are able to resolve magnetic ﬁelds with atomic spatial resolution and approach sensitivities ~ 6.1 nT Hz−1/2. Phase estimation algo- rithms allow one to enlarge the dynamical range of NV-sensors15–17 up to 3 × 105Hz1/2.W i t hm a g n e t i c

Question 972 multiple-choice

Quantum algorithms often rely on analyzing the structure of finite groups to efficiently solve problems such as the Hidden Subgroup Problem (HSP) and the hidden shift problem. The choice of measurement basis in representation theory can significantly impact the efficiency and reconstructibility of solutions in nonabelian group settings. Which statement best characterizes the role of an "adapted basis" in quantum algorithms for reconstructing hidden conjugates in nonabelian groups such as q-hedral and affine groups? 1) Adapted bases are only effective for abelian groups and do not improve reconstructibility in nonabelian settings. 2) Using an adapted basis increases the probability of error when identifying hidden subgroups in nonabelian groups. 3) Adapted bases allow efficient reconstruction only for subgroups with polynomial index in the parent group. 4) The adapted basis yields the same results as measuring in a random basis for hidden conjugate problems. 5) Measuring in an adapted basis is equivalent to applying the abelian Fourier transform in all group settings. 6) Measuring in an adapted basis enables full reconstructibility of hidden conjugates in certain exponentially large subgroups of q-hedral and affine nonabelian groups, outperforming random or abelian approaches. 7) Adapted bases are primarily used to reduce the dimension of group representations, not to aid in reconstructibility.

✓ Correct Answer:

The correct answer is 6) Measuring in an adapted basis enables full reconstructibility of hidden conjugates in certain exponentially large subgroups of q-hedral and affine nonabelian groups, outperforming random or abelian approaches..

📚 Reference Text:

“preferred” bases, those which give the matrices ρ(g) unusually structured or sparse form. In particular, Moore, Rockmore, and Russell [20]showed that so-called adapted bases yield highly eﬃcient algorithms for the quantum Fourier transform. 1.4. Contributions of this paper. As stated above, [13] and [10] leave an important open question, namely, whether there are cases where the strong standardmethod, with the proper choice of basis, oﬀers an advantage over a simple abeliantransform or the weak standard method. We settle this question in the aﬃrmative.Our results deal primarily with the q-hedral groups, i.e., semidirect products of the form Z q/multicloseleftZ p, where q|(p−1), and in particular the aﬃne groups Ap∼=Z∗ p/multicloseleftZ p. We begin in section 3 by focusing on full reconstructibility. We deﬁne the hidden conjugate problem (HCP) as follows: given a group G, a nonnormal subgroup H, and a function which is promised to be constant on the cosets of some conjugate Hb=bHb−1 ofH(and distinct on distinct cosets), determine the subgroup Hbby ﬁnding an element c∈Gso that Hc=Hb. We adopt the above classiﬁcation (fully, information- theoretically, quantum information-theoretically) for this problem in the natural way.Then we show that given a subgroup of suﬃciently small (but still exponentially large)index, hidden conjugates in A pare fully reconstructible (Theorem 1). This almost immediately implies that, for prime q=(p−1)/polylog( p), subgroups of the q-hedral groups Zq/multicloseleftZpare fully reconstructible (Theorem 2). Section 4 concerns itself with information-theoretic reconstructibility. We gener- alize the results of Ettinger and Høyer on the dihedral group and show that hiddenconjugates of any subgroup are information-theoretically reconstructible in the aﬃnegroups and, more generally, the q-hedral groups for all q(Theorem 3). We then show that we can identify the order, and thus the conjugacy class, of a hidden subgroup,and this implies that all subgroups of the aﬃne and q-hedral groups are information- theoretically reconstructible (Theorem 5). The results of sections 3 and 4 rely crucially on measuring the high-dimensional representations of the aﬃne and q-hedral groups in a well-chosen basis, namely an adapted basis that respects the group’s subgroup structure. We show in section 5 that we lose information-theoretic reconstructibility if we measure in a random basis instead. Speciﬁcally, we need an exponential number of measurements to distinguishconjugates of small subgroups of A p. This establishes for the ﬁrst time that the strong standard method is indeed stronger than measuring in a random basis: some basesprovide much more information about the hidden subgroup than others. For some nonabelian groups, the HSP can be solved with a “forgetful” approach, where we erase the group’s nonabelian structure and perform an abelian Fourier trans-form instead. In section 6 we show that this is not the case for the aﬃne groups.Speciﬁcally, if we treat A pas a direct product rather than a semidirect one, its con- jugate subgroups become indistinguishable. As an application, in section 7 we consider hidden shift problems. In the setting we consider, one must reconstruct a “hidden shift” s∈Zpfrom an oracle fs(x)= f(x−s), where fis any function that is constant on the

Question 973 multiple-choice

Quantum algorithms for the hidden subgroup problem (HSP) are central to computational group theory and cryptography, often relying on specialized measurement strategies and reductions to algebraic problems. The structure of the underlying group directly impacts the tractability of associated computational tasks in these algorithms. Which group structure allows the matrix sum problem arising in the pretty good measurement (PGM) approach for HSP to be solved efficiently using Buchberger’s algorithm for Gröbner bases? 1) Dihedral group D_N 2) Cyclic group Z_N 3) Abelian group Z_p 4) Metacyclic group Z_N⋊Z_p with N/p = poly(log N) 5) Direct product group Z_r × Z_p 6) Nonabelian simple group 7) Group of the form Z_r^p⋊Z_p for fixed r

✓ Correct Answer:

The correct answer is 7) Group of the form Z_r^p⋊Z_p for fixed r.

📚 Reference Text:

Barbara, Santa Barbara, CA 93106, USA 1 Recently, we have advocated an approach to solving hidden su bgroup problems based on a state estimation technique known as the pretty good measurement (pgm). The standard approach tosolvingthe hspwithaquantumcomputer(describedindetail inSection 2.1) reducestheproblem to the quantum mechanical task of distinguishing the member s of an ensemble of hidden subgroup states. For the dihedral hsp, we showed that the pgmis in fact the optimal measurement for distinguishing any number of copies of the hidden subgroup s tates, that it successfully identiﬁes the hidden subgroup with a polynomial number of copies of the states (for which we gave a tight lower bound), and that its implementation is closely relate d to an average-case subset sum problem [2]. Unfortunately, this subset sum problem appears to be di ﬃcult, so the approach did not yield an eﬃcient algorithm for the dihedral hsp. In this paper, we continue our study of the pgmas a tool for solving hidden subgroup problems. We apply the method to all groups that can be written as a semid irect product A⋊Zpof an abelian groupAand a cyclic group of prime order p. As in the case of the dihedral group, the pgmfor these groups is closely related to a certain kind of average- case algebraic problem, which we call the matrix sum problem . For some groups, the matrix sum problem can be solved eﬃcien tly, leading to an eﬃcient algorithm for the hsp. We demonstrate this for two classes of groups in Sections 5 a nd6. In Section 5, we give an eﬃcient algorithm for metacyclic gro upsZN⋊ZpwithN/p= poly(logN), generalizing the result of [21] for p-hedral groups, which requires Nprime. In Section 6, we give an eﬃcient algorithm for any group of the form Zr p⋊Zpfor ﬁxedr, including the Heisenberg group, the unique nontrivial semidirect product Z2 p⋊Zp. For the groups Zr p⋊Zp, the matrix sum problem is a system of polynomial equations over a ﬁnite ﬁeld, which can b esolved eﬃciently using Buchberger’s (classical) algorithm for computing Gr¨ obner bases [5]. Fo r the metacyclic groups, the matrix sum problem requires the calculation of discrete logarithms, w hich can be done eﬃciently using Shor’s algorithm [30]. In both cases, the algorithm uses abelian Fo urier transforms, but does not explicitly use a nonabelian Fourier transform. To simplify the pgmapproach to the hsp, we ﬁndit useful to focus on a speciﬁc set of subgroups rather than allowing the hidden subgroup to be arbitrary. Fo r the dihedral group, Ettinger and Høyer showed that it is suﬃcient to consider the case where th e hidden subgroup is either trivial or has order 2. In Section 3, we give an analogous reduction show ing that it is suﬃcient to consider subgroups that are either trivial or cyclic and of order p. In fact, this reduction alone is suﬃcient to solve the hspover some nonabelian groups, such as Zn 2≀Z2(cf. [28]) and the groups Pp,rof [17]. It is well known that the query complexity of

Question 974 multiple-choice

In quantum optics, the interaction between photons and electron spins in semiconductor quantum dots embedded in cavities enables precise control of quantum states for information processing. Key parameters such as coupling strength, cavity decay rate, and spin selection rules determine the behavior of the system. Which specific photon polarization and electron spin state combination allows a transition to the heavy-hole state |⇑〉 in a quantum dot-cavity system based on spin selection rules? 1) Photon |R〉 with electron spin |↑〉 2) Photon |L〉 with electron spin |↓〉 3) Photon |R〉 with electron spin |↓〉 4) Photon |L〉 with electron spin |⇓〉 5) Photon |L〉 with electron spin |↑〉 6) Photon |R〉 with electron spin |⇑〉 7) Photon |L〉 with electron spin |⇑〉

✓ Correct Answer:

The correct answer is 5) Photon |L〉 with electron spin |↑〉.

📚 Reference Text:

mode, and external field, respectively. Also, g is the coupling strength between X− and cavity mode while κ is the decay rate of the cavity mode. In the interaction between a photon and an electron in the QD, when the spin state of the excess electron is in the state |↑ 〉(|↓〉), the polariza- tion of the photon |L〉(|R〉) drives the hot cavity (the QD is coupled with the cavity: g ≠ 0), while the polarization of the photon |R 〉(|L〉) feels the cold cavity (the QD is uncoupled with the cavity: g = 0). Thus, for the resonant interaction ( ω ω=−c X ), the reflection coefficients Rh(ω) and R0(ω) can be calculated as follows: ωω ϕωωω γω ωκ κ ωω γω ωκ κ ω ωω ϕωωω κκ ωω κκ≡| |=−+ −− ++ −+ −+ ++ = ≡| |=−− + −+ +Rr iii g ii g R Rr ii ihotcavity: () ()exp[ ()][( )/ 2][( )/ 2/ 2] [( )/ 2][( )/ 2/ 2] (), cold cavity:( )( )exp[( )]() /2 /2 () /2 /2, (6)cc s cc s cs cshh rh2 2 00 r0 where ϕ ϕ ≡≡ RRarg[ ]( arg[ ])rh hr 00 is the phase shift in the hot (cold) cavity. Using these reflectances (|rh|, |r0|) and phase shifts (φrh, φr0) in the reflection coefficients (Rh, R0), we can obtain the reflection operator, R()ˆω, with the resonant interaction (c Xω ω=− ), as follows: ˆωω ω=| || 〉〈|⊗|↓〉〈↓|+| 〉〈|⊗|↑〉〈↑| +| || 〉〈|⊗|↑〉〈↑|+| 〉〈|⊗|↓〉〈↓|.ϕω ϕωre RR LL re RR LLR()( )( ) () () (7)i ih() 0()rh r0 Figure 2. Singly charged QD within a single-sided cavity with a photon ˆbin and ˆbout. Due to the spin selection rule, the interaction between a polarization of incident photon and a spin state of excess electron in QD can create the transition that the photon |L〉 drives the state as |↑〉 → |↑↓⇑〉 (or the photon |R〉 drives the state as |↓〉 → |↓↑⇓〉), where |↑〉 ≡ |+1/2〉, |↓〉 ≡ |−1/2〉 are the spin states of the excess electron while |⇑〉 and | ⇓〉 (Jz = +3/2, −3/2) represent heavy-hole spin states. 4 Scientific RepoRtS | (2019) 9:12440 | https://doi.org/10.1038/s41598-019-48695-z www.nature.com/scientificreports www.nature.com/scientificreports/When the experimental parameters are κs ≪ κ (small side-leakage rate) and g ≫ (κ, γ) (large coupling strength) with small γ (~several μ eV)71–74, two kinds of reflection operators ( R1ˆ and ˆR2), obtained by adjusting the frequencies of the external field, ω, and cavity mode, ωc, can be given as ωω κω ωω ω−= ≈⊗ ↓↓ +⊗ ↑↑ −⊗ ↑↑ −⊗ ↓↓ −= ≈⊗ ↓↓ +⊗ ↑↑ −⊗ ↑↑ −⊗ ↓↓ˆ ˆRR LL iR R iL L RR LL RR LL/2:R () , 0: R( ) , (8)c c1 2 where the reflectances and phase shifts are |rh| = |r0| ≈ 1(≈ 1) and φrh ≈ 0 (≈ 0), φr0 ≈ −π/2 (≈ π), respectively, in the detuning of frequency as ω − ωc = κ/2 (ω − ωc = 0) for g/κ = 2.4 and

Question 975 multiple-choice

Quantum algorithms often exploit group structure using the Quantum Fourier Transform (QFT) to solve problems like the hidden subgroup problem. Representation theory and character theory play a crucial role in extracting information from quantum states after applying the QFT. When solving the hidden subgroup problem with a quantum algorithm, what is the probability of measuring an irreducible representation (irrep) label μ if the hidden subgroup H is trivial? 1) $ p_\mu[H] = \frac{1}{|G|} \sum_{h \in H} \chi_\mu(h) $ 2) $ p_\mu[H] = \frac{d_\mu}{|H|} \sum_{g \in G} \chi_\mu(g) $ 3) $ p_\mu[H] = \frac{d_\mu^2}{|H|} $ 4) $ p_\mu[H] = \frac{\chi_\mu(e)}{|G|} $ 5) $ p_\mu[H] = \frac{1}{|G|} \sum_{g \in G} \chi_\mu(g) $ 6) $ p_\mu[H] = \frac{d_\mu}{|G|} \sum_{h \in H} \chi_\mu(h) $ 7) $ p_\mu[H] = \frac{d_\mu^2}{|G|} $

✓ Correct Answer:

The correct answer is 7) $ p_\mu[H] = \frac{d_\mu^2}{|G|} $.

📚 Reference Text:

educible representations that QGRL(g)Q† G=/circleplusdisplay µdµ/summationdisplay i=1|i/an}bracketri}ht/an}bracketle{ti|⊗dµ/summationdisplay j,j′=1[Dµ]j,j′[g]|j/an}bracketri}ht/an}bracketle{tj′| QGRR(g)Q† G=/circleplusdisplay µdµ/summationdisplay j,j′=1[Dµ]j,j′[g]|j/an}bracketri}ht/an}bracketle{tj′|⊗dµ/summationdisplay i=1|i/an}bracketri}ht/an}bracketle{ti|. (31) Or in other words the QFT performs exactly the change of basis whic h block diagonalizes the left and right regular representations into a irreducible irreps as de scribed in Eqs. (9). When can the QFT over a group Gbe enacted by a quantum circuit of size polynomial in log|G|? While the full answer to this question is not known, for many groups , including the important symmetric group[33] and dihedral group[34, 47], eﬃ cient quantum circuits for the QFT are known. A quite general method for performing eﬃcient QFTs over a large class of ﬁnite groups, including the Heisenberg group, has been derived b y Moore, Rockmore, and Russell[48]. We refer the reader to the latter paper for more details on the quantum Fourier transform. Finally we should note that the QFT is a change of basis which is deﬁned o nly up to the choice of the basis for the Dµirreps. For our purposes, this basis choice will not matter, and we refer the reader to the paper of Moore, Rockmore, and Russe ll[48] for details on choices of this basis for diﬀerent groups. 4.4 The Symmetry Exploiting Protocol We havenow seen that the QFT overthe group Gis the transformwhich block diagonalizesall hidden subgroup states. By the symmetry argument above we can , without loss of generality, apply this transform. Further, since ρHis an incoherent sum over the diﬀerent irreps µ, we can, without a loss of generality, perform this transform, and the n measure the irrep index |µ/an}bracketri}ht. Further, since ρHacts trivially over the space where the left regular representation acts, the|lµ/an}bracketri}htindex contains no information about the hidden subgroup and can als o be measured (resulting in a uniformly random number between 1 and dµ.) Hence we see that we can recast the single copy hidden subgroup pr oblem as 1. Perform a QFT on the hidden subgroup state. D. Bacon 13 2. Measure the irrep label µand throw away the register where the left regular represen- tation irrep acts. The probability of obtaining irrep µis given by pµ[H] =dµ |G|/summationdisplay h∈Hχµ(h), (32) whereχµ(g) = Tr[Dµ(g)] is the character of the group element gin the irrep µ. 3. One is then left with a state with support over only the space wher e the right regular representation irrep acts. This state is given by ρµ[H] =dµ |G|pµ/summationdisplay h∈HDµ(h), (33) or, in other words the state ρµ[H] =/summationtext h∈HDµ(h)/summationtext h′∈Hχµ(h′), (34) assuming that pµ/ne}ati≀nslash= 0. The above protocol has been designed based solely on the symmetr y arguments of the hidden subgroup state. Notice that there are two locations already wher e information can appear about the hidden subgroup. The ﬁrst is in the probabilities pµ[H]. The second is in the state ρµ[H]. Finally note that if the hidden subgroup is the trivial group T={e}, then we obtain the µth irrep with probability pµ[H] =d2 µ |G|. (35) Note as a sanity check that

Question 976 multiple-choice

Quantum circuit design for practical hardware often involves addressing architectural constraints, such as limiting interactions to nearest-neighbor qubits. Efficient implementation of the quantum Fourier transform (QFT) is crucial for improving the reliability and performance of quantum algorithms on such hardware. In linear nearest-neighbor (NN) quantum architectures, which improvement does a novel QFT circuit design achieve to enhance algorithmic efficiency on hardware with NN constraints? 1) It eliminates all two-qubit gate operations from the QFT circuit. 2) It allows non-neighboring qubits to interact directly, bypassing NN limitations. 3) It increases the number of single-qubit gates to compensate for hardware errors. 4) It reduces the CNOT gate count to approximately 40% of previous linear NN QFT circuit designs. 5) It replaces all CNOT gates with SWAP gates for faster execution. 6) It requires a fully connected qubit topology for optimal performance. 7) It decreases the overall circuit depth by removing all controlled gates.

✓ Correct Answer:

The correct answer is 4) It reduces the CNOT gate count to approximately 40% of previous linear NN QFT circuit designs..

📚 Reference Text:

Title: Reducing CNOT count in quantum Fourier transform for the linear nearest-neighbor architecture Year: 2023 Paper ID: 4eb3767b933c973ef593cc7cf6c0b22a1c801266 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/4eb3767b933c973ef593cc7cf6c0b22a1c801266 Abstract: Physical limitations of quantum hardware often necessitate nearest-neighbor (NN) architecture. When synthesizing quantum circuits using the basic gate library, which consists of CNOT and single-qubit gates, CNOT gates are required to convert a quantum circuit into one suitable for an NN architecture. In the basic gate library, CNOT gates are considered the primary cost of quantum circuits due to their higher error rates and longer execution times compared to single-qubit gates. In this paper, we propose a new linear NN (LNN) circuit design for quantum Fourier transform (QFT), one of the most versatile subroutines in quantum algorithms. Our LNN QFT circuit has only about 40% of the number of CNOT gates compared to previously known LNN QFT circuits. Subsequently, we input both our QFT circuits and conventional QFT circuits into the Qiskit transpiler to construct QFTs on IBM quantum computers, which necessitate NN architectures. Consequently, our QFT circuits demonstrate a substantial advantage over conventional QFT circuits in terms of the number of CNOT gates. This outcome implies that the proposed LNN QFT circuit design could serve as a novel foundation for developing QFT circuits implemented in quantum hardware that demands NN architecture.

Question 977 multiple-choice

In algebraic geometry, graded rings of theta functions play a central role in the study of abelian varieties, their moduli, and the construction of modular forms. The structure and properties of these rings are deeply connected to parity decompositions, dimensional formulas, and integrality relations. What is the dimension of the vector space Sm = Rmn(^j), where m and n are positive integers and g is the genus parameter, in the context of graded rings of theta functions associated to a positive definite matrix? 1) mn^g 2) (mn)^g 3) 2m^g 4) 2n^g 5) m^g + n^g 6) 2mng 7) (2mn)^g

✓ Correct Answer:

The correct answer is 7) (2mn)^g.

📚 Reference Text:

where 3, = (0, , 0,1,0, , 0) THEOREM 1.1.1. Let m > 0 and jδf = φ(r) if(r) 6e the 2m-parity decomposition of ££. If R<£> = βm Π if(r), tfeen: 1) JR^) is a 1-dimensional k-vector space, 2) Rm = Θ(r) JB£> and dim& Rm = (2m)^ 3) β = Θj° J?m is a graded k-algebra with Ro = fc. Proof. Suppose 2]&/^/^^m) Using the relation (P;) we see that &7 determines 67/ for I = Γ mod. 2m. Thus dim. R^ < 1. To complete the proof of 1) we exhibit a generator of R£>. Take representatives of rό in Z and by abuse of language call them rά too. If is — 2mtj + rJf set bj = f[ qW""j+rfi f] 1 j>t and let 67 = 0 if / ^έ (r) mod. 2m. Set p(X) = 2] &i^7. A calculation shows that the δ7 satisfy (P7). Also ord. bj = Σ f/mί, + r^) ord. ^ + 2] (r^ί^ + r/^ + 2mtjt£) ord. = m Σ *y*i °rd ^ + Σ ri*< ord ^^ https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press 68 HORACΣO TAPIA-RECILLAS Since the matrix (ord. 0, that we shall make constant use of. Namely: (a) θeRm(s/ij) ^θeβMβ«) (b) θ e Rm(^j) =» θ(Xn) e Rmn{^i3) . These are easily verified. Using (a) together with Theorem 1.1.1. we get: PROPOSITION 1.1.2. If n is a fixed positive integer, Sm = Rmn(^j) is a k-vector space of dimension (2mn)9, S = Θ" Sm is a graded k-algebra, Rm £Ξ Sm and R is a subring of S. Next using (b) with m replaced by mn we see that Θ(X) —> θ{Xn) defines a graded homomorphism S->R of degree n2. The restriction of this map to R is a graded endomorphism of R of degree n2. Both of these maps will be denoted by an. A dimension count shows that an(S) consists precisely of those elements of R that can be written as Laurent series in Xy, j = 1,2, , g. THEOREM 1.1.3. R is integral over an(R). Proof. We first show that S is integral over R. Let θeSm. For 1 < i < 9 let T.iθ) - CXΓ#(^). Then: Thus T^Cό1) e Sm for i = 1,2, , g and we have defined operators Ύi: Sm -> Sm. An easy induction shows that T\(0)(X) = ^aXJ for all i. Thus Γj is the identity map on Sm. Also (TtoTj)(θ) = TtiqγX Since this is symmetric in i and j, the T€ commute. For each i, the various Tt: Sm—> Sm fit together to give a graded https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press ABELIAN

Question 978 multiple-choice

In quantum information theory, quantifying nonstabilizerness is essential for understanding the resources required for universal quantum computation. Several measures have been developed to characterize the "magic" of quantum states, each with distinct mathematical properties and operational significance. Which measure of nonstabilizerness is defined via G-asymmetry, coincides with basis-minimized measurement entropy, extends to mixed states via convex roof construction, and is proven to be a strong monotone in magic-state resource theory? 1) Robustness of magic 2) Stabilizer extent 3) Stabilizer rank 4) Basis-minimized stabilizerness asymmetry (BMSA) 5) Stabilizer fidelity 6) Stabilizer Rényi entropy 7) Magic monotone via negativity

✓ Correct Answer:

The correct answer is 4) Basis-minimized stabilizerness asymmetry (BMSA).

📚 Reference Text:

Title: A nonstabilizerness monotone from stabilizerness asymmetry Year: 2024 Paper ID: b7eba732f87dfa2ae0e454cefb83b042878b1280 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b7eba732f87dfa2ae0e454cefb83b042878b1280 Abstract: We introduce a nonstabilizerness monotone which we name basis-minimized stabilizerness asymmetry (BMSA). It is based on the notion of G-asymmetry, a measure of how much a certain state deviates from being symmetric with respect to a symmetry group G. For pure states, we show that the BMSA is a strong monotone for magic-state resource theory, while it can be extended to mixed states via the convex roof construction. We discuss its relation with other magic monotones, first showing that the BMSA coincides with the recently introduced basis-minimized measurement entropy, thereby establishing the strong monotonicity of the latter. Next, we provide inequalities between the BMSA and other nonstabilizerness measures such as the robustness of magic, stabilizer extent, stabilizer rank, stabilizer fidelity and stabilizer Rényi entropy. We also prove that the stabilizer fidelity, stabilizer Rényi entropy and BMSA with index α⩾2 have the same asymptotic scaling with qubit number. Finally, we present numerical methods to compute the BMSA, highlighting its advantages and drawbacks compared to other nonstabilizerness measures in the context of pure many-body quantum states. We also discuss the importance of additivity and strong monotonicity for measures of nonstabilizerness in many-body physics, motivating the search for additional computable nonstabilizerness monotones.

Question 979 multiple-choice

In algebraic structures arising from knot theory and quantum computation, diagram monoids like Kn and algebras such as Temperley-Lieb TLn(τ) are defined through generators and specific relations. These structures facilitate representations and combinatorial constructions using tableaux and mappings between algebras. Which mapping between generators allows representations of the diagram monoid Kn to be constructed from those of the Temperley-Lieb algebra TLn(τ), by identifying the cup-cap generator ωi with a scaled projection Ei and the closed-loop diagram δ with a scalar multiple [2]ℓ? 1) ωi ↦ δ, δ ↦ Ei 2) ωi ↦ τ, δ ↦ Ei 3) ωi ↦ identity matrix, δ ↦ rank-1 projector 4) ωi ↦ [2]ℓ, δ ↦ Ei 5) ωi ↦ braid generator, δ ↦ twist operator 6) ωi ↦ rank-1 projector, δ ↦ identity matrix 7) ωi ↦ Ei, δ ↦ [2]ℓ

✓ Correct Answer:

The correct answer is 7) ωi ↦ Ei, δ ↦ [2]ℓ.

📚 Reference Text:

is advantageous because it provide s representations of the diagram monoidKn[21], deﬁned as follows. For each 1 ≤i≤n−1, deﬁne the “cup-cap” diagram ωi=··· ··· , 18The Jones Polynomial: Quantum Algorithms and Applications in Quantum Complexity Theory where the cup-caps act on strands iandi+ 1. Multiplication of cup-caps is performed pictorially (up to isotopy), amounting to stacking the gene ratorsωiin the same way as with the generators of Bn. Then,Knis generated by the cup-cap generators {ω1,...,ω n−1}, together with an extra closed-loop diagram δ, which satisfy the relations ω2 i=δωi (5.30) ωiωjωi=ωi |i−j|= 1 (5.31) ωiωj=ωjωi|i−j|>1. (5.32) These relations are perhaps best understood in terms of pict ures which we invite the reader to draw, or otherwise to consult Figure 8 of [21]. While these relations are similar to those used above to deﬁne the Temperley-Lieb algebra, a more trans parent connection is obtained by reformulating (5.27)–(5.29) in terms of the scaled proje ctionsEi≡[2]ℓei, in which case the corresponding relations E2 i= [2] ℓEi (5.33) EiEjEi=Ei |i−j|= 1 (5.34) EiEj=EjEi|i−j|>1 (5.35) are formally identical to (5.30)–(5.32). The diagram monoi dKnis then represented inside TLn(τ) via the map taking ωi/ma√sto→Eiandδ/ma√sto→[2]ℓ. As withBn, we use this map to obtain representations of Knfrom those of TLn(τ) by setting π(2,ℓ) λ(ωi) =π(2,ℓ) λ(Ei) = [2] ℓπ(2,ℓ) λ(ei). For every even number 2 p, deﬁne the rectangular tableau t2p∈T[p,p]by t2n=135 246···2p−1 2p. (5.36) We slightly abuse notation and write the projection onto the corresponding vector |t2n/an}bracketri}htas t2p≡ |t2p/an}bracketri}ht/an}bracketle{tt2p|. We pause to mention a convention; given any t∈T(2,ℓ) [p,p]andt′∈T(2,ℓ) λ, lettt′ be the tableau of shape [ p+λ1,p+λ2] obtained by adding 2 pto each box of t′and placing it to the right of t. It is easily checked that tt′∈T(2,ℓ) [p+λ1,p+λ2]. This deﬁnes a natural embedding of vector spaces V[p,p]⊗Vλ֒→V[p+λ1,p+λ2]which we denote |t/an}bracketri}ht|t′/an}bracketri}ht /ma√sto→ |tt/an}bracketri}ht. Throughout, we freely make this identiﬁcation on vectors and operators. No w consider the following lemma. Lemma 5.1 Letpandnbe positive integers satisfying 2p≤n. Ifλ= [λ1,λ2]∈Λ(2,ℓ) n, then π(2,ℓ) λ(ω1ω3···ω2p−1) =/braceleftBigg [2]p ℓt2p⊗1[λ1−p,λ2−p]ifλ2≥p 0 otherwise. Proof: Abbreviating ei=π(2,ℓ) n(ei), recall that ej|t/an}bracketri}ht=|t/an}bracketri}htiﬀjandj+ 1 are in the same column, while ej|t/an}bracketri}ht= 0 iﬀjandj+ 1 are in the same row. Because the e2i−1are mutually commuting, if 2 i−1 and 2iare in the same row of a given tableau for some 1 ≤i≤p,|t/an}bracketri}htis annihilated by e1e3···e2p−1, and thus also by ω1ω3···ω2p−1. Therefore, only vectors |t/an}bracketri}htfor Pawel Wocjan and Jon Yard 19 whicht=t2nt′for somet′∈T(2,ℓ) [λ1−p,λ2−p]survive. Clearly this is impossible if λ2< p. On the other hand, for 1 ≤i≤pthee2i−1, and therefore the ω2i−1, may only act nontrivially on the leftmost pcolumns of any given tableau. By the convention mentioned ab ove, this means that when λ2≥p, the quantity in the lemma is proportional to the tensor prod uct of the rank 1 projector t2pwith an identity matrix as required. /square It is possible to combine BnandKninto a uniﬁed structure which was called a braid monoid by Kauﬀman [21]. As this braid monoid contains diagrams cons isting of both cup- caps and twists, its generators are the union

Question 980 multiple-choice

Quantum computing utilizes the Quantum Fourier Transform (QFT) to efficiently extract periodicities from quantum states, which is essential for algorithms like Shor’s factoring algorithm. Optical implementations of QFT leverage linear components to realize quantum gates using photons in superposition and entanglement. Which feature enables optical implementations of the Quantum Fourier Transform to minimize the required number of physical components when factoring large integers? 1) Employing nonlinear optical crystals for entanglement generation 2) Utilizing Cooley-Tukey FFT matrix decomposition with 50:50 beamsplitters and phase shifters 3) Using error-correcting codes to protect quantum information 4) Introducing adaptive measurements during the QFT circuit 5) Encoding qubits in atomic energy levels rather than photons 6) Employing classical control systems for gate sequencing 7) Implementing feedback loops to enhance quantum coherence

✓ Correct Answer:

The correct answer is 2) Utilizing Cooley-Tukey FFT matrix decomposition with 50:50 beamsplitters and phase shifters.

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transforms it to the Fourier coefficients corresponding to input wavefunction , thus: 11 10 01 0011 10 01 00 X X X X  (17) where the Xn are complex coefficients corresponding to the complex Fourier components at that qubit location in the output array. However, since they comprise the overall wavefunction they will not be directly acce ssable to measurement. Rather the probability of detection, by a single photodetection event, will be given by |Ψ|2. If the input wavefunction is periodic, e.g. arises from the successful application of the gcd procedure, |Ψ|2 will have a peak in its probability distribution at the output location corresponding to this. Thus repeated application of the QFT will yield more detection events at this location and hence allow determination of the periodicity, r, of the input as required to factor the initial input sequenc e, N. Thus the QTF is more powerful than the FFT in that it can process 2N inputs in parallel with effectively the same complexity of hardware structure (and so is exponentially faster in computation), as described below. However, the FFT yields N complex frequency components at its output whereas the QFT produces a probability distribution only which collapses to a single photon detection event upon measurement. This, however, may be very useful and used to solve problems with exponential complexi ty if applied appropriately as, instance, in the Shor algorithm. Since, as has been emphasised in Section 2, the optical implemen tation of the DFT by a coherent matrix -vector multiplier is a unitary operation, this hardware configura tion could be utilis ed to realis e a QFT provided the input sequence is in the required superposition state. Also, the waveguide structure described in Section 3 for the optical implementation of the FFT could be employed in a similar way provided , again, that the inputs bits are in a quantum superposition state, thus in the optical case comprising entangled individual photons entering each input port to the FFT structure. Barak and Ben -Aryeh10 have described a QTF implementation with linear optics, following closely an Cooley -Tukey FFT based matrix decomposition to reduce the required number of optical components required. 50:50 beamsplitters and pure phase shifters are shown to be the only linear optical components required to implement the FFT structure , and an arrangemen t of these is shown for the proposed implemen tation of the 3 qubit QTF considered in detail in the paper. The basic building block for the QFT is the Ha damard gate . This can be written:   1 111 21H (18) This is the basic operation of comprising the Hadamard transform and is optically implementable with a 50:50 beamsp litter together with an addi tional π/2 phase shift in one o f the beams as shown in Figure 5 (this arrangement is also used in fibre optic communication coherent detection s ystems to realis e a 1800 hybrid). Figure 5 . Optical implementation of Hadamard gate This arrangement will act

Question 981 multiple-choice

Quantum algorithms such as the Quantum Fourier Transform (QFT) and its amplified variants are central to solving structured computational problems more efficiently than classical methods. Improvements in quantum transforms have focused on increasing success probabilities and robustness against noise, impacting practical quantum computing applications. Which quantum algorithm provides a quadratic speedup for solving the Local Period Problem by amplifying the probability of success, even in the presence of errors? 1) Standard Quantum Fourier Transform (QFT) 2) Amplified Quantum Fourier Transform (Amplified-QFT) 3) Quantum Hidden Subgroup (QHS) algorithm 4) Grover’s Search algorithm 5) Deutsch-Jozsa algorithm 6) Amplified-Haar Wavelet Transform 7) Classical Fast Fourier Transform (FFT)

✓ Correct Answer:

The correct answer is 2) Amplified Quantum Fourier Transform (Amplified-QFT).

📚 Reference Text:

Title: Amplified Quantum Transforms Year: 2014 Paper ID: 25851c8490d86ebfa537cc49e0472bc94af36bb3 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/25851c8490d86ebfa537cc49e0472bc94af36bb3 Abstract: In this thesis we investigate two new Amplified Quantum Transforms. In particular we create and analyze the Amplified Quantum Fourier Transform (Amplified-QFT) and the Amplified-Haar Wavelet Transform. First, we provide a brief history of quantum mechanics and quantum computing. Second, we examine the Amplified-QFT in detail and compare it against the Quantum Fourier Transform (QFT) and Quantum Hidden Subgroup (QHS) algorithms for solving the Local Period Problem. We calculate the probabilities of success of each algorithm and show the Amplified-QFT is quadratically faster than the QFT and QHS algorithms. Third, we examine the Amplified-QFT algorithm for solving The Local Period Problem with an Error Stream. Fourth, we produce an uncertainty relation for the Amplified-QFT algorithm. Fifth, we show how the Amplified-Haar Wavelet Transform can solve the Local Constant or Balanced Signal Decision Problem which is a generalization of the Deutsch-Jozsa algorithm.

Question 982 multiple-choice

Quantum computing has made significant progress in solving certain computational problems using algorithms that leverage group-theoretic structures. The Hidden Subgroup Problem (HSP) for non-Abelian groups is particularly challenging and remains an area of active research due to its connections with problems like graph isomorphism and cryptography. Which of the following scenarios allows quantum algorithms to efficiently find normal subgroups in non-Abelian groups? 1) When the group is a black-box permutation group with arbitrary encoding 2) When the group is a dihedral group and exponential classical post-processing is available 3) When the group elements are represented as matrices over finite fields 4) When the group is solvable but the intersection of all normalizers is trivial 5) When the non-Abelian group’s Fourier transform can be computed efficiently 6) When the group is an Abelian group with a large order 7) When the group is a wreath product with complex classical bottlenecks

✓ Correct Answer:

The correct answer is 5) When the non-Abelian group’s Fourier transform can be computed efficiently.

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found for example in Mosca’s thes is [21]. Addressing the HSP in the non-Abelian case is considered to b e the most important challenge at present in quantum computing. Beside its intrinsic mathe matical interest, the importance of this problem is enhanced by the fact that it contains as special ca se the graph isomorphism problem. Unfortunately, the non-Abelian HSP seems to be much more diﬃ cult than the Abelian case, and although considerable eﬀorts were spent on it in the last year s, only limited success can bereported. R¨ otteler and Beth [24] have presented an eﬃcient quantum al gorithm for the wreath products Zk 2≀Z2. In the case of the dihedral groups, Ettinger and Høyer [9] de signed a quantum algorithm which makes only O(log|G|) queries. However, this doesn’t make their algorithm eﬃcie nt since the (classical) post-processing stage of the results of the queries is done in exponential time in O(log|G|). Actually, this result was extended by Ettinger, Høyer and Knill [10] in the sense that they have shown that in any group, with only O(log|G|) queries to the oracle, suﬃciently statistical information can be obtained to solve the the HSP . However, it is not known how to implement eﬃciently these queries, and therefore even the “ quantum part” of their algorithm is remainingexponential. Hallgren, RusselandTa-Shma[14]p rovedthatthegenericeﬃcientquantum procedure for the HSP in Abelian groups works also for non-Ab elian groups to ﬁnd any normal subgroup, under the condition that the Fourier transform on the group can eﬃciently be computed. Grigni, Schulman, Vazirani and Vazirani could show that the HSP is solvable eﬃciently in groups where the intersection of the normalizers of all subgroups i s large [12]. A recent survey on the status of the non-Abelian HSP problem was realized by Jozsa [ 16]. In a somewhat diﬀerent line of research, recently several gro up theoretical problems have been considered in the context of black-box groups. The notion of black-box groups has been introduced by Babai and Szemer´ edi in [2]. In this model, the elements of a groupGare encoded by words over a ﬁnite alphabet, and the group operations are performed by a n oracle (the black box). The groups are assumed to be input by generators, and the encoding is not necessarily unique. There has been a considerable eﬀort to develop classical algorithms fo r computations with them [5, 3, 20], for example to identify the composition factors (especially th e non-commutative ones). Eﬃcient black- boxalgorithms giveriseautomatically toeﬃcient algorith ms whenever theblack-box operations can be replaced by eﬃcient procedures. Permutation groups, mat rix groups over ﬁnite ﬁelds and even ﬁnite matrix groups over algebraic number ﬁelds ﬁt in this mo del. In particular, Watrous [27] has recently considered solvable black-box groups in the restr icted model of unique encoding, and using some new quantum algorithmical ideas, he could construct eﬃ cient quantum algorithms for ﬁnding composition series, decomposing Abelian factors, computi ng the order and testing membership in these groups. In this paper we

Question 983 multiple-choice

W-algebras are intricate mathematical structures arising in the study of symmetries, representation theory, and integrable systems, often built from Lie algebras via reduction techniques. Finite W-algebras, constructed using Poisson geometry, provide essential insight into both classical and quantum algebraic frameworks. Which statement best characterizes a key structural property of finite W-algebras obtained via Poisson reduction of Kirillov Poisson structures on simple Lie algebras? 1) They are always commutative and generated by central elements. 2) Their representation theory is fully classified for all types of reductions. 3) They contain only linear Poisson subalgebras isomorphic to abelian Lie algebras. 4) They do not admit embeddings into any Kirillov Poisson algebra. 5) Their construction relies exclusively on infinite-dimensional Lie algebras. 6) They lack any connection to integrable systems such as Toda systems. 7) They are nonlinear, finitely generated Poisson algebras that may contain linear Poisson subalgebras isomorphic to Kirillov Poisson algebras.

✓ Correct Answer:

The correct answer is 7) They are nonlinear, finitely generated Poisson algebras that may contain linear Poisson subalgebras isomorphic to Kirillov Poisson algebras..

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The quantization of the Pois son reduction leading to Wn algebras was made more precise in [11] where the BRST formali sm was used to tackle this problem. In order to calculate the BRST cohomology Feig in and Frenkel proposed to use spectral sequences. The paper [11] is however incompl ete as we shall see in this paper and furthermore they only consider the reductions ass ociated to the principal embeddings. It was also attempted to construct the representation theor y ofWalgebras form the representation theory of aﬃne Lie algebras [12, 11]. In prin ciple the BRST procedure provides a functor from the representation theory of aﬃne Li e algebras to the repre- sentation theory of Walgebras. It turned out to be possible to obtain Wncharacters from Kac-Moody characters. The general theory of Wnrepresentations is however far from complete. Furthermore for the other reductions the qua ntizations (let alone the representation theories) are not known. Up to now the study of Walgebras has concentrated on the inﬁnite dimensional case. This situation is comparable to trying to develop the t heory of Lie algebras by starting with the inﬁnite dimensional case. As the structur e and representation theory of aﬃne Lie algebras is largely determined by that of the ﬁnit e dimensional simple Lie algebras that underly them it is our opinion that it might be helpful to look for and study the ﬁnitely generated structures underlying Walgebras. This program was initiated in [13] and will be carried out in the present paper . It will turn out that 3 the theory of ’ﬁnite Walgebras’ is remarkably rich and contains already many of th e features encountered in ordinary Walgebras. It is therefore our expectation that much of what we will say in this paper will transfer without much al teration to the inﬁnite dimensional case. The paper is roughly split up into three parts. The ﬁrst part d eals with the classical theory. Classical ﬁnite Walgebras are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras (in complete anal ogy with ordinary Walge- bras which are constructed as reductions of aﬃne Lie algebra s). The Poisson algebras thus obtained are nonlinear and ﬁnitely generated. We discu ss their structure and show that in general they do have linear Poisson subalgebras that are isomorphic to Kirillov Poisson algebras. We also derive a coordinate free expressi on for the reduced Poisson structure of an arbitrary reduction. The Miura transformat ion turns out to have a ﬁnite dimensional analogue which can in fact be extended to arbitr ary reductions. From this it follows that any ﬁnite Walgebra can be embedded into the Kirillov Poisson algebra of a certain subalgebra of the simple Lie algebra with which w e started. At the end of the ﬁrst part of the paper we investigate which theories have ﬁniteWsymmetry. It turns out that (as could have been expected) these are genera lized ﬁnite Toda systems. In deriving this however we show that ﬁnite Toda

Question 984 multiple-choice

Quantum computing algorithms can solve certain problems exponentially faster than classical algorithms, particularly for tasks involving hidden structures in functions. Simon’s problem and the Abelian hidden subgroup problem are foundational examples, and advances in quantum hardware have enabled experimental demonstrations of quantum speedup. Which technique is specifically used to extend qubit coherence time and enhance the observed quantum speedup in experimental implementations of Simon’s problem with constrained hidden periods? 1) Quantum error correction with surface codes 2) Increasing the number of qubits in the processor 3) Classical simulation of quantum circuits 4) Using photonic quantum processors 5) Dynamical decoupling 6) Employing adiabatic quantum computing 7) Variational quantum eigensolver algorithms

✓ Correct Answer:

The correct answer is 5) Dynamical decoupling.

📚 Reference Text:

Title: Demonstration of Algorithmic Quantum Speedup for an Abelian Hidden Subgroup Problem Year: 2024 Paper ID: a6c9d3d70d188cb2e2cd6db208da3842ea9e4b16 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a6c9d3d70d188cb2e2cd6db208da3842ea9e4b16 Abstract: Simon’s problem is to find a hidden period (a bitstring) encoded into an unknown 2-to-1 function. It is one of the earliest problems for which an exponential quantum speedup was proven for ideal, noiseless quantum computers, albeit in the oracle model. Here, using two different 127-qubit IBM Quantum superconducting processors, we demonstrate an algorithmic quantum speedup for a variant of Simon’s problem where the hidden period has a restricted Hamming weight w. For sufficiently small values of w and for circuits involving up to 58 qubits, we demonstrate an exponential speedup, albeit of a lower quality than the speedup predicted for the noiseless algorithm. The speedup exponent and the range of w values for which an exponential speedup exists are significantly enhanced when the computation is protected by dynamical decoupling. Further enhancement is achieved with measurement error mitigation. This case constitutes a demonstration of a bona fide quantum advantage for an Abelian hidden subgroup problem. Published by the American Physical Society 2025

Question 985 multiple-choice

Quantum algorithms for hidden subgroup problems are pivotal in advancing cryptanalysis and understanding the power of quantum computation, especially for nonabelian groups such as the dihedral and quaternion groups. Managing trade-offs between time and space complexity is critical for practical quantum algorithm implementation. Which quantum algorithm achieves a polynomial space solution to the dihedral hidden subgroup problem by increasing time complexity, thereby improving upon a previous subexponential-space approach? 1) Regev's modification of Kuperberg's algorithm 2) Shor's algorithm for abelian groups 3) Ettinger and Hoyer's original algorithm 4) Grover's search algorithm 5) The algorithm introduced for the generalized quaternion group 6) Simon's algorithm for period finding 7) Kuperberg's original algorithm

✓ Correct Answer:

The correct answer is 1) Regev's modification of Kuperberg's algorithm.

📚 Reference Text:

problem. Ettinger and Hoyer [15] developed a quantum algorithm that produces output sufﬁcient to solve the dihedral hidden subgroup problem. However, in this case exponential time is required to reconstruct the hidden subgroup from the output. The fastest known quantum algorithm for the dihedral hidden subgroup problem was intro- duced in 2003 by Kuperberg [16] . This approach uses time and query complexity of 2O(plogN), where Ncorresponds to the parameter of underlying group DN. Unfortunately, Kuperberg’s al- gorithm also requires 2O(plogN)quantum space. Another subexponential-time algorithm is due to Regev [17], whose modiﬁcation of Kuperberg’s approach resulted in a polynomial-space algorithm at a price of a slow-down in time complexity. In this dissertation, a quantum algorithm that solves the hidden subgroup problem for the generalized quaternion group is developed. 5 CHAPTER 2 A QUANTUM MODEL OF COMPUTATION 2.1 The Quantum Mechanical Framework Quantum mechanics emerged as a mathematical model for physics to address problems encountered by classical theories in their attempts to explain the results of certain experiments. The following four postulates deﬁne the content of quantum mechanics. Postulate 1 (State Space). The state space of a system is a Hilbert space, and the system is com- pletely described by its state vector, a unit vector in the system’s state space. A single qubit constitutes a two-level system whose state is described by a vector in a two- dimensional Hilbert space. Any two vectors with the same global phase are equivalent. Postulate 2 (State Evolution). The evolution of a closed system is described by a unitary trans- formation. That is, for any evolution of the closed system there exists a unitary operator U such that if the system is in the state jy1iat time t1, then its state at time t2is jy2i=Ujy1i: (2.1) A closed quantum system evolves in accordance with the Schrödinger equation i¯h¶ ¶tjy(t1)i=Hjy(t1)i; (2.2) where ¯his the reduced Planck’s constant, and His the Hamiltonian, an operator that corresponds to the total energy of the system. The eigenvalues of Hare the possible energies of the system. 6 For a time-independent Hamiltonian, the state of the system at a later time t2is given by jy(t2)i=eiHt ¯hjyi(t1): (2.3) Thus, the time evolution of the system is governed by the unitary operator U=eiHt ¯h: (2.4) The deﬁning relation for any unitary operator Uis UU†=U†U=I; (2.5) where U†denotes the adjoint of U. Since a quantum computer is an n-qubit quantum system, the time evolution of its wave function obeys the Schrödinger equation and is described by a unitary operator. In quantum com- putation, unitary operators are implemented as unitary gates. Postulate 3 (State Combination). The state space of a composite system is the tensor product of the state spaces of its component systems. Given nsystems in the states jy1i;jy2i;:::;jyni, the state of their combined system is jy1i jy2i ::: jyni=jy1y2:::yni: (2.6) IfH1andH2are Hilbert spaces of dimensions N1andN2, respectively, then their tensor product H=H1 H2 (2.7) has dimension dim(H) =N1N2: (2.8) 7 IfH1andH2are endowed with bases juiiandjvii, respectively, then the basis for His given

Question 986 multiple-choice

In advanced linear algebra and invariant theory, the concept of capacity is used to measure properties of tuples of matrices under constraints such as positivity, determinant normalization, and trace minimization. These ideas are crucial in fields like operator scaling, quantum information theory, and optimization. Which statement best characterizes the necessary and sufficient condition for the capacity of a matrix tuple to be zero under constraints involving positivity, determinant normalization, and trace minimization? 1) The capacity is zero if and only if all matrices in the tuple are singular. 2) The capacity is zero if and only if the determinant of each matrix in the tuple is negative. 3) The capacity is zero if and only if the sum of traces of the matrices diverges to infinity. 4) The capacity is zero if and only if there exists a matrix in the tuple with non-positive eigenvalues. 5) The capacity is zero if and only if the infimum over trace and determinant constraints is strictly positive. 6) The capacity is zero if and only if the product of eigenvalues is bounded away from zero for all matrix choices. 7) The capacity is zero if and only if the infimum over all positive matrices with determinant one, subject to trace constraints, yields zero, meaning no solution exists for the specified algebraic equations in the support of the basis family.

✓ Correct Answer:

The correct answer is 7) The capacity is zero if and only if the infimum over all positive matrices with determinant one, subject to trace constraints, yields zero, meaning no solution exists for the specified algebraic equations in the support of the basis family..

📚 Reference Text:

when capacityp¡0; the lemma allows us to pass to a larger set for the inﬁmum. Lemma A.5 (Adapted from [Fra18]) .LetpPPpnqwithpn¡0andbPBpnq. The character | ppbq|2: is equal to supn¹ i1detpYiqpi(35) subject to 0 Yi:CiÑCi(36) andn¸ i1pi: iYiib:diagppopqb (37) Proof.Letbethepermutationreversingtheorderofthecoordinates(thedimensionissuppressed in an abuse of notation). The lemma can be obtained from Claim 4.6 in [ Fra18] by making the changeofvariables hb:andnotingthattheprojection itotheﬁrsticoordinatesisgivenby i:. Finally one uses | ppbq|2detpdiagppq;h:hq. We have one more easy lemma. We use the shorthand Ppiq:diagppp1q Òqand P: Pp1q;:::;Ppdq : Lemma A.6 (modiﬁed capacity) . inf RPB| ppRq|2tr? PpRq0ðñ capacityppq0: Proof.IfPis nonsingular, we apply the same change of variables argument from Remark 4.8. Otherwise, from Lemma 3.5, we have pPBpqif and only if pPBpq. This is because XifX. Thus, capacityp0ðñ capacityp0: By a change of variables, capacityp0if and only if inf bPB| p pbq|2trr? P: rpbq0; but inf bPB| p pbq|2trr? P: rpbqinf bPB| ppbq|2trr? P: rprb: rq inf bPB| ppbq|2trpr? P: rqprpbqq inf bPB| ppbq|2tr? Ppbq: The last equality follows form cyclicity of trace and : rr? P? P. 56 Finally we prove Proposition A.4. Proof of Proposition A.4. Wewanttoﬁndnecessaryandsuﬃcientconditionsunderwhich capacityppq 0. By Lemma A.6, this happens if and only if inf bPB| ppbq|2tr? Ppbq inf bPB| ppbq|2trpb: 1Pp1qb1bbb: dPpdqbdq0: (38) Due to Lemma A.5, we may replace b: iPpiqbiby°ni j1: jYpiq jjand| ppbq|2by the product over iPrdsof± jPrnispdetYpiq jqppiq j, and the inﬁmum will remain the same! Thus, capacityppq0if and only if inf¸ jPrn1srndsd¹ i1ppiq jpiq tr Yp1q jp1qbbYpdq jpdq pjq 0(39) subject to 0 Ypiq j:CjÑCjfor alljPrnis(40) and¹ jPrnispdetYpiq jqppiq j1 for alliPrds:(41) Wenowprovethe“if"directionofPropositionA.4,namelythatifthevalueoftheaboveprogram is zero then there is some pd;nq-basis family Usuch that SuppCp;Uqdoes not admit a solution D to Eq. (31) and Eq. (32). The “if” direction Suppose there is a sequence Ypiq jptqsatisfying Eq. (40) and Eq. (41) such that the expression in Eq. (39) tends to zero. We can diagonalize Ypiq jptqUpiq;: jptqdiagpzpiq j;lptqqUpiq jptqsuch that 1.Upiq jptqunitary. 2.zpiq j;lptq¡0, 3.±n j1± lPrjszpiq j;lptq ppiq j1, 4.andzpiq j;lptqtends to zero if ppiq j0(in that case Ypiq jappears neither in Eq. (41) nor Eq. (39)). LetUptqbethepd;nq-basisfamilypUpiq jptq;iPrds;jPrnisq. Bycompactness,wepasstoaconvergent subsequence such that limtÑ8Upiq jptqUpiq j. LetUbe thepd;nq-basis familypUpiq j;iPrds;jPrnisq. We claim that for all pj;lqinSuppCp;Uq, lim tÑ8d¹ i1zpiq jpiq;lpiqptq0for allpj;lqPSuppCp;Uq: (42) 57 This follows from the calculation ¸ jPrn1srndsd¹ i1ppiq jpiq tr Yp1q jp1qptqbbYpdq jpdqptq pjq ¸ jPrn1srndsd¹ i1ppiq jpiq d¹ i1zpiq jpiq;lpiqptq Cp;Uptqqj;l: WehaveCp;Uptqqj;lÑCp;Uqj;l. IfCp;Uqj;l¡0,then limtÑ8±d i1zpiq jpiq;lpiqptq . Notethat we could ignore the case when some ppiq jpiq0because of our assumption that zpiq j;lptqtends to zero in that case. This implies that subject to SuppDSuppCp;Uqthere is no solution to Eq. (32). Suppose there were. Note that Eq:(32) andpPPimplies° pj;lqPSDj;l1. Now log ¸ pj;lqPSd¹ i1zpiq jpiq;lpiqptq Dj;l ¥¸ pj;lqPSd¸ i1logzpiq jpiq;lpiqptq Dj;l d¸ i1¸ jPrnisppiq j¸ lPrjslogzpiq jpiq;lpiqptq0: However, this contradicts our assumption that ¸ pj;lqPSd¹ i1zpiq jpiq;lpiqptq Dj;l0; which follows from Eq. (42). The “only if” direction We now prove the easier direction. Suppose that pd;nq-unitary family Usuch that there is no D satisfying SupppDqSuppCp;Uqand Eq. (32). We will show

Question 987 multiple-choice

Quantum computing leverages principles such as entanglement, superposition, and unitary evolution to solve certain problems more efficiently than classical computers. A recurring theme in quantum algorithms is the use of the quantum Fourier transform (QFT) and group-theoretic structures to achieve speedup. Which of the following most accurately describes a promising approach for developing new quantum algorithms that could extend computational advantages beyond problems solvable with the Abelian quantum Fourier transform? 1) Designing quantum circuits based solely on reversible classical logic gates without exploiting group structures 2) Increasing the number of qubits in a quantum computer without altering algorithmic primitives 3) Applying the classical fast Fourier transform to quantum states in non-Abelian settings 4) Implementing efficient quantum Fourier transforms over non-Abelian groups to address hidden subgroup problems 5) Replacing entanglement with classical correlations in existing quantum algorithms 6) Restricting quantum computation to only commutative (Abelian) group symmetries 7) Using random unitary operations without regard to underlying group properties

✓ Correct Answer:

The correct answer is 4) Implementing efficient quantum Fourier transforms over non-Abelian groups to address hidden subgroup problems.

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obtain an eﬃcient factoring algorithm based on the no vel idea of deter- mining an eigenvalue of a given simple unitary operation. So me of the formalism of Kitaev’s algorithm may be related to Shor’s method by usin g the decomposition of the Fourier transform given in [14] but it would be interes ting to consider other 16 problems that might be formulatable in terms of the determin ation of eigenvalues. Conclusions We have seen that the principal known quantum algorithms all revolve around one essential construction, that of the Fourier transform o n an Abelian group. Fur- thermore the quantum computational speedup provided by the se algorithms may be attributed to (non-classical) properties of entanglement operating within the imple- mentation of classical fast Fourier transform algorithms o n a quantum computer [8]. Clearly it would be of great interest to have other basic ingr edients for the construction of new quantum algorithms. Kitaev’s formalism [7] as we have illustrated, appears to involve such an ingredient. The mathematical construction of the Fourier transform also extends to non-Abelian groups and it would be interesti ng to investigate prob- lems which can be formulated in terms of non-Abelian Fourier transforms and the possiblity of their implementation on a quantum computer. T his line of development has also been advocated by Hoyer [13]. Acknowledgements The author is grateful for the hospitality of the Institute f or Theoretical Physics, University of California, Santa Barbara during the Quantum Computers and Quan- tum Coherence program during which time a preliminary versi on of this work was completed. This work was supported in part by the National Sc ience Foundation under Grant No. PHY94-07194 and by the Royal Society, London . References [1] Deutsch, D. (1985) Proc. Roy. Soc. London Ser. A 400, 97. [2] Deutsch, D. and Jozsa, R. (1992) Proc. Roy. Soc. London Ser A 439, 553-558. [3] Bernstein, E. and Vazirani, U. (1993) Proc. 25th Annual ACM Symposium on the Theory of Computing , (ACM Press, New York), p. 11-20 (Extended Abstract). Full version of this paper appears in S. I. A. M. Journal on Computing 26(Oct 1997). [4] Simon, D. (1994) Proc. of 35th Annual Symposium on the Foundations of Com- puter Science , (IEEE Computer Society, Los Alamitos), p. 116 (Extended Ab - stract). Full version of this paper appears in S. I. A. M. Journal on Computing 26(Oct 1997). [5] Shor, P. (1994) Proc. of 35th Annual Symposium on the Foundations of Compute r Science , (IEEE Computer Society, Los Alamitos), p. 124 (Extended Ab stract). 17 Full version of this paper appears in S. I. A. M. Journal on Computing 26(Oct 1997) and is also available at LANL quant-ph preprint archiv e 9508027. [6] Ekert, A. and Jozsa, R. (1996) Rev. Mod. Phys. 68, 733. [7] Kitaev, A. (1995) “Quantum Measurements and the Abelian Stabiliser Problem” , preprint available at LANL quant-ph preprint archive 95110 26. [8] Jozsa, R. (1997) “Entanglement and Quantum Computation” inGeometric Issues in the Foundations of

Question 988 multiple-choice

Quantum computing leverages specialized algorithms and error characterization techniques to achieve reliable operations on qubits. Central to this process are transformations like the Quantum Fourier Transform (QFT) and tools for analyzing the quality of quantum states. Which metric specifically quantifies the loss of quantum information, corresponding to how close a quantum state is to the maximally mixed state, when characterizing systematic errors using the isotropic index? 1) Alignment 2) Quantum State Tomography fidelity 3) Weight (ω) 4) Controlled rotation phase shift 5) Hadamard gate superposition strength 6) Density matrix trace distance 7) QFT circuit depth

✓ Correct Answer:

The correct answer is 3) Weight (ω).

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are the amplitudes on a computational basis of a wave function. It is part of many quantumalgorithms,mostnotablyShor’sfactoringalgorithm [5] which makesuse of the QFT for period ﬁnding. TheclassicalDFTmapsaninputvector (x0; :::; x N¡1)to the output vector (y0; :::; y N¡1)accordingto Eq. ( 1): yk=1p NN¡1X j=0xj!jk N; (1) where !jk N=e(2¼i N)jk: (2) Meanwhilethe QFTactsonaninputpure n-qubitquan- tum state jxi jxi=N¡1X j=0xjjji; (3) 2 C.ALLENDE, A. FONSECA DE OLIVEIRA, AND E. BUKSMAN FIGURE1.Two qubit QFT circuit for input state j00i. Illustration givenby [9]. where N= 2n, andfjjigis the standard computational ba- sis,and maps it to the output state jyiaccording to jyi=UQFTjxi=1p NN¡1X j=0N¡1X k=0!jk Nxjjki;(4) where !jk Nisthe same as the classical case, Eq. ( 2). 2.2. Circuitimplementation Experimental quantum computers nowadays generally use a small set of universal quantum gates in order to generate quantumcircuits. ThequantumIBMcomputer(IBMQ)isno exception and only works with certain quantum gates, which made translating the matrix in Eq. ( 4) to quantum gates an essentialtask. As an example, for a two qubit state, the quantum gate representation of the algorithm requires a few basic gates: the Hadamard transform and a controlled rotation of angle ¼=2around the zaxis,Rz, whose matrix representations are shownin Eq. ( 5)and (6), respectively. H=1p 2· 1 1 1¡1¸ ; (5) CRz(¼=2) = I j0ih0j+Rz j1ih1j;(6) where Rzis Rz=·1 0 0e¼i 2¸ ; (7) the QFT circuit UQFT= (IH):cRz: (HI)is shown in Fig.1. 3. Systematic errormodel 3.1. Error model using the density matrix and the isotropicindex In order to model the systematic error, the ﬁrst step is to re- constructtheresultingdensitymatrix ½exp,measuringtheex- perimental projections from the data obtained from IBM Q, using Quantum State Tomography (QST) [4]. To recover the densitymatrixstate,measurementsmustbemadeinthethree basis (axes) of space, X; YandZwhose matrices are de- ﬁned: X=· 0 1 1 0¸ ; Y=· 0¡i i0¸ ; Z=· 1 0 0¡1¸ :(8)Forexample,forany 2qubitstateitsmatrix ½isgivenby ½=1 43X i=03X j=0tr(½¾i¾j)¾i¾j; (9) where ¾0=I,¾1=X,¾2=Y,¾3=Zand the values of tr(½¾i¾j)are obtained, measuring the state several times in order to approach the expected value by the statistical av- erage. 3.2. Isotropicindex The isotropic index is useful to separate the global loss of information, called weight !, from the misalignment Awith respect to a reference pure state. While nothing can be done aboutthelossofinformation,themisalignment(comingfrom systematic errors), could be corrected, hence the importance of separating these twotype of errors. Consideringthepurereferencestate jÃi,½Ã=jÃihÃjand the decomposition of a state ½aftera noisy process, ½=!I 2n+ (1¡!)^½; (10) the double indexis deﬁned as: ²The isotropic weight != 2n¸; (11) where ¸is the smallest eigenvalueof ½, ²and the isotropic alignment A, A=Fid(^½; ½Ã)¡Fid(^½; ½Ã?);(12) whereFidistheﬁdelitybetweenstates[4],and ½Ã?= I¡½Ã=2n¡1is the isotropic mixed state orthogonal tojÃi. The misalignment takes valuesin the interval[-1,1], where 1 means the resulting state is completely aligned with the ref- erencestate ½Ãand-1meanstheresultingstateiscompletely misaligned i.e.aligned with ½Ã?. Moreover, !takes values in the interval [0,1]: when ½is a pure state != 0, and when the loss of information is total, != 1. Considernowthestate jÃiasthereferencestateofthere- sultingtheoreticalstatefromtheapplicationof QFTtosome basis state, and ½expthe obtained experimental matrix state from the same input state. Decomposing the ½exp ½exp=!I 2n+ (1¡!)^½; (13) it is alwayspossible

Question 989 multiple-choice

In quantum computing and group theory, the Hidden Translation problem involves recovering a hidden group element given functions related by a specific group action. Efficient quantum algorithms for this problem have significant implications for cryptography and computational complexity. When solving the Hidden Translation problem in elementary abelian p-groups (Zn_p) using a quantum algorithm, which key technical innovation enables polynomial-time classical postprocessing for recovering the hidden translation? 1) Utilizing non-injective hiding functions to reduce the solution space 2) Applying Shor's algorithm to factor the group order 3) Exploiting Fourier samples that yield vectors nonorthogonal to the translation, forming a system of linear inequations 4) Leveraging the structure of nonabelian groups for easier sampling 5) Employing exponential-time exhaustive search over all possible translations 6) Transforming the problem into a lattice shortest vector problem for direct solution 7) Using only classical random sampling without quantum steps

✓ Correct Answer:

The correct answer is 3) Exploiting Fourier samples that yield vectors nonorthogonal to the translation, forming a system of linear inequations.

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v e nt w o injective functions f 0andf1from a ﬁnite group Gto some ﬁnite set such that, for some group element u, the equality f1(xu)=f0(x)h o l d sf o re v e r y x. The task is to ﬁnd the translation u. In fact, whenever Gis abelian, Hidden Translation is an instance of Hidden Subgroup in the semidirect product G/multicloserightZ2, where the hiding function is f(x,b)=fb(x). The group action in G/multicloserightZ2is deﬁned as ( x1,b1)·(x2,b2)= (x1+(−1)b1x2,b1⊕b2), where + denotes the group operation in Gand⊕denotes the group operation in Z2.I nG/multicloserightZ2,fhides the subgroup H={(0,0),(u,1)}. Actually, there is an eﬃcient quantum reduction in the other direction as well, and the two problems arequantum polynomial time equivalent [13]. A nice consequence ofthis equivalence is that instead of dealing with Hidden Subgroup in the nonabelian groupG/multicloserightZ 2, we canaddress Hidden Translation in the abeliangroup G. Ettinger and Høyer [13] have shown that Hidden Translation c a nb es o l v e db yat w o - s t e p procedure when G=ZNis cyclic: a polynomial number of Fourier samplings over the abelian group ZN×Z2followed by an exponential time classicalstage without further queries. The best known quantum algorithm for Hidden Translation in cyclic HIDDEN TRANSLATION AND TRANSLATING COSET 3 (and, in general, abelian) groups is Kuperberg’s subexponential time method [31]. Its relation to certain lattice problems investigated by Regev [38] provides evidence that Hidden Translation in cyclic groups might in fact be diﬃcult. In a related work, van Dam, Hallgren, and Ip [44] gave eﬃcient solutions for three cases of what they call the hidden shift problem. They also deﬁne another problem called the hidden coset problem which generalizes hidden shift. Their hidden coset problem can be viewed as a generalization of our Hidden Translation to not necessarily injective functions. While their paper gives eﬃcient quantum algorithms for some speciﬁc hidden coset problems, in general the hidden coset problem is ofexponential query complexity even in Z n 2. Our ﬁrst result (Theorem 3.5) is an eﬃcient quantum algorithm for Hidden Translation in the case of elementary abelian p-groups, that is, groups Zn p, for any ﬁxed prime number p. The quantum part of our algorithm is the same as in Et- tinger and Høyer’s procedure [13]: it consists of performing Fourier sampling over the abelian group Zn p×Z2. But while their classical postpro cessing requires exponential time, here we are able to recover classically the translation in polynomial time from the samples. It turns out that Fourier sampling produces vectors ynonorthogonal to the translation u; that is, we obtain linear inequations for the unknown u.T h i si s diﬀerent from the situation in the standard algorithm for the abelian Hidden Sub- group, where only vectors orthogonal to the hidden subgroup are generated. We show that, after a polynomial number of samplings, the system of linear inequations has a unique solution with high probability, which we are able to determine in deter- ministic polynomial time. An immediate consequence of Theorem

Question 990 multiple-choice

In quantum physics and representation theory, the structure and properties of symmetry groups like SU(2) and SO(3) play a crucial role in understanding how quantum systems transform under rotations. The relationship between these groups influences the behavior of quantum states and operators. Which statement correctly describes the action of the adjoint representation of SU(2) on the space of 2×2 complex matrices M₂? 1) It sends every matrix in M₂ to the zero matrix. 2) It acts non-trivially only on the identity matrix, leaving traceless matrices unchanged. 3) It mixes the identity and traceless matrices together under conjugation. 4) It leaves the identity matrix invariant and transforms the traceless matrices (su(2)) among themselves, resulting in a block-diagonal structure with invariant subspaces. 5) It maps traceless matrices to the identity matrix and vice versa. 6) It acts as scalar multiplication on all elements of M₂. 7) It permutes the basis elements {X, Y, Z, 1} randomly.

✓ Correct Answer:

The correct answer is 4) It leaves the identity matrix invariant and transforms the traceless matrices (su(2)) among themselves, resulting in a block-diagonal structure with invariant subspaces..

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double cover of the group of rotations in three dimensions SO(3). A key exam- ple of this “covering” idea is intimately familiar to the QML scientist: the map 7!ei2 maps 2[0;1]and all +k, k2Zto the same rotation. In this sense, the real line is a coveroftheunitcircle. The“double” indoublecoversimply means that above each rotation R2SO(3), there are two corresponding rotations ~R1;~R22SU(2)(see Hall [51] for a proper deﬁnition and more detailed discussion). Further, SU(2)is simply connected, meaning every loop in SU(2) can be continuously contracted to a single point without leavingSU(2), whileSO(3)is not simply connected. As a sneak peek down the road, this seemingly innocuous fact will play a huge role in the representation theory of these groups: while every Lie group representation gives rise to a Lie algebra representation (Theorem 2), the converse only holds locally for “small angles” unless the group is simply connected (Theorem 3). And since representation theory for Lie algebras is essentially linear algebraic in nature, we often wish to recover Lie group representations from Lie algebra representations, so paying attention to this simply connected condition will be crucial. The following pair of examples, the adjoint representa- tion in Example 4 and the tensor representation in Ex- ample 5, may very well be the most important examples in this entire article. They are rife with representation theoretic structure, and will provide a deeper understand- ing of how representations and symmetries play a key role in QML. As such, we will be returning to these examples several times throughout our journey. Furthermore, while they may look superﬁcially distinct: these representations are in fact equivalent, and we will unravel the connection when we deﬁne equivalence of representations in Deﬁnition 16. First, let us consider the adjoint representation ofSU(2) in Example 4. As seen therein, the adjoint map admits a block diagonal representation in the basis fX;Y;Z; 1gExample 4: A7!UAUyand the adjoint rep- resentation of SU(2) LetV=M2(C)denote the set of 22complex ma- trices. We note that the linear superoperator given by conjugation A7!UgAUy g, withUg=g2SU(2), isinGL(V). Indeed, themap g7!Ug()Uy gdeﬁnesa representation of SU(2)known as the adjoint repre- sentation. Since the set fX;Y;Z; 1gVforms an orthonormal basis of Vwith respect to the Hilbert- Schmidt inner product hA;BiHS= Tr1 2[AyB], we can represent any A2Vas A=c01+c1X+c2Y+c3Z; where now (c0;c1;c2;c3)2C4. Note that the trace is invariant with respect to this representa- tion: i.e., for all g2SU(2),Tr[A] = Tr UgAUy g . Further observe that every nonzero operator in spanf1ghas nonzero trace, while every operator in spanfX;Y;Zghas trace zero. Combining these two facts, we realize that Ug()Uy gcannot map be- tween these two linear subspaces. In other words, these subspaces are invariant subspaces of the rep- resentation U(seeDeﬁnition 12). Inparticular, this means that the set of representatives fUg()Uy g:g2 SU(2)gcanbesimultaneouslyblock-diagonalizedin the basisfX;Y;Z; 1gofM2(C): Ug()Uy g=0 BB@0 0 0 0 0 01 CCA; Indeed, theinvariantsubspacespan f1gisthetrivial one, which can be seen by observing that Ug1Uy g= 1, and the invariant subspace span fX;Y;Zg= su(2)is theadjoint representation ofSU(2)(see Deﬁnition 8). In other words, V=su(2)C1as representations (see

Question 991 multiple-choice

In computational complexity theory, the study of oracle machines, counting classes, and permutation groups provides foundational insights into the power and limitations of various computational models. Concepts such as UP-like queries, the GapP and SPP classes, and the structure of permutation groups are central to understanding these domains. Which statement accurately describes a closure property of the complexity class SPP? 1) SPP is closed under UP oracles, meaning any language decidable with a UP oracle is also in SPP. 2) SPP is closed under NP oracles, so any NP oracle computation remains in SPP. 3) SPP is closed under GapP oracles, ensuring computations with GapP oracles are contained within SPP. 4) SPP is closed under group-theoretic oracle queries, such as those involving permutation groups. 5) SPP is closed under #P oracles, allowing counting computations within SPP. 6) SPP is closed under SPP oracles, so any language decidable with an SPP oracle remains in SPP. 7) SPP is closed under FPSPP oracles, meaning function computations using FPSPP oracles stay in SPP.

✓ Correct Answer:

The correct answer is 6) SPP is closed under SPP oracles, so any language decidable with an SPP oracle remains in SPP..

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machine N. We say that MAmakes UP-like queries toAif on all inputs x,MA(x) makes onlysuch queries yfor whichN(y) has at most one accepting path. Eﬀectively, it is likeMhaving access to a UP oracle. We state a useful variant of a result from [13, 14]. Theorem 2.1 ([13]) LetMbe a nondeterministic polynomial-time oracle ma- chine with oracle A∈NPsuch thatMAmakes UP-like queries to Athen the functionh(x) =gapMA(x)is in GapP . Next, we recall an important property of the class SPP shown in [7]. Theorem 2.2 ([7]) IfLis in SPPAfor some oracle A∈SPP thenL∈SPP. I.e.SPPSPP= SPP . The following lemma, which is a straightforward consequence of Theorem 2.1 and of Theorem 2.2, is in a form useful for this paper. Lemma 2.3 •SupposeLis in SPPAaccepted by the nondeterministic polynomial-time oracle machine MAwith oracle A∈NP(i.e.x∈Limplies that gapMA(x) = 1, andx/negationslash∈Limplies that gapMA(x) = 0 ), such that the machine MA makes UP-like queries to A, thenLis in SPP. •Suppose a function f: Σ∗→Σ∗is in FPA(i.e.fis computed by a polynomial-time oracle transducer MA) whereA∈NP, such that the ma- chineMAmakes UP-like queries to A, thenfis in FPSPP. 2.1 Permutation group preliminaries In general, Sym(Ω) denotes the symmetric group on the ﬁnite set Ω. A per- mutation group on Ω is a subgroup of Sym(Ω). For |Ω|=n, in this paper 3 we let Ω = [ n] and identify Sym(Ω) with the group Snof all permutations on [n] ={1,2...,n }. We use letters g,h,...,σ,τ,π,... with subscripts and superscripts to denote elements of Snandi,jandkfor the elements of the set Ω = {1,2...,n }. Sub- groups (and in general subsets) of Sym(Ω )will be usually denoted by capital lettersA,G,Hetc. We use the following notation which is standard in per- mutation group theory [22, 15]. Given g∈Snandi∈[n], we denote by ig the image of iunder permutation g. The composition g1g2of permutations g1,g2∈Snis deﬁned left to right : i.e. applying g1ﬁrst and then g2. More precisely,ig1g2= (ig1)g2for alli∈[n]. For subset A⊆Snandx∈Ω we usexA to denote the set {xg|g∈A}. In particular if Ais a subgroup of Sn,xAis the orbit ofxunder the action of Aon Ω. For a subset ∆ of [ n], letG∆denote the subgroup of Gthat ﬁxes each element of ∆. In particular, if G≤Snthen for each i∈[n], we letG(i)denote the subgroup {g∈G|jg=jfor eachj∈[i]}.G(i)is called the pointwise stabilizer of [i] inG. The identity permutation is denoted by 1 (we use 1 to denote the identity of all groups) and the subgroup consisting of only 1 is denoted 1. The permutation group generated by a subset AofSnis the smallest subgroup of SncontainingA and is denoted /angbracketleftA/angbracketright. We assume that subgroups of Snare presented by generator sets. For a generator set A⊆Sn, each permutation ψ∈Ais a list ofnordered pairs /angbracketlefti,j/angbracketright ∈[n]×[n]. For permutation groups GandH, the expression H≤Gmeans that His a subgroup of G(not necessarily a proper subgroup). For ϕ∈Gthe subset Hϕ={πϕ:π∈H}ofGis a right coset ofHinG. Two right cosets of H inGare either disjoint or identical. Thus, the right cosets of HinGform a partition of Gwritten asG=Hϕ1+Hϕ2+...+Hϕk. Each right coset of Hhas cardinality

Question 992 multiple-choice

Quantum algorithms for group-theoretic problems are central to cryptanalysis and computational mathematics. Optimizing resource usage on quantum hardware is a key challenge, especially for algorithms addressing the dihedral hidden subgroup problem (DHSP). Which of the following statements most accurately describes how distributing the DHSP algorithm across multiple quantum nodes impacts its computational complexity and resource requirements? 1) It increases circuit depth and qubit usage per node due to added communication overhead. 2) It maintains the original time complexity but reduces classical processing demands. 3) It decreases measurement success probability and exacerbates noise-related errors. 4) It provides no significant change in scalability compared to the non-distributed approach. 5) It eliminates the need for function decomposition and parallel processing. 6) It worsens the algorithm’s asymptotic performance for large problem sizes. 7) It lowers both circuit depth and qubit requirements per node, and asymptotically improves time complexity from $2^{O(\sqrt{n})}$ to $2^{o(\sqrt{n-t})}$ as more nodes are used.

✓ Correct Answer:

The correct answer is 7) It lowers both circuit depth and qubit requirements per node, and asymptotically improves time complexity from $2^{O(\sqrt{n})}$ to $2^{o(\sqrt{n-t})}$ as more nodes are used..

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Title: Distributed quantum algorithm for the dihedral hidden subgroup problem Year: 2025 Paper ID: 1f4329f082b6f0de5caa984c4e3711176bbf3285 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/1f4329f082b6f0de5caa984c4e3711176bbf3285 Abstract: To address the issue of excessive quantum resource requirements in Kuperberg's algorithm for the dihedral hidden subgroup problem, this paper proposes a distributed algorithm based on the function decomposition. By splitting the original function into multiple subfunctions and distributing them to multiple quantum nodes for parallel processing, the algorithm significantly reduces the quantum circuit depth and qubit requirements for individual nodes. Theoretical analysis shows that when $n\gg t$ ($t$ is the number of quantum nodes), the time complexity of the distributed version is optimized from $2^{O(\sqrt{n})}$ (the traditional algorithm's complexity) to $2^{o(\sqrt{n-t})}$. Furthermore, we carried out the simulation on the Qiskit platform, and the accuracy of the algorithm is verified. Compared to the original algorithm, the distributed version not only reduces the influence of circuit depth and noise, but also improves the probability of measurement success.

Question 993 multiple-choice

In photonic quantum computing, fusion operations are used to entangle qubits and generate large-scale quantum states. Quantitative modeling of errors, such as those due to photon distinguishability, is crucial for benchmarking quantum protocols and evaluating their reliability. In a fusion-based photonic quantum circuit, how does the photon distinguishability error rate (ϵ) explicitly affect the calculation of the expected number of stabilizer errors after post-processing fusions? 1) By altering the weights assigned to each Bell state outcome during error estimation 2) By changing the fusion fidelity metric used to benchmark quantum gates 3) By directly modifying the probabilities of each error configuration in the decomposition of the input state 4) By introducing additional loss channels in the linear optical processing step 5) By affecting the success rate of heralded fusion events independently of state decomposition 6) By changing the random variable used for output state modeling 7) By requiring destructive Bell measurements for error quantification

✓ Correct Answer:

The correct answer is 3) By directly modifying the probabilities of each error configuration in the decomposition of the input state.

📚 Reference Text:

if the state ηis used in a fusion-based circuit, what is the expected number of stabilizer errors on the resulting state? To quantify this, we perform post-processing fusions on the state η: we fuse the first two modes (first qubit) of ηwith an ideal Bell pair, and the final two modes (second qubit) of ηwith a distinct ideal Bell pair, post-selecting for successful heralding. (This is done without additional error, as we are attempting to benchmark the output of the BSG, not the quality of these subsequent fusion operations.) Successful post-processing fusions may be viewed as mapping the four-mode state ηto the four-mode state on the unmeasured output modes; in the absence of distinguishable photons input to the post-processing fusions, this map is in fact a non-destructive Bell measurement. Further, since the surviving qubits come from the ideal Bell pairs, the output is always a dual-rail qubit state. Then it makes sense to take this state, measure its fidelity with each of the four Bell states, and use this to calculate the expected number of errors in the stabilizers {X⊗X, Z⊗Z}: E(NS) = 0⟨B++|η|B++⟩+ 1(⟨B+−|η|B+−⟩+⟨B−+|η|B−+⟩) + 2⟨B−−|η|B−−⟩. (18) For comparison with the previous two fidelity metrics, we also calculate the fusion fidelity , the fidelity of the state obtained after the post-processing fusions. Now, let ρbe the input state to the BSG protocol with a chosen error model. Recalling the discussion above, we will break ρinto a sum ρ=P c∈CP(c)ρc, where each photon in ρchas some internal state |ψi⟩, either ideal or fully distinguishable according to the appropriate error model, and Cis the set indexing all such states. Now let Xbe a random variable on the space of four-mode photonic density matrices of the appropriate form: we assume Xis obtained by applying some linear optical circuit, measuring a subset of modes and post-selecting for a subset of heralding outcomes, applying linear optical corrections as needed, then applying some random variable X′to the resulting output state. We let HXbe the event that the heralding conditions for Xare met. We may calculate the expected value of Xgiven successful heralding as follows: E(X|HX) =P c∈CP(c)P(HX|c)E(X|c∩HX)P c∈CP(c)P(HX|c). (19) Note that for Xcorresponding to the post-selected fidelity and number of stabilizer errors, the heralding probabilities P(HX|c)take into account the reduced probabilities due to the post-selection and post-processing fusions respectively. In Appendix A 3, we explicitly calculate this expected value, for each of the error models and random variables given above. To do this, we consider each c∈ C and directly compute the appropriate heralding probabilities P(HX|c)and expected values E(X|c∩HX)corresponding to the input state ρc. Note that the only appearance of the distinguishability error rate ϵis inP(c), which may be explicitly computed for each c∈ C(depending on the choice of error model). Thus we do not need to sample different values of ϵ, rather directly computing E(X|HX)as a function of ϵ. We refer to Appendix A 3 for the exact results and Figure 7 for comparisons. We make several observations regarding the numerics in Figure 7.

Question 994 multiple-choice

Quantum algorithms for group-theoretic problems have demonstrated significant speedups over classical approaches, particularly for the Hidden Subgroup Problem (HSP). Extending efficient solutions from abelian to nonabelian solvable groups is crucial for advancing cryptanalysis and computational algebra. Which statement accurately describes a key feature of recursive quantum algorithms for the Hidden Subgroup Problem in nonabelian solvable groups? 1) They rely exclusively on classical brute-force search in all subgroup cases. 2) They are limited to abelian groups due to the lack of suitable quantum techniques for nonabelian structures. 3) They avoid the use of quotient groups by focusing solely on the entire group. 4) They implement exponential-time quantum Fourier sampling for all group types. 5) They achieve speedup only in cyclic group scenarios, not in more complex group structures. 6) They require the group to be simple and non-solvable for recursion to apply. 7) They exploit self-reducibility by recursively solving the problem in quotient groups and subgroups to simplify computation in solvable nonabelian groups.

✓ Correct Answer:

The correct answer is 7) They exploit self-reducibility by recursively solving the problem in quotient groups and subgroups to simplify computation in solvable nonabelian groups..

📚 Reference Text:

SIAM J. C OMPUT . c/circlecopyrt2014 Society for I ndustrial and Applied Mathematics Vol. 43, No. 1, pp. 1–24 HIDDEN TRANSLATION AND TRANSLATING COSET IN QUANTUM COMPUTING∗ KATALIN FRIEDL†,G´ABOR IVANYOS‡,F R´ED´ERIC MAGNIEZ§, MIKLOS SANTHA¶, ANDPRANAB SEN/bardbl Abstract. We give eﬃcient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup ina largeclassof nonabelian solvablegroups, includingsolvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve eﬃciently Hidden Translation inZn p, whenever pis a ﬁxed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation andHidden Subgroup and prove a powerful self-reducibility result: Translating Coset in a ﬁnite solvable groupGis reducible to instances of Translating Coset inG/NandN, for appropriate normal subgroups NofG. Our self-reducibility framework, combined with Kuperberg’s subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation andHidden Subgroup in any solvable group. Key words. quantum algorithms, hidden subgroup problem, solvable groups AMS subject classiﬁcation. 68Q25 DOI.10.1137/130907203 1. Introduction. Quantum computing is an extremely active research area (for introductions, see, e.g., [30, 1, 36, 35]). Many of the superpolynomial speedups achieved by quantum algorithms over their best known classical counterparts havebeen in a group theoretical setting. In this setting, we are given a ﬁnite group Gand, besides the group operations, we also have at our disposal a function fmapping G into a ﬁnite set. The function fcan be queried via an oracle. The time complexity of an algorithm is measured by the overall running time, including both the queries (counting a query as one step) and the quantum and/or classical processing of thesequeries. The most important unifying problem of group theory for the purpose of quantum algorithms has turned out to be Hidden Subgroup ,w h i c hc a nb ec a s ti n the following broad terms: Let Hbe a subgroup of Gsuch that fis constant on each left coset of Hand distinct on diﬀerent left cosets. We say that fhidesthe subgroup H. The task is to determine the hidden subgroup H. ∗Received by the editors January 24, 2013; accepted for publication (in revised form) Septem- ber 13, 2013; published electronically January 2, 2014. A preliminary version of this paper appearedin [15]. This work was partially supported by the European Commission IST STREP projects Quan-tum Computer Science (QCS) 255961 and Quantum Algorithms (QALGO) 600700, by the FrenchANR Blanc program under contract ANR-12-BS02-005 (RDAM project), and by the Hungarian Na-tional Science Fund (OTKA) through grants NK105645 and K77476. Research at the Centre forQuantum Technologies is funded by the Singapore Ministry of Education and the National ResearchFoundation, and also through the Tier 3 Grant “Random Numbers from Quantum Processes.” http://www.siam.org/journals/sicomp/43-1/90720.html †Budapest University of Technology and Economics, Budapest, Hungary (friedl@cs.bme.hu). ‡Institute for Computer Science and Control, Hungarian Academy of Sciences, Budapest, Hungary (gabor.ivanyos@sztaki.hu). §C N R S ,L I A F A ,U n i v e r s i t ´ e Paris Diderot,

Question 995 multiple-choice

In quantum optimal control, the complexity of optimizing gate operations increases with the number of levels in a quantum system, influenced by symmetries and the presence of multiple global phases. Algorithms like GRAPE and Krotov are commonly used to find optimal control pulses for high-fidelity quantum gates. For a quantum system with N=6 levels (corresponding to spin I=3), which statement best describes the behavior of Pareto fronts for different global phases? 1) Pareto fronts for global phases differing by π are always widely separated due to broken symmetry. 2) Pareto fronts for global phases differing by π should coincide, except for minor numerical inaccuracies. 3) The number of distinct Pareto fronts is always limited to two, regardless of global phase differences. 4) No secondary solutions exist for systems with N greater than 3. 5) Global phase differences never affect the optimization landscape for quantum gates. 6) GRAPE and Krotov algorithms always produce identical Pareto fronts for all values of N. 7) For N=6, only one unique solution exists with respect to global phase.

✓ Correct Answer:

The correct answer is 2) Pareto fronts for global phases differing by π should coincide, except for minor numerical inaccuracies..

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even N = 2I + 1. For integer spins (odd N) it is not true and, general ly, the gates with different global phases should have different critical times. With increasing N the number of solutions with different global phases and the number of secondary solutions increase . As was mentioned in [11], this can lead to the situation when the Pareto fronts for different solutions lie very close to eac h other. While at N = 4 we still can clearly distinguish the main and secondary solutions and easily grade them to the global phase (Fig. 2), already at N = 5 such a differentiation becomes complicated (Fig. 4). Having minimized gate error 1J (22) for N = 5 for several hundred s of random initial pulses for each global phase and several control times T, we chose the ru ns with the minimum error and plotted the Pareto fronts for the corresponding global phase s / 5 2 / 5p p    , p=0, 1, 2, 3 , 4. The result obtained is presented in Fig. 5. The shortest time cT =1.90/ q corresponds to the global phase 7π/5. The figure al so shows the Pareto fronts for the secondary solutions. It can be seen that after 12 T > 3/q the curves of different solutio ns become very close to each other. This can make certain difficulties in optimizat ion of phase -independent gate error 2J (23) for these control times, since at random initial guess es the algorithm can converge to different solutions with objective function al 2J that differ by several ord ers of magnitude. Figure 6 shows the results for I = 3 (N = 6). As was mentioned above, in this case the Pareto fronts for the global phases differ ent by π should coincid e. The minor difference in such Pareto fronts (Fig. 6 ) is caused by numerical inaccuracy and can be eliminated by enhancing S and reducing T. The shortest time cT = 2.09/ q corresponds to the global phases 2π/3 and 5π/3 Note that the secondary solutions are not observed at I = 1. We attribute this fact to the simplicity of a three -level system. All of the several thousands of runs for different T and global phases converge very quickly to the global minimum of 1J lying on a corresponding Pareto front in Fig. 1. To verif y the numerical results , some of the calculations were repeated using the Krotov algorithm with no constrain ts imposed on a control field [29, 30]. Despi te the remarkable property of monotonic convergence , the characteristics of this algorithm are similar to those of the GRAPE first-order gradient method [30]. We observed no qualitative differences between the results obtained with the use of the Krotov an d BFGS -GRAPE algorithms. Quantitative differences lie within the errors caused by discretization and features of the algorithms. In particular, the convergence rate and accuracy of the Krotov algorithm are significantly

Question 996 multiple-choice

In quantum information theory, the programmability of quantum channels using measure-and-prepare programmable quantum processors is closely tied to concepts from convex geometry and group representation theory. Programmable quantum devices leverage mathematical structures such as polytopes and commutants to determine the feasibility and limitations of exact channel implementation. For a set of quantum channels covariant under a group representation, when is exact programmability with a measure-and-prepare quantum processor guaranteed, and what determines the required dimension of the program register? 1) When the group is non-abelian and the commutant is non-abelian; the program dimension equals the group's order 2) When the set of channels is infinite and the program register has infinite dimension 3) When the commutant of the group representation is abelian; the program register dimension equals the number of irreducible representations in the tensor product decomposition 4) Only when there is a single extreme point in the polytope; the program dimension is always one 5) When the POVM has more outcomes than the number of irreps; the program dimension equals the number of POVM outcomes 6) When the multiplicity of irreps is greater than one; the program register dimension equals the multiplicity 7) Only if the set of channels forms a simplex; the program dimension is arbitrary

✓ Correct Answer:

The correct answer is 3) When the commutant of the group representation is abelian; the program register dimension equals the number of irreducible representations in the tensor product decomposition.

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Accepted in Quantum2021-06-18, click title to verify. Published under CC-BY 4.0. 9 with extreme points Tk(·). We encode the{xk}k∈{1,...,K}in the program state as follows |ψT/angbracketright=K/summationdisplay k=1√xk|k/angbracketright∈DP(HP), with an arbitrary orthonormal basis {|k/angbracketright}k∈{1,...,K}onHP. The following processor im- plementsT∈Cexactly with a program register of dimension dP=K: P(A⊗B) = trHP/bracketleftBiggK/summationdisplay k=1Tk(A)⊗/angbracketleftk|B|k/angbracketright|k/angbracketright/angbracketleftk|/bracketrightBigg ∀A∈B(H1),∀B∈B(HP) and extended by linearity. We verify that this is indeed a measure-and-prepare PQP C: P(ρ⊗|ψT/angbracketright/angbracketleftψT|) =K/summationdisplay k=1Tk(ρ)tr[Ek|ψT/angbracketright/angbracketleftψT|] =K/summationdisplay k=1xkTk(ρ) =T(ρ), where,Ek=|k/angbracketright/angbracketleftk|for allk∈{1,...,K}. Thus, we showed that if JCis isomorphic to a polytope, there is a measure-and-prepare processor Pthat implements all T∈Cexactly with program dimension dim(HP) =dP=K. Conversely, letPbe a measure-and-prepare PQP Cwith aK-outcome POVM. Then, Cis the convex hull of at most Kextreme points, since by deﬁnition for any T∈C T=P(·⊗πT) =K/summationdisplay k=1pkTk withpk= tr[EkπT], i.e.(p1,...,pK)isaprobabilitydistribution. Thus, theextremepoints ofCare a subset of{T1,...,TK}and henceCis a polytope. The following corollary assures that if we want to check whether channels T∈TUVare pro- grammable exactly by a measure-and-prepare CPQP UVwith ﬁnite-dimensional program register, we can consider the speciﬁc structure of the commutant K. Corollary 19. LetUbe an irrep of a compact group GonH1andVa representation of GonH2. Furthermore, let Kbe the commutant of ¯U⊗V. Then,Kis abelian if and only if there is a measure-and-prepare CPQP UVPthat implements all T∈TUVexactly with dP≤K, whereKis the number of irreps appearing in the direct sum decomposition of ¯U⊗V. Proof.The proof directly follows from Lemma 16 and Proposition 18. We give an example of a group for which exact programmability holds with a measure- and-prepare CPQP UV. Example20 (Unitarilycovariantchannels) .We considera measure-and-prepare CPQP UV that implements all UU-covariant channels where Uis the deﬁning representation of the groupG=U1. The tensor representation ¯U⊗Ucan be decomposed into a direct sum of a one-dimensional and a (d2−1)-dimensional irrep both with multiplicity 1. Thus, elements of the commutant consist of a one-dimensional block with multiplicity equal to one and a one-dimensional block with multiplicity d2−1, i.e. in this case K= 2,n1=b1= 1,n2= 1 andb2=d2−1. Due to Lemma 11, we know that the Choi-Jamiołkowski state cT∈JUU satisﬁes [cT,¯Ug⊗Ug] = 0. The state space therefore has the following form: K∩D (H1⊗H 1) =/braceleftbigg ˆα1 d2+ (1−ˆα)|Ω/angbracketright/angbracketleftΩ|/vextendsingle/vextendsingle/vextendsingleˆα∈/bracketleftBig 0,d2 d2−1/bracketrightBig/bracerightbigg Accepted in Quantum2021-06-18, click title to verify. Published under CC-BY 4.0. 10 with extreme points |Ω/angbracketright/angbracketleftΩ|and1 d2−1(1−|Ω/angbracketright/angbracketleftΩ|). Thus,Thas the form T(·) = ˆαtr(·)1 d+ (1−ˆα)id[26, 40]. Every cTcan be written as convex combination of the two extreme points and the set of all convex combinations K∩D (H1⊗H 1) =/braceleftbigg x|Ω/angbracketright/angbracketleftΩ|+ (1−x)1 d2−1(1−|Ω/angbracketright/angbracketleftΩ|)/vextendsingle/vextendsingle/vextendsinglex∈/bracketleftbig0,1]/bracerightbigg is isomorphic to a 1-simplex. Thus, the measure-and-prepare CPQP UVcan be implemented with program dimension dP= 2using the construction in Proposition 18. 3.2 Structure of the commutant of the tensor representation Based on the previous section, one could argue that in the case where the direct sum decomposition of ¯U⊗Vconsists of at least one one-dimensional irrep with multiplicity nk>1, it would not be possible to implement the corresponding UV-covariant channel exactly in this case. It is instructive to see why this situation never arises, which we will discuss in this section. The argument is the following: Assume d1=d2=dand

Question 997 multiple-choice

Quantum algorithms for lattice problems often rely on representing lattice points as quantum states and using techniques such as Fourier sampling to extract structural information. Managing numerical precision and encoding methods is critical when working with high-dimensional lattices and algebraic number fields. Which statement accurately describes the role of Fourier sampling in quantum algorithms for lattice reconstruction in number fields? 1) It generates superpositions over unit cell volumes, allowing direct computation of the shortest vector. 2) It produces samples in the original lattice, enabling immediate identification of all basis vectors. 3) It eliminates the need for ideal multiplication by reconstructing the lattice from classical measurements. 4) It computes the exact basis for the lattice using only a single quantum measurement. 5) It yields samples from the reciprocal lattice, and repeated measurements allow reconstruction of the lattice structure. 6) It directly solves the Hidden Subgroup Problem by measuring in the standard basis only. 7) It guarantees exact computation with algebraic numbers without concerns for floating-point precision.

✓ Correct Answer:

The correct answer is 5) It yields samples from the reciprocal lattice, and repeated measurements allow reconstruction of the lattice structure..

📚 Reference Text:

coeﬃcient vectors x∈Znthat repre- sent the lattice points relative to the basis. Then for eachx∈Znwe create the corresponding lattice point z=Bx∈Las a quantum state. In doing so, we use a straddle encoding str n,ν(to be deﬁned in Sect. 5) to ac- count for rounding errors. The original value of xmay now be erased (in a reversible way, which requires the reconstruction of xfromez≈z). The resulting state,|˜f(L)/angbracketright =cP z∈Le−π/bardblz/bardbl2/s2˛˛strn,ν(z)¸ ,does not de- pend on the basis (except for small inaccuracies in the preparation of the Gaussian superposition). Assum- ing a lower bound on the length of a shortest vector, λ1(L)/greaterorequalslantλ, and an upper bound on the unit cell vol- ume,d(L)/lessorequalslantd, we ﬁnd a Lipschitz constant of the function ˜fand estimate the inner product /angbracketleft˜f(L)|˜f(L/prime)/angbracketright when the lattices LandL/primeare far apart. •An eﬃcient quantum algorithm for ﬁnding a hidden subgroup in an elementary Abelian group (as discussed in the introduction). Such groups are quotients of Rk× Zl, therefore it is suﬃcient to consider this case. The problem is further reduced to the HSP for G=Rm. The algorithm has the usual structure. We create a superposition of points in Rmwith a suﬃciently broad wavefunction w, apply the oracle, and measure in the Fourier basis. In the ﬁrst approximation, the Fourier sampling generates a point uof the reciprocal lattice L∗with the probability distribution qu=1 d(L)2Z (Rm/L)2/angbracketleftf(x/prime)|f(x)/angbracketrighte2πi/angbracketleftx−x/prime,u/angbracketrightdxdx/prime. Repeating the procedure suﬃciently many times, we obtain a set of vectors that generate L∗. In prac- tice, the Fourier samples deviate from the lattice points by, roughly, the inverse width of the wavefunction w. Then the lattice is reconstructed from an approximate generating set (see Section 4.3). All of our results are stated for the ring of integers Oof a given number ﬁeld K, but they can easily be extended to general orders of K. 4. COMPUTING A BASIS FOR etO In this section we show how to compute an approximate basis for the lattice etO. Because etis in general doubly exponential in size and we have to use ﬂoating point com- putations, this is a non-trivial operation. The basic steps are to alternate ideal multiplication with size reduction to compute a short basis for the product of the two ideals that were multiplied. The algebraic numbers that appear in this computation would take exponentially many bits to repre- sent exactly. Instead we show that a polynomial number of bits of precision is suﬃcient. The idea is to use the fact that we are always using ideal lattices with lower bounds and up- per bounds on the vector lengths appearing throughout the computation, so that the precision loss can be bounded at each step. With this we can pick a precision high enough, some polynomial number of bits, so that we still have high precision at the end. The precision we need is that for any vector of length at most s√n, the computed vector is within 1/(2N ) of the actual vector. Here sandNare parame- ters chosen such that the lattice Gaussian superposition in Section 5

Question 998 multiple-choice

Quantum computing leverages group-theoretical structures to solve complex computational problems. The hidden subgroup problem (HSP) is central to developing efficient quantum algorithms, especially for non-Abelian groups where general solutions remain elusive. Which group-theoretic property has been shown to enable efficient quantum algorithms for solving certain non-Abelian hidden subgroup problems? 1) Existence of a large center in the group 2) Presence of a nontrivial simple subgroup 3) Having an elementary Abelian normal 2-subgroup of small index 4) Group order being a prime power 5) Existence of a maximal torus 6) Containing a non-cyclic abelian subgroup 7) Having a trivial automorphism group

✓ Correct Answer:

The correct answer is 3) Having an elementary Abelian normal 2-subgroup of small index.

📚 Reference Text:

Title: Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem Year: 2001 Paper ID: aad78939cb9b78b9acd1af54970100d6bae823f6 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/aad78939cb9b78b9acd1af54970100d6bae823f6 Abstract: In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.

Question 999 multiple-choice

In geometric group theory, groups constructed via graphs of groups can exhibit significant properties such as separability, residual finiteness, and cohomological goodness. These properties are crucial for understanding the algebraic and topological behavior of groups that generalize those arising from 3-manifold decompositions. Which of the following properties is true for extended admissible groups constructed from graphs of groups under mild conditions on the vertex groups? 1) Every subgroup is malnormal. 2) All subgroups are virtually nilpotent. 3) The group has Kazhdan’s property . 4) The group is simple and non-amenable. 5) The group is hyperbolic relative to abelian subgroups. 6) Finitely generated abelian subgroups are separable. 7) The group is virtually free.

✓ Correct Answer:

The correct answer is 6) Finitely generated abelian subgroups are separable..

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Title: Separability properties of extended admissible groups Year: 2025 Paper ID: a0487c6f260e04aadbb5e1893f5db09a5ee901fd Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a0487c6f260e04aadbb5e1893f5db09a5ee901fd Abstract: Abstract Extended admissible groups belong to a particular class of graph of groups which possess a graph of groups decomposition generalizing that of any non-geometric 3-manifold and Croke–Kleiner admissible groups. In this paper, under mild conditions on vertex groups of an extended admissible group G, we show that finitely generated abelian subgroups of G are separable (in particular G is residually finite and has solvable word problem), cohomologically good, and the profinite topology on G is efficient. These results extend those in [E. Hamilton, Abelian subgroup separability of Haken 3-manifolds and closed hyperbolic n-orbifolds, Proc. Lond. Math. Soc. (3) 83 2001, 3, 626–646] and [H. Wilton and P. Zalesskii, Profinite properties of graph manifolds, Geom. Dedicata 147 2010, 29–45] from 3-manifold groups to a wider class of graphs of groups. Finally, we establish that G has property (QT) in the sense of Bestvina–Bromberg–Fujiwara.

Question 1000 multiple-choice

Quantum algorithms offer the potential for dramatic speedups in solving algebraic problems, especially those framed within the theory of universal algebras. The Hidden Kernel Problem (HKP) is a generalization of the Hidden Subgroup Problem (HSP), extending it to broad classes of algebraic structures. Which statement accurately characterizes the classification of polynomial-time solvability for the Hidden Kernel Problem (HKP) in powers of 2-element universal algebras? 1) All powers of 2-element universal algebras admit efficient quantum solutions to HKP, but none admit classical solutions. 2) No power of any 2-element universal algebra permits a polynomial-time solution to HKP, either quantum or classical. 3) Only those powers corresponding to non-abelian group structures admit quantum speedup for HKP. 4) The classification depends solely on the existence of normal subgroups in the algebraic structure. 5) Powers of 2-element universal algebras that are rings with identity always allow quantum solutions to HKP. 6) The classification pinpoints exactly which powers of 2-element universal algebras admit polynomial-time quantum or classical solutions to HKP and which remain hard for both models. 7) HKP is always computationally equivalent to HSP for any universal algebra structure.

✓ Correct Answer:

The correct answer is 6) The classification pinpoints exactly which powers of 2-element universal algebras admit polynomial-time quantum or classical solutions to HKP and which remain hard for both models..

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The HSP for the symmetric groups and the dihe- dralgroupscorrespondtothe graphisomorphism andshort- est vector problems, respectively, and so these cases are of particularinterest. Despite focusedresearch over theyear s, LICS’20,July8–11,2020,Saarbrücken,Germany Ma/t_thew Moore and TaylorWalenczyk no general polynomial-time quantum algorithm has been found. WeproposeageneralizationoftheHiddenSubgroupProb- lem to arbitrary algebraic structures called universal alge- bras, and call the generalized problem the Hidden Kernel Problem (HKP) .Perhapsthesimplest non-trivial instance of theHSPiswhenG=Zn 2,andthisisknownasSimon’sProb- lem. Along the same lines, we consider the Hidden Kernel Problemforpowersof2-elementstructures,andgiveacom- plete classiﬁcation of which structures admit a polynomial- timequantumsolution,apolynomial-timeclassicalsolution, no polynomial-time classical solution, and no polynomial- time quantum solution. In particular, we fully classify all powers of 2-element universal algebras for which a quan- tumalgorithmachievessuper-polynomialspeedupwhencom- pared to aclassical algorithm. We begin with some necessary background in Universal Algebra and Clone theory, as well as a brief discussion of QuantumComputationinSection 2.InSection 3wedevelop and state the Hidden Kernel Problem and outline the plan of attack for analyzing the HKP on powers of 2-element universal algebras (ofwhich there are inﬁnitely many).We determine which of these algebras admit polynomial-time quantum and classical solutions to their HKP in Section 4, and in Section 5we determine which have an exponential lower-bound for classical or quantum solutions. Taken to- gether,theseresultsprovideacompleteclassiﬁcationoft he quantum as well as classical computational complexity of the HKP on powers of 2-element universal algebras. Sec- tion6concludeswithanoverviewoftheresultsaswellasa discussionofsomeopenquestionsabouttheHiddenKernel Problem. 2 Background UniversalAlgebraseekstoincludeandextendclassicalalge- braic results (for instance, from group theory and ring the- ory).Inbringingtogetherdiverse areasofresearch,Univer- sal Algebra provides a common scaﬀolding upon which to build general techniques. “What looks messy and compli- cated in a particular framework may turn out to be simple and obvious in the proper general one,” as Smith [ 20] de- scribesit. The subsections below give a brief overview of the top- icsneededforthelatersections,andthoughself-contained, they are necessarily brief. The interested reader is encour- aged to consult [ 13] and [21], which are both good refer- encesforthealgebraictopics.Thesectionconcludeswitha shortoverview ofQuantumComputation,forwhich[ 15]is a moreextensive reference. 2.1 UniversalAlgebra Given a set A, anoperation onAis a function f:An→A, wheren∈Nisthearityof f.ThesetAtogetherwithsomeoperations (fi)i∈Iisauniversal algebra ,commonlydenoted A.We specifythe set and the operationsbywriting A=/a\}⌊ra⌋ketle{tA;(fi)i∈I/a\}⌊ra⌋ketri}htorA=/a\}⌊ra⌋ketle{tA;f1,...,fk/a\}⌊ra⌋ketri}ht, dependingonwhether Iisﬁnite.Anon-emptysubset B⊆A whichisclosedunderalloperations fiofAiscalleda subal- gebra,writtenB≤A.IfC⊆Aisnon-empty,thenwedeﬁne thesubalgebrageneratedbyC tobethe smallestsubalgebra containing C,writtenSg (C).Thesegeneraldeﬁnitionsofal- gebras and subalgebras include most of the classical alge- braic structures (groups, rings, boolean algebras, etc.) and allowfor theuniﬁcation of ahost of results. Theappropriate generalization ofa normalsubgroup for a universal algebra is a congruence . Precisely, a congruence θofanalgebraAisabinaryequivalencerelationontheset Awhichis compatiblewith the operationsof A,meaning a1θ b1,...,akθ bk⇒f(a1,...,ak)θ f(b1,...,bk) for all operations fofA(inﬁx notation is used here for bi- nary relations). As with subalgebras, given a set D⊆A2, we deﬁne the congruence generated by Dto be the smallest congruencecontaining D,writtenCg (D).Similartonormal subgroups, the quotient structure A/θis well-deﬁned, and general versions of the classic isomorphism theorems for groupsandringsholdtrue in thiscontext. Two algebrasAandBare said to be similarif there is a bijectivecorrespondencebetweenthe k-aryoperationsofA andBforeachk.Inthiscase, fAiswrittenfortheoperation ofAcorresponding to the operation fBofB. When there is no chance for confusion, however, these superscripts are omitted and we simply use the same symbol ffor bothfA andfB.Ahomomorphism betweensimilarstructures Aand Bisafunction φ:A→Bsuchthatforeach k-aryoperation fAofAand alla1,...,ak∈A, φ/parenleftbigfA(a1,...,ak)/parenrightbig=fB/parenleftbigφ(a1),...,φ(ak)/parenrightbig (i.e.φiscompatiblewiththeoperationsof

Question 1001 multiple-choice

In quantum computing, the fidelity of a quantum circuit reflects how closely an experimentally prepared quantum state matches its ideal counterpart, which is crucial for evaluating performance on noisy devices such as NISQ computers. Optimizing circuit structure and depth can enhance fidelity when implementing quantum algorithms like continuous-time quantum walks on graph-based networks. Which strategy is specifically employed to reduce circuit depth and improve fidelity when simulating continuous-time quantum walks on circulant graphs using current quantum hardware? 1) Increasing the number of qubits in the quantum processor 2) Using classical random walks for pre-processing 3) Replacing quantum measurements with classical post-selection 4) Implementing error mitigation techniques after execution 5) Applying the full quantum Fourier transform without approximation 6) Utilizing the approximate quantum Fourier transform (AQFT) 7) Encoding graphs with non-circulant adjacency matrices

✓ Correct Answer:

The correct answer is 6) Utilizing the approximate quantum Fourier transform (AQFT).

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structures is highly concentrated in a small number for vertices, which means they are more susceptible to noise introduced by NISQ computers. Nonetheless, the increase of mstill has a positive impact on the ﬁdelity of these circuits. mnGG1 G2 G3 G4 G5 G6 G7 G8 0 0.470.03 0.610.02 0.780.02 0.860.01 0.860.01 0.700.04 0.540.03 0.490.04 1 0.500.03 0.630.03 0.790.03 0.870.02 0.850.03 0.700.03 0.550.05 0.500.04 2 0.550.03 0.710.03 0.830.02 0.900.01 0.890.02 0.750.02 0.620.04 0.590.06 3 0.600.03 0.700.02 0.850.01 0.920.01 0.910.01 0.800.03 0.710.04 0.690.04 Table 3. Fidelity of quantum state with N=16, backend Toronto , and t=1. (a)v1= [0;1;0;0;0;0;0;1]T (b)v2= [0;1;1;0;1;0;1;1]T Figure 5. Pretty good state transfer for different circulant graphs deﬁned by v. We implemented pretty good state transfer (PGST) which is a transport phenomon that occurs in quantum walks when the initial state is restricted to a single excitation space. PGST occurs between vertex iandjif, for any>0, there exists a time such that jU( )ijj>1: (9) It was proved by Pal et al. [Pal and Bhattacharjya 2017] that PGST occurs in some classes of circulant graphs at = 2n, wherenis a positive integer. One of those classes is cycles with 2kvertices (C2k), wherek3, and the other is the union C2k[G(2k), with G(2k)being circulant graphs with connection set given by gcd-sets of Z2k. We chose one graph of each class, v1= [0;1;0;0;0;0;0;1]Tandv2= [0;1;1;0;1;0;1;1]T, and we obtained a ﬁdelity of 0:930:01forv1and0:9060:009forv2. Figure 5 shows the comparison between the simulated model in blue and the experimental result in orange. 4. Conlusion and Remarks We presented a general framework of implementing the CTQW for an arbitrary circulant graph of any size. Previous work by [Qiang et al. 2016] implemented the CTQW through the QFT for a complete graph of small size, however our method takes advantage of the Qiskit package making it possible to create a circuit for absolutely any circulant graph while making use of the AQFT in order to reduce circuit depth. Contrasting our statistical analysis to the work of [Acasiete et al. 2020], we offer a more accurate overview of the results of running a quantum walk on a superconducting NISQ device, since we calculated the average ﬁdelity of several instances instead of choosing the best case. Finally, we presented two cases of PGST were we also explored the effects of the AQFT. References Acasiete, F., Agostini, F. P., Moqadam, J. K., and Portugal, R. (2020). Implementation of quantum walks on IBM quantum computers. Quantum Information Processing , 19(12). Ambainis, A. (2008). Quantum Algorithm for Element Distinctness , pages 686–689. Springer US, Boston, MA. Balu, R., Castillo, D., and Siopsis, G. (2018). Physical realization of topological quantum walks on IBM-q and beyond. Quantum Science and Technology , 3(3):035001. Bullock, S. S. and Markov, I. L. (2004). Smaller circuits for arbitrary n-qubit diagonal computations .Quant. Info. and Comp. , 4(27). Chakraborty, S., Luh, K., and Roland, J. (2020). How fast do quantum walks mix? Phys. Rev. Lett. , 124:050501. Childs, A. M. and Goldstone, J. (2004). Spatial search by quantum walk. Physical Review

Question 1002 multiple-choice

In early universe cosmology, scalar fields known as moduli can significantly affect the thermal history of the universe through their dynamics and decay. The thermalization process following moduli decay is crucial for preserving the conditions required for primordial nucleosynthesis. Which condition must be satisfied by the reheating temperature after moduli decay to ensure successful primordial nucleosynthesis in the early universe? 1) The reheating temperature must be less than 1 MeV. 2) The reheating temperature must be exactly equal to the modulus mass. 3) The reheating temperature must be higher than the temperature of cosmic microwave background decoupling. 4) The reheating temperature must fall within the range 1–2 MeV. 5) The reheating temperature must be lower than the temperature at which structure formation begins. 6) The reheating temperature must match the wino freeze-out temperature. 7) The reheating temperature must exceed 5 MeV.

✓ Correct Answer:

The correct answer is 7) The reheating temperature must exceed 5 MeV..

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for a hidden sector LSP with a relic density lower than that of the wino and which is small enough to evade the current bounds from indirect detection. In Section V we extend the U(1)xhidden sector to include an asymmetric dark matter candidate and nd that it is able to saturate the entire observed DM relic density while avoiding constraints from indirect detection. In Section VI we investigate a pure non-Abelian hidden sector, and show that the corresponding gaugino LSP can provide an acceptable relic density and avoid constraints from indirect detection, although it is also very strongly constrained by its e ect on structure formation and the cosmic microwave background. Finally, Section VII is reserved for our conclusions. II. MODULI REHEATING AND DARK MATTER In this section we review brie y the cosmology of moduli oscillation and decay, as well as the associated non-thermal production of dark matter. A. Moduli Reheating A modulus eld 'is very likely to develop a large initial displacement from the minimum of its potential before or during the course of primordial in ation [1, 10]. Hubble damping will trap the modulus until Hm', at which point it will start to oscillate coherently. For even moderate initial displacements, these oscillations will eventually dominate over radiation. The time evolution of the modulus oscillation energy density for H

Question 1003 multiple-choice

In quantum computation, the synthesis of arbitrary unitary operations often relies on decomposing them into sequences generated by physically available Hamiltonians. The number of steps required for such decompositions depends on geometric relationships between the corresponding rotation axes. Given two Hamiltonians generating rotations with an angle α between their axes, which of the following statements is true about the minimal number ξ of alternations required to achieve universal coverage when α = π/2? 1) ξ = 2, corresponding to a single decomposition step 2) ξ = 3, corresponding to the Euler decomposition 3) ξ = 4, due to the need for an extra alternation 4) ξ = 5, requiring additional decomposition steps 5) ξ = 6, based on the smallest angle between axes 6) ξ = k+2 for all values of k if α = π/2 7) ξ is undefined for perpendicular axes

✓ Correct Answer:

The correct answer is 2) ξ = 3, corresponding to the Euler decomposition.

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this minimal number. The reason for this is that each point of the Bloch sphere (and each point in the complex plane) can be identiﬁed pointwise with right cosets of rotations of the Blochsphere representing all unitary operations. To d istinguish the rotations in one coset in general this additional step is necessary. 1.4 Efﬁcient Decomposition of Quantum Operations into Given One-parameter Groups 7 In [Low71], ﬁgures similar to Figure 1.2 are analyzed. The analysis implies: Theorem 1.1 Letαdenote the angle between two Hamilton operators in the Bloch vector representation and ξthe order of generation. 1.α=π/2=⇒ξ=3 (Euler decomposition), 2.∀k≥2:π/(k+1 )≤α<π / k =⇒ξ≤k+2. -20-1001020 y -20 -10 10 20 x -20-1001020 y -20 -10 10 20 x Figure 1.2: As described in the text, the ﬁgure shows tangent circles correspond- ing to the alternating application of twoHamilton operators to the state space vi-sualized through stereographic projection to the complex plane. One further step in the alternating application of Hamil-ton operators is nece ssary and sufﬁcient to cover all of the complex plane. Notice that for better orientation one Apollonian circle with inﬁnite radius is pictured by avertical line. The angle between the rota-tion axes in the Bloch sphere representa- tion belonging to this ﬁgure is (2 ·π/4+ π/5)/3. In this case we get an order of generation less than or equal to 6.Figure 1.3: As in Figure 1.2 tan- gent circles corresponding to the appli- cation of three Hamilton operators to thestereographically pr ojected state space are shown in the complex plane. To cover all of the complex plane no further step in the alternating application of Hamiltonoperators is needed. Notice that for betterorientation two Apollonian circles with inﬁnite radius are shown by two lines. In this example the angles between the ro-tation axes are (2 ·π/4+π/5)/3,π/5, andπ/6. Though all angles are less than or equal to the angle in example of Fig- ure 1.2 we get a smaller upper bound of 5 steps. The next step is to generalize the analysis from two to three Hamilton operators. Since we have a third Hamilton operator we have to consider a third axis of rotation in the representationof Hamilton operators as inﬁnitesimal rotations of the Bloch sphere leading to a third systemof Apollonian circles. The three axes of rotation can be characterized by specifying the three 8 1 Algorithms for Quantum Systems — Quantum Algorithms angles α1,α2,a n dα3between them. We get a valid triple of angles if α3∈[min{π−(α1+α2),|α1−α2|},π/2]. As in the case of two Hamilton operators it is important to determine the minimal number of terms in a product of one-parameter groups necessary to transform each point in the complex plane to zero. In Figure 1.3, the tangent circles describing the application of the differentHamilton operators can be received from a similar greedy-type algorithm. As remarked inFigure 1.3 we can get with this type of analysis better bounds on the order of generation when using three Hamilton operators even though no greater angles between two rotation axes are present. Summarizing,

Question 1004 multiple-choice

Quantum programming increasingly relies on recursive techniques to design scalable and efficient circuits for advanced quantum algorithms. Formal proof systems play a critical role in verifying the correctness of these complex, recursively defined quantum circuits. Which of the following statements correctly characterizes a "sound and relatively complete" formal proof system for verifying recursively defined quantum circuits? 1) It guarantees that all quantum algorithms are efficient and error-free. 2) It ensures that all recursive circuits can be executed on any quantum hardware. 3) It provides a way to optimize quantum gates for minimal resource usage. 4) It proves that all possible quantum states can be prepared with the system. 5) It certifies that all provable circuits are correct, and the system can verify all circuits expressible within its logical framework. 6) It allows for the automatic synthesis of quantum random-access memory circuits. 7) It enables direct translation of classical recursive algorithms to quantum equivalents.

✓ Correct Answer:

The correct answer is 5) It certifies that all provable circuits are correct, and the system can verify all circuits expressible within its logical framework..

📚 Reference Text:

Title: Verification of Recursively Defined Quantum Circuits Year: 2024 Paper ID: fb5d585195641d1aa10423074ca6e23a196179d6 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/fb5d585195641d1aa10423074ca6e23a196179d6 Abstract: Recursive techniques have recently been introduced into quantum programming so that a variety of large quantum circuits and algorithms can be elegantly and economically programmed. In this paper, we present a proof system for formal verification of the correctness of recursively defined quantum circuits. The soundness and (relative) completeness of the proof system are established. To demonstrating its effectiveness, a series of application examples of the proof system are given, including (multi-qubit) controlled gates, a quantum circuit generating (multi-qubit) GHZ (Greenberger-Horne-Zeilinger) states, recursive definition of quantum Fourier transform, quantum state preparation, and quantum random-access memories (QRAM).

Question 1005 multiple-choice

In information theory, the entropy region for multiple random variables describes the set of possible joint entropy values and is vital for understanding network coding efficiency. The Ingleton inequality imposes a key constraint on the entropy region, particularly in systems involving four random variables. Which of the following developments most directly challenges the previously accepted Four-Atom Conjecture regarding the upper bound of Ingleton inequality violations in the entropy region? 1) Discovery that linear network codes can achieve negative Ingleton scores 2) Proof that all abelian groups always satisfy the Ingleton inequality 3) Demonstration that the entropy region coincides with the polymatroid region 4) Construction of new examples using only abelian group structures 5) Reporting an Ingleton score of 0.0925000777, exceeding the Four-Atom Conjecture's bound 6) Establishment that optimization methods cannot surpass existing Ingleton scores 7) Confirmation that four random variables never violate the Ingleton inequality

✓ Correct Answer:

The correct answer is 5) Reporting an Ingleton score of 0.0925000777, exceeding the Four-Atom Conjecture's bound.

📚 Reference Text:

Title: Violations of the Ingleton inequality and revising the four-atom conjecture Year: 2020 Paper ID: b58ecf5ee5baf9fe6b19adf4d6c32faf59e7387a Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b58ecf5ee5baf9fe6b19adf4d6c32faf59e7387a Abstract: The entropy region is a fundamental object of study in mathematics, statistics, and information theory. On the one hand, it involves pure group theory, governing inequalities satisﬁed by subgroup indices, whereas on the other hand, computing network coding capacities amounts to a convex optimization over this region. In the case of four random variables, the points in the region that satisfy the Ingleton inequality (corresponding to abelian groups and to linear network codes) form a well-understood polyhedron, and so attention has turned to Ingleton-violating points in the region. How far these points extend is measured by their Ingleton score, where points with positive score are Ingleton-violating. The Four-Atom Conjecture stated that the Ingleton score cannot exceed 0 . 089373, but this was disproved by Mat´uˇs and Csirmaz. In this paper we employ two methods to investigate Ingleton-violating points and thereby produce the currently largest known Ingleton scores. First, we obtain many Ingleton-violating examples from non-abelian groups. Factorizability appears in many of those and is used to propose a systematic way to produce more. Second, we rephrase the problem of maximizing Ingleton score as an optimization question and introduce a new Ingleton score function, which is a limit of Ingleton scores with maximum unchanged. We use group theory to exploit symmetry in these new Ingleton score functions and the relations between them. Our approach yields some large Ingleton scores and, using this methodology, we ﬁnd that there are entropic points with score 0 . 0925000777, currently the largest known score.

Question 1006 multiple-choice

Quantum mechanical derived force fields are increasingly used to model complex intermolecular interactions, especially in systems where traditional classical force fields lack accuracy. Halogen bonding, involving interactions between halogen atoms and electron-rich sites, is a key challenge in molecular simulation and design. Which of the following best describes the protocol for parameterizing quantum mechanical derived force fields (QMD-FFs) for halogen bond-forming molecules such as trichloro- and dichloro-acetonitrile? 1) Fitting force field parameters only to experimental thermodynamic data from bulk-phase measurements 2) Using empirical energy functions without benchmarking against quantum calculations 3) Optimizing force field parameters exclusively for isolated molecules, ignoring dimer interactions 4) Employing classical force fields that neglect polarization and anisotropy effects 5) Parameterizing intermolecular terms using the JOYCE protocol and intramolecular terms with the PICKY approach 6) Combining non-polarizable force fields with fixed-charge models for all molecular interactions 7) Optimizing intramolecular terms using the JOYCE protocol and intermolecular terms using the PICKY approach, followed by benchmarking against high-level quantum chemical methods and sampling configurations through molecular simulations

✓ Correct Answer:

The correct answer is 7) Optimizing intramolecular terms using the JOYCE protocol and intermolecular terms using the PICKY approach, followed by benchmarking against high-level quantum chemical methods and sampling configurations through molecular simulations.

📚 Reference Text:

nearest carbon atom, should be able to add part of the require d anisotropy to the descrip- tion of the system. To further investigate the behavior of QM D-FFs, based on standard 7 model potential functions ( i.e.without considering polarization nor ESH), in presence of halogen bond forming molecules, two more targets were inclu ded in the benchmark set, i.e.trichloro- and dichloro-acetonitrile. In fact, in addicti on to the interactions between halogens, already accounted for in the rest of the benchmark set, these compounds pro- vide an example of intermolecular contributions that stem f rom C-H ···N and C-Cl ···N contacts.70Standard transferable FFs parameters are not able to mimic s uch interactions, thus a more accurate FF seems required. Aside from this aspect , once the reliability of the present QMD-FF is validated, most of the thermodynamics properties that will be computed for N1andN2can be considered as reliable predictions, ﬁlling the gap ca used by limited number of experimental data available for these l ast targets. 2 Computational Details 2.1 Parameterization Protocol As previously reported,22,32,33,40the QMD-FF parameterization here proposed is based on optimizing the FF parameters which allow for a classical d escription of a given system of target molecules versus the representation obtained at a QM level. This is separately done for one isolated molecule ( A) and for an interacting dimer ( AB), exploiting the classical partition of the total FF energy ( EFFtot AB)of a pair of molecules: EFFtot AB=EFFintra A(¯bA,¯θA,¯ψA)+EFFintra B(¯bB,¯θB,¯ψB)+EFFinter AB(¯rAB) (1) EFFintra AandEFFintra Bdrive respectively molecule Aand molecule Bﬂexibility, which in turn depends only on the collection of internal coordinat es (bonds ( ¯b), angles ( ¯θ) or dihedrals ( ¯ψ)) of the related monomer. On the contrary, EFFinter ABdepends on the relative position and orientation of the two monomers, being a functi on of the set of distances ¯rABbetween all interaction sites of molecule Aand those of molecule B. According to the procedures devised in our group, the intramolecular par t of the QMD-FF, EFFintrais obtainedthroughthe Joyceprotocol,22,33whiletheintermolecularcontribution, EFFinter, isoptimizedthroughthe Pickyapproach.32,40Bothsoftwarepackagesarefreelyavailable at http://www.pi.iccom.cnr.it/joyce and at http://www.p i.iccom.cnr.it/picky. As shown in Figure 2, once parameterized, the intramolecular contri bution (blue path on the left 8 Intra-molecular FF Inter-molecular FF JOYCE PICKY DFT-D sampling of dimer ʼs IPES MC simulations PICKY DFT benchmarking vs . CCSDT(T)@cbs MD simulations Bulk phase thermodynamic properties QM data on single molecule • Equilibrium geometry • Hessian matrix • Large amplitude motion energy proﬁles Starting MM data on bulk • Starting conﬁguration from an equilibrated (MC or MD) bulk phase • Trial FF A priori info Sample of 50 dimers QMD -FF Figure 2: The parameterization route to QMD-FF proposed in this work f or halogenated hydrocarbons. Theintramolecularandintermolecularparameterizations tepsarehighlightenedwithblueandredframes, while the (optional) CCSD(T)@cbs validation is displayed i n green on the right. side) and the intermolecular term (red path on the right side ) are then joined together to buildaQMD-FF.Abriefdescriptionoftheparameterization routetoQMD-FFillustrated in Figure 2 is given in the following. Further details can be f ound in the Supporting Information and in

Question 1007 multiple-choice

In mathematical physics, matrix models with permutation symmetry are analyzed using tools from algebra and representation theory. Partition algebras play an important role in generalizing the symmetry considerations beyond those of the symmetric group. Which of the following statements accurately characterizes partition algebras in the study of matrix models with a discrete gauge group such as $S_N$? 1) Partition algebras are infinite-dimensional non-associative algebras with a basis labeled by group elements. 2) Partition algebras are subalgebras of the general linear group over complex numbers. 3) Partition algebras are commutative algebras generated by traces of permutation operators. 4) Partition algebras are isomorphic to Hecke algebras for all values of $N$. 5) Partition algebras are direct products of symmetric group algebras and matrix algebras. 6) Partition algebras are finite-dimensional, semi-simple associative algebras with bases labeled by diagrams encoding set partitions. 7) Partition algebras are defined only for systems with abelian symmetry groups.

✓ Correct Answer:

The correct answer is 6) Partition algebras are finite-dimensional, semi-simple associative algebras with bases labeled by diagrams encoding set partitions..

📚 Reference Text:

Title: Permutation invariance, partition algebras and large $N$ matrix models Year: 2023 Paper ID: 5042796b0d4ae3ba5dbcd80903ae22292bac1b33 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/5042796b0d4ae3ba5dbcd80903ae22292bac1b33 Abstract: In this thesis we will study matrix models with discrete gauge group $S_N$. We will put these matrix models into a generalized Schur-Weyl duality framework where dual algebras, known as partition algebras, emerge. These form generalizations of the symmetric group algebras -- they are semi-simple finite-dimensional associative algebras with a basis labelled by diagrams. We review the structure and representation theory of partition algebras. These algebras are then used to compute expectation values of $S_N$ invariant observables. This is a step towards studying the emergence of new geometric structures in their Feynman diagram expansion. Matrix models also appear in the form of quantum mechanical models of matrix oscillators. We explore the implications of the Schur-Weyl duality framework to matrix quantum mechanics with permutation symmetry.

Question 1008 multiple-choice

In analytic number theory and quantum computation, bounds on additive character sums over multiplicative subgroups of finite fields play a crucial role in understanding randomness and algorithmic performance. Such bounds often depend on the relative sizes of the subgroup and the field. Which of the following gives the optimal upper bound for the additive character sum ∑_{z=0}^{q-1} χ_t(a^z) over a multiplicative subgroup of F*_p when the subgroup order q satisfies q ≥ p^{2/3}? 1) O(p^{2/3}) 2) O(q^{1/2}) 3) O(p^{1/4} q^{3/8}) 4) O(p^{1/3} q^{2/3}) 5) O(p^{1/8} q^{5/8}) 6) O(p^{3/8} q^{1/4}) 7) O(p^{1/2})

✓ Correct Answer:

The correct answer is 7) O(p^{1/2}).

📚 Reference Text:

of the paper we require bounds on additive characters taken over multiplicative subgroups of F∗ p. Such sums are discussed in detail in [17]. The speciﬁc bound we require is the following. Theorem 9. Letχtbe a nontrivial additive character of Fpanda∈F∗ pan element of multiplicative order q. Then q−1/summationdisplay z=0χt(az)=⎧ ⎪⎨ ⎪⎩O(p1/2) ifq≥p2/3, O(p1/4q3/8)ifp1/2≤q≤p2/3, O(p1/8q5/8)ifp1/3≤q≤p1/2. See [17, section 2] for a proof. Note that in the body of the paper, we use Zpto denote the additive group of integers modulo pandZ∗ pto denote the multiplicative group of integers modulo p. Acknowledgments. We are grateful to Wim van Dam, Julia Kempe, Greg Kuperberg, Frederic Magniez, Martin R¨ otteler, and Miklos Santha for helpful conver- sations and to Sally Milius and Tracy Conrad for their support. REFERENCES [1]D. Bacon, A. Childs, and W. van Dam ,From optimal measurement to eﬃcient quantum algorithms for the hidden subgroup problem over semidirect product groups , in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 469–478. [2]R. Beals ,Quantum computation of Fourier transforms over symmetric groups , in Proceedings of the 29th Annual ACM Symposium on the Theory of Computing, 1997, pp. 48–53. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 958 MOORE, ROCKMORE, RUSSELL, AND SCHULMAN [3]E. Bernstein and U. Vazirani ,Quantum complexity theory (preliminary abstract) , in Pro- ceedings of the 25th Annual ACM Symposium on the Theory of Computing, 1993, pp. 11–20. [4]W. van Dam, S. Hallgren, and L. Ip ,Quantum algorithms for some hidden shift problems , in Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms,2003, pp. 489–498. [5]I. B. Damg ˚ard,On the randomness of Legendre and Jacobi sequences , in Advances in Cryptology—CRYPTO ’88, Lecture Notes in Comput. Sci. 403, Springer-Verlag, Berlin,1990, pp. 163–172. [6]M. Ettinger and P. Høyer ,On quantum algorithms for noncommutative hidden subgroups , Adv. in Appl. Math., 25 (2000), pp. 239–251. [7]M. Ettinger, P. Høyer, and E. Knill ,The quantum query complexity of the hidden subgroup problem is polynomial , Inform. Process. Lett. 91 (2004), pp. 43–48. [8]K. Friedl, G. Ivanyos, F. Magniez, M. Santha, and P. Sen ,Hidden translation and orbit coset in quantum computing , in Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 2003, pp. 1–9. [9]W. Fulton and J. Harris ,Representation Theory: A First Course , Grad. Texts in Math. 129, Springer-Verlag, London, 1991. [10]M. Grigni, L. J. Schulman, M. Vazirani, and U. Vazirani ,Quantum mechanical algorithms for the nonabelian hidden subgroup problem , in Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, 2001, pp. 68–74. [11]L. Hales and S. Hallgren ,Quantum Fourier sampling simpliﬁed , in Proceedings of the 31st Annual ACM Symposium on Theory of Computing, 1999, pp. 330–338. [12]L. Hales and S. Hallgren ,An improved quantum Fourier transform algorithm and applica- tions, in Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, 2000, pp. 515–525. [13]S. Hallgren, A. Russell, and A. Ta-Shma ,Normal subgroup reconstruction and quantum computation using group

Question 1009 multiple-choice

Quantum algorithms can efficiently solve the hidden subgroup problem in finite cyclic groups by leveraging the properties of quantum Fourier transforms and measurement. This process is foundational in quantum computing, particularly for problems related to algebraic structures and cryptography. When using a quantum algorithm to identify a hidden subgroup H in a finite cyclic group G, why does taking the greatest common divisor (GCD) of several measurement outcomes after applying the quantum Fourier transform yield the index M of H with high probability? 1) The QFT collapses the state to a single element of G, ensuring each measurement gives M directly. 2) The black-box function f outputs the generator d of H with certainty upon measurement. 3) The measurement outcomes are uniformly random elements of G, so their GCD is always 1. 4) The QFT creates entanglement between cosets, forcing all measurement results to be multiples of d. 5) The algorithm amplifies the probability of measuring the subgroup order through repeated interference. 6) Each measurement after QFT yields a random multiple of M, and the GCD of several such multiples is likely to be M. 7) The quantum procedure deterministically outputs the order of G, from which M can be trivially calculated.

✓ Correct Answer:

The correct answer is 6) Each measurement after QFT yields a random multiple of M, and the GCD of several such multiples is likely to be M..

📚 Reference Text:

LetSbe aﬁnite set, and /DoubleCapZ/DoubleCapZ()≔∕ +GN ,. Suppose that we have a function →fGS : , which separates a subgroup ⊂HG , where ⟨⟩=Hd. Let≔MH #. Assume that we have a quantum machine capable of computing the unitary transformation on two registers ∣⟩ ∣⟩ ∣⟩ ∣() ⟩ →⊕ Ux y x f x y:f (recall that we can take∣⟩ ∣⟩xy as∣⟩ ∣⟩⊗xy ). Suppose that we do not know M,d, norHand we only know Gand have a machine computing f. We want to determine a generating set for H, calling the “black-box ”functionfas few times as possible. LetFNbe the QFT for the cyclic group G. Explicitly, this is an operator on a register with ≥nNlog2qubits given by: ∣⟩ ⟨ ∣ ∑≔⎛ ⎝⎞ ⎠=− FNπijk Nkj1exp2. N jkN ,01 TheFNis a unitary transformation. If we let )(≔ω expπi N2be the primitive Nth root of unity, then () () () ()=⎛ ⎝⎜ ⎜⎜ ⎜⋯ ⋯ ⋯ ⋮⋮ ⋮ ⋮ ⋯⎞ ⎠⎟ ⎟⎟ ⎟− − −− − −FNωω ω ωω ω ωω ω111 1 1 1 1 1. NN N NN N N21 24 2 1 12 1 11 One can check that ⋅=FFI*NNN, whereINis the×NNidentity matrix. We map {}=−GN 0, 1, …, 1 onto the basis of the quantum state {∣⟩∣⟩ ∣ ⟩ }−N 0, 1,… , 1 . Suppose that the hidden subgroup is given by { ∣⟩∣⟩∣ ⟩ ∣ ( )⟩ } =−Hdd M d0, , 2 ,… , 1 . Computing on two registers: ∣⟩ ∣⟩ ∣⟩ ∣⟩ ∣⟩∣()⟩∑ ∑⟶ ⟶=− =−Nj Njfj0010 1.F jN f jNon 1st 01 apply 01N Note that we put ∣()⟩fj inside the sum since tensor product is distributive. Measuring in ∣()⟩fj0on the second register for some ≤≤−jN 0 10 collapses our state, leaving only those values ∈gG such that () ()=fg fj0in theﬁrst register. Since fseparates cosets of H, we obtain (for simplicity, we now drop our second register that remains∣()⟩fj0):4 Imin Chen and David Sun ∣⟩ ∣ ⟩ () ∣⟩ ∣⟩ ∣⟩∑∑ ∑∑ ∑∑ ∑⎜ ⎟ ⎜ ⎟ ⎜ ⎟⟶+ = + ⟶⎛ ⎝+⎞ ⎠ =⎛ ⎝⎞ ⎠⎛ ⎝⎞ ⎠ =⎛ ⎝⎞ ⎠∈=− =− =− =− =− =−MjhMjs d MNπi j sd k Nk MNπij k Nkπisdk N dπij tM NtM11 11exp2 1exp2 exp2 1exp2 ,hH sM F sM kN kN sM tdmeasure 0 01 0 apply 01 01 0 01 0 01 01 0N using the fact that ∣∑∑⎛ ⎝⎞ ⎠=⎛ ⎝⎞ ⎠=⎧⎨⎩∤ =− =−πisdk Nπik MMk MM kexp2exp2 0, if , ,i f ,sM sM s 01 01 for≤≤−kN 0 1and that =M Nd1. Now, measurement at this point gives a multiple of Min{()}−MdM0, , …, 1 with uniform probability. We repeat this whole process many times to obtain a collection of multiples of Mand take the greatest common divisor (GCD) to obtain Mwith high probability. To estimate how many trials ≥m 2we need, suppose that we have {}∈−tt d,…, 0, 1, …, 1m 1 . We want to estimate the probability that () =tt gcd , …, 1m 1

Question 1010 multiple-choice

Quantum algorithms have the potential to efficiently solve certain mathematical problems that are foundational to classical cryptography, such as the discrete logarithm problem. One approach leverages period-finding techniques and quantum Fourier transforms to extract hidden structures from function evaluations in superposition. In a quantum algorithm for solving the discrete logarithm problem using period-finding, which of the following is the key condition that enables the discrete logarithm y to be computed from measured Fourier labels (l1, l2)? 1) l2 must be divisible by l1 2) l1 must be a multiple of (p−1) 3) l1 must be coprime to (p−1) 4) l2 must be equal to zero 5) l1 must be equal to y 6) l2 must be coprime to y 7) l1 and l2 must be both even

✓ Correct Answer:

The correct answer is 3) l1 must be coprime to (p−1).

📚 Reference Text:

the function f:Zp−1× Zp−1→ Z∗ pgiven by f(a,b) =gax−bmodp which is computable in time poly( n). In terms of the discrete logarithm y= loggxwe have f(a,b) =ga−ybmodp so f(a1,b1) =f(a2,b2) if and only if ( a2,b2) = (a1,b1) +λ(y,1) forλ∈ Zp−1. Thus the pair ( y,1) is the period of fon its product domain. To determine your quantum algorithm will follow the standard period–ﬁnding procedur e of section 2, slightly generalized to deal with the fact that the domain consists of pairs rather than just single numbers. We consider a Hilbert space with an orthonormal basis {|a/an}bracketri}ht|b/an}bracketri}ht:a,b∈ Zp−1}labeled by the elements of Gand begin by computing an equal superposition of all values o ff: |f/an}bracketri}ht=1 p−1/summationdisplay a,b|a/an}bracketri}ht|b/an}bracketri}ht|f(a,b)/an}bracketri}ht. If we measure the last register and see a value k0=f(a0,b0) we obtain the periodic state |ψ/an}bracketri}ht=1√p−1p−2/summationdisplay k=0|a0+ky/an}bracketri}ht|b0+k/an}bracketri}ht. To eliminate the dependence of the labels on the randomly cho sen (a0,b0) we apply F, the Fourier transform modulo ( p−1) to each of the two registers. The calculations are very similar to those for factoring (c.f. eq. (8)). Let us introdu ce the functions χl1,l2(a,b) = exp 2πi(al1+bl2 p−1). 7 Then (similar to eq. (7)) F ⊗ F |ψ/an}bracketri}htwill yield an equally weighted superposition of those labels (l1,l2) such that χl1,l2(y,1) = 1 i.e.yl1+l2≡0 modp−1 sol2=−yl1modp−1 and l1= 0,1,...,p −2. Explicitly we have F ⊗ F |ψ/an}bracketri}ht=1√p−1p−2/summationdisplay l1=0exp 2πi(a0l1−b0yl1 p−1)|l1/an}bracketri}ht|−yl1/an}bracketri}ht. Then a measurement of the labels will provide a pair ( l1,l2) = (l1,−yl1modp−1) where l1∈ Z p−1is chosen uniformly at random. If l1happens to be coprime to p−1 we can use Euclid’s algorithm to ﬁnd l−1 1, the multiplicative inverse modulo p−1, and compute y as−l−1 1l2. Ifl1is not coprime to p−1 then we cannot uniquely determine yfrom (l1,l2). What is the probability that a uniformly chosen l1is coprime to p−1? In section 2 we saw that this probability will be of order 1 /log log(p−1) and so to determine ywith high probability we will need to repeat our algorithm a very modes tO(log logp) times (which is even exponentially smaller than our goal of poly(log p) times). As in the case of factoring there is the residual issue of eﬃci ently implementing the Fourier transform that is used. To take advantage of the FFT formalis m we would want to use FT for integers modulo a power of 2 (instead of modulo p−1). Let 2tbe the smallest power of 2 greater than p−1, sotis the smallest integer greater than log2(p−1). Then FT modulo 2tmay be implemented in O(t2) =O((logp)2) steps. If we use FT modulo 2tin place of FT modulop−1 in the above algorithm then we will obtain a larger set of pos sible output pairs (l1,l2) with varying probabilities. However as in the case of facto ring, these pairs will lie with high probability suﬃciently near to the “good” pairs ( l1,−yl1) wherel1is coprime to p−1, so thatymay still be determined. The details of dealing with the near by pairs and assessing their probabilities, are quite involved and given in

Question 1011 multiple-choice

In combinatorics and finite group theory, understanding the bounds for coset covers in abelian groups and constructing additive bases in finite fields are central problems with implications in algebra and coding theory. Progress in these areas often hinges on improving longstanding conjectural bounds and demonstrating strong existence results for certain sets and linear maps. Which of the following improvements on coset covers in abelian groups represents a breakthrough beyond the classical k! upper bound for the index of subgroup intersection in an irredundant coset cover? 1) k^2 2) e log log k 3) 2^k 4) k log k 5) k^k 6) log k 7) k/e

✓ Correct Answer:

The correct answer is 2) e log log k.

📚 Reference Text:

Title: Additive bases, coset covers, and non-vanishing linear maps Year: 2021 Paper ID: d16d61d6fbb3548160bc5db30ff995d56dd2246c Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d16d61d6fbb3548160bc5db30ff995d56dd2246c Abstract: Recently, the first two authors proved the Alon-Jaeger-Tarsi conjecture on non-vanishing linear maps, for large primes. We extend their ideas to address several other related conjectures. We prove the weak Additive Basis conjecture proposed by Szegedy, making a significant step towards the Additive Basis conjecture of Jaeger, Linial, Payan, and Tarsi. In fact, we prove it in a strong form: there exists a set A ⊂ F∗p of size O(log p) such that if B ⊂ F n p is the union of p linear bases, then A ·B = {a · v : a ∈ A, v ∈ B} is an additive basis. An old result of Tomkinson states that if G is a group, and {Hixi : i ∈ [k]} is an irredundant coset cover of G, then |G : ⋂ i∈[k] Hi| ≤ k!, and this bound is the best possible. It is a longstanding open problem whether the upper bound can be improved to e in case we restrict cosets to subgroups. Pyber proposed to study this question for abelian groups. We show that somewhat surprisingly, if G is abelian, the upper bound can be improved to e log log k) already in the case of general coset covers, making the first substantial improvement over the k! bound. Finally, we prove a natural generalization of the Alon-Jaeger-Tarsi conjecture for multiple matrices.

Question 1012 multiple-choice

In computational complexity and representation theory, the study of polytopes arising from group actions has deep connections with algorithmic tractability and quantum information theory. Understanding the complexity of deciding membership in these polytopes informs both mathematical and physical applications. Which statement accurately describes the computational complexity status of deciding membership in the moment polytope for finite-dimensional unitary representations of compact, connected Lie groups? 1) The problem is NP-complete and believed to be intractable in general. 2) The problem is in both NP and coNP, but is not known to be NP-hard. 3) The problem has a known polynomial-time algorithm for all cases. 4) The problem is only in coNP and not in NP. 5) The problem is undecidable for non-Abelian Lie groups. 6) The problem is #P-complete due to its connection with counting problems. 7) The problem is believed to be equivalent to factoring large integers.

✓ Correct Answer:

The correct answer is 2) The problem is in both NP and coNP, but is not known to be NP-hard..

📚 Reference Text:

Title: Membership in Moment Polytopes is in NP and coNP Year: 2015 Paper ID: 70ae6eae53d6328d1c40a754f11599d43ac4495d Source: semantic-scholar URL: https://www.semanticscholar.org/paper/70ae6eae53d6328d1c40a754f11599d43ac4495d Abstract: We show that the problem of deciding membership in the moment polytope associated with a finite-dimensional unitary representation of a compact, connected Lie group is in NP and coNP. This is the first nontrivial result on the computational complexity of this problem, which naively amounts to a quadratically constrained program. Our result applies in particular to the Kronecker polytopes, and therefore to the problem of deciding positivity of the stretched Kronecker coefficients. In contrast, it has recently been shown that deciding positivity of a single Kronecker coefficient is NP-hard, in general [C. Ikenmeyer, K. D. Mulmuley, and M. Walter, preprint, arXiv:1507.02955, 2015]. We discuss the consequences of our work in the context of complexity theory and the quantum marginal problem.

Question 1013 multiple-choice

In finite group theory, the structure of p-groups, especially 2-groups, is deeply influenced by the properties of their maximal normal abelian subgroups and the Frattini subgroup. The interplay between group elements and subgroup actions often reveals which group configurations are possible or impossible. Which of the following must be true for a finite 2-group where every maximal normal abelian subgroup has index 2 and the Frattini subgroup has order greater than 4? 1) Every element outside a maximal normal abelian subgroup centralizes all elements within it. 2) The group must be cyclic and abelian. 3) The group contains no involutions. 4) Certain group configurations are impossible due to contradictions arising from element inversion actions. 5) All maximal subgroups of the group are non-normal. 6) The Frattini subgroup coincides with the entire group. 7) Every group element acts trivially on all abelian subgroups.

✓ Correct Answer:

The correct answer is 4) Certain group configurations are impossible due to contradictions arising from element inversion actions..

📚 Reference Text:

¼4, so that we have A¼hy;ziW1ðAÞwith hy;ziGC4/C2C4andW 1ðAÞ¼hy2;z2i. Since jFðGÞj>4, there is an element uAG/C0Asuch that u2Bhy2;z2i. By the arguments of the previous paragraph, u acts by inversion on AandAhuiis a group in part (b6) of our theorem. Assume thatAhui0Gand let xAG/C0ðAhuiÞ.I fx2Bhy2;z2i, then xacts by inversion on Aand then xuBAandxucentralizes A, a contradiction. Hence x2Ahy2;z2iand we may set (say) x2¼z2.I fxinverts some element sof order 4 in hy;zi, then xu centralizes s, a contradiction. Since hy;xiGhyz;xiGH2, we have yx¼yz2, ðyzÞx¼ðyzÞz2and then ðyzÞz2¼ðyzÞx¼yxzx¼yz2zx; which gives zx¼z, a contradiction. It remains to consider the case where jA:W1ðAÞj ¼2 for each maximal normal abelian subgroup AofG. We have A¼hyiW1ðAÞ; note also that each maximal abe- lian subgroup of Gis normal in Gand so Ghas no subgroup isomorphic to C4/C2C4. For each xAG/C0A, we have ½x;y/C13801. Suppose that each xAG/C0Ainverts y. Then302 Z. Janko jG :Aj¼2,G¼Ahxi,hx;yiGQ8orH2andFðGÞ¼hx2;y2iwhich is of order at most 4, contrary to our assumption. Hence there is xAG/C0Asuch that yx0y/C01 and sohx;yiGH2.I fx2¼y2, then replacing xbyx0¼xy, we get ðx0Þ20y2and x0does not invert y. Writing again xinstead of x0, we may assume from the start that x20y2and yx¼yx2and we have FðAhxiÞ¼hx2;y2iGE4. Since jFðGÞj>4, there is uAG/C0ðAhxiÞwith u2Bhx2;y2i. Then we have yuAfy/C01;yu2gsince u20y2andhu;yiGH2. We have ½x;u/C13801 and so hx;uiGH2, which gives ðxuÞ2Afu2;x2gand so in any case ðxuÞ20y2. First suppose that yu¼y/C01. Then yxu¼ðyx2Þu¼yðy2x2Þ. Since hy;xuiGH2, we get yxu¼yy2oryxu¼yðxuÞ2. But from the above, yxu¼yðy2x2Þand so we must have ðxuÞ2¼y2x2, contrary to the above result that ðxuÞ2Afu2;x2g. Hence we have the second possibility yu¼yu2 and this gives yxu¼ðyx2Þu¼yðu2x2Þand so u2x2¼y2(which is not possible since u2Bhx2;y2i)o ru2x2¼ðxuÞ2. But then we have u2x2¼ðxuÞ2¼x2u2½u;x/C138, and this gives ½u;x/C138¼1, a ﬁnal contradiction. References [1] Y. Berkovich. Groups of prime power order , vol. 1 and vol. 2 (with Z. Janko) (Walter de Gruyter, 2008). [2] Y. Berkovich and Z. Janko. Structure of ﬁnite p-groups with given subgroups. Contempo- rary Math. 402(2006), 13–93. [3] N. Blackburn. Generalizations of certain elementary theorems on p-groups. Proc. London Math. Soc. (3)11(1961), 1–22. [4] N. Blackburn. Finite groups in which the nonnormal subgroups have nontrivial intersec- tion. J. Algebra 3(1966), 30–37. [5] Z. Janko. Finite 2-groups with exactly four cyclic subgroups of order 2n.J. Reine Angew. Math. 566(2004), 135–181. [6] Z. Janko. On ﬁnite nonabelian 2-groups all of whose minimal nonabelian subgroups are of exponent 4. J. Algebra 315(2007), 801–808. [7] Z. Janko. A classiﬁcation of ﬁnite 2-groups with exactly three involutions. J. Algebra 291 (2005), 505–533. Received 11 February, 2008 Zvonimir Janko, Mathematical Institute, University of Heidelberg, 69120 Heidelberg, Ger- manyE-mail: janko@mathi.uni-heidelberg.deOn minimal non-abelian subgroups in ﬁnite p-groups 303

Question 1014 multiple-choice

In quantum algebra and noncommutative geometry, H-Hopf algebroids and related structures play a central role in the generalization of symmetries and representation theory, especially when dynamical parameters and weight decompositions are involved. These constructions are foundational in the study of quantum groups and integrable systems. Which of the following statements correctly characterizes the algebra DT of difference operators on the field of meromorphic functions MT associated with an algebraic group T and a Hopf algebra H? 1) DT is a commutative algebra with trivial weight decomposition and no moment maps. 2) DT consists of differential operators indexed by the Lie algebra of T, with weight decomposition given by the dual group. 3) DT is an H-algebra of difference operators acting by shifting arguments in MT, equipped with a weight decomposition and tautological moment maps. 4) DT is a free module over MT with multiplication given by pointwise addition of operators. 5) DT represents endomorphisms of the algebraic group T and includes only constant coefficient operators. 6) DT is defined only for noncommutative base algebras and lacks any weight structure. 7) DT is a quantum group with coproduct and counit defined independently of any moment maps.

✓ Correct Answer:

The correct answer is 3) DT is an H-algebra of difference operators acting by shifting arguments in MT, equipped with a weight decomposition and tautological moment maps..

📚 Reference Text:

Jcorrepsondingtosimpleroots, and all the other coeﬃcients are then uniquely determined by Theor ems 6.1 and 6.2. /squaresolid Remark. The matrix coeﬃcients of JVV(λ) for nonsimple roots are not as easily computed directly as those for simple roots. The above appro ach allows one to avoid this calculation. 7 Quantum dynamical R-matrices and quantum groupoids In this section we will give a ”noncommutative geometric” interpreta tion of the QDYBEwhich isanalogoustothe geometricinterpretationofthe CDY BEgiven above. More precisely, to solutions of the QDYBE we will associate, f ollowing [F],[EV2], akindofquantumgroup,morepreciselya Hopf algebroid (orquantum groupoid ). ThegeneralnotionofaHopfalgebroidwasintroducedin[Lu]. Howeve r,here it will be suﬃcient to use a less general notion, that of an H-Hopf algebroid, which was introduced in [EV2]. Our exposition will follow [EV2, EV3]. 17 7.1.H-bialgebroids. LetHbe a commutative and cocommutative ﬁnitely generated Hopf algebra over C,T= SpecHthe corresponding commutative aﬃne algebraic group. Assume that Tis connected. Let MTdenote the ﬁeld of meromorphic functions on T. Let us introduce the following deﬁnitions. Deﬁnition. AnH-algebra is an associative algebra AoverCwith 1, endowed with an T-bigrading A=⊕α,β∈TAαβ(called the weight decomposition), and two algebra embeddings µl,µr:MT→A00(the left and the right moment maps), such that for any a∈Aαβandf∈MT, we have µl(f(λ))a=aµl(f(λ+α)), µr(f(λ))a=aµr(f(λ+β)).(7.1) A morphism ϕ:A→Bof twoH-algebras is an algebra homomorphism, preserving the moment maps. Example 1. LetDTbe the algebra of diﬀerence operators MT→MT, i.e. the operators of the form/summationtextn i=1fi(λ)Tβi, wherefi∈MT, and for β∈Twe denote by Tβthe ﬁeld automorphism of MTgiven by ( Tβf)(λ) =f(λ+β). The algebra DTis an example of an H-algebra if we deﬁne the weight de- composition by DT=⊕(DT)αβ, where ( DT)αβ= 0 ifα/\e}atio\slash=β, and (DT)αα= {f(λ)T−1 α:f∈MT}, and the moment maps µl=µr:MT→(DT)00to be the tautological isomorphism. Example 2. This is a generalizationof Example 1. Let Wbe a diagonalizable H-module, W=⊕λ∈TW[λ], W[λ] ={w∈W|aw=λ(a)w,for alla∈H}, and letDα T,W⊂HomC(W,W⊗DT) be the space of all diﬀerence operators on Twith coeﬃcients in End C(W), which have weight α∈Twith respect to the action of HinW. Consider the algebra DT,W=⊕αDα T,W. This algebra has a weight decom- position DT,W=⊕α,β(DT,W)αβdeﬁned as follows: if g∈HomC(W,W⊗MT) is an operator of weight β−α, thengT−1 β∈(DT,W)αβ. Deﬁnethemomentmaps µl,µr:MT→(DT,W)00bytheformulas µr(f(λ)) = f(λ),µl(f(λ)) =f(λ−h) wheref(λ−h)w=f(λ−µ)wifw∈W[µ], µ∈T. The algebra DT,Wequipped with this weight decomposition and these moment maps is an H-algebra. Now let us deﬁne the tensor product of H-algebras. Let A,Bbe twoH- algebras and µA l,µA r,µB l,µB rtheir moment maps. Deﬁne their matrix tensor product,A/tildewide⊗B, which is also an H-algebra. Let (A/tildewide⊗B)αδ:=⊕βAαβ⊗MTBβδ, (7.2) where⊗MTmeans the usual tensor product modulo the relation µA r(f)a⊗b= a⊗µB l(f)b, for any a∈A,b∈B,f∈MT. Introduce a multiplication in A/tildewide⊗B by the rule ( a⊗b)(a′⊗b′) =aa′⊗bb′. It is easy to check that the multiplication is well deﬁned. Deﬁne the moment maps for A/tildewide⊗BbyµA/tildewide⊗B l(f) =µA l(f)⊗1, µA/tildewide⊗B r(f) = 1⊗µB r(f). For any H-algebra A, the algebras A/tildewide⊗DTandDT/tildewide⊗Aare canonically iso- morphic to A. In particular, DTis canonically isomorphic to DT/tildewide⊗DT. Thus 18 the categoryof H-algebrasequipped with the product /tildewide⊗is a monoidal category, where the unit object is DT. Now let us deﬁne the notions of a coproduct and a counit on an H-algebra.

Question 1015 multiple-choice

Quantum error-correcting codes are essential for protecting quantum information from noise, but symmetries and operational constraints can limit their capabilities. The interplay between code structure, subsystem dimensions, and logical gates is a central challenge in designing robust quantum codes. Which statement most accurately characterizes the relationship between universal transversal logical gates, code infidelity, and subsystem dimension in finite-dimensional quantum error-correcting codes with continuous symmetries? 1) Code infidelity increases quadratically with subsystem dimension for codes admitting universal transversal gates. 2) Universal transversal logical gates require the code to be non-covariant under any symmetry group. 3) The existence of universal transversal gates ensures perfect error correction regardless of subsystem dimension. 4) In codes with universal transversal gates, the lower bound on infidelity scales as $ 1/\log d $, necessitating large subsystem dimension $ d $ to achieve small infidelity. 5) Code infidelity remains constant as subsystem dimension increases in covariant quantum codes. 6) Universal transversal gates can only be implemented in infinite-dimensional quantum codes. 7) Codes with universal transversal gates and small subsystem dimension exhibit arbitrarily small infidelity.

✓ Correct Answer:

The correct answer is 4) In codes with universal transversal gates, the lower bound on infidelity scales as $ 1/\log d $, necessitating large subsystem dimension $ d $ to achieve small infidelity..

📚 Reference Text:

chargeT A¼PTiis a sum of terms, each supported on a single subsystem. By measuring the charge Tiof a random subsystem, the environment can infer information about the total charge TL, distinguishing the two eigenstates. It follows that erasure of asingle subsystem cannot be perfectly corrected.PHILIPPE FAIST et al. PHYS. REV . X 10,041018 (2020) 041018-2 the code, characterized using either the entanglement fidelity for a fixed input state or the worst-case entangle- ment fidelity of the error-corrected state (Fig. 2). Our proof strategy is to show that, in the presence of a continuoussymmetry, the environment necessarily learns some infor- mation about the logical charge, which implies that the code necessarily performs imperfectly as an error-correcting code [30–33]. Some of these assumptions may FIG. 2. Overview of our main results and structure of the paper.CONTINUOUS SYMMETRIES AND APPROXIMATE QUANTUM … PHYS. REV . X 10,041018 (2020) 041018-3 be relaxed in our main technical theorem. For instance, the generating charge may be a sum of k-local terms, instead of a sum of strictly local terms as for a transversal symmetryaction, and the code needs to be only approximately rather than exactly covariant. Our lower bound on infidelity vanishes in two interesting regimes: as the dimension dof the physical subsystems gets large or as the number nof physical subsystems gets large. In these limits, we can find error-correcting codes whoseinfidelity approximately matches the scaling of our bound with dorn. We construct explicit examples based on normalized versions of the rotor code presented in Ref. [12] and note that codes considered in Ref. [8]provide further examples. We also discuss a five-rotor code that can be stacked to construct a covariant holographic code [7]. Furthermore, our results provide an approximate version of the Eastin-Knill theorem for quantum computation[12,17 –19], which states that a universal set of transversal logical gates cannot exist for a finite-dimensional encoding that protects perfectly against erasure. By applying ourbounds and exploiting the non-Abelian nature of the fullunitary group on the logical space, we derive a lower bound on infidelity which scales as 1=logd, where dis the subsystem dimension, for a code that admits universaltransversal logical gates. We also find that if a code admits a universal set of transversal logical gates, then there are strong lower bounds on the subsystem dimension dthat depend on both the code ’s infidelity and the logical system dimension d L. Using randomized code constructions, we prove (nonconstructively) the existence of codes whichapproximately achieve this relationship between dandd L. In addition, we exhibit codes with universal transversal logical gates which achieve arbitrarily small infidelity whenthe number nof subsystems becomes large with the logical dimension d Lfixed. We also provide a general framework for constructing codes that are covariant with respect to a general symmetry group, by encoding logical information into the so-calledregular representation of the group. Using this framework, we can generalize several widely known codes (bit-flip, phase-flip, ½½4;2;2/C138/C138code, etc.) to infinite-dimensional covariant codes based on oscillators or rotors. Finally, we discuss the interpretation of our

Question 1016 multiple-choice

In quantum mechanics, solving systems with radial symmetry often involves transforming differential equations into forms that can be addressed with special functions. The normalisation of wavefunctions in such systems is critical to ensure the physical interpretation of probability densities. When solving the radial part of a quantum system's partial differential equation in polar coordinates, which mathematical function typically arises as the eigenfunctions after an appropriate change of variables? 1) Associated Laguerre polynomials 2) Hermite polynomials 3) Chebyshev polynomials 4) Legendre polynomials 5) Jacobi polynomials 6) Bessel functions 7) Hypergeometric functions

✓ Correct Answer:

The correct answer is 1) Associated Laguerre polynomials.

📚 Reference Text:

us i1@t2@r1 r 3@+1 22 m! = 0;(66)which coincides with the PDE presented in Section 2. This equation conserves the normalisation condition Zpget 0 y d3x= 1; (67) whereg= det g . If one works with ( t;r; ) as coor- dinates,g=r2and this normalisation condition can be rewritten Z yd3x= 1; (68) with =pr : (69) The function obeys i 0@t+ i 1@r +i r 2 @+1 4[ 1; 2] m!1pr = 0;(70) which becomes, in the above representation: i1@t2@r1 r3@m = 0: (71) This equation coincides with the continuum limit of the free polar quantum walk introduced in this article. B: Laguerre solutions to the di erential equations To solve the di erential equations in Eq. (37), we rst make the substitutions u=re1 4Bqr2v. Assuming E6=m,vsatisfy the equations v=1 Emf(r)dv+ dr; (72) v+=1 E+mg(r)dv dr; (73) wheref(r) =r2e1 2Bqr2andg(r) =1 f(r). Now dif- ferentiate Eq. (72) and substitutedv drinto Eq. (73) to get d2v+ dr2+1 fdf drdv+ dr+ (E2m2)v+: (74) Now we focus on the case Bq > 0 and make the change of variable x=1 2Bqr2(forBq < 0, the useful change of variable reads x=1 2Bqr2). Simplifying this brings us to: xd2v+ dx2+ +1 2xdv+ dx+E2m2 2Bqv+= 0:(75) 8 Introducing n=E2m2 2Bqand =1 2, this can then be rewritten as xd2v+ dx2+ ( + 1x)dv+ dx+nv+= 0: (76) We thus write v+=CL n(x) =CL n1 2Bqr2 ; (77) whereCis a constant and Ln is an associated Laguerre polynomial. Substituting this into Eq. (72) we then ob- tain v=CBqr Emf(r)L +1 n11 2Bqr2 (78)and u(r) = Bq mECr1e1 4Bqr2L +1 n11 2Bqr2 Cre1 4Bqr2La n1 2Bqr2! :(79) Enforcing the normalisation condition Z2 0Z1 0 (r;) 2drd= 1: (80) delivers jCj2=(mE)2(Bq) +1n! 2 +1(n+ )!(2Bqn + (mE)2): (81)

Question 1017 multiple-choice

In quantum information theory, the classification and characterization of entangled states often rely on advanced mathematical frameworks, including group theory and symplectic geometry. Understanding how these structures relate to physical properties such as entanglement and local symmetries is crucial for both theoretical and practical developments. Which statement best describes Locally Maximally Entangled (LME) states in a multipartite quantum system with respect to the moment map and local unitary symmetry group K = SU(d₁) ×.. × SU(dₙ)? 1) LME states are those for which the trace of the full density matrix is zero under the action of K. 2) LME states correspond to the global minima of the Bures metric on the projective Hilbert space. 3) LME states are invariant under SLOCC transformations corresponding to invertible local operations. 4) LME states are characterized by G-invariant polynomials taking nonzero values for each generator of K. 5) LME states form the set of states where the Fubini-Study metric attains its maximal value between subsystems. 6) LME states are those that remain unchanged under all local rotations in SU(d₁) ×.. × SU(dₙ). 7) LME states are precisely the set of quantum states in the projective Hilbert space for which the moment map vanishes, i.e., μ⁻¹(0), reflecting maximal entanglement and symmetry under local unitary operations.

✓ Correct Answer:

The correct answer is 7) LME states are precisely the set of quantum states in the projective Hilbert space for which the moment map vanishes, i.e., μ⁻¹(0), reflecting maximal entanglement and symmetry under local unitary operations..

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action ofK(for example, the traces of powers of reduced density matrices f or various subsystems). The LME states that are the focus of this paper correspond to sp eciﬁc orbits of K; one way to characterize these in terms of K-invariants is to say that tr( ρ2 i) = 1/difor eachi. 4.1.2 SLOCC orbits and the action of G= SL(d1,C)×···×SL(dn,C) Sometimes, we may be interested in a coarser classiﬁcation of entan glement structure. Two states are said to be equivalent under the set of “stochastic local operations and classical communication” (SLOCC equivalent) if we can move from one to the ot her by performing reversible quantum operations on the individual subsystems. Mathematically, thesetofallowedSLOCCoperationscorrespondst othegroup G= SL(d1,C)× ···×SL(dn,C) of arbitrary invertible local transformations acting on P(H) [DVC00]. Alter- natively, we can consider ˜G= GL(d1,C)×···×GL(dn,C) acting on H. As in the above discussion, each state will be on some orbit of G. Classifying states up to 23 SLOCC equivalence means understanding how P(H) decomposes into orbits of G. As we discuss further below, we can choose a set of G-invariant polynomials (some subset of the entanglement invariants discussed above) as coordinates on the s pace of these orbits. We will see below that the classiﬁcation of SLOCC equivalence classes is intimately related to the classiﬁcation of equivalence classes of LME states up to local unitaries. 4.2SLME/Kas a symplectic manifold The space of quantum states (either HorP(H)) has a natural symplectic structure (i.e. the structureofaphasespaceinHamiltonianclassicalmechanics), ass ociatedwiththesymplectic form ω=ihijdzi∧d¯zj=i 2∂ ∂zi∂ ∂¯zjlog(|z|2)dzi∧d¯zj(4.1) onH=CDor the naturally associated pullback to P(H). The latter is known as the Fubini-Study form. The associated metric ds2=hijdzid¯zj(4.2) is (up to an overall normalization) the same as the Bures metric that gives a natural measure of distance between quantum states. The symplectic form is invariant under the action of K= SU(d1)× ··· ×SU(dn) on the phase space. Each inﬁnitesimal transformation in this group, corr esponding to some element k∈Lie(K) of the associated Lie algebra, may be associated with a vector ﬁeld Xkon phase space indicating the inﬁnitesimal form of the transformation. Via th e symplectic form, such a vector ﬁeld can be associated with a real Hamiltonian function HkonP(H) as dHk=Xk·ω. (4.3) The map from the symmetry generator kto the function Hkon phase space is precisely the usual map between symmetries and conserved quantities in Ham iltonian mechanics. Mathematically, this is referred to as a comoment map . The same relation can also be expressed as a moment map , µ:P(H)→Lie(K)∗(4.4) which associates to every state xinP(H) the function ( k→Hk(x)) fromLie(K) to real numbers (i.e. a vector in the dual space Lie(K)∗). We now show that the LME states are exactly the subset µ−1(0)⊂P(H). To see this, we note thatµ(x) = 0 if and only if Hk(x) vanishes for some basis of elements k∈Lie(K). For our groupK, these basis elements can be chosen to take the form k= 1 1⊗···⊗ki⊗···⊗1 1 (4.5) 24 wherekiis a generator of SL( di,C) (i.e. a traceless di×dimatrix). For the point x∈P(H) corresponding to

Question 1018 multiple-choice

Metasurfaces are engineered nanostructured surfaces that can manipulate light in novel ways, offering new possibilities for quantum information processing. Recent research explores integrating multiple quantum algorithm functionalities into single, programmable metasurface devices. Which technological combination enables a single metasurface device to switch between distinct quantum algorithms such as Grover’s search and quantum Fourier transform? 1) Using separate static metalens arrays for each algorithm, each with dedicated beam paths 2) Employing thermal actuators to physically reconfigure the metasurface structure 3) Sequentially replacing metasurfaces with different algorithms encoded on them 4) Modulating the input light polarization to activate specific quantum gates 5) Utilizing a spatial light modulator to selectively activate regions of the metasurface corresponding to different algorithms 6) Applying magnetic fields to reorient nanoscale structures on demand 7) Directly tuning the laser wavelength to encode algorithm operations

✓ Correct Answer:

The correct answer is 5) Utilizing a spatial light modulator to selectively activate regions of the metasurface corresponding to different algorithms.

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Title: Metasurface for programmable quantum algorithms with classical and quantum light Year: 2023 Paper ID: 4d7ccf63536d41fe2ed055c83eb0f6c64726e84a Source: semantic-scholar URL: https://www.semanticscholar.org/paper/4d7ccf63536d41fe2ed055c83eb0f6c64726e84a Abstract: Abstract Metasurfaces have recently opened up applications in the quantum regime, including quantum tomography and the generation of quantum entangled states. With their capability to store a vast amount of information by utilizing the various geometric degrees of freedom of nanostructures, metasurfaces are expected to be useful for processing quantum information. Here, we propose and experimentally demonstrate a programmable metasurface capable of performing quantum algorithms using both classical and quantum light with single photons. Our approach encodes multiple programmable quantum algorithms and operations, such as Grover’s search algorithm and the quantum Fourier transform, onto the same metalens array on a metasurface. A spatial light modulator selectively excites different sets of metalenses to carry out the quantum algorithms, while the interference patterns captured by a single-photon camera are used to extract information about the output state at the selected output directions. Our programmable quantum metasurface approach holds promising potential as a cost-effective means of miniaturizing components for quantum computing and information processing.

Question 1019 multiple-choice

Quantum computing utilizes principles of quantum mechanics to solve computational problems that are intractable for classical computers. Efficient implementation of complex quantum gates is crucial for executing algorithms on noisy, intermediate-scale quantum hardware. Which technique enables the efficient implementation of multi-controlled gates with reduced circuit depth and minimal ancilla qubits, making it suitable for NISQ quantum computers? 1) Quantum Fourier Transform (QFT)-based decomposition 2) Classical reversible logic synthesis 3) Grover's algorithm-based gate construction 4) Adiabatic quantum computation 5) Variational quantum eigensolver (VQE) 6) Quantum error correction codes 7) Quantum annealing

✓ Correct Answer:

The correct answer is 1) Quantum Fourier Transform (QFT)-based decomposition.

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Title: Implementing multi-controlled X gates using the quantum Fourier transform Year: 2024 Paper ID: 15843b5d4bfec2b41cae875608f94f72f270d63d Source: semantic-scholar URL: https://www.semanticscholar.org/paper/15843b5d4bfec2b41cae875608f94f72f270d63d Abstract: Quantum computing has the potential to solve many complex algorithms in the domains of optimization, arithmetics, structural search, financial risk analysis, machine learning, image processing, and others. Quantum circuits built to implement these algorithms usually require multi-controlled gates as fundamental building blocks, where the multi-controlled Toffoli stands out as the primary example. For implementation in quantum hardware, these gates should be decomposed into many elementary gates, which results in a large depth of the final quantum circuit. However, even moderately deep quantum circuits have low fidelity due to decoherence effects and, thus, may return an almost perfectly uniform distribution of the output results. This paper proposes a different approach for efficient cost multi-controlled gates implementation using the quantum Fourier transform. We show how the depth of the circuit can be significantly reduced using only a few ancilla qubits, making our approach viable for application to noisy intermediate-scale quantum computers. This quantum arithmetic-based approach can be efficiently used to implement many complex quantum gates.

Question 1020 multiple-choice

In the study of quantum groups and tensor categories, fusion and exchange matrices play a critical role in encoding the algebraic and categorical structure of representations. Compatibility conditions between these matrices are essential for applications in mathematical physics and topology. Which of the following compatibility conditions among exchange matrices ensures the coherence of associativity and commutativity for three modules in braided tensor categories, and is commonly represented by a hexagon diagram? 1) The hexagon relation among exchange matrices 2) The pentagon relation for fusion matrices 3) The braid relation for intertwining operators 4) The associativity condition for tensor products 5) The quantum trace condition for module homomorphisms 6) The reflection equation in integrable models 7) The fusion rule compatibility for particle states

✓ Correct Answer:

The correct answer is 1) The hexagon relation among exchange matrices.

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prove statement 3., letu∈U, v∈V, w∈Wbe homogeneous elements and consider the composition MλΦv λ−→Mλ−wt(v)⊗VΦw λ−wt(v)−→Mλ−wt(v)−wt(w)⊗W⊗V Φu λ−wt(v)−wt(w)−→ Mλ−wt(u)−wt(v)−wt(w)⊗U⊗W⊗V. The dynamical 2-cocycle condition follows from the associativity rela tion Φu λ−wt(v)−wt(w)◦(Φw λ−wt(v)◦Φv λ) = (Φu λ−wt(v)−wt(w)◦Φw λ−wt(v))◦Φv λ and from the deﬁnition of the fusion matrices. /squaresolid The fusion matrices can be viewed as the structure constants for multiplica- tion in the ”algebra” of intertwining operators. We now turn to the s tructure constants for ”commutation relations”. Let V,Wbe two ﬁnite-dimensional g- modules. Let us deﬁne RVW(λ) =JV W(λ)−1J21 WV(λ)∈Homh(V⊗W,V⊗W), whereJ21=PJPwithP(x⊗y) =y⊗x. The above deﬁnition can be rephrased in termsofintertwiningoperatorsasfollows: RVW(λ)(v⊗w) =/summationtext ivi⊗wiwhere Φw,v λ=P/summationtext iΦvi,wi λ. Deﬁnition. The operator RVW(λ) is called the exchange matrix ofVandW. Proposition 2.4. LetU,V,Wbe three ﬁnite-dimensional g-modules. Then the exchange matrices satisfy the following relation RVW(λ−h(3))RV U(λ)RWU(λ−h(1)) =RWU(λ)RV U(λ−h(2))RV W(λ) (2.1) in the algebra Homh(V⊗W⊗U,V⊗W⊗U). 5 Proof.Letu∈U, v∈V, w∈Wbe homogeneous elements and, as in Propo- sition 2.3, consider the composition Φu,w,v λ= Φu λ−wt(v)−wt(w)◦Φw λ−wt(v)◦Φv λ. The proofof relation (2.1) is obtained by rewriting Φu,w,v λas/summationtextσΦvi,wi,ui λwhere σ:U⊗W⊗V→V⊗W⊗U, x⊗y⊗z/mapsto→z⊗y⊗x, using exchange matrices in two diﬀerent ways according to the following hexagon U⊗V⊗W/d47/d47V⊗U⊗W /d40/d40/d80/d80/d80/d80/d80/d80/d80/d80/d80/d80/d80/d80 U⊗W⊗V/d54/d54/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110 /d40/d40/d80/d80/d80/d80/d80/d80/d80/d80/d80/d80/d80/d80V⊗W⊗U W⊗U⊗V /d47/d47W⊗V⊗U/d54/d54/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110/d110 /squaresolid Remark. One can also deduce this proposition from Part 3. of Proposition 2.3. Namely, one can show that if J(λ) is any element of the completion of U(g)⊗U(g) which satisﬁes the dynamical 2-cocycle condition (where JV W(λ) denotes the projection of J(λ) to the product V⊗Wof ﬁnite dimensional modules V,W) then the element R(λ) =J(λ)−1J21(λ) satisﬁes the quantum dynamical Yang-Baxter equation. Example 1. Let us evaluate the fusion and exchange matrices in the simplest example. Namely, take g=sl2=Ce⊕Ch⊕CfandV=C2=Cv+⊕Cv−with h.v±=±v±, e.v −=v+, e.v += 0, f.v −= 0, f.v +=v−. Let us compute the fusion matrix JV V(λ). By the triangularity property of JV V(λ), we have JVV(λ)(v±⊗v±) =v±⊗v±, J V V(λ)(v−⊗v+) =v−⊗v+, so it remains to compute JVV(λ)(v−⊗v+). Consider the intertwiner Φv− λ: Mλ→Mλ+1⊗V. By deﬁnition, Φv− λ(xλ) =xλ+1⊗v−+y(λ)fxλ+1⊗v+. To determine the function y(λ), we use the intertwining property: 0 = Φv− λ(exλ) = (e⊗1+1⊗e)Φv− λ(xλ) =xλ+1⊗v++y(λ)efxλ+1⊗v+ =xλ+1⊗v++y(λ)(h+fe)xλ+1⊗v+ =xλ+1⊗v++(λ+1)y(λ)xλ+1⊗v+ Hencey(λ) =−1 λ+1. A similar computation shows that Φv+ λ+1(xλ+1) =xλ⊗v+. Thus Φv+,v− λ(xλ) = Φv+ λ+1Φv− λ(xλ) =xλ⊗(v+⊗v−−1 λ+1v−⊗v+)+ lower weight terms . Therefore JV V(λ)(v+⊗v−) =v+⊗v−−1 λ+1v−⊗v+, and JV V(λ) = 1 0 0 0 0 1 0 0 0−1 λ+11 0 0 0 0 1  6 The exchange matrix is now easily computed. In the basis ( v+⊗v+,v+⊗v−, v−⊗v+,v−⊗v−) it is given by RV V(λ) = 1 0 0 0 0 1 −1 λ+10 01 λ+11−1 (λ+1)20 0 0 0 1 . 2.2. Generalization to quantum groups. Theconstructionofintertwining operators, fusion and exchange matrices admit natural quantum analogues. Let Uq(g) be the quantum universal enveloping algebra associated to g, as deﬁned in [CP], Chapter 6., and for each λ∈h∗, letMλbe the Verma module of highest weightλ. ThenProposition2.2andthedeﬁnitionofthefusionmatrices JW,V(λ) are identical to the classical case. In this situation, Proposition 2.3 , parts 2., 3. hold. However, the fusion matrices are no longer rational funct ions ofλ,

Question 1021 multiple-choice

Quantum algorithms, such as those used in phase estimation, are crucial for advancing precision measurements in metrology, including the estimation of magnetic flux. These algorithms often leverage Bayesian inference and iterative search strategies to outperform classical techniques in resolution and efficiency. In a quantum Fourier algorithm for magnetic flux estimation, what is the primary criterion for selecting the optimal time delay at each iterative step to maximize estimation efficiency? 1) Minimizing the total number of alternatives in the flux distribution 2) Ensuring equal probability among all possible flux intervals 3) Reducing the ensemble standard deviation early in the process 4) Maximizing sensitivity to external magnetic field fluctuations 5) Achieving the longest possible delay without increasing ambiguity 6) Maximizing the probability separation (ΔP) between groups of flux intervals 7) Selecting a delay that matches the classical shot-noise limit

✓ Correct Answer:

The correct answer is 6) Maximizing the probability separation (ΔP) between groups of flux intervals.

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time delay τ(s)~T2. The starting delay τ(s)is a free input parameter of the algorithm. Similarly to the Kitaev algorithm, thequantum Fourier algorithm runs the Bayesian learning subroutine until the ﬂux probability distribution P 0(Φi),i∈I0, squeezes to a twice narrower subset S1⊂I0such thatP i2S1P0ΦiðÞ /C21 1/C0ϵ0. However, in contrast to the Kitaev algorithm, the passport function Pp(τ(s),Φi) is an ambiguous function of Φiat the large delay τ(s). As a result, S1is not a single interval but rather a set of n ~τ(s)/τ0disjoint narrow intervals S1=I1∪…∪Inof almost equal lengths, see Fig. 2. Hence, after completing the ﬁrst step, the ﬂux value is distributed among nequiprobable alternatives Ii. The Fourier algorithm discriminates between these nalternatives in the next steps. First, it searches for the next optimal delay τj, where it is possible to rule out half of the remaining alternatives in the most ef ﬁcient way. At each delay τjthe algorithm splits the remaining intervals Ii,i=1,…,ninto two approximately equal-in- size groups A¼Ii1∪¼∪Ii½n=2/C138and B¼Ii½n=2/C138þ1∪¼∪Iinwhich are ordered by the passport function, Pp(τj,Φ∈A)>Pp(τj,Φ∈B). Then itﬁnds the probability distance ΔP(τj)=min i2APpτj;Φi/C0/C1 − max i2BPpτj;Φi/C0/C1 >0 separating the two sets Aand B. Repeating this procedure at all available delays τj, the algorithm ﬁnds the optimal delay τ(1)with maximal ΔP(τj) over the discrete set of delays τj. In the next step, the algorithm discriminates between A and Bby repeating the Bayesian learning subroutine approxi- mately [ ΔP(τ(1))]−2times and sets S2=AorB. Continuing in this way, the algorithm returns a single interval Ioutwhere the actual value of the ﬂuxΦ(Φi),i∈Iout, is located. Figure 2shows how the ﬂux distribution function P(Φi) develops in time during the execution of the Kitaev and Fourier algorithms. Results The superiority of our quantum metrological algorithms is clearly demonstrated by the scaling behavior of the magnetic ﬂux resolution with the total sensing time of the ﬂux measurement, see Fig.3. We run each algorithm n=25 times at every ﬂux value Φ= Φi,i∈I0, within the entire ﬂux range, and ﬁnd the corresponding arrays of estimated values ^Φji,j=1,…,n. The estimate ^Φ¼ ^ΦðPðΦÞÞis de ﬁned as the most likely value derived from the observed probability distribution P(Φ). For a probability distribu- tion Pi(Φ) measured at a known ﬂux value Φ=Φithe corresponding estimate ^Φjiis a random quantity due to statistical nature of the measurement procedure. We de ﬁne an aggregated resolution δΦas an ensemble standard deviation of the random variables ^Φji/C0Φi, δΦ2¼1 NI0ðÞX i2I01 n/C01Xn j¼1^Φji/C0Φihi2 : (5) In case of the Fourier algorithm, such a de ﬁnition is meaningful only at the ﬁnal step of the algorithm where P(Φ) becomes a single-peaked function. The sensing time tis de ﬁned through the total number of calls of the Bayesian learning subroutine m,t=Quantum-enhanced magnetometry by phase estimation algorithms S Danilin et al. 4 npj Quantum Information (2018) 29 Published in partnership with The University of New South Wales NTrepm. The scaling behavior of the measured ﬂux resolution δΦ (t) with sensing time tis shown in Fig. 3for both our algorithms and is compared with the scaling δΦstd(t) of the standard (classical) procedure, where all

Question 1022 multiple-choice

In quantum information theory and group representation, distinguishing between conjugate subgroups often involves analyzing probability distributions over high-dimensional spaces. Advanced probabilistic tools help determine the feasibility of measurement-based algorithms in such settings. Which probabilistic phenomenon is primarily responsible for making the probability distribution of observed basis vectors in a high-dimensional representation exponentially close to uniform, thereby rendering subgroup distinction information-theoretically hard? 1) Central limit theorem 2) Law of large numbers 3) Markov property 4) Concentration of measure 5) Ergodicity 6) Stationarity 7) Martingale convergence

✓ Correct Answer:

The correct answer is 4) Concentration of measure.

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In particular, there exists β>0, depending only on /epsilon1, such that /summationdisplay v|Pb(v)−Pb/prime(v)|δr d/bracketrightBig <4e−rδ2/48. Proof. We use an argument similar to [10]. We can think of a random d- dimensional complex vector vas a random 2 d-dimensional real vector of the same length, and we can think of this in turn as vi=wi/summationtext2d i=1w2 i, where the wiare independent Gaussian variables with zero mean and unit variance. By choosing a basis in which πprojects onto the ﬁrst r(complex) components of v, we have |π·v|2 2=/summationtext2r i=1w2 i/summationtext2d i=1w2 i=r d(1/2r)/summationtext2r i=1w2 i (1/2d)/summationtext2d i=1w2 i. Now we use the following Chernoﬀ bound, which can be derived from the moment generating function. For any t,w eh a v e Pr/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBigg 1 tt/summationdisplay i=1w2 i/parenrightBigg −1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>/epsilon1/bracketrightBigg <2/bracketleftBig (1 +/epsilon1) 1/2e−/epsilon1/2/bracketrightBigt . For|/epsilon1|<1/2, we have ln(1 + /epsilon1)/epsilon1/bracketrightBigg <2e −t/epsilon12/6. Now, for any a, b,i f|a/b−1|>δ, where δ<2, then either |a−1|>δ /4o r |b−1|>δ /4. Taking the union bound over these events where a=( 1/2r)/summationtext2r i=1w2 i andb=( 1/2d)/summationtext2d i=1w2 i, setting /epsilon1=δ/4 and t=2r≤2din (5) gives the stated bound. Setting d=dρandr=rkπ, Lemma 2 and the union bound imply that, for any constant A>√ 48, if (6) δ=A/radicalbigg logdρ rkπ, then, with high probability, for all dρbasis vectors vwe have /vextendsingle/vextendsingle/vextendsingle/vextendsingleP b(v)−1 dρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle<δ dρ. Summing over all v, this implies that the /lscript1distance between Pb(v) and the uniform Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE POWER OF STRONG FOURIER SAMPLING 953 distribution is at most δ. Now recall that rkπ=(p−1)/q.I f qp/epsilon1, and (6) gives δ

Question 1023 multiple-choice

Quantum algorithms often solve group-theoretic problems by exploiting properties of superposition and measurement. The hidden shift and hidden subgroup problems play key roles in both algorithmic efficiency and cryptographic security. Which parameter choice in the generalized hidden shift problem corresponds to the problem being an instance of an abelian hidden subgroup problem? 1) M = 0 2) M = 1 3) M = N/2 4) M = N-1 5) M = N 6) M = 2 7) M = √N

✓ Correct Answer:

The correct answer is 5) M = N.

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integer programming. Finally, we conclude in Section 5 with a discussion of the results and some open questions. 2. The generalized hidden shift problem It is well known that the dihedral hspis equivalent to the hidden shift problem, whichisdeﬁnedasfollows. Given twoinjective functions f0:ZN→ Sandf1:ZN→S(whereSis some ﬁnite set) satisfying f0(x) =f1(x+s) for some unknown s∈ZN, ﬁnds. For a proof of this equivalence, see Theorem 2 of [9] and Proposition 6.1 of [18]. For certain expl icit functions of interest, such as the Legendre symbol, the hidden shift pr oblem can be solved eﬃciently on a quantum computer [7]. However, for arb itrary black box functions, no eﬃcient algorithm for the hidden shift pro blem is known. A natural generalization of this problem, which we call the generalized hidden shift problem , is as follows. Considerasingle function f:{0,...,M− 1} ×ZN→Ssatisfying two conditions: (a) for ﬁxed b,f(b,x):ZN→S is injective and (b) f(b,x) =f(b+ 1,x+s) forb= 0,1,...,M−2 for some ﬁxed s∈ZN. Given such a function, our goal is again to ﬁnd the hidden shift s. ForM= 2, this problem is simply the usual hidden shift problem (with fb(x) =f(b,x) forb= 0,1), and hence is equivalent to the dihedral hsp. ForM=N, this problem is an instance of the abelian hidden subgroup problem (where the hidden subgroup is /an}⌊ra⌋k⌉tl⌉{t(1,s)/an}⌊ra⌋k⌉tri}ht ≤ZN×ZN). As a step toward understanding the dihedral hsp, we would like to investigate the diﬃculty of the problem for intermediate values of M. (Note that for intermediate values of M, the generalized hiddenshift problem is apparently not an instance of the hspfor any group.) On a quantum computer, this problem can be turned into a state distin- guishability problem in the same manner as the standard appr oach to the hidden subgroup problem. Prepare a uniform superposition o ver all values ofb∈ {0,...,M−1}andx∈ZNand then compute the value of f(b,x), 4 ANDREW M. CHILDS AND WIM VAN DAM giving the state 1√ MNM−1/summationdisplay b=0/summationdisplay x∈ZN|b,x,f(b,x)/an}⌊ra⌋k⌉tri}ht. (1) Then measure the third register, giving the state |φx,s/an}⌊ra⌋k⌉tri}ht:=1√ MM−1/summationdisplay b=0|b,x+bs/an}⌊ra⌋k⌉tri}ht (2) for some unknown x∈ZN. Equivalently, the result is the mixed state described by the density matrix ρs:=1 N/summationdisplay x∈ZN|φx,s/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tφx,s|. (3) Using a single copy of the state, we can identify sby the standard period ﬁnding algorithm (e.g., as in Shor’s algorithm) only when Mis very large. Given the state |φx,s/an}⌊ra⌋k⌉tri}ht, we can try to ﬁnd sby applying the Fourier transform overZN×ZNto the two registers, which yields the state 1 N√ M/summationdisplay y,z∈ZNωxzM−1/summationdisplay b=0ωb(y+sz)|y,z/an}⌊ra⌋k⌉tri}ht, (4) whereω:= exp(2πi/N). In the case M=N, this state equals 1√ N/summationdisplay z∈ZNωxz|−sz,z/an}⌊ra⌋k⌉tri}ht. (5) Measuring in the computational basis, we will observe ( −sz,z) for a uni- formly random z∈ZN. Ifzis invertible modulo N, which happens with probability Ω(1 /loglogN), then we can deduce simmediately from the val- ues−szandz. However, in general, for M≤N, the outcome will only be of the form ( −sz,z) with probability M/N. IfM≤Nǫwithǫ <1, this probability is exponentially small in log N, and the approach fails. A similar argument shows that an analogous approach using a Fo urier trans- form over

Question 1024 multiple-choice

Quantum algorithms leverage group-theoretic properties and number-theoretic estimates to efficiently solve the Hidden Subgroup Problem (HSP) for certain nonabelian groups. Techniques such as Gauss sum analysis, subgroup conjugacy, and probabilistic amplification play crucial roles in these algorithms. In the context of solving the Hidden Subgroup Problem for q-hedral groups where q = (p−1)/polylog(p) with p and q prime, which structural property enables the reduction of the problem to the Hidden Conjugate Problem (HCP)? 1) The existence of a cyclic normal subgroup of maximal order 2) The ability to embed the group into a finite field extension 3) The presence of a unique normal core for all subgroups 4) All nonnormal subgroups being conjugate to a subgroup isomorphic to Z_q 5) The use of probabilistic amplification to increase success probability 6) The distribution of generator powers uniformly modulo p 7) The application of Gauss sum estimates for abelian subgroups

✓ Correct Answer:

The correct answer is 4) All nonnormal subgroups being conjugate to a subgroup isomorphic to Z_q.

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(assuming without loss of generality that θis positive) φ∈[−θ(p−1),θ(p−1)]. If we again take /lscriptso that |θ|≤π/(2(p−1)), then φ∈[−π/2,π/2] and all the terms in the sum have nonnegative real parts. We will obtain a lower bound on the real partof the sum by showing that a constant fraction of the terms have φ∈(−π/3,π/3) and thus have real part more than 1 /2. This is the case whenever a tk∈(p/6,5p/6), and so it is suﬃcient to prove the following lemma. Lemma 1. Letahave order q=p/polylog( p)inZ∗ p, where pis prime. Then at least(1/3−o(1))qof the elements in the coset /angbracketlefta/angbracketrightkare in the interval (p/6,5p/6). Proof. We will prove this using Gauss sums , which quantify the interplay between the characters of Zpand the characters of Z∗ p. In particular, Gauss sums establish bounds on the distribution of powers of a. Speciﬁcally, if ahas order qinZ∗ p, then for any integer k/negationslash≡0m o d pwe have q−1/summationdisplay t=0ωatk p=O(p1/2)=o(p). (See [17] and Appendix A.) Now suppose sof the elements xin/angbracketlefta/angbracketrightkare in the set ( p/6,5p/6), for which Reωx p≥−1, and the other q−selements are in [0 ,p /6]∪[5p/6,p), for which Re ωx p≥ 1/2. Thus we have Req−1/summationdisplay t=0ωatk p≥(q/2)−(3s/2). Ifs≤(1/3−/epsilon1)qfor any /epsilon1>0, this is Θ( q), a contradiction. Now that we know that a fraction 1 /3−/epsilon1of the terms in (4) have real part at least 1 /2 and the others have real part at least 0, we can take /epsilon1=1/12 (say) and write P(/lscript)≥1 q(p−1)/parenleftBigq 8/parenrightBig2 =1 64q p−1=1 polylog( p). Thus we observe the correct frequency with polynomially small probability, and this can be boosted to high probability by a polynomial number of repetitions. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 948 MOORE, ROCKMORE, RUSSELL, AND SCHULMAN As we will now show, Theorem 1 implies that we can completely solve the HSP for certain q-hedral groups. Theorem 2. Letpandqbe prime with q=(p−1)/polylog( p). Then subgroups of the q-hedral group Zq/multicloseleftZ pare fully reconstructible. Proof. First, note that we can fully reconstruct Hif it is nontrivial and normal. We do this by reconstructing the normal core of H, C(H)=/intersectiondisplay γ∈GγHγ−1, using the techniques of [13] (the weak standard method). The q-hedral groups have the special property that nonnormal subgroups contain no nontrivial normal sub-groups; in particular, if His nonnormal, then C(H) is the trivial subgroup. Thus by reconstructing C(H), we either learn that H=C(H) or learn that His either trivial or nonnormal. Furthermore, if His trivial, we will learn this by checking our recon- struction against the oracle fand ﬁnding that it is incorrect. Therefore, it suﬃces to consider the nonnormal subgroups. Ifqis prime, then the nonnormal subgroups of Z q/multicloseleftZ pare all conjugate to a single subgroup K∼=Zq, as any such subgroup has the form {(a,(1−a)z)|a∈Zq< Aut(Zp)∼=Z∗ p}; in this case the HSP reduces to the HCP for K. While one can construct a proof similar to that of Theorem 1 directly for the q-hedral groups, it is convenient to embed them in Apusing the isomorphisms Nq∼=Zq/multicloseleftZ

Question 1025 multiple-choice

The Hidden Subgroup Problem (HSP) is central to quantum computing and computational group theory, with efficient algorithms often relying on the structure of group extensions and probabilistic methods for subgroup identification. Fractional linear transformations play a key role in these contexts due to their bijective properties over finite fields. Which statement accurately describes a significant generalization for solving the Hidden Subgroup Problem in group theory? 1) HSP can only be solved efficiently for abelian groups and their direct products with finite cyclic groups. 2) The union bound ensures zero probability of error when distinguishing subgroup cosets, regardless of sample size. 3) Fractional linear transforms over finite fields are always injective, ensuring perfect subgroup distinction. 4) Efficient solutions to HSP require that the extension group be a semidirect product of abelian groups. 5) Identifying hidden conjugates of subgroups requires exponential sample size in the order of the field. 6) HSP is solvable for any extension of a polynomial-size group K by a group H for which HSP is solvable, including non-semidirect products. 7) Hamiltonian groups and extra-special p-groups cannot be handled by generalized HSP algorithms.

✓ Correct Answer:

The correct answer is 6) HSP is solvable for any extension of a polynomial-size group K by a group H for which HSP is solvable, including non-semidirect products..

📚 Reference Text:

The function α−1(x)/β−1(x)i safractional linear transform , i.e., the ratio of two linear functions; such functions are the discrete analogues of the M¨ obius transformations in the complex plane. As in the complex case, the fractional linear transform γ(x)/δ(x) is a bijection on the projective space Zp∪ {∞} unless γandδshare a root, or, equivalently, there is a scalar z∈Z∗ psuch that γ(x)=zδ(x). If α−1(x)/β−1(x)i s injective, we can immediately conclude that Pr x[αf(x)=βf(x)]≤|M|/(p−1 )=1 /r≤1/2. Otherwise, α−1(x)/β−1(x)=zfor some scalar z. Since αf/negationslash=βf, however, in this case we must have z∈Z∗ p\M. In particular, f(zy)/negationslash=f(y) for any y/negationslash= 0, and so Pr x[αf(x)=βf(x) ]=1 /p, since this occurs only at the unique root xofα−1(x)=0 . In either case, then, αfandβfdiﬀer on at least half the elements of Zpwhenever αandβbelong to diﬀerent cosets of Hs q. It follows that if R⊂Zpconsists of m elements chosen independently and uniformly at random from Zp,w eh a v e Pr R[∀x∈R, αf(x)=βf(x)]≤1/2m for any α, β∈Apwith α−1β/∈Hq. Taking a union bound over all pairs of left cosets ofHq, Pr R/bracketleftbig ∃α, β∈Ap:α−1β/∈Hq∀x∈R, αf(x)=βf(x)/bracketrightbig ≤/parenleftbiggp(p−1) |Hq|/parenrightbigg21 2m. Selecting m= 5 log pensures that this probability is less than 1 /p. Since we showed in section 3 that we can identify a hidden conjugate of Hq whenever Hqis of polylogarithmic index in Z∗ p, this provides an eﬃcient solution to the hidden shift problem so long as p/q= polylog( p). Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 956 MOORE, ROCKMORE, RUSSELL, AND SCHULMAN 8. Closure under extending small groups. In this section we show that for any polynomial-size group Kand any Hfor which we can solve the HSP, we can also solve the HSP for any extension of KbyH, i.e., any group Gwith K/triangleleftG and G/K ∼=H. (Note that this is more general than split extensions, i.e., semidirect products H/multicloseleftK.) This includes the case discussed in [13] of Hamiltonian groups, since all such groups are direct products (and hence extensions) by abelian groups ofthe quaternion group Q 8[23]. It also includes the case discussed in [8] of groups with commutator subgroups of polynomial size, such as extra-special p-groups, since in that caseK=G/primeandH∼=G/G/primeis abelian. Indeed, our proof is an easy generalization of that in [8]. Theorem 7. LetHbe a group for which hidden subgroups are fully reconstructible andKa group of polynomial size in log|H|. Then hidden subgroups in any extension ofKbyH, i.e., any group GwithK/triangleleftG andG/K∼=H, are fully reconstructible. Proof. We assume that GandKare encoded in such a way that multiplication can be carried out in classical polynomial time. We ﬁx some transversal t(h)o ft h e left cosets of K. First, note that any subgroup L⊆Gcan be described in terms of (i) its intersection L∩K, (ii) its projection LH=L/(L∩K)⊆H, and (iii) a representative η(h)∈L∩(t(h)·K) for each h∈LH. Then each element of LHis associated with some left coset of L∩K, i.e., L=/uniontext h∈LHη(h)·(L∩K). Moreover, if Sis a set of generators for L∩KandTis a set of generators for LH, then S∪η(T) is a set of

Question 1026 multiple-choice

In theoretical physics, four-dimensional conformal fixed points are essential for understanding scale-invariant quantum field theories. Advanced analytical and computational methods are often developed to characterize these points and their physical implications. Which method, if successfully applied, is anticipated to provide deeper insight into the characterization of four-dimensional conformal fixed points and their critical behavior? 1) Renormalization Group Flow Analysis 2) Lattice Gauge Theory Simulations 3) Perturbative Expansion Techniques 4) Monte Carlo Renormalization 5) Holographic Duality Approaches 6) Functional Integral Formalism 7) QFE (Quantum Field Engineering) method

✓ Correct Answer:

The correct answer is 7) QFE (Quantum Field Engineering) method.

📚 Reference Text:

the QFE method. In the near future, we hope for the QFE method to be used to characterize the four dimensional conformal fixed points considered in the first half of this dissertation.

Question 1027 multiple-choice

Quantum algorithms have revolutionized computational approaches to group-theoretic problems, particularly those involving the hidden subgroup problem (HSP). The efficiency of these algorithms often relies on structural properties of the groups and the use of specialized quantum transformations. Which group-theoretic property most directly enables efficient quantum solutions to the Abelian hidden subgroup problem by facilitating the implementation of the quantum Fourier transform? 1) Existence of a nontrivial commutator subgroup 2) Presence of a normal 2-subgroup of small index 3) Solvability of the group 4) Commutativity of the group 5) The group being a permutation group 6) The group having a cyclic factor group 7) The group being simple

✓ Correct Answer:

The correct answer is 4) Commutativity of the group.

📚 Reference Text:

arXiv:quant-ph/0102014v1 2 Feb 2001Eﬃcient quantum algorithms for some instances of the non-Ab elian hidden subgroup problem∗ G´ abor Ivanyos†Fr´ ed´ eric Magniez‡Miklos Santha§ October 15, 2018 Abstract Inthispaperweshowthatcertainspecialcasesofthehidden subg roupproblemcanbe solved in polynomial time by a quantum algorithm. These special cases involve ﬁnding hidden normal subgroups of solvable groups and permutation groups, ﬁnding hidd en subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group. 1 Introduction A growing trend in recent years in quantum computing is to cas t quantum algorithms in a group theoretical setting. Group theory provides a unifying fram ework for several quantum algorithms, clariﬁes their key ingredients, and therefore contributes to a better understanding why they can, in some context, be more eﬃcient than the best known classica l ones. The most important unifying problem of group theory for the p urpose of quantum algorithms turned out to be the hidden subgroup problem (HSP) which can be cast in the following broad terms. Let Gbe a ﬁnite group (given by generators), and let Hbe a subgroup of G. We are given (by an oracle) a function fmapping Ginto a ﬁnite set such that fis constant and distinct on diﬀerent left cosets of H, and our task is to determine the unknown subgroup H. While no classical algorithm is known to solve this problem i n time faster than polynomial in the order of the group, the biggest success of quantum comp uting until now is that it can be solved by a quantum algorithm eﬃciently , which means in time polynomial in the logarithm of the order of G,whenever the group is Abelian. The main tool for this solutio n is the (approximate) quantum Fourier transform which can be eﬃciently implement ed by a quantum algorithm [17]. Simon’s algorithm for ﬁnding an xor-mask [26], Shor’s semin al factorization and discrete logarithm ﬁnding algorithms [25], Boneh and Lipton’s algorithm for ﬁn ding hidden linear functions [6] are all special cases of this general solution, as well as the alg orithm of Kitaev [17] for the Abelian stabilizer problem, which was the ﬁrst problem set in a gener al group theoretical framework. That all these problems are special cases of the HSP, and that an eﬃ cient solution comes easily once an eﬃcient Fourier transform is at our disposal, was realize d and formalized by several people, ∗Research partially supported by the EU 5th framework progra ms QAIP IST-1999-11234, and RAND-APX, IST- 1999-14036, by OTKA Grant No. 30132, and by an NWO-OTKA grant . †Computer and Automation Institute, Hungarian Academy of Sc iences, L´ agym´ anyosi u. 11., H-1111 Budapest, Hungary, e-mail: Gabor.Ivanyos@sztaki.hu . ‡CNRS–LRI, UMR 8623 Universit´ e Paris–Sud, 91405 Orsay, Fra nce, e-mail: magniez@lri.fr . §CNRS–LRI, UMR 8623 Universit´ e Paris–Sud, 91405 Orsay, Fra nce, e-mail: santha@lri.fr . 1 including Brassard and Høyer [7], Mosca and Ekert [22] and Jo zsa [15]. An excellent description of the general solution can be

Question 1028 multiple-choice

In quantum computing, scaling algorithms to larger problem sizes is limited by current hardware constraints, particularly the number of available qubits. Techniques such as circuit cutting have been developed to address these limitations, but practicality depends on both hardware and algorithmic considerations. Which of the following statements best explains why circuit cutting becomes impractical for large-scale quantum computations, even when fault-tolerant hardware is available? 1) Quantum error correction eliminates all classical post-processing requirements for circuit cutting. 2) The number of qubits required decreases linearly as circuit size increases. 3) Circuit cutting increases quantum runtime by a constant factor regardless of system size. 4) Subcircuit results can always be recombined without computational overhead. 5) Hamiltonian simulation and quantum Fourier transform do not benefit from circuit cutting techniques. 6) Exponential growth in the number of subcircuits leads to prohibitive increases in runtime and classical post-processing overhead. 7) Circuit cutting allows infinite scalability with current quantum hardware.

✓ Correct Answer:

The correct answer is 6) Exponential growth in the number of subcircuits leads to prohibitive increases in runtime and classical post-processing overhead..

📚 Reference Text:

Title: Understanding the Scalability of Circuit Cutting Techniques for Practical Quantum Applications Year: 2024 Paper ID: d20e5647b3d7a110ed603d3d261a431d2439e97f Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d20e5647b3d7a110ed603d3d261a431d2439e97f Abstract: Circuit cutting allows quantum circuits larger than the available hardware to be executed. Cutting techniques split circuits into smaller subcircuits, run them on the hardware, and recombine results through classical post-processing. Circuit cutting techniques have been extensively researched over the last five years and it been adopted by major quantum hardware vendors as part of their scaling roadmaps. We examine whether current circuit cutting techniques are practical for orchestrating executions on fault-tolerant quantum computers. We conduct a resource estimation-based benchmarking of important quantum applications and different types of circuit cutting techniques. Our applications include practically relevant algorithms, such as Hamiltonian simulation, kernels such as quantum Fourier transform and more. To cut these applications, we use IBM's Qiskit cutting tool. We estimate resources for subcircuits using Microsoft's Azure Quantum Resource Estimator and develop models to determine the qubit, quantum and classical runtime needs of circuit cutting. We demonstrate that while circuit cutting works for small-scale systems, the exponential growth of the quantum runtime and the classical post-processing overhead as the qubit count increases renders it impractical for larger quantum systems with current implementation strategies. As we transition from noisy quantum hardware to fault-tolerance, our work provides important guidance for the design of quantum software and runtime systems.

Question 1029 multiple-choice

Post-quantum cryptography aims to develop secure digital signature schemes that resist attacks from quantum computers, focusing on mathematical problems believed to be hard for quantum algorithms. Multivariate cryptography leverages the difficulty of solving systems of multivariate quadratic equations to build such schemes. Which feature distinguishes certain algebraic signature algorithms as practical candidates for post-quantum standards in environments with limited resources? 1) Reliance on the hidden discrete logarithm problem for signature generation 2) Use of elliptic curves over large finite fields for public key compression 3) Application of lattice-based cryptography to minimize computational overhead 4) Achieving drastically smaller public key and signature sizes through higher-dimensional algebras over small finite fields 5) Employment of hash-based techniques to ensure quantum resistance 6) Inclusion of code-based encryption mechanisms for enhanced robustness 7) Adoption of zero-knowledge proofs for signature verification

✓ Correct Answer:

The correct answer is 4) Achieving drastically smaller public key and signature sizes through higher-dimensional algebras over small finite fields.

📚 Reference Text:

Title: Signature Algorithms with a Hidden Group, Based on Difficulty of Solving Systems of Quadratic Equations Year: 2023 Paper ID: 188322529e965a0461b7c83ca783513cc4412da1 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/188322529e965a0461b7c83ca783513cc4412da1 Abstract: —This research introduces two novel algebraic signature algorithms with a hidden group, which are based on the computational difficulty of finding a solution of a large system of quadratic multivariate equations. Like signature algorithms of multivariate cryptography, the developed ones represent interest as post-quantum cryptoschemes, the latter having a significant merit consisting of a 100 or more times smaller size of public key. The introduced algorithms represent interest as candidates for a practical post-quantum signature standard with small sizes for both the signature and the public key. Their security is estimated to be vulnerable to direct attacks using the known algorithms for solving the systems of many quadratic equations. The development of structural attacks exploiting properties of the used algebraic support is estimated as an independent research task composing the next stage of the analysis of the proposed signature algorithms on finite non-commutative associative algebras. Relatively the known algebraic signature algorithms with a hidden group, which are based on the computational difficulty of the so called hidden discrete logarithm problem, the developed signature algorithms differ in the use of the signature verification equations with multiple entry of signature. This feature defines a specific technique for the signature generation. The next peculiarity is the use of algebras of higher dimensions set over the finite fields of smaller orders.

Question 1030 multiple-choice

Quantum algorithms often address hidden subgroup problems in group theory by constructing special quantum states that encode group structure. Algebraic geometry and automorphism techniques play a role in ensuring these states have properties essential for efficient computation. Which mathematical result guarantees the existence of nontrivial solutions to the quadratic system required for constructing hiding sets in quantum hidden subgroup procedures when the number of copies n exceeds three times a dimension parameter d? 1) Lagrange's theorem 2) The Fundamental Theorem of Algebra 3) Chevalley-Warning theorem 4) Noether's normalization lemma 5) Frobenius reciprocity theorem 6) Burnside's lemma 7) Cayley's theorem

✓ Correct Answer:

The correct answer is 3) Chevalley-Warning theorem.

📚 Reference Text:

5. We have 1.∀z∈Zd p,∀a∈G,∀u∈Zd p,∀j∈Zp,|aHG′ u·φj(z)/a\}bracketri}ht=ωj2|aHG′ u/a\}bracketri}ht, 2.∀h∈H,∀a∈G,∀u∈Zd p,∀j∈Zp,|aHG′ u·φj(h)/a\}bracketri}ht=ω(j−j2)|aHG′ u/a\}bracketri}ht. The principal idea now is to take several copies of the states |aiHG′ ui/a\}bracketri}htand choose the jiso that the product of the corresponding eigenvalues becomes the unity. The refore the combined actions φji(g), when gis fromHG′, will not modify the combined state. It turns out that we can achiev e this with a suﬃciently big enough number of copies. Let n=n(d) some function of dto be determined later. Fora= (a1,...,a n)∈Gn,u= (u1,...,u n)∈(Zd p)n,j= (j1,...,j n)∈(Zp)n\ {0n}andg∈G, we deﬁne the quantum state |Ψa,u,j g/a\}bracketri}htinCGnby |Ψa,u,j g/a\}bracketri}ht=n/circlemultiplydisplay i=1|aiHG′ ui·φji(g)/a\}bracketri}ht. Our purpose is to ﬁnd an eﬃcient procedure to generate triples ( a,u,j) such that for every ginHG′, |Ψa,u,j g/a\}bracketri}ht=/circlemultiplytextn i=1|aiHG′ ui/a\}bracketri}ht. We call such triples appropriate . The reason to look for appropriate triples is that they lead to hiding sets for HG′inGas stated in the next lemma. Lemma 6. If(a,u,j)is an appropriate triple then {|Ψa,u,j g/a\}bracketri}ht:g∈G}is hiding for HG′inG. Proof.To see this, ﬁrst observe that HG′is a normal subgroup of G. Ifg1andg2are in diﬀerent cosets of HG′inGthen let 1 ≤i≤nsuch that ji/\e}atio\slash= 0. The elements φji(g1) andφji(g2) are in diﬀerent cosets of HG′inGsinceφjiis an automorphism of G. Also, we have supp(|aHG′ u/a\}bracketri}ht) =supp(|aHG′/a\}bracketri}ht), and therefore supp(|aHG′ u·φj1(g1)/a\}bracketri}ht) andsupp(|aHG′ u·φj2(g2)/a\}bracketri}ht) are included in diﬀerent cosets and are disjoint. Thus the states|Ψa,u,j g1/a\}bracketri}htand|Ψa,u,j g2/a\}bracketri}htare orthogonal. Ifg1andg2are in the same coset of HG′theng1=gg2for some g∈HG′, and for all 1 ≤i≤n,we haveφji(g1) =φji(g)φji(g2). Thus|Ψa,u,j g1/a\}bracketri}ht=|Ψa,u,j gg2/a\}bracketri}ht=|Ψa,u,j g2/a\}bracketri}ht. Let us now address the question of existence of appropriate triple s and eﬃcient ways to generate them. Let (a,u,j) be an arbitrary element of Gn×(Zd p)n×(Zp)n\{0n}, and let gbe an element of HG′. Then g=hzfor some h∈Handz∈Zd p, andφji(g) =φji(h)φji(z) fori= 1,...,n. By Lemma 5, we have |aiHG′ ui·φji(z)/a\}bracketri}ht=ωj2 i|aiHG′ ui/a\}bracketri}ht, and|aiHG′ ui·φji(h)/a\}bracketri}ht=ω(ji−j2 i)|aiHG′ ui/a\}bracketri}ht, and therefore |Ψa,u,j g/a\}bracketri}ht=ωPn i=1(ji−j2 i)+j2 in/circlemultiplydisplay i=1|aiHG′ ui/a\}bracketri}ht. For a given u, we consider the following system of quadratic equations, written in vectorial form: /braceleftigg/summationtextn i=1ui(ji−j2 i) = 0d /summationtextn i=1uij2 i = 0d. 8 It should be clear that when this system has a nontrivial solution j(that is j/\e}atio\slash= 0d) then (a,u,j) is an appropriate triple, for every a. In fact, the Chevalley-Warning theorem [3, 22] implies that the fo llowing equivalent system of vectorial equations has a nontrivial solution f or every u, whenever n >3d. /braceleftigg/summationtextn i=1uij2 i= 0d /summationtextn i=1uiji= 0d.(1) Moreover, if we take a substantially larger number of variables, we c an ﬁnd a solution in polynomial time. Theorem 5. Ifn= (d+1)2(d+2)/2then we can ﬁnd a nontrivial solution for the system (1) in pol ynomial time. The proof of Theorem 5 will be given in the next section. To ﬁnish the p roof of Theorem 4 we describe the eﬃcient hiding procedure. On input |g/a\}bracketri}ht, it computes, for some a∈Gn, the superposition 1 pdn/circlemultiplydisplay i=1/summationdisplay ui∈Zp|ui/a\}bracketri}ht|aiHG′ ui/a\}bracketri}ht, which by Lemma 4 can be done eﬃciently, and then it measures the reg isters for the ui. Then, by Theorem 5 it ﬁnds eﬃciently

Question 1031 multiple-choice

Quantum computing algorithms often utilize the quantum Fourier transform (QFT) to efficiently process periodic structures and diagonalize certain matrices, such as those found in circulant graphs. To mitigate hardware limitations, the approximate quantum Fourier transform (AQFT) is employed, trading reduced circuit complexity for controlled accuracy loss. Which key property of circulant matrices enables efficient simulation of continuous-time quantum walks on circulant graphs using the quantum Fourier transform? 1) Their adjacency matrices are always sparse 2) They are characterized by eigenvalues of modulus one 3) They are diagonalizable by the quantum Fourier transform 4) They have only real-valued entries 5) Their graphs are necessarily bipartite 6) They require no controlled rotations for diagonalization 7) Their eigenvectors are always orthogonal to each other

✓ Correct Answer:

The correct answer is 3) They are diagonalizable by the quantum Fourier transform.

📚 Reference Text:

AQFT, so as to evaluate the error rate by approximating the circuit. We consider the case of 2, 3 and 4 qubits, and we perform a statistical analysis for a variety of circulant graphs. This paper is organized as follows. Section 2 gives a brief description of CTQWs, describing the circulant graphs and its spectral properties, and the quantum circuit for the AQFT. Section 3 presents our results, performing experiments for several graphs, varying the number of qubits and analysing the effect of AQFT on state ﬁdelity. Finally, we con- clude our work in section 4 giving some remarks and insights for further improvements. 2. Theoretical Background The development of new quantum algorithms can be achieved by the manipulation of the Schr ¨odinger equation, which governs the evolution of quantum mechanical systems. Given a quantum state j (x;t)i, which belongs to a Hilbert space H, and the Hamiltonian Hwhich encodes the dynamics of an isolated quantum system, we can write i~@j (t)i @t=Hj (t)i; (1) where~= 1, and its evolution occurs in a N-dimensional space. Throughout this work we consider the time-independent case of H, and this allows us to ﬁnd a solution for (1) j (t)i=Uj (0)i=eiHtj (0)i; (2) whereHis a Hermitian operator considering a closed system. Given a graph G(V;E)and its underlying adjacency matrix A, whereV;E are sets of vertices and edges, respectively, we can describe a CTQW by the equation j (t)i=eiAtj (0)i; (3) whereAis assuming the role of the Hamiltonian [Farhi and Gutmann 1998]. Throughout this work, the circulant class of graphs will be considered. This kind of graph is deﬁned by a circulant matrix, such that A=2 666664c0cN1c2c1 c1c0cN1c2 ...c1c0...... cN2......cN1 cN1cN2c1c03 777775: (4) Considering a simple graph, the adjacency matrix in (4) should have some constraints: ﬁrstly,c0= 0 by the absence of self-loops; secondly, the matrix must be symmetric, meaning that cnj=cj. In order to construct the quantum circuit, notice that the circulant matrix is diago- nalizable by the Fourier transform, such that the eigenvalues are p=c0+NX q=1cNq!pq: (5) where!=e2i=N. We can then construct an operator which diagonalizes the circulant graph and we can deﬁne the unitary operator for the CTQW as U=FyeitF (6) whereFwill be the QFT, and eit=X jeijtjjihjj (7) is a diagonal operator which is implemented using multiplexors [Shende et al. 2006,Bul- lock and Markov 2004]. Finally, we’ve implemented the approximate quantum Fourier transform (AQFT), which is similar to the Quantum Fourier Transform, but the least sig- niﬁcant controlled rotations are discarded, as depicted in ﬁgure 1, considering mn1. Figure 1. Circuit for the approximate quantum Fourier transform. 3. Simulating on a Quantum Computer In this section we present the methodology used to implement the quantum circuits, and our ﬁndings with respective analysis. Firstly, we construct our circuits, as is shown in ﬁgure 2. j 0i=nAQFTmeitAQFTy m Figure 2. Circuit for the continuous-time quantum walk. The Qiskit package is then used to submit our jobs to IBM’s Toronto backend. In order to construct this circuit, we use two main functions provided

Question 1032 multiple-choice

Quantum circuit optimization is vital for improving the efficiency and capabilities of quantum algorithms, especially as applications expand in fields like quantum machine learning and signal processing. Modern approaches now utilize advanced computational techniques to automate the design and arrangement of quantum gates. Which method enables efficient automated search for quantum circuit architectures by leveraging both probabilistic exploration and targeted sampling to optimize discrete and continuous gate arrangements? 1) Genetic algorithms combined with gradient descent 2) Simulated annealing with fixed gate sequences 3) Greedy search with exhaustive enumeration 4) Reinforcement learning with Markov decision processes 5) Monte Carlo graph search with importance sampling 6) Quantum annealing using static Hamiltonians 7) Dynamic programming for reversible circuit synthesis

✓ Correct Answer:

The correct answer is 5) Monte Carlo graph search with importance sampling.

📚 Reference Text:

Title: Monte Carlo graph search for quantum circuit optimization Year: 2023 Paper ID: fd24e175f0d9e14059a7e76b04813607c9e358a1 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/fd24e175f0d9e14059a7e76b04813607c9e358a1 Abstract: The building blocks of quantum algorithms and software are quantum gates, with the appropriate combination of quantum gates leading to a desired quantum circuit. Deep expert knowledge is necessary to discover effective combinations of quantum gates to achieve a desired quantum algorithm for solving a specific task. This is especially challenging for quantum machine learning and signal processing. For example, it is not trivial to design a quantum Fourier transform from scratch. This work proposes a quantum architecture search algorithm which is based on a Monte Carlo graph search and measures of importance sampling. It is applicable to the optimization of gate order, both for discrete gates, as well as gates containing continuous variables. Several numerical experiments demonstrate the applicability of the proposed method for the automatic discovery of quantum circuits.

Question 1033 multiple-choice

Amplitude estimation is a fundamental procedure in quantum computing, often utilized to infer probabilities of specific measurement outcomes in quantum states. Recent algorithmic advancements aim to optimize resource usage and accuracy, especially in environments with costly quantum state preparation. Which feature distinguishes the newest quantum amplitude estimation algorithm from prior approaches in terms of both practicality and output, particularly when quantum state preparation is expensive? 1) Requires a substantially larger number of ancilla qubits than earlier algorithms 2) Relies on adaptive, variable runtimes for amplitude estimation 3) Necessitates a pre-established lower bound on the amplitude to function 4) Estimates the Grover angle (arcsin(a)) rather than measurement probability 5) Only allows relative-error estimates, limiting its accuracy for small amplitudes 6) Is incompatible with singular value transformation techniques 7) Directly estimates a² with far fewer ancilla qubits and deterministic runtime

✓ Correct Answer:

The correct answer is 7) Directly estimates a² with far fewer ancilla qubits and deterministic runtime.

📚 Reference Text:

inference, thermal state preparation and partition function estimation. However, it is also quite useful in general when |Ψ/angbracketrightis expensive to prepare. For example, when estimating the expectation of an observable on the ground state of a Hamiltonian, preparing the ground state can be very expensive. Thus, it may be practical to only prepare it once. The only previously known algorithm for non-destructive amplitude estimation is given in[HW19]. Itworksviaseveralinvocationsofamplitudeestimationaccordingto[BHMT00], which is based on phase estimation. We argue that our new algorithm has several perfor- mance advantages over the prior art. However, these advantages are not very quantiﬁable, preventing us from computing a numerical constant-factor speedup. The advantages are as follows: •Our new algorithm requires dramatically fewer ancillae. This is because [HW19] relies on phase estimation with median ampliﬁcation. As argued above, median Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 43 ampliﬁcation requires O(nlog(δ−1))ancillae. In contrast, we simply use the protocol for energy estimation which requires n+O(1)ancillae. •Our new algorithm runs in a ﬁxed amount of time: one application of our energy estimation algorithm suﬃces. In contrast, [HW19]’s algorithm works by repeatedly attempting to ‘repair’ the input state. This process succeeds only with probability 1/2, so while the expected number of attempts is constant, the resulting algorithm is highly adaptive and may need a variable amount of time. •Our new algorithm does not require knowledge of a lower bound on the amplitude a. The ‘repair’ step in [HW19] itself involves another application of phase estimation, which must produce an estimate with enough accuracy to distinguish arcsin(a)and −arcsin(a). This also implies that [HW19] can only produce relative-error estimates, even when an additive-error estimate might be suﬃcient, as it is in [AHNTW20]. •While, due to the above diﬀerences, it is not really possible to perform a side-by- side constant-factor comparison of the algorithms, we do expect to inherit a modest constant-factor speedup from our energy estimation algorithm. [BHMT00] has no need to perform Hamiltonian simulation, so we expect the speedup to look more like the one in the case of phase estimation. Since no rounding promise is required, makingαtiny is not really necessary. But it isnecessary that α < 1/2since this makes traditional phase estimation round correctly. Looking at Figure 5, we already see constant factor speedups in this regime. Another minor advantage of our method over [HW19] is that it estimates a2directly, rather than going through the Grover angle θ:= arcsin(a). Coherently evaluating a sine function in superposition to correct this issue, while possible, will have an enormous ancilla overhead.a2is the probability with which a measurement of |Ψ/angbracketrightyields a state in Π, which may be useful directly. We note that an additive error estimate of ais slightly stronger than an additive error estimate of a2. Moreover this accuracy seems to be necessary for some versions of quantum meanestimation: seeforexampleAppendixCof[AHNTW20]. Iftheamplitude aisdesired rather than a2, then, since the algorithm proceeds by obtaining a block-encoding of a2, one can use singular value transformation to make a block-encoding of ainstead. A polynomial

Question 1034 multiple-choice

Quantum algorithmic techniques, such as those based on the hidden subgroup problem (HSP) and Fourier sampling, have played a significant role in analyzing the security of modern cryptosystems, particularly in the context of post-quantum cryptography. The McEliece cryptosystem and its resistance to certain quantum attacks are informed by deep results in representation theory and group analysis. Which technical strategy is crucial for demonstrating the indistinguishability of hidden subgroups in the McEliece cryptosystem under strong Fourier sampling, thereby undermining direct quantum attacks based on HSP? 1) Partitioning irreducible representations into SMALL and LARGE sets based on dimension and character properties 2) Employing Grover's algorithm to search for subgroup elements efficiently 3) Using lattice-based hardness assumptions to guarantee security 4) Applying Shor's algorithm to the automorphism group of the cryptosystem 5) Encoding subgroup information directly in the public key representation 6) Utilizing classical code equivalence tests for subgroup analysis 7) Constructing explicit isomorphisms between symmetric and general linear groups

✓ Correct Answer:

The correct answer is 1) Partitioning irreducible representations into SMALL and LARGE sets based on dimension and character properties.

📚 Reference Text:

only to weak Fourier sam- pling. Remark 1. Our results show that the natural reduction of McEliece to a hidden sub- group problem yields negligible information about the secret key. Thus they rule out the direct analogue of the quantum attack that breaks, for example, RSA. Of course, our results do not rule out other quantum (or classical) attacks. Neither do they establish that a quantum algorithm for the McEliece cryptosystem would violate a natural hardness 5The column rank of Mis understood to be over the ﬁeld Fq/lscript. Recall that the entries of the matrix Mare in Fq/lscript. 6The case of GL2(Fq)is omitted in this version for lack of space. McEliece and Niederreiter Cryptosystems 765 assumption, as do recent lattice cryptosyste m constructions whose hardness is based on the Learning With Errors problem (e.g. Regev [18]). Nevertheless, they indicate that any such algorithm would have to involve signiﬁcant new ideas beyond than those that have been proposed so far. Summary of technical ideas. LetGbe a ﬁnite group. We wish to establish general criteria for indisti nguishability of subgroups H

Question 1035 multiple-choice

Quantum machine learning explores how quantum computers can accelerate algorithms for complex tasks, such as solving parametric partial differential equations (PDEs) in physics and engineering. Quantum adaptations of classical neural operator architectures have enabled new circuit designs with improved computational efficiency. Which technical innovation specifically enables quantum Fourier neural operators to achieve logarithmic time complexity with respect to the number of PDE evaluations? 1) The use of tensor networks to compress quantum data 2) Implementing variational quantum eigensolvers for PDE solution mapping 3) Deploying amplitude encoding for efficient data representation 4) Utilizing Grover’s algorithm for PDE search 5) Adapting the quantum Fourier transform within the unary encoding basis 6) Applying quantum error correction to neural operator circuits 7) Introducing quantum convolutional layers for feature extraction

✓ Correct Answer:

The correct answer is 5) Adapting the quantum Fourier transform within the unary encoding basis.

📚 Reference Text:

Title: Quantum Fourier networks for solving parametric PDEs Year: 2023 Paper ID: df4d5ea02a97f4cde2e963bfa76f63e894305d44 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/df4d5ea02a97f4cde2e963bfa76f63e894305d44 Abstract: Many real-world problems, like modelling environment dynamics, physical processes, time series etc involve solving partial differential equations (PDEs) parameterised by problem-specific conditions. Recently, a deep learning architecture called Fourier neural operator (FNO) proved to be capable of learning solutions of given PDE families for any initial conditions as input. However, it results in a time complexity linear in the number of evaluations of the PDEs while testing. Given the advancements in quantum hardware and the recent results in quantum machine learning methods, we exploit the running efficiency offered by these and propose quantum algorithms inspired by the classical FNO, which result in time complexity logarithmic in the number of evaluations and are expected to be substantially faster than their classical counterpart. At their core, we use the unary encoding paradigm and orthogonal quantum layers and introduce a new quantum Fourier transform in the unary basis. We propose three different quantum circuits to perform a quantum FNO. The proposals differ in their depth and their similarity to the classical FNO. We also benchmark our proposed algorithms on three PDE families, namely Burgers’ equation, Darcy’s flow equation and the Navier–Stokes equation. The results show that our quantum methods are comparable in performance to the classical FNO. We also perform an analysis on small-scale image classification tasks where our proposed algorithms are at par with the performance of classical convolutional neural networks, proving their applicability to other domains as well.

Question 1036 multiple-choice

In quantum computing, the Hidden Subgroup Problem (HSP) is central to many quantum algorithms and involves determining properties of a subgroup K within a finite group G using function evaluations. Algebraic and data structure techniques, such as quantum Ordered Binary Decision Diagrams (OBDDs) and polynomial tests, are often employed to efficiently solve this problem. Which polynomial-based test is specifically designed to verify that a function is constant on each coset of a subgroup K in the Hidden Subgroup Problem framework? 1) The polynomial that checks whether coset representatives are unique 2) g1(x), which checks for constancy of the function on each coset 3) The polynomial that verifies the sum of function values over all cosets equals zero 4) g2(x), which tests if images from different cosets are non-overlapping 5) The polynomial that encodes the entire group’s Cayley table 6) The polynomial that ensures elements within a coset are pairwise distinct 7) The polynomial that computes the intersection size of two arbitrary cosets

✓ Correct Answer:

The correct answer is 2) g1(x), which checks for constancy of the function on each coset.

📚 Reference Text:

difﬁcult. Suppose we have the two conditions of the lemma satisﬁed. Let aKandbKbe twodifferent cosets with elements d∈aKand c∈bK,such that σd=σc. Weﬁxc∈bK. There are twocases possible: 1. Forall d∈aK(σd=σc); 2. Thereexists d′∈aK(σd/n⌉}ationslash=σc). Apparently in the ﬁrst case weindeed could choose any of the e lements of acoset to check inequalities. In the second case the ﬁrst condition of the lemmawould fail. Thereasoning for bKisanalogous. When the values of fare different on different cosets, obviously, the sum of the se values is the sum of numbers from 1 through G:K. Therefore, HSP G,K(σ) =1 iff both conditions of the lemma are satisﬁed. Accordingtothepreviouslemma,HSP G,K(x)hasacharacteristicconsistingoftwopolynomialsover Z2n, checking conditions of thelemma. Weshall construct them e xplicitly to show they are linear. We shall adopt another indexation of χwhen convenient: χa,qis a value of fon theq-th element of the coset aK. Therefore, for abinary input symbol xjwedeﬁne •a=a(j)for thenumber of the corresponding coset; •q=q(j)for thenumber of the corresponding element of the coset a; •r=r(j)for thenumber of bit in thebinary representation of χa,q and start indexation from 0. Thus a∈{0,...,(G:K)−1},q∈{0,...,|aK|−1}. In this notation thepolynomials are: 1.g1(x)=∑a∑q2(|K|a+q)⌈logG:K⌉(χa,q−χa,q−1 mod|K|). Thus,g1(x)=0iff for everycoset afunction fmaps all the elements of aonto the same element of X. 2.g2(x) =/parenleftBig ∑(G:K) j=1χij/parenrightBig −S, whereχijis the representative chosen from the j-th coset. Therefore, g2(x)checks whether the images of elements from different cosets are distinct. By the generalized ﬁngerprinting technique we can construc t quantum OBDD of width O(n), com- puting HSP G,K(x)withone-sided error. 10 Algorithms for Quantum Branching Programs References [1] A. Ambainis and R. Freivalds (1998): 1-way quantum ﬁnite automata: strengths, weaknesses and ge neral- ization.Proceeding of the 39th IEEE Conference on Foundation of Comp uter Science , pp. 332–342. See also arXiv:quant-ph/9802062v3. [2] F. Ablayev, A. Gainutdinova and M. Karpinski (2001): On computational power of quantum branching programs .Lecture Notes in Computer Science 2138, pp. 59–70. Springer-Verlag. See also arXiv:quant- ph/0302022v1. [3] F. Ablayev,A. Gainutdinova,M. Karpinski,C. Moore,and C. Pollette (2005): On the computationalpower ofprobabilisticandquantumbranchingprogramsofconstan twidth.InformationandComputation . [4] V.D. Agrawal,D. LeeandH. Wozniakowski(1998): Numerical Computationof CharacteristicPolynomials ofBooleanFunctionsanditsApplications .NumericalAlgorithms 17,pp.261–278. [5] A. Ambainis and N. Nahimovs (2008): Improved constructions of quantum automata . arXiv:0805.1686v1 Availableat http://xxx.lanl.gov/archive/quant-ph . [6] F. Ablayev and A. Vasiliev (2008): On the Computation of Boolean Functions by Quantum Branchin g Programs via Fingerprinting .Electronic Colloquium on Computational Complexity , TR08-059. Available athttp://www.eccc.uni-trier.de/eccc/ . [7] R. Freivalds (1979): Fast probabilistic algorithms .FCT’79, Lecture Notes in Computer Science 74, pp. 57–69. Springer-Verlag. [8] A. Khasianov (2005): Complexity Bounds On Some Fundamental Computational Probl ems For Quantum BranchingPrograms . Ph.D.thesis, BonnUniversity. Availableat http://nbn-resolving.de/urn:nbn: de:hbz:5N-05696 . [9] C. Moore and J.P. Crutchﬁeld (2000): Quantum automata and quantum grammars .Theoretical Computer Science237,pp.275–306. [10] R. MotwaniandP. Raghavan(1995): RandomizedAlgorithms . CambridgeUniversityPress. [11] M. Nakanishi, K. Hamaguchi and T. Kashiwabara (2000): Ordered quantum branchingprograms are more powerful than ordered probabilistic branching programs un der a bounded-width restriction .Proceedings of the 6th Annual International Conference on Computing and Combinatorics, Lecture Notes in Computer Science1858,pp.467–476. [12] M.SauerhoffandD.Sieling(2004): Quantumbranchingprogramsandspace-boundednonuniformq uantum complexity .TheoreticalComputerScience 334,pp.177–225. [13] I. Wegener (2000): Branching programs and

Question 1037 multiple-choice

Quantum simulation of lattice gauge theories often necessitates the replacement of continuous gauge groups, such as SU(2) and SU(3), with discrete subgroups to enable computation on quantum hardware. The choice and properties of the discrete subgroup play a critical role in preserving the physics of the original theory. Which property of the discrete subgroup Σ(360×3) is most crucial for achieving percent-level agreement with SU(3) Casimir scaling in certain irreducible representations? 1) Its ability to commute with all elements of SU(3) 2) The inclusion of all possible SU(3) representations 3) The preservation of fermionic statistics 4) The existence of irreducible representations that subduce cleanly to single Σ(360×3) irreducible representations 5) Its maximal dimension compared to other discrete subgroups 6) The invariance under time-reversal symmetry 7) Its compatibility with SU(2) symmetry

✓ Correct Answer:

The correct answer is 4) The existence of irreducible representations that subduce cleanly to single Σ(360×3) irreducible representations.

📚 Reference Text:

Title: Digitization and subduction of SU(N) gauge theories Year: 2024 Paper ID: 2f1156dad887f7318cf8e4cd513ac246fe2bb615 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/2f1156dad887f7318cf8e4cd513ac246fe2bb615 Abstract: The simulation of lattice gauge theories on quantum computers necessitates digitizing gauge fields. One approach involves substituting the continuous gauge group with a discrete subgroup, but the implications of this approximation still need to be clarified. To gain insights, we investigate the subduction of SU(2) and SU(3) to discrete crystal-like subgroups. Using classical lattice calculations, we show that subduction offers valuable information based on subduced direct sums, helping us identify additional terms to incorporate into the lattice action that can mitigate the effects of digitization. Furthermore, we compute the static potentials of all irreducible representations of Σ(360×3) at a fixed lattice spacing. Our results reveal a percent-level agreement with the Casimir scaling of SU(3) for irreducible representations that subduce to a single Σ(360×3) irreducible representation. This provides a diagnostic measure of approximation quality, as some irreducible representations closely match the expected results while others exhibit significant deviations. Published by the American Physical Society 2024

Question 1038 multiple-choice

In quantum computing, group theory underpins the design of quantum circuits, especially when implementing symmetry operations using quantum gates. The symmetric group Sn and its representations play a crucial role in manipulating qubit states and encoding permutations within quantum algorithms. When representing permutations from the symmetric group Sn on the Hilbert space of n qubits, why is it necessary to act on indices using the inverse permutation? 1) It ensures that the representation of the group is associative, properly reflecting the group operation on quantum states. 2) It allows SWAP gates to act non-locally between any pair of qubits in a circuit. 3) It guarantees that only even permutations are implemented on the quantum hardware. 4) It restricts the action to subspaces invariant under Sn, simplifying the analysis of entanglement. 5) It allows each permutation to be expressed as a product of rotation gates instead of SWAP gates. 6) It prevents the representation from being reducible, ensuring uniqueness of quantum states. 7) It ensures that the action corresponds to physical measurements rather than unitary evolutions.

✓ Correct Answer:

The correct answer is 1) It ensures that the representation of the group is associative, properly reflecting the group operation on quantum states..

📚 Reference Text:

= (12)using cycle no- tation, then P(12)ji1i2i3i=ji2i1i3i. In other words,P(12)=SWAP 1;2. So, permutation repre- sentations of the symmetric group allow us to per- mute qubits (and more generally, tensor indices). responds to the permutations that can be performed on a set of sizen. TakeS3. Now, in the search for representa- tions, a natural vector space to probe is the Hilbert space of three qubits, since we can label a basis of this space with three indices. In Example 1 we explicitly deﬁne a repre- sentation of S3acting on this basis by permuting indices as Pji1i2i3i= i1(1)i1(2)i1(2) 5. The simplicity of this representation belies its depth: it will be a frequent of- fender, playing a central role in a variety of representations on tensor spaces. It is also useful for QML tasks dealing with quantum states deﬁned over a graph (see Fig. 5(b)). In Example 2, we present an alternative representation forS3. This example serves two purposes: ﬁrstly, it illus- trates that a ﬁxed group Gcan, and will, have representa- tions which look nothing alike; secondly, it shows how to deﬁnearepresentationbyworkingongeneratorsofagroup. In particular, recall that a set of group elements generates G, writtenhg1;g2;:::;gni=G, if every element h2Gcan beexpressedasastringofgenerators gi1gi2gik=h. Of course, there are often many diﬀerent generating sets for a givenG, and particular choices are made with the intent of analysis: for example, the symmetric group Snis generated by all transpositions of two elements (i;j),16i;j6n. The attentive QML researcher’s ears should perk up at this 5One may wonder why the representation acts by the inverse per- mutation1on indices instead of . Observe that if we chose the naive deﬁnition and wrote Pji1i2i= i(1)i(2) , then associa- tivity no longer holds: P1(P2jvi)6= (P1P2)jvi. Something similar happens when deﬁning representations on group functions f:G!C: the “natural” representation RofGon the vector space of functions is given by precomposition: deﬁne Rgf(x) =f(g1x) for allx2G. 11 Example 2: Dihedral rep of S3on 1 qubit We will construct a representation of S3on a single qubitV=C2using some geometric insight: S3is alsoD3, the symmetry group of the triangle. From this perspective, we can generate the group using a rotation (123)and a reﬂection (12), and we will deﬁne a representation which acts simultaneously on two triangles given in each copy of Cas the cubic roots of unityf1;!;!1gwhere!:=e2i=3. Deﬁne these two maps in the computational basis R(123) = !0 0!1 ; R (12)= 0 1 1 0 : One way to think about these maps is the following: R(123)rotatestwotrianglesinoppositedirectionsby the angle!, by R(123)R(123)j0i=R(123)(!j0i) =!2j0i R(123)R(123)j1i=R(123)(!1j1i) =!2j1i; andR(12)is the reﬂection which swaps the two states (and thus the two triangles) R(12)j0i=j1i: second statement: this means that for index permutation representations like Example 1, all of the representatives ofSncan be expressed as products of SWAP i;jgates. 2. Representations of continuous groups Let us now accrue a smattering of representations of con- tinuous groups. Continuous groups arise very naturally within QML: the unitary groups U(d)are prototypical ex- amplesofLiegroups,andquantumcomputingisultimately all about unitaries. In what follows we will focus and give special attention to the special

Question 1039 multiple-choice

In quantum information theory, unitary k-designs are finite sets or distributions of unitary matrices that replicate the statistical properties of the Haar measure up to the k-th moment. The frame potential is a diagnostic tool used to determine whether a given ensemble forms a k-design and to establish lower bounds for their construction. For k ≤ d, what is the minimum possible size of a discrete set of unitaries S required to form an exact unitary k-design, and what is the corresponding frame potential value? 1) Size at least k! and frame potential exactly k! 2) Size at least d^k and frame potential exactly d^k 3) Size at least 2^k and frame potential exactly k^2 4) Size at least k and frame potential exactly k 5) Size at least d! and frame potential exactly d! 6) Size at least k! and frame potential exactly d^{2k} 7) Size at least d^2 and frame potential exactly k!

✓ Correct Answer:

The correct answer is 1) Size at least k! and frame potential exactly k!.

📚 Reference Text:

25. Note that although the proof of Proposition 25 was given for the Haar measure on the unitary group, it applies equally to any k-invariant measure. It is worth noting that the left and right invariance properties, and therefore Lemma 32, hold for any uniform probability distribution νdefined over a subgroup of unitaries S. In the following lemma, we show that the difference between the frame potential of a probability distribution ν(not necessarily k-invariant) and that of the Haar measure µHcan be represented by the squared 2-norm of the difference between their vectorized moment operator. Lemma 33 (Frame potential difference) .LetF(k) νandF(k) µHbe the frame potentials of the proba- bility distribution νand the Haar measure µH, respectively. Then, we have: F(k) ν−F(k) µH=/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE V∼ν/bracketleftbig V⊗k⊗V∗⊗k/bracketrightbig −E U∼µH/bracketleftbig U⊗k⊗U∗⊗k/bracketrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 2. (167) Proof. Let M(k) µH:=E U∼µH/bracketleftbig U⊗k⊗U∗⊗k/bracketrightbig and M(k) ν:=E V∼ν/bracketleftbig U⊗k⊗U∗⊗k/bracketrightbig . First, we note that M(k)† µH= M(k) µHdue to Proposition 3. Furthermore, because of the invariance of the Haar measure, we have M(k)2 µH= M(k) µHand M(k) νM(k) µH= M(k) νM(k)† µH= M(k) µH. Thus, by using these properties, we obtain: /vextenddouble/vextenddouble/vextenddoubleM(k) ν−M(k) µH/vextenddouble/vextenddouble/vextenddouble2 2= Tr/bracketleftbigg/parenleftig M(k) ν−M(k) µH/parenrightig†/parenleftig M(k) ν−M(k) µH/parenrightig/bracketrightbigg (168) = Tr/bracketleftig M(k)† νM(k) ν/bracketrightig + Tr/bracketleftig M(k)† µHM(k) µH/bracketrightig −Tr/bracketleftig M(k)† νM(k) µH/bracketrightig −Tr/bracketleftig M(k)† µHM(k) ν/bracketrightig (169) = Tr/bracketleftig M(k)† νM(k) ν/bracketrightig −Tr/bracketleftig M(k) µH/bracketrightig (170) = Tr/bracketleftig M(k) νM(k)† ν/bracketrightig −Tr/bracketleftig M(k) µHM(k)† µH/bracketrightig . (171) We can conclude by observing that for µ=µHandµ=νwe have: Tr/bracketleftig M(k) µM(k)† µ/bracketrightig =E V,U∼µTr/bracketleftbig (UV†)⊗k⊗(UV†)∗⊗k/bracketrightbig =E V,U∼µ/bracketleftig/vextendsingle/vextendsingleTr/parenleftbig UV†/parenrightbig/vextendsingle/vextendsingle2k/bracketrightig =F(k) µ. (172) 23 It follows from the previous Lemma that showing that a distribution νis ak-design can be achieved by computing its frame potential and comparing it with that of the Haar measure µH. This is due to the fact that if the two frame potentials coincide, the difference between their vectorized moment operators in 2-norm must be zero, hence they must be equal (which is an equivalent unitary design definition because of Observation 28). We can restate the result explicitly as follows: Proposition 34 (Frame potential k-design condition) .We have: F(k) ν≥F(k) µH= dim(span( Vd(π) :π∈Sk)). (173) Moreover, the equality holds if and only if νis ak-unitary design. In particular, if k≤d, then dim(span(Vd(π) :π∈Sk)) =k!. Proof. By Lemma 33, Lemma 32 and Schur-Weyl duality 9, we have: F(k) ν≥F(k) µH= dim (Comm(U( d),k)) = dim (span( Vd(π) :π∈Sk)). (174) Moreover, by Proposition 11, we have that if k≤d, then the number of linearly independent permutation matrices is k!. By utilizing this result, we can derive a straightforward lower bound on the cardinality of a discrete set Sof unitaries necessary to form a k-design. To establish this bound, let us consider the frame potential F(k) νassociated with the uniform distribution νover the discrete set S. We can then deduce that: F(k) ν=1 |S|2|S|/summationdisplay i,j=1/vextendsingle/vextendsingle/vextendsingleTr/parenleftig UiU† j/parenrightig/vextendsingle/vextendsingle/vextendsingle2k ≥1 |S|2|S|/summationdisplay i=1/vextendsingle/vextendsingle/vextendsingleTr/parenleftig UiU† i/parenrightig/vextendsingle/vextendsingle/vextendsingle2k =1 |S|d2k. (175) Furthermore, considering the fact that νconstitutes a k-design (with k≤d), by the previous proposition, we have that F(k) ν=k!. This implies

Question 1040 multiple-choice

Variational quantum circuits can be trained to implement quantum operations for diverse computational problems, including data compression and group-theoretic transformations. Quantum autoencoders leverage such circuits to achieve reversible compression of quantum information. Which feature enables a single parameterized quantum circuit to implement quantum Fourier transforms (QFTs) for multiple finite Abelian groups, such as Z₈, Z₄×Z₂, Z₂×Z₄, and Z₃², without prior knowledge of the group structure? 1) The use of tunable parameters that control gate activation and qubit swapping, allowing the circuit to adaptively realize the QFT for different group types 2) Hard-coded sequences of Hadamard and phase gates specific to each target group 3) Measurement-based quantum protocols that select the appropriate QFT after observing the input state 4) Fixed-depth circuits optimized exclusively for cyclic groups of prime order 5) Classical preprocessing to identify the group before constructing the quantum circuit 6) Post-selection techniques that filter out incorrect QFT outcomes for each group 7) Employing only universal quantum gates without parameterization or adaptive control

✓ Correct Answer:

The correct answer is 1) The use of tunable parameters that control gate activation and qubit swapping, allowing the circuit to adaptively realize the QFT for different group types.

📚 Reference Text:

on solving this variational hidden subgroup compression problem using var iational quantum circuits, without prior knowledge of the group G. This is achieved by replacing the quantum Fourier transform (QFT, see Supplementary Note C) in a standard HSPcircuit with a parameterized circuit, which recovers the QFT over any ﬁnite Abelian group Gof the form Z 2m1×Z2m2×...×Z2mq,b a s e do nt h ec h o s e n parameters. We then incorporate this parametrized hidden subgroupalgorithm into an autoencoder structure. The autoencoder compresses thedata through its bottleneck using the encoder and restores it using thedecoder, ensuring reversible compression. During training, an automaticsearch ﬁnds suitable parameters to make the decoder ’so u t p u ta ss i m i l a ra s possible to the input. Once training is complete, the decoder can be removed and the encoder acts as a compressor. Searching over different types of Abelian groups . We present here a parametrized quantum circuit that is capable of implementing the quantum Fourier transform over any Abelian group of the form Z2m1×Z2m2×...×Z2mq, with suitable choices of parameters. We willbegin with examples and then present the general architecture. The quantum Fourier transform over a group Gis deﬁned as a unitary, QFTZN¼1ﬃﬃﬃﬃ NpX i;j2Gexp2πiji N/C18/C19 ∣j/C11 ih∣:ð3Þ The quantum Fourier transform o ver a Cartesian product of two groups ZN1×ZN2is the tensor product of t he respective quantum Fourier transform24,31, i.e., QFTZN1×ZN2¼QFTZN1/C10QFTZN2. Let us now ﬁnd the patterns of the quantum circuits for quantum Fourier transform in order to formulate a parametrised circuit that generalisesthese transforms. Consider the groups GﬃZ 8,Z2×Z4,Z4×Z2,o rZ3 2 (detailed de ﬁnition of Gis in Supplementary Note A). As proved in Sup- plementary Note C, the quantum circ uit for the Fourier transform over Z8is ð4Þ where His the Hadamard gate, and Rk¼10 0e2πi 2k/C20/C21 is a phase rotation gate. The circuit for quantum Fourier transform over Z4×Z2is ð5Þ Note that we can also have the quantum Fourier transform over Z2×Z4, which is ð6Þ Finally, the quantum circuit for the quantum Fourier transform over Z3 2is ð7Þ The quantum circuit of the Fourier transform over Z4×Z2,Z2×Z4and Z3 2can be obtained by shutting down certain controlled phase rotation gates Rkin the QFT circuit of Z8, followed by some swap operations among the qubits. In fact, three parameters will be suf ﬁcient to control the circuit. Denote the parameterized swap operation over all the three qubits as SWAP θ!¼SWAPθ1 12SWAPθ2 23SWAPθ1θ2 12;ð8Þ where SWAPθ ijis the unitary that swaps qubit iand qubit jwhen θ=1 ,a n d does nothing when θ= 0. Therefore, the following circuit ð9Þ can implement the quantum Fourier transform over all considered types of groups by suitable parameters. With θ1=θ2=θ3= 1, it implements the quantum Fourier transform over Z8;w i t h θ1=1 ,θ2=θ3= 0, it implements QFT over Z4×Z2;w i t h θ1=θ3=0 ,θ2= 1, it implements QFT over Z2×Z4;https://doi.org/10.1038/s41534-024-00865-2 Article npj Quantum Information | (2024) 10:74 3 andﬁnally, with θ1=θ2=θ3=0 ,i ti m p l e m e n t st

Question 1041 multiple-choice

In knot theory and group theory, mathematical constructions can encode group relations within knot diagrams, facilitating the study of homomorphisms from knot groups to finite groups such as the alternating group An. Techniques from algebra and topology are combined to manipulate solution spaces to group equations. Which of the following is a key property of alternating groups An (for n ≥ 6) that enables the iterative amplification and suppression of homomorphisms from knot groups in group-theoretic constructions? 1) The existence of nontrivial abelian subgroups 2) The presence of central elements of order two 3) The property that all elements generate cyclic subgroups 4) The ability to represent every element as a product of three transpositions 5) The existence of multiple non-conjugate normal subgroups 6) The lack of simple quotient groups 7) The absence of nontrivial normal subgroups, making them simple groups

✓ Correct Answer:

The correct answer is 7) The absence of nontrivial normal subgroups, making them simple groups.

📚 Reference Text:

shows how to construct a knot that imposes these relati ons. Each such equation xi=xwi 1is imposed by introducing a band between the circle associated withx1and the circle associated with xi(see Figure 5b) and passing it through the circles associated with the vari ables appearing in wi; by passing such a band “through” such a circle—one over and once under as in Fi gure 5c—the generator associated with the circle operates on the band value by conjugation while le aving the generator associated with the circle unaﬀected. In general, the words wimay contain references to the variable xi, in which case one passes the band through the circle associated with xi, as in Figure 5d. This construction produces a knot we call K1 c. By construction, associating the strokes the koriginal circles with the variables xi, there is a homomorphism ψofK(K1 c) intoC⊂Gfor whichψ:xi/ma√st≀→ci. As remarked above, if φ∈Aut(G) which ﬁxes the conjugacy class C, thenφ◦ψis also an element of H(K1 c,C). However, there will, in general, be other solutions to the eq uations ( 5.3): in particular, the map which carries all strokes of the knot to a particular, ﬁxed element c∈Csatisﬁes the equations ( 5.3). Our goal is to add some strands to the knot so as to “amplify” the solution of int erest—the map ψ:xs/ma√st≀→ci—so that it appears with overwhelming majority if an element of selected at rand om fromH(·,C). To minimize the contribution from solutions not related by a n automorphism, consider a undesirable solutiond= (d1,...,d k) (for which there is an homomorphism ψ′:xi/ma√st≀→diand yetd/\e}ati≀\slash∈[c]). We now introduce a new circle, associated with the variable ynew, and as before introduce an equation of the form ynew=xw(x)ynew 1, (5.4) wherew(x) is a word in the xi. In order to “suppress” the homomorphism associated with d, we shall choose win such a way that 1.w(d) = 1: thus any homomorphism ψ′which carries xitodimust necessarily carry ynewto the same value asx1=d1. (Speciﬁcally, the only solution to the equation ynew=xw(d)ynew 1 =xynew 1isynew=x1.) 2.w(c) =α, a group constant for which the equation ynew=xαynew 1 has multiple solutions. In this case, the homomorphism ψ:xi/ma√st≀→cimay be extended in multiple ways with an assignment to ynew(any solution to ynew=xαynew 1 will suﬃce). Lemma 6below guarantees that there is a nontrivial element α∈An,n≥6 satisfying item 2above. Lemma 7 below guarantees that when Gis simple (i.e., has no nontrivial normal subgroups), there is a wordwin the variablesx1,...,x kso thatw(d) = 1 andw(c) =αfor any ﬁxed α∈G. Focusing now on a speciﬁc group G=An,n≥6, letαbe the element which maximizes the number of solutions to the equationy=cαy 1. It follows immediately that, for the new knot K′induced from this process, |{φ∈H(K(K′);C,G)|ψ(xi) =ci}|≥ 2|{φ∈H(K(K′);C,G)|ψ(xi) =di}|. 20 x1 x2···xk (a) Initial circles. (b) A band between two circles. x y z (c) A simple relation: xz=y.x y z (d) The equation xzy=y. Figure 5: Building a set of generators. Repeating this process ℓtimes for each of the oﬀending vectors dinduces a knot Kℓ cso that for any d/\e}ati≀\slash∈[c] we

Question 1042 multiple-choice

In quantum many-body physics, hybrid classical/quantum algorithms are increasingly used to simulate complex materials, particularly where classical methods excel at certain tasks while quantum computers offer advantages for others. The Anderson impurity model and its applications to correlated electron systems highlight the importance of efficient ground state preparation and accurate computation of dynamical properties. In a hybrid classical/quantum simulation of the multi-orbital Anderson impurity model within dynamical mean field theory (DMFT), which key factor allows accurate computation of Green’s functions even when the quantum circuit does not perfectly reproduce the ground state wave function? 1) The use of only single-qubit gates for time evolution 2) Increasing the number of classical tensor network layers 3) Ensuring complete entanglement in the quantum circuit 4) Maximizing the fidelity between classical and quantum ground states 5) Utilizing error mitigation strategies for quantum noise 6) Correctly reproducing the ground state energy in the quantum circuit 7) Restricting the simulation to one-dimensional impurity models

✓ Correct Answer:

The correct answer is 6) Correctly reproducing the ground state energy in the quantum circuit.

📚 Reference Text:

Title: Anderson impurity solver integrating tensor network methods with quantum computing Year: 2023 Paper ID: 5528d80aa1116443084d7896a3f6dcffa1037e66 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/5528d80aa1116443084d7896a3f6dcffa1037e66 Abstract: Solving the Anderson impurity model typically involves a two-step process, where one first calculates the ground state of the Hamiltonian and then computes its dynamical properties to obtain Green’s function. Here, we propose a hybrid classical/quantum algorithm where the first step is performed using a classical computer to obtain the tensor network ground state as well as its quantum circuit representation and the second step is executed on the quantum computer to obtain Green’s function. Our algorithm exploits the efficiency of tensor networks for preparing ground states on classical computers and takes advantage of quantum processors for the evaluation of the time evolution, which can become intractable on classical computers. We demonstrate the algorithm using 24 qubits on a quantum computing emulator for SrVO3 with a multi-orbital Anderson impurity model within the dynamical mean field theory. The tensor network based ground state quantum circuit preparation algorithm can also be performed for up to 60 qubits with our available computing resources, while the state vector emulation of the quantum algorithm for time evolution is beyond what is accessible with such resources. We show that, provided that the tensor network calculation is able to accurately obtain the ground state energy, this scheme does not require a perfect reproduction of the ground state wave function on the quantum circuit to give an accurate Green’s function. This hybrid approach may lead to quantum advantage in materials simulations where the ground state can be computed classically, but where the dynamical properties cannot.

Question 1043 multiple-choice

In algebraic number theory, the Brauer group of a field is a key structure classifying central simple algebras, and its subgroups reflect deep arithmetic properties. The relationship between torsion subgroups and relative Brauer groups for global fields involves intricate connections to field extensions and local-global principles. For a global field K and a positive integer n, which statement correctly describes the n-torsion subgroup Brn of the Brauer group Br? 1) Brn is always finite for any global field K and any n. 2) Brn can only be realized as a relative Brauer group when n is squarefree. 3) Brn equals the entire Brauer group Br if and only if K is a function field. 4) Brn is never an algebraic relative Brauer group for finite extensions L/K. 5) Brn can be realized as a relative Brauer group only if n is prime to the class number of K. 6) Brn can only be realized as a relative Brauer group for local fields, not global fields. 7) For every global field K and every n, Brn is always an algebraic relative Brauer group, i.e., there exists an algebraic extension L/K such that Br(L/K) = Brn.

✓ Correct Answer:

The correct answer is 7) For every global field K and every n, Brn is always an algebraic relative Brauer group, i.e., there exists an algebraic extension L/K such that Br(L/K) = Brn..

📚 Reference Text:

Title: Abelian extensions of global fields with constant local degree Year: 2004 Paper ID: 1716ee7f81d93c808f1794284f23f4f9635bc056 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/1716ee7f81d93c808f1794284f23f4f9635bc056 Abstract: Let K be a field, Br(K) its Brauer group. If L/K is a field extension, then the relative Brauer group Br(L/K) is the kernel of the restriction map resL/K : Br(K)→ Br(L). Relative Brauer groups have been studied by Fein and Schacher. Every subgroup of Br(K) is a relative Brauer group Br(L/K) for some extension L/K, and the question arises as to which subgroups of Br(K) are algebraic relative Brauer groups, i.e. of the form Br(L/K) with L/K an algebraic extension. For example if L/K is a finite extension of number fields, then Br(L/K) is infinite, so no finite subgroup of Br(K) is an algebraic relative Brauer group. In [1] the question was raised as to whether or not the n-torsion subgroup Brn(K) of the Brauer group Br(K) of a field K is an algebraic relative Brauer group. For example, if K is a (p-adic) local field, then Br(K) ∼= Q/Z, so Brn(K) is an algebraic relative Brauer group for all n. A counterexample was given in [1] for n = 2 and K a formal power series field over a local field. For global fields K, the problem is a purely arithmetic one, because of the fundamental local-global description of the Brauer group of a global field. In particular, for a Galois extension L/K of global fields, if the local degree of L/K at every finite prime is equal to n, and is equal to 2 at the real primes for n even, then Br(L/K) = Brn(K). In [1], it was proved that Brn(Q) is an algebraic relative Brauer group for all squarefree n. In [2], the arithmetic criterion above was verified for any number field K Galois over Q and any n prime to the class number of K, so in particular, Brn(Q) is an algebraic relative Brauer group for all n. In [3], Popescu proved that for a global function field K of characteristic p, the arithmetic criterion holds for n prime to the order of the non-p part of the Picard group of K. In this paper we settle the question completely, by verifying the arithmetic criterion for all n and all global fields K. In particular, the n-torsion subgroup of the Brauer group of K is an algebraic relative Brauer group for all n and all global fields K. The proof, an extension of the ideas in [2], reduces to the case n a prime power `. We first carry out the proof for number fields K. The proof for the function field case when ` 6= char(K) is essentially the same as the proof in the number field case. The proof for ` = char(K) appears in [3].

Question 1044 multiple-choice

In quantum computing, the hidden subgroup problem (HSP) is a central challenge that leverages group theory and representation theory to identify unknown subgroups using quantum states and measurements. Techniques such as Fourier sampling on finite groups are crucial in analyzing and solving instances of HSP. Which of the following statements best explains why weak Fourier sampling is generally insufficient to solve the hidden subgroup problem for nonabelian groups? 1) Weak Fourier sampling cannot measure any properties of the quantum state related to the hidden subgroup. 2) Weak Fourier sampling collapses the quantum state into a random group element, losing all subgroup information. 3) Weak Fourier sampling only works when the group has a unique trivial subgroup. 4) Weak Fourier sampling produces exponentially many possible outcomes, making post-processing infeasible. 5) Weak Fourier sampling is limited to detecting normal subgroups in any group. 6) Weak Fourier sampling always outputs the same irrep, providing no distinguishing power. 7) Weak Fourier sampling reveals only the irrep label, discarding critical information contained within high-dimensional irreducible representations needed to identify the hidden subgroup in nonabelian groups.

✓ Correct Answer:

The correct answer is 7) Weak Fourier sampling reveals only the irrep label, discarding critical information contained within high-dimensional irreducible representations needed to identify the hidden subgroup in nonabelian groups..

📚 Reference Text:

(sometimes called a hidden subgroup state ), whereRis theright regular representation ofG, satisfying R(g)|g′/an}bracketri}ht=|g′g−1/an}bracketri}ht (4) for allg,g′∈G. Now the hidden subgroup problem is reduced to the problem of perf orming a measurement to distinguish the states ρHfor the various possible H≤G. The symmetry of the hidden subgroup state can be exploited using F ourier analysis. In particular, the group algebra CGdecomposes under the left and right multiplication actions of G(which commute, and indeed generate each other’s commutants) a s CGG×G∼=/circleplusdisplay σ∈ˆGVσ⊗V∗ σ (5) whereˆGdenotes a complete set ofirreduciblerepresentations(or irreps) ofG, andVσandV∗ σare the (row and column, respectively) subspaces acted on by σ∈ˆG. The unitary transformation that relates the standard basis for CGand the basis for the spaces Vσ⊗ V∗ σis the Fourier transform, which can be carried out eﬃciently for most groups of in terest [9,15,28,39]. Since the hidden subgroup state ρHis invariant under the left multiplication action of G, the decomposition (5) shows that it is block diagonal in the Fourier basis, with blocks labeled by the irrepsσ∈ˆG. For eachσ, there is a dim Vσ×dimVσblock that appearsdim Vσtimes (or in other words, the state is maximally mixed in the row space). Thus, without lo ss of information (or in Kuperberg’s words, “sacriﬁcing no entropy”), we can measure t he irrep name σand discard the information about which σ-isotypic block occurred (sometimes referred to as “discarding the row label”). The process of measuring the irrep name σis referred to as weak Fourier sampling . If Gis abelian, then all its irreps are 1-dimensional. In this case, weak Fou rier sampling is a complete measurement, and all that remains is to use the statistics of the irrep name to infer the hidden subgroup. However, for the nonabelian hsp, some (indeed, typically most) irreps are more than 1-dimensional, so weak Fourier sampling is an incomplete measurement. Indeed, for most nonabelian groups, it turns out that weak Fourier sampling alone produces insuﬃcient information to identify the hidden subgroup. In particular, this is tr ue of the hsprelevant to graph isomorphism [24,30] as well as the dihedral hsp. To obtain further information about the hidden subgroup, one must perform a reﬁned measurement ins ide the (dim Vσ)-dimensional subspace that remains after weak Fourier sampling gives the result σ. The process of making such a reﬁned measurement is referred to as strong Fourier sampling . There are many possible ways to measure inside Vσ, especially when it is high-dimensional, and much work has been done on the possible measurements for particular nonabelian group s. 2 Of course, with either weak or strong Fourier sampling, a single hidde n subgroup state is not suﬃcient to determine H: one must repeat the sampling procedure to obtain statistics. However, if one simply repeats strong Fourier sampling a polynomial n umber of times, there are some groups—in particular, the symmetric group—in which hidden subgroups cannot be found, even if the measurement is allowed to be chosen adaptively [4 1]. Thus, to solve the hspin general, one must perform a

Question 1045 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) are pivotal in cryptography and computational group theory, especially for non-Abelian groups such as the dihedral group $\mathbb{D}_{2^n}$. Efficient solutions to the HSP over these groups can impact the security of lattice-based cryptosystems. Which feature distinguishes the polynomial-time quantum algorithm for the dihedral Hidden Subgroup Problem from previous subexponential-time approaches? 1) It utilizes classical random walks on the Cayley graph of $\mathbb{D}_{2^n}$ 2) It employs Grover's search algorithm to find hidden subgroups 3) It relies on solving the discrete logarithm problem in the codomain 4) It applies harmonic analysis on the subgroup lattice of $\mathbb{D}_{2^n}$ 5) It exploits structural information in the codomain and performs a directed walk down the subgroup lattice 6) It uses quantum Fourier transforms exclusively over Abelian subgroups 7) It requires exponential space due to entanglement across all group elements

✓ Correct Answer:

The correct answer is 5) It exploits structural information in the codomain and performs a directed walk down the subgroup lattice.

📚 Reference Text:

Title: A Quantum Polynomial-Time Solution to The Dihedral Hidden Subgroup Problem Year: 2022 Paper ID: c4a45e4508f239c6b2b58abec7a436c2fff9fea0 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c4a45e4508f239c6b2b58abec7a436c2fff9fea0 Abstract: We present a polynomial-time quantum algorithm for the Hidden Subgroup Problem over $\mathbb{D}_{2^n}$. The usual approach to the Hidden Subgroup Problem relies on harmonic analysis in the domain of the problem, and the best known algorithm using this approach has time complexity in $2^{\mathcal{O}(\sqrt{n})}$. By focusing on structure encoded in the codomain of the problem, we develop a polynomial-time algorithm which uses this structure to direct a"walk"down the subgroup lattice of $\mathbb{D}_{2^n}$ terminating at the hidden subgroup.

Question 1046 multiple-choice

In polynomial optimization, semidefinite programming (SDP) techniques often rely on the construction of moment or pseudomoment matrices that encode positivity and algebraic constraints. These matrices play a vital role in hierarchies that relax hard optimization problems into tractable convex formulations. Which condition ensures that a *-linear functional L sends all sums of squares polynomials of degree at most 2d to nonnegative real numbers in the context of pseudomoment matrices? 1) L vanishes on all monomials of degree greater than d 2) The pseudomoment matrix Md is diagonalizable 3) L is multiplicative on the ideal generated by polynomials 4) The Gram matrix Γh associated to h is positive definite 5) The matrix (I⊗L)(Mn d) is positive semidefinite 6) L is real-valued on all self-adjoint polynomials 7) Md has trace equal to one

✓ Correct Answer:

The correct answer is 5) The matrix (I⊗L)(Mn d) is positive semidefinite.

📚 Reference Text:

for I=Iswap, cf. Proposition 4.9 below. Consider the pseudomoment matrix pattern Mwith symbolic entries Mn d:=Vd(n)Vd(n)∗. Lemma 4.5. ForL:C⟨s⟩2d→C∗-linear ,L(SOS 2d)⊆R≥0iff(I⊗L)(Mn d)⪰0. Proof. Forp∈SOS 2d, assumep=/summationtext jp∗ jpj, and write each pj=Vd(n)∗⃗ pj, where⃗ pjis the (column) vector of coefficients of pj. Then L(p) =/summationdisplay jL(p∗ jpj) =/summationdisplay jL(⃗ p∗ jVd(n)Vd(n)∗⃗ pj) =/summationdisplay j⃗ p∗ j(I⊗L)(Md(n))⃗ pj, from which the conclusion follows. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 32 We call the entries of Md(L) := (I⊗L)(Mn d) the pseudomoments (of degree ≤2d) of L. We can now rewrite Eq. (4.7) as an SDP as follows:ν d(h) = supL(h) s.t.Md(L)⪰0 Md(L)1,1= 1 L(I∩C⟨s⟩2d) ={0}.(4.10) The objective function L(h) can be equivalently presented as L(h) =⟨Md(L),Γh⟩, (4.11) where Γhis a (not necessarily positive semidefinite) Gram matrix for h, i.e.,h= Vd(n)∗ΓhVd(n). We note that the right-hand side of Eq. (4.11) is independent of the Gram representation Γ hchosen. Remark 4.6. In general the supremum above, e.g. in Eq. (4.7) is taken all ∗-linear function- alsL:C⟨s⟩2d→C. However in the case where h∗=h∈R⟨s⟩andI∗=Iis generated by polynomials in R⟨s⟩, we can simplify this supremum and only consider ∗-linear functionals L: R⟨s⟩2d→R. To see why, assume we are in this case and note that for any function Lachiev- ing this supremum, we also have that the function L′defined byL′(p) =1 2/parenleftig L(p) +L(p)/parenrightig also achieves this supremum. Since L′:R⟨s⟩2d→Rthis simplification is justified. Example 4.7. Consider the case where I={0}. In this case the constraint L(I∩C⟨s⟩2d) = {0}is automatically satisfied. Taken= 3,d= 1. ThenV1(3) = (1,s12,s13,s23)∗and an arbitrary degree one polynomial p∈C⟨s⟩has the form p=p0+p1s12+p2s13+p3s23=V1(3)∗⃗ pwith⃗ p:= (p0,p1,p2,p3)Tand (4.12) p∗p=⃗ p∗M3 1⃗ pwith M3 1= 1s12s13s23 s12s2 12s12s13s12s23 s13s13s12s2 13s13s23 s23s23s12s23s13s2 23 (4.13) IfL:R⟨s⟩2→Ris∗-linear, then M1(L) = L(1)L(s12)L(s13)L(s23) L(s12)L(s2 12)L(s12s13)L(s12s23) L(s13)L(s13s12)L(s2 13)L(s13s23) L(s23)L(s23s12)L(s23s13)L(s2 23)  = L(1)L(s12)L(s13)L(s23) L(s12)L(s2 12)L(s12s13)L(s12s23) L(s13)L(s12s13)L(s2 13)L(s13s23) L(s23)L(s12s23)L(s13s23)L(s2 23)  is a self-adjoint matrix. Consider the Hamiltonian given by the polynomial h=s12+s13+s23. The classical first relaxation Eq. (4.10) when the ideal Iis{0}(noting that we can restrict to linear functionals L:R⟨s⟩2d→Rby Remark 4.6) becomes:ν1(h) = maxℓ12+ℓ13+ℓ23 Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 33 s.t. 1ℓ12ℓ13ℓ23 ℓ12ℓ12,12ℓ12,13ℓ12,23 ℓ13ℓ12,13ℓ13,13ℓ13,23 ℓ23ℓ12,23ℓ13,23ℓ23,23 ⪰0 =∞ whereℓ12,ℓ13,ℓ23,ℓ12,12,ℓ12,13,ℓ12,23,ℓ13,23∈Rare arbitrary, and should be thought of as giving the value of the linear functional Lapplied to the variables sijand monomials in them in the matrix M1(L) above. This gives an upper bound on the eigenvalues of the sum of three (arbitrary) self-adjoint matrices which is vacuous. However, we soon see other ideals where the first relaxation bound has substance, for example in Example 4.8. Example 4.8. We repeat Example 4.7 but with Ichanged. We now take I=ISn, which we define to be the ideal generated by the elements s2 ij−1, sijsjk−siksij, sijskl−sklsiji,j,k,l all distinct (4.14) which we can think of as enforcing the defining relations s2 ij= 1, sijsjk=siksij, sijskl=sklsiji,j,k,l all distinct (4.15) of the symmetric group Sn. We will see in a moment how enforcing the condition L(I∩C⟨s⟩2d) ={0}leads to linear constraints on the pseudomoment matrix pattern. We begin with the pseudomoment

Question 1047 multiple-choice

Factorisation homology is a method in higher category theory and topological quantum field theory for assigning algebraic data to manifolds in a way sensitive to both topology and symmetry. When applied to oriented surfaces equipped with principal bundles for a finite group, it interacts intricately with braided categories and representation theory. Which of the following best describes what factorisation homology yields when applied to a surface with boundary equipped with a principal bundle for a finite group, integrating a balanced braided category with group action? 1) The category of finite-dimensional simple objects in the braided category 2) The space of invariant tensors under the group action 3) The tensor product of quantum groups associated to each boundary component 4) The category of modules over a specific algebra in the braided category 5) The homotopy colimit of fixed points in the category under the group action 6) The direct sum of irreducible representations of the finite group 7) The set of flat connections modulo gauge equivalence

✓ Correct Answer:

The correct answer is 4) The category of modules over a specific algebra in the braided category.

📚 Reference Text:

Title: Finite symmetries of quantum character stacks Year: 2021 Paper ID: 0364828e26f9db05f068020d95334cfcfb1b4f5f Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0364828e26f9db05f068020d95334cfcfb1b4f5f Abstract: For a finite group $D$, we study categorical factorisation homology on oriented surfaces equipped with principal $D$-bundles, which `integrates' a (linear) balanced braided category $\mathcal{A}$ with $D$-action over those surfaces. For surfaces with at least one boundary component, we identify the value of factorisation homology with the category of modules over an explicit algebra in $\mathcal{A}$, extending the work of Ben-Zvi, Brochier and Jordan to surfaces with $D$-bundles. Furthermore, we show that the value of factorisation homology on annuli, boundary conditions, and point defects can be described in terms of equivariant representation theory. Our main example comes from an action of Dynkin diagram automorphisms on representation categories of quantum groups. We show that in this case factorisation homology gives rise to a quantisation of the moduli space of flat twisted bundles.

Question 1048 multiple-choice

In fault-tolerant quantum computing, optimizing resource usage in key subroutines like the quantum Fourier transform (QFT) is critical for scalability. Circuit design choices directly affect the number and depth of resource-intensive gates such as the T gate within the Clifford + T gate set. Which approach enables an approximate quantum Fourier transform (AQFT) circuit to halve its T-count without introducing extra non-Clifford gates or sacrificing approximation accuracy? 1) Employing inverse phase gradient transformations and efficient quantum adders 2) Using logarithmic-depth quantum adders instead of linear-depth adders 3) Replacing all T gates with Clifford gates through gate synthesis 4) Increasing the number of qubits to parallelize all QFT operations 5) Implementing quantum error correction at every circuit layer 6) Adding ancillary qubits and entangling gates to the circuit 7) Omitting phase rotations from the QFT circuit entirely

✓ Correct Answer:

The correct answer is 1) Employing inverse phase gradient transformations and efficient quantum adders.

📚 Reference Text:

Title: Reducing T-count and T-depth in approximate quantum Fourier transform circuits Year: 2025 Paper ID: 5c6cbc2e54a532d5552ed8f33b81c2490cd89210 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/5c6cbc2e54a532d5552ed8f33b81c2490cd89210 Abstract: The quantum Fourier transform (QFT) is a fundamental component in various quantum algorithms, including Shor’s factoring algorithm and the Harrow-Hassidim-Lloyd (HHL) algorithm for solving systems of linear equations. Efficient implementation of the QFT is essential for the practical realization of large-scale quantum algorithms, especially in fault-tolerant quantum computing. In fault-tolerant implementations, the Clifford + T gate library is the standard choice for building quantum circuits. As the most resource-intensive component within this framework, the T gate’s associated cost poses a significant challenge to the efficient implementation of the QFT and its dependent algorithms. While approximate QFT (AQFT) circuits reduce this cost, state-of-the-art implementations still require a T-count of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$8n{\text{log}}_{2}(n/\varepsilon )-O({\text{log}}^{2}(n/\varepsilon ))$$\end{document} and a T-depth of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n{\text{log}}_{2}(n/\varepsilon )+O(n)$$\end{document}. Although these results represent a notable achievement, the associated resource cost remains a primary bottleneck for practical, large-scale quantum algorithms, motivating further optimization. To address this bottleneck, this paper introduces two novel \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-qubit AQFT circuits with an approximation error of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\varepsilon )$$\end{document}. Our first design, AQFT Circuit 1, halves the T-count to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4n{\text{log}}_{2}(n/\varepsilon )-O({\text{log}}^{2}(n/\varepsilon ))$$\end{document} by constructing inverse phase gradient transformation (PGT) circuits without using additional non-Clifford gates and by implementing the inverse PGTs using quantum adders. Our second design, AQFT Circuit 2, reduces the T-depth to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2}n{\text{log}}_{2}(n/\varepsilon )+O(n)$$\end{document} through parallelization of the inverse PGTs that add only \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n)$$\end{document} additional T gates. For both AQFT circuits, the state-of-the-art linear-depth quantum adder is employed. We demonstrate that employing the linear-depth quantum adder provides advantages over the currently known logarithmic-depth quantum adder, not only in terms of T-count but also in T-depth optimization for the AQFT, particularly within the range \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3

Question 1049 multiple-choice

In the study of quantum algorithms for group-theoretic problems, the choice of basis and the type of Fourier analysis employed are crucial for extracting subgroup information, especially in nonabelian groups such as the affine group Ap. Understanding when certain techniques fail or succeed is vital for algorithm design in this domain. Which statement most accurately explains why the abelian Fourier transform fails to distinguish conjugate subgroups within the affine group Ap? 1) The abelian Fourier transform cannot handle groups with non-cyclic structure. 2) The direct product decomposition of Ap inherently merges all subgroup information. 3) Abelian character sums are always zero for nonabelian groups. 4) Random bases adapt poorly to conjugacy classes in any group. 5) The abelian Fourier transform only works for groups with prime order. 6) Conjugate subgroups in nonabelian groups are always identical regardless of representation. 7) Treating Ap as a direct product causes conjugate subgroups to become indistinguishable, erasing structural distinctions that are only revealed by nonabelian representation theory.

✓ Correct Answer:

The correct answer is 7) Treating Ap as a direct product causes conjugate subgroups to become indistinguishable, erasing structural distinctions that are only revealed by nonabelian representation theory..

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cuts across these blocks, mixing diﬀerent“frequencies” ρ kand canceling out the useful information. To be more precise, a random basis is not adapted to the subgroup structure of Ap; it does not “know” that ρdecomposes into a direct sum of the ρkwhen restricted to Nq. Second, it is worth noting that for the values of qfor which we have an algorithm for full (as opposed to information-theoretic) reconstruction, namely q=p/polylog( p), a random basis works as well, since the /lscript1distance δbecomes 1 /polylog( p). Based on the strong evidence from representation theory that some bases are much betterfor computation than others, we conjecture that, for some families of groups, adaptedbases allow full reconstruction while random bases do not; but this remains an openquestion. Third, while we focused above on distinguishing conjugate subgroups from each other, in fact our proof shows that if q

Question 1050 multiple-choice

Multipartite entangled states, such as Greenberger-Horne-Zeilinger (GHZ) states, are fundamental resources in quantum information science. Protocols for generating and analyzing these states often rely on linear optical elements, specific measurement outcomes, and post-selection techniques. In the context of linear optical quantum protocols for n-qubit GHZ state generation, which condition is critical for heralding successful entanglement creation when using beamsplitters and photon detection? 1) Simultaneous detection of two photons in the same output mode 2) Equal probability of photon detection in all possible modes 3) Complete distinguishability of all internal degrees of freedom 4) Absence of any photon detection events 5) Arbitrary measurement patterns, independent of interference 6) Detection patterns corresponding to (1,0) and (0,1) in specific modes after interference 7) Direct deterministic two-qubit gates without post-selection

✓ Correct Answer:

The correct answer is 6) Detection patterns corresponding to (1,0) and (0,1) in specific modes after interference.

📚 Reference Text:

“The Church of the Symmetric Subspace,” (2013), arXiv:1308.6595 [quant-ph]. [52] S. Bartolucci, P. Birchall, H. Bombin, H. Cable, C. Dawson, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pant, et al. , Nat. Comm. 14, 912 (2023). [53] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Reviews of Modern Physics 79, 135 (2007). [54] J. E. Humphreys, Introduction to Lie algebras and representation theory , V ol. 9 (Springer Science & Business Media, 2012). Appendix A: Resource state generation 1. Proof sketches for Sec. II D 1 We briefly sketch the proofs of the results in Sec. II D 1. Lemma 1 is immediate from the fact that success is heralded if and only if the measurement patterns (1,0),(0,1)are obtained. For Proposition 1, we begin with the terms c10|ϕ10⟩a† 0(ξ)|⃗0⟩+c01|ϕ01⟩a† 1(ξ′)|⃗0⟩and apply the beamsplitter to obtain ((c10|ϕ10⟩a† 0(ξ) +c01|ϕ01⟩a† 0(ξ′)) + ( c10|ϕ10⟩a† 1(ξ)−c01|ϕ01⟩a† 1(ξ′)))|⃗0⟩. (A1) The measurement pattern (1,0)projects onto the first term, while (0,1)projects onto the second. We consider the first case; the second will be similar. Expressing the resulting state in terms of the occupation number and internal states 20 simultaneously (referred to as the logical-internal basis in Ref. 20), we obtain (c10|ϕ10⟩|1,0⟩|ξ⟩+c01|ϕ01⟩|1,0⟩|ξ′⟩). (A2) Taking the corresponding mixed state, then taking the partial trace over the internal states in the two input modes (and dropping the now-redundant |1,0⟩factor), we obtain |c10|2⟨ξ|ξ⟩|ϕ10⟩⟨ϕ10|+|c01|2⟨ξ′|ξ′⟩|ϕ01⟩⟨ϕ01|+c10c01⟨ξ′|ξ⟩|ϕ10⟩⟨ϕ01|+c01c10⟨ξ|ξ′⟩|ϕ01⟩⟨ϕ10| =|c10|2|ϕ10⟩⟨ϕ10|+|c01|2|ϕ01⟩⟨ϕ01|+1 2Re(c10c01⟨ξ′|ξ⟩|ϕ10⟩⟨ϕ01|), giving (8) as desired. The rest of the proposition follows from considering the cases ⟨ξ′|ξ⟩= 1,0. Proposition 2 is a direct application of Proposition 1 to the Type I fusion circuit in Figure 4(b). As with Lemma 1, Lemma 2 is a direct consequence of the success conditions for the n-GHZ state analyzer, detailed in Figure 5. For Theorem 1, we simply note that the n-GHZ state analyzer is a concatenation of ngeneralized Xmeasurements. Iteratively applying Proposition 1 and carefully tracking the signs gives the result. 2. GHZ generation Here we will discuss the details of the n-GHZ state generation protocol. We begin with the ideal case, sketching the argument for the sake of exposition and later comparison. We then sketch how this argument changes for the two types of distinguishability errors studied in Theorem 4. a. Proof sketch for Lemma 3 We now briefly outline the proof of Lemma 3, showing how Protocol 2 works in the ideal case. Step 2 performs the transformation |1,1⟩ 7→1√ 2(|2,0⟩ − |0,2⟩)on each pair of modes (2i,2i+ 1). Then Step 3 distributes this state across the four modes (2i,2i+ 1,2n+ 2i,2n+ 2i+ 1) , yielding the state |β⟩=1 2(|1,0,1,0⟩ − |0,1,0,1⟩) +1 2√ 2(|2,0,0,0⟩ − |0,2,0,0⟩+|0,0,2,0⟩ − |0,0,0,2⟩), (A3) which we recognize as a linear combination of a dual-rail Bell pair1√ 2(|00⟩ − |11⟩)and “junk” terms that we hope to discard. After Step 3, then, we have prepared ncopies of |β⟩from (A3). We then feed the second half of each |β⟩into the n-GHZ state analyzer and post-select for success. Applying Lemma 2, we see that the

Question 1051 multiple-choice

In quantum gravity, particularly within the framework of AdS/CFT correspondence, the nature and role of symmetries are deeply intertwined with quantum error correction and the structure of holographic codes. The analysis of global and discrete symmetries in such codes reveals important constraints on their exactness and compatibility with error correction. Which statement most accurately reflects the restriction on exact global symmetries in AdS/CFT quantum error-correcting codes at leading order in Newton's constant $ G_N $? 1) Exact continuous global symmetries are always allowed if the code corrects for erasure. 2) Approximate discrete symmetries are forbidden even when corrections are small. 3) Exact discrete symmetries are ruled out by the Eastin-Knill theorem for all quantum codes. 4) Exact bulk global symmetries are prohibited as long as corrections to subregion duality remain small. 5) Boundary global symmetries necessarily induce nontrivial logical operators in correctable regions. 6) Higher-order corrections in $ G_N $ restore exact global symmetry actions in the bulk. 7) Subregion duality permits unrestricted global symmetry actions in the bulk.

✓ Correct Answer:

The correct answer is 4) Exact bulk global symmetries are prohibited as long as corrections to subregion duality remain small..

📚 Reference Text:

the identity acting on the code space. In this argument, we assume that subregion duality and complementary recovery are exact properties of the AdS/ CFT code and, thus, infer that erasure of boundary region Akisexactly correctable. In fact, though, these properties of the code hold precisely only in the leading order of asystematic expansion in Newton ’s gravitational constant G Nand can be modified when corrections higher order in GNare included. Nevertheless, the conclusion that bulk global symmetries are disallowed continues to hold evenwhen these higher-order corrections are taken into account, assuming the corrections are small. A nontrivial global symmetry operation (if one were allowed), acting on a bulklocal operator ϕ, should modify ϕby an amount δϕwhich isOð1Þ, independent of G N. But we argue that δϕ¼0to leading order in GN(since exact correctability of Akholds to this order). Higher-order corrections might make an OðGNÞcontribution to δϕ, but these small corrections do not suffice to restore the proper nontrivial action on ϕof the putative global symmetry. Now, we have found that exact bulk local symmetries cannot occur in AdS/CFT. But what can we say aboutwhether approximate discrete global symmetries are allowed? As we discuss, finite-dimensional quantum error-correcting codes canhave exact discrete symmetries, even though the AdS/CFT code does not. In this respect, discrete symmetries are essentially different than continu- ous symmetries, which are disallowed by the Eastin-Knilltheorem for any finite-dimensional quantum code that can correct erasure of subsystems exactly. Therefore, we cannot expect to make general statements which are directlyanalogous to Theorem 2 about limitations on approximate discrete symmetries that apply to general codes. Nevertheless, it may be instructive to study further the properties of approximate quantum error-correcting codes which are approximately covariant with respect to adiscrete symmetry. In the setting of AdS/CFT, it is ofparticular interest to consider the case where the localtransformations fW kgin Eq. (82) are either precisely or approximately logical. B. Bulk time evolution A natural symmetry arising in AdS/CFT is the time- translation invariance of the boundary CFT, which is governed by a local Hamiltonian. Time evolution in the bulk AdS space is a bit subtle because of the generalcovariance of the bulk theory, but, if we fix the gauge bychoosing a preferred sequence of bulk time slices, then time evolution in the bulk corresponds to time evolution on the boundary. From the perspective of quantum error correc-tion, this correspondence is puzzling, because covariance ofthe AdS/CFT code with respect to time evolution seems to be incompatible with perfect correctability of erasure on the boundary [12,25] . Indeed, the analysis of bulk global symmetries in Sec. IX A , which is applicable to both discrete and continuous symmetries, builds on the obser- vation that a boundary global symmetry operator, when restricted to a correctable boundary subregion, preservesthe code space and, therefore, must be a trivial logicaloperator. Why can we not apply similar reasoning to the action of the boundary Hamiltonian, concluding (incor- rectly) that bulk time evolution is trivial? The answer hinges on a crucial distinction, emphasized

Question 1052 multiple-choice

Quantum algorithms have shown promise in solving hidden subgroup problems (HSPs) for various group-theoretic structures, with significant implications for cryptography and computational complexity. The non-Abelian case presents notable challenges compared to the Abelian case, leading researchers to focus on certain tractable group classes. Which group-theoretic condition enables quantum algorithms to efficiently solve the non-Abelian hidden subgroup problem in polynomial time? 1) The group contains a large non-normal subgroup. 2) The group is simple and non-solvable. 3) The group has an infinite center. 4) The group is neither solvable nor has a cyclic factor group. 5) The group is nilpotent of class greater than two. 6) The group has a large commutator subgroup. 7) The group contains an elementary Abelian normal 2-subgroup of small index.

✓ Correct Answer:

The correct answer is 7) The group contains an elementary Abelian normal 2-subgroup of small index..

📚 Reference Text:

Title: Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem Year: 2001 Paper ID: aad78939cb9b78b9acd1af54970100d6bae823f6 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/aad78939cb9b78b9acd1af54970100d6bae823f6 Abstract: In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.

Question 1053 multiple-choice

In quantum information theory, distinguishing quantum states often involves analyzing the probability distributions that result from measurements in random orthonormal bases, with group representation theory playing a key role in structuring the underlying Hilbert space. The ℓ1 (total variation) distance is commonly used to quantify how distinguishable two such distributions are. When measuring quantum states associated with subspaces of a Hilbert space structured by irreducible unitary representations of a group G, which of the following expressions gives a lower bound on the ℓ1 distance between the resulting probability distributions in terms of representation-theoretic quantities, for the case where the states have ranks r1 and r2 and restricted ranks r'1 and r'2? 1) Ω((r'_1 + r'_2) / (r_1 + r_2)) 2) Ω(log(r_1/r'_1) + log(r_2/r'_2)) 3) Ω(r_1^2 + r_2^2) 4) Ω(1/(r_1 r_2)) 5) Ω(r'_1/r_1 + r'_2/r_2) 6) Ω(√(r'_1/r_1) + √(r'_2/r_2)) 7) Ω(1/(r'_1 + r'_2))

✓ Correct Answer:

The correct answer is 6) Ω(√(r'_1/r_1) + √(r'_2/r_2)).

📚 Reference Text:

ors of all vectors inW′are/radicalbig (1±ǫ)d, whereǫ=O(δ).E0occurs with probability at least 1−2exp(−α))over the 17 choice of the Gaussians. Let E1denote the event that |/an}bracketle{tw′i|w′j/an}bracketri}ht|<2δfor allw′i,w′j∈W′,i/ne}ationslash=j.E1 occurs withprobability atleast 1−2exp(−α)over thechoice oftheGaussians. Callacoordinate jgoodif/summationtextp i=1|α′i j|2>p+√p. LetE2denote the event that more thandθ 2coordinatesjare good.E2occurs with probability at least 1−exp(dθ/8)over the choice of theGaussians. Now suppose that all three events E0,E1,E2occur. LetS′be the analogous probability vector deﬁned withrespect to W′instead ofW. Then,/bardblS−S′/bardbl1=O(p−1/2). For agoodcoordinatej /vextendsingle/vextendsingle/vextendsingle/vextendsingleS′ j−1 d/vextendsingle/vextendsingle/vextendsingle/vextendsingle=1 (1+ǫ)d−1 d+1√p(1+ǫ)d =−ǫ (1+ǫ)d+1√p(1+ǫ)d =−O/parenleftbigg−1 dp/parenrightbigg +Ω/parenleftbigg1 d√p/parenrightbigg = Ω/parenleftbigg1 d√p/parenrightbigg . Thethird step above follows from the fact that ǫ=O(δ) =O(p−1). Hence, /bardblS′−U/bardbl1≥/summationdisplay j:jgood|S′ j−T′ j|>dθ 2·Ω/parenleftbigg1 d√p/parenrightbigg = Ω/parenleftbigg1√p/parenrightbigg . Finally, /bardblS−U/bardbl1≥ /bardblS′−U/bardbl1−/bardblS−S′/bardbl1= Ω(p−1/2)−O(p−1) = Ω(p−1/2). Theconﬁdence bound is Pr[E0∧E1∧E2]>1−4exp(−α)−exp(dθ/8) = 1−exp(−Ω(α)), sinceδ=O(1). Thiscompletes the proof of the lemma. Wearenow in aposition toﬁnally prove Theorem 1. Proof:(of Theorem 1) Letρbe an irreducible dρ-dimensional unitary representation of G. We follow the notation of Deﬁnition 1 for ρ. LetV′′ 2denote the orthogonal complement of V′ 1inV′ 1+V′ 2. Letσi denote the totally mixed state in Viandσ′′ 2denote the totally mixed state in V′′ 2+ (V1∩V2). By Proposi- tion 2,/bardblσ2−σ′′ 2/bardbltr<2δ1/2(1−δ2)−1/4. LetBρbe a random orthonormal basis for ρ. LetPi:=PG,Bρ Hi denote the probability distributions on the vectors of Bρgot by Fourier sampling the states |Hi/an}bracketri}htrespec- tively, conditioned on ρbeing observed. Then Piis the probability distribution got by measuring σiin the basis Bρ. LetP′′ 2denote the probability distribution got by measuring σ′′ 2in the basis Bρ. Then, /bardblP2−P′′ 2/bardbl1<2δ1/2(1−δ2)−1/4. Deﬁner1,2= rank(Π 1,2). Notethatri=r′ i+r1,2. Suppose case 1 of Deﬁnition 1 applies. Let Wbe a random orthonormal set of r′ 1+r′ 2+r1,2vectors inCdρ. Deﬁne probability distributions S,Twith respect to Was in Lemma2. Bysymmetry, P1=Sand P′′ 2=T. Note thatr′ 1+r′ 2+r1,2≤r1+r2andr′ 1+r′ 2+r1,2≤dρ

Question 1054 multiple-choice

Quantum algorithms such as Shor's leverage the unique properties of periodic states and the Quantum Fourier Transform (QFT) to achieve computational speedup. Understanding the relationship between entanglement measures and the QFT is crucial for analyzing these algorithms' efficiency. Which statement best describes the effect of the Quantum Fourier Transform on the Groverian entanglement measure of periodic states during the preprocessing stage of Shor’s algorithm? 1) The QFT completely eliminates the Groverian entanglement of periodic states. 2) The QFT transforms periodic states into product states with zero entanglement. 3) The QFT causes a dramatic increase in the Groverian entanglement of periodic states. 4) The QFT reverses the periodic structure, leading to maximal entanglement. 5) The QFT converts periodic states into maximally mixed states. 6) The QFT only slightly alters the Groverian entanglement of periodic states. 7) The QFT randomizes entanglement, making it unpredictable in periodic states.

✓ Correct Answer:

The correct answer is 6) The QFT only slightly alters the Groverian entanglement of periodic states..

📚 Reference Text:

Title: Entanglement of Periodic States, the Quantum Fourier Transform and Shor's Factoring Algorithm Year: 2010 Paper ID: 1bbadae55d07978f1dd61e04170651f8b103cd98 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/1bbadae55d07978f1dd61e04170651f8b103cd98 Abstract: The preprocessing stage of Shor's algorithm generates a class of quantum states referred to as periodic states, on which the quantum Fourier transform is applied. Such states also play an important role in other quantum algorithms that rely on the quantum Fourier transform. Since entanglement is believed to be a necessary resource for quantum computational speedup, we analyze the entanglement of periodic states and the way it is affected by the quantum Fourier transform. To this end, we derive a formula that evaluates the Groverian entanglement measure for periodic states. Using this formula, we explain the surprising result that the Groverian entanglement of the periodic states built up during the preprocessing stage is only slightly affected by the quantum Fourier transform.

Question 1055 multiple-choice

In quantum graph theory and computational physics, analyzing the spectral properties of graph Hamiltonians often involves advanced combinatorial and algebraic techniques, such as tree clique decompositions and representation theory. These methods are particularly useful for simplifying the calculation of eigenvalues in complex quantum systems. Which statement accurately describes the role of Young’s Branching Rule in the recursive computation of a graph Hamiltonian's eigenvalues using a tree clique decomposition? 1) It assigns weights to each vertex based solely on its degree in the graph. 2) It determines the adjacency matrix for each clique in the tree decomposition. 3) It identifies which cliques contribute nontrivial eigenvalues by checking for leaf nodes. 4) It specifies the sign alternation in the sum of clique Hamiltonians according to tree depth. 5) It calculates the dimension of each subgraph using the number of its edges. 6) It selects which singleton subgraphs should be excluded from the eigenvalue sum. 7) It guides how symmetry representations decompose at each tree depth, enabling recursive calculation of eigenvalues through representation labels and parameters η[m,k].

✓ Correct Answer:

The correct answer is 7) It guides how symmetry representations decompose at each tree depth, enabling recursive calculation of eigenvalues through representation labels and parameters η[m,k]..

📚 Reference Text:

the tree clique decomposition of G. Details of this use of ncSoS is given in Section 5.5. 6.4 Example of Clique Decomposition, its Hamiltonians and their Eigenvalues In this subsection we apply the algorithms discussed previously in this section to analyze the graphGshown in Figure 4. 1 24 63 5 Figure 4: The graph Gto be analyzed in this example. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 59 6.4.1 The tree clique decomposition of Gand the Hamiltonian decomposition v1 v2 v9 v3v8 v4v7 v5v6 Figure 5: The output tree, T 1 24 63 5 G(v1)1 24 3 5 G(v2)1 24 3 G(v3)1 23 G(v4) 1 2 G(v5)3 G(v6)4 G(v7)5 G(v8)6 G(v9) Figure 6: The output graphs. The second row of graphs corresponds to the leaves LofT, the first row toR. NoteG(v1)cis the union of its children, the two depth 1 graphs G(v9)andG(v2). Likewise G(v2)cis the union of two depth 2 graphs and G(v3)cis the union of two depth 3 graphs and G(v4)cis the union of two depth 4 graphs. This can be read off from the tree. To write down the Hamiltonian clique decomposition for G, first see that the leaves of treeTareL:={v9,v5,v6,v7,v8}and the rest are R:={v1,v2,v3,v4}. The corresponding depthsd(vj) are read off from Tand listed in Table 3 G(v1)G(v2)G(v3)G(v4)G(v5) Depth inT 0 1 2 3 4 SignHam + - + - DimK(G(v)) 6 5 4 3 2 Table 3: Clique decomposition of graph G, ignoring single vertex graphs and force the signs of the decomposing Hamiltonians in the table. By Theorem 6.5 Item (2) Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 60 the Hamiltonian HGhas the decomposition: HG= +HK(G(v1))−HK(G(v2))+HK(G(v3))−HK(G(v4)) (6.37) +HG(v5)+HG(v6)−HG(v7)+HG(v8)−HG(v9) (6.38) Next use that G(vj) forj= 6,7,8,9 are singletons, to see the corresponding Hamil- tonians are 0; also K(G(v5)) =G(v5). Thus HG= +HK(G(v1))−HK(G(v2))+HK(G(v3))−HK(G(v4))+HK(G(v5)), (6.39) a sum/difference of clique Hamiltonians. We mention that for this special example, one sees from Figure 4 the containments V(G(vj))⊇V(G(vj+1) forj= 1,2,3,4, so K(G(v1))⊇K(G(v2))⊇K(G(v3))⊇K(G(v4))⊇K(G(v5)) =K(G(v5)). (6.40) 6.4.2 Eigenvalues Now we see how one can use Algorithm 8 to compute max/min eigenvalues of HG; we re- strict to one case eigmax(H[3,3] G), since the min eigenvalues and other irreps behave similarly. Firstly, we list the data needed for eigenvalues (and also above). K6K5K4K3K2 Sign + , max−, min + , max−, min + , max η[3,3]= 24η[3,2]= 16η[2,2]= 12η[2,1]= 6η[2,0]= 0 x xη[3,1]= 8η[2,1]= 6η[1,1]= 4 x x xη[3,0]= 0η[2,0]= 0 Table 4:GandIrrep [3,3]. Data for computing eigmax(H[3,3] G). The number next to [m,k]isηm,k. Entries to the right are needed in light of those to the left because of Young’s Branching Rule. We shall apply Algorithm 7 repeatedly to compute eigmax(H[3,3] G). eigmax(H[3,3] G) =η[3,3]−/parenleftigg min ξ∈Rstrct(5,[3,3])/parenleftig eigmin(Hξ G(v2)/parenrightig/parenrightigg . (6.41) =η[3,3]−eigmin(H[3,2] G(v2))and (6.42) eigmin(H[3,2] G(v2)) =η[3,2]− max ξ∈Rstrct(4,[3,2])/parenleftig eigmax(Hξ G(v3)/parenrightig (6.43) =η[3,2]−max/parenleftig eigmax(H[3,1] G(v3),eigmax(H[2,2] G(v3))/parenrightig (6.44) Combine these to get eigmax(H[3,3] G) =η[3,3]−η[3,2]+ max/parenleftig eigmax(H[3,1] G(v3),eigmax(H[2,2] G(v3))/parenrightig (6.45) Next eigmax(H[3,1] G(v3)) =η3,1−min/parenleftig eigmin(H[3,0] G(v4),eigmin(H[2,1] G(v4))/parenrightig (6.46) eigmax(H[2,2] G(v3)) =η2,2−eigmin(H[2,1] G(v4)) (6.47) Before combining again we compute

Question 1056 multiple-choice

In lattice gauge theory, the structure of entanglement and the ability to distinguish quantum states often depend on the choice of observables and the properties of the underlying symmetry group. The interplay between gauge invariance, unitary decompositions, and topological effects is critical in understanding ground state degeneracy and quantum computation. Which statement best characterizes the limitation of Wilson loop operators in lattice gauge theories with finite non-Abelian groups? 1) Wilson loops always uniquely determine gauge-invariant states for any group. 2) Wilson loops fail to detect nontrivial holonomy in Abelian groups. 3) The expectation values of Wilson loops are identical for all possible excited states. 4) Wilson loops cannot be defined in the presence of boundaries. 5) For certain finite non-Abelian groups, Wilson loops do not form a complete set of observables, allowing orthogonal gauge-invariant states with identical Wilson loop values. 6) Wilson loops commute with all magnetic plaquette operators in any lattice gauge theory. 7) The completeness of Wilson loops is guaranteed for all topologically ordered phases.

✓ Correct Answer:

The correct answer is 5) For certain finite non-Abelian groups, Wilson loops do not form a complete set of observables, allowing orthogonal gauge-invariant states with identical Wilson loop values..

📚 Reference Text:

U∂Aas a product of a unitary acting on Aand a unitary acting on ¯A, i.e.U∂A=VAV¯A. From the deﬁnition of |ξg∂A/angbracketrightA, we can see VA|ξg∂A/angbracketrightA=|ξg/prime ∂A/angbracketrightA. (38) Accepted in Quantum2020-09-17, click title to verify. Published under CC-BY 4.0. 11 Then from the invariance of |ψ/angbracketrightunderU∂A, and using decomposition (36), we have |φg/prime ∂A/angbracketright¯A=A/angbracketleftξg/prime ∂A|ψ/angbracketright =A/angbracketleftξg/prime ∂A|U∂A|ψ/angbracketright =V¯A(A/angbracketleftξg/prime ∂A|VA|ψ/angbracketright) =/angbracketleftV¯Aξg∂A||ψ/angbracketright =V¯A|φg∂A/angbracketright¯A.(39) Because|φg∂A/angbracketright¯Aand|φg/prime ∂A/angbracketright¯Aare related by a unitary V¯A, it follows they must have the same norm. We conclude that decomposition of (36)is a Schmidt decomposition, with uniform Schmidt coeﬃcients that have been absorbed into the non-normalized states |φg∂A/angbracketright¯A. Thus we can imme- diately calculate the reduced density matrix ρA= Tr ¯A|ψ/angbracketright/angbracketleftψ| =1 |G||∂A|−1/summationdisplay g∂Awith trivial holonomy on ∂A|ξg∂A/angbracketrightA/angbracketleftξg∂A|A (40) where|∂A|is the number of boundary edges, and the normalization factor is calculated by noting there are|G||∂A|−1terms in the sum. (The latter is the number of assignments g∂Aof group elements to boundary edges such that the boundary has trivial holonomy: the group elements on all but one boundary edge may be chosen freely, and the element on the ﬁnal edge is uniquely determined by requiring trivial holonomy.) This reduced state on Ais manifestly invariant of the state |ψ/angbracketright, depending only on our original assumptions that |ψ/angbracketrighthas zero energy-density on A. 3.3 Wilson loops are not a complete set of observables It is standard in gauge theory to think of Wilson loop operators as the basic gauge-invariant observables. In typical models, such as gauge theory based on special unitary groups, or in Kitaev’s lattice model based on the group Z2, these observables are suﬃcient to completely characterize a gauge-invariant state; see Sengupta [23] for a discussion. It is therefore tempting to think that this holds quite generally, e.g. for lattice gauge theory or Kitaev’s lattice model based on any ﬁnite groupG. Such a result would seem to suggest that Theorem 3.1 is “morally obvious”: if gauge invariant states are determined by their Wilson loops, our main result would simply be an easy corollary of a local version of this statement. In fact, we will show that this naive intuition fails for certain choices of G. That is, for judicicously chosen G, we will exhibit a pair of orthogonal gauge-invariant states with the same Wilson loops. This result emphasizes that it is a property only of the ground space—where the Wilson loops are not only locally the same, but also locally trivial—that states are determined by their non-Abelian ﬂuxes. The question of completeness for Wilson loop observables was addressed by Sengupta [23] for continuous gauge groups. To study quantum double models, we will be interested in the case of ﬁnite gauge groups and spatial lattices, which Sengupta comments on only brieﬂy. Let us state our claim more precisely. We will work in the gauge invariant subspace Hgauge ={|ψ/angbracketright∈H|Av|ψ/angbracketright=|ψ/angbracketright,∀v∈V}. (41) The magnetic plaquete operators B(v,p)(h)do not in general preserve the gauge invariant subspace, so we will consider combinations which do: B(v,p)([h])≡|[h]| |G|/summationdisplay g∈GB(v,p)(ghg−1) (42) where|[h]|is the order of the conjugacy class of h. These operators depend on

Question 1057 multiple-choice

In group theory, the study of nilpotency and automorphisms of abelian p-groups often involves analyzing direct products of cyclic groups and their extensions. The structure and rank conditions of these groups have significant implications for the nilpotency class of associated split extensions. For a sparse direct product of cyclic p-groups A, if b is an automorphism of order p and G = A ⋊ ⟨b⟩ is the split extension, what is the nilpotency class of G? 1) r_1 + r_{n+1} 2) p - 1 3) r_n + 1 4) 2p 5) r_1 6) r_1 + 2 = p 7) p + 1

✓ Correct Answer:

The correct answer is 6) r_1 + 2 = p.

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group of orderp n + ] by one of order/?. Then as is easy to see, for t ^ p the factor group Wn/Zt(Wn) of W n by the /-th term of its upper central series is an extension of CP n + 1 X C pn https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press 1034 J. BUCKLEY AND J. WIEGOLD by C and, by [6], it has class (n + \)p — n — t, which is at least n(p — 1). Thus we can state: Example 4.1. Suppose that A is a restricted direct product of cyclic /^-groups and that the ranks r n of the homocyclic components A n of the exponent p" satisfy the inequality rn + rn + \ =P ~ l for infinitely many n. Then there is an automorphism b of A of order /? such that the split extension (A, b) is not nilpotent. For the conditions give that A has a direct factor X expressible as a direct product X x X X 2 . . ., where X, is of the form for suitable /, ^ /?, and n i —» oo as / —» oo. Thus each A^- has an automorphism 6 / of order /? such that the split extension (X h bt) is isomorphic with W„/Z ti(W„) and so has class going to infinity with /. The b i extend to an automorphism of order p of A in the obvious way, and the extension (A, b) is not nilpotent since it contains subgroups of unbounded class. Let A be any direct product of cyclic /^-groups, A n the homocyclic component of exponent p f\ and r n the rank of A n. We introduced the adjective "thin" in Section 3 in dealing with the case of divisible B\ in the case of cyclic B the example just made indicates that we need a more stringent condition. Definition 4.1. A is said to be sparse if rn + rn+\ = P ~ 2 f° r aU n = i; eventually sparse if there exists n 0 such that rn + rn + x ^ p — 2 for all w ^ H 0- As an aside, we could observe that a sparse 2-group is trivial and that an eventually sparse 2-group is of finite exponent, so that 2-groups fit into the general picture correctly. However we have kept the cases separate, since it was so easy to deal with 2-groups by themselves. The rest of this section is devoted to the proof of the following result: THEOREM 4.2. Let A be a sparse direct product of cyclic p-groups and let r n be the rank of the homocyclic component of exponent p". For every automorphism b of A of order p 7 the split extension G = A XI (b) is nilpotent of class c = r x + 2 = p. Proof. The main thrust will be to prove that c ^ p and then a few

Question 1058 multiple-choice

In quantum optics, managing photon distinguishability is essential for accurate quantum interference and robust quantum information processing. Various models and mathematical frameworks help simulate, analyze, and mitigate the effects of internal photon state differences on experimental outcomes. Which mathematical approach allows simulation and error analysis to treat imperfections from mutually orthogonal photon states as independent, non-interfering sectors within a quantum optical system? 1) Employing entanglement purification across all photon modes 2) Using tensor product decomposition of the overall system 3) Modeling photon errors solely via classical probability distributions 4) Applying non-orthogonal state mixing in the Hilbert space 5) Averaging measurement outcomes over all photon rails 6) Restricting simulation to the ideal photon source only 7) Decomposing the Hilbert space into a direct sum of orthogonal subspaces associated with different photon types

✓ Correct Answer:

The correct answer is 7) Decomposing the Hilbert space into a direct sum of orthogonal subspaces associated with different photon types.

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is motivated by the fact that in linear optics, orthogonal internal states will never interfere with each other and so the relative magnitude of each ‘sector’ remains unchanged. This can also be straightforwardly generalized to handling different numbers of ‘error mode’, e.g.P iϵi|ψi⟩⟨ψi|, where ⟨ψi|ψj⟩=δij. Two common models of multi-photon distinguishability are known as the Orthogonal Bad Bits (OBB) model and the Random Source model [20]. We discuss various error models in more detail in Sec. II D 3 below: namely, we consider the OBB model and two others that we call the Same Bad Bits (SBB) and Orthogonal Bad Pairs (OBP) models. Under these same assumptions, we can decompose the Hilbert space as a direct sum of non-interacting pieces, H= ⊕k i=0Hi, where there are kdifferent error types. In particular, Hiis the Hilbert space for photons of ‘type’ i(and we leti= 0 index the ‘ideal’ photon subspace). Operations in this framework are of the form M=⊕imi, which for state agnostic operations [21] reduces to mi=m,∀i. In this case, a simulation with kerrors can be visualized by copying the circuit of interest ktimes, i.e., each physical rail can host k+ 1internal modes and represented in the basis {|n0, . . . , n k⟩}. A simple example with a single beamsplitter and one error mode is shown in Fig. 2. To handle PNR measurements, we perform a POVM over allmodes, i.e., {|n0, . . . , n k⟩⟨n0, . . . n k|}, for a single PNRD. However, we must in general do further post-processing, since the internal mode index sampled is typically not visible to the experimenter6(e.g., in a single photon experiment, we would not learn which i= 0, . . . , k was detected). Indeed, we’ll assume only the total number of photons N=Pk i=0niis accessible. Post-selection on a particular pattern can therefore have many associated ‘microscopic’ configurations that must be traced out, in general leading to mixedness. In terms of simulation, we can treat each ‘error subspace’ entirely separately, and simulate independently the photons in each class. For example, if there is one error injected along some optical fibre, we will simulate separately the n−1‘ideal’ photons, and then the 1 photon error state. At the measurement step we must perform post-processing to determine which configurations on each state can yield a desired measurement outcome, as discussed above. So far we have mostly been considering the case where distinguishability between photons is introduced at the source(s). In LOQC however, it is typical that certain photons may propagate several ‘clock cycles’, whereas others are introduced later. Via dispersion, this can introduce distinguishability, even if the different photon sources are perfect [17]. As such it is important to also consider introducing photon distinguishability during the execution of a simulation, for example transferring photons from one internal mode to another. C. Distinguishability distillation It is highly desirable to have a heralded source of highly pure and indistinguishable photons. There are two main methods to mitigate distinguishabiilty effects; engineering the source,

Question 1059 multiple-choice

In particle physics, the decomposition of tensor products of group representations is crucial for building models that describe how particles interact and acquire mass. Computational algebra systems are often used to perform these complex group-theoretical calculations, especially in constructing viable flavor models. Which condition guarantees the existence of a solution to an equation involving symmetry representations when analyzing the direct product AL⊗Al in model-building? 1) The decomposition yields only one-dimensional representations. 2) The direct product contains the trivial representation. 3) The product AL⊗Al includes the adjoint representation. 4) The decomposition results in a completely reducible set. 5) The direct product AL⊗Al contains a two-dimensional representation AΦ among the irreducible components. 6) The direct product AL⊗Al is symmetric under all group operations. 7) The direct product AL⊗Al yields only conjugate representations.

✓ Correct Answer:

The correct answer is 5) The direct product AL⊗Al contains a two-dimensional representation AΦ among the irreducible components..

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the second part of Eq.(4) it is easy to notice that we get the equivalence if we assume D= (AΦ)∗. This theorem indicates that in order to ﬁnd out if any solution for Eq.( 4) exists, all we need to do, is the investigation of the Clebsch-Gordan decompositio ns: AL⊗Al=⊕DD, (19) AL⊗Aν=⊕DD. (20) Therefore, if one ﬁnds the representations ALandAl, direct product of which gives AΦ (one of the 2-dimensional representations among D) in the decomposition, then we have the guarantee that some solution to Eq.(4) exists .The similar situatio n takes place, when it comes to AL⊗Aν. The only exception lies in the fact, that we need to look for ( AΦ)∗in the decomposition. All these theorems can be easily veriﬁed by GAP. V. CONCLUSIONS To sum up, it is necessary to replace TBMmixing, which according to the experiments carried out in 2012, is no longer valid. Basing on the literature, we hav e developed the tools which are indispensable to search for the answer in the models with mo re Higgs doublets (thesemethodscanbetriviallygeneralizedintothecaseofmoreHigg sdoublets). Ourresults 10 will be presented in the forthcoming paper. In the meantime, the pr eliminary outcomes can be found in [12]. This work has been supported by the Polish Ministry of Science and Hig her Education under grant No. UMO-2013/09/B/ST2/03382. [1] S.F.King, A.Merle, S. Morisi, Y. Shimizu, M.Tanimoto, N ew Journal of Physics, Volume 16, April (2014) [2] P. F. Harrison, D. H. Perkins and W. G. Scott, Phys. Lett. B 530, 167 (2002) [3] F.G. Canales et al., Journal of Physics Conference Serie s 07/2013 [4] B. Dziewit, S.Zajac, M.Zralek, Acta Physica Polonica B, Vol.44 (2013) [5] P.O.Ludl, [arXiv:0907.5587v2 [hep-ph] ] [6] C.Giunti, C.W.Kim, Fundamentals of neutrino physics and astrophysics (Oxford University Press, 2007) [7] http://www.gap-system.org/ [8] H.U.Besche, B.Eick, E.O’Brien, Small Group Library [9] V.Dabbaghian, The REPSN package [10] W.Grimus, O.Ludl, J.Phys. A45 (2012) [11] K. M. Parattu, A. Wignerter, Phys.Rev.D84:013011,(20 11) [12] B.Dziewit et al., Acta Physica Polonica B, Vol(46),(20 15)

Question 1060 multiple-choice

Quantum algorithms have significantly advanced computational group theory, particularly in the study and manipulation of nilpotent black-box groups. These techniques enable efficient solutions to problems such as subgroup generation, element decomposition, and the analysis of group structure. Which of the following statements accurately describes a core feature of quantum algorithms applied to nilpotent black-box groups of known order? 1) They guarantee classical polynomial-time solutions for all hidden subgroup problems in non-nilpotent groups. 2) They enable the computation of all automorphism groups associated with any finite group. 3) They always construct a chief series without using commutators or knowledge of subgroup structure. 4) They can efficiently determine the conjugacy classes of any solvable group. 5) They allow polynomial-time quantum membership tests that reduce the size of generating sets and efficiently compute the lower central series. 6) They exclusively produce generating sets for abelian groups without reference to subgroup intersections. 7) They eliminate the need for oracle-based group operations when decomposing group elements.

✓ Correct Answer:

The correct answer is 5) They allow polynomial-time quantum membership tests that reduce the size of generating sets and efficiently compute the lower central series..

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also be given by sets of generators. One can use the exact polynomial time quantum membership test of [ 16] to reduce the size of generating sets to at most log |G|. During the rest of this part, we assume that Gis a nilpotent black-box group of class cand the prime factors of |G|are known. For a normal subgroup NofG, the subgroup [G,N]is a normal subgroup of G contained in N.I f/Gamma1and/Delta1are sets of generators for GandN, respectively, a generating set for [G,N]can be obtained by taking the commutators [x,y]forx∈/Gamma1,y∈/Delta1and then adding iterated commutators with elements of /Gamma1until the subgroup generated by the elements stabilizes. For testing stabilization, one can use the exact quantum subgroup membership algorithm of [ 16]. This gives a polynomial time exact method in particular to compute the lower central series. Below we describe efﬁcient solutions to some further group theoretic tasks that we use in our hidden subgroup algorithm. The p-Sylow subgroup of Gcan be computed as follows. Let /Gamma1be a generating set for G. Then for each g∈/Gamma1we compute the order ogofgand decompose ogas the product pαog/primewhere og/primeis coprime with p. Then thegog/prime(g∈/Gamma1) generate the (unique) p-Sylow subgroup of G. We shall compute hidden subgroups in Gby computing the intersections with the Sylow subgroups. As an example, let us consider the Heisenberg group modulo 12. It consists of the 123 Zero sum subsequences and hidden subgroups Page 7 of 23 14 matrices of the form⎛ ⎝1ab 1c 1⎞ ⎠, where a,b,care integers taken modulo 12. (Here and throughout the paper entries of a matrix left empty stand for entries having value zero.) The 2-Sylow subgroup consists of the matrices whose off-diagonal entries are divisibleby 3, while the matrices having off-diagonal entries divisible by 4 form the 3-Sylow subgroup. If the hidden subgroup Hconsists of matrices of the form⎛ ⎝1d3e 1 1⎞ ⎠, then its 2-Sylow subgroup is the set of matrices of the form⎛ ⎝13g3h 1 1⎞ ⎠, while the 3-Sylow consists of matrices of the form⎛ ⎝14k 1 1⎞ ⎠. The normalizer of a subgroup of Gcan be computed using the deterministic poly- nomial method of Kantor and Luks [ 28]. It was originally described for nilpotent permutation groups, but it also ﬁnds normalizers in any nilpotent black-box group oforder having small prime factors only. Assume that Lis a subgroup of G. It will be useful to decompose elements xof Gas products of the form α L(x)βL(x)where βL(x)∈LandαL(x)depend only on the coset xL. (Thus the range of αLis a transversal of LinG.) To this end, compute a chief series (a series of normal subgroups with cyclic factors of prime order) G=K0>K1>...> Kr=1. Perhaps the easiest way to obtain such series is taking a reﬁnement of the lower central series. By taking the subgroups KiL, and removing repeated elements, we obtain a subnormal series G=M0>M1>...> Ms=Lwith cyclic factors of prime order. Also take elements ai∈Mi−1\Miand denote by pithe order of Mi−1/Mi(i=1,..., s). This generalizes the frequently used technique of linear algebra when, given a

Question 1061 multiple-choice

Quantum algorithms have significantly advanced the computational efficiency of group-theoretic problems, especially within black-box groups where operations are accessed via oracles. The structure of normal and commutator subgroups plays a pivotal role in determining which problems admit polynomial-time quantum solutions. In which scenario is the hidden subgroup problem (HSP) in a black-box group G guaranteed to be solvable by a polynomial-time quantum algorithm, based on structural properties of G and its subgroups? 1) When G is non-Abelian and has a commutator subgroup of exponential size 2) When the normal subgroup N of G is neither solvable nor of polynomial size 3) When G's elements are encoded non-uniquely and group operations are not accessible via oracle 4) When the normal subgroup N of G is not normal 5) When G has unique encoding, and the commutator subgroup G′ is small or N is solvable or of polynomial size 6) When G has a non-trivial center but an unsolvable normal subgroup of exponential size 7) When the quantum Fourier transform must be computed exactly for all group elements

✓ Correct Answer:

The correct answer is 5) When G has unique encoding, and the commutator subgroup G′ is small or N is solvable or of polynomial size.

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for distinct g∈A/H, therefore an observation of the ﬁrst register will give a uniform probability distribut ion onH⊥. After suﬃcient number of iterations, this will give a set of generators for H⊥, which leads then to a set of generators for H. Note that in the above steps it is suﬃcient to compute only the approximate quantum Fourier transform on Awhich can be done in polynomial time. Theorem 10. Assume that Gis a black-box group with a unique encoding of group elements . Suppose that Nis a normal subgroup given by generators. Assume further tha tNis either solvable or of polynomial size. Then all the tasks listed in Corollary 5 f orG/Ncan be solved by a quantum algorithm in running time polynomial in the input size +ν(G/N). Proof.For applyingCorollary 5, onehastoverify that wecan perfor mtasks(a)–(b)ofthecorollary. IfNis of polynomial size, it is trivial. Therefore we suppose th atNis solvable. We will closely follow the approach indicated by Watrous in [27] for dealing with factor groups. First, let g∈G. To compute the order of ginG/N, we compute the period of the quantum function f(k) =|gkN/a\}bracketri}ht, wherek∈ {1,...,m}for some multiple mof the order. This function can be computed eﬃciently since one can prepare the superpositi on|N/a\}bracketri}htby Theorem 2, and for example we can take mas the order of ginG. Therefore by Lemma 9 one can ﬁnd this period. Second, let g∈Gandleth1,...,h r∈Gbepairwisecommuting elements modulo N. generating some Abelian subgroup H≤G/N. We compute the orders of the underlying elements on G/N using the previous method. Let the orders of h1,...,h randgbes1,...,srands, respectively. Then for a tuple ( α1,...,α r,α) fromZs1×···×Zsr×Zs, setφ(α1,...,α r,α) =|hα1 1···hαrrg−αN/a\}bracketri}ht. Thenφis a homomorphism from Zs1× ··· ×Zsr×ZsintoCG/N. From Lemma 9, the kernel of φcan be computed in polynomial time by a quantum algorithm. Mo reover it contains an element the last coordinate of which is relatively prime to sif and only if gis representable as a product of powers of his. Also, from such an element an expression for gin the desired form can beconstructed eﬃciently using elementary number theory. 8 5 Groups with small commutator subgroups Assumethat Gis ablack-box groupwith uniqueencodingof elements, andsu pposethat a subgroup His hidden by a function f. Our next result states that one can solve the HSP in time poly nomial in the input size+ |G′|, whereG′is the commutator subgroup of G. Let us recall the commutator subgroup is the smallest normal subgroup of Gcontaining the commutators xyx−1y−1,for every x,y∈G. Theorem 11. LetGbe a black-box group with unique encoding of elements. The hid den subgroup problem in Gcan be solved by a quantum algorithm in time polynomial in the input size +|G′|. Proof.LetHbe a hidden subgroup of Gdeﬁned by the function f. We start with the following observation. If Nis a normal subgroup of GandH1≤His such that H1∩N=H∩Nand H1N=HN, then by the isomorphism theorem, H1/(H∩N)∼=H1N/N∼=H/(H∩N) which impliesH1=H. We will generate such a subgroup H1≤HforN=G′. As the commutator subgroup G′ofGconsists of products conjugates of commutators of the generators of Gwe can enumerate G′, and therefore

Question 1062 multiple-choice

The Hidden Subgroup Problem (HSP) is central to quantum algorithms, especially in non-abelian groups such as the Heisenberg group $H_p$. Efficient solutions for HSP require careful analysis of both quantum and classical resource requirements. Which of the following statements best characterizes the classical post-processing component of the quantum algorithm for solving HSP in the Heisenberg group $H_p$? 1) It requires $O(\log^4 p)$ queries and has a running time of $O(\log^3 p)$. 2) It uses no queries and completes in $O(\log^2 p)$ time. 3) It performs subgroup enumeration in $O(p)$ time and requires no probability table precomputation. 4) It computes the quantum Fourier transform using $O(\log^3 p)$ gates during post-processing. 5) It relies solely on quantum measurements with no classical computation. 6) It requires no queries and has a running time of $\widetilde{O}(p^4)$, primarily due to probability table precomputation for all subgroups. 7) It implements measurement in a random basis with $\widetilde{O}(p^2)$ gates during classical post-processing.

✓ Correct Answer:

The correct answer is 6) It requires no queries and has a running time of $\widetilde{O}(p^4)$, primarily due to probability table precomputation for all subgroups..

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conﬁdence bound in applying the above random basis argu ments is at least1−exp(−Ω(log2|G|)). Since there are at most |G|representations, the total conﬁdence bound is at least1−|G|exp(−Ω(log2|G|)) = 1−exp(−Ω(log2|G|)). Thiscompletes the proof of the theorem. Wenowhaveallthetoolstoprovethat r(Hp) = Ω(1). Infact,wecannowprovethefollowingtheorem. Theorem 2 Therandomstrongmethodissufﬁcient tosolvethehiddensub groupproblemintheHeisenberg groupHp. The query complexity of this algorithm is O(logp). The quantum part of the algorithm consists of a circuit of size O(log4p)followed by a circuit of size ˜O(p2)for implementing the measurement in a random orthonormal basis. The classical post-processing d oes not make any queries and has a running timeof˜O(p4). Proof: First, wecharacterize thenormal core families inthe Heis enberg group. Wehave that ncf(Hp) ={Hp},ncf(ζ(Hp)) ={ζ(Hp)},ncf(Ni) ={Ni},fori∈ {0,...,p−1,∞} 19 are families of size 1each. Forthe trivial group weget that ncf({1}) ={Ai,j:i∈ {0,...,p−1,∞},j∈ {0,...,p−1}}∪{{1}}. IfH1andH2are candidate hidden subgroups from different normal core f amilies, then by Proposition 4 we get thatr(Hp;H1,H2)≥w(Hp;H1,H2)≥1/2. We now consider the situation where both H1,H2∈ ncf({1}). Weﬁxanirreducible representation ρ=ρk(fork= 1,...,p−1) ofdegree degρ=p. Now,we distinguish twocases: 1.|H1|=|H2|=p, i.e., there are i,j,i′,j′,(i,j)/ne}ationslash= (i′,j′)such thatH1=Ai,jandH2=Ai′,j′. UsingthenotationofDeﬁnition1wehavethat r1=r2= 1,sincebyLemma1theranksof Pk;i,jand Pk;i′,j′arebothone. Also, /bardblPk;i,jPk;i′,j′/bardbl ≤1√p. Thisalsoimpliesthat r′ 1=r1= 1andr′ 2=r2= 1, since the one-dimensional projectors Pk;i,j,Pk;i′,j′are linearly independent. Hence, P′ k;i,j=Pk;i,j andP′ k;i′,j′=Pk;i′,j′. So,δ=/bardblP′ k;i,jP′ k;i′,j′/bardbl ≤1√p. Since|H1|r1=|H2|r2=p,ˆh=pand˜h= 0. As /radicalbig dρ log|Hp|=√p 3logp= Ω((r1+r2)3/2) = Ω(1), weare inthe ﬁrst case of Deﬁnition 1and obtain that r(Hp;H1,H2;ρ) =p 2·/parenleftBig Ω(1)−2(p−1)−1/4/parenrightBig = Ω(p). 2.|H1|=pand|H2|= 1, i.e., we have to distinguish H1=Ai,jfrom the trivial subgroup H2={1}. In this caser1= 1andr2= rank(ρ({1})) =pwhichimplies that ˆh=p,˜h= 0. Since /radicalbig dρ log|Hp|= Ω(r1)andr2 r1=p= Ω(log2|Hp|), weare inthe second case of Deﬁnition 1and obtain that r(Hp;H1,H2;ρ) =p 2·Ω(1) = Ω(p). Overall weobtain that for H1,H2∈ncf({1}), r(Hp;H1,H2) =/summationdisplay ρdρ p3·r(Hp;H1,H2;ρ)≥p−1/summationdisplay k=1p p3·r(Hp;H1,H2;ρk)≥(p−1)p p3·Ω(p) = Ω(1). Hence,r(Hp) = Ω(1) . Recall that s(Hp) =O(p2). Now Corollary 1 shows that with probability at least 2/3over the choice of random representation bases, the HSP for Hpcan be solved using O(logp)random strong Fourier samplings. AsshowninProposition 1,the QFToverHpcanbeimplementedusing O(log3p)elementary quantum gates. Since there are O(logp)Fourier samplings, the initial part of the quantum circuit h as sizeO(log4p). The claimed statements about the number of quantum operatio ns necessary to implement a measurement in a random orthonormal basis follow from general upper boun ds of˜O(p2)on the number of gates in a factorization of a unitary operation U∈U(p)into elementary gates. The classical time to generate the random Uis˜O(p3)since we can start with a set of prandom unit vectors and apply Gram-Schmidt orthonormalisation toobtainarandomunitary matrix. Fort heclassical post-processing wehavetocompute 20 a table of probability distributions with respect to the ran dom measurement bases for all subgroups. Since there areO(p2)subgroups and each probability distribution computation t akes time ˜O(p2)we can upper boundthisby ˜O(p4). Afterthistablehasbeenprecomputed theactualalgorithm toﬁndthehiddensubgroup is‘minimum-ﬁnding-like’ inwhich we‘compare’ twosubgrou ps at atime. Thistakes time ˜O(p2). Overall, weobtain that the running time of the classical part of this a lgorithm can be upper bounded by ˜O(p4). Acknowledgments We thank Fr´ ed´ eric Magniez, Leonard Schulman, Cris Moore, Alex Russell and Avery Miller for useful discussions. References [AS00] N.Alonand J. Spencer. Theprobabilistic method . John Wileyand Sons,

Question 1063 multiple-choice

Infinite abelian groups can be classified by properties such as thinness, exponent, divisibility, and decomposition into reduced and divisible components. These structural features play a key role in understanding group extensions and nilpotent p-groups. In an infinite reduced nilpotent p-group B decomposed as E × M, with E reduced and M divisible, which of the following must be true regarding the exponent of B? 1) B always has infinite exponent, regardless of E and M. 2) B has finite exponent only if M is trivial. 3) B has finite exponent provided E has finite exponent, regardless of M. 4) B has finite exponent only if both E and M have finite exponent. 5) B can have infinite exponent if E is infinite cyclic. 6) B has finite exponent only if B is cyclic. 7) B has finite exponent only if B is homocyclic.

✓ Correct Answer:

The correct answer is 3) B has finite exponent provided E has finite exponent, regardless of M..

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be taken as a basic subgroup of R and since JV(R, B) it follows from Theorem 2.8 that L must be thin. Otherwise R would have as direct factor an infinite homocyclic group, namely some Ln. Note that R must have infinite exponent here and so L does also. Next B is not divisible, so has C as an image. Thus JV(R, C ) and we see from Theorem 5.3 that L is eventually sparse. Now let Bab = E X M where E is reduced and M divisible. We claim that E has finite exponent. Assume not. Since E maps onto its basic subgroup, it follows that B has a homomorphic image K isomorphic with the restricted direct product C X C p2 X ... X C pn X . . . . Further JV(R, K). Since L has infinite exponent, we can write L = X X Y, where both X and Y are of infinite exponent. By Theorem 5.2, ^ = 1X5" for some S and we see that ^V(X, K). However, ^V(X, K) is false in this situation, for the following reason. For any n > 1 the group V„ = (x,y\x p" = y p"~l = 1, x v = x1+/? > has class n, and obviously it is possible to make an extension of X by K containing V n for infinitely many n. This completes the proof of (ii) in one direction. We begin the proof of the converse by considering the case where B is reduced. LEMMA. If B is an infinite reduced nilpotent p-group and B ah = E X M where E has finite exponent and M is divisible, then B has finite exponent (and therefore M = 1). Proof. We have E = FIB' and M = N/B f for suitable F, N. From a result of Zaleskii [7], F' = B' and F is a basic subgroup of B (in his terminology). Since FIF' has finite exponent, so does F. Zaleskii also https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press NILPOTENT EXTENSIONS 1051 showed in [7] that B = FZ where Z is the center of B. With bars denoting images mod B', we have B ah = FZ and thus Z is a direct product of a group of finite exponent and a divisible group. But a basic subgroup K of Z maps to one of Z, so that K has finite exponent, and also so does B f. Thus Z has finite exponent and finally B does. Assume now that B is reduced in the statement of Theorem 5.4. Because of Theorem 5.1, we may assume that A is reduced. Case 1. A is a direct product of cyclic groups. Then A = L and so A is thin and eventually sparse. Let the exponent of B bep e. Then from Theorem 5.3 we see that the split extension (A, b) has class depending only on A and/?* 7, for each b in B. Thus by Lemma

Question 1064 multiple-choice

Quantum algorithms such as Deutsch-Jozsa demonstrate how quantum computation can provide exponential speedup over classical approaches, even in systems without entanglement. Extensions to qudit systems allow these algorithms to operate with quantum states having more than two levels, expanding their applicability and functionality. In the qudit generalization of the Deutsch-Jozsa algorithm, which outcome is enabled by measuring the circuit output, apart from determining if the function is constant or balanced? 1) The complete reconstruction of the global phase factor of the measured quantum state 2) Recovery of the coefficients of the underlying affine function, except for a universal phase factor 3) Identification of entanglement between qudit subsystems 4) Direct computation of the function’s classical truth table 5) Determination of the minimum number of function evaluations required classically 6) Measurement of the error correction capability of the circuit 7) Extraction of the probabilities for all possible output states

✓ Correct Answer:

The correct answer is 2) Recovery of the coefficients of the underlying affine function, except for a universal phase factor.

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is also realized on a linear optic system [ 163]. Although the model problem has no signi ﬁcant applications and the speedup in the higher dimensional cases is not exponential,this proposed algorithm provides an elegant yet simple examplefor quantum computation without entanglement. 3.1.2 Qudit Deutsch-Jozsa Algorithm Deutsch algorithm (with its origin in Ref. 44and improved in Ref. 33is one of the simplest examples to show the speed advantage of quantum computation. Deutsch-Jozsa algorithm is n-qubits generalization of the Deutsch algorithm. Deutsch-Jozsaalgorithm can determine if a function f(x)isconstant , with constant output, or balanced , that gives equal instances of both outputs [ 125]. The process itself consists of only one evaluation of the function f(x). In this algorithm, Alice sends Bob Nqubits in the query register and one in the answer register where Bobapplies the function to the query register qubits and stores the results in the answer register. Alice can measure the qubits in the query register to determine whether Bob ’s function is constant or balanced. This algorithm makes use of the superposition propertyof the qubit and reduces the minimum number of the functioncall from 2 n/2+1 classically to only 1 with quantum algorithm. This gives another example of the advantages of quantumalgorithms. The Deutsch-Jozsa algorithm can be performed in the qudit system with a similar setup. Furthermore, with the qudit system,Deutsch-Jozsa algorithm can also ﬁnd the closed expression of an afﬁne function accurate to a constant term [ 53]. The constant and balanced function in the ndimensional qudit case have the following de ﬁnition: “Anr-qudit multi-valued function of the form f:{0,1,... ,n−1} r→{0,1,... ,n−1} (80) isconstant when f(x)/equalsf(y)∀x,y∈{0,1,... ,n−1}rand is balanced when an equal number of the nrdomain values, namely nr−1, is mapped to each of the nelements in the co- domain ”[53]. It can be shown that all of the af ﬁne functions of rqudits f(x1,... ,xr):/equalsA0⊕A1x1⊕/⊕Arxr,A0,... ,Ar∈Zn,(81) can be categorized to either constant or balanced functions [ 53]. If all the coef ﬁcients Ai≠0/equals0 then the function is constant. For afﬁne function with non-zero coef ﬁcient Ai≠0, every element in its domain {0,1,... ,n−1}ris reducible modulo nto a uniqueelement m∈{0,1,... ,n−1}.A s f(p)/equalsf(q)ifp≡q(mod n), each of the elements in the codomain {0,1,... ,n−1}is mapped tonr−1different elements in the domain. To ﬁnish the proof of the n-nary Deutsch-Jozsa algorithm, another trivial lemma is needed: Primitive nthroots of unity satisfy ⎥summationtextn−1 k/equals0ωαk/equals0 for nonzero integers α. The circuit of the Deutsch-Jozsa algorithm in qudits is shown inFigure 9 . This algorithm of rqudits can both distinguish whether a function Ufis balanced or constant and verify a closed expression for an af ﬁne function in Ufwithin a constant term which is a universal phase factor of the x-register and thus is lost during the measurement. The other coef ﬁcients of the af ﬁne function A1,... ,Arare determined by measuring the state of the x-register at the output, |A1,... ,Ar〉. A detailed derivation of the circuit has been shown [ 53]; but the reasoning is an analogy to the

Question 1065 multiple-choice

Advanced mathematical frameworks such as geometric constructions over complex numbers and finite fields are integral to modeling quantum systems. These approaches are deeply connected to representation theory, which underpins the classification of quantum entanglement and the study of symmetries in multipartite quantum systems. Which mathematical concept provides a unified classification scheme for entanglement classes in various tripartite quantum systems, including three qubits, three fermions, and three bosonic qubits? 1) Boolean algebra 2) Topological invariants 3) Group cohomology 4) Differential geometry 5) Tensor networks 6) Representation theory of semi-simple Lie groups and algebras 7) Measure theory

✓ Correct Answer:

The correct answer is 6) Representation theory of semi-simple Lie groups and algebras.

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Title: Geometric Constructions over ℂ $${\mathbb {C}}$$ and 𝔽 2 $${\mathbb {F}}_2$$ for Quantum Information Year: 2018 Paper ID: f6535b01d5cffdc027a04963b827d865acd4601a Source: semantic-scholar URL: https://www.semanticscholar.org/paper/f6535b01d5cffdc027a04963b827d865acd4601a Abstract: In this review paper I present two geometric constructions of distinguished nature, one is over the field of complex numbers ${\mathbb {C}}$ and the other one is over the two elements field $\mathbb {F}_2$. Both constructions have been employed in the past 15 years to describe two quantum paradoxes or two resources of quantum information: entanglement of pure multipartite systems on one side and contextuality on the other. Both geometric constructions are linked to representation of semi-simple Lie groups/algebras. To emphasize this aspect one explains on one hand how well-known results in representation theory allows one to see all the classification of entanglement classes of various tripartite quantum systems (three qubits, three fermions, three bosonic qubits…) in a unified picture. On the other hand, one also shows how some weight diagrams of simple Lie groups are encapsulated in the geometry which deals with the commutation relations of the generalized N-Pauli group.

Question 1066 multiple-choice

Quantum combinatorial optimization often leverages advanced algebraic techniques, such as representation theory and polynomial algebras, to analyze and solve problems involving graph Hamiltonians. In particular, the Quantum Max Cut problem utilizes structures from symmetric groups and swap matrices, alongside algorithmic tools like Gröbner bases and semidefinite programming relaxations. Which statement accurately describes the role of Young’s branching rule in determining eigenvalues of Quantum Max Cut Hamiltonians for graphs decomposable into signed sums of cliques? 1) It provides a method for approximating eigenvalues using random walks on unitary groups. 2) It generates the swap matrices necessary for constructing Quantum Max Cut Hamiltonians. 3) It establishes the polynomial hierarchy levels necessary for semidefinite programming relaxations. 4) It identifies residual subgraphs to bound eigenvalues in general quantum graphs. 5) It enables exact calculation of eigenvalues by decomposing representations when restricting to subgroups corresponding to cliques. 6) It defines the Gröbner bases needed for symbolic manipulation of swap variables. 7) It proves the isomorphism between symbolic swap algebra and matrix algebra.

✓ Correct Answer:

The correct answer is 5) It enables exact calculation of eigenvalues by decomposing representations when restricting to subgroups corresponding to cliques..

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given by Schur-Weyl, lets us reproduce a well known characterization of eigenvalues of the QMC Hamiltonian associated with a clique.) Then we consider graphs whose associated QMC Hamiltonians decompose as a signed sum of cliques, and show how the previously discussed exact arithmetic solutions to the clique irrep Hamiltonians coupled with Young’s branching rule can be used to find the k max and min eigenvalues of these Hamiltonians. When graphs do not decompose entirely into a signed sum of cliques, we show similar techniques can be used to bound the max and min eigenvalues of these graphs in terms of the max and min eigenvalues of smaller “residual” graphs. 1.3 Reader’s Guide In Section 2 we introduce some basic definitions related to the Quantum Max Cut problem, swap matrices, and the representation theory of the symmetric group algebra. We then introduce the concept of Quantum Max Cut Irrep Hamiltonians and give some basic results concerning them, including an exact arithmetic formula for the single (integer) eigenvalue of any Quantum Max Cut Irrep Hamiltonian associated with a clique. In Section 3 we introduce the symbolic swap algebra, give a presentation for it, and show it is isomorphic to the swap matrix algebra. We then prove some basic properties about polynomials in this algebra, to include how one produces a “simple” polynomial q equivalent to a given polynomial pevaluated on swaps. In Section 4 we discuss how the non-commutative Sum of Squares (ncSoS) algorithm can be applied to the symbolic swap algebra to produce a semidefinite programming hierarchy analogous to the Quantum Lasserre Hierarchy. We also construct an explicit linear algebra basis for polynomials of degree ≤4in the symbolic swap algebra. This shows the first two levels of the swap-ncSoS hierarchy can be run in polynomial time. Then, in Section 5, we discuss the theory of Gr¨ obner bases and show it is possible to find rewrite rules for polynomials of any fixed degree in the swap variables. This lets us conclude that any finite level of the swap-ncSoS hierarchy can run in, in principle, polynomial time. In Section 6 we discuss graphs for which we can compute, in exact arithmetic, any constant number of max and min eigenvalues associated to the QMC Hamiltonian using representation theoretic techniques. We give an algorithm which identifies these graphs, then we show how to compute their eigenvalues (or bound the eigenvalues of other graphs). Finally, we give some simple examples of these algorithms in practice. There are 3 online appendices. Appendix A contains relationships on Swaps, all used for proofs in the paper. Appendix C gives some examples and properties of Gr¨ obner bases for ideals involving Swaps. Finally, in Appendix E we compare and contrast this paper with the independent and simultaneously released paper [TRZ+]. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 8 Acknowledgements The authors thank Claudio Procesi for bringing the notion of a “good permutation” to their attention, and for sharing his expertise in invariant and representation theory.

Question 1067 multiple-choice

Quantum error correction plays a central role in protecting information against noise in quantum computing, especially when systems possess continuous symmetries. Understanding how symmetry requirements impact code design is crucial for both practical quantum technologies and theoretical physics. Which statement best characterizes the approximate Eastin-Knill theorem in the context of quantum codes covariant under continuous symmetry groups? 1) It guarantees perfect error correction for codes with universal transversal gates. 2) It allows universal transversal gates without any increase in physical resources. 3) It asserts that the accuracy of erasure correction is independent of subsystem size. 4) It requires the number of physical qubits per subsystem to scale inversely with the desired error parameter for fixed-accuracy and universal transversal gates. 5) It permits universal transversal gates only for codes encoding classical information. 6) It states that error-correction infidelity increases as subsystem dimension increases. 7) It claims that subsystem erasure can always be corrected regardless of the symmetry group.

✓ Correct Answer:

The correct answer is 4) It requires the number of physical qubits per subsystem to scale inversely with the desired error parameter for fixed-accuracy and universal transversal gates..

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Title: Continuous Symmetries and Approximate Quantum Error Correction Year: 2019 Paper ID: 05ced1e48b0db96fd8c592496ffb33aa93898edc Source: semantic-scholar URL: https://www.semanticscholar.org/paper/05ced1e48b0db96fd8c592496ffb33aa93898edc Abstract: Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here, we study the compatibility of these two important principles. If a logical quantum system is encoded into n physical subsystems, we say that the code is covariant with respect to a symmetry group G if a G transformation on the logical system can be realized by performing transformations on the individual subsystems. For a G-covariant code with G a continuous group, we derive a lower bound on the error-correction infidelity following erasure of a subsystem. This bound approaches zero when the number of subsystems n or the dimension d of each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with n or d as the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem for quantum computation: If a code admits a universal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, we need a number of physical qubits per subsystem that is inversely proportional to the error parameter. We construct codes covariant with respect to the full logical unitary group, achieving good accuracy for large d (using random codes) or n (using codes based on W states). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.

Question 1068 multiple-choice

Quantum-inspired algorithms in linear algebra leverage classical data structures and sampling strategies to efficiently solve high-dimensional problems that were previously considered suitable mainly for quantum computation. These methods often replicate key quantum subroutines using randomized classical techniques and efficient access to data. In quantum-inspired algorithms for estimating inner products like ⟨b, Q^m b⟩, which of the following strategies is used to efficiently replace the quantum Hadamard test while maintaining classical computational feasibility? 1) Using direct matrix multiplication for inner product estimation 2) Implementing the quantum Fourier transform on classical hardware 3) Applying brute-force computation over all vector entries 4) Utilizing fast matrix inversion algorithms 5) Employing sample and query access techniques with data structures like binary trees to randomly select indices and retrieve values 6) Storing precomputed permutation outcomes for all possible shifts 7) Using neural networks to approximate inner products

✓ Correct Answer:

The correct answer is 5) Employing sample and query access techniques with data structures like binary trees to randomly select indices and retrieve values.

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of measurements in the proposition can be expressed as O(K+T)TB4κ2 C ε5 . (17) Further, in order to achieve an ε-close solution to the optimal x∗where x∗= argminx∈CN∥Cx−b∥2, we note by Proposition 4 that T∈O(K·κClogκC ε). Hence the total number of measurements needed is OK2B4κ4 C ε5logκC ε . (18) V. QUANTUM-INSPIRED ALGORITHM We now elaborate on the implementation of Algorithm 1 using purely classical computers that achieve the same performance as the hybrid implementation. To do so, we simply replace the Hadamard test subroutine with a classical subroutine that utilizes techniques used by Tang [8] to estimate inner products. To use these techniques, we require sample and query access to the vector b. Definition 7 (Sample and query access [8, 33]) .For vector x∈CN, letDxbe the distribution over [N]such that a sample X∼ D xsatisfies Pr(X=k) =|xk|2 ∥x∥2. A sample access of xreturns k∈[N]overDx. A query access of xwith supplied index kreturns the k-th element of x. We then make the following assumption: Assumption 8. Assume that there exists a classical data structure Sbcontaining sufficient information of vector b such that command Sample( Sb)achieves sample access to band command Query( Sb,i)achieves query access to thei-th element of b. Further, assume that the input of vector bto Algorithm 1 is given in the form of Sb. A binary tree implementation of such a data structure has been shown by Chia et al. [9]. The leaf nodes of the tree stores the square of the absolute value of each entry, in addition to its original value, while the internal nodes store the sum of their children. As a result, the root node stores the square of the ℓ2norm of the vector. Each sample and query access then requires O(logN) time. Although the hybrid algorithm partially implemented on quantum hardware decomposes Qinto a QFT operation and phase shift operations, the quantum-inspired version does not perform any Fourier-related operations due to its high classical time complexity. Instead, we use the fact that the application of Qon vector bperforms a cyclic 9 FIG. 3. We illustrate how sample and query access to vector bprovides the same access to Qb. Sample access to Qbis equivalent to shifting the output of sample access to b, while query access to Qbis equivalent to shifting the input to query access to b. Note that sample access to Qbis not needed for Algorithm 2. permutation of vector b. Observe that the query access to Qmbretrieves the elements of bshifted mtimes to the right and sample access to Qmboutputs an index that corresponds to the index mtimes to the right of the original vector b, as illustrated in Figure 3. We estimate the value of ⟨b, Qmb⟩by Algorithm 2, which is modified from Algorithm 1 of [33], and whose perfor- mance guarantee of requiring O(1 ε2log1 δ) sample and query accesses to execute is ensured by Proposition 3 of the same paper. Such a process can effectively replace the use of Hadamard tests, allowing us to perform Steps 1 to

Question 1069 multiple-choice

In group theory and algebraic combinatorics, elementary Abelian p-groups are studied for their vector space structure over finite fields, and their properties are analyzed using linear maps, subspaces, and group actions. Concepts such as Schur rings, Cayley isomorphism, and 2-closedness are important in understanding group symmetries and graph classifications. Which statement accurately describes a key structural property of elementary Abelian p-groups of rank n ≥ 4p−2 in relation to the CI(2)-group concept? 1) They are always CI(2)-groups regardless of rank. 2) They become CI(2)-groups only when n is less than 2p. 3) They fail to be CI(2)-groups when n ≥ 4p−2. 4) They are CI(2)-groups if and only if the group order is prime. 5) Their CI(2)-group property depends on the existence of a non-trivial Schur ring. 6) They are CI(2)-groups only for odd values of p. 7) The CI(2)-group property is independent of rank for elementary Abelian p-groups.

✓ Correct Answer:

The correct answer is 3) They fail to be CI(2)-groups when n ≥ 4p−2..

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6we have H(ei+ej)= /angbracketleftgC(ei+ej)|C∈Xp−2∪Xp−1/angbracketright. It is routine computation in linear algebra to prove that/angbracketleftgC(ei+ej)|C∈Xp−2∪Xp−1/angbracketright=/angbracketleftεX\(C∪{i,j}),εX\(C/prime∪{i})+εX\(C/prime∪{j})|C∈ Xp−2,C/prime∈Xp−1,i,j /∈C∪C/prime/angbracketright=(εi−εj)⊥. /square Lemma 8 H(e)=ε⊥. Proof LetCbe a set of size p−i, where 1 ≤i≤p−1. Then ϕ(gC(e),ε) =/summationdisplay B∈Xi,C∩B=∅ϕ(εX\(B∪C),ε)=/summationdisplay B∈Xi,C∩B=∅(p−1) =−/parenleftbiggp+i−1 i/parenrightbigg =0. IfC∈Xp−1, then we have gC(e)=/summationtext i/∈CεX\(C∪{i})=−εX\C. Thus ε⊥⊇H(e)⊇ /angbracketleftgC(e)|C∈Xp−1/angbracketright=/angbracketleftεX\C|C∈Xp−1/angbracketright=ε⊥. /square Lemma 9 H(e−ei)=(ε+εi)⊥for any i∈X. Proof LetCbe a set of size p−j, where 1 ≤j≤p−1. Ifi/∈C,B∈Xj, i/∈B,B∩C=∅, then ϕ(εX\(C∪B),ε+εi)=0. Ifi∈C,B∈Xj,C∩B=∅, then ϕ(εX\(C∪B),ε+εi)=p−1. The function e∗ Ais not 0 on the element e−eiif and only if i/∈A. Thus (∗)ϕ ( g C(e−ei),ε+εi)=/summationdisplay B∈Xj,i/∈B,C∩B=∅ϕ(εX\(B∪C),ε+εi). So, ifi/∈C, then(∗)is equal to 0. If i∈C, then(∗)is equal to /summationdisplay B∈Xj,C∩B=∅(p−1)=(p−1)/parenleftbiggp+j−1 j/parenrightbigg =0. IfC∈Xp−1andi/∈C, then gC(e−ei)=/summationtext j/∈C,j/negationslash=iεX\(C∪{j})=−εX\C− εX\(C∪{i}).I fC∈Xp−1andi∈C, then gC(e−ei)=/summationtext j/∈CεX\(C∪{j})=−εX\C. Now,(ε+εi)⊥⊇H(e−ei)⊇/angbracketleftgC(e−ei)|C∈Xp−1/angbracketright=/angbracketleftεX\C,εX\C/prime+εX\(C/prime∪{i})| C/prime,C∈Xp−1,i∈C,i /∈C/prime/angbracketright=(ε+εi)⊥. /square Lemma 10 f( ei)=0,f(e i+ej)=0,f(e) =ε,f(e −ei)=ε,for any i,j∈X, i/negationslash=j. Proof Clearly the ﬁrst two identities are trivial. We have f( e)=/summationtext A∈XpεX\A=/parenleftbig2p−2 p/parenrightbig/summationtext i∈Xεi=ε. Finally f( e−ei)=/summationtext A∈Xp,i /∈AεX\A=/parenleftbig2p−3 p/parenrightbig/summationtext j∈X,j/negationslash=iεj+/parenleftbig2p−2 p/parenrightbig εi=ε. /square J Algebr Comb (2007) 26: 343–355 351 Letgbe the linear map deﬁned by 2−1/summationdisplay i∈XεX\{i}e∗ i. We have g(ei)=2−1εX\{i},g(ei+ej)=2−1(εX\{i}+εX\{j}),g(e)= 2−1/summationtext j∈XεX\{j}=−εandg(e−ei)=−ε−2−1εX\{i}. Lemma 11 There exists no linear map /Lambda1:W→Vsuch that (f−/Lambda1)(w) ∈H(w) for every w∈W. Proof By the preliminary remark on the map gand by Lemmas 6,7,8,9,10we have that (f+g)(ei)∈H(ei),(f+g)(ei+ej)∈H(ei+ej),(f+g)(e)∈H(e) . Therefore it sufﬁces to prove that there exists no linear map xsuch that x(ei)∈ε⊥ i, x(ei+ej)∈(εi−εj)⊥,x(e)∈ε⊥andϕ(x(e −en)−2−1εX\{n},ε+εn)=0. By way of contradiction let us assume that we have such an x=/summationtext i∈Xaie∗ i,f o rai∈V. In particular we have ϕ(aj,εi)=ϕ(ai,εj)for any i/negationslash=jandϕ(ai,εi)=0. This yields ϕ(x(e),ε i)=/summationdisplay j∈Xϕ(aj,εi)=/summationdisplay j∈Xϕ(ai,εj)=ϕ(ai,ε). Furthermore ϕ(x(e −en),ε+εn)=ϕ(x(e),ε) +ϕ(x(e),ε n)−ϕ(an,ε)−ϕ(an,εn)=0. Soϕ(x(e −en)−2−1εX\{n},ε+εn)=−2−1ϕ(εX\{n},ε+εn)=1/negationslash=0, a contradic- tion. Thus the result is proved. /square Theorem 1An elementary Abelian p-group of rank n≥4p−2is not a CI(2)-group . Proof By construction the element fnormalizes the 2-closure G(2)ofG.W eh a v e G(2)=WL(2)and clearly GandG(2)induce the same map H. Therefore [−,f]∈ Der(W,L) andθ([−,f])∈Hom H(W,V ) .W eh a v e θ([−,f])(w)=[w,f](0)= −f(−w). Thus f∈Hom H(W,V ) . The group W×Vhas rank 4 p−2. Therefore, by Lemma 4and Lemma 11, the group W×Vis not a CI(2)-group. It is an easy ap- plication of the ﬁrst paragraph of Sect. 5 .1i n[ 5] to see that any elementary Abelian p-group of rank n≥4p−2i sn o taC I(2)-group. /square Based on some computer evidence we present the following conjecture. Conjecture The group Gis 2-closed. 4 Proof of Theorem 2 As in Section 3, we assume that p>2. In this section we shall repetitively use Propo- sition 22 .1 (Schur-Wielandt principle), Propositions 22 .4,23.5 and Theorem 23 .9 352 J Algebr Comb (2007) 26: 343–355 in [9]. For the computations inside the group algebra Q[W×V]we shall stick to the notation and to the terminology of Sect. 2 and 4 in [ 6]. In the proof of Theorem 1we have shown that f∈Hom H(W,V ) . Moreover, it was proved in [ 6] (Proposition 5 page 173) that the linear span, AH,o ft h es i m p l e quantities {(w,H(w) +v)}w∈W,v∈Vis a Schur ring. LetEbe{ei}i∈X∪{ei+ej}i/negationslash=j∪{e,e−en}. Note that in the proof of Lemma 11

Question 1070 multiple-choice

In computational group theory and quantum algorithms, efficiently solving the Hidden Subgroup Problem (HSP) over finite Abelian groups is foundational for applications in cryptography and mathematics. These groups are often represented as direct products of cyclic groups of prime power order, which enables specialized algorithmic approaches. When solving the Hidden Subgroup Problem over a cyclic group of the form Z_{p^k}, which element is always present in every non-trivial subgroup and thus can be used for an efficient decision algorithm? 1) The identity element 0 2) The element 1 3) The element p 4) The element p^{k-2} 5) The element p^{k/2} 6) The element p^k 7) The element p^{k-1}

✓ Correct Answer:

The correct answer is 7) The element p^{k-1}.

📚 Reference Text:

paper. Let [ m] = {1,2,...,m}andZpkdenote the additive group of elements {0,1,...,pk−1}with addition modulo pk. For two groups G1,G2, letG1×G2denote the direct product of G1andG2. For an arbitrary ﬁnite Abelian group G, we have G∼= {0}×Zpk1 1×···×Zpkl l2. Fora,b∈G, suppose a= (0,a1,...,al),b= (0,b1,...,bl). 2Generally, we say G∼=Zpk1 1×···×Zpkl l. In this paper, we add {0}for the simplicity of algorithms discription. 5 Then we deﬁne the group operation “+” as follows: a+b= (0,a1+b1modpk1 1,...,al+blmodpkl l). Correspondingly, we deﬁne a−b= (0,a1−b1modpk1 1,...,al−blmodpkl l). ForW1,W2⊆G, we have W1+W2={a+b|a∈W1,b∈W2}, W1−W2={a−b|a∈W1,b∈W2}. W1,W2is called a generating pair ofVifV=W1−W2.W1,W2is called to have a collision if there exist x∈W1,y∈W2such that f(x) =f(y). We denote that His a subgroup of GbyH≤G. ForH≤G, a subset Sis said to be a generating set for Hif all elements in Hcan be expressed as the ﬁnite sum of elements in Sand their inverses, i.e., H=/a\}⌊ra⌋k⌉tl⌉{tS/a\}⌊ra⌋k⌉tri}ht={l1a1+l2a2+···+ lnan|ai∈S,li=±1,n∈N}. Forg∈G, we have/a\}⌊ra⌋k⌉tl⌉{tg/a\}⌊ra⌋k⌉tri}ht={g+···+g/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright n|n∈Z}and gH={g+h|h∈H}. Ifgis a nonzero element, we let Igdenote the maximum nonzero coordinate. For example, if g= (0,g1,...,gi,0,...,0) andgi/\⌉}atio\slash= 0, then Ig=i. Ifg= 0, then we let Ig= 0. For g1,g2∈G,g1H=g2His equivalent to g1−g2∈H. The group with only one element, the identity element, is called thetrivial group . Additionally, we use W1×···×Wito represent W1×···×Wi×{0}···×{0}/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright l−i+1 for simplicity. In the description and analysis of Algorithm 3 and 4, let Gi= {0}×Zpk1 1×···×Zpki i,Hi={h∈H|Ih≤i}. 3. Algorithms for HSPoverG=Zpk In this section, we propose algorithms to solve the hidden subgroup problem overG=Zpk(pis a prime), which establishes Theorem 1. 3.1. Decision version The algorithm of the decision version is as follows: Query f(0) andf(pk−1). Iff(0) =f(pk−1), thenHis non-trivial; otherwise, His trivial. The correctness of the algorithm relies on the following fact. Fact 1. The subgroups of Zpkconsist of{0},/a\}⌊ra⌋k⌉tl⌉{tpk−1/a\}⌊ra⌋k⌉tri}ht,/a\}⌊ra⌋k⌉tl⌉{tpk−2/a\}⌊ra⌋k⌉tri}ht,...,/a\}⌊ra⌋k⌉tl⌉{t1/a\}⌊ra⌋k⌉tri}ht. Since all non-trivial subgroups of Gcontain the element pk−1, we only need to check whether pk−1∈Hor not to decide the triviality of H. 6 3.2. Identiﬁcation version In the following, we give an algorithm to identify H(Algorithm 1). The idea of Algorithm 1 is to check whether pi∈Hfor 0≤i≤k−1. If there exists someisuch that pi∈H, thenH=/a\}⌊ra⌋k⌉tl⌉{tpi/a\}⌊ra⌋k⌉tri}ht; otherwist, His trivial. Algorithm 1 FindHinZpk 1:Query 0; 2:fori= 0→k−1do 3:Querypi; 4:iff(0) =f(pi)then 5:return/a\}⌊ra⌋k⌉tl⌉{tpi/a\}⌊ra⌋k⌉tri}ht; 6:end if 7:end for 8:return{0}; Speciﬁcally, we analyze the correctness and query complexity of Alg orithm 1 as follows. If H=/a\}⌊ra⌋k⌉tl⌉{tpj/a\}⌊ra⌋k⌉tri}htfor some 0≤j≤k−1, thenf(0) =f(x) iﬀx∈H, i.e., pj|x. Thus, the algorithm will perform the loop repeatly for 0 ≤i≤j. Finally, the algorithm returns /a\}⌊ra⌋k⌉tl⌉{tpj/a\}⌊ra⌋k⌉tri}ht. Since|G|=pk,|H|=pk−j, the query complexity of the algorithm is 1+( j+1) =O(log|G| |H|). IfH={0}, then the algorithm will perform the loop entirely since f(0)/\⌉}atio\slash= f(pi) for any 0≤i≤k−1. Finally, the algorithm returns {0}. Since|H|= 1, the query complexity of the algorithm is 1+ k=O(log|G| |H|). 4. General algorithms for HSPover ﬁnite Abelian groups In this section, we will discussthe generalAbelian hidden subgroupp roblem. Since any ﬁnite Abelian group is isomorphic to {0}×Zpk1 1×···×Zpkl l, where pi’s are primes and ki≥1 for any i, we only need to consider the case G= {0}×Zpk1 1×···×Zpkl l. The decision and identiﬁcation versions are discussed in Section 4.2 and 4.3, respectively.

Question 1071 multiple-choice

Quantum simulation using cold atoms in optical lattices leverages advanced protocols for engineering complex Hamiltonians and exploring exotic physical phenomena. Recent innovations enable the realization of quantum systems with long-range interactions and robust topological properties. Which technological advancement most directly enables the implementation of the quadratic quantum Fourier transform (QQFT) protocol for programmable Hamiltonian engineering in cold atom optical lattices? 1) Digital-micromirror-device (DMD) programmable laser potentials 2) Superconducting qubits with tunable capacitive coupling 3) Ion trap arrays with individual addressing 4) Photonic crystal waveguides 5) Cryogenic dilution refrigerators 6) Spin-polarized electron sources 7) High-resolution magnetic field gradient coils

✓ Correct Answer:

The correct answer is 1) Digital-micromirror-device (DMD) programmable laser potentials.

📚 Reference Text:

Title: Programmable Hamiltonian engineering with quadratic quantum Fourier transform Year: 2022 Paper ID: 24c0be313859f103d110702eb1884e8f93d073de Source: semantic-scholar URL: https://www.semanticscholar.org/paper/24c0be313859f103d110702eb1884e8f93d073de Abstract: Quantum Fourier transform (QFT) is a widely used building block for quantum algorithms, whose scalable implementation is challenging in experiments. Here, we propose a protocol of quadratic quantum Fourier transform (QQFT), considering cold atoms conﬁned in an optical lattice. This QQFT is equivalent to QFT in the single-particle subspace, and produces a di ﬀ erent unitary operation in the entire Hilbert space. We show this QQFT protocol can be implemented using programmable laser potential with the digital-micromirror-device techniques recently developed in the experiments. The QQFT protocol enables programmable Hamiltonian engineering, and allows quantum simulations of Hamiltonian models, which are di ﬃ cult to realize with conventional approaches. The ﬂexibility of our approach is demonstrated by performing quantum simulations of one-dimensional Poincar´e crystal physics and two-dimensional topological ﬂat bands, where the QQFT protocol e ﬀ ectively generates the required long-range tunnelings despite the locality of the cold atom system. We ﬁnd the discrete Poincar´e symmetry and topological properties in the two examples respectively have robustness against a certain degree of noise that is potentially existent in the experimental realization. We expect this approach would open up wide opportunities for optical lattice based programmable quantum simulations.

Question 1072 multiple-choice

In quantum computing and computational group theory, functions that are injective under group actions are crucial for solving problems like the Hidden Subgroup and Hidden Shift Problems. Techniques such as Quantum Fourier Sampling leverage representation theory to uncover hidden group structures. Which statement correctly characterizes the relationship between a function f's injectivity under right multiplication and its ability to distinguish right cosets of a subgroup in the context of group-based quantum algorithms? 1) f is injective under right multiplication only if it is constant on every subgroup. 2) f being injective under right multiplication means it cannot distinguish left cosets of any subgroup. 3) f is injective under right multiplication if and only if it annihilates all non-trivial group elements. 4) f is injective under right multiplication if and only if it can distinguish right cosets of a subgroup. 5) f is injective under right multiplication only when the group is abelian. 6) f is injective under right multiplication if and only if its image forms a subgroup. 7) f is injective under right multiplication only if it is invariant under all group automorphisms.

✓ Correct Answer:

The correct answer is 4) f is injective under right multiplication if and only if it can distinguish right cosets of a subgroup..

📚 Reference Text:

injective under right multiplication and G |f=H. Conversely, if f is injective under right multiplication, then G |fis a subgroup and f distinguishes the right cosets of the subgroup G |f. Hence, the function fdeﬁned in (2) can distinguish the right cosets of some subgroup if and only if it is injectiv e under right multiplication. Lemma 1. The function f deﬁned in (2)is injective under right multiplication if and only if (1) f 0is injective under right multiplication and (2) f 1(x)=f0(sx)for some s. McEliece and Niederreiter Cryptosystems 767 The proof of this lemma is straightforward, so we omit it here. Proposition 2. Assume f 0is injective under right multiplication. Let H 0=G|f0and s be a shift. Then the function f deﬁned in (2)distinguishes right cosets of the following subgroup of G ≀Z2: G≀Z2|f=/parenleftbig (H0,s−1H0s),0/parenrightbig ∪/parenleftbig (H0s,s−1H0),1/parenrightbig , which has size 2|H0|2. The set of all shifts from f 0to f 1is H 0s. If we can determine the hidden subgroup K=G≀Z2|f, we can ﬁnd a shift by select- ing an element of the form ((g1,g2),1)from K.T h e n g1must belong to H0s,a n ds oi s a shift from f0tof1. Application to the Scrambler-Permutation problem. Returning to the Hidden Shift Problem over G=GLk(Fq)×Sncorresponding to the Scrambler-Permutation problem, it is clear that the function f0deﬁned in (1) is injective unde r right multiplication, and that H0=GLk(Fq)×Sn|f0=/braceleftbig (S,P)∈GLk(Fq)×Sn|S−1MP=M/bracerightbig . The automorphism group of Mis the projection of H0onto Sn, i.e., Aut(M)=/braceleftbig P∈Sn|∃S:S−1MP=M/bracerightbig . Note that each P∈Aut(M)has the same number of preimages S∈GLk(Fq)in this projection. 3 Quantum Fourier sampling (QFS) 3.1 Preliminaries and Notation Fix a ﬁnite group G, abelian or non-abelian, and let /hatwideGdenote the set of irreducible unitary representations, or “irreps” for short, of G. For each irrep ρ∈/hatwideG,l e tVρdenote a vector space over Con which ρacts so that ρis a group homomorphism from Gto the general linear group over Vρ,a n dl e t dρdenote the dimension of Vρ. For each ρ,w eﬁ x an orthonormal basis Bρ=/braceleftbig b1,..., bdρ/bracerightbig forVρ. Then we can represent each ρ(g)as a dρ×dρunitary matrix whose jthcolumn is the vector ρ(g)bj. Viewing the vector space C[G]as the regular representation of G, we can decom- poseC[G]into irreps as the direct sum/circleplustext ρ∈/hatwideGV⊕dρ ρ. This has a basis {|ρ,i,j/angbracketright:ρ∈ /hatwideG,1≤i,j≤dρ},w h e r e{|ρ,i,j/angbracketright|1≤i≤dρ}is a basis for the jthcopy of Vρ.U pt o normalization, |ρ,i,j/angbracketrightcorresponds to the i,jentry of the irrep ρ. Deﬁnition 4. TheQuantum Fourier transform over G is the unitary operator, denoted FG, that transforms a vector in C[G]from the point-mass basis {|g/angbracketright|g∈G}into the basis given by the decomposition of C[G]. For all g ∈G, FG|g/angbracketright=∑ ρ,i,j/radicalBigg dρ |G|ρ(g)i,j|ρ,i,j/angbracketright, where ρ(g)ijis the (i,j)-entry of the matrix ρ(g). Alternatively, we can view F G|g/angbracketrightas a block diagonal matrix consisting of the block/radicalbig dρ/|G|ρ(g)for each ρ∈/hatwideG. 768 H. Dinh, C. Moore, and A. Russell Notation. For each subset X⊆G,d e ﬁ n e|X/angbracketright=(1//radicalbig |X|)∑x∈X|x/angbracketright, which is the uniform superposition over X. For each X⊆Gandρ∈/hatwideG, deﬁne the operator Πρ Xdef=1 |X|∑x∈Xρ(x), and let/hatwideX(ρ)denote

Question 1073 multiple-choice

In computational group theory and complexity theory, efficient algorithms for problems involving permutation groups and graph symmetries are central to understanding the tractability of various computational tasks. The relationship between group-theoretic problems and complexity classes informs the design and limits of such algorithms. Which statement correctly describes the significance of Theorem 4.1 regarding the FIND-GROUP problem and its implications for computational complexity? 1) It shows that membership testing for any subgroup of Sn can be performed in deterministic logarithmic time. 2) It proves that the Graph Isomorphism problem is NP-complete for all input sizes. 3) It demonstrates that the hidden subgroup problem cannot be solved in polynomial time for permutation groups. 4) It establishes that every automorphism group can be generated by just two permutations. 5) It proves that coset representatives in permutation groups are unique for any mapping. 6) It indicates that AUTO and FIND-GROUP are not related in terms of polynomial-time computability. 7) It claims the existence of an FPSPP algorithm for FIND-GROUP, implying polynomial-time tractability for strong generators and related group-theoretic problems.

✓ Correct Answer:

The correct answer is 7) It claims the existence of an FPSPP algorithm for FIND-GROUP, implying polynomial-time tractability for strong generators and related group-theoretic problems..

📚 Reference Text:

1 ≤k≤i, for eachg∈G(i)andh∈G(i+1), we havekhπi+1=kπi+1=kgiπi=kπi=kgπi. In particular if ρis the lex- least element of G(i)πi, every element in G(i+1)πi+1agrees with ρon the ﬁrst i elements. Furthermore, for each g∈G(i+1)notice that ( i+ 1)gπi+1= (i+ 1)πi+1= (i+ 1)giπi=x, wherexis deﬁned above. It is clear that G(i+1)πi+1is precisely the subset of G(i)πieach of whose elements maps i+ 1 tox. Together with the fact that ( i+ 1)ρ=x(by the lex-least property of ρ), we get the desired conclusion. By induction and the above Claim it follows that the lex-least element of Gσ=G(0)π0is inG(n)πn={πn}. Henceπnis the desired lex-least element of Gσ. We can easily extend the above result to show the following. Theorem 3.3 There is a polynomial-time algorithm that takes as input a per- mutation group /angbracketleftA/angbracketright=G≤Snand two permutations τ,σ∈Sn, and computes the lexicographically least element of τGσ. Remark. The above theorem implies, in particular, that the lexicographically least element of a left coset τGcan be computed in polynomial time. 4 Graph Isomorphism in SPP We are ready to prove the main theorem of the paper. Recall that the Graph Isomorphism problem is the following decision problem: GI = {(X1,X2)|X1 andX2are isomorphic }. A related problem is AUTO which is a functional problem: given a graph Xas input the problem is to output a strong generator set for Aut( X). It is well-known from the result of Mathon [16] (see e.g. [14]) that GI and AUTO are polynomial-time Turing equivalent. Thus, in order to show that GI ∈SPP it suﬃces to show that AUTO ∈ FPSPP. In other words, it suﬃces to show that there is a deterministic polynomial- time Turing machine Mwith oracle A∈SPP that takes a graph Xas input and outputs a strong generator set for Aut( X). We observe here that the problem AUTO itself is one among a class of prob- lems, each of which we will show is in FPSPPby giving such an algorithm for the following generic problem FIND-GROUP which we formally describe below: To each instance /angbracketleftx,0n/angbracketrightofFIND-GROUP there is associated an unknown subgroupGx≤Snfor which there is polynomial time membership test. More precisely, a polynomial-time function MEMB( x,g) is given, that takes xand 6 g∈Snas input and evaluates to true if and only if g∈Gx. The FIND-GROUP problem is to compute a strong generator set for Gxgiven /angbracketleftx,0n/angbracketrightas input. Notice that AUTO is an example of the generic FIND-GROUP problem, as checking whether g∈Snis an automorphism of a graph Xonnnodes can be done in time polynomial in n. Remark. The advantage of solving the generic problem FIND-GROUP is that it allows us to show at one stroke that several group-theoretic problems apart from GI are all in SPP. In particular, as a corollary to Theorem 4.1 we will show in the next section that the hidden subgroup problem (of interest in quantum computing) in the case of permutation groups is also in the class FPSPP. Theorem 4.1 There is an FPSPPalgorithm for the FIND-GROUP problem. Proof . Let /angbracketleftx,0n/angbracketrightbe an input instance of FIND-GROUP . The

Question 1074 multiple-choice

Computational algebra and quantum optimization often rely on systematic methods to handle polynomial expressions and non-commutative structures. Tools such as Grӧbner Bases, linear algebraic bases, and semidefinite programming (SDP) relaxations are central to these domains. Which statement accurately describes the role of the trace functional in establishing strong duality for semidefinite programs (SDPs) over the swap algebra? 1) The trace functional guarantees strict feasibility for the dual SDP, ensuring strong duality holds. 2) The trace functional always yields the minimum eigenvalue, which is necessary for primal feasibility. 3) The trace functional is used exclusively in commutative algebras to determine strict feasibility. 4) The trace functional serves as a generator for the swap algebra ideal in Grӧbner basis computations. 5) The trace functional ensures that only the first SDP relaxation converges to the optimal value. 6) The trace functional eliminates the need for monomial ordering in polynomial reduction. 7) The trace functional determines the linear independence of the swap algebra basis elements.

✓ Correct Answer:

The correct answer is 1) The trace functional guarantees strict feasibility for the dual SDP, ensuring strong duality holds..

📚 Reference Text:

we can similarly asso- ciate a list grlex based of rules. Then for some p, after applying rules, ˜pmay depend on the order in which the rules are applied and even worse p∈Imight not reduce to˜p= 0. Later, in Section 5, we show how this problem can be fixed by selecting rules corresponding to a set of generators for Icalled a Gr¨ obner Basis (GB). Indeed the rules Equation (4.19) do correspond to a GB for S3w.r.t. the grlex monomial order. A longer discussion of Gr¨ obner Basis can be found in Section 5. (2)Linear algebra basis. An alternate approach is possible if a good understanding of the quotient algebra AIS3 3:=C⟨s⟩/IS3is available. Namely, if a linear basis for the quo- tient space can be obtained (at least for images of polynomials up to a certain degree), then simple linear algebra allows us to form (lower) levels of the relaxation hierarchy. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 35 For instance, a basis for the image of C⟨s⟩2inAIS3 3is 1, s12, s13, s12s13, s13s12, since all monomials in C⟨s⟩2are a linear combination of these modulo the IS3, as can be derived from Equation (4.14). Thus producing a moment matrix pattern for the idealS3is done by using the trivial linear expansion of monomials in terms of this basis, s23s13=s12s23, s 13s12=s12s23, s 23s12=s12s13, s 13s23=s12s13. The symmetric group S3is so simple that the distinction between the rule approach and the linear basis approach is merely semantic. However, for swaps the difference has substance. Indeed, Section 4.3 takes the linear basis approach while Section 5 takes the Gr¨ obner Basis approach. 4.3 Non-Commutative Sum Of Squares Hierarchy for the Swap algebra In this section we restrict our attention to the swap algebra. We begin with a brief discussion of convergence of Sum of Squares relaxations in this algebra. Then we turn our attention to implementations of the 1st and 2nd relaxations in the swap algebra using linear algebraic bases. We begin in Section 4.3.1 with an example of the 1st relaxation using a linear algebraic basis, then in Section 4.3.2, construct a linear algebraic basis for monomials of degree at most two in the swap algebra. In Section 4.3.3 we discuss an implementation of the 2nd swap relaxation using a linear algebraic basis for swap algebra monomials of degree at most four which we construct in Appendix B. We end this section with a discussion of some numerical results obtained using the 2nd swap relaxation. Proposition 4.9. Strong duality holds between the primal-dual pair of SDPs in Eq. (4.5) and Eq. (4.10) . Proof. It suffices to find a strictly feasible point Lfor the dual SDP in Eq. (4.10). Equiv- alently, a strictly positive linear functional ℓon the Swap algebra Aswap n, or equivalently, onMswap n. But this is easy: simply take the trace. Theorem 4.10. Forh=h∗∈Aswap n, the sequences νd(h)andν d(h)defined in Eq. (4.5) and Eq. (4.10) converge monotonically to eigmax(h(Swap)) . More precisely, for d≥⌈n 2⌉,

Question 1075 multiple-choice

In group theory, finite 2-groups are groups where every element has order a power of 2, and the structure of their maximal subgroups and commutator subgroups plays a central role in classification. Properties such as minimal generation, subgroup normality, and derived subgroup structure provide key insights into the group's overall architecture. Which of the following statements is true regarding a finite 2-group G minimally generated by two elements, with one maximal subgroup H that is neither abelian nor minimal nonabelian, and two maximal subgroups H1 and H2 that are minimal nonabelian with distinct derived subgroups? 1) All maximal subgroups of G are abelian. 2) The derived subgroup G′ can be isomorphic to C4 × C2. 3) The Frattini subgroup of G must be nonabelian. 4) Every maximal subgroup of G is minimal nonabelian. 5) The commutator element v in G necessarily has order 2. 6) The center Z is never contained in the Frattini subgroup Φ. 7) G is necessarily a metacyclic group.

✓ Correct Answer:

The correct answer is 2) The derived subgroup G′ can be isomorphic to C4 × C2..

📚 Reference Text:

[h1, h2] follows s= [h1, h2] = [h1, xh1] = [h1, h1][h1, x]h1= [h1, x]h1 and so [ h1, x] =sz. We have Φ( G) =G′/an}bracketle{th2 1/an}bracketri}htis abelian and H=F/an}bracketle{th2 1/an}bracketri}ht, where [h2 1, x] = [h1, x]h1[h1, x] = (sz)h1(sz) = (szz)(sz) =z. Since H//an}bracketle{tz/an}bracketri}ht=H/H′is abelian of type (2 ,2,2n−1), we have d( H) = 3. Re- placing swiths′=sz, we get [ h1, x] =s′and writing again sinstead of s′, we may write [ h1, x] =s∈G′− /an}bracketle{tz/an}bracketri}ht. Also, writing hinstead of h1we obtain the relations of part (c) of our theorem. Theorem 2.3.LetGbe a2-group with d(G) = 2which has exactly one maximal subgroup Hwhich is neither abelian nor minimal nonabelian. If the other two maximal subgroups H1andH2are minimal nonabelian with H′ 1/ne}ati≀nslash=H′ 2, then one of the following holds: (a)Gis a group of order 26withn= 2given in Theorem 87.15(a) in [2]. (b)Gis a group of order 2m+4(m≥2)withn= 2given in Theorem 87.16 in [2]. (c) G=/an}bracketle{th, x|[h, x] =v, v4= 1, vh=vz1, vx=v−1, v2=z1z2, z2 1=z2 2= 1, [z1, h] = [z1, x] = [z2, h] = [z2, x] = 1, x2∈ /an}bracketle{tz1, z2/an}bracketri}ht, h2m= (z1z2)ǫ/an}bracketri}ht, where ǫ= 0,1,m≥2,|G|= 2m+4,G′=/an}bracketle{tv, z1/an}bracketri}ht∼=C4×C2, K3(G) = [G, G′] =/an}bracketle{tz1, z2/an}bracketri}ht∼=E4and/an}bracketle{tz1, z2/an}bracketri}ht ≤Z(G). Moreover, maxi- mal subgroups of GareH1=G′/an}bracketle{th/an}bracketri}ht,H2=G′/an}bracketle{thx/an}bracketri}htandH=G′/an}bracketle{tx, h2/an}bracketri}ht, where H1andH2are both nonmetacyclic minimal nonabelian with H′ 1=/an}bracketle{tz1/an}bracketri}ht,H′ 2=/an}bracketle{tz2/an}bracketri}htandH′=/an}bracketle{tz1, z2/an}bracketri}ht∼=E4(d(H) = 3). Proof. Set/an}bracketle{tz1/an}bracketri}ht=H′ 1and /an}bracketle{tz2/an}bracketri}ht=H′ 2so that W=/an}bracketle{tz1, z2/an}bracketri}ht∼=E4, W≤Z(G) and W≤A=H1∩H2= Φ(G), where Ais abelian. Here {H1//an}bracketle{tz1/an}bracketri}ht, H2//an}bracketle{tz1/an}bracketri}ht, H//an}bracketle{tz1/an}bracketri}ht}is the set of maximal subgroups of G//an}bracketle{tz1/an}bracketri}htand H1//an}bracketle{tz1/an}bracketri}htis abelian and two-generated, H2//an}bracketle{tz1/an}bracketri}htis minimal nonabelian and so H//an}bracketle{tz1/an}bracketri}htmust be nonabelian. If H//an}bracketle{tz1/an}bracketri}htis minimal nonabelian, then by a re- sult of N. Blackburn (Theorem 44.5 in [1]), G//an}bracketle{tz1/an}bracketri}htwould be metacyclic. But then, by another result of N. Blackburn (Lemma 44.1 and Corol lary 44.6 in [1]),Gis also metacyclic, contrary to W≤G′andW∼=E4. Hence H//an}bracketle{tz1/an}bracketri}ht is neither abelian nor minimal nonabelian. By Theorem 2.1, d (H//an}bracketle{tz1/an}bracketri}ht) = 3, (H//an}bracketle{tz1/an}bracketri}ht)′∼=C2andG′//an}bracketle{tz1/an}bracketri}ht∼=C4. Similarly, considering G//an}bracketle{tz2/an}bracketri}ht, we get (H//an}bracketle{tz2/an}bracketri}ht)′∼=C2andG′//an}bracketle{tz2/an}bracketri}ht∼=C4. It follows that G′is abelian of type (4 ,2) with ℧(G′) =/an}bracketle{tz1z2/an}bracketri}ht. On the other hand, {H1/W, H 2/W, H/W }is the set of maximal subgroups of the nonabelian group G/W, where both H1/Wand FINITE 2-GROUPS 69 H2/Ware abelian. Hence H/W is also abelian and so H′≤W. By the above, H′is distinct from /an}bracketle{tz1/an}bracketri}htand/an}bracketle{tz2/an}bracketri}htand so either H′=WorH′=/an}bracketle{tz1z2/an}bracketri}ht. Suppose that H′=/an}bracketle{tz1z2/an}bracketri}ht. Then {H1//an}bracketle{tz1z2/an}bracketri}ht, H2//an}bracketle{tz1z2/an}bracketri}ht, H//an}bracketle{tz1z2/an}bracketri}ht}is the set of maximal subgroups of G//an}bracketle{tz1z2/an}bracketri}ht, where H1//an}bracketle{tz1z2/an}bracketri}htandH2//an}bracketle{tz1z2/an}bracketri}ht are minimal nonabelian and H//an}bracketle{tz1z2/an}bracketri}htis abelian. By results of §71 in [2], G//an}bracketle{tz1z2/an}bracketri}htis metacyclic, contrary to G′//an}bracketle{tz1z2/an}bracketri}ht∼=E4. We have proved that H′=W=/an}bracketle{tz1, z2/an}bracketri}ht. Take h1∈H1−A,h2∈H2−Aso that we have /an}bracketle{th1, h2/an}bracketri}ht=G. Ifv= [h1, h2]∈W, then vis an involution in Z( G) and G//an}bracketle{tv/an}bracketri}ht is abelian, G′≤ /an}bracketle{tv/an}bracketri}ht, a contradiction. Hence v/ne}ati≀nslash∈Wand so v∈G′−Wis of order 4 with v2=z1z2and/an}bracketle{tv/an}bracketri}htis not normal in G(and so also /an}bracketle{tvz1/an}bracketri}htis not normal in G). Indeed, if /an}bracketle{tv/an}bracketri}htis normal in G, then G//an}bracketle{tv/an}bracketri}htwould be abelian since [ h1, h2] =vand/an}bracketle{th1, h2/an}bracketri}ht=G. In particular, /an}bracketle{tv/an}bracketri}htcannot be normal in bothH1andH2and so we may assume without loss of generality that /an}bracketle{tv/an}bracketri}htis not normal in H1. Hence [

Question 1076 multiple-choice

In computational group theory and quantum computing, the complexity of problems such as the Hidden Subgroup Problem (HSP) and related group-theoretic decision problems plays a crucial role in understanding the limitations and capabilities of classical and quantum algorithms. Complexity class relationships further clarify the boundaries of efficient computation. Which of the following statements is TRUE regarding the complexity and algorithmic status of group-theoretic problems involving permutation groups? 1) Efficient quantum algorithms are known for HSP over all nonabelian groups, including permutation groups. 2) The CONJ-GROUP problem is strictly harder than the NORM problem for permutation groups. 3) The Hidden Subgroup Problem over permutation groups is known to be solvable in quantum polynomial time. 4) Membership testing for normalizers in permutation groups requires exponential time in the worst case. 5) BQP is strictly more powerful than PP for solving group-theoretic problems. 6) There exists an FPSPP algorithm for HSP over permutation groups, and CONJ-GROUP and NORM are polynomial-time equivalent. 7) SPP and FPSPP do not contain any group-theoretic problems related to permutation groups.

✓ Correct Answer:

The correct answer is 6) There exists an FPSPP algorithm for HSP over permutation groups, and CONJ-GROUP and NORM are polynomial-time equivalent..

📚 Reference Text:

eﬃcient quantum algorithm for all these. Based on suitable generalizations of Shor’s technique [19], the above problem has eﬃcient quantum algorithms for the case when Gis an abelian group (see e.g. [17] for an exposition). However, the status of HSP is open for general nonabelian groups, except for some special cases where it is settled (see, e.g. [10, 12]). In particular, even when we restrict attention to Gbeing the permutation group Sn, it is not known if HSP has quantum polynomial time algorithms except in special cases. Independently, it is shown by Fortnow and Rogers [8] that the class BQP of languages that have polynomial-time quantum algorithms is closely connected 10 with language classes that are low for PP. In particular, it is shown in [8] that BQP ⊆AWPP where AWPP is a language class that generalizes both BPP and LWPP. Theorem 5.2 [8] BQP ⊆AWPP and hence BQP is low for PP. In this section we show as a corollary to Theorem 4.1 that there is an FPSPP algorithm for the HSP problem over permutation groups. Theorem 5.3 There is an FPSPPalgorithm for the HSP problem over permu- tation groups, and hence HSP over permutation groups is low for PP,GapP , ⊕P,C=Petc. Proof Sketch . We are given (in the form of an oracle) a function ffromSnto a ﬁnite set Xsuch thatfis constant and distinct on diﬀerent right cosets of a hidden subgroup HofSn. The FPSPPwill ﬁrst compute f(1) with one query tof. Now, notice that fgives a membership test for the unknown subgroup H, because a permutation g∈Snis inHif and only if f(g) =f(1). Thus we essentially have a membership test as required for the FIND-GROUP problem of Theorem 4.1. The result now follows by invoking the algorithm described in the proof of Theorem 4.1. Lowness for PP also follows as SPP is low for PP. 5.1 Other applications Using the FPSPPalgorithm for the FIND-GROUP problem we can show that other algorithmic problems on permutation groups [15] which are not known to have polynomial-time algorithms are also in SPP. Among the diﬀerent problems mentioned in [15] we pick the following two examples as most other problems are known to be polynomial time reducible to these. The input instance to the CONJ-GROUP problem consists of three subgroups /angbracketleftS/angbracketright=G,/angbracketleftS1/angbracketright=H1, and /angbracketleftS2/angbracketright=H2ofSn, and the problem is to determine if there is ag∈Gsuch thatgH1g−1=H2(i.e.H1andH2areG-conjugate). A closely related problem NORM has input instance two subgroups Gand HofSn, and the problem is to determine a generator set for the normal- izer subgroup NG(H) ={g∈G|gHg−1=H}. Just as GI and AUTO are polynomial-time equivalent, it turns out that CONJ-GROUP and NORM are also polynomial-time equivalent [15]. Theorem 5.4 The problem NORM is inFPSPPand the problem CONJ-GROUP is in SPP. Proof . We show that NORM is an example of the generic problem FIND-GROUP . The theorem will follow as a direct consequence of Theorem 4.1. It suﬃces to observe that given subgroups /angbracketleftS/angbracketright=Gand/angbracketleftT/angbracketright=HofSn, testing ifg∈NG(H) 11 (i.e.gHg−1=H) can be carried out in polynomial time. More precisely,

Question 1077 multiple-choice

Tensor scaling algorithms are widely used in computational mathematics and quantum information to adjust tensor marginals so their spectra match specified targets. This process involves matrix decompositions, probability measures, and inequalities relating divergence and norm distances. Which mathematical inequality is used to relate the Kullback-Leibler (KL) divergence between probability distributions derived from tensor marginals to the total variation distance or L2 norm, thereby supporting progress in each scaling step of tensor scaling algorithms? 1) Pinsker’s inequality 2) Cauchy-Schwarz inequality 3) Jensen’s inequality 4) Holder’s inequality 5) Markov's inequality 6) Chebyshev's inequality 7) Triangle inequality

✓ Correct Answer:

The correct answer is 1) Pinsker’s inequality.

📚 Reference Text:

exists a homogeneous G-invariant polynomial QPCrSym`pVqbVspmqfor somem¡0such thatQpZq0. But then PpZqQpZbkbvqis a highest weight vector of highest weight k`msuch thatPpYq0. SincePis an eigenvector of the B-action andrYsPBrXs, it follows that also PpXq0. 25 3 Analysis of Algorithm 1 Inthissectionweanalyzeourtensorscalingalgorithm,Algorithm1. Section3.1containsananalysis of the progress made per step (Proposition 3.1). Section 3.2 contains the proof of Theorem 1.13. Section 3.3 contains a sketch of bit-complexity analysis of Algorithm 1. Section 3.4 contains a reduction from the singular spectra setting to the setting of non-singular spectra. 3.1 Scaling step Considerasinglescalingstep(3inAlgorithm1). Givenatensor Y(ofunit`2norm)withnonsingular marginals, this amounts to the update YÐY1:diag ppiq Ò 1{2 Rpiq 1 Y wherepiq YRpiqpRpiqq:is the Cholesky decomposition of the i-th marginal. Here ppiq Òdenotes ppiq ni;:::;ppiq 1 . Thefollowingpropositionshowsthatthehighestweightvectorsgrowbyaconstantfactorin each scaling step. Proposition 3.1 (Progress per step) .LetPPHWVpCrVspkqqandp:{k. Then, jPpY1qj¥2k 32 ln 2 diag ppiq Ò piq 2 trjPpYqj: The following crucial lemma is needed for the proof. We delay its proof until after the proof of Proposition 3.1. Lemma 3.2. Letbe a PSDnn-matrix with unit trace such that RR:, whereRis an arbitrary nn-matrix. Then, qpjR1;1j2;:::; jRn;nj2qis a subnormalized probability distribution, and, for any probability distribution p, DKLpp}qq¥1 16 ln 2kdiagppqk2 tr: whereDKLpp}qq:°n j1pjlog2ppj{qjqis the KL-divergence. Proposition 3.1 follows straightforwardly from Lemma 3.2: Proof of Proposition 3.1: SincePisahighest-weightvectorand diagpppiq Òq1{2pRpiqq1isupper-triangular, Eq. (12) shows that jPpY1qj2ni¹ j1pppiq ni1jqkppiq ni1jjRpiq j;jj2kppiq ni1j jPpYqj2 ni¹ j1pppiq ni1jqppiq ni1jjRpiq j;jj2ppiq ni1j k jPpYqj2 2kDKLpppiq Ò}qpiqqjPpYqj2¥2k 16 ln 2 diag ppiq Ò piq 2 trjPpYqj2: The inequality is Lemma 3.2, which applies because Yis unit norm and hence piqis unit trace. 26 Proof of Lemma 3.2: To see thatqis subnormalized, observe that n¸ j1qjn¸ j1jRj;jj2¤trrRR:strrs1: Ontheonehand,wecannowapplyPinsker’sinequalityintheform[ Wil13,Thm.10.8.1],where the second distribution is allowed to be subnormalized: DKLpp}qq¥1 2 ln 2kpqk2 11 2 ln 2kdiagppqdiagpqqk2 tr: On the other hand, DKLpp}qqDKLpp}q{kqk1qlog2kqk1¥log2kqk1¥1 ln 2p1kqk1q; since lnx¤x1for allx¡0. Further, 1 ln 2p1kqk1q1 ln 2n¸ jkjRj;kj21 ln 2kRdiagprqk2 F¥1 4 ln 2kdiagpqqk2 tr; whererpR1;1;:::;R;n;nqisthediagonalof R. Inthelaststep,weusedthatforanytwomatrices A andB,kAA:BB:ktr¤kABkFkABkF¤pkAkFkBkFqkABkF(see [Bha13, Proof of X.2.4]). Averaging both inequalities, we ﬁnd that DKLpp}qq¥1 4 ln 2kdiagppqdiagpqqk2 tr1 8 ln 2kdiagpqqk2 tr ¥1 8 ln 2 kdiagppqdiagpqqk2 trkdiagpqqk2 tr ¥1 16 ln 2pkdiagppqdiagpqqktrkdiagpqqktrq2 ¥1 16 ln 2kdiagppqk2 tr: The following lemma can be found in [Bha13]. Lemma 3.3 (Lemma IV.3.2 in [Bha13]) .LetAandBbe Hermitian matrices. Then kABktr¥kspecpAqspecpBqk1 Here specpAqdenotes the vector of eigenvalues of Aarranged in decreasing order. As a consequence, we get that at the end of Algorithm 1, the output tensor YgXsatisﬁes kspec piq Y diag ppiq k1¤" for alli1tod. 27 3.2 Proof of Theorem 1.13 In this section, we prove Theorem 1.13 which we restate for convenience. Theorem 1.13 (Tensor scaling) .LetXPTenpn0;n1;:::;ndqbe a (nonzero) tensor whose entries are Gaussian integers of bitsize no more than b. Also, letpPPpn1;:::;ndqwith rational entries of bitsize no more thanbsuch thatppiq ni¡0for alli1;:::;d. Finally, let "¡0. Then,withprobabilityatleast1/2,Algorithm1eithercorrectlyidentiﬁesthat pRpXq,oritoutputs gPGsuch that the marginals of YgXare"-close top. In fact, we have kpiq Ydiagpppiq Òqktr¤"fori1;:::;d (5) in the latter case, where kAktrtrr? A:Asis thetrace norm . Proof of Theorem 1.13. LetXPTenpn0;n1;:::;ndqbeatensorwhoseentriesareGaussianintegers, pPPpn1;:::;ndqarationalspectrumsuchthat ppiq ni¡0foralli1;:::;d,and"¡0. Assumethat pPpXq. We need to show

Question 1078 multiple-choice

Efficient implementation of quantum algorithms is critical due to limitations in current quantum hardware, such as restricted qubit connectivity and short coherence times. Methods that optimize both gate count and circuit depth enhance the practicality of running complex quantum algorithms on real devices. Which circuit design approach utilizes connectivity-adapted CNOT-based building blocks known as Parity Twine chains to achieve optimal gate count and circuit depth across various hardware topologies? 1) Using SWAP networks on linear qubit arrays 2) Implementing Quantum Random Access Memory circuits 3) Employing Parity Twine chains tailored to device connectivity 4) Decomposing multi-qubit gates via Trotterization 5) Utilizing gate teleportation for non-local operations 6) Applying ancilla-assisted modular arithmetic circuits 7) Mapping circuits with unoptimized all-to-all connectivity

✓ Correct Answer:

The correct answer is 3) Employing Parity Twine chains tailored to device connectivity.

📚 Reference Text:

Title: Connectivity-aware Synthesis of Quantum Algorithms Year: 2025 Paper ID: 61580f8197e7a7dc718fd30fda5f2a3e98223e76 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/61580f8197e7a7dc718fd30fda5f2a3e98223e76 Abstract: We present a general method for the implementation of quantum algorithms that optimizes both gate count and circuit depth. Our approach introduces connectivity-adapted CNOT-based building blocks called Parity Twine chains. It outperforms all known state-of-the art methods for implementing prominent quantum algorithms such as the quantum Fourier transform or the Quantum Approximate Optimization Algorithm across a wide range of quantum hardware, including linear, square-grid, hexagonal, ladder and all-to-all connected devices. For specific cases, we rigorously prove the optimality of our approach.

Question 1079 multiple-choice

Quantum complexity theory studies the relationships between quantum algorithms, computational hardness, and group structures underlying quantum problems. PromiseBQP-completeness characterizes problems that are representative of the maximal difficulty for quantum polynomial-time algorithms subject to input promises. Which property ensures that k-fold Forrelation is PromiseBQP-complete when k grows polynomially with the number of qubits n? 1) It uses the cyclic group structure Z/2^nZ for all quantum transformations. 2) The problem remains as hard as any in PromiseBQP when k is polynomial in n, allowing reductions from any PromiseBQP problem. 3) The measurement outcome probability is always exactly 1/2 for the all-zero state. 4) Classical algorithms can solve k-fold Forrelation efficiently without quantum speedup. 5) The oracles encode non-Boolean functions over the qubit register. 6) Hadamard gates are replaced with quantum Fourier transforms in the algorithm. 7) The circulant matrix used is diagonal and independent of the input functions.

✓ Correct Answer:

The correct answer is 2) The problem remains as hard as any in PromiseBQP when k is polynomial in n, allowing reductions from any PromiseBQP problem..

📚 Reference Text:

implemented by the tensor product of nHadamard gates instead of DFT implemented by the QFT circuit due to the consideration of the n-qubit system as the boolean group ( Z/2Z)ninstead of the cyclic group Z/2nZ. Given the promise that |Φf,g|is either less than1 100or greater than3 5,Forrelation is a decision problem that differentiates between the two cases. In the k-foldForrelation variant introduced by Aaronson and Ambainis [40], kfunctions f1, f2,···fk∈ {0,1}n→ {−1,1}are given. The forrelation is calculated as follows: Φf1,f2,···,fk:=1 2(k+1)n/2X x1,x2,···,xk∈{0,1}nf1(x)(−1)x1·x2f2(x2)(−1)x2·x3···(−1)xk−1·xkfk(xk) (C2) Given the promise that |Φf1,f2,···,fk|is either less than1 100or greater than3 5, the k-foldForrelation is a decision problem that differentiates between the two cases. Suppose that we are given oracles Uf1, Uf2,···, Ufksuch that for any input computational basis state |x⟩where x∈ {0,1}n,Ufj|x⟩=fj(x)|x⟩. Using these oracles, a quantum algorithm can be given such that Forrelation can be solved with a bounded probability of error using only kquantum queries to f1, f2,···fk, as illustrated in Figure 12. First, prepare a uniform superposition state over all x∈ {0,1}n. Then alternate between querying Ufj in order of Uf1toUfkand applying WHT until Ufkis queried. Lastly, measure the probability of recovering the uniform superposition state, i.e. measuring on the Pauli-X basis for all qubits, or measuring on the Pauli-Z basis after applying the Hadamard transform on all qubits. We note that the probability of obtaining the measurement result 00···0 is|Φf1,f2,···,fk|. Aaronson and Ambainis [40] also proposed an alternate algorithm that replaces the direct measurement on the Pauli-Z basis with the modified Hadamard test. We omit this algorithm as we do not utilize the algorithm in the later proof. However, it is easy to see in the Hadamard test setup that k-foldForrelation is in PromiseBQP due to its single qubit measurement. Further, it has been proved that when k= poly n,k-foldForrelation isPromiseBQP - complete. We now show in the following that k-foldForrelation can be reduced to QuantumCirculantOptim , hence showing that QuantumCirculantOptim isPromiseBQP -hard. Proof of Proposition 11. Given a k-foldForrelation problem acting on nqubits, we assume we have access to oracles Uf1, Uf2,···, Ufk. We construct a banded circulant linear system of dimension 2n+1, with the qubit representing the least significant bit in the computational basis being the extra qubit. Let the matrix Cbe the cyclic permutation matrix Qover dimension 2n+1. We also give the following three quantum states: |b⟩=H⊗nUfkH⊗nUfk−1H⊗n···H⊗nUf1H⊗n|0n⟩ ⊗ |1⟩,|u1⟩=|0n+1⟩,|u2⟩=|1n+1⟩ (C3) We see that when applying matrix Cto the Ansatz set, we obtain the following: C|u1⟩=|0n⟩ ⊗ |1⟩, C|u2⟩=|0n⟩ ⊗ |0⟩ (C4) We can see that ⟨b|C|u1⟩evaluates to Φ f1,f2,···,fk·1 = Φ f1,f2,···,fkwhile ⟨b|C|u2⟩evaluates to Φ f1,f2,···,fk·0 = 0. Further note that ⟨u1|C†C|u2⟩=⟨u1|u2⟩= 0. Suppose there exists an algorithm Athat solves the QuantumCirculantOptim problem with error rate ζ≤1 16. We now show that ˆ α1can be used to infer ψf,g. By expansion, we have ∥C(α1|u1⟩+α2|u2⟩ − |b⟩∥2=|α1|2+|α2|2−2 Re{α1⟨b|C|u1⟩+α2⟨b|C|u2⟩}+ 1 (C5) 20 =|α1−Φf1,f2,···,fk|2+|α2−0|2+ (1− |Φf1,f2,···,fk|2) (C6) We plug in the solution ˆ α1and ˆα2obtained by algorithm A, and find that |ˆα1−Φf1,f2,···,fk|2+|ˆα2−0|2+ (1− |Φf1,f2,···,fk|2)≤1− |ψf,g|2+ζ⇒ |ˆα1−Φf1,f2,···,fk|2+|ˆα2−0|2≤ζ(C7) Note that this gives us |ˆα1−Φf1,f2,···,fk|2≤ζ.

Question 1080 multiple-choice

Factoring polynomials over finite fields and solving hidden polynomial problems are central topics in computational algebra and quantum computing, with significant implications for cryptography and algorithmic efficiency. Advanced quantum algorithms can sometimes reduce complex multivariate problems to simpler univariate cases, optimizing computational resources. When solving the Hidden Polynomial Problem for quadratic polynomials in n variables over a finite field Fq using quantum algorithms, which statement accurately describes the reduction process and its computational complexity? 1) The reduction requires solving exponentially many instances of the univariate problem, making it infeasible for large n. 2) The reduction focuses on recovering absolute values of coefficients rather than their ratios. 3) The reduction process cannot handle cases where certain coefficients are zero. 4) The reduction is only applicable to polynomials of degree greater than two. 5) The reduction relies solely on classical algorithms for efficiency. 6) The reduction efficiently maps the multivariate quadratic case to O(n²) instances of the univariate problem, with overall time complexity polynomial in n and log q. 7) The reduction yields a time complexity that is linear in both n and q, regardless of the polynomial's structure.

✓ Correct Answer:

The correct answer is 6) The reduction efficiently maps the multivariate quadratic case to O(n²) instances of the univariate problem, with overall time complexity polynomial in n and log q..

📚 Reference Text:

factoring univariate polynomials over Fq; see [25]. This is because a set Bwhich separatestwo(unknown) ﬁeld elements uandvcanbe used to ﬁnd aproper decompo- sition of a polynomial having both uandvas roots. In fact, an eﬃcient deterministic construction of strong bases for such aﬃne groups over prime ﬁelds would imply an eﬃcient deterministic algorithm for factoring polynomials over ﬁnite ﬁelds. 4.3. Multivariate quadratic hidden polynomials. In this part, we reduce theHPPfor multivariate polynomials ofdegreeat most two to the univariate HQPP. As already noted, adding a constant term does not change the level sets; therefore we consider polynomials with zero const ant term. Thus, we assume that the hidden polynomial is of the form (4.1) P(x1,...,x n)=/summationdisplay 1≤i≤j≤naijxixj+/summationdisplay 1≤k≤nbkxk. 1998 T. DECKER, G. IVANYOS, M. SANTHA, AND P. WOCJAN Also, as the partition πPremains the same when we multiply all coeﬃcients with the same nonzero element from Fq, we consider that the HPPhas been solved if we determine the ratios between all the pairs of the n(n+1)/2 coeﬃcients aijandbk. Proposition 4.8. The problem HPP(Fq,n,2)can be reduced on a quantum computer to O(n2)instances of HQPP(Fq)in time(n+logq)O(1). Proof. In order to simplify the following discussions we deﬁne ajito beaijfor j>i. Additionally, if q= 2, then we also assume aii= 0 because x2=xholds overF2. We assume that we have a procedure Rfor determining the coeﬃcients of a univariate quadratic polynomial up to a common factor. Its oracle input is a functiononF qthat has the same level set structure as a polynomial of the form ax2+bx.W e assume that Rdecides whether ais zero, and if a/negationslash=0 ,t h e n Rreturns the quotient b/a. We start with the case n=2 .W eh a v ea no r a c l ew i t ht h es a m el e v e ls e t sa st h e polynomial P(x1,x2)=a11x2 1+a22x2 2+a12x1x2+b1x1+b2x2. We use the oracle with the inputs ( x1,x2): =(x,0). This way, we obtain an instance ofHQPPfor the univariate polynomial a11x2+b1x.W eu s e Rto decide whether a11 is zero or not, and if a11/negationslash= 0, then we compute the quotient b1/a11.F u r t h e r m o r e , we set (x1,x2): =(x,1) for the inputs of the oracle to compute ( a12+b1)/a11in the second step. From this result we c an easily compute the quotient a12/a11. Similarly, using the substitutions ( x1,x2): =( 0,x)a n d(x1,x2): =( 1,x), we decide whether a22 is zero or not. If a22/negationslash= 0, then we obtain the quotients a12/a22andb2/a22.W en o w consider the following diﬀerent cases. •a11,a22/negationslash=0 :I fa12/negationslash= 0, then we have determined all coeﬃcients of Pup t oac o m m o nf a c t o r . I f a12= 0, then we use the inputs ( x1,x2): =(x,x) and obtain HQPPfor (a11+a22)x2+(b1+b2)x. WithRwe can determine whether a11+a22is zero or not. If it is nonzero, then we ﬁnd an element r∈Fqsuch that b1+b2=r(a11+a22). When we write bi/aii=ci, then the equation ( r−c1)a11=(c2−r)a22follows. Since aii/negationslash= 0, we can compute

Question 1081 multiple-choice

In advanced set theory and group theory, properties of infinite cardinals such as strong limits, cofinality, and admissibility play a crucial role in determining the structure of large Abelian groups constructed via direct sums with finite constituents. Understanding the interplay between cardinal arithmetic and group-theoretic properties is essential for analyzing these algebraic systems. Which of the following is a necessary and sufficient condition for a strong limit cardinal γ to be admissible? 1) The cofinality of γ satisfies cf(γ) > ω. 2) γ is of the form 2^α for some cardinal α. 3) γ satisfies γ = 2^logγ. 4) γ has countable cofinality, cf(γ) = ω. 5) γ is a successor cardinal. 6) For every δ < γ, δ is regular. 7) γ is less than the first inaccessible cardinal.

✓ Correct Answer:

The correct answer is 1) The cofinality of γ satisfies cf(γ) > ω..

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7 then from SCH and cf((log7)+) > ω follows 7 > (Iog7)+ = (Oog7)+r > (1θΕ7)ω· α Bereitgestellt von | Technische Informationsbibliothek Hannover Angemeldet Heruntergeladen am | 27.03.18 14:50 330 W. W. Comfort, D. Remus It occurs for many cardinals y that 7 = 2logv. (Every cardinal y of the form y = 2a satisfies that relation.) Clearly, such y satisfy the condition of Theorem 3.5. Although strong limit cardinals y never satisfy y = 2logy, it is nevertheless easy to determine which limit cardinals are, and which are not, admissible. 3.9. Theorem (cf. also [17]). Lei ybea strong limit cardmal. Then γ is admissible ifand only if cf (y) > ω. Proof. If 7 is admissible then from 3.4 and 2.3 (c) follows 7 = loglogy < w(log7) < 7; thus 7 = m(log7), and cf(7) > ω follows from 2.3 (b). Conversely if cf (7) > ω then again from 2.3 follows m(log7) < (Ioglog7r = 7ω = Σ δ<° ^ Σ 2* = 7 ^ 2l°gr>δ<γ δ<γ so \ogye<$#(y). E The following lemma, which shows that for every infinite admissible cardinal 7 there exists α such that m (a) < 7 < 2a, will allow us to extend (in part) the results of 3.3 to groups of the form @y F 0 φ( < y Gf with each Gi c F. Our notation is intended to admit the possibility that Gf = {0} for some or all of the groups Gf, so 3.11 and 3.12 apply in particular to the groups (f)y F. 3.10. Lemma. Lei α and y be infinite cardinals such that α 6 <$tf(y). Then either (i) m(a)<2a, or (ii) (m(cf) = 7 = 2a and) every cardinal κ such that y < κ < 2y satisfies κ e ^(7) and m(K)<2K. Proof. Suppose that (i) fails, and let 7 < κ < 2y. Then from 2.2 follows m(ic) < m(2y) < Gog(20)e < Γ = 7 < κ < 2K , so (ii) holds. D 3.11. Theorem. Let α 6 £#(i) with y > ω, let Fbe afinite Abelian group with \ F\ > l, and let G = (J)yF® ®i

Question 1082 multiple-choice

In combinatorial group theory, the study of subsets avoiding certain pairwise relationships, such as inverse pairs, reveals deep connections to group structure and order. The presence or absence of small prime divisors in a group's order can critically affect the existence of such subsets. In a finite group $G$, under what condition does it become impossible to construct a large subset of fixed size $k$ that contains no inverse pairs? 1) When $G$ is non-abelian 2) When $G$ has a trivial center 3) When $G$ is simple 4) When the order of $G$ has no small prime divisors 5) When $G$ is cyclic of composite order 6) When $G$ is nilpotent 7) When $G$ is generated by involutions

✓ Correct Answer:

The correct answer is 4) When the order of $G$ has no small prime divisors.

📚 Reference Text:

k$ that contain no inverse pairs. This gives a negative answer to a question of Erdős. The construction, which is surprisingly simple, makes heavy use of the fact that their set lives in a group with order divisible by a small prime. They go on to show that this is necessary: if the order of $G$ has no small prime divisors, then Erdős's question has a positive answer.

Question 1083 multiple-choice

Metasurface-based photonic devices are emerging as platforms for programmable quantum information processing, leveraging nanoscale engineering to manipulate light in quantum computational tasks. The integration of metalens arrays and advanced nanofabrication enables precise control over quantum operations such as Grover’s search and the quantum Fourier transform. Which feature of the metalens array design is primarily responsible for suppressing unwanted diffraction and ensuring efficient transmission at the operational wavelength of 810 nm? 1) Use of geometric-phase manipulation through nanoslot orientation 2) Fabrication on silver-coated glass using focused ion beam technology 3) Selection of nanoslot array periodicity smaller than the wavelength 4) Implementation of quantum operations via a Spatial Light Modulator 5) Embedding resource-sharing between different quantum algorithms 6) Generation of entangled photons with a BBO crystal 7) Assignment of each metalens to different columns of the unitary matrix

✓ Correct Answer:

The correct answer is 3) Selection of nanoslot array periodicity smaller than the wavelength.

📚 Reference Text:

on the lens array. This process can set 𝜓(in) jwithboth amplitude and phase [ 35]. The metalens then imple- ments the jth column of the unitary matrix Uthrough a transmission proﬁle at each lens. Figure 1(b) illustrates a typicaltransmissionproﬁle(amplitudeandphaseinacolormap)ofthemetasurface,comprisingatotalof9metalenses.Ourmetalensarrayisprogrammedwithtwodifferentquan-tum operations: Grover’s search (GS) or quantum Fouriertransform (QFT), achieved by illuminating a 2-qubit input stateonthe4upper-left(dashedredsquare)orlower-right lenses (dashed blue square) in Figure 1(b) , referred to as R-space, using the SLM. The lens labels R 0toR3for the GS areindicatedin Figure1(a) ,whilethesamelenslabelsare 930—R.S.Tanuwijayaetal.:Metasurfaceforprogrammablequantumalgorithms deﬁnedinasimilarrow-majororderforthelensesusedin QFT.Ontheotherhand,thefouroutputdirections k0tok3 for the GS or QFT are shown in Figure 1(c) , referred to as k-space,againusingdashedredsquareorbluesquare.Forconvenience,wehaveadoptedthesamelabelorderasthe conventionusedforR-space,butotherchoicesarepossible aslongas4outofthe9directionsareselectedasoutputsindeﬁningthe Umatrix.Thecorrespondingunitarymatrices of GS ( U GS)a n dQ F T( UQFT)a r es h o w ni n Figure 1(d) (nor- malizationfactoromitted).Takingthecenterlens, R3,asan example, it generates {1,1,1,−1}at its 4 output directions forUGS,or{−1,−1,−1,−1}(forthesamelensnowlabeled asR0)atits4outputdirectionsfor UQFT(aglobal𝜋phaseis addedtotheentirematrixwithoutalteringunitarity).Thus, R3→k3forUGSandR0→k0forUQFTboth implement the same matrix element −1, indicating that this resource is shared between the two operators. The matrix elements, with one of them shared and implemented by the centerlens, are shown in Figure 1(c) . The sharing of resources allowsforamorecompactimplementationofthemetalens arraygivenlimitedresources. 2.2 Geometric-phase metasurface and experimental setup Figure 2(a) showcases the Scanning Electron Microscope (SEM) image of the metasurface, with a close-up view pro- videdinFigure2(b) .Thedesignconsistsofninemetalenses, each comprising 32-unit cells along the diameter (approxi-mately 20 μm in diameter). Each metalens is composed of nano-slots with variable orientations. In particular, each unit cell, with a periodicity of p=620nm, contains two pairs of rotated nanoslots (195nm long, 50nm wide) withrotation angles 𝜃1and𝜃2,a sd e p i c t e di n Figure 2(c) .T h e chosenperiodicityislessthantheoperationalwavelengthof 𝜆=810nmtosuppresshigher-orderdiffraction.Regarding the two pairs of rotating slots within a unit cell, we focuson achieving cross-circular polarization (CP) from LCP toRCP in our scheme, contributing to a transmission withanamplitudeproportionalto|||cos( 𝜃1−𝜃2)|||andanoverall geometric phase of 𝜃1+𝜃2[46]. By selecting appropriate values of 𝜃1and𝜃2at each unit cell within the metalens atRj,theresultantcomplextransmissionamplitudeproﬁle is designed to match the speciﬁcations outlined in Eq. (1), wherenowthe U-matrixrepresentsthewhole9 ×9trans- fer matrix obtained by combining all the unitary matrices of the programmed quantum operators. We note that theamplitude control accompanies a power loss in transmis-sion. Here, iranges from 0 to 8, corresponding to the 9 outputdirectionsfromthesamelensat R j.Moreover,forthe entire (9 lenses) ×(9 directions) matrix, there are unused matrix elements where we can embed more operations ifneeded (for furtherdiscussion, see Supplementary Materi- als). In addition, an overall normalization constant and a quadratic focusing phase proﬁle are actually added to themetasurfacetransmissionproﬁle,integratingtheschematicFourier lens shown in Figure 1(a) onto the metalens array fortheconvenienceofcharacterization(see Supplementary Materials).Thedesignofthemetalensarray,incorporating the nanoslots with the orientation proﬁle, is fabricated onsilver-coated glass (50nm coating) using the focused ionbeam(FIB)technique. The experimental setup, incorporating an entangled quantum light source, is illustrated in Figure 2(d) .A 405nm laser (CrystaLaser DL-405-400) pumps a type-II β- barium borate (BBO) crystal to generate two photons at a Figure 2:Geometric-phasemetalensarrayandheraldingphotonexperimentalsetup.(a)SEMimageofthemetasurfacedesignedforoperational wavelength 𝜆=810nm.(b)EnlargedSEMviewofthemetalensarrayshowcasingthequalityofthemetalens.(c)Unitcellforcontrollingtheamplitude andphaseresponse,composedoftwopairsofrotatednano-air-slots(50nm ×195nm)onasilverfilm(50nmthick)coatedglasssubstrate,with variableorientations 𝜃1and𝜃2.(d)Experimentalsetupusingaheraldingphotonsourceforimagingthefar-fieldinterferencepatternsresultingfrom thequantumoperation. R.S.Tanuwijayaetal.:Metasurfaceforprogrammablequantumalgorithms — 931 wavelength of 810nm in a polarization-entangled state of 1∕√ 2(|HV⟩+|VH⟩),withonephotonsenttotheheralding arm and the other to the signal arm. A polarizer on theheralding arm selectively ﬁlters horizontally (H) polarizedphotons,whicharethendetectedbyaSinglePhotonCount-ing

Question 1084 multiple-choice

Quantum error correction codes are crucial for protecting quantum information from noise and physical imperfections, especially when systems exhibit continuous symmetries represented by Lie groups. Covariant codes are specifically designed to respect these symmetries during encoding and error correction. Which property must an isometric covariant quantum code satisfy in order to preserve the symmetry generator's eigenvalue during encoding, aside from a possible constant offset? 1) The code words must be orthogonal to the environment’s states. 2) All physical subsystems must have identical symmetry generators. 3) The encoded states must commute with all possible group actions on the environment. 4) The physical generator must act trivially on the code space. 5) The encoding must be invariant under arbitrary unitary transformations unrelated to the symmetry. 6) The code words must have the same eigenvalue under the physical symmetry generator as the logical state, up to a constant offset. 7) The complementary noise channel must erase all information about the logical state.

✓ Correct Answer:

The correct answer is 6) The code words must have the same eigenvalue under the physical symmetry generator as the logical state, up to a constant offset..

📚 Reference Text:

noise channel (11) a complementary channel is given by dN∘E L→C0EFð·Þ¼X qαjαihαjC0⊗trAnAα½Vð·ÞV†/C138; ð15Þ where the register C0now remembers which combination of systems are lost. This channel provides the environmentwith the systems that are erased, where each erasure combination αappears with probability q α. D. Covariant codes The final ingredient we introduce is covariance with respect to a symmetry group (Fig. 1). Let Gbe any Lie group acting unitarily on the logical and physical systems,with representing unitaries U LðgÞandUAðgÞ, respectively, for any g∈G. A code EL→Aiscovariant if it commutes with the group action:EL→A½ULðgÞð·ÞU† LðgÞ/C138 ¼UAðgÞEL→Að·ÞU† AðgÞ: ð16Þ On either logical and physical systems, we can expand the unitary action of Gin terms of generators of the corresponding Lie algebra; i.e., for a given g, there is a generator TLonLand a generator TAonAsuch that ULðgÞ¼e−iθTL;U AðgÞ¼e−iθTA; ð17Þ for some θ∈Rthat we can choose to normalize our generators. The generators are Hermitian matrices, and theycan be interpreted as physical observables. (For instance, the generators of the rotations in 3D space are the angular momenta.) If the encoding map is isometric, E L→Að·Þ¼VL→Að·ÞV†, then any eigenstate jtiofTLwith eigenvalue tmust necessarily be encoded into an eigenstate of TAwith the same eigenvalue t(up to a constant offset). This constraint can be seen as follows. Expanding the condition (16) for small θyields V½TL;ð·Þ/C138V†¼½TA;Vð·ÞV†/C138: ð18Þ Letfjt; jiLgbe a basis of eigenstates of TLwhere tis the eigenvalue and where jis a degeneracy index. Inserting in the place of ð·Þthe operator jt; jiht0;j0j, we obtain ðt−t0Þjψt;jihψt0;j0j¼½TA;jψt;jihψt0;j0j/C138; ð19Þ where jψt;ji¼Vjt; ji. Setting t¼t0;j¼j0, we see that jψt;jiis necessarily an eigenstate of TA; let ut;jbe its corresponding eigenvalue. Setting t¼t0;j≠j0in Eq. (19) implies 0¼ðut;j−ut;j0Þjψt;jihψt;j0jand, hence, ut;j¼ ut;j0≕ut. Now Eq. (19) tells us for any t; t0;j ;j0that t−t0¼ut−ut0. It follows that ut¼t−νfor all t, for some constant offset ν; in other words, the code words must have the same charge as the logical state, except for a possibleconstant offset ν. We may condense this condition into the constraint ½T A;VV†/C138¼0along with the identity V†TAV¼TL−ν1L: ð20Þ Equivalently, acting with Von Eq. (20),w eh a v e TAV¼VðTL−ν1Þ: ð21Þ This result is a crucial property of covariant codes and is a central ingredient of the proof of our main result. Our main result, in its simplified form, further assumes that the action of the group is transversal on the physical systems, meaning that UAðgÞ¼U1ðgÞ⊗U2ðgÞ⊗/C1/C1/C1UnðgÞ. In this case, the corresponding generator is strictly local, TA¼T1þT2þ/C1/C1/C1þ Tn, where each of the Ti’s act only onAi.PHILIPPE FAIST et al. PHYS. REV . X 10,041018 (2020) 041018-8 As opposed to covariant isometries, covariant channels, in general, do not conserve charge, since they may exchange charge with the environment. For instance, thefully depolarizing channel is covariant with respect to any symmetry, but it changes the charge of its input. Our main result in its fully general form is formulated for approxi-mately charge-conserving channel encodings, which is asuperset of covariant isometries. IV. INFIDELITY OF COVARIANT CODES FOR A CONTINUOUS SYMMETRY Our first main result is a general characterization of how

Question 1085 multiple-choice

Quantum groups and integrable systems are deeply interconnected through algebraic constructions like the ABRR equation and transfer matrices. These structures are fundamental for studying solvable models and their spectral properties in mathematical physics. In the context of quantum groups, which statement correctly describes the role of difference operators D_V,W associated with quantum groups Uq(g)? 1) They provide explicit formulas for the universal R-matrix in classical Lie algebras. 2) They classify simple modules over Hopf algebras via their eigenvalues. 3) They generalize fusion matrices to non-semisimple algebras with non-commuting operators. 4) They encode the deformation of classical trace functions in the semiclassical limit. 5) They form a commuting family of operators acting on spaces of meromorphic functions valued in representations, crucial for integrable systems. 6) They determine the structure constants of the quantum group in the absence of spectral parameters. 7) They serve as intertwiners between irreducible modules without dependence on dynamical variables.

✓ Correct Answer:

The correct answer is 5) They form a commuting family of operators acting on spaces of meromorphic functions valued in representations, crucial for integrable systems..

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i)3)Φw λ−wt(v)Φv λvλ +Φw λ−wt(v)Φv λx2 ivλ = Φw λ−wt(v)Φv λx2 ivλ+((x2 i)3−2λ3)Φw λ−wt(v)Φv λvλ. Summing up these equations, we ﬁnally obtain F= (−2ρ|V+2(ρ,λ)+(/summationdisplay ix2 i)|V−2λ|V+λ2−2/summationdisplay α∈∆+e−α⊗eα)J(λ) (8.4) 24 Combining (8.3) and (8.4) yields (/summationdisplay ix2 i−2(λ+ρ))|VJ(λ)−(wt(v)2−2(λ+ρ,wt(v)))J(λ) =−2(/summationdisplay α∈∆+e−α⊗eα)J(λ) which is equivalent to (8.2). /squaresolid Example. Let us use the recursive procedure in the proof above to com- puteJ(λ) forU(sl2). Setting J(λ) = 1 +/summationtext n≥1J(n)(λ) where J(n)(λ)∈ U′(b−)[−2n]⊗U′(b+)[2n], the ABRR equation reads 1 2[1⊗((λ+1)h−h2 2),J−1] =−(f⊗e)J, which gives the recurrence relation 1⊗((λ+1)n−nh+n2)J(n)= (−f⊗e)J(n−1). Hence J(n)(λ) =(−1)n n!fn⊗(λ−h+2)−1...(λ−h+n+1)−1en. This formula and its quantum analogue were obtained in the pioneering paper [BBB], which was a motivation to the authors of [ABRR]. 8.2. Classical limits of fusion and exchange matrices. The ABRR re- lations can also be used to derive the classical limits of the fusion (and thus of the exchange) matrices. Setting q=e−γ/2, rescaling λ/mapsto→λ γand considering the limit γ→0 yields the following classical version of the ABRR equation: Ad(1⊗e−λ)j(λ)−j(λ) =/summationdisplay α∈∆+e−α⊗eα, which admits the unique lower triangular solution j(λ) =−/summationdisplay α∈∆+e−α⊗eα 1−e−(α,λ). From this we deduce r(λ) =r21+j(λ)−j21(λ) =1 2Ω+1 2/summationdisplay α>0cotanh/parenleftbigg1 2/a\}bracketle{tα,λ/a\}bracketri}ht/parenrightbigg (eα⊗e−α−e−α⊗eα) which is consistent with Proposition 3.3. ThecaseofasimpleLiealgebra giscompletelyanalogous;theclassicalversion of the ABRR equation is [j(λ),λ⊗1] =/summationdisplay α∈∆+e−α⊗eα, which admits the unique solution j(λ) =−/summationdisplay α∈∆+e−α⊗eα (λ,α). This yields Corollary 3.1. 25 9 Transfermatrices andGeneralized Macdonald- Ruijsenaars equations 9.1. Transfer matrices. We ﬁrst recall the well-known transfer matrix con- struction. LetAbe a Hopf algebra with a commutative Grothendieck ring, and let R ∈A⊗Abe an element such that (∆ ⊗1)(R) =R13R23. A basic example is: Ais quasitriangular, Ris its universal R-matrix. For any ﬁnite-dimensional representation πV:A→EndVofA, set TV= Tr|V(πV⊗1)(R)∈A. These elements are called transfer matrices. Lemma 9.1. For any ﬁnite-dimensional A-modules V,Wwe have TVTW= TV⊗W=TWTV. Proof.By deﬁnition we have (πV⊗W⊗1)R= (πV⊗πW⊗1)(∆⊗1)R= (πV⊗1)R13(πW⊗1)R23, which implies the ﬁrst equality. The second equality follows from the co mmu- tativity of the Grothendieck ring. /squaresolid Thetransfermatrixconstructiongivesrisetointerestingexample sofquantum integrable systems which arise in statistical mechanics. For example , ifAis the quantum aﬃne algebra or the elliptic algebra, one gets transfer mat rices of the 6-vertex and 8-vertex models, respectively. We adapt the notion of transfer matrices in our dynamical setting in the following way. Let gbe a simple Lie algebra and let Uq(g) be the associated quantum group. For any two ﬁnite-dimensional Uq(g)-modules VandWlet RV W(λ) be the exchange matrix. It is more convenient to work with the shif ted exchange matrix R(λ) =R(−λ−ρ). LetFVbe the space of V[0]-valued meromorphic functions on h∗. Forν∈h∗ letTν∈End(FV) be the shift operator ( Tνf)(λ) =f(λ+ν). As pointed out in [FV3], the role of the transfer matrix is played by the following diﬀere nce operator DV W=/summationdisplay νTr|W[ν](RWV(λ))Tν. It followsfrom the dynamical 2-cocyclecondition forfusion matrice s(see Propo- sition 2.3) that for any Uq(g)-modules U,V,Wwe have DU V⊗W=DU VDU W=DU WDU V. Hence{DU W}span a commuting family of diﬀerence operators acting on FU. 26 9.2. Weighted trace functions. LetVbeaﬁnite-dimensional Uq(g)-module. Recall that, for any homogeneous vector v∈V[ν] and for generic µ∈h∗there exists a unique intertwiner Φv µ:Mµ→Mµ−ν⊗Vsuch that /a\}bracketle{tv∗

Question 1086 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) on finite groups utilize the group Fourier transform and representation theory to extract information about hidden subgroups. Measurement strategies, such as weak and strong sampling, play a pivotal role in determining the efficiency and efficacy of these algorithms. In quantum algorithms addressing the Hidden Subgroup Problem for nonabelian groups such as those arising in Graph Automorphism, which of the following choices best explains why strong sampling may offer advantages over weak sampling, and what essential factor determines its success? 1) Strong sampling always distinguishes between all subgroups regardless of basis due to higher dimensionality. 2) Weak sampling provides full information when combined with classical post-processing for any group. 3) The probability of observing representations in strong sampling is independent of the chosen measurement basis. 4) Weak sampling can efficiently reconstruct non-normal subgroups in any nonabelian group. 5) Random selection of measurement basis in strong sampling guarantees efficient HSP solutions. 6) The projection operator rank is irrelevant when measuring quantum states in the Fourier basis. 7) Strong sampling can potentially solve the HSP for nonabelian groups if the measurement basis is carefully chosen to align with subgroup structure and extract distinguishing information.

✓ Correct Answer:

The correct answer is 7) Strong sampling can potentially solve the HSP for nonabelian groups if the measurement basis is carefully chosen to align with subgroup structure and extract distinguishing information..

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ρ,ˆφ(ρ) denotes the Fourier transform of φatρand is deﬁned by ˆφ(ρ)=/radicalBigg dρ |G|/summationdisplay gφ(g)ρ(g). Note that φtakes values in Cwhile ρis matrix-valued. It is a fact that a ﬁnite group has a ﬁnite number of distinct irreducible representations up to isomorphism(i.e., up to a unitary change of basis). The Fourier transform of a function φ:G→ Cis then the collection of matrices ˆφ(ρ), taken over all nonisomorphic irreducible representations ρ. Fixing a group Gand a subgroup H, we shall focus primarily on the functions Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 942 MOORE, ROCKMORE, RUSSELL, AND SCHULMAN ϕc:G→Cof the form ϕc(g)=/braceleftBigg 1//radicalbig |H|ifg∈cH, 0 otherwise, corresponding to the ﬁrst register of the state ψ3resulting from Step 3 above, which is a uniform superposition over the coset cH. The Fourier transform of such a function is /hatwiderϕc(ρ)=/radicalBigg dρ |G||H|ρ(c)·/summationdisplay h∈Hρ(h). Note, as above, that /hatwiderϕc(ρ)i sa dρ×dρmatrix. For any subgroup H, the sum/summationtext hρ(h) is precisely |H|times a projection operator (see, e.g., [13]); we write /summationdisplay hρ(h)=|H|πH(ρ). With this notation, we can express /hatwiderϕc(ρ)a s√nρρ(c)·πH(ρ), where nρ=dρ|H|/|G|. For a d×dmatrix M,w el e t /bardblM/bardbldenote the matrix norm given by /bardblM/bardbl2=tr/parenleftbig M†M/parenrightbig =/summationdisplay ij|Mij|2, where M†denotes the conjugate transpose of M. Then the probability that we observe the representation ρis /bardbl/hatwiderϕc(ρ)/bardbl2=/vextenddouble/vextenddouble√nρρ(c)πH(ρ)/vextenddouble/vextenddouble2 =nρ/bardblπH(ρ)/bardbl2 =nρrkπH(ρ), (1) where rkπH(ρ) denotes the rank of the projection operator πH(ρ). See [13] for more discussion. 1.3. Weak vs. strong sampling and the choice of basis. Hallgren, Russell, and Ta-Shma [13] show that by measuring only the names of representations—the so- called weak standard method in the terminology of [10]—it is possible to reconstruct normal subgroups (and thus solve the HSP for Hamiltonian groups , all of whose subgroups are normal). More generally, this method reconstructs the normal core of a subgroup, i.e., the intersection of all its conjugates. On the other hand, they showthat this is insuﬃcient to solve Graph Automorphism, since even in an information-theoretic sense this method cannot distinguish between the trivial subgroup of S nand subgroups of order 2 consisting of the identity and an involution. Therefore, in order to solve the HSP for nonabelian groups, we need to measure not just the name of the representation we are in but also the row and column. Inorder for this measurement to be well deﬁned, we need to choose a basis for U( d ρ) for each ρ. Grigni, Schulman, Vazirani, and Vazirani [10] call this the strong stan- dard method . They show that if we measure using a uniformly random basis, then trivial and nontrivial subgroups are still information-theoretically indistinguishable.However, they leave open the question of whether the strong standard method with Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE POWER OF STRONG FOURIER SAMPLING 943 a clever choice of basis, rather than a random one, allows us to solve the HSP in nonabelian groups, yielding an algorithm for Graph Automorphism. Indeed, from a computational perspective the representation theory of a ﬁnite group Gdoes distinguish certain

Question 1087 multiple-choice

Quantum algorithms often leverage group theory and Fourier analysis to solve problems such as integer factoring and discrete logarithms, threatening classical cryptographic schemes. The structure and properties of groups like Z*_p and choices in algorithm parameters are foundational for their efficiency and correctness. For a prime p, which statement correctly describes the group Z*_p under multiplication and its connection to the discrete logarithm problem? 1) Z*_p under multiplication includes zero and has no inverses for its elements. 2) Z*_p under addition is isomorphic to Z_p and used for the discrete logarithm problem. 3) Z*_p forms a ring but not a group under multiplication when p is prime. 4) Every element x in Z*_p can be written as x = g^y for some generator g, but y is not unique. 5) Z*_p under multiplication is a group when p is prime; each nonzero element has a unique discrete logarithm with respect to a generator g. 6) Z*_p is not closed under multiplication for any prime p. 7) The discrete logarithm problem in Z*_p is efficiently solvable by classical algorithms for large primes.

✓ Correct Answer:

The correct answer is 5) Z*_p under multiplication is a group when p is prime; each nonzero element has a unique discrete logarithm with respect to a generator g..

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someq. Ifqis not an exact multiple of (the unknown) r i.e.q=Ar+tfor some 0

Question 1088 multiple-choice

Quantum algorithms have demonstrated notable efficiency in solving problems involving knot invariants, particularly through the use of algebraic structures such as Hecke algebras and braid groups. These approaches leverage representation theory and combinatorial methods to perform computations that are classically intractable. In the context of quantum algorithms for approximating knot invariants, how are irreducible representations of Hecke algebras Hn(q) typically labeled to organize the structure of braid group representations? 1) By tensor product rules 2) By eigenvalues of braid generators 3) By Schur polynomials 4) By root systems of Lie algebras 5) By Dynkin diagrams 6) By cyclic subgroups 7) By partitions represented as Young diagrams

✓ Correct Answer:

The correct answer is 7) By partitions represented as Young diagrams.

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Jones Closure) Given is a braid b∈Bnof lengthm, two braidsx,y∈Bnof lengthO/parenleftbig poly(m)/parenrightbig , positive integers ℓ,pandrwhich satisfy 2p+r= n, andδ >0. The task is to sample from a random variable 0≤Z≤1which is an additive approximation of the absolute value of the Jones po lynomial of the generalized closure, evaluated at e2πi/ℓ, in the sense that Pr/braceleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 [2]p+r−1 ℓ/vextendsingle/vextendsingle/vextendsingleJ/parenleftbig χp,r x,y(b),e2πi/ℓ)/vextendsingle/vextendsingle/vextendsingle−Z/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤δ/bracerightBigg ≥3/4. In Section 6, we prove: Theorem 2.1 Each of the above two problems can be solved in O(poly(m,1/δ))time on a quantum computer. 3 Unitary Jones-Wenzl representations of the braid group Here we summarize the representation theory of the braid gro up that we will need, following [38]. The relevant representations are inherited from repr esentations of a certain family of Hecke algebras Hn(q) which generalize the group algebra CSnof the symmetric group Sn. In Section 5, we will show how these representations give rise t o useful formulae for evaluations of the Jones and one-variable HOMFLYPT polynomials at roots of unity. 8The Jones Polynomial: Quantum Algorithms and Applications in Quantum Complexity Theory Fig. 3.1. Young diagram for the partition λ= [5,3,2]. The Hecke algebra It is well-known that the group algebra CSnof the symmetric group Snon{1,2,...,n }objects has a presentation in terms of generators {1,s1,s2,...,s n−1}, wheresiis the involution which swapsiandi+ 1. These satisfy the relations s2 i= 1 (3.1) sisjsi=sjsisj,|i−j|= 1 (3.2) sisj=sjsi,|i−j|>1. (3.3) For everyq∈C×, theHecke algebra Hn(q)of typeAn−1is deﬁned to be the algebra over C generated by 1 and {g1,...,g n−1}satisfying the relations g2 i=gi(q−1) +q (3.4) gigjgi=gjgigj |i−j|= 1 (3.5) gigj=gjgi |i−j|>1. (3.6) Note that this is a deformation of the group algebra CSnof the symmetric group Sn, which is obtained when q= 1. While it is known that Hn(q)≃CSnwheneverqis not a root of unity, we will rather be interested in the cases when qis a primitive root of unity, where the representation theory is slightly more subtle than that of CSn. One may represent the braid groupBninside the Hecke algebra Hn(q) by simply mapping each generator σiofBnto the generatorgiofHn(q). Indeed, the representations of Bnwe will use in this paper are induced by representations of Hn(q) on suitable ﬁnite-dimensional Hilbert spaces. As with the braid groups, for each n,Hn(q) is isomorphic to the subalgebra of Hn+1(q) generated by 1 and {g1,...,g n−1}. We use the same notation for the inclusion maps as with the br aid groups, writingι:Hn(q)→Hn+1(q). One important distinction is that, writing 1 nfor the identity inHn(q), we have ι(1n) = 1 n+1. We will therefore often omit the subscript for the identity element of each Hn(q) whenever there is no cause for confusion. Young diagrams and tableaux As withSn, the unitary irreps of Bnare labeled by partitionsλ= [λ1,λ2,...,λ n] satisfying λ1≥λ2≥ ··· ≥λn≥0 and/summationtext iλi=n. Throughout, we consider all partitions obtained by adding or deleting trailing zeros to be equivalent. We will i dentify any such partition with its Young diagram as pictured in Figure 3.1 for the partition λ= [5,3,2]. Note that the diagram has one row for each partof the

Question 1089 multiple-choice

Quantum algorithms have demonstrated significant advantages in solving problems related to hidden structures in finite groups, which are fundamental in cryptography and computational complexity theory. The hidden shift problem explores the identification of a shift between functions defined on finite abelian groups using quantum resources. Which property of a bent function most directly enables a quantum algorithm to determine the hidden shift with probability one when given oracle access to a shifted function and knowledge of the Fourier transform of the original function? 1) The function's invariance under all group automorphisms 2) The flatness of its Fourier spectrum across all group elements 3) Its periodicity with respect to the shift value 4) Its support limited to a proper subgroup of the group 5) The function's ability to factor through a quotient group 6) Its constant magnitude on the group domain 7) The existence of a classical algorithm that finds the shift efficiently

✓ Correct Answer:

The correct answer is 2) The flatness of its Fourier spectrum across all group elements.

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Title: Hidden shift problem for complex functions Year: 2025 Paper ID: c83c6468893a80487eec5cc3db61d26bea5b02d3 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c83c6468893a80487eec5cc3db61d26bea5b02d3 Abstract: We study quantum algorithms for the hidden shift problem of complex scalar- and vector-valued functions on finite abelian groups. Given oracle access to a shifted function and the Fourier transform of the unshifted function, the goal is to find the hidden shift. We analyze the success probability of our algorithms when using a constant number of queries. For bent functions, they succeed with probability 1, while for arbitrary functions the success probability depends on the `bentness'of the function.

Question 1090 multiple-choice

Classical simulation of quantum circuits is essential for benchmarking algorithms and hardware, especially in the era of Noisy Intermediate-Scale Quantum (NISQ) devices. Techniques for reducing the resource demands of simulations are crucial as quantum circuits grow in complexity and depth. Which compression approach enables high-fidelity simulation of the Quantum Fourier Transform using only 7 significand bits per amplitude's real and imaginary parts, and also provides a systematic method to control simulation fidelity based on bits per amplitude? 1) Lossless Huffman coding 2) Run-length encoding applied to sparse state vectors 3) Scalar quantization without block grouping 4) Principal component analysis on amplitudes 5) Vector quantization of amplitude blocks 6) Error-correcting code overlays on state vectors 7) Adaptive floating-point precision per gate layer

✓ Correct Answer:

The correct answer is 5) Vector quantization of amplitude blocks.

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Title: Lossy Compression for Schr\"odinger-style Quantum Simulations Year: 2024 Paper ID: 6ffca7d7fe0e57866ed8d1357b0d809b8498a64e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/6ffca7d7fe0e57866ed8d1357b0d809b8498a64e Abstract: Simulating quantum circuits on classical hardware is a powerful and necessary tool for developing and testing quantum algorithms and hardware as well as evaluating claims of quantum supremacy in the Noisy Intermediate-Scale Quantum (NISQ) regime. Schr\"odinger-style simulations are limited by the exponential growth of the number of state amplitudes which need to be stored. In this work, we apply scalar and vector quantization to Schr\"odinger-style quantum circuit simulations as lossy compression schemes to reduce the number of bits needed to simulate quantum circuits. Using quantization, we can maintain simulation fidelities $>0.99$ when simulating the Quantum Fourier Transform, while using only 7 significand bits in a floating-point number to characterize the real and imaginary components of each amplitude. Furthermore, using vector quantization, we propose a method to bound the number of bits/amplitude needed to store state vectors in a simulation of a circuit that achieves a desired fidelity, and show that for a 6 qubit simulation of the Quantum Fourier Transform, 15 bits/amplitude is sufficient to maintain fidelity $>0.9$ at $10^4$ depth.

Question 1091 multiple-choice

Finite group lattice models play a foundational role in understanding topological quantum order and quantum error correction in two-dimensional systems. These models use group theory and topology to construct exactly solvable Hamiltonians on lattices embedded in surfaces. In a finite group lattice model defined on a closed oriented surface Σ with a finite group G and a lattice Λ, which of the following best describes the dimension of the ground space of the frustration-free Hamiltonian constructed from commuting vertex and plaquette projectors? 1) The number of elements in the group G 2) The number of edges in the lattice Λ 3) The number of irreducible representations of G 4) The genus of the surface Σ 5) The total number of plaquettes on the lattice 6) The number of G-conjugacy classes of homomorphisms from π₁(Σ) to G 7) The Euler characteristic of Σ

✓ Correct Answer:

The correct answer is 6) The number of G-conjugacy classes of homomorphisms from π₁(Σ) to G.

📚 Reference Text:

ﬁnite group lattice models [13], which we will see instantiate the concepts of the previous sections. Let Gbe a ﬁnite group, Σbe an oriented 2D surface with no boundary, andΛ = (V,E,F )5be an arbitrary oriented lattice on Σ, whereV,E, andFare the sets of vertices, oriented edges, and plaquettes of the lattice, respectively. Then, for every e∈E, setHe=C[G] the group algebra of G, i.e.Heis spanned by the basis {|g/angbracketright:g∈G}. The overall Hilbert space is given by H≡/circlemultiplytext e∈EHe. A natural basis for this Hilbert space consists of tensor products of the form|g/angbracketright≡/circlemultiplytext e∈E|ge/angbracketright; we refer to this as the group basis . We deﬁne the sitesofΛto be the set of pairs s= (v,p)∈V×Fsuch thatpis adjacent tov. Given a site s= (v,p)and two elements g,hinG, we deﬁne two sets of operators: gauge transformations Av(g)and magnetic operators B(v,p)(h). Their action is most readily seen in the group basis. For example, Av(g)acts on the edges which touch vby multiplication by gon the left, or multiplication by g−1on the right, depending on whether the edge is oriented away from or towardsv. The magnetic operator B(v,p)(h)computes the product of the group elements sitting on the edges of p, and compares it to h, annihilating the state if there is a discrepancy, while stabilizing it if the group elements agree. The prescription for computing the product is to start atvand move around pcounter-clockwise, inverting the group element associated to an edge if 3SoBhas size (/lscript+ 2)×(/lscript+ 2). 4For deﬁning TQO-1, we do not need that His frustration free. 5We abbreviate V≡V(Λ),E≡E(Λ), andF≡F(Λ). Accepted in Quantum2020-09-17, click title to verify. Published under CC-BY 4.0. 5 that edge is oriented opposite relative to the direction of travel. For example, Av(g)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglevg1 g2g3g4/angbracketrightBigg ≡/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglevgg1 gg2gg3g4g−1/angbracketrightBigg B(v,p)(h)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle vh1h2h3 h4p/angbracketrightBigg ≡δh,h1h−1 2h3h4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle vh1h2h3 h4p/angbracketrightBigg(10) whereδg,his the Kronecker delta symbol. Some basic facts follow: Av(g)Av(h) =Av(gh) B(v,p)(g)B(v,p)(h) =δg,hB(v,p)(h) Av(g)B(v,p)(h) =B(v,p)(ghg−1)Av(g).(11) We can now deﬁne the vertex and plaquette operators as Av≡1 |G|/summationdisplay g∈GAv(g) Bp≡B(v,p)(1),(12) wherevis any vertex adjacent to pand1∈Gis the identity element.6It is easily veriﬁed that for allv∈V,p∈F,AvandBpare commuting projectors. The Hamiltonian of this system is deﬁned in terms of these projectors: H=/summationdisplay v∈V(1−Av) +/summationdisplay p∈F(1−Bp). (13) This Hamiltonian is frustration-free and the ground space is simply given by Vg.s.≡{|ψ/angbracketright∈H:Av|ψ/angbracketright=Bp|ψ/angbracketright=|ψ/angbracketright,∀v∈V, p∈F}. (14) In gauge-theoretic language, where we think of a state as specifying the ﬁeld conﬁguration of aGvector potential, the condition that Av|ψ/angbracketright=|ψ/angbracketrightmeans that|ψ/angbracketrightis gauge invariant, while B(v,p)(h)|ψ/angbracketright=|ψ/angbracketrightmeans that the connection is ﬂat. Now, due to the identities Av(g)Av=Av B(v,p)(h)Bp=δh,1Bp,(15) the action of the Av(g)andB(v,p)(h)operators on the ground space is simply Av(g)|ψ/angbracketright=|ψ/angbracketright B(v,p)(h)|ψ/angbracketright=δh,1|ψ/angbracketright(16) for all|ψ/angbracketright∈Vg.s.. In Section 2.4, we show that the dimension of Vg.s.is the number of orbits of Hom(π1(Σ),G)under the action of Gby conjugation, where π1(Σ)is the fundamental group of Σ. We recall that the toric code is the ground space of the above Hamiltonian for Σ =T2the two-torus, ΛanL×Lperiodic square lattice, and G=Z2. In this case, the orientations of the edges inEdoes not matter and we can identify C[G]with a qubit, with the two elements 0,1of Z2corresponding to |0/angbracketright,|1/angbracketrightof

Question 1092 multiple-choice

In quantum computing, the Hidden Subgroup Problem (HSP) for non-Abelian groups such as the symmetric group Sn and projective linear groups PSL(2, Fq) has significant implications for algorithms addressing problems like graph isomorphism. Representation theory and combinatorial estimates play a critical role in determining the quantum resources required to solve these problems efficiently. For algorithms solving the Hidden Subgroup Problem in the group PSL(2, Fq) using coset states, what is the asymptotic lower bound on the number of joint entangled measurements required, in terms of the field size parameter q? 1) k = Ω(q) 2) k = Ω(n) 3) k = Ω(n log n) 4) k = Ω(log n) 5) k = Ω(√q) 6) k = Ω(q^2) 7) k = Ω(log q)

✓ Correct Answer:

The correct answer is 7) k = Ω(log q).

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of irreps of Gis|/hatwideG|=/parenleftbigp(n) 2/parenrightbig +2p(n)≤p(n)2,wherep(n)denotes the number of partitions of n. Inorder toapplyTheorem 2wechoose ε=n−αnforsomeconstant α>0tobedetermined later. Then Sε=/braceleftBig σ∈/hatwideG:|χσ(h)| dσ≥ε}={ϑλ,ϑ′ λ:dλ≤nαn/bracerightBig .Hence weobtain that /summationdisplay σ∈Sεdσ·|χσ(h)| ≤2/summationdisplay λ∈/hatwiderSn,dλ≤nαnd2 λ·dλ≤p(n)n2αn·nαn≤n3αneν√n. Herewehave estimated the partition number as p(n) =O(eν√n),whereν=π/radicalBig 2 3. Wealso compute that /summationdisplay σ∈Sεd2 σ≤2/summationdisplay λ∈/hatwiderSn,dλ≤nαnd4 λ≤p(n)n4αn≤n4αneν√n. Inorder toapply Theorem 2,wenow deﬁne α:= 1/4and obtain that δ1≤ε+/parenleftBigg/summationdisplay σ∈Sεdσ|χσ(h)|/parenrightBigg/parenleftBigg |/hatwideG| |G|/parenrightBigg1/2 ≤n−αn+n3αneν√n/parenleftbiggp(n)2 2(n!)2/parenrightbigg1/2 ≤n−1/4n+n3/4ne2ν√n √ 2n!=n−Ω(n), where wehave used the fact that n!≥(n/e)nfor largen. For theparameter δ2in Theorem 2 weobtain δ2= 2k(1+2kε)δ1/2 1+3kε+3k/summationtext σ∈Sεd2 σ |G| ≤2k/parenleftBig 1+2kn−1/4n/parenrightBig n−Ω(n)+3kn−1/4n+3knneν√n 2(n!)2= 2kn−Ω(n). 16 Hence, wehave proved the following corollary toTheorem 2: Corollary 14. Any algorithm operating on coset states that solves the hidd en subgroup problem in G= Sn≀S2in polynomial time has to make joint measurements on k= Ω(nlogn)coset states. The same is true for any algorithm that solves the hidden subgroup probl em inSnusing coset states. Also, any efﬁcient algorithm for isomorphism of two n-vertex graphs that uses the standard reduction to HSP in S2nand then uses coset states to solve the HSP needs to make measurements entangled across k= Ω(nlogn)coset states. Finally, we remark that if we apply Theorem 2 to all the full-s upport involutions in S2n, we only get a lower bound of k= Ω(n). This is because we use Roichman’s [Roi96] upper bound on the normalized characters of S2nin order to deﬁne Sǫ, as in [MRS05], and Roichman’s bound is always at least e−O(n). Since the involutive swaps form an exponentially small frac tion of all the full-support involutions, it is possible that an average hidden full-support involution ma y be distinguishable from the hidden identity subgroup by an O(n)-entangled POVM acting on nO(1)-coset states. However, no such POVM is known and the best upper bound for this problem continues to bethe k=O(nlogn)information-theoretic one. 4.2 The projective lineargroups PSL(2,Fq) The representation theory of the projective linear groups G= PSL(2,Fq)over any ﬁnite ﬁeld Fqis well- known. The following is a summary of the necessary results, f or more details we refer to Appendix B. We treat the cases qeven andqodd separately. In case qodd we have that |PSL(2,Fq)|=q(q2−1) 2. There is oneconjugacy class ofq(q±1) 2involutions (depending onwhether q≡1or3modulo4); lethdenote aﬁxed member of this conjugacy class. The degrees of the irreps are given by1,q,q±1, andq±1 2. The character values|χ(h)|can be upper bounded by 1,1,2, and1, respectively. There is a total number of |/hatwideG|=q+5 2 irreps. In order to apply Theorem 2,wechoose ε=2 q−1. Then Sε={σ∈/hatwideG:|χσ(h)| dσ≥ε}={11} contains only the trivial irrep. With this choice of thepara meterεwehave that /summationdisplay σ∈Sεdσ·|χσ(h)|= 1,/summationdisplay σ∈Sεd2 σ= 1,and/parenleftBigg |/hatwideG| |G|/parenrightBigg1/2 =/parenleftbigg(q+5)/2 q(q2−1)/2/parenrightbigg1/2 =O(q−1). Hence, wecan bound the parameter δ1used in Theorem 2asfollows: δ1≤ε+/parenleftBigg/summationdisplay σ∈Sεdσ|χσ(h)|/parenrightBigg/parenleftBigg |/hatwideG| |G|/parenrightBigg1/2 ≤2 q−1+1·O(q−1) =O(q−1). For theparameter δ2weobtain δ2= 2k(1+2kε)δ1/2 1+3kε+3k/summationtext σ∈Sεd2 σ |G| ≤2k/parenleftbigg 1+2k2 q−1/parenrightbigg O(q−1/2)+3k2 q−1+3k1 q(q2−1)/2≤2kO(q−1/2). 17 The caseq= 2n, where|PSL(2,F2n)|=|SL(2,F2n)|=q(q2−1), can be treated similarly. There we useε=1 q−1which implies that δ2≤2kO(q−1/2). Hence, using Theorem 2 we have shown the following result: Corollary 15. Letqbe a prime power. Then any algorithm operating on

Question 1093 multiple-choice

In theoretical physics and mathematics, BRST quantization is a standard approach for handling gauge symmetries in systems described by Lie algebras. The cohomology of the BRST complex plays a crucial role in identifying physical and mathematical invariants in these systems. For a principal sl2 embedding in a Lie algebra, which statement best describes the cohomology of the associated BRST complex? 1) All cohomology groups vanish, indicating no gauge-invariant content. 2) Only the first cohomology group is non-zero and corresponds to the universal enveloping algebra. 3) The BRST complex has non-vanishing cohomology in all positive degrees. 4) The zeroth and first cohomology groups are both isomorphic to the center of the universal enveloping algebra. 5) The cohomology is trivial for all embeddings, including non-principal cases. 6) Higher cohomology groups classify non-trivial gauge anomalies in the system. 7) Only the zeroth cohomology group is non-vanishing and isomorphic to the center of the universal enveloping algebra.

✓ Correct Answer:

The correct answer is 7) Only the zeroth cohomology group is non-vanishing and isomorphic to the center of the universal enveloping algebra..

📚 Reference Text:

ts is more cumbersome than in the presence of ﬁrst class constraints, it requires t he introduction of extra aux- iliary ﬁelds to change the second class constraints into ﬁrs t class constraints. However, it was shown above that for arbitrary embeddings of sl2one can always choose a set of constraints that is completely ﬁrst class and leads to the sameWalgebras as the set{φj,m (µ)}. To perform a BRST quantization of the ﬁnite Walgebras we use these alternative systems of ﬁrst class constraints. 5.2. The BRST Complex Consider the map χ:g+→C deﬁned by χ(El+sk−1,l+sk) = 1 forl≥1,1≤k≤nl−1 andχ(Eij) = 0 otherwise. Because the constraints {Jl+sk−1,r+sk−δr,l}l≥1;1≤k≤nl−1;r>0 are ﬁrst class, χdeﬁnes a one-dimensional representation of g+. In terms of χ, the constraints can be written as π+(J) =χ(π+(J)), whereπ+again denotes the projection g→g+. It is this form of the constraints that we will use. Furtherm ore we will take ¯h= 1 for simplicity; the explicit ¯ hdependence can be determined afterwards. As before latin indices will be supposed to run over a basis taof the Lie algebra g, greek indices run over a basis of g+and barred greek indices (like ¯ α) run over a basis of g−⊕g0. Indices can again be raised and lowered by use of the Cartan K illing metric. The basis elements ta,tαandt¯αare as before so chosen that they have a well deﬁned degree with respect to adδ. To set up the BRST framework we need to introduce anticommuti ng ghosts and antighostscαandbα, associated to the constraints that we want to impose [25]. T hey satisfybαcβ+cβbα=δα βand generate the Cliﬀord algebra Cl(g+⊕g∗ +). The quantum Kirillov algebra is just the universal enveloping algebra Ug, and the total space on which the BRST operator acts is Ω = Ug⊗Cl(g+⊕g∗ +). AZgrading on Ω is deﬁned by deg(Ja) = 0, deg( cα) = +1 and deg( bα) =−1, and we can decompose Ω = ⊕kΩk accordingly. The BRST diﬀerential on Ω is given by d(X) = [Q,X], whereQis the BRST charge Q= (Jα−χ(Jα))cα−1 2fαβ γbγcαcβ. (5.2) 37 and [.,.] denotes the graded commutator (as it always will from now on ) [A,B] =AB−(−1)deg(A).deg(B)BA (5.3) Note that deg( Q) = 1. This is the standard BRST complex associated to the ﬁrst clas s constraints of the previous section. Of interest are the cohomology groups of t his complex, Hk(Ω;d). The zeroth cohomology group is the quantization of the classica l ﬁniteWalgebra. Because the gauge group Hin (39) acts properly on gc, we expect the higher cohomologies of the BRST complex to vanish, as they are generically related t o singularities in the quotientgc/H. In the mathematics literature the cohomology of the BRST co mplex is called the Hecke algebra H(g,g+,χ) associated to g,g+,χ. Hecke algebras related to arbitrary sl2embeddings have not been computed, apart from those related to the principalsl2embeddings. In that case it was shown by Kostant [26] that the only nonvanishing cohomology is H0(Ω;d) and that it is isomorphic to the center of the universal enveloping algebra. Recall that

Question 1094 multiple-choice

In universal algebra, congruences, homomorphisms, and clones play a central role in understanding the structure and properties of algebras and their connections to computational models such as quantum circuits. Lattice-theoretic notions provide a powerful framework for investigating these algebraic constructs. Which property must an algebra A possess if its lattice of congruences Con contains only the trivial congruence (which relates only identical elements) and the universal congruence (which relates all pairs of elements)? 1) It must be distributive. 2) It must be simple. 3) It must be a Boolean algebra. 4) It must be congruence modular. 5) It must have a functionally complete clone. 6) It must admit only unitary operations. 7) It must be a quantum algebra.

✓ Correct Answer:

The correct answer is 2) It must be simple..

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AandB).Thisis indicatedbywriting φ:A→B.Similartogroupsandrings, congruencesare precisely the kernelsofhomomorphisms ker(φ)=/braceleftbig (a,b) ∈A2|φ(a)=φ(b)/bracerightbig , but in contrast to groupsthe kernelis a binary relation. Every algebraAhas at least two (possibly non-distinct) congruences — the trivial congruence 0and the universal congruence 1, 0=/braceleftbig (a,a) ∈A2|a∈A/bracerightbig and 1=A2=/braceleftbig (a,b) ∈A2|a,b∈A/bracerightbig , Aswithgroups,ifthesearetheonlycongruencesof A,then we callAsimple. The set of congruencesof Ais denoted Con (A), and can beordered by inclusion, in whichcase 1and0are the max- imal and minimal elements, respectively. More than this, The Hidden SubgroupProblem forUniversalAlgebras LICS’20,July8–11,2020,Saarbrücken,Germany Con(A)has the structure of a lattice, where the operations of∧and∨are deﬁned θ∧ψ≔θ∩ψandθ∨ψ≔Cg/parenleftbigθ∪ψ/parenrightbig forθ,ψ∈Con(A). Perhaps surprisingly, diﬀerent kinds of lattice-theoretic structure in Con (A)are in correspondence with algebraic structure in A, and this correspondence is a central area of study in the ﬁeld. Of particular interest for usis whenCon (A)isadistributive lattice , meaning α∧/parenleftbigβ∨γ/parenrightbig=/parenleftbigα∧β/parenrightbig∨/parenleftbigα∧γ/parenrightbig for allα,β,γ∈Con(A). In this case, we say that Aiscon- gruencedistributive . 2.2 Clones Given a set Fof operations on a common domain A, new operationsmaybeproducedbycomposingoperations.For- mally, iff∈Fisn-ary and/afii10069.itali∈Fiski-ary fori∈ [n], then h(x11,...,xnkn) ≔f/parenleftbig/afii10069.ital1(x11,...,x1ki),...,/afii10069.italn(xn1,...,xnkn)/parenrightbig is a(/summationtext.1ki)-ary operationof A. New operations of Amay also be produced by variable manipulations. Given n-ary operation f∈Fand any func- tionσ:[n] → [m], h(x1,...,xm)≔f/parenleftbigxσ(1),...,xσ(n)/parenrightbig isanm-aryoperationof A.Lessformally,given n-aryf∈F we may deﬁne operations of equal arity by permuting vari- ables, operations of smaller arity by identifying variables, and operations of larger arity by introducing extraneous variables. Acloneoveradomain Aisasetofoperationsof Awhich isclosedunder compositionandvariablemanipulations(as detailedabove).The clonegeneratedby asetofoperations F, written Clo (F)is the smallest clone containing F. Likewise, thecloneoftermoperations ofthealgebraA,writtenClo (A), is theclone generatedbythe operationsof A. Given a ﬁxed domain A, the set of clones over Acan be ordered by inclusion. As with the set of congruences of an algebra, the set of clones over Aformsa lattice with opera- tions∧and∨deﬁned A ∧B≔A ∩B andA ∨B≔Clo/parenleftbigA ∪B/parenrightbig forclones AandBoverthedomain A.Weextendtheorder- ingonclonestoalgebras BandDoverthecommondomain Aby writingB/√re⌋edesequalDif and only if Clo (B) ⊆Clo(D). IfCis a clone over the domain A, then we will slightly abuse this notation by writing C /√re⌋edesequalBifC ⊆Clo(B), and similarly for B/√re⌋edesequal C.2.3 QuantumComputation Qubitsaretheinformation-theoreticfoundationofquantum computation. A qubit can have a purestate of either 0 or 1, butmayalsobeina superposition of0and1states.Formally, thisisrepresentedbya2-dimensionalnormedcomplexvec- torspace (i.e. a Hilbertspace), B≔C-span/braceleftbig |0/a\}⌊ra⌋ketri}ht,|1/a\}⌊ra⌋ketri}ht/bracerightbig . We adopt the notation ofDirac [ 4] where|α/a\}⌊ra⌋ketri}htdenotes a col- umnvectorand /a\}⌊ra⌋ketle{tα|≔|α/a\}⌊ra⌋ketri}ht†denotesarowvector(“ †”repre- sentsthe Hermitian adjoint). A system of kqubits is represented by the tensor power ofB, writtenB⊗k≔B⊗ ··· ⊗B. This vector space has dimension 2kwith basis/braceleftbig |x1/a\}⌊ra⌋ketri}ht ⊗ ··· ⊗ |xk/a\}⌊ra⌋ketri}ht |xi∈ {0,1}/bracerightbig . Using thenotation |x1x2/a\}⌊ra⌋ketri}ht≔|x1/a\}⌊ra⌋ketri}ht ⊗ |x1/a\}⌊ra⌋ketri}htgives us B⊗k=C-span/braceleftbig |x1···xk/a\}⌊ra⌋ketri}ht |xi∈ {0,1}/bracerightbig . Aquantum state isa vectorinB⊗k, |α/a\}⌊ra⌋ketri}ht=/summationdisplay.1 xi∈{0,1}λx1···xk|x1···xk/a\}⌊ra⌋ketri}htsuch that /⌊ard⌊lα/⌊ard⌊l=1. Theconditionthat /⌊ard⌊lα/⌊ard⌊l=1isequivalentto/summationtext.1|λx1···xk|2=1, and so/parenleftbig|λx1···xk|2/parenrightbig xi∈{0,1}is regarded asa probability distri- bution: when the quantum state |α/a\}⌊ra⌋ketri}htis measured, the pure state|/y.alt1···/y.altk/a\}⌊ra⌋ketri}htwill beobserved with probability |λ/y.alt1···/y.altk|2. The evolution over time of a quantum state |α/a\}⌊ra⌋ketri}htis rep- resented by the action of a unitary operator Uon|α/a\}⌊ra⌋ketri}ht. A givenunitaryoperatorcanbeapproximatedusingproducts and tensor products of operators from the standard quan- tum gate set . Aquantum circuit for the operator Uis this decomposition, typically given as a diagram. Figure 2is an example of such a diagram and Deﬁnition 3.2is the inline formof the samecircuit. One particular

Question 1095 multiple-choice

Quantum Phase Estimation (QPE) is a central algorithm in quantum computing, enabling the extraction of eigenvalues for unitary operators and playing a key role in applications such as factoring and quantum simulation. Implementing QPE on current Noisy Intermediate-Scale Quantum (NISQ) devices presents significant challenges due to hardware limitations. Which strategy is most effective for improving the accuracy of Quantum Phase Estimation on NISQ quantum devices characterized by short coherence times and high gate error rates? 1) Increasing the number of ancillary qubits to boost computational parallelism 2) Replacing all controlled gates with non-controlled gates to reduce complexity 3) Using longer quantum circuits to enable more precise measurements 4) Relying exclusively on classical post-processing to correct errors 5) Reducing the number of controlled rotation gates and phase shift operations in the quantum circuit 6) Performing phase estimation exclusively with the inverse quantum Fourier transform 7) Eliminating entanglement to simplify the quantum system

✓ Correct Answer:

The correct answer is 5) Reducing the number of controlled rotation gates and phase shift operations in the quantum circuit.

📚 Reference Text:

arXiv:1910.11696v1 [quant-ph] 23 Oct 2019An Improved Implementation Approach for Quantum Phase Estimation on Quantum Computers Hamed Mohammadbagherpoor1, Young-Hyun Oh2∗, Patrick Dreher3, Anand Singh2, Xianqing Yu2, Andy J. Rindos2 1,3Department of ECE & CSC, North Carolina State University, Raleigh, NC 27606, USA 2Cloud and Cognitive Software, IBM, Durham, NC 27703, USA Email: ohy@us.ibm.com∗ Abstract —Quantum phase estimation (QPE) is one of the core algorithms for quantum computing. It has been extensiv ely studied and applied in a variety of quantum applications suc h as the Shor’s factoring algorithm, quantum sampling algori thms and the calculation of the eigenvalues of unitary matrices. The QPE algorithm has been combined with Kitaev’s algorithm and the inverse quantum Fourier transform (IQFT) which are util ized as a fundamental component of such quantum algorithms. In th is paper, we explore the computational challenges of implemen ting QPE algorithms on noisy intermediate-scale quantum (NISQ) machines using the IBM Q Experience (e.g., the IBMQX4, 5- qubit quantum computing hardware platform). Our experimen tal results indicate that the accuracy of ﬁnding the phase using these QPE algorithms is severely constrained by the NISQ computer ’s physical characteristics such as coherence time and error r ates. To mitigate these physical limitations, we propose impleme nting a modiﬁed solution by reducing the number of controlled rotat ion gates and phase shift operations, thereby increasing the ac curacy of the ﬁnding phase in near-term quantum computers. I. INTRODUCTION The ﬁrst generation of noisy intermediate-scale quantum (NISQ) [1] computers now provides a framework for re- formulating algorithms originally optimized for digital c om- puters into a form suitable for the new quantum computing hardware platforms. These re-formulations hold the promis e of potentially being able to solve particular problems expo - nentially faster than classical computers and also to explo re regions that are inaccessible using even the most powerful digital high performance computers. A key difference between an algorithm that is formulated for a digital computer versus a quantum computer is that digital computations are modeled on a Load-Run-Read cycle while quantum computers operate on a Prepare-Evolve-Measure cycle. The information ﬂow for digital algorithms assumes that input data in digital bit format is inserted into the sys tem, the program runs and then the output of the program is read. However, in quantum computers the qubit states are prepared as the input, manipulation of the input states is done using t he operators and then the results are measured [2]. As part of th e design for information ﬂow is a quantum computer developer will also incorporate the quantum mechanical properties of both of superposition and entanglement of the qubits [3], [4 ] in order to strive for a quantum advantage over their digital counterparts.Quantum phase estimation (QPE) is the critical building block for various quantum algorithms. In QPE the main ob- jective of quantum phase estimation is to determine the eige n- values of an unitary matrix with an unchanged eigenvector. This technique was described by Kitaev [5]. This procedure is

Question 1096 multiple-choice

In quantum control theory, the implementation of unitary gates such as the quantum Fourier transform (QFT) is subject to constraints arising from both the physical system and the mathematical properties of quantum mechanics. The role of global phase and Hilbert space dimension is central to understanding which gate implementations are feasible and how control strategies can be optimized. Which statement best describes the relationship between the Hilbert space dimension and allowable global phase values in unitary quantum gate implementation? 1) The Hilbert space dimension determines the number of accessible eigenstates but does not influence the global phase. 2) Allowable global phase values are continuous and independent of the Hilbert space dimension. 3) Only spin-½ systems have discrete allowable global phase values due to their two-dimensional Hilbert space. 4) Global phase is irrelevant for all quantum gates and does not affect their implementation time. 5) The Hilbert space dimension sets a discrete set of allowable global phase values for unitary quantum gates. 6) Global phase values are always restricted to real numbers in quantum mechanics. 7) The global phase can take any arbitrary value regardless of Hilbert space properties.

✓ Correct Answer:

The correct answer is 5) The Hilbert space dimension sets a discrete set of allowable global phase values for unitary quantum gates..

📚 Reference Text:

functional to be minimized has the only global minimum and the only global maximum, with the rest of the functional critical points being saddle points [15, 16]. However , the problem of suboptimal minima in optimi zation at certain control field constraint s remains under -investigat ed [17, 18]. Here, we demonstrate that there are local minima related not only to the global phase problem but also to the existence of solutions with the same global phase but different critica l time s cT. As is known [7, 11] , for a system with the traceless Hamiltonian , the unitary quantum gate GU can be implemented just up to the global phase factor ()pi G U T e U , (1) where ()UT is the operator of the system evolution for time T. The global phase can be chosen from the set of values [7] 02 / , 0,1, ..., 1p p N p N       , (2) where N is the Hilbert space dimension for th e system under consideration and 0 is the smallest angle 0[0, ] at which 0det 1i GeU . The numerical calculation s for the quantum Fourier transform (QFT) gate in the system of spin s ½ showed that the minimum time for implementing the gate strongly depends on the global phase [7]. The same result was obtained for the QFT and SWAP gates on two spins ½ [11]. Finally, in study [19], the numerical simulation of the QFT on qudits with N = 3 and 4 represented by quadrupole nuclei with the spins I = 1 and 3/2 , respectively, also showed the strong dependence of the minimum gate duration on the global phase value . In this study , we explore in more detail the effect of the global phase on the minimum time for the system with the spins I = 1, 3/2, 2, and 5/2. Note that the QFT gate is often chosen for testing various control methods , since it plays a key role in many quantum algorithms [1, 2] and has rather a nontrivial form , i.e. is not straightforward to implement [7]. The paper is organized as follows. In Section II, we derive theoretical foundation s for the existence of many solutions in the optimal control task for the case of a traceless Hamiltonian. By an example of the QFT gate on a qutrit (N = 3) represented by a quadrupole nucleus with the spin I = 1 and controlled by rf magnetic field, we find approximate analytical solutions for the control field that correspond to different global phases (2) and the same global phase but with 3 different critical times cT . In Section III, we present numerical data on the effect of the global phase on the minimum time of the QFT gate implementation on the quadrupole nucleus with the spins I = 1, 3/2, 2 , and 5/2 (N = 2I +1). Section IV contains the conclusion s. II.

Question 1097 multiple-choice

Quantum algorithms are increasingly used to solve scientific problems modeled by partial differential equations, such as heat conduction, often relying on discretization techniques and specialized matrix structures. Understanding the properties of circulant matrices and their role in numerical solution methods is essential for analyzing algorithm performance. In the numerical solution of heat conduction problems using discretization on a periodic grid, why is a circulant matrix particularly advantageous for computational efficiency? 1) Because it guarantees the system is always well-conditioned regardless of grid parameters 2) Because it can be diagonalized efficiently by the Fourier transform, enabling fast linear system solutions 3) Because it eliminates the need for boundary condition specification 4) Because its eigenvalues are all equal, simplifying inversion 5) Because it allows direct use of Hadamard gates for state preparation 6) Because it always represents stochastic processes 7) Because it is inherently sparse for any grid size

✓ Correct Answer:

The correct answer is 2) Because it can be diagonalized efficiently by the Fourier transform, enabling fast linear system solutions.

📚 Reference Text:

Rz(π 16) |0⟩H Rz(π 8) |0⟩H Rz(π 4) |0⟩H Rz(π 2) |0⟩H Rz(π 32) FIG. 4. State preparation circuit of |b⟩modeled after the QAOA circuit [35, 36]. (a) /s49 /s50 /s51 /s52 /s53 /s54/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48/s76/s111/s115/s115 /s84/s114/s117/s110/s99/s97/s116/s105/s111/s110/s32/s116/s104/s114/s101/s115/s104/s111/s108/s100/s32/s40/s84/s41/s32/s77/s97/s116/s114/s105/s120/s32/s109/s117/s108/s116/s105/s112/s108/s105/s99/s97/s116/s105/s111/s110 /s32/s72/s121/s98/s114/s105/s100/s32/s97/s108/s103/s111/s114/s105/s116/s104/s109/s32/s45/s32/s113/s105/s115/s107/s105/s116/s32/s97/s101/s114/s32/s115/s105/s109/s117/s108/s97/s116/s111/s114 /s32/s81/s117/s97/s110/s116/s117/s109/s45/s105/s110/s115/s112/s105/s114/s101/s100/s32/s97/s108/s103/s111/s114/s105/s116/s104/s109 (b) /s49 /s50 /s51 /s52/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57/s48/s46/s49/s50/s48/s46/s49/s53/s76/s111/s115/s115 /s84/s114/s117/s110/s99/s97/s116/s105/s111/s110/s32/s116/s104/s114/s101/s115/s104/s111/s108/s100/s32/s40/s84/s41/s32/s77/s97/s116/s114/s105/s120/s32/s109/s117/s108/s116/s105/s112/s108/s105/s99/s97/s116/s105/s111/s110 /s32/s72/s121/s98/s114/s105/s100/s32/s97/s108/s103/s111/s114/s105/s116/s104/s109/s32/s45/s32/s113/s105/s115/s107/s105/s116/s32/s105/s98/s109/s45/s112/s101/s114/s116/s104/s32/s104/s97/s114/s100/s119/s97/s114/s101 (c) /s49/s48/s48 /s49/s48/s49 /s49/s48/s50 /s49/s48/s51 /s49/s48/s52/s49/s48/s48/s49/s48/s49/s49/s48/s50/s84/s114/s117/s110/s99/s97/s116/s105/s111/s110/s32/s116/s104/s114/s101/s115/s104/s111/s108/s100/s32/s40/s84/s41 /s67/s111/s110/s100/s105/s116/s105/s111/s110/s32/s110/s117/s109/s98/s101/s114/s32/s40 /s41/s32/s65/s109/s112/s108/s105/s116/s117/s100/s101/s32/s101/s110/s99/s111/s100/s101/s100/s32/s115/s116/s97/s116/s101 /s32/s71/s72/s90/s32/s115/s116/s97/s116/s101 /s32/s90/s101/s114/s111/s32/s115/s116/s97/s116/s101 FIG. 5. Numerical results of hybrid and quantum-inspired algorithms proposed in our paper for solving heat conduction linear systems. First, we show that simulation results of our hybrid and quantum-inspired algorithms can provide decent approximations to the results obtained by matrix multiplication in Panel 5a, with errors noticeable only in the logarithmic scale. Further, our hybrid algorithm is executable on IBM quantum hardware as shown in Panel 5b. Lastly, Panel 5c shows that truncation thresholds fall within our provided upper bound in Proposition 4 in practice. where ∆=∇2is the Laplacian operator, Φ( x, t) and f(x, t) are the temperature and a known heat source at the grid position of xat time trespectively. Consider an initial time constraint and a xR∈Rperiodic spatial boundary condition, Φ(x,0) = ϕ(x); Φ( x, t) = Φ( x+xR, t), (20) where ϕ(x) is the initial distribution function. With the finite difference method [23], we can obtain a numerical solution to the heat transfer problem by solving the corresponding linear systems of equations in the form of Cx=b, where xis the temperature distribution and bis a known vector [34]. In this scenario, the matrix Cis given by C= −2−ξ 1 0 ··· 0 1 1−2−ξ1...... 0 0 1............ ............... 0 0............ 1 1 0 ··· 0 1 −2−ξ , (21) where ξ∈Ris a grid parameter. As Cis a circulant matrix, it has the form of a linear combination of cyclic permutation matrices QandQ−1, in addition to the identity matrix, C= (−2−ξ)I+Q+Q−1. (22) The condition number of the matrix in Equation 21 is shown to be κC=ξ+4 ξby Guseynov et al. [34]. Note that when ξ→0, the condition number κC→+∞. 12 Numerical results. To demonstrate our algorithm in practice, we perform three numerical experiments, the results of which are summarized in Figure 5. We first examine the performance of our algorithm on simulations of solving the heat transfer problem. We set the grid parameter ξ= 0.2 and the system size N= 25. The input state bis prepared by a single-layer QAOA circuit [35, 36] with the rotation parameters of {π 2,π 4,π 8,π 16,π 32}as shown in Figure 4. Note that we do not tune the parameters of the circuit, but merely use its structure to prepare our quantum state. We solve the corresponding circulant linear system of equations using three methods—direct matrix multiplication, simulation of the hybrid algorithm via the qiskit-aer simulator with 6 ×104shots per Hadamard test, and simulation of the quantum-inspired algorithm with 6 ×104accesses per sampling. The results are shown in Figure 5a. We note that our algorithms can provide a close estimate to that of the true value obtained by matrix multiplication, with errors noticeable only on the

Question 1098 multiple-choice

Cryptographic protocols often rely on the properties of algebraic groups to provide security guarantees, particularly regarding the difficulty of extracting roots and the composition of group orders. The design of such groups is critical for ensuring protocol soundness and resistance to various attacks. In the context of a group G suitable for secure cryptographic protocols, which property is essential to prevent efficient probabilistic algorithms from finding non-trivial roots of random group elements? 1) G must have a large subgroup whose order consists only of large prime factors, making the root problem hard. 2) G must support only abelian group operations to ensure root-finding is difficult. 3) G must be cyclic with order equal to a fixed small integer for root extraction to be infeasible. 4) G must have an unknown order to all protocol participants for security. 5) G must forbid inversion operations to prevent root-finding. 6) G must be constructed over elliptic curves with a low embedding degree. 7) G must include a subgroup of order two to complicate root extraction.

✓ Correct Answer:

The correct answer is 1) G must have a large subgroup whose order consists only of large prime factors, making the root problem hard..

📚 Reference Text:

Gis an RSA key generation algorithm – in this case it is indeed possible to generate a group with known and controlled order. Given descr(G), we assume that one can compute eﬃciently some estimates on the order, 2 A≤ord(G)≤2B, whereAandBare polynomial in k.W e also assume that elements can be sampled randomly from the group and thatinversion and group operation can be computed eﬃciently. In order for our protocols to work, we need, loosely speaking, that it is hard to ﬁnd non-trivial roots of elements in G. Furthermore, we need a condition on the structure of G’s output by G. Loosely speaking, we need that Ghas a AStatistically-HidingIntegerCommitmentScheme 129 large subgroup with only large prime factors in its order. To make this more precise, we assume that two functions are associated with G:C(·),l(·), that map positive integers to positive integers. Typically, C(k) is super-polynomially large as a function of k, whereasl(k) is always a polynomial. For any Gproduced byGon input 1k, we will consider primes greater that C(k) as being “large”. By the structure theorem for Abelian groups, we can always write G=U×H, where the order of Hhas only prime factors larger than C(k), and the order of Uhas only primes factors at most C(k). We say |H|isC(k)-rough, as opposed to beingC(k)-smooth, which means a number has only prime factors less than C(k). ThuslG:=|U|isC(k)-smooth. We are now ready to state our assumptions about groups output by G: Group Assumption. For anyGgeneratedby Goninput1kthefollowinghold: 1. WriteG=U×Has above, with C(k)-smooth,lG=|U|andC(k)-rough |H|. ThenlG≤l(k) and descr(G) includeslG. 2. For any string Y, when given descr(G), it can be decided in (determin- istic) polynomial-time in kwhetherYrepresents an element in G. Root Assumption. LetAbe a probabilistic algorithm. We run Gon input 1kto get descr(G). We give descr(G) and a random Y∈Gas an input to A. We say that Asolves the root problem if Aoutputs an integer e(>1), X∈G, andµ∈Usuch thatY=µXe(whereµ∈Ucan be veriﬁed by checking that µlG=1∈G). In particular, we say that the root problem is (t(k),zepsilon<(k))-secure if for k, any adversary Athat runs in time at most t(k), solves the root problem with probability of at most zepsilon<(k). The probability is taken over the coin tosses of GandA, as well as the random choice in G. Some remarks on the assumptions: The condition that lG≤l(k) says that Ghas many elements with only large prime factors in their orders: If Yis chosen randomly in G, then there is a signiﬁcant probability, 1 /l(k), that the order of YisC(k)-rough. We want to stress that it is essentially important that membership in Gcan be decided eﬃciently – although this property is often ignored and forgotten, this was onereason why proofs of soundness for the earlier protocols suggested in [14] wereincomplete. We will assume throughout that when a party receives an elementthat is supposed to be in G, membership in Gis always checked. The assumption that l Gis known and is part of the description can be re- moved, if in the protocols to follow, one replaces exponentiations to the powerl

Question 1099 multiple-choice

In quantum information theory and quantum algorithms, understanding the properties of random unit vectors, orthonormal sets, and the role of Fourier transforms over groups is crucial for analyzing quantum states and designing algorithms. Probabilistic estimates and concentration inequalities are key tools in this domain. Which property is characteristic of the empirical distribution of coordinates when sampling a sufficiently large set of random orthonormal vectors in complex space? 1) The coordinates follow a standard normal distribution. 2) The coordinates form a sparse distribution with most entries near zero. 3) The coordinates are concentrated at the edges of the unit sphere. 4) The coordinates exhibit heavy-tailed behavior with high variance. 5) The coordinates are always perfectly correlated. 6) The empirical distribution of coordinates approximates the uniform distribution. 7) The coordinates are exclusively real-valued.

✓ Correct Answer:

The correct answer is 6) The empirical distribution of coordinates approximates the uniform distribution..

📚 Reference Text:

get the unit vector(x1,...,x 2d), wherexi=yi√ y2 1+···+y2 2d(note that any yi= 0with zero probability). We thus get a random unit vector in R2d. Identifying a pair of real numbers with a single complex num ber, we get a ran- dom unit vector (z1,...,zd)inCd. To generate a random orthonormal ordered set {v1,...,vm}of vectors inCd,wecanﬁrstsample munitvectors {v′ 1,...,v′ m}inCdandthendoGram-Schmidtorthonormalisation on them to get {v1,...,vm}(note that withprobability 1,{v′ 1,...,v′ m}arelinearly independent). The following fact can be proved by combining Theorem 14.3.2 and Proposition 14.3.3 of [Mat02, Chapter 14] and using the concavity of the square-root funct ion. Fact 2Lett>0, and|v/an}bracketri}ht,|w/an}bracketri}htindependent random unit vectors in Cd. Then, Pr/bracketleftbigg |/an}bracketle{tv|w/an}bracketri}ht|>t+10√ d/bracketrightbigg ≤2exp(−t2d). We will require the following upper and lower bounds on the ta ils of the chi-square distribution (the chi-square distribution with ddegrees of freedom is the sum of squares of dindependent Gaussians with mean0and variance 1). The upper bound can be proved via Chernoff-style argument s on the moment generating function of the chi-square distribution. The lo wer bound follows, for example, from the central limittheoreminprobability theory. Onecanalsogiveadire ctproofofthelowerboundusingtheprobability density function of the chi-square distribution and estima ting it via Stirling’s approximation of the gamma function. Fact 3Let(X1,...,X d)beindependent random variables such that Xiisone-dimensional Gaussian with mean0and variance 1. LetX2=X2 1+···+X2 d. Let0≤ǫ <1/2. There exists a universal constant γ >0such that 1.Pr[|X2−d|>dǫ]<2exp(−dǫ2/6), 2.Pr[X2>d+√ d]>γ,Pr[X2γ. Thefollowing result follows easily from Fact3. Asimilar re sult appears as Lemma2in [MRRS04]. 11 Fact 4LetV={a1,...,ap}be a random orthonormal set of pvectors in Cd. Letai jdenote thejth coordinate of vector ai. Deﬁne the d-dimensional probability vector Sas follows:Sj=1 p/summationtextp i=1|ai j|2. Let 0≤ǫ<1/2. Supposep= Ω(ǫ−2logd). LetUdenote the uniform probability distribution on {1,...,d}. Then, with probability at least 1−exp(−Ω(ǫ2p))over thechoice of V,/bardblS−U/bardbl1≤ǫ. WewillalsoneedthefollowingChernoffupperboundsonthet ailofthesumof dindependentidentically distributed binary random variables. Fact 5([AS00,Cor. A.7,Theorem A.13]) Let(X1,...,X d)beindependentbinaryrandomvariablessuch thatPr[Xi= 1] =p. LetX=X1+···+Xd. Let0≤ǫ<1/2. Then, 1.Pr/bracketleftbig/vextendsingle/vextendsingleX d−p/vextendsingle/vextendsingle>ǫ/bracketrightbig <2exp(−2ǫ2d), 2.Pr[X

Question 1100 multiple-choice

Knot theory and topological quantum field theory are deeply interconnected, with quantum field models producing knot invariants that help classify and distinguish knots. Chern-Simons theory, particularly with various gauge groups, plays a central role in this mathematical and physical framework. Which classical topological invariant is used in quantum field theory to compute the expectation values of Wilson loops and measures how two loops are linked? 1) Jones polynomial 2) Gauss linking number 3) HOMFLY polynomial 4) A-polynomial 5) Super A-polynomial 6) Colored superpolynomial 7) Categorified invariant

✓ Correct Answer:

The correct answer is 2) Gauss linking number.

📚 Reference Text:

Title: Knot Invariants and Topological Quantum Field Theory Year: 2021 Paper ID: 15a12db57050ba7b8290a08e1914ddc41cf0ee31 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/15a12db57050ba7b8290a08e1914ddc41cf0ee31 Abstract: An elementary introduction to knot theory and its link to quantum field theory is presented with an intention to provide details of some basic calculations in the subject, which are not easily found in texts. Study of Chern-Simons theory with gauge group G, along with the Wilson lines carrying some representation is explained in generality, and a vital calculation of the Chern-Simons propagator is done. Explicit calculation for U(1) Chern-Simons theory is presented, which leads to the topological invariants, and finally to knot invariants. Further, using this result along with the Gauss linking number formula, the expectation value of Wilson loops are calculated. Colored knot invariants are also discussed along with more advanced knot invariants which are obtained using Homology theory, i.e., categorification of Jones and HOMFLY polynomials. Various knot invariants for SU(N) gauge group are also introduced, along with a brief introduction to A-polynomials and super A-polynomials. Recent developments in the field are explored, and we discuss a conjectured formula for colored superpolynomials, closed-form expression for HOMFLY polynomials, and conjectured expression for 6j symbol for Uq(slN) for multiplicity free case. Also, a MATHEMATICA program based on the conjectured formula had been developed, which can compute the 6j-symbols and the desired duality matrices which are needed to use the closed-form expression for HOMFLY polynomials. 1shoaib.akhtar@iitb.ac.in ar X iv :2 11 2. 13 64 3v 1 [ he pth ] 4 D ec 2 02 1

Question 1101 multiple-choice

Quantum journalism integrates concepts from quantum theory, such as indeterminacy and holism, with the study and practice of modern journalism, particularly in complex, interconnected digital environments. The field grapples with defining rigorous units of analysis and measurement suitable for its dynamic and synergetic nature. Which proposed unit is designed to measure the impact of quantum energy on information flow within quantum journalism, embodying the principle of holism? 1) 1 hall (one holism) 2) Quantum bit (qubit) 3) Audience reach index 4) Sentiment score 5) Synergy coefficient 6) Kyuzhur value 7) Observer factor

✓ Correct Answer:

The correct answer is 1) 1 hall (one holism).

📚 Reference Text:

Title: QUANTUM JOURNALISM: PROLEGOMEN OF MEASUREMENT (identification of criteria for the formation of representation about the unit of analysis and the unit of measurement) Year: 2022 Paper ID: 50ca88c6339f4988b20623d105dad01247133810 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/50ca88c6339f4988b20623d105dad01247133810 Abstract: The article aimed to search for and identify the criteria for forming an idea of the unit of analysis and the unit of measurement in quantum journalism (Kyushu).The research methodology involved the use of methods of three groups: general philosophical (the method of quantum monism; the ideas of realism; the concept of Unity); scientific-practical methods –observation, description and measurement; scientific theoretical methods –formalization, axiomatization, analysis and synthesis, deduction and induction, generalization, analogy, and abstraction.Results. Since today in the philosophical fields (metaphysics, nonclassical physics (quantum), and classical physics) there are no units of analysis and measurement of synergetic processes and phenomena in quantum journalism (quantum communication), further within holism and the author's concept of Unity it is proposed to create a new "1 hall", or "one holism". The range of analysis and measurement that form the unit "1 hall" should be considered the manifestations of the impact on the environment of a portion of energy (quantum) in the form of intensity and strength of changes that such energy (quantum) causes information flow.Conclusions. According to the results of the analytical review of existing ideas on measurement, it is appropriate to state ambivalent tendencies to identify the criteria for forming an idea of the unit of analysis and the unit of measurement in quantum journalism (more broadly -in the theory of quantum communication). On the one hand, measurement in quantum journalism is inexpedient. On the other hand, such a measurement is possible. From the standpoint of materialism, measurement in quantum journalism is possible thanks to digital indicators. From the point of view of metaphysics, measurement is possible taking into account the synergetic unity as a result of the combination of synergetic-convergent and cross-transmedia characteristics of information processes due to the unit "hall" (holism).Key words: quantum journalism, kyuzhur, measurement, unit of analysis, unit of measurement, identification of criteria.

Question 1102 multiple-choice

In the representation theory of algebras associated with quivers, especially cyclic and type A quivers, properties such as semisimplicity, the structure of the Jacobson radical, and module decomposition play crucial roles. Hopf algebras built from quivers also exhibit interesting behaviors regarding tensor products and commutativity. Which statement accurately describes the Jacobson radical of the algebra Λn(q) constructed from a cyclic quiver Zn with an nth root of unity q of order d? 1) It is always trivial, meaning Λn(q) is semisimple for all n and d. 2) It is generated by the vertices of the quiver and has dimension n. 3) It coincides with the entire algebra, making Λn(q) nilpotent. 4) It is generated by the idempotents and has dimension d(n – 1). 5) It is zero if and only if d = 1. 6) It is generated by the arrows of Zn and has dimension n(d – 1). 7) Its dimension depends only on n, independent of d.

✓ Correct Answer:

The correct answer is 6) It is generated by the arrows of Zn and has dimension n(d – 1)..

📚 Reference Text:

quiver is still A™. Using analogous considerations than in Proposition 3.10 we obtain that 2tf is not quasicommutative. The only Hopf algebra Jίf(q)/Id which is quasicommutative (and is in fact quasitriangular) is Notice that the free abelian group with one generator acts on J#*(q)/Id by translations of amplitude n provided d divides n. The quotient by this action is clearly the Hopf algebra kZn(q)/Id attached to the cyclic quiver Zn and to an nth root of unity q of order d. 4. Modules Let k be a field, and let Λn(q) be the wrf-dimensional Hopf algebra kZn(q)/Id, where as before Zn is the cyclic quiver of length n and q is a nth root of unity of order d. This algebra is not semisimple since its Jacobson radical is not zero; in fact its radical is of dimension n(d — 1) and is generated by the arrows of the quiver. Therefore there exist indecomposable modules which are not simple. Any finitely generated module M is a direct sum of indecomposable submodules, and by the Krull-Schmidt Theorem such decompositions only differ by an automorphism of M. Actually Λn(q) is a uniserial algebra, meaning that each indecomposable Quiver Quantum Group 467 projective module has a unique composition series; consequently this algebra is of finite representation type, i.e. there is only a finite number of iso-classes of indecomposable modules (each indecomposable module is a quotient of an in- decomposable projective). More precisely let P^ be the left ideal generated by the primitive idempotent s,-. This ideal is a projective module and any iso-class of a projective indecomposable module is obtained in this way. Let M" be the quotient of Pt by the submodule generated by the path y". In other words, M" is the u + 1 dimensional cyclic module with generator sf = γf subject to y"+1 = 0. The vector space M" has a canonical basis {y?, γl, . . . , γ "} and the composition of paths provides the module structure. Notice that M?"1 = Pt and that any simple module is isomorphic to M? = St for some ieZ/ftZ. From this discussion we infer the well known fact that {M"}iez/nz, i^u^d provides the complete list of iso-classes of finite dimensional indecomposable An(q) left modules. The module category of a k-Hopf algebra A is provided with an associative tensor product obtained via the comultiplication; notice that the tensor product is not necessarily commutative, the canonical switch map is not in general a map of vi-modules. Actually the switch map from A (χ)fc A to itself is a A-map if and only if the Hopf algebra is cocommutative. However if there exists an invertible element R such that the comultiplication A is quasicocommutative (i.e. A' = RAR'1) and if M and N are arbitrary A- modules, the modules M ®kN and N ®kM are isomorphic; indeed, the action of R followed by the switch map provides a A isomorphism which is natural in M and N. The following Clebsch- Jordan-like

Question 1103 multiple-choice

In mathematical physics, symmetries of physical systems are often formalized using abstract groups, with group representations describing how these symmetries act on concrete states or observables. Understanding the classification and identification of these groups is fundamental to modern theoretical physics and quantum information science. Which statement best describes the connection between the bit-flip symmetry in quantum systems and abstract mathematical group theory? 1) The bit-flip symmetry is modeled by the cyclic group Z2, consisting of two elements where the Cayley table reveals identity and flip operations. 2) The bit-flip symmetry corresponds to the permutation group S3, due to its three-element structure. 3) The bit-flip symmetry is represented by the Lie group SO(3), describing rotations in three dimensions. 4) The bit-flip operation forms a non-abelian group with more than two elements. 5) The bit-flip symmetry cannot be matched to any known abstract mathematical group. 6) The bit-flip group is identical to the continuous group U(1), associated with phase rotations. 7) The bit-flip symmetry requires a Cayley table with at least four elements to characterize its structure.

✓ Correct Answer:

The correct answer is 1) The bit-flip symmetry is modeled by the cyclic group Z2, consisting of two elements where the Cayley table reveals identity and flip operations..

📚 Reference Text:

an element of the Lie group eX=g. That is, g=fX2CddjeX2Gg. The correspondences between Lie algebras and Lie groups will be ﬂeshed out in Section VIC, but we will ﬁrst spend some time investigat- ing the connection between symmetries and groups. V. ABSTRACTIFYING PHYSICAL SYMMETRIES TO GROUPS Once the symmetries of the data have been identiﬁed, it is extremely useful to “ abstractify ” them:connect a phys- ical symmetry group with some familiar abstract mathe- matical group . To bridge this language with the rest of the article, recall the earlier motif: groupsencode abstract symmetries, and representations describe concrete symme- tries. The main utility of this abstractiﬁcation procedure is that groups as mathematical objects have been thoroughly studied since the early 19th century, and a wealth of infor- mation is readily available for scores of them. Moreover, in theeyesofphysics,thelistofabstractgroupsissurprisingly short, thanks in large part to classiﬁcation programs for ﬁ- nite groups and semisimple Lie groups and nature’s seem- ingpreferentialtreatmentofthesegroups—thismeansthat identiﬁcation is direct in many cases. It is worth highlight- ing, however, that this procedure is highly heuristic and 2Geometry note: technically, we are abusing the term “coordinates”, because we are thinking of (c0;c1;c2;c3)as parametrizing R4and identifying U(2)as a submanifold. Figure 3. Example of another discrete symmetry group. In this QML task the goal is to classify real valued data x= (x1;x2)living in a two-dimensional plane. The data xiwith labelyi= 0(blue circles) is sampled from the green region of the plane, while the data xiwith labelyi= 1(orange triangles) is sampled from the yellow region of the plane. As indicated by the pink dashed arrow, the symmetry operation (x1;x2)! (x2;x1)preserves the label. We encode this classical data in a two-qubit quantum state by initializing the qubits to the state j+i j+iand performing a rotation about the y-axis with a anglex1 ifor the ﬁrst qubit, and x2 ifor the ﬁrst qubit. Now, the symmetry group preserving the labels of the quantum states is GSWAP =f11;SWAPgof Eq. (9). fully general approaches do not exist: one physical symme- try group (representation) can be identiﬁed with several ambient abstract groups. But there is commonly a “sim- ple” choice of group to make, and this often suﬃces for the goals of abstractiﬁcation. As an example, consider the aforementioned problem where the labels are invariant under a bit-ﬂip transforma- tion, i.e., under the group Gb ipof Eq. (6). To abstractify a small discrete group, it can be useful to construct the group multiplication table, or Cayley table. Cayley tables show the multiplication of all group elements, which con- stitutes a type of ﬁngerprint for a group. For instance, for the bit-ﬂip symmetry group the Cayley table is as follows: 11X 1111X XX11: (8) Theonlygroupconsistingof2elementsobeyingthisCayley table is the cyclic group Z2, so we can identify this as the group underlying bit-ﬂip symmetry. Interestingly, we can also consider the QML task of Fig. 3 where we want to 7 classify real-valued two-dimensional vectors x= (x1;x2). Here the symmetry that preserves the labels is (x1;x2)! (x2;x1). We can verify that after encoding

Question 1104 multiple-choice

Operator scaling and tensor scaling algorithms play a fundamental role in fields such as invariant theory, quantum information, and algebraic complexity. Recent advances have unified various generalizations, linking algorithmic efficiency, complexity theory, and applications in high-dimensional settings. Which of the following statements accurately describes a main algorithmic breakthrough for scaling tensors to general marginals? 1) The algorithm only works for tensors with real-valued entries and uniform marginals. 2) A deterministic polynomial-time algorithm was presented for tensor scaling with arbitrary marginals. 3) The algorithm requires entries to be positive integers and does not handle complex values. 4) A randomized polynomial-time algorithm was developed for tensor scaling to general marginals, accepting tensors with Gaussian integer entries and rational target spectra. 5) The algorithm is limited to matrix scaling and cannot be applied to higher-order tensors. 6) Only uniform marginals are supported by the algorithm, restricting its applicability to specific cases. 7) The algorithm is exponential-time and not practical for high-dimensional tensors.

✓ Correct Answer:

The correct answer is 4) A randomized polynomial-time algorithm was developed for tensor scaling to general marginals, accepting tensors with Gaussian integer entries and rational target spectra..

📚 Reference Text:

analysis, that brought on the connectiontoinvarianttheory(andotherﬁelds,somementionedabove)wasachievedin[ GGOW16 ]. In the language we use here, it deals with d2(operator scaling) and uniform marginals, and resultsinpolynomialtimealgorithmsforproblemsindiverseareasdiscussedthere. /one.superior⁸Theoperator scaling problem was then extended in two directions, which we mention next: one direction being general values of d(tensor scaling) and the other being d2and arbitrary marginals. Forgeneral d,adeterministicalgorithmwasgivenin[ BGO17](basedonaproposalin[ VDDM03 ] forn01). Very recently, a randomized polynomial time algorithm for operator scaling to general marginalswasgivenin[ Fra18]. Thetwopapers[ BGO17,Fra18]studytwodiﬀerentgeneralizations of the operator scaling problem in [ GGOW16 ]. The present paper completes a natural square by studying a common generalization of the problems studied in [ BGO17,Fra18]. All these algorithmscanbeseenasnoncommutativegeneralizationsoftheSinkhorn-Knoppalgorithmfor ‘matrix scaling‘ [Sin64, SK67]. ItwasshownrecentlyknownthatProblem1.1isinNP XcoNP[BCMW17 ]. InviewofEq.(4), this should be contrasted with the NP-hardness of deciding whether a single Kronecker coeﬃcient is zero or not [IMW17]. 1.4 Summary of results Our main result in this paper is a randomized algorithm for tensor scaling to general marginals (Problem 1.6). As a consequence, we obtain algorithms for all other problems. Theorem 1.7. There is a randomized algorithm running in time polypN;1{"q, that takes as input XP Tenpn0;n1;:::;ndqwith Gaussian integer entries (speciﬁed as a list of real and complex parts, each encoded inbinary,withbitsize ¤b)andpPPpn1;:::;ndqwithrationalentries(speciﬁedasalistofnumerators anddenominators,eachencodedinbinary,withbitsize ¤b). Thealgorithmeithercorrectlyidentiﬁesthat pRpXq,oritoutputsascaling gPGsuchthatthemarginalsof gXare"-closetothetargetspectra p. HereNis the total bit-size of the input, N2n0n1ndb2pn1ndqb. As a consequenceof Theorem 1.7, we obtaina randomized algorithm for apromise version of the membership Problem 1.4 (and hence for Problem 1.1, see Corollary 2.6). /one.superior⁸The underlying algebraic problem associated with operator scaling, namely non-commutative singularity and rank of symbolic matrices found a diﬀerent, algebraic algorithm in the works of [IQS17, DM15] 7 Corollary 1.8. There is a randomized algorithm running in time polypN;1{"q, that takes as input XPTenpn0;n1;:::;ndqwith Gaussian integer entries (speciﬁed as a list of real and complex parts, each encoded in binary, with bit size ¤b) andpPPpn1;:::;ndqwith rational entries (speciﬁed as a list of numerators and denominators, each encoded in binary, with bit size ¤b). The algorithm distinguishes between the following two cases: 1.pPpXq. 2.pis"-far (in`1norm) from any point qPpXq. HereNis the total bit-size of the input, N2n0n1ndb2pn1ndqb. This yields the following corollary. Corollary 1.9. There is a randomized algorithm running in time polypn0n1nd;b;1{"q, that takes as inputpPPpn1;:::;ndqwith rational entries (speciﬁed as a list of numerators and denominators, each encoded in binary, with bit size ¤b). The algorithm distinguishes between the following two cases: 1.pPp1;n1;:::;ndqi.e. there exists YPTenpn1;:::;ndqsuch that specppiq Yqppiqfor alli. 2.pis"-far (in`1norm) from any point qPp1;n1;:::;ndq. Asdescribedbefore,Problem1.1capturestheasymptoticvanishingofKroneckercoeﬃcients. Hence we get the following corollary which describes a randomized polynomial time algorithm for a promise version of the asymptotic Kronecker problem. Corollary 1.10. There is a randomized algorithm running in time polypn;b;1{"q, that takes as input three partitions;;PZn ¥0withentriesdescribedinbinarywithbit-sizeatmost b. Thealgorithmdistinguishes between the following two cases: 1. There exists and integer s¥1s.t.gps;s;sq¡0. 2.For all1;1;1s.t.gp1;1;1q¡0, it holds thatp1;1;1q{|p1;1;1q|is"-far (in`1-norm) fromp;;q{|p;;q|. Heregdenotes the Kronecker coeﬃcient and |p;;q|° jj° jj° jj. Inmanyapplications,thetensor Xcanbemoresuccinctlyrepresentedthanbyits n0n1nd manycoordinates. Iftherepresentationispreservedbyscalingsandallowsforeﬃcientcomputation of the marginals, then this yields a useful optimization of Algorithm 1. A prime example of which are the so called matrix-product

Question 1105 multiple-choice

In quantum information theory, optimal measurement strategies often employ group symmetries and representation theory to distinguish quantum states associated with subgroups. The construction of measurement operators can leverage properties of conjugate subgroups and irreducible representations to ensure both normalization and optimality. Which choice for the coefficients $ c_{\mu,C} $ in the construction of measurement operators guarantees satisfaction of both normalization and optimality conditions in symmetric group-based quantum state discrimination? 1) Assign nonzero $ c_{\mu,C} $ only to the largest subgroup $ H $ where the irrep projector is nonzero, and zero elsewhere. 2) Assign equal nonzero values to $ c_{\mu,C} $ for all subgroups regardless of their size. 3) Assign nonzero $ c_{\mu,C} $ only to the smallest subgroup in the group. 4) Assign nonzero $ c_{\mu,C} $ to every subgroup for which any irrep projector vanishes. 5) Assign nonzero $ c_{\mu,C} $ only to subgroups outside the conjugacy class containing the identity. 6) Assign nonzero $ c_{\mu,C} $ to subgroups chosen at random, irrespective of their properties. 7) Assign $ c_{\mu,C} $ proportional to the number of elements in each subgroup without considering irreps.

✓ Correct Answer:

The correct answer is 1) Assign nonzero $ c_{\mu,C} $ only to the largest subgroup $ H $ where the irrep projector is nonzero, and zero elsewhere..

📚 Reference Text:

these sets of conjugate subgroups. Then a sum over all subgroups HofGcan be broken down into a sum over diﬀerent sets of conjugate subgroups, i.e. /summationdisplay H∈Sub(G)=/summationdisplay C∈C(G)/summationdisplay H∈C. (23) Applying this to the condition that the EHare valid measurement operators yields, /summationdisplay C∈C(G)/summationdisplay H∈Ccµ,HPµ,H=Idµ (24) or, explicitly performing the sum over conjugate subgroups, /summationdisplay C∈C(G)/summationdisplay g∈G|C| |G|cµ,gHCg−1Pµ,gHCg−1=Idµ (25) where|C|is the number of the subgroups in C,HCis a representative subgroup of CandPµ,gHCg−1is theµth irrep projector onto the subgroup gHCg−1={ghg−1,h∈ HC}. We now make the further assumption that cµ,Hdepends only on which C∈C(G) theHbelongs to. Then we obtain /summationdisplay C∈C(G)|C| |G|cµ,C/summationdisplay g∈GPµ,gHCg−1=Idµ (26) Now exploit the fact that we have symmetrized the projectors ove r the group G. This is the trick relevant to the construction of Moore and Russell where they demonstrate that the PGM is optimal for subgroups selected uniformly from a set of conjugate subgroups [18]. While the case we consider in cludes a symmetrized sum like that used in [18], our more general setting also has a sum over the diﬀerent sets of c onjugate subgroups. Using Eq. (19) we ﬁnd that 1 dµ/summationdisplay C∈C(G)|C| |HC|cµ,C/summationdisplay h∈HCχµ(h) = 1, (27) where|HC|is the number of elements in the subgroup HC. Now back to the nontrivial inequality for the optimal measurement. This is, for all H′∈Sub(G) and for all irrep labelsµ, /summationdisplay H∈Sub(H)|H|cµ,HPµ,H≥ |H′|Pµ,H′. (28) 6 Turning the sum on the left hand side into a sum over conjugate subg roups this becomes /summationdisplay C∈C(G)/summationdisplay H∈C|H|cµ,CPµ,H≥ |H′|Pµ,H′ (29) or /summationdisplay C∈C(G)/summationdisplay g∈G|HC||C| |G|cµ,CPµ,gHCg−1≥ |H′|Pµ,H′ (30) Using our expression for the sum over the conjugate projectors , Eq. (19), we obtain /summationdisplay C∈C(G)|C|cµ,C dµ/summationdisplay h∈HCχµ(h)Idµ≥ |H′|Pµ,H′ (31) Deﬁning sµ(HC) :=|C| dµ|HC|/summationdisplay h∈Hχµ(h), (32) we can then write the condition that the generalized measurement is a valid measurement as /summationdisplay C∈C(G)sµ(HC)cµ,C= 1, cµ,C≥0 (33) and the optimality condition as /summationdisplay C∈C(G)|HC|sµ(HC)cµ,CIdµ≥ |H′|Pµ,H′ (34) forallH′∈Sub(G). Notethat sµ(H)Cisanon-negativerealnumber. Equations(33)and(34) represen tthe optimality criteria for our ansatz. We will now show that it is possible to choose a particular cµ,Cthat satisﬁes both of these equations and thus we will have identiﬁed the optimal measurement. Begin by examining the condition of Eq. (34) for a ﬁxed H′. IfPµ,H′= 0, then this condition is automatically satisﬁed. Thus we can assume Pµ,H′/ne}ationslash= 0. Now we will choose the constant cµ,Cthat yields the optimal measurement. Pick the largest subgroup Hsuch that Pµ,H/ne}ationslash= 0 (break ties arbitrarily) and call it Hmax. This subgroup will belong to a particular set of conjugate subgroups, call this Cmax(µ). Then set cµ,C=/braceleftbigg1 sµ(HC)ifC=Cmax(µ) 0 if C /∈Cmax(µ)(35) (Note that sµ(HC) is not zero because of our condition that Pµ,H′/ne}ationslash= 0.) Certainly this expression for cµ,Cobeys the normalized measurement condition and is positive. Further, since Hmaxis the largest subgroup with Pµ,H/ne}ationslash= 0 all of the inequalities of Eq. (34) will also hold. This is because on the left han d side we have the identity operator and on the right hand side we have a projector and since these two are b oth diagonal

Question 1106 multiple-choice

Efficiently loading classical data into quantum memory is crucial for quantum computing applications such as quantum machine learning and signal processing. Algorithm selection depends on balancing resource requirements like circuit depth, qubit count, classical runtime, statevector representation, and circuit alterability. Which concept represents a set of algorithms in multi-objective optimization where improvement in one evaluation metric cannot occur without deterioration in at least one other metric? 1) Pareto set 2) Quantum superposition 3) Grover's oracle 4) Qubit entanglement 5) Circuit parallelism 6) Hilbert space basis 7) Noisy intermediate-scale quantum (NISQ) devices

✓ Correct Answer:

The correct answer is 1) Pareto set.

📚 Reference Text:

Title: Comparing Algorithms for Loading Classical Datasets into Quantum Memory Year: 2024 Paper ID: d30624a527bbd5d6d285b9e3cdc6cfd8cede0537 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d30624a527bbd5d6d285b9e3cdc6cfd8cede0537 Abstract: Quantum computers are gaining importance in various applications like quantum machine learning and quantum signal processing. These applications face significant challenges in loading classical datasets into quantum memory. With numerous algorithms available and multiple quality attributes to consider, comparing data loading methods is complex. Our objective is to compare (in a structured manner) various algorithms for loading classical datasets into quantum memory (by converting statevectors to circuits). We evaluate state preparation algorithms based on five key attributes: circuit depth, qubit count, classical runtime, statevector representation (dense or sparse), and circuit alterability. We use the Pareto set as a multi-objective optimization tool to identify algorithms with the best combination of properties. To improve comprehension and speed up comparisons, we also visually compare three metrics (namely, circuit depth, qubit count, and classical runtime). We compare seven algorithms for dense statevector conversion and six for sparse statevector conversion. Our analysis reduces the initial set of algorithms to two dense and two sparse groups, highlighting inherent trade-offs. This comparison methodology offers a structured approach for selecting algorithms based on specific needs. Researchers and practitioners can use it to help select data-loading algorithms for various quantum computing tasks.

Question 1107 multiple-choice

Character varieties are important objects in algebraic geometry and representation theory, often revealing intricate relationships between topology, group representations, and geometric invariants. One advanced method for studying their properties involves the application of Topological Quantum Field Theory (TQFT) to compute invariants such as Hodge structures and Deligne-Hodge polynomials. Which mathematical invariant is explicitly computed for parabolic $\mathrm{SL}_2(\mathbb{C})$-character varieties using a formalism that combines geometric and algebraic data through TQFT? 1) Chern-Schwartz-MacPherson class 2) Poincaré polynomial 3) Euler characteristic 4) Fundamental group 5) Deligne-Hodge polynomial 6) Kodaira dimension 7) Todd class

✓ Correct Answer:

The correct answer is 5) Deligne-Hodge polynomial.

📚 Reference Text:

Title: Topological Quantum Field Theories for character varieties Year: 2018 Paper ID: 09e39f5fc32f690b369612c7ce951efe1ab9076f Source: semantic-scholar URL: https://www.semanticscholar.org/paper/09e39f5fc32f690b369612c7ce951efe1ab9076f Abstract: This PhD Thesis is devoted to the study of Hodge structures on a special type of complex algebraic varieties, the so-called character varieties. For this purpose, we propose to use a powerful algebro-geometric tool coming from theoretical physics, known as Topological Quantum Field Theory (TQFT). With this idea in mind, in the present Thesis we develop a formalism that allows us to construct TQFTs from two simpler pieces of data: a field theory (geometric data) and a quantisation (algebraic data). As an application, we construct a TQFT computing Hodge structures on representation varieties and we use it for computing explicity the Deligne-Hodge polynomials of parabolic $\mathrm{SL}_2(\mathbb{C})$-character varieties.

Question 1108 multiple-choice

Young diagrams and standard tableaux play a central role in combinatorics and representation theory, providing graphical and numerical frameworks for studying integer partitions and symmetric group representations. Restricted classes, such as (k, ℓ)-Young diagrams, introduce constraints on diagram shapes that are important for classifying representations of certain algebras. Which condition must a Young diagram satisfy to be considered a (k, ℓ)-Young diagram for given integers k and ℓ with ℓ > k > 0? 1) Each row contains at least k boxes, and the first row contains exactly ℓ boxes 2) The total number of boxes is equal to k + ℓ 3) The diagram has at most k rows, and the difference between the first and k-th row is at most ℓ - k 4) Each column has at most ℓ boxes, and the difference between adjacent rows is at most k 5) The sum of the lengths of the first k rows is equal to ℓ 6) All rows contain the same number of boxes, and there are exactly k rows 7) The total number of boxes is divisible by ℓ but not by k

✓ Correct Answer:

The correct answer is 3) The diagram has at most k rows, and the difference between the first and k-th row is at most ℓ - k.

📚 Reference Text:

partition, and that the i’th row contains λiboxes. We allow for the empty partition and diagram, which we denote ∅. Let Λ nbe the collection of n-box Young diagrams. Given two diagrams λ∈Λnandµ∈Λm, their sumλ+µ∈Λn+mis always well-deﬁned, having parts equal to ( λ+µ)i=λi+µi. When it is well-deﬁned as a partition, we will similarly deﬁne the diﬀerenceλ−µ∈Λn−mto have parts ( λ−µ)i=λi−µi. Pawel Wocjan and Jon Yard 9 ∅ /d15/d15 /d28/d28/d57/d57/d57/d57/d57 /d0/d0/d2/d2/d2/d2/d2 /d2/d2/d4/d4/d4/d4/d4/d4 /d27/d27/d55/d55/d55/d55/d55 /d4/d4/d9/d9/d9/d9/d9 /d26/d26/d53/d53/d53/d53/d53 /d124/d124/d121/d121/d121/d121/d121/d121/d121/d121/d121/d121 /d15/d15/d123/d123/d118/d118/d118/d118/d118/d118/d118/d118/d118 /d15/d15 /d33/d33/d68/d68/d68/d68/d68/d68/d68/d68 /d15/d15/d30/d30/d61/d61/d61/d61/d61/d61/d61/d61 Fig. 3.2. The Young graph Λ 0,4. Deﬁne the sets of diagrams Λ m,n≡/unionmultitextn i=mΛiand abbreviate Λ ≡Λ0,∞. We may consider elements of Λ to be the nodes of an inﬁnite directed graph, whi ch we will call the Young graph, which contains an edge from a diagram λ′to a diagram λwhenever a single box can be added to λ′to obtainλ. We remark that we depart slightly from common terminology, where this graph, with each arrow reversed, is called a Bratteli diagram . With a slight abuse of notation, we write Λ for both the set of diagrams and for the corresponding Young graph, making a similar identiﬁcation between subsets of Λ such as Λ 0,nand the associated subgraph. In Figure 3.2, we give a picture of Λ 0,4. By anumbering of ann-box diagram, we mean any assignment of the numbers {1,2,...,n } to the boxes of the diagram, where each number appears only on ce. For a given diagram λ∈Λn, we writeTλfor the set of standard tableaux with shapeλ, corresponding to numberings which are strictly increasing along each row and column. We o ften call a standard tableau just a tableau . We denote the set of all n-box tableaux as Tn=/unionmultitext λ∈ΛnTλand the set of all tableaux as T=/unionmultitext∞ n=1Tn. We identify members of Tnwith length- npaths starting at ∅ in Λ0,n, pictured in Figure 3.2, with elements of Tλcorrespond to paths which end at the diagramλ. In this paper, we shall rather require certain restricted cl asses of Young diagrams and tableaux. To each pair ( k,ℓ) of integers satisfying ℓ>k> 0 corresponds a class of irreducible unitary representations of Bn. These representations are labelled by the n-box (k,ℓ)-Young diagrams , deﬁned as Λ(k,ℓ) n={λ∈Λn:λk+1= 0,λ1−λk≤ℓ−k}. Note that these are the diagrams with at most krows for which the diﬀerence between the numbers of boxes in the ﬁrst and k’th rows is at most ℓ−k. We will often just say that such a λis an (k,ℓ)-diagram. For a given diagram λ, we refer to λ1−λkas the levelofλ, so thatℓ−k is the maximum level of all diagrams in Λ(k,ℓ) n. As before, we make the similar abbreviations Λ(k,ℓ) m,nand Λ(k,ℓ). We also identify any of these sets of diagrams with the appro priate subgraph of Λ. We refer to such a graph as a ( k,ℓ)-Young graph . In Figure 3.3, we show the graph Λ(2,5) 0,4. For a given λ∈Λ(k,ℓ) n, deﬁne the ( k,ℓ)-tableauxT(k,ℓ) λ⊂Tλto be those standard tableaux of shape λfor which successively deleting the largest

Question 1109 multiple-choice

Quantum computing with qudits leverages multi-level quantum systems to encode more information per quantum unit, enhancing computational capabilities and algorithmic flexibility. High-fidelity characterization of quantum states and gates is crucial for validating the accuracy and reliability of quantum operations in these systems. Which metric specifically quantifies the similarity between an experimentally prepared quantum state and its ideal theoretical counterpart in multi-level quantum computing? 1) Classical statistical fidelity (Fc) 2) Quantum process fidelity (Fq for processes) 3) Detection rate 4) Quantum state fidelity (Fq for states) 5) Gell-Mann matrix decomposition score 6) Path interference visibility 7) Quantum parallelism index

✓ Correct Answer:

The correct answer is 4) Quantum state fidelity (Fq for states).

📚 Reference Text:

the detection rate depends on the performance and loss of the quantum devices as well as their pumping and measurement apparatuses)58. Details of device fabrication, state evolution and experimental setup are provided in Supplementary Notes 1 and 3. Characterisation of d-ary multi-value controlled-unitary gates . Before reporting experimental results, we ﬁrst de ﬁne classical statistic ﬁdelity ( Fc) and quantum state (process) ﬁdelity ( Fq), used in this work to quantify the qudit states, logic gates and algorithm implementations. The Fcis de ﬁned as ð∑iﬃﬃﬃﬃﬃﬃﬃpiqipÞ2,where pi,qiare theoretical and measured distributions, respec- tively; the state Fqis deﬁned as ðTr½ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃρ0p/C1ρ/C1ﬃﬃﬃﬃ ﬃρ0p p/C138Þ2, where ρ0,ρ are ideal and measured states, respectively; the process Fqis deﬁned as Tr[ χ0χ], where χ0,χare ideal and reconstructed process matrices, respectively. Weﬁrst characterised the single-q uquart and two-ququart logic gates. As examples, two single-ququart gates are characterised: the generalised d-level Pauli- Xdgate that is de ﬁned as Xdki/C12/C12/C11 ¼kði/C8d1Þ/C12/C12/C12E where ⊕ dis addition module of d,a n dt h e d-level quantum Fourier gateFdthat transforms the c omputational basis of ki/C12/C12/C11 to the Fourier basis fi/C12/C12/C11 of1ﬃﬃ dp∑d/C01 j¼0ωijj/C12/C12/C11 where i,j∈Ndandω=expði2π dÞ.W h e n dis two, they return to the standard Pauli and Fourier (Hadamard) gates for qubits. In Fig. 2ci n s e t ,i ts h o w st h em e a s u r e dm e a n Fcof 0.988(13) for the ﬁveX4gates and 0.967(19) for the ﬁveF4gates, where the values in parentheses are uncertainty from photon statistics. Next, we characterise d the two-qudit entangling gate: MVCU xjiy/C12/C12/C11 ¼xjiOxy/C12/C12/C11 ; ð1Þ where Ocan be arbitrarily operated59on the xjiand y/C12/C12/C11 registers. Notably, the MVCU gate presents a coherent entanglement between the auxiliary x-register and the data y-register. The processing of d- ary quantum algorithms relies on th e multiple path interference in thed-dimensional Fourier gate to obt ain the desired solutions. Such coherent superposition of qudits ensures quantum parallelism, that is function evaluations for multiple in puts are executed in parallel. The MVCU is thus a core logic enabling the quantum parallel evaluation o ft h ef u n c t i o n .F o re x a m p l e ,a st h e d-ary generalisation of the CNOT gate24,t h eM V C X dgate allows the creation of a complete set of four- level Bell states Ψjii;jdeﬁned as1 2∑3 m¼0ωmimjim/C8dj/C12/C12/C11 , by inputing thefi/C12/C12/C11 /C10kj/C12/C12/C12E states into the logic, i,j=0,1,2,3. Figure 3as h o w st h e reconstructed Ψji 12state, and Fig. 3cs h o w sm e a s u r e d Fqfor the 16 Bell states with an averaged ﬁdelity of 0.967(31). The state matrices (ρ) represented as a linear combination of Gell-Mann matrices were reconstructed by implementing compressed sensing quantum state tomography techniques60. In addition, a fully product state was created in Fig. 3b, given an input of f0f0/C12/C12/C11 .F i g u r e 3ds h o w st h e experimental process matrix ( χ)o ft h eM V C

Question 1110 multiple-choice

Self-harm in adolescents and adults is a complex phenomenon influenced by psychological, social, and cultural factors. Understanding the patterns, risk factors, and outcomes associated with various methods is crucial for effective intervention. Which method of self-harm is rare, associated with high lethality, and has a notably high mortality rate among adult patients admitted to burn units? 1) Jumping from heights 2) Self-shooting 3) Drug overdose 4) Hanging 5) Self-cutting 6) Self-poisoning 7) Self-burning (setting fire to oneself)

✓ Correct Answer:

The correct answer is 7) Self-burning (setting fire to oneself).

📚 Reference Text:

In a clinical study by Hawton et al. , 14,892 self-harm patients were identified in a United Kingdom general hospital gathered over a period of 23 years with 428 self- cutters versus 11,065 self-poisoners.25 The self-cutters were more often male, single, unemployed, had a history of deliberate self-harm, lived alone, and misused alco - hol.25 One study of adolescents (11-18 years of age with a mean age of 15) who were self-cutters and were seen in intensive psychiatric programs revealed a range of self-cutting behavior.26 Those who are frequent self-cut - ters (i.e., over three times) were typically females who engage in high-risk behavior, such as sexual activity without consistent condom use and who had a history of sexual abuse.26 Matsumoto and colleagues showed that the prevalence for self-cutting and suicidal ideation in junior and senior high school students in Japan were 9.9% and 40% respectively.27 Setting fire to oneself is an unusual means of self-harm and suicide attempt that is seen more often in adults than in adolescents.28, 29 Over an 11 year period (1994 to 2005), 1,745 patients were admitted to a tertiary care burn unit in the United Kingdom; this included 41 pa - tients with a mean age of 29 ± 12 years who had a burn following an assault and 86 patients (mean age of 37 ± 12) who had self-induced burns.30 Gasoline, accelerants, and other flammable liquids were the means used for the burns and the rate of mortality was 29% in the self-burn patients. In summary, deliberate self-harm is a widespread yet often hidden problem in adolescents, especially females, which shows both similarities and differences interna - tionally.31 Etiology of Deliberate Self-harm Ecological models have been commonly used to un - derstand suicide and related behavior, where an individ - ual’s behavior is seen as a result of the interaction of dif - ferent ecological systems ( Table 5 ). For example, Shagle and Barber used a social-ecological framework to ana - lyze adolescent suicidal ideation where family, school, peers and religion were proposed to influence adolescent suicidal ideation via self-derogation.32 Jackson and Nut - tall33 and Perkins and Hartless 34 similarly supported the use of an ecological risk-factor model for examining sui - cidal behavior in children and adolescents. With reference to the personal system, research find - ings show that hopelessness was related to suicide and related behavior.35, 36 Previous research also showed that adolescents with affect dysregulation,37 problem solving deficits,38 and low self-esteem39 had increased chance of suicide. With specific reference to the role of emotions in deliberate self-harm, adolescents engage in non-sui - cidal self-injury to influence behaviors of others and to deal with internal emotions.40 Two pathways to self- wounding have been identified: cutting may result when Table 3 Metho ds of self-harm 9-14 1. Drug Overdose 2. Self-Poisoning 3. Self-Cutting 4. Other forms of self-mutilation a. Self-hitting (battery) b. Pinching c. Scratching d. Biting e. Burning 5. Self-shooting 6. Hanging 7. Jumping from high places 8. Jumping

Question 1111 multiple-choice

Quantum Fourier Transform (QFT) is a fundamental algorithm in quantum computing that generalizes the classical discrete Fourier transform for quantum systems. Qudits, which are quantum units with d levels, enable more compact representation and processing compared to qubits. For an n-qudit (d-level) quantum system with a state space dimension dⁿ, what is a principal advantage of using qudits over qubits in the implementation of the QFT, assuming the same total state space size? 1) The output of QFT on qudits can be directly accessed for classical signal processing tasks 2) The gate count for QFT is reduced by a factor of log₂d compared to a qubit system 3) The QFT on qudits requires only CNOT gates instead of controlled-phase gates 4) The QFT on qudits decreases the number of measurements needed to extract information 5) The quantum state after QFT on qudits is not subject to collapse upon measurement 6) The computational complexity of QFT becomes sub-linear in n with qudits 7) The QFT on qudits eliminates the need for matrix decomposition during circuit synthesis

✓ Correct Answer:

The correct answer is 2) The gate count for QFT is reduced by a factor of log₂d compared to a qubit system.

📚 Reference Text:

Just like before, these diagonal operators can be synthesiz ed in controlled- Rjoperators acting on two qudits by Theorem 6 and Lemma 8: q1 Ik⊕Ωk⊕···⊕Ωd−1 kq1• q1•R2···• q2 q2 Ωkq2 •··· ...=...=...... qk qk qk ···Rk qk+1 qk+1 qk+1Rk+1 ··· The representation is only one of the many possible implemen tations of the diagonal matrix under the freedom allowed by the commutativity and swap property (Lemma 8) of c ontrolled-Rgates. We conclude that the gate count for a quantum Fourier transfo rm on annquditd-level system is still O(n2), but now for a QFT of a state vector of dimension dn. This results in a log2dreduction in gates for a qudit system compared to a qubit system that has a state space of the same di mension at the cost of more complex controlled operations. 7 Conclusion In this paper, we showed how the quantum Fourier transform al gorithm can be derived as a decomposition of the discrete Fourier matrix. The derivation starts from the rad ix-2 decomposition of the DFT matrix that yields the FFT algorithm, and only makes use of Kronecker products to fu rther decompose a diagonal matrix into a product of simpler unitary matrices, each of which can be written as t he sum of two Kronecker products of 2 ×2 matrices. This alternative approach to the derivation of the quantum F ourier transform in [14, 4] requires little knowledge of quantum computing. Weshowed insection 5.1 thattheKroneckerstructureofthe Q FTdecomposition does notlead toan immediate reduction in complexity compared to that of the FFT algorith m on a classical computer, even when the QFT is applied to arank-1tensor. We explained why the complexity o f the QFTon aquantumcomputer can be evaluated by counting the number of 2 ×2 matrices produced by the QFT matrix decomposition, which i sO((logN)2) for 16 a transformation of a vector of size N= 2n. We made the connection between the matrix decomposition of the QFT and the quantum circuit representation widely used in th e quantum computing literature. We pointed out that the QFT decomposition of the DFT matrix and the correspo nding quantum circuit is not unique. The non- uniqueness of the decomposition can potentially provide so me ﬂexibility in the quantum circuit topology in terms of the placement of controlled- Rgates. We also generalized the radix-2 decomposition to a ra dix-ddecomposition which requires the QFT to be implemented on a quantum compute r equipped with qudits. Although the QFT can be performed eﬃciently on a quantum comp uter, the result of the transform is not easily accessible. This is a key diﬀerence between the QFT an d FFT, and between a quantum and classical algorithm in general. Because of this, one cannot use the QFT in the same way the FFT is used. For example, it is not straightforward to perform a fast convolution of two v ectors [12], which is the most widely used application of the FFT, on a quantum computer using QFTs. On the other hand , the QFT

Question 1112 multiple-choice

In quantum computing, the hidden subgroup problem (HSP) plays a central role in the development of efficient algorithms for group-theoretic problems, especially in non-abelian groups. Recent advances have leveraged group structural properties and sophisticated mathematical techniques to tackle HSP in broader classes of groups. Which of the following statements accurately describes a key feature of the efficient quantum algorithm for the hidden subgroup problem in nil-2 groups? 1) It utilizes automorphisms of nil-2 groups and solves systems of quadratic and linear equations, with solution existence ensured by the Chevalley-Warning theorem. 2) It reduces the problem to abelian groups and applies classical Fourier analysis exclusively. 3) It avoids the use of group actions by employing only direct search methods over subgroup lattices. 4) It depends on the assumption that all hidden subgroups are normal in arbitrary groups. 5) It relies purely on probabilistic sampling without any use of algebraic structure. 6) It applies only to groups of exponent 2 and cannot be extended to exponent p for p > 2. 7) It is limited to solving HSP where the hidden subgroup is always non-cyclic.

✓ Correct Answer:

The correct answer is 1) It utilizes automorphisms of nil-2 groups and solves systems of quadratic and linear equations, with solution existence ensured by the Chevalley-Warning theorem..

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arXiv:0707.1260v1 [quant-ph] 9 Jul 2007An eﬃcient quantum algorithm for the hidden subgroup proble m in nil-2 groups∗ G´ abor Ivanyos†Luc Sanselme‡Miklos Santha§ October 24, 2018 Abstract In this paper we extend the algorithm for extraspecial group s in [12], and show that the hidden subgroup problem in nil-2 groups, that is in groups of nilpot ency class at most 2, can be solved eﬃciently by a quantum procedure. The algorithm presented here has sev eral additional features. It contains a powerful classical reduction for the hidden subgroup probl em in nilpotent groups of constant nilpotency class to the speciﬁc case where the group is a p-group of exponent pand the subgroup is either trivial or cyclic. This reduction might also be useful for dealing wi th groups of higher nilpotency class. The quantum part of the algorithm uses well chosen group actions based on some automorphisms of nil-2 groups. The right choice of the actions requires the solutio n of a system of quadratic and linear equations. The existence of a solution is guaranteed by the Chevalley-W arning theorem, and we prove that it can also be found eﬃciently. ∗Research supported by the European Commission IST Integrat ed Project Qubit Applications (QAP) 015848, the OTKA grants T42559 and T46234, the NWO visitor’s grant Algebraic Aspects of Quantum Computing, and by the ANR Blanc AlgoQP grant of the French Research Ministry. †SZTAKI, Hungarian Academy of Sciences, H-1111 Budapest, Hu ngary.ivanyos@sztaki.hu ‡UMR 8623 Universit´ e Paris–Sud 91405 Orsay, France. sanselme@lri.fr §CNRS–LRI, UMR 8623 Universit´ e Paris–Sud 91405 Orsay, Fran ce.santha@lri.fr 1 1 Introduction Eﬃcient solutions to some cases of the hidden subgroup problem (HS P), a paradigmatic group theoretical problem, constitute probably the most notable success of quantu m computing. The problem consists in ﬁnding a subgroup Hin a ﬁnite group Ghidden by some function which is constant on each coset of H and is distinct in diﬀerent cosets. The hiding function can be accesse d by an oracle, and in the overall complexity of an algorithm, a query counts as a single computational step. To be eﬃcient, an algorithm has to be polylogarithmic in the order of G. While classically not even query eﬃcient algorithms are known for the HSP, it can be solved eﬃciently in abelian groups by a quantum algor ithm. A detailed description of the so called standard algorithm can be found for example in [18]. The main q uantum tool of this algorithm is Fourier sampling, based on the eﬃciently implementable Fourier trans form in abelian groups. Factorization and discrete logarithm [21] are special cases of this solution. After the settling of the abelian case, substantial research was d evoted to the HSP in some ﬁnite non- abelian groups. Beside being the natural generalization of the abelia n case, the interest of this problem is enhanced by the fact, that important algorithmic problems, such as graph isomorphism, can be cast in this framework. The standard algorithm has been extended to so me non-abelian groups by R¨ otteler

Question 1113 multiple-choice

In quantum information theory, integrating over the space of unitary operators is essential for understanding typical properties and randomness in quantum systems. Specialized mathematical tools and norms are used to rigorously analyze quantum operations and states. Which of the following best describes a unitary design in the context of quantum information? 1) A set of Hermitian operators that diagonalizes all quantum states 2) A collection of tensors representing entangled states 3) A method for vectorizing matrices to simplify computations 4) A group of operators that maximize the trace norm 5) A structured set of unitary operators that mimics the statistical properties of the Haar measure up to certain moments 6) A subspace invariant under all permutations of basis elements 7) An approximate algorithm for computing average gate fidelity

✓ Correct Answer:

The correct answer is 5) A structured set of unitary operators that mimics the statistical properties of the Haar measure up to certain moments.

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fundamental tool in quantum information, its study can be challenging for beginners due to its reliance on advanced concepts from representation theory. In order to make this topic more accessible, this tutorial provides an introduction to Haar measure tools using only concepts from linear algebra. Throughout the tutorial, we intentionally avoid delving into representation theory to enhance accessibility. For further reading, one can refer, for example, to [39, 40, 45–51]. The tutorial is structured as follows. Section 2 offers an overview of the notation and pre- liminary concepts that will be used throughout the tutorial. Section 3 focuses on introducing the Haar measure, with a specific emphasis on characterizing the moment operator—a crucial quantity for computing integrals over the Haar measure. In Section 4, the symmetric and antisymmetric subspaces are introduced, highlighting their properties and their connection to the Haar measure over pure states. Sections 5 and 6 introduce tools that facilitate calculations in Haar measure applications. Section 5 presents the vectorization formalism, while Section 6 introduces the tensor networks diagrammatic notation, providing visual representations that enhance comprehension and streamline computations. Section 7 introduces the concept of unitary designs—a method employed to mimic some properties of the Haar measure, facilitating efficient protocols in quantum comput- ing. Building upon this foundation, Section 8 explores approximate notions of unitary designs, elucidating the relationships between the different introduced notions. In Section 9, the tutorial showcases practical examples of Haar measure calculations. We begin by deriving well-known for- mulas such as the twirling of a quantum channel, average gate fidelity, average purity of reduced states in bipartite systems, and Haar averages of observables expectation values. Furthermore, we demonstrate how these expected value calculations can be translated into probability statements using concentration inequalities. Finally, we delve into two in-depth applications: Barren Plateaus phenomena in Quantum Machine Learning and Classical Shadow Tomography, both of which rely on the theory of k-designs. 2 Notation and Preliminaries We use the following notation throughout this tutorial. L(Cd)is the set of linear operators that act on thed-dimensional complex vector space Cd, while Herm( Cd)is the set of Hermitian operators onCd. The identity operator is denoted by I, and we define the operator I:=I⊗Ias the tensor product of two identity operators. The unitary group is denoted by U(d)and is defined as the set of operators U∈L(Cd)such thatU†U=I. Furthermore, we use the notation [d]to denote the set of integers from 1 to d, i.e., [d]:={1,...,d}. Letv∈Cdbe a vector, and let p∈[1,∞]. Thep-norm ofvis denoted by∥v∥p, and is defined as∥v∥p:= (/summationtextd i=1|vi|p)1/p. The Schatten p-norm of a matrix A∈L/parenleftbig Cd/parenrightbig is given by ∥A∥p:= Tr((√ A†A)p)1/p, which corresponds to the p-norm of the vector of singular values of A. The trace norm and the Hilbert-Schmidt norm are important instances of Schatten p-norms and are respectively denoted as ∥·∥ 1and∥·∥ 2. The Hilbert-Schmidt norm is induced by the Hilbert- Schmidt scalar product ⟨A,B⟩HS:= Tr/parenleftbig A†B/parenrightbig forA,B∈L/parenleftbig Cd/parenrightbig . The infinite norm, denoted as ∥·∥∞, of a matrix is defined as its largest singular value.

Question 1114 multiple-choice

Quantum algorithms for hidden subgroup problems leverage group-theoretic structures and quantum measurement techniques to efficiently distinguish between different subgroup conjugates, particularly in nonabelian groups such as q-hedral and affine groups. The use of transforms and distance metrics is critical for achieving efficient reconstruction of subgroup information. In the context of quantum algorithms for nonabelian hidden subgroup problems, which statement accurately characterizes the reconstructibility of conjugate subgroups in q-hedral groups using quantum measurements? 1) Only normal subgroups of q-hedral groups can be efficiently reconstructed using quantum measurements. 2) Quantum measurements yield indistinguishable outcomes for all conjugate subgroups in q-hedral groups. 3) The total variation distance between measurement outcomes for different conjugates in q-hedral groups is always zero. 4) Subgroups of q-hedral groups cannot be distinguished information-theoretically, regardless of quantum resources. 5) Only subgroups of order p in q-hedral groups are reconstructible with quantum algorithms. 6) Efficient reconstruction of conjugate subgroups in q-hedral groups is impossible due to high sample complexity. 7) Every nonnormal subgroup of order q′ in a q-hedral group is a conjugate of a maximal subgroup, and all such conjugates can be reconstructed information-theoretically using quantum measurements.

✓ Correct Answer:

The correct answer is 7) Every nonnormal subgroup of order q′ in a q-hedral group is a conjugate of a maximal subgroup, and all such conjugates can be reconstructed information-theoretically using quantum measurements..

📚 Reference Text:

has rank ( p−1)/q, and the total probability that we observe ρis (p−1)/p=1−1/pas before. Then these four partial measurements determine k, remove the eﬀect of the coset, and determine that the row has one of two values, uoru+ 1. Up to an overall phase we can write this as a two-dimensional vector: 1√ 2/parenleftbiggωkaub p ωkau+1b p/parenrightbigg . Our only goal in doing this is to create a one-qubit state where the relative phase between the two basis vectors depends on the conjugate b. Moreover, the relative phase is multiplied by the irrep label k, which is uniformly random. As a result, the typical angle between the states corresponding to any two distinct cosets b, b/primewill be Ω(1), and a simple measurement yields a constant variation distance between them. To make this precise, apply the Hadamard transform 1√ 2/parenleftbigg11 1−1/parenrightbigg and measure according to the resulting basis. The probability that we observe the ﬁrst or second basis vector is then cos2θand sin2θ, respectively, where θ=(kau(a− 1)bπ)/p. Now when we observe a q-dimensional representation ρkofNk, the observed label kis uniformly distributed over Z∗ p/Zq. Moreover, when we perform the POVM in step (iv) above, the uwe observe is uniformly distributed over Zq. It follows that the coeﬃcient m=kau(u−1) is uniformly distributed over Z∗ p. For any two distinct b,b/prime, the total variation distance is then 1 2(p−1)/summationdisplay m∈Z∗p/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos 2πmb p−cos2πmb/prime p/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsinglesin 2πmb p−sin2πmb/prime p/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg . Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 950 MOORE, ROCKMORE, RUSSELL, AND SCHULMAN This we rewrite 1 p−1/summationdisplay m∈Z∗ p/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos 2πmb p−cos2πmb/prime p/vextendsingle/vextendsingle/vextendsingle/vextendsingle =1 2(p−1)/summationdisplay m∈Zp/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos2πmb p−cos2πmb/prime p/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≥1 4(p−1)/summationdisplay m∈Zp/parenleftbigg cos2πmb p−cos2πmb/prime p/parenrightbigg2 =p 4(p−1)>1 4. (Adding the m= 0 term contributes zero to the sum in the second line. In the third line we use the facts that |x|≤x2/2 for all |x|≤2, the average of cos2xis 1/2, and the two cosines have zero inner product.) Since the total variation distance between any two distinct conjugates is bounded below by a constant, we can distinguish between the pdiﬀerent conjugates with only O(logp) = poly( n) samples. Thus, hidden conjugates in Apare information- theoretically reconstructible, completing the proof. By embedding the q-hedral groups in Apas in Theorem 2, we can generalize Theorem 3 to the q-hedral groups ( qnot necessarily prime) as follows. Let pbe prime, letqbe a divisor of p−1, and let q/primebe a divisor of q. The q-hedral group Zq/multicloseleftZ p has a unique normal subgroup K∼=Zq/prime/multicloseleftZ p, in which there is a maximal subgroup Hq/primeof order q/prime. Moreover, all nonnormal subgroups of order q/primeare conjugates of Hq/prime. Theorem 4. Ifp, q, q/prime, and Hq/primeare as above, then the hidden conjugates of Hq/prime inZq/multicloseleftZ pare information-theoretically reconstructible. We now wish to information-theoretically reconstruct all subgroups of the aﬃne andq-hedral groups. We can do this by using the fact that, except for normal sub- groups, there is a unique conjugacy class of each order q/prime, namely the conjugates of Hq/prime. Thus if we can determine the order of H, this determines

Question 1115 multiple-choice

W-algebras are mathematical structures that generalize the Virasoro algebra and play an important role in integrable systems and conformal field theory. Their construction often involves deep connections to Lie algebra theory, geometry, and quantization methods. Which statement correctly describes the classification of inequivalent reductions leading to W-algebras for the Lie algebra sln? 1) They are classified by the number of simple roots of sln. 2) Inequivalent reductions correspond to the rank of sln. 3) The number is given by P(n), the number of partitions of n. 4) Each reduction is uniquely determined by the center of sln. 5) Only principal embeddings yield inequivalent reductions. 6) Classification depends solely on the dimension of sln. 7) Reductions are equivalent for different embeddings of sl2 into sln.

✓ Correct Answer:

The correct answer is 3) The number is given by P(n), the number of partitions of n..

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their Hamiltonian structures were however already show n to be reductions of a diﬀerent class of integrable hierarchies which have a secon d Hamiltonian structure that is equal to the Kirillov Poisson structure on the dual of an aﬃ ne Lie algebra [4]. This means on an algebraic level that classical Wnalgebras can be obtained from aﬃne Lie algebras by Poisson reduction. This picture was worked out i n [5] where it was shown that a classical Wnalgebra is nothing but the Dirac bracket algebra on a submani fold of the aﬃne Lie algebra. In the meantime many new Walgebras were constructed by what can be called the ’direct method’, i.e. by imposing Jacobi identities on g eneral nonlinear extensions of the Virasoro algebra. Since the Jacobi identities are the mselves nonlinear algebraic equations the construction of Walgebras in this manner is rather cumbersome. It was therefore a natural question to ask (also from the point of vi ew of Poisson reduction of Poisson manifolds) whether more of these algebras can be o btained via Poisson re- duction from aﬃne Lie algebras. That this is the case was show n in [6] where the construction of [4, 5] was generalized to include many more Walgebras besides Wn. Motivated by a similar situation encountered in the theory o f dimensional reductions of selfdual Yang-Mills equations it was shown that to every e mbedding of sl2into the simple Lie algebra underlying the aﬃne algebra there is asso ciated a Poisson reduction leading to a Walgebra.sl2embeddings that are related to one another by inner auto- morphisms lead to the same reductions, so in order to ﬁnd out h ow many inequivalent 2 reductions there are one needs to count the number of equival ence classes of sl2em- beddings. For slnthis number is P(n), the number of partitions of n. The standard reductions leading Wnalgebras turned out to be associated to the so called ’princi pal embeddings’. The fact that one knows that these Walgebras have a linear origin helps a lot when one tries to analyse them. For example the construction of in variant chiral actions is facilitated as was shown in [7]. Also the construction of the classical covariant W gravities and their moduli spaces have been made possible by this [8, 9]. The procedure of Poisson reduction is of course purely class ical. In order to really make contact with CFT one would like to quantize the Walgebras obtained by Poisson reduction. The Wnalgebras were quantized in [10] by (naively) quantizing the well known Miura transformation. In essence what one does is clas sically express the Wn generators in terms of classical harmonic oscillators via t he Miura transform. One then quantizes the Wnalgebra by quantizing the harmonic oscillators and normal o rdering. More or less by accident this gives a quantum algebra that clo ses forAn. It does not work for all algebras however [2], as was to be expected since this is in general not a valid quantization procedure.

Question 1116 multiple-choice

Image interpolation is a key technique in both classical and quantum computing for resizing and enhancing digital images. Advances in quantum algorithms have introduced new methods for efficient and accurate image reconstruction. Which image interpolation algorithm combines quantum processing restricted to small subspaces, utilizes ancilla qubits for efficient circuit operation, and is computationally least expensive among quantum approaches while achieving high accuracy? 1) Bicubic interpolation 2) Quantum Fourier Transform (QFT) interpolation 3) Full QCT with s=n 4) OpenCV classical interpolation 5) JPEG-inspired block interpolation 6) Classical nearest neighbor interpolation 7) s=3-QCT quantum interpolation

✓ Correct Answer:

The correct answer is 7) s=3-QCT quantum interpolation.

📚 Reference Text:

s-QCT withs=nis equivalent to the full QCT approach pre- sented in the previous section. We compare the performance of the presented algo- rithms for image interpolation. A trial image is ﬁrst down-scaled using the recommended classical algorithm asprovidedbytheOpenCVlibrary[30]. Then, thedown- scaled image is interpolated back into its original size, and compared to the base image. Two metrics are com- monly used in order to assess the accuracy of image in- terpolation [31], the Peak Signal-to-Noise Ratio (PSNR) and Structural SIMilarity (SSIM) [32], detailed in App. E. The interpolation is performed using 4 algorithms, classical bicubic interpolation, QFT interpolation, QCT interpolation and s=3-QCT interpolation. In Table I we compare the two metrics for the diﬀer- ent interpolation methods. Bicubic interpolation is one of the go-to methods for classical image interpolation, it uses the adjacent 44pixels to compute each new value. This method achieves the best results by a slight margin when compared to the better of the quantum algorithms presented. However, it incurs a computational cost of O N2 p [33], where Npis the number of pixels in the image, exponentially more expensive than the quantum counterparts. Alternatively, the best performing quan- tum method is the s=3-QCT, which is also the least ex- pensive in terms of computational complexity. By focus- ing on small subsets of the image, the algorithm is both faster, due to processing data in superposition, and more accurate in reproducing the original data. 6 a) b) c) Figure 4. Interpolation of a grayscale image using the JPEG-inspired, s=3-QCT method. a) Depicts the original image embedded in the larger subspace. b) Showcases the quantum circuit that performs s=3-QCT interpolation for two dimensional data. The decomposition of the QCT blocks are shown in Fig. 3, and the two ancilla needed are implied. The interpolation algorithm for two dimensions can be applied simultaneously, therefore not incurring extra depth cost. The algorithm is independent on the original system size, as it only acts on the subspace sand the added ancilla m. c) Shows the result of the interpolated image using m= 2ancilla for each of the two dimensions. The original image is shown for scale. By using techniques of classical image processing we have enhanced the quantum interpolation algorithm, both in terms of the accuracy of the reconstruction and complexity. The ideas presented in the previous sections about computing in superposition have been extended to small subsets of the same image. By ﬁxing the subspace where the QCT acts on to 3 qubits, the interpolation algorithm no longer scales with the size of the original image, allowing for a very eﬃcient circuit for interpola- tion of natural data in a quantum computer. V. CONCLUSION We have proposed eﬃcient quantum techniques to interpolate distributions, all exploiting the Quantum Fourier Transform. We show that this QFT interpola- tion techniques achieves favourable results on data with negligible high-frequency components, as well as in prac- tical examples using natural images. Thistechniquescanbeusedtoenhancecurrentupload- ing algorithms by focusing on smaller scale and accurate uploading techniques that can then

Question 1117 multiple-choice

Quantum computing seeks efficient algorithms for the Hidden Subgroup Problem (HSP), which is central to solving problems in cryptography and group theory. Advances in quantum algorithm design have explored a variety of group structures, particularly semi-direct product groups and measurement strategies. Which quantum algorithmic technique leverages entangled measurements, specifically the "pretty good measurement," to efficiently solve the Hidden Subgroup Problem for certain semi-direct product groups? 1) Application of the Abelian Fourier transform followed by classical post-processing 2) Use of Kuperberg’s sieve method for sub-exponential time algorithms 3) Information-theoretic solutions based on state distinguishability 4) Classical brute-force search using group multiplication tables 5) Polynomial-time algorithms for q-hedral groups using separable measurements 6) Quantum algorithms based solely on direct product group decompositions 7) Entangled measurement strategies employing the pretty good measurement developed by Bacon, Childs, and van Dam

✓ Correct Answer:

The correct answer is 7) Entangled measurement strategies employing the pretty good measurement developed by Bacon, Childs, and van Dam.

📚 Reference Text:

rsaries using quantum computers. That is why an important part of the research on the HSP focused on the case where Gis the dihedral group. Notice that although no polynomial-time quantum algorithmis known so lvingthis case, a quantum algorithm running in sub-exponential time has been discovered by Kuperberg [15], and then improved by Regev [24]. The dihedral group can actually be deﬁned as the semi-direct produ ctDn=Zn⋊ Z2. Ettinger and Høyer [8] showed that considering the group Dnas the Abelian group Zn×Z2, and applying the Abelian Fourier transform over it is suﬃcient to obtain relevant information about the hidden subgroup. However, ∗E-mail: psi@is.s.u-tokyo.ac.jp †E-mail: legall@qci.jst.go.jp the post-processing proposed in [8] requires exponential-time to extract a set of generators of this subgroup from this information and thus the global algorithm is not eﬃcient. If , for other values of nandq, the groups Zn⋊Zqis suﬃciently Abelian, this method or other methods that failed to solv e completely the dihedral case may work and it is one of the motivation for considering this class of se mi-direct product groups. Indeed, Moore, Rockmore, Russell and Schulman [16] proposed a polynomial- time quantum algorithm based on the non-Abelian Fourier sampling method solving the HSP over the q-hedral groups Zp⋊Zqwherepandqare two primes such that qdividesp−1 andp/q=poly(logp). Other quantum algorithms are known solving the HSP over some class es of semi-direct product groups that are not semi-direct product of cyclic groups. Using the Fourie r transform over Zn pk×Z2, Friedl, Ivanyos, Magniez, Santha and Sen [9] solved in polynomial time the HSP over the groupsZn pk⋊Z2whenpkis a ﬁxed prime power. Radhakrishnan, R¨ otteler and Sen [21] haveshown th at it is possible to solvein polynomialtime, information-theoretically, the HSP over the Heisenberg groups Z2 p⋊Zp. Another class of semi-direct product groups for which eﬃcient quantum algorithms are known correspon ds to some wreath product groups [20]. A new promising method has been recently proposed by Bacon, Childs and van Dam [5], leading to eﬃcient quantum algorithms solving the HSP over some groups of the form A⋊ Zq, whereAis an Abelian group. This method is fundamentally diﬀerent from previous quantum algorit hms for the HSP: it uses entangled measurements, corresponding to the so-called pretty good meas urement, to identify the hidden subgroup. In particular, Bacon, Childs and van Dam’s algorithm solves in polynomial t ime the HSP over the groups of the form Zn⋊Zq, for any integer nand prime qsuch that n/q=poly(logn), thus improving the result [16]. They also present an eﬃcient quantum algorithm solving the HSP over theZr p⋊Zp, with ﬁxed r, improving the result [21] (and solving completely the problem, not only informat ion-theoretically). In this paper, we consider the HSP over the class of semi-direct pro duct groups Zpr⋊Zq, wherepandq are prime. The deﬁnition of the semi-direct product depending on th e choice of a homomorphism, we ﬁrst analyze, in Section 3, the diﬀerent possibilities for this homomorphism in function of p,randq. Then, in Section 4, we present a polynomial-time quantum algorithm so lving the HSP

Question 1118 multiple-choice

Quantum algorithms leverage unique properties such as superposition and entanglement to solve mathematical problems more efficiently than classical algorithms. One prominent approach is the hidden subgroup problem (HSP), which has significant implications for cryptography and computational complexity. Which step in the quantum algorithm for solving the hidden subgroup problem involves performing a quantum Fourier transform on a coset state to extract information about the hidden subgroup? 1) Preparing the quantum registers in a uniform superposition over group elements 2) Initializing the second quantum register to zero 3) Using the function to entangle the quantum registers 4) Measuring the second register to collapse the first into a coset state 5) Applying the quantum Fourier transform to the coset state and measuring 6) Selecting a random element from the group 7) Discarding the entangled state and resetting the system

✓ Correct Answer:

The correct answer is 5) Applying the quantum Fourier transform to the coset state and measuring.

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revised form) December 20, 2006; published electronically August 24, 2007. Support for this work was provided by theCalifornia Institute of Technology’s Institute for Quantum Information (IQI), the Mathematical Sci-ences Research Institute (MSRI), the Institute for Advanced Study (IAS), NSF grants ITR-0220070,ITR-0220264, CCR-0093065, CCR-0049092, EIA-0218443, QuBIC-0218563, and CCF-0524613, theCharles Lee Powell Foundation, and the Bell Fund. http://www.siam.org/journals/sicomp/37-3/44717.html †University of New Mexico, Albuquerque, NM 87131, and Santa Fe Institute, Santa Fe, NM 87501 (moore@cs.unm.edu) ‡Dartmouth College, Hanover, NH 03755 (rockmore@cs.dartmouth.edu). §University of Connecticut, Storrs, CT 06269 (acr@cse.uconn.edu). ¶California Institute of Technology, Pasadena, CA 91125 (schulman@caltech.edu). 938 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE POWER OF STRONG FOURIER SAMPLING 939 subgroup problem is the problem of determining the subgroup Hfrom such a function. Algorithms for these problems typically adopt the approach detailed below, calledFourier sampling [3]. Step1. Prepare two registers, the ﬁrst in a uniform superposition over the ele- ments of a group Gand the second with the value zero, yielding the state ψ 1=1/radicalbig |G|/summationdisplay g∈G|g/angbracketright⊗|0/angbracketright. Step2. Calculate (or, if it is an oracle, query) the function fdeﬁned on Gand XOR it with the second register. This entangles the two registers and results in thestate ψ 2=1/radicalbig |G|/summationdisplay g∈G|g/angbracketright⊗|f(g)/angbracketright. Step3. Measure the second register. This produces a uniform superposition over one of f’s level sets, i.e., the set of group elements gfor which f(g) takes the measured value f0. As the level sets of fare the cosets of H, this puts the ﬁrst register in a uniform distribution over superpositions on one of those cosets, namely cH, where f(c)=f0for some f0. Moreover, it disentangles the two registers, resulting in the state ψ3⊗|f0/angbracketright, where ψ3is a so-called coset state , ψ3=|cH/angbracketright=1/radicalbig |H|/summationdisplay h∈H|ch/angbracketright. Alternately, since the value f0we observe has no bearing on the algorithm, we can use the formulation in which the environment, rather than the user, measures f.I n that case, tracing over fyields a mixed state with density matrix ρH=1 [G:H]/summationdisplay f0|ψ3/angbracketright/angbracketleftψ3|=1 |G|/summationdisplay c|cH/angbracketright/angbracketleftcH|, i.e., a classical mixture consisting of one pure state ψ3for each coset. Kuperberg refers to this as the coherent hidden subgroup problem [18]. Step4. Carry out the quantum Fourier transform on ψ3orρHand measure the result. For example, in Simon’s algorithm [26], the “ambient” group Gover which the Fourier transform is performed is Zn 2,fis an oracle with the promise that f(x)=f(x+ y) for some y, and H={0,y}is a subgroup of order 2. In Shor’s factoring algorithm [25]Gis the group Z∗ n, where nis the number we wish to factor, f(x)=rxmod n for a random r

Question 1119 multiple-choice

Quantum algorithms for signal and image processing often rely on frequency-domain techniques like the Quantum Fourier Transform (QFT) and the Discrete Cosine Transform (DCT). These methods are particularly effective for encoding and manipulating natural image data on quantum computers. Which property of natural images makes the Discrete Cosine Transform (DCT) especially suitable for image compression algorithms such as JPEG? 1) Natural images are composed primarily of sharp discontinuities and high-frequency noise. 2) The DCT spreads image energy uniformly across all frequency components. 3) Natural images have most of their information in high-frequency DCT coefficients. 4) Natural images exhibit suppressed high-frequency content, concentrating energy in low-frequency DCT coefficients. 5) The DCT cannot represent smooth image gradients accurately. 6) JPEG compression relies on storing all frequency components with equal precision. 7) Most natural images require complex, non-linear transformations for effective compression.

✓ Correct Answer:

The correct answer is 4) Natural images exhibit suppressed high-frequency content, concentrating energy in low-frequency DCT coefficients..

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low-pass ﬁlter to the signal, so that the high frequencies are cut oﬀ while maintaining the band-limited frequen- cies intact. This, however, is not as straightforward with quantum states, as they have to maintain their `2norm throughout the process, and a low-pass ﬁlter is not a uni- tary operation. Therefore, see App. C for the detailed derivation, the optimal distance between states will be j i~j i 2 =2kj outik2 1 +q 1kj outik2; 2kj outik2;(2) wherej outidenotes the Fourier components of target statej ioutside of the n-qubit band-limit. The distance between the target state and the best possible interpo- lation will depend on the `2norm of the Fourier com- ponents that are ﬁltered out. Using this result, we can bound the trace-distance between the target and inter- polated state by distTr(j i;~j i)p 2kj outiks 1kj outik2 2; p 2kj outik:(3) That being said, many applications do not have access to the the ﬁltered distribution, as only the low-resolution values of the original distribution are available. The interpolated state will then suﬀer from aliasing eﬀects, where the Fourier components of subsampled distribu- tions are mixed with its own high-frequency modes due to the periodic nature of the discrete Fourier transform. In other words, the frequency components outside the n- qubit band limit, j outi, are added to the low frequency components, which we will call j ini, creating artifacts in the interpolated distribution. The accuracy of the QFT interpolation algorithm will be worse in this approach, but we can still provide ana- lytical bounds on the distinguishability due to the eﬀects of aliasing. Now the distance between the target and interpolated state will be, refer to App. D for details, j i~j i 2 =2kj outik2 N(N1)2 N 2kj outik2;(4) whereN1is the normalization factor of the target state~j idue to the eﬀects of aliasing. Therefore, theupper bound on the trace-distance under aliasing eﬀects remains the same as Eq. 3. We have shown how the QFT interpolation algorithm approaches the target quantum state for a probabil- ity distribution. The smaller the norm of the high- frequency components, the better the interpolation will be. When the distribution is n-qubit band-limited, that iskj outik= 0, it captures all the information of the un- derlying function and can be interpolated with as many qubits as needed. Even for non band-limited distribu- tions, we show in Fig. 1 that with an initial uploading of 4 qubits, a Gaussian distribution can be interpolated with high ﬁdelity to an exponentially larger space. III. EFFICIENT QCT RESAMPLING OF NATURAL IMAGES Interpolation methods, referred to also as resampling, are very common techniques in image processing. More- over, natural images tend to have suppressed high- frequency components, and in particular, algorithms de- signed to process natural images tend to utilize the DCT [3]. This real-to-real variant of the Fourier Transform is particularly suited for this type of signals. So much so, that some of the most prominent image processing tech- niques, like the image compression scheme JPEG [4], use

Question 1120 multiple-choice

In quantum computing, qudits are quantum systems characterized by having more than two discrete levels, generalizing the concept of qubits. This allows for novel gate formulations and the possibility of more efficient quantum algorithms and error correction schemes. Which key advantage does a universally applicable qudit gate formulation provide for quantum computation? 1) It enables the design and simulation of quantum circuits with arbitrary dimension, allowing universal computation beyond binary qubits. 2) It confines quantum computation exclusively to two-level systems, restricting algorithmic flexibility. 3) It eliminates the need for error correction in quantum hardware by making gates inherently fault-tolerant. 4) It allows only binary operations, limiting the exploration of higher-dimensional entanglement. 5) It prohibits the implementation of essential quantum algorithms such as the Quantum Fourier Transform. 6) It requires separate gate sets for each possible qudit dimension, complicating circuit design. 7) It prevents the use of photonic systems and trapped ions for physical realization of quantum circuits.

✓ Correct Answer:

The correct answer is 1) It enables the design and simulation of quantum circuits with arbitrary dimension, allowing universal computation beyond binary qubits..

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Title: Generalised Quantum Gates for Qudits and their Application in Quantum Fourier Transform Year: 2024 Paper ID: 7078292cf717ff9ce6265147f55d7831a11b580d Source: semantic-scholar URL: https://www.semanticscholar.org/paper/7078292cf717ff9ce6265147f55d7831a11b580d Abstract: Quantum computing with qudits, quantum systems with $d>2$ levels, offers a powerful extension beyond qubits, expanding the computational possibilities of quantum systems, allowing the simplification of the implementation of several algorithms and, possibly, providing a foundation for optimised error correction. In this work, we propose a novel formulation of qudit gates that is universally applicable for any number of levels $d$, without restrictions on the dimensionality. By extending the mathematical framework of quantum gates to arbitrary dimensions, we derive explicit gate operations that form a universal set for quantum computation on qudits of any size. We demonstrate the validity of our approach through the implementation of the Quantum Fourier Transform (QFT) for arbitrary $d$, verifying both the correctness and utility of our generalized gates. This novel methodology broadens the design space for quantum algorithms and fault-tolerant architectures, paving the way for advancements in qudit-based quantum computing.

Question 1121 multiple-choice

Quantum phase estimation (QPE) is a key algorithm in quantum computing, used for tasks such as quantum simulation and factoring. Implementing QPE efficiently on noisy intermediate-scale quantum (NISQ) hardware requires minimizing error-prone operations and adapting algorithms to hardware constraints. Which modification to the iterative quantum Fourier transform (QFT) in QPE most directly reduces the overall number of controlled-rotation gates required for a given phase estimation accuracy on NISQ devices? 1) Replacing controlled-rotation gates with unitary rotation gates 2) Increasing the number of ancilla qubits in the algorithm 3) Adding extra measurement calibration steps before each gate 4) Using analog quantum simulation instead of digital quantum algorithms 5) Doubling the circuit depth by repeating QFT operations 6) Implementing error correction codes for all qubits 7) Reducing the total number of physical qubits in the hardware

✓ Correct Answer:

The correct answer is 1) Replacing controlled-rotation gates with unitary rotation gates.

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decreased by sacriﬁcing some amount of the overall accuracy of the phase determination. This paper showed that it was possible to remove the ancilla qubit from the iterative QFT without the loss of its functionality, thereby removing the unnecessary controll ed- rotation gates replacing them with unitary rotation gates f or each qubit. Making this change did reduce the number of controlled rotation gates required for a given level of accu racy. All of these approaches were implemented on NISQ com- puters. One of the properties of these machines is that there are multiple sources of measurement errors that do occur and can be attributed to the physical system and the overall environment. For superconducting qubits coupled to readou t cavities the state of the qubit is determined by measurement the response of a microwave tone incident on the readout cavity. Quantum computing hardware platforms do contain classical sources of noise that lead to readout errors of the qubit state. In addition it can also happen that not only the relaxation time T1decays the state of qubit during the measure- ment but also crosstalk between resonantors on chip and on th e lines changes the probability distribution of the qubit sta tes. These types of errors can be addressed by various techniques such as measurement calibration and error mitigation. In this paper, the proposed approach was tested and ana- lyzed using IBMQX4 which contains 5 qubits. The experimen- tal results showed that using the proposed method the phase can be correctly estimated with reasonably distinct probab ility to other probabilities. The proposed approach can be easily applied to the large number of qubit systems to estimate the unknown phase of the complicated inputs. However, errors in the system described in this section can also increase as the number of qubits increases. Thus, it is critical that hig her ﬁdelity, longer coherence time, and lower readout errors sh ould be followed by increasing the number of qubits to take fulladvantage of the presented technique. V. CONCLUSIONS This paper demonstrates how to implement existing quan- tum phase estimation (QPE) algorithms on the state-of-the- art IBM quantum computers. Our work also has documented the challenges of implementing QPE algorithms on real quantum processors. We have proposed modiﬁed solutions of these algorithms by minimizing the number of controlled-rotation gates and b y utilizing the digital computer’s capabilities. Our experi mental results can guide researchers to consider these challenges when they implement their quantum algorithms on noisy intermediate-scale quantum (NISQ) computers. Using these methodologies, substantial progress has been achieved app ly- ing QPE in various subject domains. The experimental results show that our solutions signiﬁ- cantly increase the accuracy for ﬁnding correct phase. Re- searchers can now implement these techniques using publicl y available NISQ quantum computers such as the IBMQX4 [2] and Rigetti QPU [29] in order to take better advantage of exis t- ing NISQ machines and advance toward the longer term goals of quantum advantage and ultimately quantum supremacy. REFERENCES [1] John

Question 1122 multiple-choice

The study of finite 2-groups, especially those with restricted subgroup structures, involves understanding the interplay between their maximal subgroups, centers, and commutator subgroups. Groups such as the quaternion group $ Q_8 $, the dihedral group $ D_8 $, and cyclic groups frequently serve as building blocks in their classification. Which of the following must be true for a finite 2-group $ G $ that has exactly three abelian maximal subgroups, three minimal nonabelian maximal subgroups, and derived subgroup of order 2? 1) The group $ G $ is isomorphic to $ D_8 \times D_8 $. 2) The group $ G $ has a derived subgroup of order 8. 3) The group $ G $ must have order 24. 4) The group $ G $ contains a maximal subgroup isomorphic to $ C_2 \times C_4 $. 5) The group $ G $ is a non-split extension of $ Q_8 $ by a cyclic group. 6) The group $ G $ has a Frattini subgroup of order 4. 7) The group $ G $ is isomorphic to one of: the central product $ Q_8 \ast C_{2n} $ (with $ n \geq 3 $), the direct product $ Q_8 \times C_{2n} $ (with $ n \geq 2 $), or $ D_8 \times C_{2n} $ (with $ n \geq 2 $).

✓ Correct Answer:

The correct answer is 7) The group $ G $ is isomorphic to one of: the central product $ Q_8 \ast C_{2n} $ (with $ n \geq 3 $), the direct product $ Q_8 \times C_{2n} $ (with $ n \geq 2 $), or $ D_8 \times C_{2n} $ (with $ n \geq 2 $)..

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subgroup, then one o f the following holds: (a)G=Q∗Z, where Q∼=Q8,Z∼=C2n,n≥3andQ∩Z= Z(Q). (b)G=Q×Z, where Q∼=Q8andZ∼=C2n,n≥2. (c)G=D×Z, where D∼=D8andZ∼=C2n,n≥2. Proof. By our assumption, Ghas exactly three abelian maximal sub- groups. This implies |G′|= 2 and Gpossesses exactly three maximal sub- groups which are minimal nonabelian. Let Hbe a minimal nonabelian max- imal subgroup of G. Since H′=G′∼=C2, we get G=HZ(G), where Z(G)∩H= Z(H) = Φ( H) = Φ( G). All three maximal subgroups of G containing Z( G) are abelian. IfGis a title group with |G′|= 2, then the similar arguments (as above) imply that Gpossesses more than one abelian maximal subgroup. In what follows Hwill denote a ﬁxed maximal subgroup of Gwhich is minimal nonabelian. Suppose that there is an involution c∈Z(G)−Z(H). Then G=H× /an}bracketle{tc/an}bracketri}htand so each maximal subgroup of Gwhich does not contain /an}bracketle{tc/an}bracketri}htis isomorphic to G//an}bracketle{tc/an}bracketri}ht∼=Hand so is minimal nonabelian, a contradiction. Hence there are no involutions in Z( G)−Hwhich implies that Ω 1(Z(H)) = Ω 1(Z(G)) so that d(Z( H)) = d(Z( G)). It follows that for eachx∈Z(G)−H,x2∈Z(H)−Φ(Z(H)). Suppose that |G|= 24. Then each nonabelian maximal subgroup of Gis isomorphic to D 8or Q8and so is FINITE 2-GROUPS 71 minimal nonabelian, a contradiction. Hence |G| ≥25and in particular His not isomorphic to Q 8or D8. (i) First assume that His metacyclic. Since His not isomorphic to Q 8, it follows that His a ”splitting” metacyclic group and so we may set: H=/an}bracketle{ta, b|a2m=b2n= 1, ab=az, z =a2m−1/an}bracketri}ht, where m≥2,n≥1,m+n≥4,H′=/an}bracketle{tz/an}bracketri}ht,|H|= 2m+nand|G|= 2m+n+1. We have Z( H) =/an}bracketle{ta2/an}bracketri}ht × /an}bracketle{tb2/an}bracketri}ht= Φ(H) = Φ( G) and for each x∈Z(G)−H, x2∈ /an}bracketle{ta2, b2/an}bracketri}ht − /an}bracketle{ta4, b4/an}bracketri}htsince /an}bracketle{ta4, b4/an}bracketri}ht= Φ(Z( H)). Suppose that n= 1 so that bis an involution, m≥3,H∼=M2m+1and Z(H) = /an}bracketle{ta2/an}bracketri}ht. Hence Z( G)∼=C2mis cyclic and therefore we may choose c∈Z(G)−Hsuch that c2=a−2which gives ( ca)2=c2a2= 1. Since [ca, b] =z, we get D=/an}bracketle{tca, b/an}bracketri}ht∼=D8,/an}bracketle{tca, b/an}bracketri}ht ∩Z(G) =/an}bracketle{tz/an}bracketri}htwhich together with |G: Z(G)|= 4 gives G=DZ(G). But D∗Ω2(Z(G)) contains a subgroup Q∼=Q8and so G=Q∗Z(G) with Z( G)∼=C2m,m≥3,Q∩Z(G) = Z( Q) =/an}bracketle{tz/an}bracketri}ht and we have obtained the groups stated in part (a) of our theor em. It remains to treat the case n≥2. Suppose that there is an involution x∈G−H. We know that x/ne}ati≀nslash∈Z(G) and so [ a, x]/ne}ati≀nslash= 1 or [ b, x]/ne}ati≀nslash= 1. Obviously, /an}bracketle{ta, b, x /an}bracketri}ht=G. Suppose that [ a, x]/ne}ati≀nslash= 1. Since /an}bracketle{ta/an}bracketri}ht/triangleleftequalG,/an}bracketle{ta, x/an}bracketri}htis minimal nonabelian of order 2m+1. Because |G|= 2m+n+1andn≥2, Φ(G)/an}bracketle{ta, x/an}bracketri}ht=/an}bracketle{ta, x, b2/an}bracketri}htis a maximal subgroup of Gwhich is neither abelian nor minimal nonabelian. Assume at the moment that [ b, x]/ne}ati≀nslash= 1. In that case /an}bracketle{tb/an}bracketri}ht×/an}bracketle{tz/an}bracketri}htis normal in Gand so/an}bracketle{tb, x/an}bracketri}htis a nonmetacyclic minimal nonabelian subgroup of Gof order 2n+2. It follows that /an}bracketle{tb, x/an}bracketri}htmust be maximal in Gwith |G|= 2n+3(and so m= 2). But the case of a nonmetacyclic minimal nonabelian maximal s ubgroup in G will be studied in part (ii) of this proof. Hence we may assume [x, b] = 1 so that [x, ab] = [x, a][x, b] =zwhich implies that /an}bracketle{tx, ab/an}bracketri}htis minimal nonabelian and so /an}bracketle{tx,

Question 1123 multiple-choice

Quantum walks are mathematical models that generalize classical random walks to the quantum domain and have become powerful tools for simulating quantum systems, including relativistic particles and condensed matter phenomena. The structure of quantum walks and their coupling to external fields allow the investigation of fundamental effects such as Landau levels and topological phases. For a quantum walk simulating a Dirac fermion on a polar grid, which requirement is imposed by the topological properties of spinors to ensure a globally well-defined evolution? 1) The radial coordinate must be discretized using non-uniform steps 2) The polar angle must be extended from 0 to 4π instead of the usual 0 to 2π 3) The walk must be restricted to square lattice geometry 4) The spinor components must be real-valued functions 5) The momentum operator must be replaced by its classical counterpart 6) The coupling to gauge fields must be purely scalar 7) The evolution operator must be non-unitary

✓ Correct Answer:

The correct answer is 2) The polar angle must be extended from 0 to 4π instead of the usual 0 to 2π.

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Dirac quantum walks with conserved angular momentum Gareth Jay1,Pablo Arnault2 3,yand Fabrice Debbasch4z 1Physics Department, The University of Western Australia, Perth, WA 6009, Australia 2Departamento de Fsica Terica and IFIC, Universidad de Valencia and CSIC, Dr. Moliner 50, 46100 Burjassot, Spain 3Institute for Quantum Computing, 200 University Ave W, Waterloo, ON N2L 3G1, Canada 4Sorbonne Universit e, Observatoire de Paris, Universit e PSL, CNRS, LERMA, F-75005, Paris , France (Dated: August 16, 2021) A Quantum Walk (QW) simulating the at (1 + 2)D Dirac Eq. on a spatial polar grid is con- structed. Because fermions are represented by spinors, which do not constitute a representation of the rotation group, but rather of its double cover, the QW can only be de ned globally on an extended spacetime where the polar angle extends from 0 to 4 . The coupling of the QW with arbitrary electromagnetic elds is also presented. Finally, the cylindrical relativistic Landau levels of the Dirac Eq. are computed explicitly and simulated by the QW. INTRODUCTION First proposed by Feynman as possible discretizations of Dirac path integrals [1, 2], Quantum Walks (QWs) are unitary quantum automata that can be viewed as formal generalizations of classical random walks. Rein- troduced later by Aharonov et al. [3], and then studied systematically by Meyer [4], QWs, like classical random walks in classical computing, have found application in quantum information and algorithmic development [5{ 7]. They can also be used as quantum simulators [8{16], where the lattice represents a discretization of continuous space, that could potentially represent a realistic discrete spacetime underlying the apparently continuous physical universe [17]. It has been shown that several discrete-time quan- tum walks de ned on regular square lattices simulate the Dirac dynamics in various spacetime dimensions and that these Dirac Quantum Walks (DQWs) can be coupled to various discrete gauge elds [18{28]. Extensions to regu- lar non-square lattices have also been proposed [29{31]. More recently, a discrete action principle has been con- structed for quantum automata [32]. In this context, the charge current of 1D DQWs has been recovered and a stress-energy `tensor' for DQWs has been constructed. In particular, a `true' Hamiltonian (as opposed to an e ec- tive Hamiltonian), and a linear momentum for 1D DQWs have been proposed and their conservation has been es- tablished for free 1D QWs. These results extended to QWs de ned on higher dimensional square lattices. It is however not obvious that angular momenta can be built for QWs. This question is not purely of fundamental, but also of practical interest, to simulate problems with axial symmetry. For example, the Landau levels of an electron in a constant uniform magnetic eld are degenerated and the conserved angular momentum can be used to dis- tinguish between states sharing the same energy. An experimental proposal based on magnetic discrete-time quantum walks has been made to construct anomalous Floquet Chern topological insulators, that exhibit edge charge currents similar to those observed in the quantumHall e ect [33]. The aim of this

Question 1124 multiple-choice

Quantum groups generalize classical symmetries by introducing q-deformed algebraic structures, leading to novel definitions of numbers, derivatives, and special functions. These frameworks are central in the study of integrable systems, representation theory, and non-perturbative phenomena in physics. In the context of quantum groups and integrable systems, which property is specifically associated with the more sophisticated version of the q-exponential function Eq(x) as opposed to the simple q-exponential eq(x)? 1) It satisfies the equation Dx eq(x) = eq(x) only. 2) It is defined exclusively for integer values of x. 3) It has no relevance to matrix elements in group representations. 4) It possesses invariance under all quantum group automorphisms. 5) It features a "summation rule" and a "decoupling" property for tensor structures. 6) It reduces to a linear function in the classical limit q → 1. 7) It can only be constructed for groups with rank one.

✓ Correct Answer:

The correct answer is 5) It features a "summation rule" and a "decoupling" property for tensor structures..

📚 Reference Text:

group case. Of course, they are primarily needed in order to obtain the Hir ota equations in the nice form of diﬀerence equations. This, however, does not ﬁx the choic e completely. Indeed, there are various ways to deﬁne q-numbers and thus q-derivatives and q-special functions. These ways are in correspondence with the choices of comultiplication in qua ntum algebra. Through this text we are using “symmetric” coproduct (13), thus q-numbers need to be deﬁned in symmetric way as well. For given deﬁnition of q-numbers there is still an ambiguity in the choice ofq-exponent. In (1) and in section 3 we used the simplest deﬁnition12 eq(x) =/summationdisplayxn [n]!,satisfyingDxeq(x) =eq(x). (111) 12One can also keep in mind the possibility to consider α-dependent qin (1) (see for example [15]). The reasonablechoicesof qαcouldbeqα=qα2/2orqα=q||α||, where||α||isthe“height”oftheroot α,||α||=αρ. 28 However, it can appear more reasonable in some situations to use mo re sophisticated choices, for example, Eq(x) =/summationdisplay nqn(n−1)/2xn [n]!,satisfyingDxEq(x) =M+ xEq(x) =Eq(qx).(112) The advantage of such q-exponent is that it satisﬁes the “summation rule” Eq(x)Eq(y) =Eq(x+y),forxy=q2yx. (113) With this deﬁnition we have, for example, the following “decoupling” pr operty: Eq(¯ti∆k(T−i)) =Eq(¯tiT−i⊗q−Hi⊗...⊗q−Hi)×Eq(¯tiqHi⊗T−i⊗...⊗q−Hi)× ×...×Eq(¯tiqHi⊗qHi⊗...T−i)(114) for eachi= 1,...r(but not for the sum over i). 6 Conclusion To summarize, we suggest to begin investigation of the concept of τ-function in the very general framework, deﬁning it as a generating function for all the matrix elements of a group in a given representation. In physical language this notion is of cour se very close to that of the “non-perturbative partitionfunction”, which contains informatio nabout all the amplitudes in the theory. We argued that the bilinear Hirota-like identities, mixing v arious representations, are a very general feature of such generating functions, as are the analogues of Virasoro constraints and “string equation”. This opens the road to a very g eneral group-theoretical interpretation of non-perturbative partition functions in quantu m mechanics, ﬁeld and string theories. As a byproduct of this general formalism we derived an example of Hir ota equations for quantum groups, which incorporate properly the non-commut ative (q-number) nature of the corresponding τ-functions. This q-numberτ-function can be further transformed into ac-number one in a particular representation of coordinate ring A(G). Such procedure is 29 of course reminiscent of the general ideology of the second quant ized string theory (when the partition function of the ﬁrst-quantized model is further sub stituted into the functional integral over the space of theories). The most important example which remains to be discussed in order to demonstrate all the new features of “generalized τ-functions” is the case of non-fundamental representations ofG=SL(3), where for the ﬁrst time the existence of “non-Cartanian” tim e-variables will be essential. As to the fundamental representations of G=SL(n), there exists a closed subsystem ofHirotaequations forthem, which doesnotrequire mo rethatn−1 time-variables, and all of these can be associated with commuting generators. This example is described in some detail in section 5, though its quantum analogue is to be worked out in more details. In particular, the interpretation within the terms of quantum

Question 1125 multiple-choice

In computational mathematics and cryptography, the LLL algorithm is used to reduce lattice bases, producing vectors with special structural and length properties that are crucial for various applications. Understanding how these reduced bases are partitioned and interpreted is essential for tasks such as integer relation finding and ideal multiplication. When an LLL-reduced lattice basis is partitioned into the first k−r vectors and the last r vectors, what is the principal role of the first k−r vectors in applications such as number theory and cryptography? 1) They represent linearly independent relation vectors among the original lattice generators. 2) They provide the shortest possible basis for the entire lattice. 3) They approximate the orthogonal complement of the lattice space. 4) They contain the original input generators, unchanged. 5) They maximize the determinant of the lattice. 6) They encode the error bounds for basis approximations. 7) They are used solely for polynomial interpolation purposes.

✓ Correct Answer:

The correct answer is 1) They represent linearly independent relation vectors among the original lattice generators..

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coordinate of the vec- tor. The LLL-algorithm is then applied to the lattice basis 297 ˜a1,...˜akto obtain an LLL-reduced basis ˜b1,..., ˜bk. For the ﬁrstk−rof these vectors we will denote the top and bottom components as ˜bj= (m j,ˆzj)T(1≤j≤k−r), and for the last rvectors as ˜bk−r+j= (m/prime j,ˆbj)T(1≤j≤r). Note that the vectors m1,...,mk−r∈Zkare the co- eﬃcient vectors transforming the vectors ˆa1,..., ˆakto the vectors ˆz1,..., ˆzk−r, respectively, and the vector m/prime jtakes ˆa1,..., ˆaktoˆbj(for 1 ≤j≤r). In [BP87] and [BK93] it is shown that the resulting ba- sis has the following two properties. First, the top left k−rcolumns contain a linearly independent set of relations m1,...,mk−r∈Zkfor the exact vectors a1,...,ak. A rela- tion vector mj= (m 1j,...,m kj)tsatisﬁesPk i=1mijai= 0. In the exact matrix the vectors z1,...,zk−rwould be zero, so the approximate vectors ˆziwill be small. The second property of the resulting basis is that the bottom right of the matrix contains approximations ˆb1,..., ˆbrto a basis for the lattice generated by a1,...,ak. In [BK93] it is shown that given good enough approxima- tions of a,,...,ak, the vectors bj=kX i=1m/prime i,jai(1≤j≤r) (4.1) form a basis for Land satisfy ||bj|| ≤(√ kn+ 2)2k−1 2λj(Lr) (1 ≤j≤r). (4.2) This holds for every sublattice Lrwhich is spanned by a subset ofrlinearly independent vectors of a1,...,ak. Lemma 4.2. Chooseqas is [BK93, Theorem 4.1]. The square length of the coeﬃcient vectors m/prime j(1≤j≤r) that transform the generators aiofLinto a basis vector bjof Las in Equation 4.1 is bounded by (αr−1 d(L)·√r∆1,j)2+ ∆2 2. Here ∆1,jis the right-hand-side of Equation 4.2, ∆1,j= (√ kn+ 2)·2k−1 2·λj(Lr), andLris any sublattice of Lwhich is spanned by a subset of rlinearly independent vectors of a1,...,ak. The quantity ∆2is ∆2=√ k−r·2k−1 2„k√n 2+√ k« ·αr d(L). Lemma 4.2 can be used to prove the following theorem which shows that a basis of bounded length for Lexists, for which we can eﬃciently compute a good approximation. Theorem 4.3. Leta1,...,ak∈Rngenerate a lattice L of rankr, letµbe a lower bound on the shortest vector, letα= max ||ai||, and letqbe such that 2q≥(k2(k+1)/2· max||ai||)r/(µdet(L)2). Given approximations of a1,..., ak∈Rnwithqbits of precision, a basis ˆc1,..., ˆcrforLcan be computed in time polynomial in q, where the exact vectors cjsatisfy ||cj|| ≤(√ kn+ 2)2k−1 2·λj(L).The absolute error on each output vector ˆciis bounded by rkγ 1γ3√n/2q+1. Hereγ1=r“ αr−1 det(L)√r∆1,n”2 + ∆2 2, and γ3=s„(α/prime)r d(L)·√r· ||bj||«2 +„√ k−r√ k·(α/prime)r d(L)«2 , withα/prime= (√ kn+ 2)2k−1 2α. 4.4 The algorithm for computing etO Givent, compute a basis for etO: 1. Choose a polynomial q. 2. For each bit index jdo the following: 3. Compute the diagonal matrix Tj, where (T j)i,i=eri,j fori

Question 1126 multiple-choice

In quantum information theory, understanding random unitary matrices and their statistical properties is essential for analyzing quantum processes and developing new algorithms. Specialized mathematical tools have been developed to facilitate calculations involving random quantum operations. Which mathematical construct is specifically designed to provide a rigorous and uniform way to average over all possible unitary transformations in quantum mechanics, enabling unbiased sampling of quantum operations? 1) Tensor network diagram 2) Symmetric subspace 3) Haar measure 4) Classical shadow tomography 5) Concentration inequality 6) Moment operator 7) Unitary design

✓ Correct Answer:

The correct answer is 3) Haar measure.

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Introduction to Haar Measure Tools in Quantum Information: A Beginner’s Tutorial Antonio Anna Mele Dahlem Center for Complex Quantum Systems, Freie Universit ¨at Berlin, 14195 Berlin, Germany The Haar measure plays a vital role in quantum information, but its study often re- quires a deep understanding of representation theory, posing a challenge for beginners. This tutorial aims to provide a basic introduction to Haar measure tools in quantum information, utilizing only basic knowledge of linear algebra and thus aiming to make this topic more accessible. The tutorial begins by introducing the Haar measure with a specific emphasis on characterizing the moment operator, an essential element for computing integrals over the Haar measure. It also covers properties of the symmetric subspace and introduces helpful tools like tensor network diagrammatic notation, which aid in visualizing and simplifying calculations. Next, the tutorial explores the concept of unitary designs, providing equivalent definitions, and subsequently explores approx- imate notions of unitary designs, shedding light on the relationships between these different notions. Practical examples of Haar measure calculations are illustrated, in- cluding the derivation of well-known formulas such as the twirling of a quantum channel. Lastly, the tutorial showcases the applications of Haar measure calculations in quantum machine learning and classical shadow tomography. Contents 1 Introduction 1 2 Notation and Preliminaries 1 3 Haar measure and moment operator 2 4 Symmetric subspace 9 5 Vectorization formalism 14 6 Tensor network diagrams 17 7 Unitary designs 22 8 Approximate unitary designs 26 9 Examples and applications 30 9.1 Examples of moment calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 9.2 Concentration inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 9.3 Applications in Quantum Machine Learning . . . . . . . . . . . . . . . . . . . . . . 37 9.4 Classical shadow tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 10 Acknowledgments 43 Antonio Anna Mele: a.mele@fu-berlin.de Accepted in Quantum2024-02-14, click title to verify. Published under CC-BY 4.0. 1arXiv:2307.08956v4 [quant-ph] 9 Apr 2024 1 Introduction The notion of Haar measure formalizes the fundamental concept of drawing unitary matrices uniformly at random. In the context of quantum information, unitary matrices often represent quantum evolutions that are commonly described using quantum circuits. The applications of the Haar measure are widespread across various domains within quantum information, such as quantum tomography [1–8], computational advantage of random quantum circuits sampling [9– 16], benchmarking of quantum devices [17–20], quantum foundations and communication [21–29], quantum machine learning [30–33], quantum many-body and high energy physics [34–44]. Although the Haar measure serves as a

Question 1127 multiple-choice

In quantum information theory, representation theory of groups such as the symmetric group plays a central role in understanding the structure of multipartite quantum states. Classical notions like frequency typicality are being generalized to quantum types through invariant subspaces and advanced measurement techniques. Which concept most directly enables the construction of a quantum analog of classical typical sets by selecting invariant subspaces under permutations, allowing the decomposition of tensor products into irreducible components relevant for quantum Shannon theory? 1) Quantum error correction codes 2) Entanglement entropy 3) Unitary group representations 4) Von Neumann measurement 5) Representation theory of the symmetric group 6) Quantum channel capacity 7) Bell inequalities

✓ Correct Answer:

The correct answer is 5) Representation theory of the symmetric group.

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theory a nd quantum states [14], [15], [16]. They are well suited to derive e.g. estimates of the spectra of states th rough the basic estimate described in [44] or entropic inequalities. On a more general level, recent results [40, 17, 18] show that there is an enormously fruitful connection between representation theory and the structure of multiparty states. Unfortunately, the quantum method of types as such was not app lied or developed any further and could thus not even be used to give proofs for such basic tasks as hypot hesis testing, which are both simple and at the very heart of quantum Shannon information theory. At this point, we clearly distinguish between applying the structure given by representation theory as was suc cessfully done in [25] and a reasoning in terms of anything comparable to classical types. We thus now mak e an explicit choice of ’types’ that singles out certain invariant subspaces of the symmetric group, an d then show that with this choice we can prove our Theorem 1. We hope that this will give new thrust to th e idea of ’quantum types’. At present, it seems not totally clear what such a theory should look like, the relevant parameters will have to be found by subsequently proving the important theorems in quantum Shannon information theory. For the time being, we shall restrict to considering decomp ositions /BD⊗n=/summationtext iPion (Cd)1⊗n where each Piis an orthogonal projection onto an irreducible subspace of the na tural action of Snon (Cd)1⊗nand further tr {PiPj}= 0 fori/ne}ationslash=j. We then consider the set {Pi}ias a POVM, which is to re- placethefrequencytypicalsubsetsfromtheclassicaltheory. A noutlineofourapproachwillnowbe given. Despite above mentioned drawbacks, the methods from classical in formation theory were taken over to the quantum case with great success, e.g. in the form of ’fr equency typical subspaces’ that arise from choosing a particular orthonormal basis that is labelled by a clas sical alphabet and then looking at the linear span of all elementary tensor products of nof these basis vectors that are built up from the same number of each of the basis vectors. Closely related is the study of isotypical subspaces in the n-fold tensor product of Cd. Like typical sets in the classical case, both of the above classes o f subspaces are permutation invariant. This work combines the two approaches by exploiting the fact that e very frequency typical subspace can be decomposed into subspaces that are irreducible under the actio n of the permutation group. 3 As mentioned already, the action of the commutant of Snon the frequency typical subspaces will be insigniﬁcant in every single one of the proofs given here. Its exact r ole will therefore also be addressed in future work. The basic idea behind the paper is to look, for a given orthonormal ba sis{ei}d i=1and a ’fre- quency’ or ’type’ (a nonnegative function f:{1,...,d} →Nsatisfying/summationtext if(i) =n), at representations of the symmetric group Snon the frequency typical subspaces Vf:= span({ei1⊗...⊗ein:|{k:ik=i}|=f(i) for alli}. Since these

Question 1128 multiple-choice

Variational quantum algorithms rely on optimizing parameters in quantum circuits, but their effectiveness can be severely limited by phenomena such as barren plateaus. Advances in mathematical frameworks have extended the understanding of when and why barren plateaus occur in quantum computing, especially in relation to circuit expressiveness and entanglement. In the analysis of parametrized matchgate circuits, which concept is generalized from the dimension of the Lie algebra to provide a broader measure of circuit expressiveness, and is directly related to the structure of barren plateaus? 1) The number of qubits in the quantum circuit 2) The rank of the measurement operator 3) The dimension of Lie group modules 4) The depth of the quantum circuit 5) The eigenvalue spectrum of the Hamiltonian 6) The number of Majorana operators in the system 7) The classical runtime complexity of simulation

✓ Correct Answer:

The correct answer is 3) The dimension of Lie group modules.

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Title: Showcasing a Barren Plateau Theory Beyond the Dynamical Lie Algebra Year: 2023 Paper ID: 5b06f89e5c4854e291aef2522a670b8bb00acafd Source: semantic-scholar URL: https://www.semanticscholar.org/paper/5b06f89e5c4854e291aef2522a670b8bb00acafd Abstract: Barren plateaus have emerged as a pivotal challenge for variational quantum computing. Our understanding of this phenomenon underwent a transformative shift with the recent introduction of a Lie algebraic theory capable of explaining most sources of barren plateaus. However, this theory requires either initial states or observables that lie in the circuit's Lie algebra. Focusing on parametrized matchgate circuits, in this work we are able to go beyond this assumption and provide an exact formula for the loss function variance that is valid for arbitrary input states and measurements. Our results reveal that new phenomena emerge when the Lie algebra constraint is relaxed. For instance, we find that the variance does not necessarily vanish inversely with the Lie algebra's dimension. Instead, this measure of expressiveness is replaced by a generalized expressiveness quantity: The dimension of the Lie group modules. By characterizing the operators in these modules as products of Majorana operators, we can introduce a precise notion of generalized globality and show that measuring generalized-global operators leads to barren plateaus. Our work also provides operational meaning to the generalized entanglement as we connect it with known fermionic entanglement measures, and show that it satisfies a monogamy relation. Finally, while parameterized matchgate circuits are not efficiently simulable in general, our results suggest that the structure allowing for trainability may also lead to classical simulability.

Question 1129 multiple-choice

In quantum computing, query models frequently utilize oracles to reveal hidden information about a string, with the structure of the oracle and its response register influencing computational success. Group theory, particularly the distinction between abelian and non-abelian groups, plays a crucial role in how quantum algorithms exploit interference to identify hidden strings. When using a quantum Hamming distance oracle with response register dimension r = 2, which statement correctly describes the relationship between the parity of the hidden string length n and the possibility of perfect identification in a single query? 1) Perfect identification is possible for both even and odd n with any mapping. 2) Perfect identification is possible only for odd n with an appropriate mapping. 3) Perfect identification is possible only for even n with an appropriate mapping. 4) Perfect identification is impossible for all n, regardless of mapping. 5) Perfect identification depends solely on the choice of response register dimension, not on n's parity. 6) Perfect identification is possible for even n only if the oracle acts via non-abelian permutations. 7) Perfect identification is possible for odd n if the response register dimension r is increased beyond 2.

✓ Correct Answer:

The correct answer is 3) Perfect identification is possible only for even n with an appropriate mapping..

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Title: Single query learning from abelian and non-abelian Hamming distance oracles Year: 2009 Paper ID: ffe096791c919932a5fad7c033aa7c9a0746e1ad Source: semantic-scholar URL: https://www.semanticscholar.org/paper/ffe096791c919932a5fad7c033aa7c9a0746e1ad Abstract: We study the problem of identifying an n-bit string using a single quantum query to an oracle that computes the Hamming distance between the query and hidden strings. The standard action of the oracle on a response register of dimension r is by powers of the cycle (1...r), all of which, of course, commute. We introduce a new model for the action of an oracle--by general permutations in S_r--and explore how the success probability depends on r and on the map from Hamming distances to permutations. In particular, we prove that when r = 2, for even n the success probability is 1 with the right choice of the map, while for odd n the success probability cannot be 1 for any choice. Furthermore, for small odd n and r = 3, we demonstrate numerically that the image of the optimal map generates a non-abelian group of permutations.

Question 1130 multiple-choice

Quantum optimization problems often rely on sophisticated mathematical frameworks, including group theory and operator algebra, to develop efficient algorithms and relaxations for Hamiltonian models. The Quantum Max Cut (QMC) problem serves as a testbed for exploring these advanced techniques in quantum computation. Which of the following statements accurately describes a key feature of the new hierarchy of relaxations for the Quantum Max Cut (QMC) problem introduced using swap operators? 1) It relies exclusively on polynomials in Pauli matrices to construct relaxations. 2) Its efficacy diminishes on graphs with uniform edge weights and fewer than 8 vertices. 3) It does not utilize any algebraic structure from the symmetric group. 4) It cannot be manipulated using computer algebra due to infinite operator relations. 5) It is based on non-commutative Sum of Squares optimization using qubit swap operators, leveraging a finite algebraic presentation for efficient computer algebraic manipulation. 6) It generalizes the Lieb-Mattis theorem exclusively to classical spin systems. 7) It cannot yield near-exact solutions at low hierarchy levels for small QMC instances.

✓ Correct Answer:

The correct answer is 5) It is based on non-commutative Sum of Squares optimization using qubit swap operators, leveraging a finite algebraic presentation for efficient computer algebraic manipulation..

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Title: Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators Year: 2023 Paper ID: 7f586e37f16e1f1e0e3197ae3a880c99d18ec608 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/7f586e37f16e1f1e0e3197ae3a880c99d18ec608 Abstract: The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for local Hamiltonian problems. In this paper we attack this problem using the algebraic structure of QMC, in particular the relationship between the quantum max cut Hamiltonian and the representation theory of the symmetric group.The first major contribution of this paper is an extension of non-commutative Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of relaxations to Quantum Max Cut. The hierarchy we present is based on optimizations over polynomials in the qubit swap operators. This is in contrast to the "standard" quantum Lasserre Hierarchy, which is based on polynomials expressed in terms of the Pauli matrices. To prove correctness of this hierarchy, we exploit a finite presentation of the algebra generated by the qubit swap operators. This presentation allows for the use of computer algebraic techniques to manipulate and simplify polynomials written in terms of the swap operators, and may be of independent interest. Surprisingly, we find that level-2 of this new hierarchy is numerically exact (up to tolerance 10−7) on all QMC instances with uniform edge weights on graphs with at most 8 vertices.The second major contribution of this paper is a polynomial-time algorithm that computes (in exact arithmetic) the maximum eigenvalue of the QMC Hamiltonian for certain graphs, including graphs that can be "decomposed" as a signed combination of cliques. A special case of the latter are complete bipartite graphs with uniform edge-weights, for which exact solutions are known from the work of Lieb and Mattis \cite{lieb1962ordering}. Our methods, which use representation theory of the symmetric group, can be seen as a generalization of the Lieb-Mattis result.

Question 1131 multiple-choice

The double-scaled Sachdev-Ye-Kitaev (DSSYK) model is a quantum mechanical system notable for its connections to quantum chaos and holography. Quantum group symmetries, particularly SUq(1,1), have been used to model its dynamics and explore its potential gravitational duals. Which gravitational theory is identified as most relevant for describing the bulk dual of the double-scaled SYK model, with the capacity to interpolate between AdS and dS regimes? 1) Einstein-Hilbert gravity 2) Jackiw-Teitelboim gravity 3) Liouville gravity 4) Sine dilaton gravity 5) Chern-Simons gravity 6) Gauss-Bonnet gravity 7) Brans-Dicke gravity

✓ Correct Answer:

The correct answer is 4) Sine dilaton gravity.

📚 Reference Text:

Title: Dynamical actions and q-representation theory for double-scaled SYK Year: 2023 Paper ID: edd781b8f1d023c1e55603427118cd88ac69d132 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/edd781b8f1d023c1e55603427118cd88ac69d132 Abstract: We show that DSSYK amplitudes are reproduced by considering the quantum mechanics of a constrained particle on the quantum group SUq(1, 1). We construct its left-and right-regular representations, and show that the representation matrices reproduce two-sided wavefunctions and correlation functions of DSSYK. We then construct a dynamical action and path integral for a particle on SUq(1, 1), whose quantization reproduces the aforementioned representation theory. By imposing boundary conditions or constraining the system we find the q-analog of the Schwarzian and Liouville boundary path integral descriptions. This lays the technical groundwork for identifying the gravitational bulk description of DSSYK. We find evidence the theory in question is a sine dilaton gravity, which interestingly is capable of describing both AdS and dS quantum gravity.

Question 1132 multiple-choice

In quantum computing, qudits are quantum systems with d possible states, generalizing the binary qubit model. The design of universal gate sets for qudits is important for enabling arbitrary quantum operations in higher-dimensional Hilbert spaces. Which statement accurately describes a necessary condition for a set of gates to be universal for qudit-based quantum computing? 1) The gate set must include only diagonal gates acting on single qudits. 2) Universality can be achieved using exclusively two-qudit gates with commuting operations. 3) The gate set must consist of gates that preserve the computational basis of qudits. 4) Universality requires only a single non-entangling gate with fixed parameters. 5) The gate set must include noncommuting single-qudit gates and at least one entangling two-qudit gate. 6) Any gate set containing U_d(α) transformations without controlled gates is sufficient for universality. 7) Universality is guaranteed if the gate set can implement only spectral decompositions.

✓ Correct Answer:

The correct answer is 5) The gate set must include noncommuting single-qudit gates and at least one entangling two-qudit gate..

📚 Reference Text:

is transformed by qudit gates. This section gives a review of various qudit gates and their applications. Section 2.1 provides criteria for the qudit universality and introduces several fundamental qudit gatesets. Section 2.2 presents examples of qudit gates and illustrates their advantages compared to qubit gates. In the lastsection, Section 2.3 , a quantitative discussion of the circuit efﬁciency is included to give a boundary of the number of elementary gates needed for decomposing an arbitrary unitarymatrix. 2.1 Criteria for Universal Qudit Gates This subsection describes the universal gates for qudit-basedquantum computing and information processing. We elaborateon the criteria for universality in Section 2.1.1 and give examples inSection 2.1.2 . 2.1.1 Universality In quantum simulation and computation, a set of matrices Uk∈U(dn)is called the universal quantum gate set if the product of its elements can be used to approximate anyarbitrary unitary transformation Uof the Hilbert space H ⊗n d with acceptable error measured in some appropriate norm [ 153]. This idea of universality not only applies to the qubit systems [47]; but can also be extended to the qudit logic [ 24, 39, 65, 102, 114, 164 ]. Several discussions of standards and proposals for a universal qudit gate set exist. Vlasov shows that the combination of two noncommuting single qudit gates and a two-qudit gate are enough to simulate any unitary U∈U(dn)with arbitrary precision [ 153]. Qudit gates can themselves be reduced to, and thus simulated by, sequences of qudit gates of lower-dimensionalqudit gates [ 135, 137 ] Brylinski and Brylinski prove a set of sufﬁcient and necessary conditions for exact qudit universality which needs some random single qudit gates complemented byone two-qudit gate that has entangled qudits [ 23]. Exact universality implies that any unitary gate and any quantumprocess can be simulated with zero error. Neither of thesemethods is constructive and includes a method for physical implementation. A physically workable procedure is given by Muthukrishnan and Stroud using single- and two-qudit gates todecompose an arbitrary unitary gate that operates on Nqudits [118]. They use the spectral decomposition of unitary transformations and involve a gate library with a group ofcontinuous parameter gates. Brennen et al. [ 21] identify criteria for exact quantum computation in qudit that relies on the QRdecomposition of unitary transformations. They generate a libraryof gates with a ﬁxed set of single qudit operations and “one controlled phase ”gate with single parameter as the components of the universal set. Implementing the concept of a coupling graph, they proved that by connecting the nodes (equivalently logical basis states) they can show the possibility of universal computation. 2.1.2 Examples of Universal Gate Sets An explicit and physically realizable universal set comprisingone-qudit general rotation gates and two-qudit controlledextensions of rotation gates is explained in this section [ 108]. Weﬁrst de ﬁne U d(α):⎥summationdisplayd−1 l/equals0αl|l〉1|d−1〉,α:/equals(α0,α1,... ,αd−1). (2) as a transformation in the d-dimension that maps any given qudit state to |d−1〉. Complex parameters of Udmay not be unique Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504

Question 1133 multiple-choice

Quantum dot (QD)-cavity systems are widely utilized for implementing photonic quantum gates due to their strong coupling between photons and QD electron spins. Accurate modeling of these systems is essential for achieving high-fidelity quantum operations in the presence of practical imperfections and noise. Which of the following most accurately describes the role of the Jaynes-Cummings Hamiltonian in modeling QD-cavity quantum gate systems? 1) It simulates photon loss due to sideband leakage and absorption in the cavity. 2) It provides a framework for calculating output field operators influenced by vacuum noise. 3) It models the nonlinear effects resulting from QD saturation in strong excitation regimes. 4) It describes the dynamics of external driving fields and cavity dissipation. 5) It accounts for single-qubit rotations implemented via half-wave plates in quantum circuits. 6) It evaluates the fidelity reduction caused by imperfections in the cavity mirrors. 7) It characterizes the interaction between a two-level quantum dot and a quantized cavity mode, forming the basis for conditional quantum gate operations.

✓ Correct Answer:

The correct answer is 7) It characterizes the interaction between a two-level quantum dot and a quantized cavity mode, forming the basis for conditional quantum gate operations..

📚 Reference Text:

for the experimental realization of our CRk gate, we should analyze the interaction of the QD-cavity system (QD within a single-sided cavity) under vacuum noise, sideband leakage, and absorption, in practice. Analysis of QD-cavity System under Vacuum noise and Sideband Leakage In our CRk gate, in order to implement a controlled-rotation k (CRk) operation, the critical component is the interaction of the QD-cavity system inducing a difference in reflectance |r h| [|r0|] with phase shift φrh [φr0] from the reflection coefficient R h [R 0] shown in Eq. 6 with respect to the hot [cold] cavity. Thus, the interaction between a photon and an electron spin state in QD should be analyzed to quantify the efficiency and reliable performance of the QD-cavity system under vacuum noise, N (ω), for operation of the QD-dipole and leaky modes, S(ω) (sideband leakage and absorption) 40,42,53–56. For this analysis, we introduce the Jaynes-Cummings Hamiltonian (HJC) in the rotating frame at the input field ( ˆbin) frequency. ∫ ∫ωω ωω σσ σσ ωω πωκ= −+− +− +′ ′ +′ +.+ +− ++ −+ ++−   ˆˆ ˆˆ ˆˆ ˆˆ ˆˆˆˆ ˆˆ ()Ha ai ga ad bb da ba b() () () 2 (15)c JC X Using the solutions of the Heisenberg equations of motion (Bloch equation with damping, γ ), we can calculate the quantum Langevin equations of a cavity field operator, ˆa, a dipole operator, ˆσ−, of X−, and the input-output relations with vacuum noise, N(ω), and leaky modes, S(ω), as follows: Figure 4. Three-qubit DQFT (the theoretical circuit of DQFT shown in Fig. 1), based on three CRk (two CR2 and a CR3) operations and three Hadamard operations, can be implemented in the three-photon DQFT scheme, which consists of three CPk (two CR2 and a CR3) gates using the QD-cavity systems, and three HWPs. The HWP , which can rotate the polarization of a photon, performs the Hadamard operation. 7 Scientific RepoRtS | (2019) 9:12440 | https://doi.org/10.1038/s41598-019-48695-z www.nature.com/scientificreports www.nature.com/scientificreports/ ˆˆˆ ˆ ˆ ˆ ˆˆˆ ˆˆ ˆˆ ˆ ˆˆ ˆˆˆ ˆωωκ κσκ κ σσω ωγσσ γσ σ ρρ κκ=− =− −+ +− −− =− =− −+ −+ =−= =+ =+− − −−−da dtiaH ia gb S d dtiHi ga N d dtd dt bb aS Sa[, ][ () 22], [, ][ () 2], ()0, ,, (16)cs s ZZ ZJC in in JC X 22 11 outi no ut in where SS()in outˆˆ is the input (output) field operator from leaky modes due to sideband leakage and absorption in the cavity mode, and ˆN is the vacuum noise operator for ˆσ−. In Sec. 3.1, we assumed the approximation of weak excitation with the ground state, ˆσ=−1Z (no saturation), in the QD for the steady state42,45,46,50–52. Based on this condition and Eq. 16, the output field operator, ˆbout, can be given by ωω ω =+ +. bR bN NS S () () () (17) outi ni nˆˆ ˆˆ Here, we consider the vacuum noise, N(ω), and leaky mode, S(ω) (sideband leakage and absorption) in the

Question 1134 multiple-choice

Coherent optical systems and quantum algorithms share similarities in their implementation of the Discrete Fourier Transform (DFT), particularly regarding hardware structure and computational techniques. Advances in optical hardware have enabled direct realization of the Fast Fourier Transform (FFT) using unitary operations and wavefront superpositions. Which statement best describes the unique advantage of coherent optical FFT hardware when processing data encoded in quantum superposition states? 1) It enables measurement of quantum entanglement without the need for photon counting detectors. 2) It allows direct manipulation of intensity patterns, providing faster classical readout. 3) It utilizes incoherent addition and amplitude modulation, ensuring efficient classical data encoding. 4) It collapses quantum superposition states during computation for accurate frequency analysis. 5) It achieves analog-to-digital conversion through acousto-optic modulators only. 6) It depends exclusively on digital input methods, preventing phase information loss. 7) It performs computation using only unitary (coherent) additions and phase rotations, allowing processing of superpositions without state collapse, making it directly comparable to quantum FFTs.

✓ Correct Answer:

The correct answer is 7) It performs computation using only unitary (coherent) additions and phase rotations, allowing processing of superpositions without state collapse, making it directly comparable to quantum FFTs..

📚 Reference Text:

carrier wave holographic methods has led to the development of the two lens 4 - focal length coherent optical correlator which has been extensively investigated now for several decades2. In this paper , by con trast, we concentrate on the one dimensional discrete Fourier transform (DFT) of a discretely sampled data sequence (which, in the most general case, can be complex ). We also review the scenario in which the data is in a quantum superposition state , as the DFT forms a central component of one of the most important quantum computing algorithms, namely the Shor factoris ation algorithm3. Firstly, w e briefly review implementation of the DFT by vector matrix multiplication approach es. However, a major component of the paper is to highlight that a more elegant realisation of the DFT can be accomplished by employing the superpositio n of coherent wavefronts to naturally implement the Butterfly operation which forms the basic computation involved in computing the DFT via the fast Fourier transform (FFT) algorithm4,5,8. Based on this, a waveguiding structure can be realis ed employing an optical hybrid device developed for combining the signal and local oscillator waves in optical communicati on coherent detection systems6. Combining these hybrids together with waveguides to control the relative paths le ngths between the elements, thus allow s a coherent implementation of the FFT by, effectively, embedding the algorithm directly into an optical hardware structure. Further, since this operation has been shown to involve only coherent additions of the wavefronts, together with pure phase rotations, this computation can be performed on a superposition state without its collapse since it involv es only unitary operations. This means that the coherent optical implementation of the FFT can be compared directly to the quantum FFT which forms a central component of the Shor algorithm . Thus very similar hardware structure s can be used for the implementation of both the coherent optical and quantum optical FFTs , although the data entry and readout will be very different, as described below. 2. OPTICAL IMPLEMENTATION OF THE DISCRET E FOURIER TRANSFORM The most direct way in which to implement a Fourier transform with coherent light is to directly employ a converging lens but since this is so well known no further discussi on of this will be made here1. Slightly more inconvenient is the implementation of the Fourier transformation of a 1 -D signal since this must in addition be imaged along the orthogonal axis to the 1 -D Fourier transformation to avoid diffraction effects. The data input is by means of some form of spatial light modulator (SLM) which may be analogue or discrete. For instance, an acousto -optic cell has been employed to input high frequency time domain data from radar systems for real-time spectral analysis7. Alternatively , a 1 -D SLM could be employed for a similar purpose, mos t commonly based on a nematic or ferro-electric liquid crystal1,7. If this device is capable of full complex modulation of the signal, real -only

Question 1135 multiple-choice

In the study of quantum information, group theory and representation theory play crucial roles in constructing covariant quantum channels, especially when dealing with non-abelian symmetries such as those of the alternating group $ A_4 $. Understanding the decomposition of representations and the structure of invariant sets allows for advanced programmable quantum processor designs. Which of the following statements accurately describes the decomposition of the tensor product representation $ \vartheta \otimes \vartheta $ for the group $ A_4 $? 1) It decomposes into four copies of the one-dimensional irreducible representation. 2) It decomposes into one three-dimensional irrep and three one-dimensional irreps. 3) It decomposes entirely into one-dimensional irreps. 4) It decomposes into two one-dimensional irreps and one six-dimensional irrep. 5) It decomposes into one copy each of the one-, two-, and three-dimensional irreps. 6) It decomposes into three one-dimensional irreps and two copies of the three-dimensional irrep. 7) It decomposes into two three-dimensional irreps and a four-dimensional irrep.

✓ Correct Answer:

The correct answer is 6) It decomposes into three one-dimensional irreps and two copies of the three-dimensional irrep..

📚 Reference Text:

higher-dimensional subspace in d×dthat merely consists of maximally entangled states (see, for instance Ref. [12, Proposition 6]). After this discussion, we give a diﬀerent construction of covariant programmable quantum processors which is more widely applicable. 3.3 Covariant programmable quantum processors from teleportation IfKis not abelian, we need a diﬀerent method to show exact programmability. This case appears for example for the ﬁnite group A4[16, p. 20], which is outside the scope of Ref. [30]. Example23. The alternating group A4is a subgroup of the symmetric group S4consisting of only the even permutations: A4={e,(12)(34),(13)(24),(14)(23),(123),(134),(243),(142),(321),(431),(342),(241)}. Note thatK4={e,(12)(34),(13)(24),(14)(23)}is an abelian subgroup of A4, in fact iso- morphic to the Klein group, and since it can be described as the set of all elements of order at most 2, it is a normal subgroup. Its cosets are (123)K4and(321)K4= (123)2K4, which are conjugacy classes, and the quotient A4/K4is the cyclic group Z/3of three elements. Accepted in Quantum2021-06-18, click title to verify. Published under CC-BY 4.0. 12 The group has four irreps ϕ(0),ϕ(1),ϕ(2), andϑ. The ﬁrst three, ϕ(0),ϕ(1),ϕ(2), are one- dimensional, while ϑis three-dimensional [16, Section 2.3]. Indeed, with the third root of unityζ=e2πi/3, we haveϕ(j)/parenleftbig(123)kK4/parenrightbig=ζjk. On the other hand, ϑhas a very intuitive form as the real SO (3)symmetries of a regular tetrahedron in three-space: concretely, let H= (1,1,1,1)⊥=/braceleftbig(a1,a2,a3,a4) :/summationtext iai= 0/bracerightbig⊂C4, which is spanned by the vertices v1= (3,−1,−1,−1),v2= (−1,3,−1,−1),v3= (−1,−1,3,−1),v4= (−1,−1,−1,3)of the tetrahedron. The group action of ϑis simply by permutation of the coordinate axes, and so all its matrices are real; as a consequence, ϑ=ϑ. Consider now the set of ϑϑ-covariant channels. Note that they form a semigroup (since they are closed under composition), and at the same time a closed convex set, isomorphic to the 3×3-states invariant under ϑ⊗ϑ. The latter representation decomposes as ϑ⊗ϑ/similarequalϕ(0)⊕ϕ(1)⊕ϕ(2)⊕2ϑ, (3.1) whereϑappears with multiplicity 2. Hence, there are three one-dimensional blocks on the diagonal and two three-dimensional ones, i.e. the commutant consists of three one- dimensional blocks and a two-dimensional block of multiplicity three. Thus, Lemma 16 shows that Jϑϑis not a polytope; indeed its extreme points are three maximally entangled pure states, as well as a two-dimensional family of rank-three states, parametrized by the surface of the Bloch sphere. This implies that Corollary 19 does not cover all cases of interest. We can give a more explicit description of the extreme points and the associated quantum channels. To start, because of the semigroup property, the unitary covariant channels form a group, and since by the preceding representation-theoretic analysis it has three elements, corresponding to the three maximally entangled pure states in Jϑϑ, we are looking for a unitaryVof degree 3that commutes with ϑup to a phase given by the one-dimensional irreps:ϑgVϑ∗ g=ϕ(1) gV. Then, the unitary channels Tj(ρ) =VjρV∗j(j= 0,1,2) are ϑϑ-covariant. For instance, Vcan be given as V|v1/angbracketright=1 2/parenleftbig|v2/angbracketright+ζ|v3/angbracketright+ζ2|v4/angbracketright/parenrightbig, V|v2/angbracketright=1 2/parenleftbig|v1/angbracketright +ζ2|v3/angbracketright+ζ|v4/angbracketright/parenrightbig, V|v3/angbracketright=1 2/parenleftbigζ|v1/angbracketright+ζ2|v2/angbracketright +|v4/angbracketright/parenrightbig, V|v4/angbracketright=1 2/parenleftbigζ2|v1/angbracketright+ζ|v2/angbracketright+|v3/angbracketright/parenrightbig, keeping in mind that this description is redundant (but consistent), as |v1/angbracketright+|v2/angbracketright+|v3/angbracketright+ |v4/angbracketright= 0and the sum of the r.h.s. vectors vanishes, too. This deﬁnes a unitary because the

Question 1136 multiple-choice

In quantum information theory, analyzing multipartite systems often involves decomposing operator algebras using group representation theory. The structure and dimensions of irreducible representations (irreps) depend on both the number of particles and the Hilbert space dimension. For a system of n=4 particles, which of the following statements correctly describes the dimension of the irreducible representation associated with the partition (1,1) when the Hilbert space dimension is d=2? 1) The matrix representation is 4x4 due to full symmetry. 2) The matrix representation reduces to 2x2 because the Hilbert space dimension is insufficient for higher-dimensional irreps. 3) The matrix representation remains 3x3, regardless of d. 4) The partition (1,1) does not appear for n=4. 5) The dimension of the irrep increases to 5x5 for d=2. 6) The representation becomes trivial (1x1) for any d. 7) The matrix representation splits into two 2x2 blocks for d=2.

✓ Correct Answer:

The correct answer is 2) The matrix representation reduces to 2x2 because the Hilbert space dimension is insufficient for higher-dimensional irreps..

📚 Reference Text:

for our paper (case study examples). Of course our method works efﬁciently for an arbitrary number of particles nand dimensions of Hilbert space d, but here we present them only for n=3, 4, because for these cases we are able to represent our results graphically. 2See also Appendix IV A for a short review on this topic. 4 • Case when n=3. In this case in algebra Mwe have only one irrep labeled by trivial partition a= (1). V0 a(13) =1 2 d+1p d21 p d21 d1 ,V0 a(23) =1 2 d+1p d21p d21d1 (8) • Case when n=4. In this case in algebra Mwe have two irreps labeled by partitions a1= (2)anda2= (1, 1). For partition a1we deal with matrices 3x3 for any d1: V0 a1(14) =1 3Da10 BB@1 61 2p 31 3p 2 1 2p 31 21p 6 1 3p 21p 61 31 CCADa1,V0 a1(24) =1 3Da10 BB@1 61 2p 31 3p 2 1 2p 31 21p 6 1 3p 21p 61 31 CCADa1, V0 a1(34) =1 3Da10 B@2 302 3p 2 0 0 0 2 3p 201 31 CADa1,(9) where Da1=0 @p d1 0 0 0p d1 0 0 0p d+21 A (10) and#2=1. For partition a2situation is more complicated. Dimension of irrep a2depends on dimension of local Hilbert space d. Namely for any d3 we have V0 a2(14) =1 3Da20 BB@1 21 2p 31p 6 1 2p 31 61 3p 2 1p 61 3p 21 31 CCADa2,V0 a2(24)) =1 3Da20 BB@1 21 2p 31p 6 1 2p 31 61 3p 2 1p 61 3p 21 31 CCADa2, V0 a2(34) =1 3Da20 B@0 0 0 02 3p 2 3 0p 2 31 31 CADa2,(11) where Da2=0 @p d+1 0 0 0p d+1 0 0 0p d21 A. (12) For every d<3 (in our case only d=2 is interesting) we deal with matrices 2x2: V0 a2(14) =3 1 21 2p 3 1 2p 31 6! ,V0 a2(24) =3 1 21 2p 3 1 2p 31 6! ,V0 a2(34) =30 0 02 3 . (13) The full knowledge about irreps of V0(sab), where sab2S(n)(see Notation 8 in the Appendix IV A) allows us to decompose these operators and density operators r1...nwhich are U U . . .Uinvariant into block diagonal form V0(sab) =M a1r(a) V0 a(sab),r1...n=M a1r(a) era, (14) where the direct sum runs over all inequivalent irreps a,r(a)denotes the dimension of irrep aanderais a representation of operator r1...non irrep a. In the next paragraph we present how to use the decomposition from formula (14) and explicit matrix form of irreps of V0(sab)to calculate ﬁdelities. 5 C. Method of calculations Since, in principle, calculations techniques are similar to those from [19], in most cases, proofs are skipped and unless speciﬁed otherwise, we refer to the above-mentioned work for them. In this section we provide a general formula for an allowed region of N-tuples of ﬁdelities in terms of overlaps of pure states with irreducible representations from the previous section. This is contained in Theorem 3. Lemma 1. Fidelity F

Question 1137 multiple-choice

Quantum algorithms for hidden subgroup problems exploit quantum parallelism and measurement to extract algebraic structure from functions defined over groups. A generalized version of Simon's algorithm employs distributed computation and specialized operators to identify orthogonal complements of subgroups within quantum systems. In the distributed quantum algorithm for the generalized Simon's problem, which operator is specifically responsible for lexicographically organizing quantum register states and performing XOR operations essential for extracting hidden subgroup information? 1) BI operator 2) Hadamard transform 3) O′fw query operator 4) Measurement operator 5) Gaussian elimination 6) USort operator 7) Swap gate

✓ Correct Answer:

The correct answer is 6) USort operator.

📚 Reference Text:

have G(u) =G(v). (2) =⇒. SinceG(u) =G(v), we have ∀z∈G(u),z∈G(v). Then we have ∃w,w′∈ {0,1}t such that z=f(uw) andz=f(vw′). So we have f(uw) =f(vw′). According to the deﬁnition of the generalized Simon’s problem, we have uw⊕vw′= (u⊕v)(w⊕w′)∈S. Further, by the deﬁnitions of the group SandSl, we have u⊕v∈Sl. □ 4 Distributed quantum algorithm for generalized Simon’s problem In the following section, we will start by introducing the relev ant notations, functions, and operators used in the distributed quantum algorithm for the generali zed Simon’s problem, which we present as Algorithm 1. Let [N] represent the set of integers {0,1,···,2t−1}, and let BI :{0,1}t→[N] be the function to convert a binary string of tbits to an equal decimal integer. The query operators O′ fwin Algorithm 1are deﬁned as O′ fw|u⟩|b⟩|c⟩=|u⟩|b⟩|c⊕fw(u)⟩, (11) whereu∈{0,1}n−t,w∈{0,1}t,b∈{0,1}BI(w)andc∈{0,1}m. 5 The operator USort:{0,1}2t+1m→{0,1}2t+1min Algorithm 1is deﬁned as USort /circlemultiplydisplay w∈{0,1}t|fw(u)⟩ |b⟩= /circlemultiplydisplay w∈{0,1}t|fw(u)⟩ |b⊕S(u)⟩,(12) where/circlemultiplytext w∈{0,1}t|fw(u)⟩≜|f0t(u)⟩|f0t−11(u)⟩···|f1t(u)⟩andb∈{0,1}2tm. Intuitively, in Algorithm 1, the eﬀect of USortis to arrange the values in the 2t control registers in lexicographical order and perform an XOR operation wit h the target register. In addition, for any operator Awwithw∈ {0,1}t, we let/producttext w∈{0,1}tAw≜ A1tA1t−10···A0t,/producttext′ w∈{0,1}tAw≜A0tA0t−11···A1t. Algorithm 1 Distributed quantum algorithm for ﬁnding the elements in S⊥ l 1:procedure DSL(integern, integert, integerm, operatorOfw) 2:Y←{0n−t}; 3:|ψ0⟩=|0n−t⟩/vextendsingle/vextendsingle/vextendsingle02t+1m/angbracketrightBig ; 4:|ψ1⟩=/parenleftBig H⊗n−t⊗I⊗2t+1m/parenrightBig |ψ0⟩; 5:|ψ2⟩=/producttext w∈{0,1}t/parenleftBig O′ fw⊗I⊗2t+1m−BI(w)·m−m/parenrightBig |ψ1⟩; 6:|ψ3⟩= (I⊗n−t⊗Usort)|ψ2⟩; 7:|ψ4⟩=/producttext′ w∈{0,1}t/parenleftBig O′ fw⊗I⊗2t+1m−BI(w)·m−m/parenrightBig |ψ3⟩; 8:|ψ5⟩=/parenleftBig H⊗n−t⊗I⊗2t+1m/parenrightBig |ψ4⟩; 9:Measure the ﬁrst register, and get an element z; 10:ifz /∈⟨Y⟩then 11:Y←Y∪{z}. 12:end if 13:end procedure Next, we prove the correctness of Algorithm 1. The state after the third step of Algorithm 1is |ψ1⟩=/parenleftBig H⊗n−t⊗I⊗2t+1m/parenrightBig |ψ0⟩=1√ 2n−t/summationdisplay u∈{0,1}n−t|u⟩/vextendsingle/vextendsingle/vextendsingle02t+1m/angbracketrightBig .(13) 6 Then, Algorithm 1queries each of the oracles to get the following state. |ψ2⟩=/productdisplay w∈{0,1}t/parenleftBig O′ fw⊗I⊗2t+1m−BI(w)·m−m/parenrightBig |ψ1⟩ =1√ 2n−t/summationdisplay u∈{0,1}n−t|u⟩ /circlemultiplydisplay w∈{0,1}t|fw(u)⟩ /vextendsingle/vextendsingle/vextendsingle02tm/angbracketrightBig .(14) After sorting by using USort, we have the following state. |ψ3⟩=/parenleftbig I⊗n−t⊗Usort/parenrightbig |ψ2⟩ =1√ 2n−t/summationdisplay u∈{0,1}n−t|u⟩ /circlemultiplydisplay w∈{0,1}t|fw(u)⟩ |S(u)⟩.(15) Then, we query each oracle again and obtain the following state. |ψ4⟩=/productdisplay′ w∈{0,1}t/parenleftBig O′ fw⊗I⊗2t+1m−BI(w)·m−m/parenrightBig |ψ3⟩ =1√ 2n−t/summationdisplay u∈{0,1}n−t|u⟩/vextendsingle/vextendsingle/vextendsingle02tm/angbracketrightBig |S(u)⟩.(16) After using Hadamard transform on the ﬁrst register, in the light of Theore m1, we get the following state. |ψ5⟩=/parenleftBig H⊗n−t⊗I⊗2t+1m/parenrightBig |ψ4⟩ =1 2n−t/summationdisplay u,z∈{0,1}n−t(−1)u·z/vextendsingle/vextendsingle/vextendsinglez,02tm,S(u)/angbracketrightBig =1 2n−t|Sl|/summationdisplay u,z∈{0,1}n−t/summationdisplay sl∈Sl(−1)(u⊕sl)·z/vextendsingle/vextendsingle/vextendsinglez,02tm,S(u⊕sl)/angbracketrightBig =1 2n−t+kl/summationdisplay u,z∈{0,1}n−t/summationdisplay sl∈Sl(−1)u·z(−1)sl·z/vextendsingle/vextendsingle/vextendsinglez,02tm,S(u)/angbracketrightBig =1 2n−t+kl/summationdisplay u,z∈{0,1}n−t/parenleftBigg/summationdisplay sl∈Sl(−1)sl·z/parenrightBigg (−1)u·z/vextendsingle/vextendsingle/vextendsinglez,02tm,S(u)/angbracketrightBig .(17) 7 Note that if there exists s′ l∈Slsuch thats′ l·z= 1, then we have /summationdisplay sl∈Sl(−1)sl·z=1 2/summationdisplay sl∈Sl/parenleftBig (−1)sl·z+(−1)(sl⊕s′ l)·z/parenrightBig =1 2/summationdisplay sl∈Sl/parenleftBig (−1)sl·z+(−1)sl·z(−1)s′ l·z/parenrightBig =1 2/summationdisplay sl∈Sl(−1)sl·z/parenleftBig 1+(−1)s′ l·z/parenrightBig = 0.(18) Ifz∈S⊥ l, then/summationtext sl∈Sl(−1)sl·z= 2kl, so we have |ψ5⟩=1 2n−t/summationdisplay u∈{0,1}n−t/summationdisplay z∈S⊥ l(−1)u·z/vextendsingle/vextendsingle/vextendsinglez,02tm,S(u)/angbracketrightBig =1 2n−t/summationdisplay z∈S⊥ l|z⟩/summationdisplay u∈{0,1}n−t(−1)u·z/vextendsingle/vextendsingle/vextendsingle02tm,S(u)/angbracketrightBig .(19) Thus, in line 9 of Algorithm 1, after measuring the ﬁrst register of the state |ψ5⟩, we can obtain an element z∈Sl⊥. In line 9 of Algorithm 1, the result we measure may not be linearly independent of priormeasurements.Consequently,thereisnoguaranteethatwecanob tain⟨Y⟩⊥=Sl for the setY, even after running Algorithm 1multiple times iteratively. After running Algorithm 1iteratively many times, we denote S′ l=⟨Y⟩⊥. Denote k′ l=rank(S′ l),kl≤k′ l≤n−t, and the basis of S′ las{e′ i|e′ i∈{0,1}n−t,1≤i≤k′ l}. Note thatS′ lmay not be

Question 1138 multiple-choice

In the representation theory of symmetric groups, central elements and their associated normalized characters play a key role in classifying representations via Young diagrams. Understanding the minimal number of central generators required to distinguish all representations has significant implications for both computational efficiency and theoretical insights in mathematical physics. Which asymptotic estimate describes the minimal number k*(n) of central generators needed to distinguish all Young diagrams of the symmetric group Sn as n becomes large? 1) k*(n) ~ log n 2) k*(n) ~ n^{1/2} 3) k*(n) ~ n / log n 4) k*(n) ~ n^{1/4} log n 5) k*(n) ~ n 6) k*(n) ~ e^{\sqrt{n}} 7) k*(n) ~ n^{2}

✓ Correct Answer:

The correct answer is 4) k*(n) ~ n^{1/4} log n.

📚 Reference Text:

as required by the QPE procedure. The set of central elements {T2,T3,···,Tn}multiplicatively generate the centre, i.e. linear combinations of these and their powers linearly span the centre Z(C(Sn)), while in fact subsets{T2,···,Tk∗(n)}withk∗(n)< ngenerically form multiplicative sets of generators [ 25]. This is related to the fact that the ordered list of normalized characters (/hatwideχR(T2),/hatwideχR(T3),···,/hatwideχR(Tk∗(n)))uniquely identiﬁes the Young diagram R. The connection between such ordered lists and sets of multiplicative generators is reviewed and extended to general groups in [ 42]. The sequence k∗(n)was explicitly computed, with the help of character formulae in [ 43,44]. Here are some values of k∗(n)for lower values of nup to 79: k∗(n) = 2forn∈{2,3,4,5,7} k∗(n) = 3forn∈{6,8,9···,14} k∗(n) = 4forn∈{15,16,···,23,25,26} k∗(n) = 5forn∈{24,27,···,41} k∗(n) = 6forn∈{42,···,78,79,81} (2.9) 2.1.1 A heuristic argument for the asymptotics of k∗(n)at largen A necessary condition for the eigenvalues of {T2,T3,···,Tk}to be able to distinguish all Young diagrams is that the number of distinct lists of eigenvalues exceeds the number of Young diagrams. The maximum eigenvalue of T2isn(n−1)/2∼n2. These eigenvalues are integers. The number of possible eigenvalues is order n2. ForT3we have order n3, and Tkordernkas long ask/lessmuchn, see subsection 3.2.1 for a justiﬁcation. The total number of distinct eigenvalue lists is approximately n2×n3···×nk∼nk(k+1)/2(2.10) A heuristic estimate for k∗(n)is given by equating the number of distinct eigenvalue lists and the number of Young diagrams (number of partitions of n) in the large nlimit. Thus, we have: nk∗(k∗+1)/2∼ec√n(2.11) for a constant c, where the growth of number of partitions is given by Hardy-Ramanujan formula [ 45]. Then k∗(n)(k∗(n) + 1)/2 logn∼√n (2.12) – 5 – JHEP05(2023)191Approximately at large n, k∗(n)∼n1/4 logn∈n1/4(2.13) Based on this heuristic argument, we conjecture that k∗(n)∈O(n1/4). For our main claim of a polynomial complexity of the QPE-based algorithms we describe subsequently the precise exponent is not important. It suﬃces that k∗(n)∈O(nα)withα<1/2. Notation: asymptotic behaviours. In the above and henceforth, we use the following notation: for two positive functions f,g:N→R+,f∈O(g), if it exists a constant c>0, and an integer n0∈N, such that∀n≥n0,f(n)≤cg(n). We will equally use, for that relation, the shorthand notation f∼g. Using a loose language, sometimes, we will say f isO(g). 2.1.2k∗(n), Casimirs of U(N)and multipole moments This problem of identifying a multiplicatively generating set of cycle central elements of the form{T2,T3,···,Tk∗(n)}inZ(C(Sn))isrelatedtotheproblemofidentifyingYoungdiagrams withnboxes. There is a transformation giving a sequence of Casimirs of U(N)of the form {C2,C3,···,Ck}and in terms of a set of cycle central operators {T2,T3,···,Tk}. This transformation has been described in the literature on large N2d Yang Mills theory [ 5,46] and has been used to show that elements in the list {C2,C3,···,Ck}can be expressed in terms of cycle operators with a cut-oﬀ at k[25]. The work [ 24] relates the Casimirs of U(N)to multi-pole moments of LLM geometries. Therefore the task of distinguishing LLM geometries corresponding to Young diagrams with nboxes can formally be accomplished with a cut-oﬀ Λ(n)on the Casimirs with Λ(n) =k∗(n). We are assuming N >nwhere the Young diagram detection problem involves all Young diagrams with nboxes: the U(N) constraint that

Question 1139 multiple-choice

Quantum sieve algorithms employ advanced routines to process and combine labeled quantum states for solving problems such as the hidden subgroup problem. These algorithms rely on precise manipulation of qubits and quantum gates to efficiently extract information. Which mathematical extension is proposed as a means to potentially generalize quantum computation and enable novel algorithms by leveraging a four-dimensional number system beyond complex numbers? 1) Dual numbers 2) Octonions 3) Hyperreal numbers 4) Surreal numbers 5) Matrix numbers 6) Real numbers 7) Quaternions

✓ Correct Answer:

The correct answer is 7) Quaternions.

📚 Reference Text:

any j2f1;::;mg:The input to the ﬁrst routine is the output of CSG , i.e., single-qubit states of the form (4.77), each with a corresponding label z chosen uniformly at random from f0;1;:::;2n1g. The output of routine jare qubits such that their labels have jmleast signiﬁcant bits equal to 0 and njmbits chosen uniformly at random. Each routine generates qubits whose labels have madditional bits set to 0. The process of combining the qubits is implemented as follows. Each routine jmaintains a batch of qubits, which is initially empty. When a new qubit comes in, the routine compares the m bits in positions (j1)m+1;:::;jmof its label to the bits in the same label positions in all of the qubits already in its batch. If no qubit already accumulated in the batch has matching label bits, the newly arrived qubit is added to the batch. If there is a match, the new qubit is combined with the matching one. 61 Proposition 4.5.5. There exists a procedure that with probability1 2implements the mapping (z1;jyz1i)+(z2;jyz2i)7!(z1z2;jyz1z2i) (4.86) Proof. Suppose CSG generates states jyz1i=1p 2 j0i+e2piz1b 2nj1i jyz2i=1p 2 j0i+e2piz2b 2nj1i (4.87) with labels z1andz2, respectively. The tensor product of these states is given by jyz1;yz2i=1 2 j00i+e2piz1b 2nj10i+e2piz2b 2nj01i+e2pi(z1+z2)b 2nj11i : (4.88) Application of a CNOT gate with the ﬁrst qubit playing the role of a control results in the stat jyz1;yz2i7!1 2 j00i+e2piz1b 2nj11i+e2piz2b 2nj01i+e2pi(z1+z2)b 2nj10i =1p 2 j0i+e2pi(z1+z2)b 2nj1i j0i +1p 2h e2piz2b 2nj0i+e2piz1b 2nj1i j1ii (4.89) =1p 2 jyz1+z2;0i+e2piz2b 2njyz1z2;1i : Measurement of the second qubit yields j1iwith probability1 2, leaving the ﬁrst qubit in the state jyz1z2i, up to an inconsequential global phase. The Kuperberg sieve discards the combined qubits of the form jyz1+z2i, and by iterating the combination process ensures that the last routine in the sequence outputs the desired state jy2n1i. Kuperberg showed [16] that given an input of 2O(pn)qubits, the sieve generates the desired state with high probability. 62 Quaternions’ potential in the world of quantum computation is just beginning to be ex- plored. Perhaps a quaternionic formulation of quantum mechanics applied to a quantum computa- tional framework could lead to an improvement of existing quantum algorithms or to the emergence of new ones. And if everything three-dimensional is ultimately quaternionic, quaternions could make signiﬁcant, even profound, contributions in subjects as diverse as cosmology and biology, shedding new light on the unfolding cosmos and the origins of life. As Hamilton put it [43], If some new truth, O Friend! thy toil discover, If thine eyes ﬁrst by some fair form be blest, Love it for what it is, and as a lover Gaze, or with joy receive thine honoured guest: The new-found thought set free, awhile may hover Gratefully near thee, but it cannot rest. 63 REFERENCES [1] Yu. I. Manin. The computable and incomputable. Sovetzkoe Radio, 1980. [2] R. Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 21:467-488, 1982. [3] R. Feynman. Quantum Mechanical Computers. Optics News, 11:11, 1985. [4] D. Deutsch. Quantum Theory, the

Question 1140 multiple-choice

Quantum algorithms often rely on group-theoretic structures, especially when addressing the hidden subgroup problem in non-abelian groups. Semidirect products of abelian groups with cyclic groups are a key class of groups studied in this context. In the analysis of the hidden subgroup problem for a group G = A ⋊ϕ Zp, where A is abelian and p is prime, why is determining whether the subgroup H1 = H ∩ (A × {0}) is normal in G important for algorithmic reduction? 1) It determines if the group G is simple and cannot be decomposed further. 2) It reveals whether G possesses a nontrivial center that simplifies the problem. 3) It directly identifies if the subgroup H is cyclic of order p. 4) It allows H1 to be factored out, simplifying G by quotienting and narrowing the search for the hidden subgroup. 5) It ensures that G is nilpotent, enabling the use of abelian quantum algorithms. 6) It guarantees that all elements of G have order p, reducing computational complexity. 7) It confirms that the automorphism ϕ is trivial, making subgroup identification straightforward.

✓ Correct Answer:

The correct answer is 4) It allows H1 to be factored out, simplifying G by quotienting and narrowing the search for the hidden subgroup..

📚 Reference Text:

state hiding the subgroup H. In the standard approach, one attempts to determine H using samples of hidden subgroup state ρH. 2.2 Semidirect product groups Thesemidirect productof two groups A,Bis deﬁnedin terms of a homomorphism ϕ:B→Aut(A), whereAut( A) denotes theautomorphismgroupof A. Thesemidirect productgroup A⋊ϕBconsists of the elements ( a,b) witha∈Aandb∈B. With group operations in AandBwritten additively, the group operation in A⋊ϕBis deﬁned as ( a,b)(a′,b′) = (a+ϕ(b)(a′),b+b′). It is not hard to show that group inversion satisﬁes ( a,b)−1= (ϕ(−b)(−a),−b). We consider the hidden subgroup problem for semidirect prod uct groups G=A⋊ϕZp, where Ais an abelian group and pis prime. In this case, since ϕis a homomorphism of a cyclic group, it is determined entirely by ϕ(1) (in particular, ϕ(p) is the identity map); hence, with a slight abuse of notation, we let ϕ:=ϕ(1) (an automorphism of A) henceforth, and deﬁne ϕb:= (ϕ◦···◦ϕ) (b times), giving ϕ(b) =ϕb. We will be especially interested in cyclic subgroups of such a semidirect product group. For any a∈A, we have /an}bracketle{t(a,1)/an}bracketri}ht={(0,0),(a,1),(a+ϕ(a),2),(a+ϕ(a) +ϕ2(a),3),...}. For convenience, we introduce the function Φ(b):A→Adeﬁned by Φ(b)(a) :=b−1/summationdisplay i=0ϕi(a) (3) such that the elements of /an}bracketle{t(a,1)/an}bracketri}htare (a,1)b= (Φ(b)(a),bmodp) forb∈N. 3 3 Reduction to cyclic subgroups To simplify the hidden subgroup problem over semidirect pro duct groups, we can reduce the prob- lem to that of ﬁnding either the trivial hidden subgroup or a h idden cyclic subgroup of order p. This reduction generalizes a result of Ettinger and Høyer, w ho reduced the dihedral hspto the problem of ﬁnding a trivial hidden subgroup or a hidden reﬂec tion (an order 2 subgroup). Lemma 1 (cf. Theorem 2.3 of [8]). To ﬁnd an eﬃcient algorithm for the hspoverA⋊Zpwith pprime, it suﬃces to ﬁnd an eﬃcient algorithm for the hspoverA2⋊Zpfor anyA2≤Awith the promise that either His trivial or H=/an}bracketle{t(d,1)/an}bracketri}htfor somed∈A2with|H|=p. Proof.The proof proceeds along the lines of the Ettinger-Høyer red uction. Their idea is to factor out the part of the hidden subgroup that lies within A. For generalized dihedral groups ( p= 2, ϕ(a) =−a), their reduction goes through unchanged, as noted in [10]. However, in some cases there is an additional complication coming from the fact tha t theA-part of the hidden subgroup may not be normal in G. Fortunately, we can deal with this case separately. Letf:G→Shide a subgroup H. LetG1:=A×{0}, and letH1:=H∩G1=:A1×{0}. Since frestricted to the abelian group G1hides the subgroup H1, we can eﬃciently ﬁnd generators for H1by solving an abelian hidden subgroup problem. We would like to factor out H1, so we check whether it is normal in G. Forg= (a,b)∈Gand h1= (h,0)∈H1, we havegh1g−1= (a,b)(h,0)(ϕ−b(−a),−b) = (ϕb(h),0). Therefore H1/triangleleftequalGif and only if ϕ(H1) =H1, whereϕ(H1) :={(ϕ(h),0) :h∈A1}. IfH1/ne}ati≀nslash=H, thenϕ(H1) =H1: since pis prime, there must be some ( d,1)∈H, and for any h∈A1, (d,1)(h,0)(d,1)−1= (ϕ(h),0)∈H, henceϕ(h)∈A1. IfH1=H, it could be that ϕ(H1)/ne}ati≀nslash=H1. However, given a generating set for H1, we can check whether H1/triangleleftequalG(for example, we can use the results of [31]; note that Gis solvable since its commutator subgroup is abelian1). If we

Question 1141 multiple-choice

In quantum computing, hidden subgroup and hidden shift problems play a foundational role in designing algorithms for tasks such as factoring and cryptographic analysis. The generalized hidden shift problem extends these concepts by considering functions with specific shift properties and varying parameters. Which parameter regime allows an efficient quantum algorithm for the generalized hidden shift problem using the pretty good measurement technique and Lenstra’s integer programming algorithm as a subroutine? 1) When M = 2 and N is arbitrary 2) Only when M is strictly less than N 3) When M is greater than or equal to N raised to a fixed positive ε 4) Only when N is a prime and M is arbitrary 5) Whenever M divides N exactly 6) Only when M = N 7) For all values of M and N regardless of their relationship

✓ Correct Answer:

The correct answer is 3) When M is greater than or equal to N raised to a fixed positive ε.

📚 Reference Text:

Title: Quantum algorithm for a generalized hidden shift problem Year: 2005 Paper ID: 8de18683ac62aeeacde78d82a9d69977480dbd55 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/8de18683ac62aeeacde78d82a9d69977480dbd55 Abstract: Consider the following generalized hidden shift problem: given a function f on {0,..., M - 1} x ZN promised to be injective for fixed b and satisfying f(b, x) = f(b + 1,x + s) for b = 0, 1,..., M - 2, find the unknown shift s ε ZN. For M = N, this problem is an instance of the abelian hidden subgroup problem, which can be solved efficiently on a quantum computer, whereas for M = 2, it is equivalent to the dihedral hidden subgroup problem, for which no efficient algorithm is known. For any fixed positive ε, we give an efficient (i.e., poly(log N)) quantum algorithm for this problem provided M ≥ N^ε. The algorithm is based on the "pretty good measurement" and uses H. Lenstra's (classical) algorithm for integer programming as a subroutine.

Question 1142 multiple-choice

In the study of W-algebras constructed from simple Lie algebras via quantum Hamiltonian reduction, the structure and properties of the resulting algebras are deeply influenced by the chosen nilpotent element and the associated grading. Principal embeddings of $ sl_2 $ into simple Lie algebras provide particularly tractable cases with well-understood algebraic properties. For a principal embedding of $ sl_2 $ into $ sl_n $, what is a defining property of the resulting finite W-algebra? 1) It is a non-commutative associative algebra with $ n $ generators. 2) Its center coincides with the Cartan subalgebra of $ sl_n $. 3) It contains exactly $ n $ irreducible representations of $ sl_2 $. 4) It is Poisson commutative and has $ n-1 $ generators. 5) Its Poisson structure is inherited from the multiplication in $ sl_n $. 6) The number of generators equals the rank of $ sl_n $ plus one. 7) It is always isomorphic to the universal enveloping algebra of $ sl_2 $.

✓ Correct Answer:

The correct answer is 4) It is Poisson commutative and has $ n-1 $ generators..

📚 Reference Text:

on as before. The only diﬀerence is that we now know of a system of ﬁrst class constra ints that in the end leads to the same reduction. Note that the number of generators of the ﬁnite Walgebra is equal to dim( g0) = (/summationtext im2 i)−1. This is indeed the same as the number of irreducible repres entations of sl2, minus one, one obtains from ( ⊕ini)⊗(⊕ini), as the latter number equals (/summationtext j(2j− 1)nj)−1, and one easily checks that/summationtext im2 i=/summationtext j(2j−1)nj. The generalized Miura transformation can now be formulated as Theorem 5 There exists an injective Poisson homomorphism from W(i)toK(g0). Proof: First we show that for every element x∈t++g0there exist a unique element hin the gauge group ˆHsuch thath·x·h−1∈gfix. For this note that there exists a unique element h′∈ˆHsuch thath′.x.h′−1∈gc. This follows from the previous theorem and the remarks made after it. In fact it follows from those remarks that h′is an element of that subgroup of ˆHthat is generated by the ’ex’ second class constraints (the ones that were made into ﬁrst class constraints by not im posing the other half). It was shown earlier however that gfix×H≃gc (3.71) which means that there exists a unique element h′′∈Hsuch that h′′h′.x.(h′′h′)−1∈gfix (3.72) We conclude from this that there is a surjective map from x∈t++g0togfixgiven by (3.72). The pull back of this map C∞(gfix)≡ W(i)→C∞(t++g0) (which will therefore be injective) is then the Miura map. Of course we st ill have to check whether this map is a Poisson homomorphism. This we address next. What is the Poisson structure on C∞(t++g0) ? Since the constraints of which the spacet++g0is the zero set are obviously second class the Kirillov brack et ong 27 induces a Dirac bracket on it. It is not diﬃcult to see that the Dirac term in the Dirac bracket cancels in this case which means that the Poisson alg ebraC∞(t++g0) with the induced Poisson structure is isomorphic to the Poisson a lgebraK(g0). Since the transformation from W(i) toK(g0) corresponds to a gauge transformation (3.72) the map is necessarily a homomorphism. 3.4. Examples The simplest examples of ﬁnite Walgebras are those associated to the so called ’principalsl2embeddings’. These embeddings are associated to the trivia l partition of the number n:n=n. The fundamental representation of g=slntherefore becomes an irreducible representation of the sl2subalgebra, i.e. nn≃n2 (3.73) The branching rule for the adjoint representation of gtherefore reads adn≃32⊕52⊕...⊕2n−12 (3.74) From this follows immediately that the ﬁnite Walgebra will have n−1 generators (since there are n−1sl2multiplets). Without going into details we can immediately predict the Poisson relations between these generators fro m the generalized Miura transformation for in this case the subalgebra g(0)=g0coincides with the Cartan subalgebra of sln. Since the Cartan subalgebra is an abelian algebra, and sinc e the Kirillov algebra of an abelian Lie algebra is a Poisson commu tative algebra we ﬁnd that a ﬁnite Walgebra associated to a principal embedding must also be Poi sson commutative since it is isomorphic to a Poisson subalgebra

Question 1143 multiple-choice

In topological quantum computation, anyons are modeled using algebraic structures known as ribbon fusion categories, which rely on consistency equations to define their fusion and braiding properties. Computational tools help generate data such as F-matrices, essential for constructing quantum gates based on anyonic systems. Which property must a fusion ring possess for a pentagon equation solver to generate F-matrices that unambiguously encode associativity in anyon systems? 1) Non-abelian braiding statistics 2) Modular invariance in fusion outcomes 3) Presence of conformal field theory symmetry 4) Multiplicity-rich fusion rules 5) Existence of quantum double algebra 6) Multiplicity-free fusion rules 7) Inclusion of non-trivial hexagon equations

✓ Correct Answer:

The correct answer is 6) Multiplicity-free fusion rules.

📚 Reference Text:

Title: Quantum computing with anyons: an $F$-matrix and braid calculator Year: 2022 Paper ID: 7056216d55ed9dfe62a3675fc41818b915b299f8 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/7056216d55ed9dfe62a3675fc41818b915b299f8 Abstract: We introduce a pentagon equation solver, available as part of SageMath, and use it to construct braid group representations associated to certain anyon systems. We recall the category-theoretic framework for topological quantum computation to explain how these representations describe the sets of logical gates available to an anyonic quantum computer for information processing. In doing so, we avoid venturing deep into topological or conformal quantum field theory. Instead, we present anyons abstractly as sets of labels together with a collection of data satisfying a number of axioms, including the pentagon and hexagon equations, and explain how these data characterize ribbon fusion categories (RFCs). In the language of RFCs, our solver can produce $F$-matrices for anyon systems corresponding to multiplicity-free fusion rings arising in connection with the representation theory of quantum groups associated to simple Lie algebras with deformation parameter a root of unity.

Question 1144 multiple-choice

Quantum phase estimation is a key algorithmic technique used in quantum computing, with accuracy and resource efficiency critically impacting practical implementations. Techniques such as median amplification and extra bit estimation are employed to reduce error probability and optimize resource scaling. In quantum phase estimation, what method allows for achieving optimal α⁻¹ scaling of resource cost with respect to estimation accuracy, overcoming the quadratic scaling limitation of median amplification alone? 1) Increasing the number of quantum gates in the circuit 2) Applying amplitude amplification repeatedly without rounding 3) Estimating fewer bits and post-selecting measurement outcomes 4) Estimating r extra bits and rounding them away before median amplification 5) Using only the Chernoff-Hoeffding bound without additional bit estimation 6) Implementing classical post-processing after phase estimation 7) Relying solely on the success probability near bin edges

✓ Correct Answer:

The correct answer is 4) Estimating r extra bits and rounding them away before median amplification.

📚 Reference Text:

ηthen the median of ⌈ln(δ−1)/2η2⌉estimates is incorrect with probability ≤δ. However, median ampliﬁcation alone is not suﬃcient to accomplish phase estimation as we have deﬁned it in Deﬁnition 1. This is because any λithat is≈10%·2−n−1close to a multiple of 1/2n, the probability of correctly obtaining ﬂoor (2nλj)is actually lessthan 1/2! This means that no matter how much median ampliﬁcation is performed, this approach cannot achieve αbelow≈10%. (For reference, a lower bound γon the probability is plotted in Figure 1, although reading this ﬁgure demands some notation from the proof of the following proposition. We see that when |λ(x) j−1/2|.10%/2the probability of obtaining ﬂoor (2nλj)is less than 1/2.) Furthermore, even if the signal obtained from the QFT could obtain arbitrarily small α, it would not achieve the desired α−1scaling. This is because the ampliﬁcation gap η scaleslinearlywith α, andmedianampliﬁcationscalesas η−2duetotheChernoﬀ-Hoeﬀding theorem. Therefore, we would obtain a scaling of α−2. Fortunately, achieving ∼α−1is possible, using a diﬀerent method for reducing α: we perform estimation for radditional bits which are then ignored. This makes use of the fact that αis independent of the number of estimated bits, but the gap away from the multiples of 1/2n, which isα/2n+1, is not. Every time we increase n, the width of Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 10 each disallowed interval is chopped in half, but to compensate the number of disallowed intervals is doubled. If we simply round away some of the bits, then we do not care about the additional disallowed regions introduced. If we estimate and round radditional bits, we suppress αby a factor of 2−rwhile multiplying our runtime by a factor of 2r- so the scaling is∼α−1. We still need median ampliﬁcation, though. In order to use the above strategy we need rounding to be successful: if an eigenvalue λjfalls between two bins ﬂoor (2nλj) and ﬂoor (2nλj) + 1, then it must be guaranteed to be rounded to one of those bins with failure probability at most δ. Before ampliﬁcation, the probability that rounding succeeds is≥8/π2, a constant gap above 1/2corresponding to α= 1/2. We cannot guarantee a success probability higher than 8/π2using the rounding trick alone, and thus need median ampliﬁcation to ﬁnish the job. Below, we split our analysis into two regimes: the α≤1/2regime and the α > 1/2 regime. When α>1/2then we do not really need the rounding trick described above, and we can achieve this accuracy with median ampliﬁcation alone. Otherwise when α≤1/2 we use median ampliﬁcation to get to the point where rounding is likely to succeed, and then estimate radditional bits such that α2r≥1/2. Proposition 5. Standard phase estimation. There exists an phase estimator with phases and garbage. Consider a unitary satisfying an (n,α)-rounding promise. Let: γ(x) :=sin2(πx) π2x2η0:=8 π2−1 2δmed:=δ2 6.25(36) Ifα≤1/2,letr:=/ceilingleftBig log2/parenleftBig 1 2α/parenrightBig/ceilingrightBig . Then the phase estimator has query complexity: (2n+r−1)·/ceilingleftBigg ln(δ−1 med) 2η2 0/ceilingrightBigg (37) and has (n+r)/ceilingleftbigg ln(δ−1 med) 2η2 0/ceilingrightbigg qubits of garbage. Otherwise, if α > 1/2, letη:=γ/parenleftBig 1−α 2/parenrightBig −1 2.

Question 1145 multiple-choice

The Quantum Fourier Transform (QFT) is a key building block in quantum algorithms, relying on efficient circuit constructions using modular gate decompositions. The structure and properties of the gates used in QFT circuits greatly affect their scalability and hardware implementation. Which statement accurately describes a property of the controlled-R (controlled rotation) gates within the QFT circuit that enables flexible circuit arrangement and optimization? 1) Controlled-R gates require non-diagonal matrix representations and must be applied before Hadamard gates. 2) All controlled-R gates in the QFT circuit are diagonal operators and commute, allowing their application order to be freely rearranged. 3) Controlled-R gates cannot be further decomposed into single-qubit gates, limiting scalability. 4) The action of controlled-R gates depends on the sequence of SWAP operations performed before them. 5) Controlled-R gates always require the control qubit to precede the target qubit in circuit layout. 6) Controlled-R gates can only be implemented using three CNOT gates and a single Hadamard. 7) Controlled-R gates act non-locally on all qubits and cannot be reordered without changing the final state.

✓ Correct Answer:

The correct answer is 2) All controlled-R gates in the QFT circuit are diagonal operators and commute, allowing their application order to be freely rearranged..

📚 Reference Text:

qubits 2 ,...,k+ 1 as target. It follows from (8) that a bit reversal operator Pncan be written in terms of a sequence of SWAP operations. Ther efore,Pncan be implemented using sequence of CNOT gates. In general, a unitary transformation Unapplied to an n-qubit stateψncan be drawn as q1 Unq2 ... qn−1 qn 11 An alternative and simpliﬁed way to draw this transformatio n is |ψn/an}⌊ra⌋ketri}ht/Un|φn/an}⌊ra⌋ketri}ht, where the line with a ‘/’ through indicates a wire representi ngnqubits. 5.4 Quantum circuit for the quantum Fourier transform WenowshowhowtheQFTmatrixdecomposition derivedinTheor em2andTheorem3canbeexpressedsuccinctly by using the quantum circuit diagrams introduced in section 5.3. We start from Theorem 2 and write down the circuit representation of the decomposition of the DFT matr ixFn∈C2n×2nas follows q1 Bn··· Pnq2 Bn−1··· |ψn/an}⌊ra⌋ketri}ht/Fn=...... qn−1 ··· B2 qn ··· B1(28) Note that the gate associated with Bk+1is applied to the last k+1 qubits. Each multi-qubit gate block of this quantum circuit can be de composed into a sequence of single or two-qubit gates. For example, the bit-reversal permutation matrix Pncan ﬁrst be written in terms of ⌊n/2⌋SWAP gates as follows q1×··· q2×··· |ψn/an}⌊ra⌋ketri}ht/Pn=...... qn−1×··· qn×···(29) Since each SWAP can be implemented with 3 CNOT gates, Pnrequires ⌊3n/2⌋CNOT gates. It follows from (10) that the Bk+1gates can be decomposed further as q1H Ik⊕Ωkq2 |ψk+1/an}⌊ra⌋ketri}ht/Bk+1=... qk qk+1(30) Furthermore, the decomposition given in Theorem 3 can be use d to rewrite the Ik⊕Ωkblock in terms of controlled operations involving two qubits, as shown in the rightmost circuit below q1 Ik⊕Ωkq1• q1• q1••···• q2 q2 Ωkq2R2 q2 ···R2 ...=...=......=...... qk qk qkRk qk Rk··· qk+1 qk+1 qk+1Rk+1 qk+1Rk+1 ···(31) The circuit on the left is a dense k+ 1 qubit diagonal gate Ik⊕Ωk, which we can rewrite as a controlled- Ωk with the ﬁrst qubit as control and the next kqubits as target based on (17). From Lemma 4 we have that Ωk can be decomposed as a Kronecker product of Rmatrices, which gives the third circuit above, where the con trol qubit controls the whole group R2⊗···⊗Rk+1. This circuit can evidently be expanded as a product of separ ate controlled-Rgates to obtain the rightmost circuit, which gives a compact visual proof of Theorem 3 only using 12 circuit diagrams. The decomposition of Ik⊕Ωkinkgates is exceptionally eﬃcient, because the implementatio n of ann-qubit diagonal operator requires O(2n) gates in general [21, 1]. All the controlled- Rgates in this decomposition are diagonal matrices and commu te so the order in which they are applied does not change the outcome. Furthermore, we hav e from Lemma 5 that the roles of the control and target qubits can be swapped without changing the result. Th is leads us to a whole class of equivalent quantum circuits for Ik⊕Ωk. Finally combining (28)–(31) gives the following complete Q FT quantum circuit q1HRnRn−1···R2 ··· × q2 ···•HRn−1Rn−2··· × ......... qn−1 •··· •···HR2 × qn • ··· • ··· •H×(32) In this QFT circuit, we swapped the control and target qubits compared to Theorem 3 for all controlled- Rgates. This particular

Question 1146 multiple-choice

In finite group theory, especially within the study of groups of Lie type, involutions and their centralizers are critical for understanding subgroup structure and conjugacy classes. The classification of elementary abelian subgroups and analysis of their properties play a key role in these investigations. Which of the following statements correctly describes a group E that is an elementary abelian subgroup consisting entirely of involutions, where all involutions are conjugate to a fixed element z? 1) E is isomorphic to (Z2)^5 and every involution in E is conjugate to z. 2) E is isomorphic to SL3(2) and contains both involutions and elements of order 4. 3) E is an abelian group of order 8, with involutions forming multiple conjugacy classes. 4) E is cyclic of order 4, and its involutions are not all conjugate. 5) E is isomorphic to AGL3(2), with involutions split into distinct classes by the normalizer. 6) E is non-abelian of order 32, with a trivial center. 7) E is a direct product of two SO4 groups, each containing involutions of different types.

✓ Correct Answer:

The correct answer is 1) E is isomorphic to (Z2)^5 and every involution in E is conjugate to z..

📚 Reference Text:

~ i ~ 8. Take F ~ D, so that z E Z(F). The group N0(F)/F induces 23.SL3(2) on F (see [2J), and Z(F) = (e1, ez, e7, e8) has two N0(F)-classes of involutions: eG n Z(F) = {e;ll~i.;;:8}, andz0nZ(F)={e;ei\i::f.j}. Takee=e 1 andz=e 1e8• Throughout, a1 and b; denote elements of order 4 in l; satisfying af = b7 = [a;, bi]= e;. We divide the proof into steps. Step 1. Involutions in D. We make some observations concerning involutions in D. If x is an involution in D such that C0(,x) = S1S2 with S1 and S2 of type D4, then Z(Co(z, x)) = (z, x) and z, x, xz are all conjugate. We may take J1J2J118 ~ S1 and l4lsl6f1,.;: Sz. Then Z(S1) = Z(Sz) = (e1e2, e2e3) = (e4e7, e4e5). . Next, J? contains two classes of subgroups A7. In one class the groups A7 are 1somorph1c to SLs, with central involution z. The groups A7 in the other class have centre of order 4 and do not contain z; the central involutions in these latter groups A1 are conjugates of e. Note that a subgroup A7 of E7 has centre of order 4. LOCAL MAXIMAL SUBGROUPS 37 Step 2. Involutions in F. Let L\. = { {i, j, k, l} I e;eieke 1=1}. The 4-sets in L\. are determined by Lemma 2.14(ii). We have IL\.I = 14 and Nc(F)/ F = AGL3(2) is transitive on L\.. Each 4-set in L\. corresponds to a product of two groups S04 in an N(F)-conjugate of D. For an involution t ED corresponding to an element of so,6 with a eigenvalues -1 and b eigenvalues + 1 (with a+ b = 16), we say that t is of type (-lt(l)b. Thus e=e1 isoftype(-1)4(1)12, e,e8 = z is of type (-1)8(1)8• The involutions in F\Z(F) are a1 ... a8 and a;aiaka1e;, where {i, j, k, /}EL\. and r $. {i, j, k, !}, s = 0 or 1. Calculation in D gives: a1 ••• a8 is of type (-1)8(1)8, conjugate to e1e8, a;aiaka1 is of type ( -1)4(1)12, conjugate to e 1, a;aiaka1er (r

Question 1147 multiple-choice

In quantum computing, the Approximate Quantum Fourier Transform (AQFT) is an optimized version of the Quantum Fourier Transform, crucial for algorithms like Shor’s factoring and quantum phase estimation. Circuit efficiency often depends on minimizing gate counts, especially costly gates such as T gates. When implementing AQFT circuits for scalable quantum algorithms, which key improvement yields both asymptotic and constant-factor reductions in T-gate count while maintaining efficiency for lower approximation errors? 1) Replacing all controlled-NOT gates with Toffoli gates 2) Using only single-qubit rotations instead of entangling gates 3) Increasing the depth of the circuit to reduce error rates 4) Applying error correction codes to each qubit individually 5) Optimizing the AQFT with controlled-Z gates to reduce T-count complexity from O(n log² n) to O(n log n), while moving ε-dependence to a lower-order term 6) Removing all phase gates from the AQFT design 7) Implementing the AQFT with only classical post-processing steps

✓ Correct Answer:

The correct answer is 5) Optimizing the AQFT with controlled-Z gates to reduce T-count complexity from O(n log² n) to O(n log n), while moving ε-dependence to a lower-order term.

📚 Reference Text:

AQFT implementation AQFT with controlled-zaper30(Fig. 10) Optimized AQFT37 Circuit nq CNOT T nq CNOT T nq CNOT T AQFT 8 25 390 303 9 336 1083 8 56 1821 AQFT 16 51 1798 1162 17 1404 6309 16 234 7815 AQFT 32 67 4654 2698 33 3900 19,261 32 650 22,683 AQFT 64 99 10,366 5770 65 8892 47,099 64 1482 54,269 AQFT 128 163 21,790 11,914 129 18,876 106,631 128 3146 123,333 AQFT 256 291 44,638 24,202 257 38,844 229,729 256 6474 267,007 AQFT 512 547 90,334 48,778 513 78,780 476,873 512 13,130 553,277 AQFT 1024 1059 181,726 97,930 1025 158,652 993,727 1024 26,442 1,148,497 AQFT 2048 2083 364,510 196,234 2049 318,396 2,084,983 2048 53,066 2,427,081 AQFT 4096 4131 730,078 392,842 4097 637,884 4,316,993 4096 106,314 4,993,035 Complexity O(n) OðnlogðnÞÞ OðnlogðnÞÞ O(n) OðnlogðnÞÞ Oðnlog2ðnÞÞ O(n) OðnlogðnÞÞ Oðnlog2ðnÞÞY. Nam et al. 4 npj Quantum Information (2020) 26 Published in partnership with The University of New South Wales rotations, and term Oðlogðn logðn=εÞεÞÞcoming from the fault- tolerance overhead. In this paper we reported an improved approximation of the QFT by a quantum Clifford +T circuit with the T-count of Oðnlogðn=εÞþlogðn=εÞlogðlogðn=εÞ εÞÞ. Our improvement is twofold: ﬁrst, we reduce the dependence on nfrom Oðnlog2ðnÞÞtoOðnlogðnÞÞ, and second, we moved the dependence on εfrom the leading term into a lower order additive term. This means that the smaller the desired approxima- tion error the more ef ﬁcient our construction is compared to those known previously. Our implementation includes constant factor improvements that are not captured by the asymptotics. We report signi ﬁcant practical advantages from applying our construction, as is evidenced by the numbers in Table 1, showing the improvement by a factor of 10 to 12 in the T-count for values of nof the size expected in practical applications of quantum computers. This shows that our result carries both theoretical and practical value. METHODS Descriptions of the methods used to construct the AQFT circuit, the central result of our paper, are available in Section II. See Section II A for thedetailed methods of the circuit construction. See Section II B for furthercircuit optimization methods used to improve the T-gate counts. DATA AVAILABILITY The AQFT circuits that use our improved circuit design are available in the online repository38https://github.com/y-nam/QFT . CODE AVAILABILITY Our improved AQFT circuits, which are the quantum programs, are available in the online repository38https://github.com/y-nam/QFT . Received: 18 July 2019; Accepted: 13 February 2020; REFERENCES 1. Shor, P. Polynomial-time algorithms for prime factorization and discrete loga- rithms on a quantum computer. SIAM J. Comput. 26, 1484 –1509 (1997). 2. Kitaev, A. Quantum measurements and the Abelian Stabilizer Problem. Preprint at https://arxiv.org/abs/quant-ph/9511026 (1995). 3. Brassard G. & Hoyer P. An exact quantum polynomial-time algorithm for Simon ’s Problem. In Proc. of Fifth Israeli Symposium on Theory of Computing and Systems , 12–23 (IEEE, Ramat-Gan, Israel, 1997). https://arxiv.org/quant-ph/9704027 . 4. Grover, L. Quantum computers can search rapidly by using almost any trans- formation. Phys. Rev. Lett. 80, 4329 (1998). 5. Kassal, I., Whit ﬁeld, J.

Question 1148 multiple-choice

Finite exceptional simple groups of Lie type, such as E6(q) and G2(q), play a central role in both group theory and algebraic geometry, particularly in the study of their maximal subgroups and automorphism structures. Understanding the classification of local maximal subgroups is essential for grasping the intricate symmetries and subgroup embeddings of these groups. In the classification of local maximal subgroups of finite exceptional simple groups of Lie type over a finite field Fq, which statement accurately describes the situation when an elementary abelian r-subgroup E satisfies r ≠ p, where p is the characteristic of Fq, and E is not contained in a maximal torus? 1) E must be conjugate to a subgroup occurring only in classical groups, not in exceptional ones. 2) E is always contained in a parabolic subgroup regardless of r and p. 3) E corresponds to a subgroup of maximal rank classified via root subsystems. 4) Only specific pairs (L, E) can occur, each admitting a unique conjugacy class and being enumerated via detailed tables of centralizer and normalizer structures. 5) E necessarily generates the entire automorphism group Ga of L. 6) E is never associated with twisted versions of exceptional groups. 7) E is classified exclusively by the field size q and not by group type.

✓ Correct Answer:

The correct answer is 4) Only specific pairs (L, E) can occur, each admitting a unique conjugacy class and being enumerated via detailed tables of centralizer and normalizer structures..

📚 Reference Text:

21-48. 22 A. M. COHEN, M. W. LIEBECK, J. SAXL AND G. M. SEITZ l. Statement of results and notation Let L be a finite exceptional simple group of Lie type over Fq, where q =pa and p is prime. As described in [27], there is a simple ad joint algebraic group G over the algebraic closure of Fq, and a surjective endomorphism a of G such that L = ()P'( Ga)· Also Ga= lnndiag(L), the group generated by all inner and diagonal automorphisms of L. Let X be a group such that L ""'X ""'Aut L, and let M be a local maximal subgroup of X. Then M = Nx(E) for some elementary abelian r-subgroup E of X with r prime. Choosing E minimal, we may assume that M normalizes no proper non-trivial subgroup of E. ( *) Assume that E""' Ga (the case where E 'f Ga is discussed in the note after Theorem 1 below). If r = p then by [5, 3.12] (see also [8]), M lies in a parabolic subgroup of X. Otherwise, we may take it that one of the following holds: (I) E lies in a a-stable maximal torus of G; or (II) M normalizes no non-trivial subgroup of a torus of Ga. In Case (I), let D = Ca(E)0. Then D is a a-stable closed connected reductive subgroup of G containing a maximal torus, and M = Nx(Da n L). In the situation of the previous sentence, we say that Mis a subgroup of maximal rank in X (and also that D is a subgroup of maximal rank in G). The subgroup D has a root system relative to the maximal torus which is a subsystem of the root system of G. The possibilities for such subsystems are given in Tables A and B of [23, § 2], and the results of [23] include a complete determination of the maximal subgroups of maximal rank in exceptional groups of Lie type. Theorem 1 determines all the subgroups Min Case (II). THEOREM 1. Let L, X, G be as above, and let M = Nx(E) be a local maximal subgroup of X, with E""' Ga, E an elementary abelian r-group. Then either (I) M is a parabolic subgroup or a subgroup of maximal rank (determined in [23]), or (II) the pair (L, E) is as in Table l; in each case r =i= p and there is just one Ga-conjugacy class of such subgroups E. In Table 1 we use the notation E~(q) for E6(q) if E = + 1, and 2 E6(q) ifs= -1. Also for a prime r, we write just re for an elementary abelian group of that order. TABLE 1 L E CG0(£) NG0(E)/CaJE) Conditions Gz(p) 23 E SL3(2) zG2(3)' 23 E Z1 F4(p) 33 E SL3(3) p'35 E~(p) 33 special, of order 36 SLD) e=±l,3\p-e,p'35 Elq) 22 Ex (PQ;(q).22) S3 PQ;(q ).22 = Inndiag(D 4(q)) Ea(P) 25 special, of order 215 SL5(2) Es(P") 53 E SL3(5) p oF2, 5,

Question 1149 multiple-choice

In algebraic topology and mathematical physics, topological quantum field theories (TQFTs) provide powerful tools for studying invariants of topological spaces using algebraic structures such as Hecke algebras connected to Coxeter groups. These theories often employ graphical techniques and explicit algebraic formulas to compute invariants of specialized surfaces. Which statement correctly describes a property of the invariants assigned by one-dimensional non-commutative TQFTs associated with Hecke algebras of finite Coxeter systems? 1) For all surfaces, the invariant is always a symmetric polynomial. 2) The invariants are only defined for closed surfaces without marked points. 3) The computation of invariants requires the use of commutative algebra structures. 4) For punctured surfaces, the invariant takes the form of a Laurent polynomial. 5) Positivity properties of the invariants are exclusive to type A Coxeter groups. 6) Minimal colored graphs are never used in the graphical computation of the invariants. 7) Schur elements do not appear in explicit formulas for the invariants.

✓ Correct Answer:

The correct answer is 4) For punctured surfaces, the invariant takes the form of a Laurent polynomial..

📚 Reference Text:

Title: Topological quantum field theories from Hecke algebras Year: 2021 Paper ID: a871d801f3ffc8b966abc22a49640a134dfcab91 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a871d801f3ffc8b966abc22a49640a134dfcab91 Abstract: We construct one-dimensional non-commutative topological quantum field theories (TQFTs), one for each Hecke algebra corresponding to a finite Coxeter system. These TQFTs associate an invariant to each ciliated surface, which is a Laurent polynomial for punctured surfaces. There is a graphical way to compute the invariant using minimal colored graphs. We give explicit formulas in terms of the Schur elements of the Hecke algebra and prove positivity properties for the invariants when the Coxeter group is of classical type, or one of the exceptional types H 3 H_3 , E 6 E_6 and E 7 E_7 .

Question 1150 multiple-choice

Efficient algorithms for transforming and compressing large datasets are essential in areas such as quantum simulation and signal processing. The effectiveness of these algorithms often depends on whether the underlying data can be represented compactly, such as with low bond dimension in tensor networks. Which statement correctly describes why the Quantum Fourier Transform (QFT) implemented via Matrix Product States (MPS) can achieve linear scaling in computational time for certain datasets, compared to the classical Fast Fourier Transform (FFT)? 1) Because QFT always has lower asymptotic complexity than FFT, regardless of data structure 2) Because the QFT-MPS method exploits fast hardware acceleration that is unavailable to FFT algorithms 3) Because the bond dimension in the MPS representation is fixed and independent of the data being compressed 4) Because data that is highly compressible allows the MPS bond dimension to remain small, resulting in overall O scaling for QFT-MPS compression and transformation 5) Because QFT does not use singular value decomposition and thus avoids expensive matrix operations 6) Because FFT requires explicit knowledge of quantum entanglement, leading to inefficiencies 7) Because the preprocessing cost of constructing the QFT operator dominates the total runtime for all datasets

✓ Correct Answer:

The correct answer is 4) Because data that is highly compressible allows the MPS bond dimension to remain small, resulting in overall O scaling for QFT-MPS compression and transformation.

📚 Reference Text:

involve the singular value decomposition, which in the worst case involves√ N×√ Nmatrices, where N=2n, and taking a time proportional to N3/2=23n/2. Assuming the data are compressible, we instead use randomized sin- gular value decompositions [ 48], a fairly simple and very robust method, for more eﬃcient compression. For an MPS with maximum bond dimension χm, the compression time for the central bond becomes O(χmN+χ2 m√ N). Here the dominant χmNterm consists of a matrix-matrix multipli- cation, one of the most eﬃcient linear algebra operations, of the√ N×√ Ndata matrix and a√ N×χ/prime mmatrix of random numbers, where χ/prime m−χm∼5–10. The subdomi- nant term corresponds to a subsequent standard SVD on the product√ N×χ/prime mmatrix. Moving from the center to both sides of the MPS, one decomposes matrices with sizes√ N/2×2χm,√ N/4×2χm, etc., which contribute to the prefactor of the subdominant term only. The overall com- pression time for the entire MPS is thus O(χmN+χ2 m√ N). For constant χm, we have the complexity O(N)=O(2n). This is to be compared with the O(NlogN)=O(2nn)time of a standard FFT for the same size of data. The timings we obtain are shown in Fig. 4. In the back- ground of the plot, we show the time taken by the FFT as a function of nfor data of length 2n. Note that the FFT operates as a “black box” insensitive to the data being transformed, whereas the QFT approach is fast only if the data compress well into an MPS of small or moderate bond dimension. We determine the MPS bond dimension adaptively by setting a truncation error threshold of 10−12 in the randomized SVD compression procedure. The con- struction of the QFT MPO is done in advance and is not included, but its construction time of O(n2)is negligible anyway. To apply the QFT MPO to the data MPS, we usethe “ﬁtting” approach based on a single-site, DMRG-style algorithm [ 38,39]. Use of the QFT approach on random data is much slower than use of the FFT, as shown in Fig. 4. This is because the 040318-12 QUANTUM FOURIER TRANSFORM... PRX QUANTUM 4,040318 (2023) Random data RSVD + QFT QFT RSVD + QFT QFT20 cosines RSVD + QFT QFTOne cosine FFTRSVD+QFT QFTOne cosine + cuspsTime (s) Number of qubits n FIG. 4. Time to compute the discrete Fourier transform with use of either the FFT algorithm for data of size 2nor the QFT MPO acting on various functions represented as an MPS with nsites (qubits). Dashed curves include the time to convert the data to the MPS format with use of a randomized SVD (RSVD) algorithm plus the application of the QFT, while solid curves represent the timeof the QFT step only. Timings were performed with a 2021 ten-core MacBook Pro with an M1 Max processor with use of FFTW and ITensor [ 41]. MPS bond dimensions saturate their maximum values for this case, which are powers of 2 up to the central bond having a dimension of 2n/2. Therefore, the QFT and MPS

Question 1151 multiple-choice

Coherent optical computing leverages the wave properties of light, such as phase and amplitude, to perform mathematical operations at high speed. Both classical and quantum algorithms, including the fast Fourier transform (FFT) and quantum Fourier transform (QFT), can be implemented using specialized optical hardware. Which feature of coherent optical implementations enables direct hardware realization of the "Butterfly operation" fundamental to the FFT algorithm? 1) The use of incoherent light sources for parallel processing 2) The application of digital logic gates for signal routing 3) The employment of nonlinear crystals for frequency conversion 4) The coherent addition and subtraction of wavefronts with controlled phase shifts 5) The measurement of photon counts for state initialization 6) The amplification of light through optical pumping 7) The utilization of single-photon detectors for error correction

✓ Correct Answer:

The correct answer is 4) The coherent addition and subtraction of wavefronts with controlled phase shifts.

📚 Reference Text:

Coherent optical implementations of the fast Fourier transform and their Coherent optical implementations of the fast Fourier transform and their comparison to the optical implementation of the quantum Fourier transform comparison to the optical implementation of the quantum Fourier transform Rupert Young, Phil Birch, Chris Chatwin Publication date Publication date 29-04-2013 Licence Licence This work is made available under the Copyright not evaluated licence and should only be used in accordance with that licence. For more information on the specific terms, consult the repository record for this item. Document Version Document Version Published version Citation for this work (American Psychological Association 7th edition) Citation for this work (American Psychological Association 7th edition) Young, R., Birch, P., & Chatwin, C. (2013). Coherent optical implementations of the fast Fourier transform and their comparison to the optical implementation of the quantum Fourier transform (Version 1). University of Sussex. https://hdl.handle.net/10779/uos.23418620.v1 Link to external publisher version Link to external publisher version http://dx.doi.org/10.1117/12.2014042 Copyright and reuse: Copyright and reuse: This work was downloaded from Sussex Research Open (SRO). This document is made available in line with publisher policy and may differ from the published version. Please cite the published version where possible. Copyright and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners unless otherwise stated. For more information on this work, SRO or to report an issue, you can contact the repository administrators at sro@sussex.ac.uk . Discover more of the University’s research at https://sussex.figshare.com/ Coherent optical implementations of the fast Fourier transform and their comparison to the optical implementation of the quantum Fourier transform Rupert C. D. Young, Philip M. Birch, Chris R. Chatwin Department of Engineering and Design, School of Engineering and Informatics University of Sussex, Brighton, United Kingdom, BN1 9QT E-mail: r.c.d.young@sussex.ac.uk ABSTRACT Optical structures to implement the discrete Fourier transform (DFT) and fast Fourier transform (FFT) algorithms for discretely sampled data sets are considered. In particular, the decomposition of the FFT algorithm into the basic Butterfly operations is described, as this allows the algorithm to be fully implemented by the successive coherent addition and subtracti on of two wavefronts (the sub traction being performed after one has been appropriately phase shifted), so facilitating a simple and robust hardware implementation based on waveguided hybrid devices as employed in coherent optical detection modules. Further, a comparison is made to the optical structures proposed for the optical implementation of the quantum Fourier transform a nd they are shown to be very similar. Keywords : coherent optical Fourier t ransform, quantum Fourier transform, discrete optical Fourier transform, fast Fourier transform 1. INTRODUCTION The manipulation of the complex amplitude a coherent wave front forms the basis of coherent optical processing techniques. Central to these methods has been the ability of a converging lens to form in its back focal plane the complex two-dimensional Fourier transform of an aperture function placed in its front focal plane1. In addition, the recording of the Fourier domain complex field via

Question 1152 multiple-choice

Quantum algorithms for lattices rely on the encoding of lattice points into quantum states and use superposition and Fourier sampling to solve problems such as the hidden subgroup problem. Precision and error management are critical when working with lattice bases and algebraic structures in computational number theory. In quantum algorithms for finding hidden subgroups in elementary Abelian groups represented as quotients of $ \mathbb{R}^k \times \mathbb{Z}^l $, which process yields samples from the reciprocal lattice $ L^* $ that can be used to reconstruct the original lattice? 1) Measuring the quantum wavefunction in the Fourier basis after applying the oracle 2) Computing the Gram-Schmidt orthogonalization of the lattice basis vectors 3) Applying a Grover search algorithm to the lattice points 4) Executing classical lattice reduction algorithms such as LLL 5) Measuring the wavefunction in the computational basis before oracle action 6) Multiplying ideals in the number field followed by quantum phase estimation 7) Repeatedly applying Hadamard gates to the basis states

✓ Correct Answer:

The correct answer is 1) Measuring the quantum wavefunction in the Fourier basis after applying the oracle.

📚 Reference Text:

coeﬃcient vectors x∈Znthat repre- sent the lattice points relative to the basis. Then for eachx∈Znwe create the corresponding lattice point z=Bx∈Las a quantum state. In doing so, we use a straddle encoding str n,ν(to be deﬁned in Sect. 5) to ac- count for rounding errors. The original value of xmay now be erased (in a reversible way, which requires the reconstruction of xfromez≈z). The resulting state,|˜f(L)/angbracketright =cP z∈Le−π/bardblz/bardbl2/s2˛˛strn,ν(z)¸ ,does not de- pend on the basis (except for small inaccuracies in the preparation of the Gaussian superposition). Assum- ing a lower bound on the length of a shortest vector, λ1(L)/greaterorequalslantλ, and an upper bound on the unit cell vol- ume,d(L)/lessorequalslantd, we ﬁnd a Lipschitz constant of the function ˜fand estimate the inner product /angbracketleft˜f(L)|˜f(L/prime)/angbracketright when the lattices LandL/primeare far apart. •An eﬃcient quantum algorithm for ﬁnding a hidden subgroup in an elementary Abelian group (as discussed in the introduction). Such groups are quotients of Rk× Zl, therefore it is suﬃcient to consider this case. The problem is further reduced to the HSP for G=Rm. The algorithm has the usual structure. We create a superposition of points in Rmwith a suﬃciently broad wavefunction w, apply the oracle, and measure in the Fourier basis. In the ﬁrst approximation, the Fourier sampling generates a point uof the reciprocal lattice L∗with the probability distribution qu=1 d(L)2Z (Rm/L)2/angbracketleftf(x/prime)|f(x)/angbracketrighte2πi/angbracketleftx−x/prime,u/angbracketrightdxdx/prime. Repeating the procedure suﬃciently many times, we obtain a set of vectors that generate L∗. In prac- tice, the Fourier samples deviate from the lattice points by, roughly, the inverse width of the wavefunction w. Then the lattice is reconstructed from an approximate generating set (see Section 4.3). All of our results are stated for the ring of integers Oof a given number ﬁeld K, but they can easily be extended to general orders of K. 4. COMPUTING A BASIS FOR etO In this section we show how to compute an approximate basis for the lattice etO. Because etis in general doubly exponential in size and we have to use ﬂoating point com- putations, this is a non-trivial operation. The basic steps are to alternate ideal multiplication with size reduction to compute a short basis for the product of the two ideals that were multiplied. The algebraic numbers that appear in this computation would take exponentially many bits to repre- sent exactly. Instead we show that a polynomial number of bits of precision is suﬃcient. The idea is to use the fact that we are always using ideal lattices with lower bounds and up- per bounds on the vector lengths appearing throughout the computation, so that the precision loss can be bounded at each step. With this we can pick a precision high enough, some polynomial number of bits, so that we still have high precision at the end. The precision we need is that for any vector of length at most s√n, the computed vector is within 1/(2N ) of the actual vector. Here sandNare parame- ters chosen such that the lattice Gaussian superposition in Section 5

Question 1153 multiple-choice

Commitment schemes are essential tools in cryptography, providing mechanisms for securely concealing values while enabling protocols that require verification without disclosure. Advanced schemes increasingly support efficient zero-knowledge proofs and operations on committed values across diverse mathematical groups. Which property distinguishes statistically-hiding integer commitment schemes using groups with hidden order from earlier single-base modular commitment schemes? 1) They require the committer to know the group order during setup. 2) They only support commitments to small fixed-range integers. 3) They inherently lack the binding property needed for cryptographic applications. 4) They offer computational hiding based solely on hardness assumptions. 5) They provide hiding guarantees even against adversaries with unlimited computational power. 6) They cannot be applied to RSA groups or class groups. 7) They restrict zero-knowledge proofs to additive relations only.

✓ Correct Answer:

The correct answer is 5) They provide hiding guarantees even against adversaries with unlimited computational power..

📚 Reference Text:

AStatistically-HidingIntegerCommitment SchemeBasedonGroupswithHiddenOrder Ivan Damg˚ ard1and Eiichiro Fujisaki2 1BRICS,Dept.ofComputerScience,AarhusUniversity, NyMunkegadeAARHUSCDK-8000Denmark, ivan@daimi.au.dk 2NTTLabs,1-1Hikarinooka,Yokosuka-shi239-0847Japan, fujisaki@isl.ntt.co.jp Abstract.We present a statistically-hiding commitment scheme allow- ing commitment to arbitrary size integers , based on any (Abelian) group with certain properties ,mostimportantly,thatitishardforthecommit- tertocompute its order.Wealsogiveeﬃcientzero-knowledgeprotocols for proving knowledge of the contents of commitments and for verify- ing multiplicative relations over the integers on committed values. Theschemecanbeseenasageneralization,withaslightmodiﬁcation,oftheearlierschemeofFujisakiandOkamoto[14].Thereasonswerevisittheearlierschemeandgivesomemodiﬁcationtoitareasfollows: –The earlier scheme [14] has some gapsin the proof of soundness of the associated protocols, one of which presents a non-trivial prob- lemwhich, to the best of our knowledge, has remained open until now.Weﬁllallthegapshereusingadditionalideasincludingminormodiﬁcationoftheformofacommitment. –Although related works such as [8, 3, 10, 4] do not suﬀer from the main problem we solve here, the reason for this is that they use“commitments”with a single base(i.e.,ofform c=g smod n).Such commitments,however,cannotsatisfythestandardhidingpropertyforcommitments,andhenceprotocolsusingthem cannot in general be (honest-veriﬁer) zero-knowledge nor witness indistinguishable . –Inacomputationallyconvincingproofofknowledgewheretheproverproducesthecommoninput(whichisthetypeofprotocolwelookathere),onecannotcompletelyexcludethepossibilitythataprovermanagestoproduceacommoninputonwhichhecancheateasily.Thismeansthatthestandarddeﬁnitionofproofsofknowledgecan-notbesatisﬁed.Thereforeweintroducea new deﬁnition forcomputa- tionallyconvincingproofsofknowledge,designedto handle the case where the common input is chosen by the (possibly cheating) prover . –Ourresultsapplyto any groupwithsuitableproperties.Inparticular, theyapplytoamuchlargerclassofRSAmodulithanthesafeprimeproducts proposed in [14] – Potential examples include RSA mod-uli, class groups and, with a slight modiﬁcation, even non-Abeliangroups. Our scheme can replace the earlier one in various other constructions,suchastheeﬃcientintervalproofsofBoudot[4]andtheeﬃcientproofsfortheproductoftwosafeprimesproposedbyCamenischandMichels[9]. Y. Zheng (Ed.): ASIACRYPT 2002, LNCS 2501, pp. 125–142, 2002. c/circlecopyrtSpringer-Verlag Berlin Heidelberg 2002 126 IvanDamg˚ ardandEiichiroFujisaki 1 Introduction 1.1 Statistically-Hiding Commitment and Associated Protocols The notion of commitment is at the heart of many cryptographic protocols. The basic functionality one wants from a commitment is that the committermay choose in private a secret sfrom some set Sand release some information, the commitment to a veriﬁer, such that: even though the scheme is hiding, i.e., the veriﬁer cannot compute anything about sfrom the commitment, it is also binding, i.e., the committer cannot change his mind after having committed, but he can later open the commitment to reveal s, and convince the veriﬁer that this was indeed the original value committed to. In many applications, one wants extra functionality from a commitment scheme, for instance that the committer can prove in zero-knowledge that heknows how to open a given commitment, in particular that he knows the valuecommitted to. Also, if Shas an algebraic structure, say as a ring or a group, it can be very useful to have a multiplication protocol , i.e., a zero-knowledge protocol in which the committer can prove that committed values a,b,csatisfy ab=c.I fSis a ring, one can often, in addition, achieve that from commitments toa,b∈S, the veriﬁer can compute a commitment to a+bwithout interacting with the committer. One example of such a scheme where S=Z/qZ, whereqis a prime, is the scheme of Pedersen [17]. For the associated protocols and additional examples,see [7]. In the vast majority of examples known, the set SisZ/mZfor some m, wheremmay or may not be a prime. A multiplication protocol for such a scheme is a protocol by which one can demonstrate that for committed numbersa,b,c,ab=cmodmholds. However, there are several important cases where what you actually need is something stronger, namely to be able to prove thatab=choldsover the integers . One example

Question 1154 multiple-choice

Exceptional algebraic groups such as G2, F4, E6, E7, and E8 are fundamental objects in modern group theory, with complex subgroup structures that are categorized using concepts like maximal rank, parabolic subgroups, and automorphism types. The classification of local maximal subgroups and their normalizers is crucial for understanding symmetry and subgroup lattices in these groups. In the classification of local maximal subgroups of simple exceptional algebraic groups over algebraically closed fields, which condition guarantees that a subgroup D is maximal, aside from being a parabolic or of maximal rank? 1) D is a finite cyclic subgroup of order two 2) D contains a nontrivial centralizer of a maximal torus 3) D is a non-parabolic subgroup with a disconnected normalizer 4) D is a Levi subgroup with trivial center 5) D is a subgroup normalized by a field automorphism only 6) D is isomorphic to D4 in E7, with its normalizer given by (D4 × D4):2 7) D is a Borel subgroup containing a nontrivial abelian r-group

✓ Correct Answer:

The correct answer is 6) D is isomorphic to D4 in E7, with its normalizer given by (D4 × D4):2.

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a={l, if5 \p2-! 2£6(2) 32 Ex G2(2) NL(E) = Ul2) X G2(2) 2, if 5 \ p2 + I £7(3) 22 Ex F4(3) NL(E) = L1(3) x F4(3) LOCAL MAXIMAL SUBGROUPS 23 REMARK. In the maximal rank case in (I), our proof in fact shows that M normalizes a non-trivial subgroup of a torus of Ga (see Lemma 2.2). Constructions of the subgroups in Table 1 can be found in the proof of Theorem 1 in § 2. It is of interest to note that the condition that p 1:-2, when E is 33 or 53 and L is F4(p) or E8(pa), occurs because of the embeddings Li3) < f4(2) and L4(5) < £8(4); the subgroups L4(3) and L4(5) contain NL(E) in each case. The embedding L4(3) < Fl2) is known [U, 25], but the fact that L4(5) < £8( 4) is new, and we give a proof in§ 5. Note. In Theorem 1 we assume that E ~Ga. When E ~ G0, we have EnG 0=l by, and so MnL=CL(a) for some automorphism aE (Aut L)\G 0 of prime order r. Then a is a field, graph-field or graph automorph ism (see [16, § 7]). The conjugacy classes of such automorphisms are known, by [16, § 7; 3, § 19] and Proposition 2.7 of this paper. The centralizers arising are subgroups of the same type as Lover a maximal subfield of Fq, subgroups 2Gz(q), 2f4(q), 2E6(q~) in G2(q), F4(q), E6(q), and subgroups C4(q) (with q odd) and F4(q) in E~(q). Next we turn to algebraic groups. We prove Theorems 2 and 3, which determine all local subgroups (that is, normalizers of finite abelian subgroups) of simple algebraic groups of exceptional type over algebraically closed fields, subject to certain maximality conditions. Theorem 2 handles the subgroups of positive dimension and Theorem 3 those of dimension zero. THEOREM 2. Let G be a simple adjoint algebraic group of exceptional type (G2, F4, £6, £7 or E8) over an algebraically closed field of characteristic 1. Let S be a subgroup of Aut G such that D =(Sn G)0 is a non-trivial closed connected subgroup of G, and assume that (1) Nc(D)/ Dis finite, (2) for some primer, O,(C0(D)) i= I, and (3) Dis maximal among closed connected S-invariant subgroups of G. Then either D is parabolic or a subgroup of maximal rank, or G = E7, l i= 2, D = D4 and NG(D) = (22 x D4).S3 (with Cu(D) = 22 as in Table 1). THEOREM 3. Let G be as in Theorem 2, and suppose A is a subgroup of G satisfying the following conditions: (a) A is an elementary abelian r-group, where r is prime and r 1:-l, (b) NG(A) is finite, (c) N0(A) normalizes no proper non-trivial subgroup of A, (d) Nc;(A) is maximal with respect to (a), (b) and (c), ( e) there is no proper non-trivial connected NG(A)-invariant subgroup of G. Then A is given in Table 2, and is uniquely determined up to G-conjugacy. Theorem

Question 1155 multiple-choice

Quantum computing leverages group theory to solve hidden structure problems, with the Hidden Subgroup Problem (HSP) serving as a foundational challenge in the field. The difficulty of the HSP varies significantly depending on whether the underlying group is Abelian or non-Abelian, impacting algorithmic approaches and computational complexity. Which structural property of a non-Abelian group most directly enables efficient quantum algorithms for finding hidden normal subgroups, provided an efficient quantum Fourier transform is available? 1) The group has a cyclic subgroup of large order 2) The group admits an efficiently computable quantum Fourier transform and possesses large intersections of normalizers 3) The group is simple and non-solvable 4) The group is generated by involutions 5) The group contains a large number of conjugacy classes 6) The group embeds into a symmetric group 7) The group is nilpotent of class greater than two

✓ Correct Answer:

The correct answer is 2) The group admits an efficiently computable quantum Fourier transform and possesses large intersections of normalizers.

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found for example in Mosca’s thes is [21]. Addressing the HSP in the non-Abelian case is considered to b e the most important challenge at present in quantum computing. Beside its intrinsic mathe matical interest, the importance of this problem is enhanced by the fact that it contains as special ca se the graph isomorphism problem. Unfortunately, the non-Abelian HSP seems to be much more diﬃ cult than the Abelian case, and although considerable eﬀorts were spent on it in the last year s, only limited success can bereported. R¨ otteler and Beth [24] have presented an eﬃcient quantum al gorithm for the wreath products Zk 2≀Z2. In the case of the dihedral groups, Ettinger and Høyer [9] de signed a quantum algorithm which makes only O(log|G|) queries. However, this doesn’t make their algorithm eﬃcie nt since the (classical) post-processing stage of the results of the queries is done in exponential time in O(log|G|). Actually, this result was extended by Ettinger, Høyer and Knill [10] in the sense that they have shown that in any group, with only O(log|G|) queries to the oracle, suﬃciently statistical information can be obtained to solve the the HSP . However, it is not known how to implement eﬃciently these queries, and therefore even the “ quantum part” of their algorithm is remainingexponential. Hallgren, RusselandTa-Shma[14]p rovedthatthegenericeﬃcientquantum procedure for the HSP in Abelian groups works also for non-Ab elian groups to ﬁnd any normal subgroup, under the condition that the Fourier transform on the group can eﬃciently be computed. Grigni, Schulman, Vazirani and Vazirani could show that the HSP is solvable eﬃciently in groups where the intersection of the normalizers of all subgroups i s large [12]. A recent survey on the status of the non-Abelian HSP problem was realized by Jozsa [ 16]. In a somewhat diﬀerent line of research, recently several gro up theoretical problems have been considered in the context of black-box groups. The notion of black-box groups has been introduced by Babai and Szemer´ edi in [2]. In this model, the elements of a groupGare encoded by words over a ﬁnite alphabet, and the group operations are performed by a n oracle (the black box). The groups are assumed to be input by generators, and the encoding is not necessarily unique. There has been a considerable eﬀort to develop classical algorithms fo r computations with them [5, 3, 20], for example to identify the composition factors (especially th e non-commutative ones). Eﬃcient black- boxalgorithms giveriseautomatically toeﬃcient algorith ms whenever theblack-box operations can be replaced by eﬃcient procedures. Permutation groups, mat rix groups over ﬁnite ﬁelds and even ﬁnite matrix groups over algebraic number ﬁelds ﬁt in this mo del. In particular, Watrous [27] has recently considered solvable black-box groups in the restr icted model of unique encoding, and using some new quantum algorithmical ideas, he could construct eﬃ cient quantum algorithms for ﬁnding composition series, decomposing Abelian factors, computi ng the order and testing membership in these groups. In this paper we

Question 1156 multiple-choice

Quantum algorithms have significantly advanced the solution of hidden subgroup problems (HSP) in group theory, impacting fields such as cryptography and computational mathematics. The efficiency of these algorithms often relies on specific structural properties of the groups involved. Which of the following statements accurately describes a scenario where quantum algorithms can solve the hidden subgroup problem efficiently in polynomial time? 1) When the group is a simple non-Abelian group with no normal subgroups. 2) When the group has a large cyclic normal subgroup of prime order, and the quotient is non-Abelian. 3) When the group is a direct product of two non-Abelian groups with trivial intersection. 4) When the group is a symmetric group of degree greater than five with no Abelian quotient. 5) When the group has a non-normal subgroup of order two, and the quotient is non-Abelian. 6) When the group has a nontrivial commutator subgroup that is not contained in the center. 7) When the group is a black-box group with unique encoding and contains an elementary Abelian normal 2-subgroup N, such that G/N is small or cyclic.

✓ Correct Answer:

The correct answer is 7) When the group is a black-box group with unique encoding and contains an elementary Abelian normal 2-subgroup N, such that G/N is small or cyclic..

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also G′∩H, in time polynomial in the input size + |G′|. We consider the function F:x/ma√st≀→ {f(xG′)}={f(xg)|g∈G′}which can be computed by querying |G′|times the function f. The function Fhides the subgroup HG′. Note that HG′is normal since G/G′is Abelian. Thus by Theorem 8, we can ﬁnd generators for HG′by a quantum algorithm in time polynomial in the size of the input + |G′|sinceν(G/HG′) = 1, because G/HG′is Abelian. For each generator xofHG′, we enumerate all the elements of coset xG′and select an element ofxG′∩H. The cost of this step is again polynomial in the input size+ |G′|. We take for H1the subgroup of Ggenerated by the selected elements and H∩G′. We get H1∩G′=H∩G′, and by the deﬁnition of the selected elements H1G′=HG′. A group Gis anextra-special p-groupif its commutator subgroup G′coincides with its center, |G′|=p, and moreover G/G′is an elementary Abelian p-group. Therefore we get the following corollary from the previous theorem. Corollary 12. The hidden subgroup problem in extra-special p-groups can be solved by a quantum algorithm in time polynomial in input size +p. 6 Groups with a large elementary Abelian normal 2-subgroup Assume that Nis an elementary Abelian normal 2-subgroup of a group G,and it is given by generators as part of the input. Our aim is to solve the HSP in Gin the cases where G/Nis either small or cyclic. Typical examples of groups of the latter typ e are matrix groups over a ﬁeld of characteristic 2 of degree k+1 generated by a single matrix of type (a), where the k×ksub-matrix in the upper left corner is invertible, together with severa l matrices of type (b): (a) ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ ∗ 0 0 0 0 0 1 ,(b) 1 0 0 0 ∗ 0 1 0 0 ∗ 0 0 1 0 ∗ 0 0 0 1 ∗ 0 0 0 0 1 . Note that the class of groups of this kind include the wreath p roductsZk 2≀Z2in which the hidden subgroup problem has been shown to be solvable in polynomial time by R¨ otteler and Beth in [24]. Based on a technique inspired by the idea of Ettinger and Høye r used for the dihedral groups in 9 [9], we solve the hidden subgroup problem in quantum polynom ial time in this more general class of groups. Theorem 13. LetGbe a black-box group with unique encoding of elements and Nbe a normal elementary Abelian 2-subgroup of Ggiven by generators. Then the hidden subgroup problem in G can be solved by a quantum algorithm in time polynomial in the input size +|G/N|. IfG/Nis cyclic then the hidden subgroup problem can be solved in poly nomial time. Proof.LetHbe a subgroup of Ghidden by the function f. The main line of the proof is like in Theorem 11: we will generate H1≤Hwhich satisﬁes H1∩N=H∩NandH1N/N=HN/N(or equivalently H1N=HN). Again we start the generation of H1withH∩Nwhich can be computed in polynomial time in the input

Question 1157 multiple-choice

Quantum stochastic thermodynamics investigates energy fluctuations and work statistics in quantum systems, particularly during nonequilibrium processes. Group-theoretical methods are increasingly used to analyze systems with quadratic Hamiltonians due to their symmetries and computational advantages. Which key advantage does group-representation theory provide for calculating quantum work distributions in systems governed by quadratic Hamiltonians? 1) It enables efficient and exact computation of work distributions for arbitrarily driven quantum systems. 2) It restricts the analysis to only static, noninteracting systems. 3) It eliminates the need for any knowledge of the system's symmetry properties. 4) It replaces the concept of work distribution with only average energy calculations. 5) It approximates work distributions using perturbative expansions valid for small systems. 6) It requires the Hamiltonian to be diagonalizable by numerical methods only. 7) It is limited to classical thermodynamic processes and cannot address quantum coherence.

✓ Correct Answer:

The correct answer is 1) It enables efficient and exact computation of work distributions for arbitrarily driven quantum systems..

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Title: Group-theoretical approach to the calculation of quantum work distribution Year: 2019 Paper ID: b056b653dbd232fe323749a2b96b72f8f9a3befe Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b056b653dbd232fe323749a2b96b72f8f9a3befe Abstract: Usually the calculation of work distributions in an arbitrary nonequilibrium process in a quantum system, especially in a quantum many-body system is extremely cumbersome. For all quantum systems described by quadratic Hamiltonians, we invent a universal method for solving the work distribution of quantum systems in an arbitrary driving process by utilizing the group-representation theory. This method enables us to efficiently calculate work distributions where previous methods fail. In some specific models, such as the time-dependent harmonic oscillator, the dynamical Casimir effect, and the transverse XY model, the exact and analytical solutions of work distributions in an arbitrary nonequilibrium process are obtained. Our work initiates the study of quantum stochastic thermodynamics based on group-representation theory.

Question 1158 multiple-choice

In advanced algebraic geometry and representation theory, the interplay between group actions, graded algebras, and dimension formulas is key to understanding the structure of varieties and their function spaces. Properties such as parity invariants, generation of graded components, and the absence of common zeros among certain elements play crucial roles in constructing projective models of abelian varieties. Which statement accurately describes a property of elements in $ R_{m,u} $ with fixed $ n $-parity in relation to the variety $ G_g $? 1) Elements of $ R_{m,u} $ with fixed $ n $-parity have no common zero in $ G_g $. 2) Elements of $ R_{m,u} $ with fixed $ n $-parity always vanish on $ G_g $. 3) Elements of $ R_{m,u} $ with fixed $ n $-parity generate only the even-degree components of $ G_g $. 4) Elements of $ R_{m,u} $ with fixed $ n $-parity necessarily intersect at a single point in $ G_g $. 5) Elements of $ R_{m,u} $ with fixed $ n $-parity form a basis for the coordinate ring of $ G_g $. 6) Elements of $ R_{m,u} $ with fixed $ n $-parity are invariant under all group actions on $ G_g $. 7) Elements of $ R_{m,u} $ with fixed $ n $-parity always have common zeros in $ G_g $.

✓ Correct Answer:

The correct answer is 1) Elements of $ R_{m,u} $ with fixed $ n $-parity have no common zero in $ G_g $..

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££ fixed by the Γ?, so βm,w c Sm. Also the i?m,w are just the subspaces of Sm corresponding to the various irreducible representa- tions of the group T. So Sm — Θ(a) Rm>u. The proof that dimRm}U ==(2m)° is similar to that of Theorem 1.1.1. We omit it. PROPOSITION 1.2.4. With the same notation as above, the elements of Rm,u of pre-assigned n-parity have no common zero in Gg. Proof. Let (r) = (rί9 , rg) be a given n-parity. Suppose θ e Rn with trivial ^-parity. By extending k we can get Cj e k* such that Cf = If (0 = ^, .-., Cg\ set ^(Z) = (Π?-iX;θ 0(GY). Then = (Πί^i^i^ΓOgj^C. Z^-^CZ) and it follows that u with ti- parity (r). The zeroes of φ are just translates of the zeroes of θ by C"1. But, by Theorem 1.2.2, the θ eRm with trivial ^-parity have no common zero. COROLLARY 1.2.5. // m is a multiple of n, the elements of Rm>u of pre-assigned n-parity have no common zero in Gg. Proof. If (r) is the given ^-parity and x e Gg9 choose θ1 e Rn with trivial ^-parity such that θλ(x) Φ 0 and θ2 β Rn>u with 7^-parity (r) such that Θ2(x) Φ 0. If m = ^p, θl'xθ2^RWjyU and has n-parity (r). The following simple lemma will be used to prove the finite genera- tion of i?. LEMMA 1.2.6. Let M be a graded algebra over a field k. Assume: Mm = 0 for all negative m, Mm is finite dimensional over k for all m and there is a polynomial P such that dim Mm = P(m) for all large m. Then, if Mx generates Mm for infinitely many m, it generates Mm for all large m. https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press ABELIAN VARIETIES 73 Proof, Let M be the subalgebra of M generated by Mx and P be the Hubert polynomial of Mm. By assumption, Mm = Mm for infinitely many m. Thus P and P are equal at infinitely many m, P = P and dim Mm = dim Mm for large m. Suppose now that we are in the situation of Proposition 1.2.3 with n = 2. In other words, we assume that char, k Φ 2. PROPOSITION 1.2.7. // char. & Φ 2 and n = 2, then ^ generates Sm for all large m. Proof. It suffices to show that each βeS2t is in St St. For then Sx generates S2r for all r and we can use Lemma 1.2.6. By Proposition 1.2.3 we may assume θ e R2t,u for some u e Uξ, and that θ has a definite 2-parity. Choose θλ e R2t,u with the same 2-parity as θ so that ^(1) Φ 0. (see Cor. 1.2.5). Let φ(X, Y) = Θ(XY)Θ1(XY~1). It is easy to see that φeRU Since 0 and θx have the same 2-parity, φ is a power series in X\, Y\ and therefore is in a2(St) ® a2(St), (cf.

Question 1159 multiple-choice

The Jones polynomial is a knot invariant from topology whose computation is closely linked to quantum complexity theory. The classification of problems involving the Jones polynomial reveals deep connections between knot theory, quantum algorithms, and complexity classes such as QCMA and PSPACE. Which statement accurately describes the relationship between the Increase Jones Plat problem and the complexity class QCMA? 1) The Increase Jones Plat problem is solvable in polynomial time by classical algorithms. 2) The Increase Jones Plat problem is QMA-complete, requiring a quantum witness for verification. 3) The Increase Jones Plat problem is NP-complete, with solutions verifiable by classical means alone. 4) The Increase Jones Plat problem is PSPACE-complete, encoding the hardest problems in PSPACE. 5) The Increase Jones Plat problem is BQP-complete, capturing all problems efficiently solvable by quantum computers. 6) The Increase Jones Plat problem is QCMA-complete, meaning it is among the hardest problems in QCMA and every problem in QCMA can be reduced to it. 7) The Increase Jones Plat problem can be solved by a deterministic polynomial-space Turing machine.

✓ Correct Answer:

The correct answer is 6) The Increase Jones Plat problem is QCMA-complete, meaning it is among the hardest problems in QCMA and every problem in QCMA can be reduced to it..

📚 Reference Text:

isQCMA -complete. Proof: To begin, we demonstrate that Increase Jones Plat is in QCMA. Given a braid b∈B2nof lengthO/parenleftbig poly(n)/parenrightbig , Merlin begins by sending Arthur the classical description of the “witness” braid c. Next, Arthur checks to see if c∈ C2n, which can be done in polynomial-time by deﬁnition. Because of the promise, Arthur only needs to le arn the value of 1 [2]n−1/vextendsingle/vextendsingle/vextendsingleJ(/tildewiderc−1bc,e2πi/ℓ)/vextendsingle/vextendsingle/vextendsingle2 with an accuracy of <1/2 with probability ≤3/4. For this, it is suﬃcient for him to runApproximate Jones Closure on a quantum computer with δ= 1/4, which takes O/parenleftbig poly(n)/parenrightbig -time. To see that this problem is QCMA-hard, and is thus QCMA-compl ete, suppose that L∈ QCMA. Let Ube then’th circuit from the uniform family of quantum veriﬁers for L, and suppose that Uacts onn′qubits while accepting witnesses of size m. Lettingǫ >0 be a constant to be determined at the end of the proof, Arthur prec omputes a set of four braids C8⊂B8whichǫ/m-approximate the circuits/braceleftbig Xy1⊗Xy2:y1,y2∈ {0,1}/bracerightbig (obviously, there is nothing to compute when y1=y2= 0). This can be done in polynomial-time. Deﬁne C4n′⊂B4n′to consist of braids which act as the identity everywhere exc ept for the strands 4n+ 1,...4n+ 4m, where only a single braid from C8can act on each group of eight strands 8i+ 1,...,8i+ 8 within those strands. If we deﬁne Vymin an analogous manner to Wxnin the proof of Theorem 8.0, where it acts only by ﬂipping the wit ness qubits, it follows by (7.2) that for each braid cinC4n′, the corresponding unitary π(2,ℓ) [2n′,2n′](c)ǫ-approximates Vymfor one and only one ym(this is because ||Vym−Vzm||∞= 2 whenever ym/ne}ationslash=zm). Arthur then Pawel Wocjan and Jon Yard 29 tellsxnto Merlin, who sends him a witness braid cfromC4n′. DeﬁneWxnas in the proof of Theorem 8.0, except that it acts as the identity on the witn ess and ancillary qubits. By the ﬁrst half of Theorem 7.1, Arthur may compute a braid bfor which the circuit WxnUWxn isǫ-approximated by π(2,ℓ) [2n′,2n′](b). Therefore, we obtain by (7.2) that π(2,ℓ) [2n′,2n′](c−1bc) is a 3ǫ-approximation of WxnVymUWxnVym. Since /an}bracketle{t00···0|WxnVynUWxnVyn|00···0/an}bracketri}ht=/an}bracketle{txnym00···0|U|xnym00···0/an}bracketri}ht, we may apply the rest of Theorem 7.1 to obtain that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/an}bracketle{txnym00···0|U|xnym00···0/an}bracketri}ht/vextendsingle/vextendsingle2−1 [2]2n′−1/vextendsingle/vextendsingle/vextendsingleJ(/tildewiderc−1bc,e2πi/ℓ)/vextendsingle/vextendsingle/vextendsingle2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤3ǫ. In order to complete the proof, notice that because of the pro mise, it is suﬃcient to choose ǫ= 1/8./square A rather large complexity class is PSPACE, which contains al l languages which can be decided using only polynomial space. In [41], PSPACE is characteriz ed as the class of languages which can be decided by applying the same polynomial-size ci rcuit (possibly exponentially) many times. Formally, it is shown there that if L∈PSPACE, then there exists a uniform family of polynomial-size quantum circuits {Un}, together with a sequence of polynomial-time computable natural numbers en, for which, if f:{0,1}∗→ {0,1}is such that f−1(1) =L, then /an}bracketle{txn100···0|Uen n|xn100···0/an}bracketri}ht=f(xn). Using this characterization, we will show that the followin g problem is PSPACE-complete. Problem 8.2 (Approximate Concatenated Jones Plat) Given a braid b∈B2nwith O/parenleftbig poly(n)/parenrightbig crossings, a constant ℓsatisfyingℓ≥5,ℓ/ne}ationslash= 6, and a positive integer ewhich is describable by O/parenleftbig

Question 1160 multiple-choice

In quantum computing, the hidden subgroup problem (HSP) is a central challenge that underlies powerful algorithms, such as those for factoring and discrete logarithms. Effective quantum measurements are essential for efficiently identifying hidden subgroups from quantum states generated through oracle queries. Which measurement strategy generalizes the single-copy optimal measurement for HSP when all subgroups are equally likely, thereby removing the requirement that subgroups are conjugate? 1) The standard Fourier sampling measurement 2) The quantum phase estimation algorithm 3) The majority-vote measurement over multiple copies 4) The adaptive measurement based on classical post-processing 5) A hybrid of Ip’s measurement and the pretty good measurement (PGM) 6) The measurement designed for Gel’fand pairs only 7) The measurement using random coset representatives

✓ Correct Answer:

The correct answer is 5) A hybrid of Ip’s measurement and the pretty good measurement (PGM).

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cases the optimal measurement wasthe so-called pretty good measurement (PGM) [17]. Building on this result, Moore a nd Russell [18] showed that the PGM is the optimal single copy measurement for the non-abelian HSP when all of the subgroups are conjugate to each other and is the optimal multi-copy measurement for the non-abelian HSP when the subgroups are all Gel’fand pairs with the group [18]. Here we derive the optimal single copy measurement for t he HSP when all of the subgroups are given with equal probability, removing the restriction of Moore and Russell th at these subgroups are conjugate to each other. The measurement we derive can be seen as a hybrid between the mea surement of Ip [15] and the PGM [16, 18]. II. THE HIDDEN SUBGROUP PROBLEM AND OPTIMAL MEASUREMENTS We begin by deﬁning the hidden subgroup problem. A detailed review of this problem can be found in [19]. Hidden Subgroup Problem (HSP) LetGbe a group and fa function from elements of this group to some set S. This function is promised to be constant and distinct on diﬀerent lef t cosets of a subgroup H ofG. The hidden subgroup problem is to identify, by querying f, the subgroup H. In all of the cases we work with, Gwill be ﬁnite, and elements of Gand ofSwill be expressed as O(polylog |G|) bitstrings. An algorithm for the HSP is considered to be eﬃcient when the running time is polynomial in log |G|. By identifying Hwe do not require that all of the elements of Hbe returned, but only that a generating set or any succinct description of the subgroup be returned. When Gis abelian, we call this problem the abelian HSP and similarly when Gis non-abelian we call this problem the non-abelian HSP. 2 In the standard query model of the hidden subgroup problem, one approaches a quantum algorithm for the HSP by querying the function fin superposition over the group elements. This produces the state 1/radicalbig |G|/summationdisplay g∈G|g/an}bracketri}ht⊗|f(g)/an}bracketri}ht. (1) The second register of this state (whose range is the set S) is then discarded and we obtain a mixed state depending only on the subgroup Hhidden by f, ρH=|H| |G|/summationdisplay g∈bG|gH/an}bracketri}ht/an}bracketle{tgH|, (2) where we have deﬁned the left coset states |gH/an}bracketri}ht:=1/radicalbig |H|/summationdisplay h∈H|gh/an}bracketri}ht, (3) and/hatwideGis a set of coset representatives for the subgroup H. We call ρHthe hidden subgroup state. In the standard approach to solving the hidden subgroup problem on a quantum comp uter, one is given access to one or many copies ofρH. The goal is then to perform a set of quantum gates and a measure ment that eﬃciently identiﬁes H. In the optimal measurement approach to the HSP, one examines th e HSP from a state distinction perspective. Suppose that the function fhidesHwith probability pH. Then in the single copy case, we can assume that we have been given ρHwith probability pH. One then attempts to ﬁnd a generalized measurement that maximiz es the probability of successfully identifying the hidden subgroup H. A

Question 1161 multiple-choice

Finite W algebras are quantum algebras that arise in mathematical physics, particularly in the study of symmetries and integrable systems. Their structure can be analyzed through cohomological and homological methods, and quantum effects often manifest for larger algebras. In the quantization of finite W algebras, what is the primary role of the tic-tac-toe construction within the cohomology H∗(Ω;d)? 1) To classify irreducible representations of the algebra 2) To construct explicit representatives for the generators as cohomology classes 3) To compute the partition function for integrable systems 4) To determine the embedding of sl2 subalgebras 5) To calculate higher-order quantum corrections in the algebra 6) To identify the center of the quantum algebra 7) To evaluate the physical observables of conformal field theories

✓ Correct Answer:

The correct answer is 2) To construct explicit representatives for the generators as cohomology classes.

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correspondence with the components of gthat made up the highest weight gauge in section 1. Therefore H∗(Ω;d) really is a quantization of the ﬁnite W algebra. What remains to be done is to compute the algebra str ucture ofH∗(Ω;d). The only thing that (5.24) tells us is that the product of two elem entsaandbof bidegree (−p,p) and ( −q,q) is given by the product structure on Uglw, modulo terms of bidegree (−r,r) withr < p +q. To ﬁnd these lower terms we need explicit representatives o f the generators of H0(Ω;d) in Ω. Such representatives can be constructed using the so-called tic-tac-toe construction [28]: take some φ0∈glw, of bidegree ( −p,p). Then d0(φ) is of bidegree (1 −p,p). Because d1d0(φ) =−d0d1(φ) = 0, and there is no d1 cohomology of bidegree (1 −p,p),d0(φ) =d1(φ1) for someφ1of bidegree (1 −p,p−1). Now repeat the same steps for φ1, giving aφ2of bidegree (2 −p,p−2), such that d0(φ1) =d1(φ2). Note that d1d0(φ1) =−d0d1(φ1) =−d2 0(φ) = 0. In this way we ﬁnd a sequence of elements φlof bidegree ( l−p,p−l). The process stops at l=p. LetW(φ) =/summationtextp l=0(−1)lφl. ThendW(φ) = 0, and W(φ) is a representative of φ0in H0(Ω;d). The algebra structure of H0(Ω;d) is then completely determined by looking at the commutation relations of W(φ) in Ω, where φ0runs over a basis of glw. This is the quantum ﬁnite Walgebra. Let us now give an example of the construction described abov e. 45 5.7. Example Consider again the embedding associated to the following pa rtition of the number 3: 3 = 2 + 1. We constructed the classical Wassociated to this embedding earlier. We shall now quantize this Poisson algebra by the methods dev eloped above. Take the following basis of sl3: rata= r4 6−r5 2r2r1 r6−r4 3r3 r8r7r4 6+r5 2 . (5.28) Remember that (in the present notation) the sl2embedding is given by t+=t1,t0= −t5andt−=t8. The nilpotent subalgebra g+is spanned by {t1,t3},g0by{t2,t4,t5,t6} andg−by{t7,t8}. Thed1cohomology of Ω redis generated by {ˆJ4,ˆJ5,ˆJ6,ˆJ8}, and using the tic-tac-toe construction one ﬁnds representatives for these generators in H0(Ωred;d): W(ˆJ4) = ˆJ4, W(ˆJ6) = ˆJ6, W(ˆJ7) = ˆJ7−1 2ˆJ2ˆJ5−1 2ˆJ4ˆJ2+1 2ˆJ2, W(ˆJ8) = ˆJ8+1 4ˆJ5ˆJ5+ˆJ2ˆJ6−ˆJ5. (5.29) Let us introduce another set of generators C=−4 3W(ˆJ8)−1 9W(ˆJ4)W(ˆJ4)−1, H=−2 3W(ˆJ4)−1, E=W(ˆJ7), F=4 3W(ˆJ6). (5.30) The commutation relations between these generators are giv en by [H,E] = 2E, [H,F] = −2F, [E,F] =H2+C, [C,E] = [C,F] = [C,H] = 0. (5.31) These are precisely the same as the relations for the ﬁnite W(2) 3algebra given in [13]. Notice that in this case the quantum relations are identical to the classical ones. The explicit ¯hdependence can be recovered simply by multiplying the right hand sides of (5.31) by ¯h. 46 In the appendix we also discuss the explicit quantization of all the ﬁnite Walgebras that can be obtained from sl4. There one does encounter certain quantum eﬀects, i.e the quantum relations will contain terms of order ¯ h2or higher. 6. The Representation Theory of Finite Walgebras The next important topic in the

Question 1162 multiple-choice

Topological quantum computing encodes quantum information in the fusion spaces of anyons or symmetry defects, and quantum gates are implemented via braiding operations described by braid group representations. The mathematical properties of these representations play a key role in the computational power and robustness of this paradigm. Which statement most accurately characterizes the significance of projective representations of the braid group in topological quantum computing with symmetry defects? 1) They restrict available quantum gates to classical reversible operations only. 2) They allow direct measurement of global phase differences between quantum states. 3) They prevent the implementation of fault-tolerant quantum gates due to non-unitarity. 4) They ensure that quantum computation is insensitive to global phase, making the representation suitable for encoding quantum gates. 5) They imply that only abelian anyons can be used for universal quantum computation. 6) They require that all braid operations commute with each other for reliable computation. 7) They necessitate the use of non-deterministic fusion outcomes for encoding quantum information.

✓ Correct Answer:

The correct answer is 4) They ensure that quantum computation is insensitive to global phase, making the representation suitable for encoding quantum gates..

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defects) {xgi}i≥1with total charge xg0. Typically for applications to quantum computing one takes all topological charges to be identical, and considers vector spaces of the f ormVyhn,xg:=Hom(yh,x⊗n g). Then Vyhn,xgis the Hilbert space of states associated to a collection of nsymmetry defects xg∈ C gwith total charge yh∈ C hon a surface with trivial topology. We identify a d- dimensional qudit Cdwith Vyhn,xg. Vyhn,xg= xgxg··· xg yh As we will see in the next subsection these vector spaces give rise to a projective repre- sentation of the n-strand braid group Bn. Bn=/angbracketleftig σ1,σ2,..., σn−1/vextendsingle/vextendsingle/vextendsingleσiσj=σjσifor|i−j|≥2 σiσi+1σi=σi+1σiσi+1,i=1,2,..., n−1/angbracketrightig . In general Vxg0,...,xgnwill not be a representation of the braid group, or even the pu re braid group. However, it is a representation of the subgroup ofBnthat preserves the data of the topological charges xgi. As with anyons, for physical reasons one is typically intere sted in state spaces with trivial total charge. When all nsimple objects xgilive inC0, i.e. are anyons, then the formalism reduces to TQC with anyons, see [8]. 3.2. Projective braid group representations. Next we deﬁne the representations associ- ated to nsymmetry defects xgwith total charge yh. When the ntopological charges are identical, there is an action of the n-strand braid group on Hom (yh,x⊗n g). An orthonormal basis of qudit states is given by fusion trees whose internal edges are admissibly labeled by simple objects in C× G, normalized with respect to the Markov trace inner product. ···xgxgxgxg yh In the remainder of the paper we suppress the arrows indicati ng the orientations when drawing fusion trees, and assume all diagrams are oriented f rom the bottom up. The action of the braid group is deﬁned exactly the same as for anyons. Abstractly, each element of the braid group is interpreted as a morphism b:x⊗n g→x⊗n g. Then elements of the braid group act on a fusion tree basis element f∈Vyhn,xgby postcomposition: SYMMETRY DEFECTS AND THEIR APPLICATION TO TOPOLOGICAL QUAN TUM COMPUTING 11 b·f=b◦f. On the level of diagrams, elements of the braid group b∈ B nact on an admissibly labeled fusion trees by stacking the diagram bofbon the fusion tree. b ··· By extending linearly to all of Vyhn,xgthis gives a projective representation ρn,xg,yh, which we refer to as the representation corresponding to nsymmetry defects of type xgwith total charge yh. The matrix representation in the fusion tree basis is then gi ven by resolving b·fas a linear combination of admissibly labeled fusion trees usin g the graphical calculus. (Or in practice, by taking inner products as in Section 4.) However , this deﬁnes a braid group representation which is not linear in general, only project ive. This is due to symmetry defects being conﬁned, see [2]. However, since quantum mech anics is insensitive to a global phase already, even when analyzing representations coming from UMTCs one is only interested in the projective image. That means we can me aningfully compare quantum computers designed using either anyons or symmetry defects . 3.2.1. Universality. For applications to quantum

Question 1163 multiple-choice

Quantum algorithms frequently tackle group-theoretic problems, such as the Hidden Subgroup Problem (HSP) and its generalizations, which are fundamental in quantum computing and cryptography. Understanding the relationships and reductions between variants like the Hidden Multiple Shift (HMS) problem is crucial for analyzing quantum algorithmic efficiency. In the context of the Hidden Multiple Shift (HMS) problem HMS(N, n, r), which statement best characterizes how the problem’s difficulty changes as the parameter r, representing the size of the subset H, increases? 1) The problem becomes exponentially harder with increasing r because the search space expands. 2) The problem becomes easier with larger r, due to increased redundancy exploited by quantum algorithms. 3) The complexity remains constant regardless of the value of r. 4) The problem transitions to a classical problem for large r, making quantum algorithms unnecessary. 5) Larger r leads to the problem being equivalent to the standard hidden shift problem. 6) Increasing r causes quantum algorithms to fail unless additional structure is available. 7) The problem becomes equivalent to factoring when r reaches N.

✓ Correct Answer:

The correct answer is 2) The problem becomes easier with larger r, due to increased redundancy exploited by quantum algorithms..

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Borel HSP for arbitrary N. The main idea of the quantum algorithmisbasedontheobservationthattheproblemcanbereduc edtoanother restrictedhiddensubgroupproblemin the group G= (Z/NZ)2⋊(Z/NZ)∗where the hidden subgroup is promised to be a complement of the normal su bgroup (Z/NZ)2. The latter restricted HSP can be cast as an instance of the multiple shift problem considered in [ IPS18], which can itself be seen as a generalization of the hidden shift problem. Problem 2.4. The hidden multiple shift problem HMS( N,n,r) is parameterized by three positive integers N,nandr,whereN >1 and 2≤r≤N−1. Assume that we have a set H⊆Z/NZof cardinality rand a function fs: (Z/NZ)n× H→ {0,1}l, deﬁned as fs(x,h) =f(x−hs) wheres∈(Z/NZ)nandf: (Z/NZ)n→ {0,1}lis an injective function. Given fsby an oracle, the task is to ﬁndsmodN δ(H,N), whereδ(H,N) is deﬁned as the largest divisor of Nsuch thath−h′is divisible by δ(H,N) for every h,h′∈H. A special case of the HMS problem was ﬁrst considered by Childs and v an Dam [CVD05]. They presented a quantum polynomial time algorithm for the case whenn= 1 andHis a contiguous interval of size NΩ(1). For general n, an algorithmin [ IPS18] solvesHMS( N,n,r) inO(poly(n)(N r)n+O(1)). Fora setHof small size, HMS is close to the hidden shift problem. Speciﬁcally, HMS( N,n,2) is the standard hidden shift problem. On the other extreme, for r=N, HMS is an abelian hidden subgroup problem in the group ( Z/NZ)n+1. Intuitively, the largerris, the easier HMS( N,n,r) becomes. Below we restate the above mentioned result from [ IPS18] for the special case n= 1. Theorem 2.5. There is a quantum algorithm that solves the HMS( N,1,r) in time/parenleftbigN r/parenrightbigO(1)with high probability. 2.7 The malleability oracle In [KMPW21 ] the authors introduce a general framework dubbed the malleabil- ity oracle. Let Gbe a group acting on a set Xand letf:X→Ibe an injective function where Iis some set. The input of the malleability oracle is an element g∈Gand a value f(x) (xis not provided) and the output is f(g∗x). It is shown in [KMPW21 , Theorem 3.3] that if Gis abelian and the action of GonXis free and transitive then inverting f(x) can be reduced to an abelian hidden shift problem. The idea of the proof is as follows. One takes an arbitrary k nownx0 and the corresponding f(x0). Then one can deﬁne two functions f0,f1fromGto Iwheref0(g) =f(g∗x0) andf1(g) =f(g∗x). These functions are well-deﬁned asfwas injective. Now since the action of Gis transitive there is an element sthat takesx0tox. One can easily see that fandf0are shifts of each other and the shift is realized by that element s. Since the action is free, fandf0will be injective functions themselves hence one can apply Kuperberg’s algorithm to ﬁndsand ﬁnally that is enough to compute x. Title Suppressed Due to Excessive Length 11 Remark 2.6. It follows from the proof that it is not strictly necessary for the action to be transitive. It is enough if we know any element in the orbit of the secretx. For instance if suﬃces if there are only a few orbits and we have a representative of each

Question 1164 multiple-choice

Quantum Machine Learning (QML) utilizes quantum computers to achieve new capabilities in data analysis. Advances in the field increasingly focus on leveraging mathematical frameworks to encode symmetries and inductive biases directly into quantum algorithms. Which approach in Quantum Machine Learning specifically employs group theory and representation theory to encode and exploit symmetries in data, thereby restricting the function space and enhancing model performance? 1) Quantum Support Vector Machines 2) Quantum Reinforcement Learning 3) Variational Quantum Algorithms 4) Quantum Generative Adversarial Networks 5) Geometric Quantum Machine Learning (GQML) 6) Quantum Annealing Optimization 7) Quantum Decision Trees

✓ Correct Answer:

The correct answer is 5) Geometric Quantum Machine Learning (GQML).

📚 Reference Text:

one of the most promising candidates to make practical use of quantum computers [1–4]. By leveraging quantum systems to process information, QML presents the ultimate framework for data analysis [4]. In the near- term [5], QML aims to achieve a quantum advantage, i.e., to solve certain tasks exponentially faster than any classi- cal supercomputer. In the fault-tolerant era, QML will be the most natural choice to learn from the data created by large-scale quantum devices. Despite some promising results [6–12], the ﬁeld of QML is in its infancy. One of the main challenges for QML is de- veloping quantum-aware and problem-speciﬁc models, as it has been shown that simply translating classical architec- tures into quantum ones, or using problem-agnostic archi- tectures, can lead to serious issues that hinder the perfor- mance of the QML model [13–21]. Recently, and inspired by the tremendous success of clas- sical geometric deep learning [24], there have been several eﬀorts to create QML models with strong inductive biases that respect the underlying structure and symmetries of the data over which they act [25–31]. The goal of induc- tive biases is to restrict the space of functions explored by the QML model by imposing task-speciﬁc knowledge or as- sumptions. It is expected that models tailored to a given task will have better performance, both in training and generalization, thanthosewithoutinductivebiases[24–32]. These eﬀorts have led to the inception of the ﬁeld of Geo- metric Quantum Machine Learning (GQML). We remark that the scope of GQML is quite broad and is relevant to quantum deep learning, quantum kernels, quantum gener- ative modeling, and other related topics. A rich source of inductive biases in GQML arises from the analysis of symmetries in datasets. Symmetries are mathematically captured by group theory and represen- tation theory, and so GQML researchers will need to be armed with their fundamentals to attack symmetric tasks. Group theory studies abstract algebraic objects called groups: sets equipped with a binary operation for combin- ing elements satisfying certain properties. Representation theory, on the other hand, studies groups (and other al- gebraic structures) by representing them as collections of linear transformations acting upon a vector space. The two are intimately related, and a common motif is: groups en- code abstract symmetries, and representations describe the concrete actions of these symmetries . Their joint impact within physics has been deep and pervasive. Noether’s the- orem rests fundamentally upon continuous symmetries to extract conserved physical quantities [33]. The analysis of central potentials, notably the hydrogen atom, uses sym-metries to perform a block-diagonalization which turns an intimidating 3D problem into a morally 1D problem [34]. The development of quantum mechanics as a ﬁeld was pro- foundly shaped by groups and their representations, and they continue to be central tools across quantum infor- mation theory [35–40], the study of phase transitions and critical phenomena in condensed matter systems [41–43], quantum ﬁeld theory [44–46], and even the work towards understanding the exotic behavior of black holes [47–50]. In the program of GQML, group and

Question 1165 multiple-choice

Quantum optimization problems often utilize hierarchies of polynomial relaxations to bound quantities like the quantum max cut, with techniques involving pseudoexpectations, ideals, and swap matrices. Algebraic tools such as Gröbner bases and the structure of the generated ideals are crucial in determining the strength and exactness of these relaxations. In the context of polynomial hierarchies for quantum max cut, which of the following statements is true regarding the swap relaxation at level ⌈n/2⌉ for polynomials of degree ≤ 2d? 1) The swap relaxation at level ⌈n/2⌉ always underestimates the quantum max cut value for all graphs. 2) The swap relaxation at level ⌈n/2⌉ exactly solves the quantum max cut problem, closing the gap at this finite level. 3) At level ⌈n/2⌉, the relaxation matches only the minimum eigenvalue and not the maximum. 4) The swap relaxation at level ⌈n/2⌉ depends on the choice of polynomial generators and is not canonical. 5) Numerical evidence shows that the swap relaxation at level ⌈n/2⌉ fails to match the quantum max cut for graphs with more than 8 vertices. 6) The swap relaxation at level ⌈n/2⌉ only provides upper bounds, never tight values. 7) At level ⌈n/2⌉, the swap relaxation cannot be implemented using Gröbner basis methods.

✓ Correct Answer:

The correct answer is 2) The swap relaxation at level ⌈n/2⌉ exactly solves the quantum max cut problem, closing the gap at this finite level..

📚 Reference Text:

for any degree≤2dpolynomial in the two sided ideal generated by {ηj}, so, ˜E[p] = 0 if deg( p)≤2dand∃monomials{βij},{γij}s.t.p=/summationdisplay i,jβijηjγij. (E.6) Contrastingly, the pseudoexpectation for the d-th level hierarchy constructed in [ TRZ+] only enforces the constraints in the truncated ideal generated by{ηj}, meaning ˜E[p] = 0 if deg( p)≤2dand∃monomials{βij},{γij}s.t.p=/summationdisplay ijβijηjγij Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 86 with deg(βijηjγij)≤2dfor alli,j. (E.7) In the case of swaps the polynomials pdefined in Equation (E.6) above could also be defined to be those pof degree≤2dwhich annihilate all of the swap matrices, namely the set of polynomials pin the variables sijsuch that p({Swapij}) = 0 (E.8) where Swapijis the matrix obtained by replacing each sijvariable with the corresponding Swap matrix. This follows from Proposition 3.5 in our paper or [ TRZ+, Theorem 3.8]. The ideal corresponding to the swaps can be defined by different sets of constraint polynomials. Equation (E.8) implies the value obtained by our dth relaxation depends only on the ideal and does not depend on the choice of defining constraints {ηj}. On the other hand, this property may not be shared by d-th relaxed value of [TRZ+]. Further discussion of this difference is given in Remark 4.3 in this paper, where the bound obtained at level dof our hierarchy is denoted νd(h) and the bound obtained by the hierarchy in [ TRZ+] is denoted νd. While both hierarchies converge after finitely many steps, this difference means that at each level the hierarchy presented in this paper gives an upper bound that may be tighter than the one presented in [TRZ+]. We found numerically (up to SDP accuracy in Mathematica, 10−7) for graphs with up to 8 vertices and uniform edge weights the value of our second swap relaxation equals the numerically exact max/min eigenvalue of the QMC Hamiltonian. In contrast, [ TRZ+] do an impressive numerical study on a sampling of these same graphs and find the performance of their first relaxation in swaps is not numerically exact (to the same tolerance) on numerous graphs with 6, 7 and 8 vertices, for details see [TRZ+, Figure 4 of Section 5]. As mentioned in Section 1.2, computing all constraints of the form given in Equa- tion (E.6) at higher levels drequires considerable technical work. This is the content of the later part of Section 4 of our paper (which gives an explicit linear algebraic basis for the degree dmonomials in the Symbolic Swap Algebra when d≤4) and of Section 5 (which applies Gr¨ obner Basis to swap polynomials). These techniques for simplifying swap polynomials are not present in [ TRZ+]. One consequence of our swap algebra theory, see Theorem 4.10, is that the level ⌈n 2⌉swap relaxation in our hierarchy solves the quantum max cut problem exactly. The corresponding result for the hierarchy present in [ TRZ+] is stated as an open question after Proposition 3.13 in [ TRZ+], which gives the bound/parenleftbign 2/parenrightbig. The present paper and [ TRZ+] both theoretically analyze specific instances where the

Question 1166 multiple-choice

Perverse sheaves play a central role in modern algebraic geometry and representation theory, providing a link between geometric, topological, and algebraic methods. Their applications span the study of singularities, group actions, and advanced algebraic structures such as quantum groups and Lie algebras. Which mathematical object is critical for computing intersection cohomology on singular spaces, encoding stratified geometric data, and facilitating connections to representation theory through varieties like Grassmannians, Schubert varieties, and flag manifolds? 1) Symplectic forms 2) Differential graded algebras 3) Topological K-theory classes 4) Homotopy groups 5) Vector bundles 6) Chow rings 7) Perverse sheaves

✓ Correct Answer:

The correct answer is 7) Perverse sheaves.

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Title: Perverse Sheaves and Applications to Representation Theory Year: 2021 Paper ID: 112bff363b7a3260bcad311cafdc31a7e5ff3859 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/112bff363b7a3260bcad311cafdc31a7e5ff3859 Abstract: | Linear algebraic Several complex variables and analytic spaces – Complex singularities – Stratiﬁcations; constructible sheaves; intersection cohomology (complex-analytic aspects). | Group theory and generalizations – Linear algebraic groups and related topics – Representation theory for linear algebraic groups. | Algebraic geometry – Special varieties – Grassmannians, Schubert vari- eties, ﬂag manifolds. | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Coadjoint orbits; nilpotent varieties. | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Quantum groups (quantized enveloping algebras) and related deformations.

Question 1167 multiple-choice

Modular tensor categories (MTCs) are algebraic structures at the intersection of mathematics and physics, playing a key role in topological quantum computing due to their fusion and braiding properties. Their construction often involves advanced representation theory and deep connections to quantum groups and topology. Which property of modular tensor categories is required for physical implementations in topological quantum computing to ensure well-defined quantum evolution and probability conservation? 1) Symmetry under tangle transformations 2) Invariance under link polynomials 3) Unitarity 4) Complete reducibility of representations 5) Existence of a modular S-matrix 6) Fusion rules derived from finite group algebras 7) Compatibility with three-manifold invariants

✓ Correct Answer:

The correct answer is 3) Unitarity.

📚 Reference Text:

of Hopf algebra constructions. A ge ometric construction using link invariants and tangle categories was introduced in [T2], advanced by Turaev and Wenzl in [ TW2 ] and somewhat simpliﬁed by Blanchet and Beliakova in [BB]. However, all examples that have been carried out lead to MT Cs also obtainable from quantum groups. Yet another construction o f MTCs from repre- sentations of vertex operator algebras has recently appear ed in a paper by Huang [Hu]. See Subsection 3.2 for further discussion of these approa ches. Although it is expected that there are non-trivial examples of MTCs that do not arise from Hopf algebras ( e. g.quantum groups and ﬁnite group algebras), none have been rigorously produced. This is probably due to the hi ghly advanced state of the theory of representations of quantum groups at roots o f unity provided by the pioneering work of many including Lusztig ([ L]) and Andersen and his co-authors ([A], [AP] [APW ]). The description of the MTCs derived from quantum groups can be understood with little more than a ﬁrm grasp on the theo ry of representations of simple ﬁnite-dimensional Lie algebras found in Humphrey ’s book [ Hm] or any other introductory text. The purpose of this paper is two-fold: to survey what is known about the mod- ularity and unitarity of categories arising from quantum gr oups at roots of unity, and to give useful combinatorial tools for explicit computa tions in these categories. For more in-depth developments the reader is directed to two references: 1) Bakalov and Kirillov’s text ([ BK], Sections 1.3 and 3.3) contains concise constructions and examples of quantum group MTCs, and 2) Sawin’s paper ([ S2]) gives a thorough treatment of the representation theoretic details necessa ry to construct MTCs from quantum groups. The modularity results described below par tially overlap with Section 6 of [ S2]. 1.2. Motivation. There are two fairly well-known motivations for studying MTCs. They are applications to low-dimensional topology (s ee [T2]), and confor- mal ﬁeld theory (see [ Hu] and references therein). Recently, an application of unitary MTCs to quantum computi ng has been proposed by Freedman and Kitaev and advanced in the series of papers ([ FKW ], [FKLW ], [FLW] and [ FNSWW ]). Their topological model for quantum com- puting has a major advantage over the “classical” qubit mode l in that errors are corrected on the physical level and so has a higher error thre shold. For a very read- able introduction to topological quantum computing see [ FKW ]. In this model unitary MTCs play the role of the software, while the hardwar e is implemented via a quantum physical system and the interface between them is achieved by a 3-dimensional TQFT. The MTCs encode the symmetries of the co rresponding phys- ical systems (called topological states, see [ FNSWW ]), and must be unitary by physical considerations. Aside from the problem of constructing unitary MTCs, there a re several open problems currently being

Question 1168 multiple-choice

In the theory of finite groups of Lie type, understanding the interplay between involution centralizers, maximal tori, and subgroup structure is crucial for classification and analysis. Examining how abelian subgroups embed within larger groups often reveals constraints on possible configurations. Which subgroup type is valid for Cv(f) when the centralizer Ca(e, f) is calculated under the assumption that only direct product types are allowed? 1) T1E6 2) A7 3) A1D6 4) A1A1D6 5) D4A1 6) E7 7) T2A5

✓ Correct Answer:

The correct answer is 4) A1A1D6.

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nJ, = T1 and b1 inverts Ti. Moreover, C(e, a1y)0 is T1A1 or T, T;E6 (see Lemma 2.15· for the involution classes in Y), and y' acts as a graph automorphism on (C(e, a1y)0)'. Thus by Proposition 2. 7, ( C(E)")' = K is of type h D4 or C4. Clearly K centralizes T1, so T1 ~ C(K)0• Moreover En C(K)0 contains e, so E ~ C(K)°. Since b1y' inverts T1, E does not centralize C(K)0. Hence Lemma 2.l(ii) applies to give a contradiction. Step 4. The subgroup Cc,(e, f). In this step we calculate Ca(e, f) (where f is as given by Step 3). As Y is simply connected, Cv(f) is A1D6, A7 or T1E6• The last possibility does not hold, by Lemma 2.5(i); and in the second case Cy(f) = SJ,,8/Z2, forcing/=e, which is false. Hence Cc(e,f)=J 1Cy(f)=A 1A1D6• We may take Cc;(e, f) = 1118R where R = D6• Then (e, !) = Z(Cc;(e, f)) = (e1, e8), and z EE. Also lif8 n R = (e1> e8). 38 A. M. COHEN, M. W. LIEBECK, J. SAXL AND G. M. SEITZ Step 5. We have En R = (e1> e8). Suppose that the claim is false, and pick g e (En R)\(e 1, e8). Then CR(g) =A1A1D4• We may take g = e1, CR(g) = J2/3S2, so that C0(e1, e8, g) = 111213lsS2 (recall that S2 = D4). First assume that there also exists he En (S2\Z(S2)). We may take Cs2(h) = J4J5J6J7, so that E::;:;; F and Z = Z(F).;:; E. For 1.;:; i.;:; 8 choose a;, b; El; of order 4 with [a;, b;] = e;. For a 4-set A= {a, /3, y, c5} E .6., write aA = a"'apayat,, bA = b()(bpbybe,. Define the subgroups U, YA (A e .6.) of F as follows: V=(Z,a 8,b1 ••• b8\ al!Be.6.), YA= (Z, aA, bA, ll,4, bA), where A denotes the complement of A in {l, ... , 8}. Thus U = (Z2)9 and YA= (Z2)8• Moreover, if V0 = (Z, as j BE Li}, the coset Vob1 ••• b8 in U consists entirely of conjugates of z. We claim now that for some A e .6. and choice of a;, b;, the group E is contained in either U or YA. To see this, note that if E contains as and ac with B, C e Li and IB n Cl= 2, then E can contain no element bA (as Eis abelian), and hence E::;:;; U. Otherwise, the set {a8 I a8 EE} is contained in {a A, a.4} for some A e 6., and then clearly E ~YA, proving the claim. If E::;:;; U then (e0 n E) ~ V0, and hence E ~ U0 by Lemma 2.4; but clearly V0 lies in a maximal torus of F, so this contradicts our hypothesis on E. Hence E.;:; YA for some A e 6. and choice of a;, b;. Now E centralizes no l; by Lemma 2.3, and hence we may assume that aAaA =

Question 1169 multiple-choice

The unit group of an integral group ring ℤ[G] plays a central role in algebra, particularly in understanding invertible elements and their structure for non-abelian groups. Semi-direct products and torsion-free normal complements offer powerful tools in describing these unit groups. In the group ring ℤ(S₃ × C₃), which statement accurately describes a key structural feature of its unit group? 1) The unit group decomposes into a direct product of abelian groups only. 2) The unit group contains a subgroup structured as a semi-direct product involving free groups, such as F₅₅ ⋊ F₃ ⋊ (S₃* × C₂). 3) The unit group does not contain any torsion-free subgroup. 4) Only cyclic groups appear in the decomposition of the unit group. 5) The unit group is always finite for any group ring ℤ[G]. 6) The subgroup structure excludes any free group components. 7) The unit group is simple and cannot be decomposed further.

✓ Correct Answer:

The correct answer is 2) The unit group contains a subgroup structured as a semi-direct product involving free groups, such as F₅₅ ⋊ F₃ ⋊ (S₃* × C₂)..

📚 Reference Text:

Title: Unit group of integral group ring ℤ(G × CPresenting an explicit descryption of unit group in the integral group ring of a given non-abelian group is a classical and open problem. Let S3 be a symmetric group of order 6 and C3 be a cyclic group of order 3. In this study, we firstly explore the commensurability in unit group of integral group ring ℤ(S3 × C3) by showing the existence of a subgroup as (F55 ⋊ F3) ⋊ (S3∗× C2) where Fρ denotes a free group of rank ρ. Later, we introduce an explicit structure of the unit group in ℤ(S3 × C3) in terms of semi-direct product of torsion-free normal complement of S3 and the group of units in RS3 where R = ℤ[ω] is the complex integral domain since ω is the primitive 3rd root of unity. At the end, we give a general method that determines the structure of the unit group of ℤ(G × C3) for an arbitrary group G depends on torsion-free normal complement V (G) of G in U(ℤ(G × C3)) in an implicit form. As a consequence, a conjecture which is found in [21] is solved.

Question 1170 multiple-choice

Quantum groups are algebraic structures that generalize classical symmetry groups, playing a significant role in contemporary theoretical physics and mathematics. Their construction often involves advanced concepts such as R-matrices, Hopf algebras, and braid group representations. Which algebraic structure is used to formalize both non-commutative and non-cocommutative properties in the study of quantum groups, enabling the definition of operations like multiplication, comultiplication, and antipode? 1) Group algebra 2) Lie algebra 3) Matrix algebra 4) Hopf algebra 5) Weyl algebra 6) Clifford algebra 7) Symplectic algebra

✓ Correct Answer:

The correct answer is 4) Hopf algebra.

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Title: Quantum groups and Yang-Baxter equations Year: 2025 Paper ID: d5880de78e7ba8b14dbda56ee51d1a3217af7335 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d5880de78e7ba8b14dbda56ee51d1a3217af7335 Abstract: This introductory review is devoted to the newest section of the theory of symmetries - the theory of quantum groups. The principles of the theory of quantum groups are reviewed from the point of view of the possibility of their use for deformations of symmetries in physics models. The R-matrix approach to the theory of quantum groups is discussed in detail and is taken as the basis of the quantization of classical Lie groups, as well as some Lie supergroups. We start by laying out the foundations of non-commutative and non-cocommutative Hopf algebras. Much attention has been paid to Hecke and Birman-Murakami-Wenzl (BMW) R-matrices and related quantum matrix algebras. Noncommutative differential geometry on quantum groups of special types is discussed. Trigonometric solutions of the Yang-Baxter equations associated with the quantum groups GLq(N), SOq(N), Spq(2n) and supergroups GLq(N|M), Ospq(N|2m), as well as their rational (Yangian) limits, are presented. Rational R-matrices for exceptional Lie algebras and elliptic solutions of the Yang-Baxter equation are also considered. The basic concepts of the group algebra of the braid group and its finite dimensional quotients (such as Hecke and BMW algebras) are outlined. A sketch of the representation theories of the Hecke and BMW algebras is given, including methods for finding idempotents (quantum Young projectors) and their quantum dimensions. Applications of the theory of quantum groups and Yang-Baxter equations in various areas of theoretical physics are briefly discussed.

Question 1171 multiple-choice

Nuclear Magnetic Resonance (NMR) quantum computing utilizes radio frequency pulses to manipulate nuclear spins, enabling experimental implementation of quantum algorithms. The Quantum Fourier Transform (QFT) is a central algorithm in quantum information science, often demonstrated in NMR systems using molecules with multiple spin-½ nuclei. In an NMR quantum computing experiment using a molecule with three spin-½ nuclei, which technique enables entangling operations necessary for the Quantum Fourier Transform, and what type of interaction does it exploit? 1) Applying magnetic field gradients to each nucleus independently 2) Using J-coupling between pairs of nuclear spins to implement entangling gates 3) Modulating the chemical environment to induce collective spin resonance 4) Employing optical pumping to synchronize spin states across nuclei 5) Utilizing microwave irradiation to target electron spin transitions 6) Directly measuring spin projections without intermediate rotations 7) Applying static magnetic fields to shift resonance frequencies only

✓ Correct Answer:

The correct answer is 2) Using J-coupling between pairs of nuclear spins to implement entangling gates.

📚 Reference Text:

to all Lbits asjis indexed from L−1 to 0 will complete the QFT. This sequence of quantum logic gates can be realized NMR. The idea of using nuclear spins as the basic unit of a quantum computer was proposed by Lloyd [18], and detailed schemes for using NMR as a method of quantum computing were proposed by Cory et al[19] and Gershen- feld and Chuang [20]. In NMR a series of ra- dio frequency pulses are used to control the ex- cess magnetization of an ensemble of quantum states. NMR experiments are easily visualized by picturing the excess magnetization as a vec- tor pointing in some direction and the pulses as rotations about the various axes. In addition, a bilinear coupling term in the Hamiltonian allows for quantum superposition. The Hamiltonian of a three spin (qubit) NMR sample with J-coupling is H=ω1Iz 1+ω2Iz 2+ω3Iz 3+ 2π(J1,2Iz 1Iz 2+J1,3Iz 1Iz 3+J2,3Iz 2Iz 3)(8) where Ii=σi/2. The three bit QFT was im- plemented via NMR using the three carbon-13 spins of an alanine sample. The resonant fre- quency of carbon-13 at 9.4 Tesla is approxi- mately 100.617MHz. The carbonyl was labeled spin 1, Cαwas labeled spin 2, and Cβspin 3. The chemical shift of the three alanine carbons are 12587Hz, 0Hz, and -3435Hz respectively. Cou- pling constants between the three spins are J12 = 54Hz, J23= 35Hz, and J13= 1.2Hz. Relax- ation time T1for alanine is approximately 1.56s while T2is about 420ms. TheAjmatrix described above can be broken up into idempotents E+−E−+σx(E++E−). 2 The pulse sequence of the Ajgate can now be de- termined using the geometric algebra formalism [21], Aj=/parenleftbiggπ 2/parenrightbiggj y−(π)j x. (9) This pulse program reads: apply a pulse along they-axis that rotates spin j90 degrees, apply a pulse along the x-axis that rotates j180 degrees. Magnetization on the z-axis would be rotated to the positive x-axis. Since this experiment starts with the spins at thermal equilibrium (pointing along the z-axis) the above sequence for the Aj gate can be replaced by the simplerπ 2pulse along the positive y-axis. TheBjkgate, which can be constructed using the coupling between qubits, In terms of idempo- tents reduces to 1 −E−1E−2+eiθE−1E−2. Again using geometric algebra this yields the following pulse sequence: Bjk=(π)j φ−/parenleftig θ 2πJjk/parenrightig −(π)j φ /parenleftbigπ 2/parenrightbigj,k y−/parenleftig θ 2/parenrightigj,k x−/parenleftbigπ 2/parenrightbigj,k −y.(10) The notation θ/2πJjkrepresents an interval of spin evolution under the coupling Hamiltonian. The ﬁnal three pulses eﬀectively perform a ro- tation around the z-axis. These pulses are not necessary, however, since the same eﬀect may be achieved by rotating the prior pulses of the ex- periment. The complete pulse program is the combina- tion of AjandBjkgates described above. In this implementation, the necessity of performing a swap gate has been removed by reordering the bits at the appropriate interval. The complete pulse program is,QFT 3=/parenleftbigπ 2/parenrightbig1 −sin(3π 8)x+cos(3π 8)y−(π)2 x /parenleftig 1 8J12/parenrightig −(π)3 x−/parenleftig 1 8J12/parenrightig −(π)2 −x /parenleftbigπ 2/parenrightbig2 −x+y√ 2−/parenleftig 1 16J13/parenrightig −(π)2 x− /parenleftig 1 16J13/parenrightig −(π)2 −x−/parenleftig 1 8J23/parenrightig −(π)1 x− /parenleftig

Question 1172 multiple-choice

In group theory, automorphism orbits describe how the elements of a finite group are grouped under the action of its automorphism group. The number of automorphism orbits is an important invariant that reveals the level of symmetry and structure within the group. Which statement is true regarding finite non-solvable groups and their automorphism orbits? 1) There are no finite non-solvable groups with exactly six automorphism orbits. 2) All finite groups with at most seven automorphism orbits are solvable. 3) The presence of a non-trivial abelian normal subgroup always increases the number of automorphism orbits beyond seven. 4) Every finite simple group has fewer than six automorphism orbits. 5) For any fixed number n, there are infinitely many finite groups with at most n automorphism orbits, regardless of subgroup structure. 6) Finite non-solvable groups with seven automorphism orbits necessarily have abelian normal subgroups. 7) There exist infinitely many finite non-solvable groups with exactly seven automorphism orbits.

✓ Correct Answer:

The correct answer is 7) There exist infinitely many finite non-solvable groups with exactly seven automorphism orbits..

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Title: Finite groups with six or seven automorphism orbits Year: 2015 Paper ID: 51c1badc7cb55a2271cadbad412fba881fd222ad Source: semantic-scholar URL: https://www.semanticscholar.org/paper/51c1badc7cb55a2271cadbad412fba881fd222ad Abstract: Abstract Let G be a group. The orbits of the natural action of Aut ⁡ ( G ) {\operatorname{Aut}(G)} on G are called “automorphism orbits” of G, and the number of automorphism orbits of G is denoted by ω ⁢ ( G ) {\omega(G)} . In this paper the finite non-solvable groups G with ω ⁢ ( G ) ≤ 6 {\omega(G)\leq 6} are classified – this solves a problem posed by Markus Stroppel – and it is proved that there are infinitely many finite non-solvable groups G with ω ⁢ ( G ) = 7 {\omega(G)=7} . Moreover, it is proved that for a given number n there are only finitely many finite groups G without non-trivial abelian normal subgroups and such that ω ⁢ ( G ) ≤ n {\omega(G)\leq n} , generalizing a result of Kohl.

Question 1173 multiple-choice

Braided tensor categories are fundamental structures in modern algebra and mathematical physics, providing frameworks for studying symmetries, quantum groups, and topological phenomena. The classification and construction of new families of these categories often involve intricate algebraic tools and conditions. Which technical condition on fusion ring morphisms guarantees that certain braided tensor categories can be realized from the semisimplified representation category of a quantum group, enabling the construction of new series of twisted categories not originating from known Hopf algebras? 1) Ribbon equivalence 2) Monoidal closure 3) Local isomorphie 4) Dualizability 5) Modular invariance 6) Tannakian reconstruction 7) Central extension

✓ Correct Answer:

The correct answer is 3) Local isomorphie.

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Title: On braided tensor categories Year: 1994 Paper ID: 52a782c11d417caad6f6b32c4b59ec4495c89502 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/52a782c11d417caad6f6b32c4b59ec4495c89502 Abstract: We investigate invertible elements and gradings in braided tensor categories. This leads us to the definition of theta-, product-, subgrading and orbitcategories in order to construct new families of BTC's from given ones. We use the representation theory of Hecke algebras in order to relate the fusionring of a BTC generated by an object $X$ with a two component decomposition of its tensorsquare to the fusionring of quantum groups of type $A$ at roots of unity. We find the condition of `local isomorphie' on a special fusionring morphism implying that a BTC is obtained from the above constructions applied to the semisimplified representation category of a quantum group. This family of BTC's contains new series of twisted categories that do not stem from known Hopf algebras. Using the language of incidence graphs and the balancing structure on a BTC we also find strong constraints on the fusionring morphism. For Temperley Lieb type categories these are sufficient to show local isomorphie. Thus we obtain a classification for the subclass of Temperley Lieb type categories.

Question 1174 multiple-choice

Topological quantum field theories (TQFTs) utilize algebraic structures such as quantum groups and Hopf algebras to model the symmetries of surfaces and their invariants. In non-semisimple settings, quantum doubles and bicrossed product structures play a crucial role in addressing challenges related to degeneracy and modularity. Which of the following statements accurately describes the decomposition of the projective SL(2,Z) action on the center of Uq(sl2) when the deformation parameter q is a 2m+1-th root of unity? 1) It splits into two infinite-dimensional irreducible representations. 2) It is entirely reducible and consists only of one-dimensional representations. 3) It does not admit any tensor product decomposition. 4) It decomposes into a direct sum of infinite-dimensional representations. 5) It decomposes into a finite-dimensional part and an irreducible tensor product of a two-dimensional SL(2,Z) representation with a finite m-dimensional representation from truncated TQFT of the semisimplified category. 6) It is equivalent to a trivial representation with no modular relations. 7) It consists solely of projective phases without any representation-theoretic decomposition.

✓ Correct Answer:

The correct answer is 5) It decomposes into a finite-dimensional part and an irreducible tensor product of a two-dimensional SL(2,Z) representation with a finite m-dimensional representation from truncated TQFT of the semisimplified category..

📚 Reference Text:

arXiv:hep-th/9402017v1 3 Feb 1994Mapping Class Group Actions on Quantum Doubles December 1993 Thomas Kerler Department of Mathematics Harvard University Cambridge, MA, USA kerler@math.harvard.edu Abstract: We studyrepresentations of themappingclass gro upof thepuncturedtoruson the doubleofaﬁnitedimensionalpossiblynon-semisimpleHopf algebrathatariseintheconstruction of universal, extended topological ﬁeld theories. We discu ss how for doubles the degeneracy problem of TQFT’s is circumvented. We ﬁnd compact formulae f or theS±1-matrices using the canonical, non degenerate forms of Hopf algebras and the bic rossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular rel ations and the computation of the projective phases is supplied using Radford’s relati ons between the canonical forms and the moduli of integrals. We analyze the projective SL(2,Z)-action on the center of Uq(sl2) forqanl= 2m+ 1-st root of unity. It appears that the 3 m+ 1-dimensional representation decomposes into an m+1-dimensional ﬁnite representation and a 2 m-dimensional, irreducible representation. The latter is the tensor product of the two d imensional, standard representation ofSL(2,Z) and the ﬁnite, m-dimensional representation, obtained from the truncated TQFT of the semisimpliﬁed representation category of Uq(sl2). 1.Introduction Since the seminal paper of Atiyah [A] on the abstract deﬁnition of a t opological quantum ﬁeld theory (TQFT) much progress has been made in ﬁnding non trivia l examples and extended structures. The most interesting developments took p lace in three dimensions where actual models of quantum ﬁeld theory, like rational conform al ﬁeld theories and Chern-Simons theory led to the discovery of new invariants. See [Cr ] and [Wi]. In an attempt to counterpart these heuristic theories by mathem atically rigorous con- structions the ﬁeld theoretical machinery had been replaced by qu asitriangular Hopf al- gebras, or quantum groups. The resulting invariants are describe d in [TV] and [RT]. From here it is not hard to understand how to associate a TQFT to a r igid, abelian, monoidal category and an extended TQFT to a braided tensor cate gory (BTC). In order for these theories to be well deﬁned one has to make a few more ass umptions. One is that the category shall contain only a ﬁnite number of inequivalent, simple objects, i.e., it is rational. The other is a technical non degeneracy condition, calle d “modularity” in [T], which is to assure that elementary cobordisms are associated to identiﬁcations rather than projections. Alternatively, if the modularity condition fails to h old, it is standard in the Atiyah [A] description to deﬁne a truncated TQFT by reducing th e vectorspaces to the images of the projections. All of the mentioned TQFT’s aresemisimple, i.e., they rely onthe decomp ositions into simple objects. Clearly, semisimplicity cannot be an assumption of fun damental but only of techncial nature. In seeking universal constructions of TQFT’s, which do not refer to decompositions, one should thus not only generalize the existing one s to non semisimple theories but also gain a deeper understanding of the structure un derlying them. A partial answer for the genus one case of how a universal TQFT sh ould look

Question 1175 multiple-choice

Hopf algebras are important structures in algebra that generalize group symmetries and play a key role in quantum group theory. Their properties, such as selfduality and quasitriangularity, depend on parameters like field characteristic and specific algebraic relations. Which of the following conditions is necessary for the Hopf algebra kZn(q)/Id to be quasitriangular? 1) n is odd and q is a primitive nth root of unity 2) The characteristic of the field divides n 3) n is even and q = -1 4) n is prime and q^n = 1 5) q is a nontrivial square root of unity for any n 6) The algebra is commutative and cocommutative 7) n is even and q = 1

✓ Correct Answer:

The correct answer is 3) n is even and q = -1.

📚 Reference Text:

simple presentation, the module theory cannot be recovered from the representation theory of the tensorands; in fact the latter are of finite representation type while their tensor product is of wild representation type. 464 C. Cibils Proposition 3.8. Let n be a positive integer and let k be afield of characteristic not dividing n. Ifq is a primitive nth root of unity, the Hopf algebras kZn(q) and kZn(q)/Id are selfdual. Proof. The dual Hopf algebra has the following structure with respect to the dual basis {7"}: j + sj+u (x) yjf , u + υ The elements σt = -ΣieZ/nzffii and αΐ = <( ΣteZ/nZ#(ί + 1)l#/ can be considered as the Fourier transforms of the given ones. They satisfy the multiplication formulas of kZn(q) and the map which assigns σf to st and α/ to at is a Hopf algebra isomorphism. The same map gives also an isomorphism at the level of the finite dimensional quotients. Remark 3.9. If the characteristic of k divides n the Hopf algebra kZn(q)/Id is not selfdual. Indeed, y ? is an element of order exactly n in the dual algebra while the original one has no such elements. For q a primitive nth root of unity Taft has considered in [28] the Hopf algebra generated by X and Y subjected to the relations Xn = 1, Yn = 0 and YX = qXY. The comultiplication is given by Δ(X) = X®X and Δ(Y)= Y®X + 1®X. When q is a primitive nih root of unity, this algebra is isomorphic to (kZn(q)/In)* by sending X to y? = Si and 7 to yj = αo In case n = 2 and g = — 1 the Hopf algebra is isomorphic to Sweedler's 4-dimensional neither commutative nor cocom- mutative example constructed in [27] and quoted in [8]. Recently these algebras have been considered by several authors. It has been shown that their Drinfeld double (see [8]) for n odd has a quotient isomorphic to a finite dimensional quotient of Uq(sl2)^ introduced by Reshetikhin and Turaev in [25]. Simple L^(s/2)-modules have been described in [26]; in the next section we consider the finitely generated indecomposable Λπ(g)-modules and study their tensor product. In [23] D. Radford has shown that Sweedler's 4-dimensional Hopf algebra is quasitriangular (see [8] for the definition). In fact there is a narrow subfamily of the Hopf algebras kZn(q)/Id sharing this property, as the next result shows. Proposition 3.10. The Hopf algebra kZn(q)/Id is quasitriangular if and only ifn is even and q = — 1. In all other cases the Hopf algebra is not even quasίcocommutative. Proof. Let R = £#;/?? ® yvj be an invertible element such that Δ'R = RA, where A' stands for the opposite comultiplication. This quasicommutativity condition expressed on the vertices of the quiver translates into the fact that the coefficient of β ® α in R has to be zero if the product of the sources of the paths β and α differs from the product of

Question 1176 multiple-choice

Superconducting quantum circuits utilize precise control of qubit parameters and engineered interactions to optimize two-qubit and multi-qubit gate operations. The minimum achievable time for executing a quantum gate, known as the quantum speed limit (QSL), is a critical benchmark for advancing quantum computation. In superconducting circuits, which control field configuration most reliably achieves both the quantum speed limit and best convergence for the CZ gate, and what is the corresponding QSL value? 1) "No-X" configuration; QSL of 12 ns 2) "Interaction" configuration; QSL of 10 ns 3) "Full" configuration; QSL of 14 ns 4) "No-X" configuration; QSL of 10 ns 5) "Interaction" configuration; QSL of 14 ns 6) "Full" configuration; QSL of 12 ns 7) "No-X" configuration; QSL of 14 ns

✓ Correct Answer:

The correct answer is 2) "Interaction" configuration; QSL of 10 ns.

📚 Reference Text:

/lessorsimilarω n(t)/lessorsimilar7100 MHz and their anharmonicities given by αn≈200 MHz, with exact values depending on n. The tunable coupling strength is given by−40 MHz /lessorequalslantgnm(t)/lessorequalslant5 MHz, as reported in Ref. [ 8]. Moreover, we assume the (auxiliary) X-type control ﬁelds ¯/Omega1n,re(t) and ¯/Omega1n,im(t), which encode the physical X-type con- trol ﬁelds /Omega1n(t) and their tunable driving frequencies ¯ ωn(t), to satisfy −50 MHz /lessorequalslant¯/Omega1n,re(t),¯/Omega1n,im(t)/lessorequalslant50 MHz. 023026-11 BASILEWITSCH, DLASKA, AND LECHNER PHYSICAL REVIEW RESEARCH 6, 023026 (2024) FIG. 5. Overview of different QSLs similar to Fig. 3but exclusively for the three- and four-qubit constraint gates ZZZ( γ) and ZZZZ( γ), cf. Eq. ( A6). The left column shows results for neutral atoms, obtained using the “phase” conﬁguration of Fig. 3, for the pseudo-2D architecture (triangles) and an actual, planar 2D architecture (stars). The right column compares the results for superconducting circuits using the “no-X” conﬁguration from Fig. 3. The data correspond to the physical architecture of Ref. [ 8] where only nearest-neighbor (NN) couplings between qubits are present (triangles) and where diagonal couplings, i.e., next-to-nearest-neighbor (NNN) couplings, are added (stars). 1. Two-qubit gates Figure 3(f)shows results for a CZ gate on superconducting circuits using three different conﬁgurations of control ﬁelds—different, of course, from those used for neutral atoms. Inthe “full” conﬁguration (circles), all available control ﬁelds,ω n(t),gnm(t),¯/Omega1n,re(t), and ¯/Omega1n,im(t), are time dependent and being optimized. In the “no-X” conﬁguration (crosses), onlyω n(t) and gnm(t) are optimized while the X-type control ﬁelds are set to zero, ¯/Omega1n,re(t)=¯/Omega1n,im(t)=0. At last, in the “in- teraction” conﬁguration only ωn(t) is time dependent and optimized while gnm(t)=− 40 MHz =gmaxis set to its maxi- mum magnitude and ¯/Omega1n,re(t)=¯/Omega1n,im(t)=0. For the CZ gate in Fig. 3(f), we observe all three conﬁgurations to indicate the same QSL of TCZ QSL=10 ns. In terms of convergence behavior, the “interaction” conﬁguration shows the best performanceas indicated by the probability density on the right side ofFig. 3(f). In general, the three conﬁgurations show slightly worse convergence behavior than the three conﬁgurations forthe neutral atoms, cf. Fig. 3(a). Nevertheless, since all three conﬁgurations indicate the same QSL and, in general, yieldthe same achievable error ε Tdepending on T, we believe our method to determine the QSL to yield reliable results also forsuperconducting circuits and conjecture T CZ QSL=10 ns to be the fundamental QSL for CZ gates. While there are referenceimplementations for CZ gates on similar architectures withtunable couplers [ 71,73,99,100], none of these architectures matches our architecture and parameter regime. Hence, wecompare our QSL to the gate time of the fastest two-qubit gate,the Sycamore gate, reported in Ref. [ 8]. We ﬁnd T CZ QSL=10 ns to be slightly faster than TSyc. lit=12 ns [ 37]. In Fig. 3(g), the results for a CNOT gate are shown using the same three conﬁguration as for the CZ gate. We ﬁnd onlythe “full” conﬁguration to be capable of realizing a CNOTgate, yielding the QSL T CNOT QSL=14 ns, while the other two conﬁguration are not capable of it. Among the available con-trol knobs,

Question 1177 multiple-choice

In quantum computing, exchange-only architectures utilize only swap operations between particles to implement computation. Representation theory and combinatorial methods play a key role in determining when such architectures achieve universal quantum computation with qudits (systems with d states, d > 2). Which combinatorial object provides necessary and sufficient conditions for exchange-only universality in systems with multiple qudits by determining how irreducible representations combine? 1) Young tableaux 2) Gelfand–Tsetlin patterns 3) Symmetric group character tables 4) Schur functions 5) Littlewood–Richardson coefficients 6) Weyl group elements 7) Clebsch–Gordan coefficients

✓ Correct Answer:

The correct answer is 5) Littlewood–Richardson coefficients.

📚 Reference Text:

Title: Universality of swap for qudits: a representation theory approach Year: 2021 Paper ID: 4c8ca3ef19560a2e1b58fd4d3fa09d4bf9377182 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/4c8ca3ef19560a2e1b58fd4d3fa09d4bf9377182 Abstract: An open problem of quantum information theory has been to determine under what conditions universal exchange-only computation is possible for qudits encoded on $d$-state systems for $d>2$. This problem can be posed in terms of representation theory by recognizing that each quantum mechanical swap, generated by exchange-interaction, can be identified with a transposition in a symmetric group, each $d$-state system can be identified with the fundamental representation of $SU(d)$, and each encoded qudit can be identified with an irreducible representation of a Lie algebra generated by transpositions. Towards this end we first give a mathematical definition of exchange-only universality in terms of a map from the special unitary algebra on the product of qudits into a representation of a Lie algebra generated by transpositions. We show that this definition is consistent with quantum computing requirements. We then proceed with the task of characterizing universal families of qudits, that is families of encoded qudits admitting exchange-only universality. This endeavor is aided by the fact that the irreducible representations corresponding to qudits are canonically labeled by partitions. In particular we derive necessary and sufficient conditions for universality on one or two such qudits, in terms of simple arithmetic conditions on the associated partitions. We also derive necessary and sufficient conditions for universality on arbitrarily many such qudits, in terms of Littlewood--Richardson coefficients. Among other results, we prove that universal families of multiple qudits are upward closed, that universality is guaranteed for sufficiently many qudits, and that any family that is not universal can be made so by simply adding at most five ancillae. We also obtain results for 2-state systems as a special case.

Question 1178 multiple-choice

Quantum algorithms often exploit group-theoretic structures to solve complex computational problems. The Hidden Symmetry Subgroup Problem (HSSP) extends these techniques to address challenges involving symmetries of partitions induced by oracles. Which statement accurately describes a notable consequence of reducing the Hidden Symmetry Subgroup Problem (HSSP) to the Hidden Subgroup Problem (HSP) within quantum algorithm design? 1) It guarantees polynomial-time quantum algorithms for all subgroup problems. 2) It eliminates the need for oracles when recovering subgroup symmetries. 3) It enables efficient quantum solutions for problems involving polynomial symmetries by leveraging established HSP techniques. 4) It proves that all instances of HSSP have exponential quantum query complexity. 5) It restricts the framework to factorization problems only. 6) It removes connections between hidden polynomial problems and algebraic structures. 7) It ensures classical algorithms outperform quantum algorithms for symmetry-related problems.

✓ Correct Answer:

The correct answer is 3) It enables efficient quantum solutions for problems involving polynomial symmetries by leveraging established HSP techniques..

📚 Reference Text:

Title: Hidden Symmetry Subgroup Problems Year: 2011 Paper ID: 026ed560c487dd3d8ffb6f6d16d08b467e8d6cd5 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/026ed560c487dd3d8ffb6f6d16d08b467e8d6cd5 Abstract: We advocate a new approach for addressing hidden structure problems and finding efficient quantum algorithms. We introduce and investigate the hidden symmetry subgroup problem (HSSP), which is a generalization of the well-studied hidden subgroup problem (HSP). Given a group acting on a set and an oracle whose level sets define a partition of the set, the task is to recover the subgroup of symmetries of this partition inside the group. The HSSP provides a unifying framework that, besides the HSP, encompasses a wide range of algebraic oracle problems, including quadratic hidden polynomial problems. While the HSSP can have provably exponential quantum query complexity, we obtain efficient quantum algorithms for various interesting cases. To achieve this, we present a general method for reducing the HSSP to the HSP, which works efficiently in several cases related to symmetries of polynomials. The HSSP therefore connects in a rather surprising way certain hidden polynomial problems with the HSP. Using this co...

Question 1179 multiple-choice

Quantum algorithms have significantly impacted the analysis of algebraic problems such as inverting permutations and constructive membership in semigroups, particularly when these are accessed as black-box operations. Understanding the query complexity of these problems reveals both the potential and limits of quantum speedup in computational algebra. What is the quantum query complexity lower bound for solving the constructive semigroup membership problem in a black-box semigroup S generated by k elements, where |S| = Θ(n^k)? 1) Ω(log n) 2) Ω(n) 3) Ω(|S|^{1/2}) 4) Ω(n^{k}) 5) Ω(|S|^{1/k}) 6) Ω(|S|^{1-1/k}) 7) Ω(|S|^{1/2 - 1/2k})

✓ Correct Answer:

The correct answer is 7) Ω(|S|^{1/2 - 1/2k}).

📚 Reference Text:

black-box permutation W!(i.e., computing 1./for any ﬁxed 2given a black box for ) reduces to constructive semigroup membership in a black-box version of Swith respect to the generators g1;:::;gk. Since inverting a permutation of mpoints requires .pm/quantum queries [1],jjDnCk1 k1 D.nk1/, andjSjDnCk k D.nk/, this shows that constructive semigroup membership requires p nk1 D jSj1 21 2k queries. To construct the black-box semigroup, we specify an encoding encWS!® .a1;:::;ak/2Nk 0W1a1CCakjSj. Then, by the pi- geonhole principle, there must exist c1;:::;ck;d1;:::;dk2N0withci;diai (for alliD1;:::;k ) such thatgc1 1gck kDgd1 1gdk kand.c1;:::;ck/¤ .d1;:::;dk/. Suppose without loss of generality that .c1;:::;ck/is lexicograph- ically smaller than .d1;:::;dk/. Letbi´aiCcidifor alli, and note that aidi0. Thus ga1 1gak kDgd1 1gdk kga1d1 1gakdk k; gb1 1gbk kDgc1 1gck kga1d1 1gakdk k: This implies gb1 1gbk kDx. Also, for the ﬁrst index iwithci¤di, we have ci1 . ClearlyjD.1;L/jDL. We have.a1;:::;arC1/2D.rC1;L/ if and only if.a1;:::;ar/2D.r;bL=.arC1C1/c/. Therefore jD.rC1;L/jDLX aD1jD.r;bL=ac/jLX aD1bL=ac.3 2log2bL=ac/r1 LX aD1.L=a/.3 2log2L/r1L.3 2log2L/r; where we used the fact that for every integer L>1 ,PL aD11 a<3 2log2L. 414 A. M. Childs and G. Ivanyos We are now ready to prove the upper bound for constructive semigroup mem- bership. Proof of Theorem 5.2.By Lemma 5.3, there

Question 1180 multiple-choice

Quantum processors based on qudits, which are quantum systems with more than two levels, offer increased computational capabilities but introduce unique scalability and resource challenges, especially when implemented using photonic and silicon technologies. Efficient control and entanglement generation are vital for realizing large-scale, programmable quantum devices using these platforms. In a silicon photonic quantum processor employing qudits of dimension d, which statement accurately describes how the success probability of the entanglement-assisted quantum processing scheme changes as the qudit dimension increases, assuming a fixed number of qudits? 1) The success probability increases linearly with d. 2) The success probability remains constant for all values of d. 3) The success probability increases exponentially as d increases. 4) The success probability decreases logarithmically with d. 5) The success probability is proportional to the square of d. 6) The success probability decreases inversely with d, following a 1/d dependence. 7) The success probability is independent of both d and the number of qudits.

✓ Correct Answer:

The correct answer is 6) The success probability decreases inversely with d, following a 1/d dependence..

📚 Reference Text:

y( s e eS u p p l e m e n t a r yF i g .3 b ) . As the multi-value quantum controlled gates are the result of the entanglement in the generation stage and the gates are instead local operations that steer the state to collapse in the desired outputs, our scheme can be straightforwardly generalised to multi- qudit quantum computiation. Its scalability is naively dependent on the number ( n) of qudits and the dimensionality ( d) of eachqudit. Regarding the dimension of units, though the ququart states are implemented as an example in this work, it is straightforward to extend to a larger- ddevice36, which can be fabricated using the same CMOS fabrication techniques. Remarkably, thisentanglement-assisted d-QPU scheme works with a success probability of 1/ dregardless of n(Supplementary Note 3). The scaling of d-QPU therefore strongly relies on the generation of the qudit GHZ entangled states. Combing the state-of-the-art tech- nologies, including the techniques of generating multi-photon qudit GHZ states 45,46, on-chip high- ﬁdelity control of qudit states36, high-quality photon-pair sources75,76, low-loss ﬁbre-chip interface75,77, and large-scale quantum integration57, we estimate a 10-photons d-QPU is achievable in near term. Its further scaling requires high-ef ﬁciency heralded multiplexing photon-pairs sources78and multiplexing qudit GHZ generators31. That being said, given the ef ﬁcient generation of the multi-photon multi- qudit GHZ states, the d-QPU scheme is scalable. Calculations and analysis are provided in Supplementary Note 9 and Supplemen- tary Fig. 3d. Moreover, when scaling up the d-QPU, an interesting concern is the required resources, in particular the number of classical controls. As an example, let us consider a processor with one qudit in the auxiliary register and nqudits in the data register (see Fig. 2a). It requires a number of ( n+1) single-qudit gen- erators for state preparation, ( nd) local single-qudit operators for multi-qudit MVCU operation, and ( dn+1) single-qudit projec- tors. The physical resources, i.e, the number of phase-shifters, scale with ( d2−d) for the qudit operators50,59,a n d2 ( d−1) for the qudit generators and projectors36, as shown in Supplementary Note 9 and Supplementary Fig. 2. Importantly, the required resources for classical controls scale polynomially with the number of particles. In Supplementary Fig. 3c, it is shown that thousands of phase-shifters are required for a 10-photon d-QPU. This large amount of phase-shifters can be individually addressed and con- trolled, by using a co-integration technology of photonic and electronic circuits in silicon. The highly ﬂexible and reliable programmability of the qudit processor, that is enabled by technological advances in a mono-lithic integration of all key functionalities and capabilities in a silicon chip, has allowed the implementations of more than one million qudit generators, operators and projectors (see Supple- mentary Note 8), and also the benchmarking of different gen- eralised quantum algorithms. Such programmability can transition the advanced technologies in controlling qudit states and gates 36–48to algorithm implementations, playing an enabling role in the roadmap of qudit-based quantum computations. The full chip-scale integration technologies also perfectly match

Question 1181 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) use specialized state preparation and measurement techniques to reveal hidden parameters within group structures. Advanced methods such as the Kuperberg sieve enable efficient extraction of information that would be exponentially rare using naive sampling. Which technique is critical for efficiently combining quantum states to amplify the probability of extracting bits of a hidden parameter in non-Abelian Hidden Subgroup Problem algorithms? 1) Grover's search algorithm 2) Kuperberg sieve 3) Classical brute-force search 4) Quantum teleportation 5) Simon's algorithm 6) Shor's factoring algorithm 7) Quantum error correction

✓ Correct Answer:

The correct answer is 2) Kuperberg sieve.

📚 Reference Text:

ˆH=H=eH=h(b;1)i, and it sufﬁces to consider subgroups of the formh(b;1)i: 57 We have (b;1)2= (b;1)(b;1) = b+(1)b+2n2;2 = 2n2;0 : (4.72) Since 2n2;0 corresponds to j2=1, the QHSP reduces to ﬁnding a hiddenp1, and ulti- mately to determining b. 4.5.2 Coset Sample Generation Procedure LetjREG 1i,jREG 2idenote the top (input) register and the bottom (output) register, re- spectively. The algorithm starts with a coset sample generation procedure (CSG)that produces single-qubit states of a particular form, with bencoded in a phase factor. The routine CSG is given by the following sequence. 1. Create the uniform superposition over all elements of Q2n, i.e., the state jy0i=1p 2nå (x;y)2Q2njx;yi: (4.73) 2. Query the black-box function fto obtain the state jy1i=1p 2nå (x;y)2Q2njx;yijf(x;y)i: (4.74) 3. MeasurejREG 2ito collapse the state (4.74) to jy2i=1p 2(jx;0i+jx+b;1i): (4.75) 58 4. Apply the QFT†tojREG 1ito produce the state jy3i=1p 2n+12n1 å z=0 e2pizx 2njz;0i+e2piz(x+b) 2njz;1i =1p 2n2n1 å z=0e2pizx 2njzi 1p 2 j0i+e2pizb 2nj1i : (4.76) 5. MeasurejREG 1ito obtain the state jyzi=1p 2 j0i+e2pizb 2nj1i (4.77) for some zchosen uniformly at random from f0;1;:::;2n1g. We now exploit the form of jyzito ﬁnd the least signiﬁcant bit of b. Proposition 4.5.3. All the bits of b can be determined with n iterations of an algorithm that ﬁnds the least signiﬁcant bit of b. Proof. The group Q2ncontains two subgroups isomorphic to Q2n1, namely, G1=f(2a;0);(2a;1)ja2Z2n1g (4.78) G2=f(2a;0);(2a+1;1)ja2Z2n1g: (4.79) Ifbis even, then HG1, where His the hidden subgroup. On the other hand, if bis odd, then HG2. Once an algorithm determines the parity of b, the hiding function fis restricted to G1or G2, and the algorithm is called with a modiﬁed oracle. The algorithm outputs the least signiﬁcant bit of bwithin an appropriate subgroup, and this result corresponds to the second least signiﬁcant bit of binQ2n. Therefore, niterations of the algorithm determine all bits of b. 59 4.5.3 The Kuperberg Sieve The goal is to prepare the state jy2n1i=1p 2 j0i+e2pi(2n1)b 2nj1i! =1p 2 j0i+epibj1i =1p 2 j0i+(1)bj1i : (4.80) which is subsequently subjected to a measurement in the Hadamard basis ji=j0ij1ip 2(4.81) to extract the least signiﬁcant bit of b. The extraction works as follows. Proposition 4.5.4. The parity of b can be determined by measuring the state (4.80) in the Hadamard basis. Proof. The Hadamard gate Happlied to an arbitrary state jji=aj0i+bj1iis Hjji=aHj0i+bHj1i=aj0i+j1ip 2 +bj0ij1ip 2 =a+bp 2 j0i+abp 2 j1i: (4.82) Then application of Hadamard to the state (4.80) yields Hjy2n1i= 1+(1)b 2! j0i+ 1(1)b 2! j1i: (4.83) Ifbhas even parity, the measurement results in H1p 2(j0i+j1i) =j0i: (4.84) 60 If the parity of bis odd, the result is H1p 2(j0ij1i) =j1i: (4.85) However, the routine CSG produces the desired state (4.80) with an exponentially small probability of1 2n. The Kuperberg sieve resolves this difﬁculty by combining states of the form (4.77) to generate states of the same form but with preferred values of z. The Kuperberg sieve is essentially a sequence of m=pn1 routines such that the output of routine jis input to routine j+1 for

Question 1182 multiple-choice

Quantum computing poses significant risks to classical public-key cryptography, prompting the development of new cryptographic methods designed to resist quantum attacks. Mathematical problems such as the Hidden Subgroup Problem and Hidden Shift Problem play a crucial role in evaluating the security of these postquantum cryptosystems. Which mathematical problem's hardness most directly influences the design of lattice-based postquantum cryptographic schemes and their resistance against quantum computers? 1) The Traveling Salesman Problem 2) The Graph Isomorphism Problem 3) The Integer Factorization Problem 4) The Discrete Logarithm Problem 5) The Learning With Errors Problem 6) The Subset Sum Problem 7) The Linear Programming Problem

✓ Correct Answer:

The correct answer is 5) The Learning With Errors Problem.

📚 Reference Text:

Title: Shaping Postquantum Cryptography: The Hidden Subgroup and Shift Problems Year: 2022 Paper ID: b5f8f25fd81bddc8802357278278a46386c8fd6b Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b5f8f25fd81bddc8802357278278a46386c8fd6b Abstract: The security of popular public key-cryptographic protocols, such as RSA, Diffie–Hellman key exchange and the digital signature algorithm (DSA), is endangered by the advent of quantum computers. Shor brought a big breakthrough with his quantum algorithm that can be used to factor an arbitrarily large integer into the product of its prime factors, hence jeopardizing the security of RSA, and that at the same time also solves the Discrete Logarithm Problem, which raises issues for certain Diffie–Hellman-based cryptosystems and digital signatures. It is hence crucial to upgrade our current tools for postquantum cryptography: it should be infeasible, even using quantum algorithms, to break the new cryptosystems. Popular candidates include for example elliptic curve or lattice-based cryptography, but they share something in common: they are specific cases of the more general Hidden Subgroup and connected Hidden Shift Problem.

Question 1183 multiple-choice

In quantum computing, the hidden subgroup problem (HSP) is a central framework for designing algorithms that solve group-theoretic problems efficiently. The efficiency of such quantum algorithms is often described in terms of polynomial time relative to a logarithmic measure of group order. Which of the following most accurately describes the input size n for quantum algorithms addressing the hidden subgroup problem in a group G with subgroup K? 1) n is defined as log₂[G:K], the base-2 logarithm of the index of K in G. 2) n is defined as the order of subgroup K. 3) n is defined as the number of generators required for group G. 4) n is defined as the total number of quantum gates in the circuit. 5) n is defined as the dimension of the Hilbert space used for computation. 6) n is defined as the order of the quotient group G/K. 7) n is defined as the bit-length of the group identity element.

✓ Correct Answer:

The correct answer is 1) n is defined as log₂[G:K], the base-2 logarithm of the index of K in G..

📚 Reference Text:

lem is to ﬁnd K(that is, a generating set for K), given a way of computing f. When Kis normal in G, we could in fact decompose fash◦g, where gis a homomorphism fromGto some ﬁnite group H, and his some 1-to-1 mapping from Hto the setX. In this case, Kcorresponds to the kernel of gandHis isomorphic to G/K. We will occasionally refer to this decomposition, which we illustrate in Fig. 1. Deﬁne the input size ,n, to be of order log2[G:K]. We will count the number of operations, or the running time , in terms of n. An algorithm is considered eﬃcient if its running time is polynomial in the input size. By elementary quantum operations , we are referring to a ﬁnite set of quantum logic gates which allow us to approximate any unitary operation. S ee [BBCDMSSSW] for a discussion and further references. Our running times w ill always refer to expected running times, unless explicitly stated otherwis e. Byexpected running 2 timewe are referring to the expected number of operations for anyinput (and not just an average of the expected running times over all inp uts). We should be clear about what it means to have a ﬁnitely genera ted group G, and to be able to compute the function f. This is diﬃcult without losing some generality or being dry and technical, or both. The algo rithms we de- scribe only apply for groups Gwhich are represented as ﬁnite tuples of integers corresponding to the direct product of cyclic groups (conse quently, Gis ﬁnitely generated and Abelian). Conversely, for any ﬁnitely genera ted Abelian G, there is a temptation to point out that Gis isomorphic to such a direct product of cyclic groups, and assume that we can easily access this product structure. This is not always the case, even in cases of practical interest. F or example, Z∗ N, the multiplicative group of integers modulo Nfor some large integer N, which is Abelian of order φ(N) (the Euler φ-function) and thus isomorphic to a product of cyclic groups of prime power order. We will not necessaril y know φ(N) or have a factorisation of it along with a set of generators for Z∗ N. However, in light of the quantum algorithms described in this paper, we c ould eﬃciently ﬁnd such an isomorphism, thereby increasing the number of ﬁn itely generated Abelian groups which can be eﬃciently expressed in a manner w hich allows us to employ these algorithms. We will however leave further di scussion of these details to another note [EM]. When we talk about computing f, we assume that we have some unitary operation Ufwhich takes us from state |x/an}brack⌉tri}ht |0/an}brack⌉tri}htto |x/an}brack⌉tri}ht |f(x)/an}brack⌉tri}ht. It could, for example, take |x/an}brack⌉tri}ht |y/an}brack⌉tri}htto|x/an}brack⌉tri}ht |y+f(x)/an}brack⌉tri}ht, where + de- notes an appropriate group operation, such as addition modu loNwhen the second register is used to represent the integers modulo N. Various cases of the hidden subgroup problem are described i n [Si], [Sh], [Ki], [BL], [Gr], [Jo],

Question 1184 multiple-choice

In group theory and harmonic analysis, the concept of product-free subsets in compact Lie groups intersects with representation theory, quasirandomness, and geometric inequalities. Advances in these areas have led to sharper bounds on the size and structure of such subsets, with implications for probability, geometry, and quantum information. In the study of product-free subsets within compact, D-quasirandom Lie groups, which of the following best describes the improved upper bound on the measure of measurable product-free subsets in the special unitary group SU(n), and its significance? 1) The upper bound is polynomial in n and is not attainable for large n. 2) The upper bound is linear in n and improves on previous exponential bounds. 3) The upper bound is independent of n and applies only to finite groups. 4) The upper bound is given by n^{-1/2}, using classical combinatorial techniques. 5) The upper bound is subexponential in n and relies solely on random matrix theory. 6) The upper bound is exp(-c n^{1/3}), where c > 0 is an absolute constant, and this exponent is proven to be sharp. 7) The upper bound depends only on the group's center and has no relation to representation theory.

✓ Correct Answer:

The correct answer is 6) The upper bound is exp(-c n^{1/3}), where c > 0 is an absolute constant, and this exponent is proven to be sharp..

📚 Reference Text:

Title: Product Mixing in Compact Lie Groups Year: 2024 Paper ID: 493363c00cd3ff6335ce3a57ce499e1088ce0236 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/493363c00cd3ff6335ce3a57ce499e1088ce0236 Abstract: If G is a group, we say a subset S of G is product-free if the equation xy=z has no solutions with x,y,z ∈ S.In 1985, Babai and Sós [] asked, for a finite group G, how large a subset S⊆ G can be if it is product-free. The main tool (hitherto) for studying this problem has been the notion of a quasirandom group. For D ∈ ℕ, a group G is said to be D-quasirandom if the minimal dimension of a nontrivial complex irreducible representation of G is at least D. Gowers showed that in a D-quasirandom finite group G, the maximal size of a product-free set is at most |G|/D1/3. This disproved a longstanding conjecture of Babai and Sós from 1985. For the special unitary group, G=(n), Gowers observed that his argument yields an upper bound of n−1/3 on the measure of a measurable product-free subset. In this paper, we improve Gowers’ upper bound to exp(−cn1/3), where c>0 is an absolute constant. In fact, we establish something stronger, namely, product-mixing for measurable subsets of (n) with measure at least exp(−cn1/3); for this product-mixing result, the n1/3 in the exponent is sharp. Our approach involves introducing novel hypercontractive inequalities, which imply that the non-Abelian Fourier spectrum of the indicator function of a small set concentrates on high-dimensional irreducible representations. Our hypercontractive inequalities are obtained via methods from representation theory, harmonic analysis, random matrix theory and differential geometry. We generalize our hypercontractive inequalities from (n) to an arbitrary D-quasirandom compact connected Lie group for D at least an absolute constant, thereby extending our results on product-free sets to such groups. We also demonstrate various other applications of our inequalities to geometry (viz., non-Abelian Brunn-Minkowski type inequalities), mixing times, and the theory of growth in compact Lie groups. A subsequent work due to Arunachalam, Girish and Lifshitz uses our inequalities to establish new separation results between classical and quantum communication complexity.

Question 1185 multiple-choice

Quantum process tomography (QPT) is a powerful tool for characterizing the detailed dynamics of quantum operations, especially in experimental quantum computing platforms like nuclear magnetic resonance (NMR) processors. Understanding the nature of errors and the properties of quantum processes is crucial for advancing quantum control and improving gate performance. Which category of error is most directly associated with deviations from complete positivity in reconstructed quantum operations when using QPT on an NMR quantum processor? 1) Coherent errors caused by systematic miscalibrations 2) Decoherent errors due to environmental noise 3) Statistical sampling errors from limited measurement data 4) Gate fidelity loss from imperfect pulse sequences 5) Hardware instability during experimental runs 6) Incoherent errors originating from ensemble inhomogeneities 7) Unitary errors from residual unwanted qubit interactions

✓ Correct Answer:

The correct answer is 6) Incoherent errors originating from ensemble inhomogeneities.

📚 Reference Text:

arXiv:quant-ph/0406239v1 30 Jun 2004Quantum Process Tomography of the Quantum Fourier Transfor m Yaakov S. Weinstein,1,∗Timothy F. Havel,1Joseph Emerson,1,† Nicolas Boulant,1Marcos Saraceno,2Seth Lloyd,3and David G. Cory1 1Massachusetts Institute of Technology, Dept. of Nuclear Engineering, Cambridge, MA 02139, USA 2Unidad de Actividad Fisica, Tandar, Comisi´ on Nacional de Energia At´ omica, 1429, Buenos Aires, ARGENTINA 3d’Arbeloﬀ Laboratory for Information Systems and Technolo gy, Dept. of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Abstract The results of quantum process tomography on a three-qubit n uclear magnetic resonance quan- tum information processor are presented, and shown to be con sistent with a detailed model of the system-plus-apparatus used for the experiments. The quant um operation studied was the quantum Fourier transform, which is important in several quantum al gorithms and poses a rigorous test for the precision of our recently-developed strongly modulati ng control ﬁelds. The results were ana- lyzed in an attempt to decompose the implementation errors i nto coherent (overall systematic), incoherent (microscopically deterministic), and decoher ent (microscopically random) components. This analysis yielded a superoperator consisting of a unita ry part that was strongly correlated with the theoretically expected unitary superoperator of the qu antum Fourier transform, an overall at- tenuation consistent with decoherence, and a residual port ion that was not completely positive - althoughcompletepositivity isrequiredforanyquantumop eration. Bycomparisonwiththeresults of computer simulations, the lack of complete positivity wa s shown to be largely a consequence of the incoherent errors during the quantum process tomogra phy procedure. These simulations further showed that coherent, incoherent, and decoherent e rrors can often be identiﬁed by their distinctive eﬀects on the spectrum of the overall superopera tor. The gate ﬁdelity of the experimen- tally determined superoperator was 0 .64, while the correlation coeﬃcient between experimentall y determined superoperator and the simulated superoperator was 0.79; most of the discrepancies with the simulations could be explained by the cummulative e ﬀect of small errors in the single qubit gates. PACS numbers: 03.67.Lx, 82.56.-b, 83.85.Ns ∗Present address: Center for Computational Materials Science, N aval Research Laboratory, Washington, DC 20375, USA †Present address: Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2J 2W9, CANADA 1 I. INTRODUCTION In order to develop larger and more powerful quantum information processing devices, it is essential to quantify the precision with which they can be controlle d. This information is generally reported as a single number, the ﬁdelity of the operation [1, 2, 3]. Although ﬁdelity is a reasonable measure of control, it gives experimentalists lit tle useful informa- tion about what went wrong or how to improve their control over th e quantum system. Quantum process tomography (QPT) [5, 6, 7] provides additional in formation that may be useful in this regard, by yielding an estimate of the quantum operat ion that was actually implemented. This in turn provides a stringent check on the complete ness and accuracy of the system-plus-apparatus model used to design the implementat ion. It is nevertheless a challenging task to interpret the deviations of this estimate

Question 1186 multiple-choice

Quantum Markov semigroups (QMS) are central to the mathematical modeling of open quantum systems and quantum noise, leveraging advanced concepts such as functional inequalities and operator theory. Understanding the distinction between various functional inequalities is key to analyzing mixing times, decoherence, and resource preservation in quantum information science. Which functional inequality is particularly important for estimating short-time behavior in quantum Markov semigroups before spectral gap properties dominate? 1) Cheeger’s inequality 2) Poincaré inequality 3) Fannes-Audenaert inequality 4) Triangle inequality 5) Pinsker’s inequality 6) Golden-Thompson inequality 7) Hypercontractivity, logarithmic Sobolev, and ultracontractivity inequalities

✓ Correct Answer:

The correct answer is 7) Hypercontractivity, logarithmic Sobolev, and ultracontractivity inequalities.

📚 Reference Text:

estimate decoherence times of these channels independently from the number of qubits, and similar estimates hold for their capacities. Moreover, we will derive some estimates that are new even in the classical literature. This will be the case for quantum channels that randomly swap subsystems. Inspired by the techniques of [6], we also show a smorgasbord of functional inequalities for these semigroups, such as spectral gap, hypercontractivity, logarithmic Sobolev inequalities and ultracontractive estimates. The latter three are particularly important to obtain good estimates for small times, before the spectral gap kicks in and closes the deal. We also exemplify why in the non-transference case, the correct notion for applications of the decoherence-time is a complete version, i.e. where we consider the semigroup tensorized with the identity on a matrix algebra. Finally, let us point out that in contrast to [6], which focuses on Lie groups and Hormänder systems (where very good estimates are available from the fundamental work of Rothschild and Stein [7]), we are also interested in ﬁnite groups and jump processes,as mixed unitary channels with unitaries arising from a representation of a ﬁnite group play an important role in quantum information theory. a) Layout of the paper:: In Section II, we introduce the framework of quantum Markov semigroups and explain their connection to classical diffusions and jump processes on groups via the so-called transference technique . In Section III, we explain the technical tools that allow us to bound various norm estimates of a quantum Markov semigroup in terms of the kernel of an associated classical process: namely, noncommutative Lp spaces and the norm transference technique. Section IV is devoted to some examples of transferred semigroups to illustrate the technique. We then illustrate how to use our techniques to estimate capacities and apply them to other resource theories in Section V. Then, in Section VI, we show how contractivity properties (in particular, ultracontractivity) of the quantum Markov semigroups can also provide a way to estimates some capacities in the quantum case, without the use of transference. II. Q UANTUM MARKOV SEMIGROUPS VIA GROUP TRANSFERENCE Aquantum Markov semigroup (QMS) (Tt)t≥0 onB(H)is a uniformly continuous semigroup of completely positive maps such that T0=id andTt(IH)=IHfor allt≥0[8]. The limit L=limt→0(id−Tt)/slash.lefttexists and is called the Lindblad generator . We insist on our convention that consequently Tt=e−tLwith a minus sign ! This is not the most often used convention in the quantum case but it is more consistent with the classical situations we will consider. The QMS (Tt)t≥0models the evolution of observables in the Heisenberg picture . In the dual Schrödinger picture , one is instead interested in the evolution of states of density matrices. We recall that a density matrix ∈B(H)onHis a trace-one positive semi-deﬁnite operator. We denote by D(H)the set of density matrices on Hand by D+(H)the set of invertible (full-rank) density matrices. We shall mainly (but not only) study self-adjoint (or symmetric) QMS for the Hilbert Schmidt scalar product: Tr[Tt(x∗)y]=Tr[x∗Tt(y)]∀x;y∈B(H);∀t≥0: This is equivalent to the fact that Tt=T† t, whereT† is

Question 1187 multiple-choice

Quantum algorithms such as Shor’s and Grover’s are analyzed by their group-theoretic structures, particularly through the framework of the hidden subgroup problem (HSP). The symmetric group SN and its stabilizer subgroups play a key role in understanding the limitations of quantum speedup for unstructured search. Which group-theoretic property most directly explains why non-abelian quantum hidden subgroup algorithms cannot efficiently find the hidden item in Grover’s search problem when formulated over SN? 1) The stabilizer subgroups are cyclic 2) SN is a simple group for large N 3) The coset representatives form an abelian subgroup 4) The index of the stabilizer subgroup is always prime 5) The stabilizer subgroups are unique and normal in SN 6) The stabilizer subgroups in SN are not normal and are mutually conjugate 7) SN lacks a well-defined transversal mapping

✓ Correct Answer:

The correct answer is 6) The stabilizer subgroups in SN are not normal and are mutually conjugate.

📚 Reference Text:

1 if i=j. Proposition 1. The set {(0j0),(1j0),(2j0),...,((N−1)j0)} is a complete set of distinct coset representatives for the h idden subgroup Stabj0 ofSN, i.e., the coset space SN/Stabj0is given by the following complete set of mutually distinct cosets. SN/Stabj0={(0j0)Stabj0,(1j0)Stabj0,(2j0)Stabj0,...,((N−1)j0)Stabj0} Proof.Since (kj0)Stabj0= (ℓj0)Stabj0⇐⇒(ℓj0)−1(kj0)∈Stabj0⇐⇒k=l, it follows that (0j0)Stabj0,(1j0)Stabj0,(2j0)Stabj0,...,((N−1)j0)Stabj0 are mutually distinct cosets of Stabj0inSN. It now follows from Lagrange’s theorem that the above collection of mutually distinct cosets is comp lete. /square IS GROVER’S ALGORITHM A QUANTUM HIDDEN SUBGROUP ALGORITHM ? 9 8.A comparison of Grover’s and Shor’s algorithms Now let us compare Shor’s algorithm with Grover’s. LetSbe the set of integers S={0,1,2,...,N−1}, whereN= 2n, and letj0∈Sdenote the unknown label to be found by Grover’s algorithm. Shor’s algorithm solves the HSP ϕ:Z−→Z× Nwith hidden subgroup structure Zϕ−→Z× N νց ր ι Z/PZ, whereZ× Ncan be thought of as the result of the unknown (”malicious”) relabe ling ι:k+PZ/ma√sto−→akmodN ofZ/PZ. In likemanner, Grover’salgorithmsolvesan HSP, namely, the HSP ϕ:SN−→S with hidden subgroup structure SNϕ−→S νց ր ι SN/Stabj0, whereS={0,1,2,...,N−1}denotes the set resulting from the unknown (”mali- cious”) relabeling (bijection) ι: (j j0)Stabj0/ma√sto−→j ofSN/Stabj0. For Shor’s algorithm, Shor’s oracle /tildewideϕ:ZQ−→Z× Nis created by pushing the HSPϕ:Z−→Z× Nusing Zϕ−→Z× N µցտτր/tildewideϕ Z/QZ, thereby producing /tildewideϕ=Push(ϕ) =ϕ◦τwith the transversal τ:kmodQ/ma√sto−→k. In likemanner, for Grover’salgorithm, Grover’soraclecan be creat ed by pushing the HSPϕ:SN−→Susing SNϕ−→S µցտτր/tildewideϕ SN/Stab0, thereby producing /tildewideϕ=Push(ϕ) =ϕ◦τwith the transversal τ: (0j)Stab0/ma√sto−→SN of the natural surjection µ. 10 SAMUEL J. LOMONACO, JR. AND LOUIS H. KAUFFMAN Although it is not immediately apparent, the resulting push /tildewideϕ(forj0/ne}ationslash= 0) is actually Grover’s oracle relabelled by the injection ι:SN/Stabj0−→S. For /tildewideϕ=ϕ◦τ= (ι◦ν)◦τ=ι◦(ν◦τ) and (ν◦τ)[(0j)Stab0] =  (0j0)Stabj0ifj=j0 Stabj0 otherwise which is informationally the same as Grover’s oracle f(j) =  1 ifj=j0 0 otherwise Hence, we can conclude that Grover’s algorithm is an quantum algorit hm very much like Shor’s algorithm, in that it is a quantum algorithmthat solvest he Grover hidden subgroup problem. 9.However However,... this appearstobe wherethe similaritybetweenthesetw oalgorithms ends. For, the standard non-abelian QHS algorithm on SNfor the HSP ϕ(or/tildewideϕ) can not ﬁnd the hidden subgroup Stabj0for each of the following two reasons: •Since the subgroups Stabjare not normal subgroups of SN, it follows from the work of Hallgren et al [11] that the standard non-abelian hidden sub- group algorithm will ﬁnd the largest normal subgroup of SNlying inStabj. But unfortunately, the largest normal subgroup of SNlying inStabj. is the trivial subgroup of SN. •The subgroups Stab0,Stab1, ... ,StabN−1are mutually conjugate sub- groups of SN. We should also mention that this hidden subgroup approach can not p ossibly lead to a quantum algorithm that is faster than Grover’s.. For Zalka[3 1] has shown that Grover’s algorithm is asymptotically optimal. IS GROVER’S ALGORITHM A QUANTUM HIDDEN SUBGROUP ALGORITHM ? 11 A Comparison of Two Quantum Algorithms Shor’s Algorithm Grover’s Algorithm Similarities Shor’s algorithm solves an HSP, namely: Zϕ−→Z× N νց ր ι Z/PZGrover’s algorithm solves an HSP, namely: SNϕ−→S νց ր ι SN/Stabj0 Pushingϕusing Zϕ−→Z× N µցտτր/tildewideϕ Z/QZ produces /tildewideϕ=Push(ϕ) =ϕ◦τ which is Shor’s oraclePushingϕusing SNϕ−→S µցտτր/tildewideϕ SN/Stab0

Question 1188 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) rely on group representation theory and the properties of coset states to distinguish hidden subgroups within finite groups. The efficiency of these algorithms is closely tied to bounds derived from the analysis of group characters and irreducible representations. For the symmetric group wreath product Sn≀S2, which is relevant to the graph isomorphism problem, what is the asymptotic lower bound on the number of coset states required to distinguish between trivial and non-trivial hidden subgroups with constant probability? 1) Ω(n) 2) Ω(log n) 3) Ω(n log n) 4) Ω(n^2) 5) Ω(n^3) 6) Ω(log log n) 7) Ω(√n)

✓ Correct Answer:

The correct answer is 3) Ω(n log n).

📚 Reference Text:

|G|+Dε |G|  ≤kε+kDε |G|. Theﬁrst inequality follows from kapplications of the triangle inequality. Finally, /vextenddouble/vextenddoubleMHg−M{1}/vextenddouble/vextenddouble 1≤/vextenddouble/vextenddoubleMHg−M′ Hg/vextenddouble/vextenddouble 1+/vextenddouble/vextenddouble/vextenddoubleM′ Hg−M′′ {1}/vextenddouble/vextenddouble/vextenddouble 1+/vextenddouble/vextenddouble/vextenddoubleM′′ {1}−M{1}/vextenddouble/vextenddouble/vextenddouble 1 ≤2k(1+2kε)µg+3kε+3k |G|·/summationdisplay τ∈Sεd2 τ. Wearenow ready to prove themain theorem of the paper. Proof of Theorem 2. Thetheorem follows from Lemmas10and 11,using theconvexit y of the square func- tion. The upper bound on δ1follows from the observation that Cauchy-Schwartz implies that/summationtext ν∈/hatwideGdν≤ |/hatwideG|1/2/parenleftbig/summationtext ν∈/hatwideGd2 ν/parenrightbig1/2=|/hatwideG|1/2|G|1/2. 14 Finally, we prove a simple lower bound, irrespective of the o rder of entanglement, on the total number of coset states trequired to distinguish a hidden subgroup Hgfrom the identity hidden subgroup. For that, weneed the following theorem. Theorem 12. LetGbe a ﬁnite group and H:={1,h}be an order two subgroup of G. Lett≥1be an integer. Then,/vextenddouble/vextenddouble/vextenddoubleEg/bracketleftbig σ⊗t Hg/bracketrightbig −σ⊗t {1}/vextenddouble/vextenddouble/vextenddouble tr<2t |G|/summationdisplay τ∈/hatwideGdτ|χτ(h)|. Proof.Letρ∈/hatwideG⊗t,I⊆[t],I/\e}ati≀\slash={}. Using arguments similar to those above, it is easy tosee tha t /vextenddouble/vextenddoubleEg[2tρ((Hg)t)]−ρ({1}t)/vextenddouble/vextenddouble tr=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleEg 11dρ+/summationdisplay I/\e}atio\slash={}ρI(ghg−1) −11dρ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble tr=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay I/\e}atio\slash={}Eg/bracketleftbig ρI(ghg−1)/bracketrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble tr ≤/summationdisplay I/\e}atio\slash={}/vextenddouble/vextenddoubleEg[ρI(ghg−1)]/vextenddouble/vextenddouble tr=/summationdisplay I/\e}atio\slash={}/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/circleplusdisplay τ∈/hatwideGχτ(h) dτaρI τ/circleplusdisplay j=111dτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble tr=/summationdisplay I/\e}atio\slash={}/summationdisplay τ∈/hatwideGaρI τ|χτ(h)|. Writing thedensity matrices in the Fourier basis and using F act6weget, /vextenddouble/vextenddouble/vextenddoubleEg/bracketleftbig σ⊗t Hg/bracketrightbig −σ⊗t {1}/vextenddouble/vextenddouble/vextenddouble tr =/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleEg 2t |G|t/circleplusdisplay ρdρ/circleplusdisplay i=1|ρ∗,i/a\}bracketri}ht/a\}bracketle{tρ∗,i|⊗ρ((Hg)t) −1 |G|t/circleplusdisplay ρdρ/circleplusdisplay i=1|ρ∗,i/a\}bracketri}ht/a\}bracketle{tρ∗,i|⊗ρ({1}t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble tr =/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1 |G|t/circleplusdisplay ρdρ/circleplusdisplay i=1|ρ∗,i/a\}bracketri}ht/a\}bracketle{tρ∗,i|⊗(Eg[2tρ((Hg)t)]−ρ({1}t))/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble tr =1 |G|t/summationdisplay ρdρ/vextenddouble/vextenddoubleEg[2tρ((Hg)t)]−ρ({1}t)/vextenddouble/vextenddouble tr≤1 |G|t/summationdisplay ρdρ/summationdisplay I/\e}atio\slash={}/summationdisplay τ∈/hatwideGaρI τ|χτ(h)| =/summationdisplay I/\e}atio\slash={}/summationdisplay τ∈/hatwideG|χτ(h)|/parenleftBigg/summationdisplay ρd2 ρ |G|taρI τ dρ/parenrightBigg =/summationdisplay I/\e}atio\slash={}/summationdisplay τ∈/hatwideGdτ|χτ(h)| |G| <2t |G|/summationdisplay τ∈/hatwideGdτ|χτ(h)|. Corollary 13. Any algorithm using a total of tcoset states that distinguishes with constant probability betweenthecasewhenthehiddensubgroup istrivialandthec asewhenthehiddensubgroupis Hgforsome g∈Gmust satisfy t= Ω(log(1/η)). Proof.Thealgorithm canbeviewedasatwo-outcome POVMthat output s1withprobability atleast 2/3if thehiddensubgroup isnon-trivial, and 0withprobability atleast 2/3ifthehiddensubgroup istrivial. Thus, the POVMdistinguishes between the states Eg/bracketleftbig σ⊗t Hg/bracketrightbig andσ⊗t {1}with constant total variation distance. Since the trace distance is always an upper bound on the total varia tion distance, invoking Theorem 12 completes the proof. 15 Theabovecorollary shows,for example, thatanycoset state basedalgorithm solving theHSPin Sn≀S2 needs atotal number of Ω(nlogn)coset states. Inthenext section, weapply Theorem 2toshow a stronger result,namely,anyalgorithmsolvingtheHSPin Sn≀S2usingpolynomiallymanycosetstatesneedstomake measurements entangled across Ω(nlogn)coset states. However, Corollary 13 can sometimes prove non - trivial lower bounds on the total number of coset states for s olving the HSP in groups Gwhere Theorem 2 can only prove a constant lower bound on the order of entangle ment. For example, the HSP in groups G:=A⋊Z2,whereAisanabeliangroupand Z2actsonAbyinversioncanbesolvedbyanalgorithmusing atotalnumberof O(log|G|)cosetstatesthatmeasuresonecosetstateatatime[EH00]. U singCorollary13, one can show a matching Ω(log|G|)lower bound on the total number of coset states when Ais the cyclic groupZn,i.e.,Gis the dihedral group Dn. Using a different technique, Childs and Wocjan [CW05] in fa ct show anΩ(log|G|)lower bound on thetotal number of coset states for the above g roups for all abelian A. 4 Limitationsof quantum cosetstatesfor HSP: Examples 4.1 The wreath product Sn≀S2and graphisomorphism The representation theory of the wreath product G=Sn≀S2is well-known. The following is a summary of the necessary results, for more details we refer to Append ix A: the wreath product has irreps κλ,λ′of dimension 2dλdλ′,whereλ,λ′∈/hatwiderSn,λ/\e}ati≀\slash=λ′. Deﬁneh:= (e,e,1)∈G,whereeistheidentity permutation inSn. Thecharacter valueof hontheseirrepsiszero. Furthermore, thereareirreps ϑλandϑ′ λofdimension d2 λ,whereλ∈/hatwiderSn. Thecharacter values of ϑλandϑ′ λonharegiven by dλand−dλ,respectively. Thetotal number

Question 1189 multiple-choice

Quantum computing with high-dimensional systems (qudits) requires specialized gate constructions to enable robust, fault-tolerant operations. The π/8 gate and SWAP gates play crucial roles in universal quantum computation and efficient circuit design for qutrits and higher-dimensional qudits. Which gate is essential for magic-state distillation protocols in prime-dimensional qudit systems due to its non-Clifford nature and enables fault-tolerant universal quantum computation? 1) π/8 gate 2) Hadamard gate 3) Controlled-phase gate 4) Qudit SWAP gate 5) C~X gate (generalized CNOT) 6) GXOR gate 7) K_d gate (modulo complement)

✓ Correct Answer:

The correct answer is 1) π/8 gate.

📚 Reference Text:

isω/equalse2πi/9and, from Eq. 29 we derive that det⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ω2c’C ⎥parenleftbigg⎥bracketleftbigg10 c’1⎥bracketrightbigg⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥bracketleftbigg1 z’⎥bracketrightbigg⎥parenrightbigg⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠/equals1. (41) The qutrit version of U π/8has a more general form than in Eq. 30 ; i.e. Uυ/equalsU(υ0,υ1,...)/equals⎥summationdisplay2 k/equals0ωυk|k〉〈k|,υk∈Z9. (42)Then the general solution is υ/equals⎥parenleftbig0,6z′+2c′+3ε′,6z′+c′+6ε′⎥parenrightbigmod9 . (43) For example, choosing z′/equals1,c′/equals2 and ε′/equals0, Uυ/equals⎛⎜⎜⎝ω000 0ω10 00 ω−1⎞⎟⎟⎠. (44) The π/8 gate, with its close relation to the Clifford group, has many applications and utilities in teleportation-based UQC [ 66]; transversal implementation [ 50, 162 ]; learning an unknown gate [105]; or securing assisted quantum computation [ 28]. The generalized qudit version of the π/8 gate, Uυ, is shown to be identical to the maximally robust qudit gates for qudit fault- tolerant UQC discussed in reference [ 150]. This gate also plays an important role in the magic-state distillation (MSD) protocols for general qudit systems, whichwasﬁrst established for qutrits [ 6] and then extended to all prime- dimensional qudits [ 25]. 2.2.2 Qudit SWAP Gate A SWAP gate is used to exchange the states of two qudit such that: SWAP |ϕ〉|ψ〉/equals|ψ〉|ϕ〉 (45) Various methods to achieve the SWAP gate use different variants of qudit controlled gates [ 4, 58, 112, 131, 155, 158, 159 ] as shown inFigure 2A,B . The most used component of the SWAP gate is a controlled-shift gate CXdthat perform the following operation: CX d|x〉|y〉/equals|x〉|x+y〉 (46) with a modulo daddition. Its inverse operation is CX† d|x〉|y〉/equals|x〉|y−x〉 (47) In some approaches, the operation Kdis required to complete the circuits, where Kd|x〉/equals|d−x〉/equals|−x〉, (48) which outputs the modulo dcomplement of the input. These circuits are more complex and less intuitive then the qubit SWAP gate [ 58] because they are not Hermitian, i.e., CXd≠CX† d. One way to create a Hermitian version of the qudit CNOT uses the GXOR gate FIGURE 3 | Qudit SWAP circuits with the GXOR and the Kdgates [ 4, 155]. FIGURE 4 | (A) is the qudit SWAP gate with the C~Xgate. (B)is the decomposing C~Xgate. The QFT represents the quantum Fourier transform while CZdis the selective phase shift gate. Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504 6Wang et al. Qudits and High-Dimensional Quantum Computing GXOR |x〉|y〉/equals|x〉|x−y〉. (49) However, this SWAP gate needs to be corrected with an Kd[4]a s shown in Figure 3 . A partial SWAP gate Sp[39] works on a hybrid system where |i〉is a qudit of dimension dcand |j〉is a qudit of dimension dt Sp|i〉⊗|j〉/equals⎥braceleftBigg|j〉⊗|i〉fori,j∈Zdp |i〉⊗|j〉otherwise(50) where dp≤dmin/equalsmin(dc,dt). In the rest of this section, we present a Hermitian generalization of the qudit CNOT gate with a symmetryconﬁguration and a qudit SWAP circuit with a single type of qudit gate as shown in Figure 4A [61]. Compared with all the previously proposed SWAP gate for qudit, this method is easier toimplement since there is only one type of gate C~Xneeded. To begin with, we de ﬁne a gate C~Xacting on d-level qudits |x〉and |y〉such that C~X|x〉|y〉/equals|x〉|−x−y〉, (51) where |−x−y〉represents a state |i/equals−x−y〉in the range i∈{0,... ,d−1}mod d. Notice that, for d/equals2, the C~Xgate is equivalent

Question 1190 multiple-choice

Quantum computing leverages phenomena such as superposition and entanglement to perform operations that are intractable for classical computers. The quantum Fourier transform (QFT) is a critical subroutine in many quantum algorithms, including those for finding periodicities and factoring large numbers. Which of the following best characterizes the experimental significance of implementing the quantum Fourier transform (QFT) on a three-qubit nuclear magnetic resonance (NMR) quantum computer? 1) It demonstrated that NMR systems can accurately perform complex quantum operations required for algorithms like Shor’s by extracting periodicity and quantitatively measuring efficiency via state tomography. 2) It showed that NMR quantum computers are universally scalable for all quantum algorithms without practical limitations. 3) It proved that classical computers could efficiently simulate all quantum algorithms using NMR techniques. 4) It revealed that quantum state tomography is unnecessary for assessing quantum algorithm performance. 5) It established that the QFT is only applicable to two-qubit systems and cannot be extended to higher dimensions. 6) It indicated that quantum error correction is not needed when using NMR quantum computers for any algorithm. 7) It confirmed that the QFT has no relevance to periodicity extraction or integer factorization problems.

✓ Correct Answer:

The correct answer is 1) It demonstrated that NMR systems can accurately perform complex quantum operations required for algorithms like Shor’s by extracting periodicity and quantitatively measuring efficiency via state tomography..

📚 Reference Text:

Title: Implementation of the quantum Fourier transform. Year: 1999 Paper ID: 69721134c51df06033990560b5a530201288de00 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/69721134c51df06033990560b5a530201288de00 Abstract: A quantum Fourier transform (QFT) has been implemented on a three qubit nuclear magnetic resonance (NMR) quantum computer to extract the periodicity of an input state. Implementation of a QFT provides a first step towards the realization of Shor's factoring and other quantum algorithms. The experimental implementation of the QFT on a periodic state is presented along with a quantitative measure of its efficiency measured through state tomography. Experimentally realizing the QFT is a clear demonstration of the ability of NMR to control quantum systems.

Question 1191 multiple-choice

In quantum computation, algorithms that operate on black-box groups often rely on group-theoretic structures and quantum superpositions to efficiently solve problems such as the Hidden Subgroup Problem. Translating Coset Superposition (TCS) is a quantum procedure central to these approaches, especially when a normal subgroup and its quotient group have specific properties. Which of the following is crucial for the correctness and efficiency of a quantum algorithm implementing Translating Coset Superposition (TCS) for a black-box group G with normal subgroup N, when G/N is abelian? 1) The use of classical group multiplication exclusively 2) Access to the full multiplication table of G 3) The restriction of G to be a cyclic group 4) The ability to perform quantum Fourier transform over nonabelian groups 5) The assumption that N is a non-normal subgroup 6) Encoding elements of G and N independently without consistency 7) Compatibility of encoding between G and N to ensure consistent representation in quantum states

✓ Correct Answer:

The correct answer is 7) Compatibility of encoding between G and N to ensure consistent representation in quantum states.

📚 Reference Text:

|N·ϕ0/angbracketright⊗t|N·ϕ1/angbracketright⊗t|1G/N/angbracketright: |N·ϕ0/angbracketright⊗t|φ0/angbracketright⊗s|N·ϕ1/angbracketright⊗t|φ1/angbracketright⊗s|1G/angbracketright|1G/angbracketright⊗2t|uHN/N /angbracketright, whereH=G|φ1/angbracketrightand|u·φ1/angbracketright=|φ0/angbracketright, if there is any; |N·ϕ0/angbracketright⊗t|φ0/angbracketright⊗s|N·ϕ1/angbracketright⊗t|φ1/angbracketright⊗s|1G/angbracketright|1G/angbracketright⊗2t|∅/angbracketright otherwise 4. Undo step 1: |φ0/angbracketright⊗s(t+1)|φ1/angbracketright⊗s(t+1)|1G/angbracketright|N/angbracketright⊗2t|uHN/N /angbracketright, or |φ0/angbracketright⊗s(t+1)|φ1/angbracketright⊗s(t+1)|1G/angbracketright|N/angbracketright⊗2t|∅/angbracketright. In the second case, Stop the algorithm here. 5. Perform sapplications of inverse of group element in the last register to registers |φ0/angbracketright(viewed as an element of Gthanks to the compatible encoding of G/N ): /summationdisplay zN∈uHN/N|z−1·φ0/angbracketright⊗s|φ0/angbracketright⊗st|φ1/angbracketright⊗s|φ1/angbracketright⊗st|1G/angbracketright|N/angbracketright⊗2t|encN(zN)/angbracketright.3 6. Perform Translating Coset Superposition (N,α,s )on |z−1·φ0/angbracketright⊗s|φ1/angbracketright⊗s|1G/angbracketright: /summationdisplay zN∈uHN/N|z−1·φ0/angbracketright⊗s|φ0/angbracketright⊗st|φ1/angbracketright⊗s|φ1/angbracketright⊗st|nz(H∩N)/angbracketright|N/angbracketright⊗2t|encN(zN)/angbracketright. (See the proof of Theorem 4.11 for notation and justiﬁcation.) 7. Apply the group element in the last register to the first s registers |z−1·φ0/angbracketright: /summationdisplay zN∈uHN/N|φ0/angbracketright⊗s(t+1)|φ1/angbracketright⊗s(t+1)|nz(H∩N)/angbracketright|N/angbracketright⊗2t|encN(zN)/angbracketright. 8. Left multiply by the group element in the last register the group element in the 2s(t+1 )+1 st register /summationdisplay zN∈uHN/N|φ0/angbracketright⊗s(t+1)|φ1/angbracketright⊗s(t+1)|znz(H∩N)/angbracketright|N/angbracketright⊗2t|encN(zN)/angbracketright. 9. Inverse in the last and the 2s(t+1 )+1 st registers the computation of the compatible encoding |zn/angbracketright|0/angbracketright/mapsto→|zn/angbracketright|encN(zN)/angbracketrightfor every n∈N: |φ0/angbracketright⊗s(t+1)|φ1/angbracketright⊗s(t+1)|uH/angbracketright|N/angbracketright⊗2t|0/angbracketright. 3We explicitly mention herethe encoding used for the lastregister inorder to avoid any ambiguity in the notation. Observe also that zhas the same encoding as zN(encN(zN)=enc(z)). 20 FRIEDL, IVANYOS, MAGNIEZ, SANTHA, AND SEN Theorem 4.11. LetGbe a black-box solvable group with unique encoding of length/lscriptand letNbe a normal subgroup of Gsuch that G/Nis abelian. Let αbe a group action of Gand lets,tbe positive integers. Then TCS(G,α,s(t+1))implements Translating Coset Superposition (G,α,s(t+1))using(4t+1)implementations of Translating Coset Superposition (N,α,s),2implementations of Translating Coset Superposition (G/N,(αs)N,t),2tcopies of |N/angbracketrightas ancilla, and quantum time poly(/lscriptts)for the remaining computation. Proof. The complexity analysis of the algorithm is direct; only its analysis needs to be detailed. First observe that when the translating coset of |φ0/angbracketrightand|φ1/angbracketrightis empty, the algorithm sets H=∅, and its correctness is clear. From now on, we assume that the translating coset is not empty, and it is uH,f o r some unknown u∈G,w h e r eHis the unknown stabilizer of |φ0/angbracketrightand|φ1/angbracketright.N o t et h a t the translating coset of |N·φ0/angbracketrightand|N·φ1/angbracketrightforαNinG/NisuHN/N, and therefore nonempty. At step 1, Theorem 4.10 is applied, then after step 4, the algorithm has therefore computed the state |uHN/N/angbracketrightin its last register. In step 5, the group element in the last register is encoded using encN.B u tw h e n its inverse is applied to the ﬁr st register as an element of G,w em e a nt ou s et h e encoding enc. Thanks to our deﬁnition of compatible encoding, this makes sense as long aszsatisﬁes encN(zN)= enc(z). That is why the computed state becomes a uniform superposition of states |z−1·φ0/angbracketright⊗s...|φ1/angbracketright⊗s...|encN(zN)/angbracketright, where the su- perposition is over zN∈uHN/N,a n dzis chosen such that encN(zN)= enc(z). For each such z, we prove that states |z−1·φ0/angbracketrightand|φ1/angbracketrighthave the translating coset nz(H∩N) over the subgroup Nfor some nz∈Nsuch that |nz·φ1/angbracketright=|z−1·φ0/angbracketright, meaning that znz∈uH. Indeed, since |u·φ1/angbracketright=|φ0/angbracketright,w eg e t|(z−1u)·φ1/angbracketright=|z−1·φ0/angbracketright. Therefore |z−1·φ0/angbracketright and|φ1/angbracketrighthavethe translatingcoset z−1uHoverG.S i n c ezN∈uHN/N,on ec anw ri t e znz=uhzfor some hz∈Handnz∈N. Note that both hzandnzare uniquely deﬁned up to some element in H∩N. Then the translating coset can be rewritten asnzH, implying that |z−1·φ0/angbracketrightand|φ1/angbracketrighthave a nonempty translating coset over N, which is nz(H∩N). Now set H1=/uniontext z/parenleftbig hz(H∩N)/parenrightbig . Then after step

Question 1192 multiple-choice

In computational complexity and representation theory, the moment polytope plays a central role in problems related to group actions and quantum information. Understanding the algorithmic tractability of membership testing for these polytopes has deep implications across mathematics and physics. Which of the following statements correctly characterizes the computational complexity of deciding whether a point lies in the moment polytope associated with a finite-dimensional unitary representation of a compact, connected Lie group? 1) It is solvable in polynomial time for all group representations. 2) It is contained in both NP and coNP, but may not be known to be in P. 3) It is known to be NP-complete for all representations. 4) It requires solving a quadratically-constrained program with no complexity classification. 5) It is only in coNP and not in NP. 6) It is undecidable for compact, connected Lie groups. 7) It is in P only for representations indexed by single-row Young diagrams.

✓ Correct Answer:

The correct answer is 2) It is contained in both NP and coNP, but may not be known to be in P..

📚 Reference Text:

MEMBERSHIP IN MOMENT POLYTOPES IS IN NP AND CONP PETER BÜRGISSERy, MATTHIAS CHRISTANDLz, KETAN D. MULMULEYx,AND MICHAEL WALTER{ Abstract. We show that the problem of deciding membership in the moment polytope associated with a ﬁnite-dimensional unitary representation of a compact, connected Lie group is in NP\coNP. This is the ﬁrst non-trivial result on the computational complexity of this problem, which naively amounts to a quadratically-constrained program. Our result applies in particular to the Kronecker polytopes, and therefore to the problem of deciding positivity of the stretched Kronecker coeﬃcients. In contrast, it has recently been shown that deciding positivity of a single Kronecker coeﬃcient is NP-hard in general [ 23]. We discuss the consequences of our work in the context of complexity theory and the quantum marginal problem. Key words. computational complexity, moment polytope, asymptotic representation theory, Kronecker coeﬃcients, quantum information theory, quantum marginal problem AMS subject classiﬁcations. 68Q25, 53D20, 22E46, 81R05 1. Introduction and summary of results. Moment polytopes are convex polytopes that describe invariants of Hamiltonian manifolds. Their study has a long and rich history in mathematics [ 39,22,28,3,35,25] and in physics [ 19], most recently in quantum information theory in the context of the quantum marginal problem [ 13,14,26,12,2,43,10,42]. Moment polytopes and their underlying representation-theoretic data have also become of interest in computer science, since they are possible sources of representation-theoretic obstructions that may lead to new complexity-theoretic lower bounds [34, 7, 9]. In this paper, we consider the computational complexity of deciding membership in agivenmomentpolytope. Weshowthat, forabroadclassofgroupsandrepresentations, the problem of deciding membership in the associated moment polytope is in the complexity classes NPandcoNP. Before presenting our general results, we discuss the important special case of the moment polytopes associated with the Kronecker coeﬃcients, which are of particular interest in several of the applications mentioned above. Recall that a Young diagram withkboxes is an integer partition 1++m= kwhere1m>0. We write`kfor a Young diagram with kboxes. Young diagrams can be visualized as arrangement of blocks with iblocks in the i-th row; thusmis called the height of the diagram. Any Young diagram with height no larger thanmcan be understood as a highest weight of the unitary group U(m)and, by restriction, of SU(m). Now consider the irreducible representation of G=SU(m)3on Submitted to the editors on Nov 18, 2015. Funding: PB was partially supported by DFG grant BU 1371/3-2. MC was supported by a Sapere Aude grant of the Danish Council for Independent Research, an ERC Starting Grant, an SNSF Professorship, and the VILLUM FONDEN via the QMATH Centre of Excellence (grant no. 10059). KM was supported by NSF grant CCF-1017760. MW was supported by the Simons Foundation, FQXI, and AFOSR (grant no. FA9550-16-1-0082). yTechnische Universität Berlin, Institut für Mathematik, MA 3-2, Straße des 17. Juni 136, 10623 Berlin, Germany (pbuerg@math.tu-berlin.de). zQMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark (christandl@math.ku.dk). xThe University of Chicago, Computer Science Dept., 1100 East 58th Street, Chicago, IL 60637 (mulmuley@cs.uchicago.edu). {Stanford University, Department of Physics, 382 Via Pueblo Mall, Stanford, CA 94305

Question 1193 multiple-choice

In advanced group theory, the structure of exceptional Lie groups such as $E_8$ and their subgroups plays a key role in classification, symmetry, and representation theory. Understanding centralizers, normalizers, and the relationships between subgroup actions provides insight into algebraic and geometric properties. Which of the following describes the quotient of the normalizer of an elementary abelian subgroup of order $2^5$ within the exceptional Lie group $E_8$? 1) The symmetric group $S_5$ 2) The alternating group $A_5$ 3) The unitary group $U_5(2)$ 4) The special linear group $SL_5(2)$ 5) The spin group $\mathrm{Spin}(12)$ 6) The orthogonal group $O_8^+(2)$ 7) The projective general linear group $PGL_5(2)$

✓ Correct Answer:

The correct answer is 4) The special linear group $SL_5(2)$.

📚 Reference Text:

])0=11l2l3R, where 11, 12, 13 are conjugate fundamental subgroups SL2, and R = D4. Letting (e;) = Z(J;), we have Z(JJ213R) = (e1) x (e2) x (e3) and Z(J1l2l3) n Z(R) = (e1e2, e2e3). By Lemma 2.3, C.1;(£)0 = 1 for i = 1, 2, 3, so there exist elements a1a2r1, b1b2r2, a;a3r3, b;b3r4E E such that a; and b; (also a; and bD are elements of order 4 in l; with [a;, b;] = e; (also [a;, b;] = e2), and r1, r2, r3, r4ER. But then either [a2, a;]* 1 or [a2, b;] * 1, say the latter. This forces [a1a2r1, b;b3r4] * 1, a contradiction. Thus there is no such element/. By Lemmas 2.3 and 2.5, there exist x1=a1y,, x2=b1Y2EE with a1, b1El1, y1, y2 E Y and ai =bi= yi = y~ = [a1, bi]= [y1, y2] = e1• The previous paragraph implies that 36 A. M. COHEN, M. W. LIEBECK, J. SAXL AND G. M. SEITZ Now Y/(z> is a half-spin group: to see this, regard Y/~z) as a_Levi factor of G; the corresponding unipotent radical is a 32-dimen~ional spm modul~ for a half-spin group of type D6. Since z is conjugate to e1 m NE,(Y), YI (e1) is also a half-spin group, and hence Y/(ze 1) =S012• Moreover, Y1(ze,) and Y2(ze1) are elements of order 4 squaring to the central involution. Consequently Cc(e, z, x1)0 = T2As. Further, x2 = b1y2 inverts T2 a~d acts as a graph automorphism on~;· ~hus by Proposition 2.7, K = Co(E)0 is of type C3 or D3. Also T2 ~ C(K) . Smee Xz inverts T-, E does not centralize C(K) 0, so by Lemma 2.1, q =3 and (C(K)0)a has facto;~ L2(3). But the rank of Eis 3, which contradicts Lemma 2. l(ii). For the final lemma, handling the case where G = £8 and r = 2, we require the following elementary proposition. PROPOSITION 2.16. Let D be an elementary abelian group of order 2,,. ~ 4, and let V be a module for D over a field of odd characteristic. Then V = Cv(D) E9 2: CU.Do). where the sum is over the subgroups D0 of index 2 in D, and c

Question 1194 multiple-choice

Modular tensor categories (MTCs) are mathematical structures central to the construction of 3-dimensional topological quantum field theories (TQFTs) and have significant applications in areas such as quantum computing, knot theory, and low-dimensional topology. Quantum groups, especially at roots of unity, are a primary source of MTCs, but other algebraic constructions also yield examples. Which of the following statements best characterizes the role of quantum groups at roots of unity in the theory of modular tensor categories and topological quantum field theories? 1) They provide only classical solutions to the Yang-Baxter equation, unrelated to braid group representations. 2) Their representation categories satisfy the axioms of modular tensor categories, enabling the construction of 3D topological quantum field theories. 3) They are exclusively used for constructing invariants of 4-manifolds and do not relate to knot theory. 4) They define modular Hopf algebras that lack braiding and fusion structures required for TQFTs. 5) Their q-deformations are incompatible with applications in quantum computing due to non-unitary representations. 6) They are not involved in the development of knot invariants like the Jones polynomial or the HOMFLY polynomial. 7) Their structure is limited to finite group algebras and cannot be generalized to Lie algebras.

✓ Correct Answer:

The correct answer is 2) Their representation categories satisfy the axioms of modular tensor categories, enabling the construction of 3D topological quantum field theories..

📚 Reference Text:

arXiv:math/0503226v3 [math.QA] 13 Mar 2006Contemporary Mathematics 2000]Primary 20G42; Secondary 20F46, 57R56 From Quantum Groups to Unitary Modular Tensor Categories Eric C. Rowell Abstract. Modular tensor categories are generalizations of the repre sentation categories of quantum groups at roots of unity axiomatizing the properties nec- essary to produce 3-dimensional TQFTs. Although other cons tructions have since been found, quantum groups remain the most proliﬁc sou rce. Recently proposed applications to quantum computing have provided a n impetus to un- derstand and describe these examples as explicitly as possi ble, especially those that are “physically feasible.” We survey the current statu s of the problem of producing unitary modular tensor categories from quantum g roups, emphasiz- ing explicit computations. 1. Introduction We outline the development of the theory of modular tensor ca tegories from quantum groups with an eye towards new applications to quant um computing that motivate our point of view. In this article, we take quantum group to mean the “classical” q-deformation of the universal enveloping algebra of a simpl e complex ﬁnite dimensional Lie algebra as in the book by Lusztig [ L], rather than the broader class of Hopf algebras the term sometimes describes. 1.1. Background. The representation theory of quantum groups has proven to be a useful tool and a fruitful source of examples in many ar eas of mathematics. The general deﬁnition of a quantum group (as a Hopf algebra) w as given around 1985 by Drinfeld [ D] and independently Jimbo [ Ji] as a method for ﬁnding solutions to the quantum Yang-Baxter equation. These solutions led to new representations of Artin’s braid group Bnand connections to link invariants. In fact, special- izations of the famous polynomial invariants of Jones [ J], the six-authored paper [HOMFLY ] and Kauﬀman [ Kf] have been obtained in this way. Reshetikhin and Turaev [ RT] used this connection to derive invariants of 3-manifolds f rommodular Hopf algebras , examples of which can be found among quantum groups at roots of unity (see [ RT] and [ TW1 ], much simpliﬁed by constructions in [ A]). When Witten [ Wi] introduced the notion of a topological quantum ﬁeld theory (TQFT) relating ideas from quantum ﬁeld theory to manifold invaria nts, non-trivial exam- ples were immediately available from the constructions in [ RT] (after reconciling notation). Modular Hopf algebras were replaced by the more g eneral framework of modular tensor categories (MTCs) by Turaev [ T1] (building on deﬁnitions in [Mc] and [ JS]), axiomatizing the conditions necessary (and suﬃcient, s ee [T2], Introduction) to construct 3-dimensional TQFTs. 1991Mathematics Subject Classiﬁcation. [. Key words and phrases. ribbon category, modular tensor category, quantum groups a t roots of unity. c/circlecopyrt0000 (copyright holder) 1 2 ERIC C. ROWELL Aside from the quantum group approach to MTCs, there are seve ral other gen- eral constructions. Representation categories of Hopf alg ebra doubles of ﬁnite group algebras are examples of MTCs that are often included with qu antum groups in the more general discussion

Question 1195 multiple-choice

In quantum computing, the Hidden Subgroup Problem (HSP) is fundamental to algorithms for tasks like factoring and graph isomorphism. Group-theoretic properties and algorithmic strategies are leveraged to efficiently identify hidden subgroups within finite groups. Which statement accurately describes a group-theoretic condition ensuring an algorithm can detect non-trivial elements of a hidden subgroup H in a finite group G using a subgroup G1? 1) G1 must be a normal subgroup of G and H must be cyclic. 2) The intersection G1∩H is trivial if and only if G1H is not equal to G. 3) G1 must have order exactly equal to |H|. 4) The union G1 ∪ H must form a direct product. 5) G1 must be abelian and H must be non-abelian. 6) If |G1| ≥ |G|/|H|, then either G1H = G or |G1∩H| contains non-identity elements. 7) G1 must be a subgroup of H with index two.

✓ Correct Answer:

The correct answer is 6) If |G1| ≥ |G|/|H|, then either G1H = G or |G1∩H| contains non-identity elements..

📚 Reference Text:

the function is injective on the set of points queried, we say fis injective. Otherwise, we say that fis not injective, and output/braceleftbig xy:x∈S1,y∈S2,f(x−1) = f(y)/bracerightbig . If the function is injective, the above algorithm outputs th e correct answer. Suppose fhides the non-trivial subgroup H. Let h∈Hbe any element of the hidden subgroup. We have h=h1h2for some h1∈S1and h2∈S2. By deﬁnition of Hwe have f(h−1 1) = f(h2). When his not the identity, we have h−1 1/\⌉}atio\slash=h2and the algorithm detects a collision. Moreover, the algorit hm computes H correctly. We can obtain explicit, optimal algorithms for certain clas ses of non-abelian groups. For exam- ple, if a group Gof order nhas a subgroup Hof order Θ(√n), then we can construct a generating pair S1,S2of size Θ(√n)by taking S1:=Hand S2to be a complete set of coset representatives ofH. In fact, in this case, the group satisﬁes a stronger property ; see, for example, Ref. [3]. Not all groups have a subgroup of such size. For example, the abelian group Zpfor a prime pdoes not have such a subgroup, and yet admits a suitable generating pa ir. 5 Subgroup-dependent query complexity There is a randomized algorithm for the Hidden Subgroup Prob lem that has expected query com- plexity O (/radicalbig |G|/|H|)when the oracle fhides H, and has worst-case query complexity O (/radicalbig |G|). 7 In this section we use the notion of a generating pair in a more sophisticated manner to match this performance with a deterministic algorithm for all abelian groups, and some classes of non-abelian groups. The algorithm rests on the following observation. Lemma 5.1. Suppose G 1and H are subgroups of the possibly non-abelian ﬁnite group G s uch that G 1has cardinality at least|G|/|H|. Then either G 1H=G or|G1∩H|>1. Proof. The intersection G1∩His a subgroup of Gand contains the identity element. There are exactly|G|/|H|distinct left cosets of H. IfG1H/\⌉}atio\slash=G, by the Pigeon-Hole Principle, there are two distinct elements g1,g2∈G1such that g1H=g2H, i.e., g−1 2g1∈H. Since g−1 2g1is also an element ofG1, and is not the identity element, we have |G1∩H|>1. Suppose that Gis abelian, we know the order mof the hidden subgroup H, and m>1. Then we may ﬁnd a non-trivial element of Has follows. Consider any subgroup G1ofGof order n/m, where n:=|G|. Such a subgroup exists since Ghas a subgroup with order dfor any positive divisor dofn[8, Corollary 2.4, page 77]. Let S1,S2be a generating pair for G1, and let gbe any element of G\G1. By Lemma 5.1, either (i) there is a non-identity element h∈Hthat is also inG1, or (ii) G1H=G. In case (i), let h=a+b, where a∈S1and b∈S2. If we query the oracle fat the elements−xand y, for all x∈S1and y∈S2, we will ﬁnd f(−a) = f(b), and can compute h. In case (ii), we have g=g1+hfor some g1∈G1and h∈H. Since gis not in G1, the element his not the identity. Suppose g1=a+b, with a∈S1and b∈S2. Then g−a=b+h, and f(g−a) = f(b). If we query the oracle fat the elements g−xand y, for all

Question 1196 multiple-choice

In representation theory of Lie algebras, geometric constructions such as the Borel-Weil theorem and Fock space realizations play a crucial role in connecting algebraic structures to spaces of holomorphic functions. These frameworks utilize concepts like dominant weights, principal bundles, and differential operator realizations. Which statement correctly describes Kostant's realization of a Lie algebra representation in terms of differential operators on the Schubert cell associated with the identity element in the Weyl group? 1) Each element of the Lie algebra acts as a multiplication operator involving the Weyl group elements on the flag variety. 2) The representation is realized solely through right translations on the space of holomorphic sections over the whole group G. 3) For each positive root, the representation assigns a constant scalar operator on the corresponding coordinate. 4) The representation is given by second-order differential operators involving only the Cartan subalgebra elements. 5) The representation is given by first-order differential operators on Y, combining Killing vector fields from the group action and additional terms determined by the highest weight. 6) The cell Y is mapped isomorphically to the negative root subgroup, and the Lie algebra acts via automorphisms of the associated line bundle. 7) Each line bundle over Y corresponds to a direct sum of reducible representations of the Lie algebra.

✓ Correct Answer:

The correct answer is 5) The representation is given by first-order differential operators on Y, combining Killing vector fields from the group action and additional terms determined by the highest weight..

📚 Reference Text:

was already said above that Γ χis a representation space of g. In fact the Borel- Weil theorem states that for Λ χan dominant weight of gthe space of holomorphic sections Γ χofLχis isomorphic to the ﬁnite dimensional irreducible represe ntation ofg with highest weight Λ χ. If Λ χis not a dominant weight then the complex line bundle Lχhas no non-trivial holomorphic sections. The construction that leads to the Fock realization of the Li e algebragis very similar to the Borel-Weil construction. First one makes use of the fact that B−\G possesses a canonical decomposition, called the Bruhat dec omposition B−\G=/uniondisplay w∈WCw (6.8) whereWis the Weyl group of g. The spaces Cware called ’Schubert cells’ and are isomorphic to Cn. Denote from now on the Schubert cel corresponding to the uni t element ofWbyY. SinceY⊂B−\Gthe groupGacts onYby right multiplication. It can also be shown that Yis diﬀeomorphic to N+. This diﬀeomorphism is deﬁned by assigning to [ g]∈Ythe unique element x∈N+such that [x] = [g] (i.e. such that x=bgfor someb∈B−). The right G-action onYinduces a right G-action onN+via this diﬀeomorphism. Let’s from now on identify YandN+. The setB−N+is a submanifold of Gand the triple ( B−N+,Y,B −) is again a principle bundle (in fact it is nothing but the restriction o f the bundle ( G,B −\G,B −) to the cellY⊂B−\G. Let again χbe a character of B−and let ¯Lχbe the associated complex line bundle over Y. Denote the space of sections of this line bundle by ¯Γχ. It is a standard result (and easy to prove) that there exists a 1- 1 correspondence between ¯Γχand the set ¯Rχof holomorphic functions on B−N+such that f(b.g) =χ−1(b)f(g) (6.9) whereb∈B−andg∈N+. Since obviously the space of holomorphic functions on N+and¯Rχare isomorphic (a holomorphic function fonN+uniquely determines a holomorphic function ˆfonB−N+with the property (6.9) if we set ˆf(b.g) =χ−1(b)f(g)) we ﬁnd that ¯Γχ≃Hol(Y) (6.10) 50 In [30] Kostant showed that there exists a representation of the Lie algebra gofGin terms of ﬁrst order smooth diﬀerential operators on the mani foldY. Since the space Yis isomorphic to C|∆+|where |∆+|is the number of positive roots of gwe can choose a set of global coordinates z≡ {zα}α∈∆+onY. The representation σΛ:g→Diﬀ(Y) (6.11) ofgon the set of diﬀerential operators Diff(Y) onYis then given by σΛ(x) =ξ(x) +hΛ(x)x∈g (6.12) whereξ(x) is a Killing vector of the right G-action onY (ξ(x)f)(z) =d dtf(z.etx)|t=0z∈Y (6.13) and (hΛ(x)f) (z) =/a\}bracketle{tΛ∗,Ad zx/a\}bracketri}htf(z) (6.14) Here Λ ∗is the trivial extension of Λ ∈H∗togandfis some holomorphic function of Y. Note that in (6.14) we made use of the fact we have identiﬁed YandN+or else Adzwould not make sense. The subspace C[{zα}] ofHol(Y) is obviously invariant under σΛ(g) which means that we have obtained Fock realizations of the Lie algebra g(one for each Λ ∈H∗). Forsl2the Fock realizations σΛhave the form σΛ(e) =d dz σΛ(f) = (Λ,α)z−z2d dz σΛ(h) =1 2(Λ,α)−zd dz(6.15) whereαis the root of sl2. The expressions for arbitrary slncan be found in [14] and will not be given here. Using the results described

Question 1197 multiple-choice

In group theory, the study of nilpotent group extensions often involves analyzing the interplay between group elements, automorphisms, and commutator identities. Understanding how subgroups behave under the action of automorphisms is key to controlling the structure of such groups. In a split extension G = (A, b) where b satisfies b^p = 1 and A is a sparse group with homocyclic components, which of the following is the primary reason why proving [A₂, b]^p ⊆ Ap₁ is significant for understanding the nilpotency class of G? 1) It guarantees that every subgroup of G is cyclic. 2) It shows that the derived subgroup generated by commutators involving b and A₂ becomes trivial modulo Ap₁, ensuring bounded complexity. 3) It implies that b acts trivially on all elements of A. 4) It establishes that A is a simple group. 5) It ensures that G is abelian. 6) It proves that every element of G has order p. 7) It demonstrates that all commutator identities in G are linear.

✓ Correct Answer:

The correct answer is 2) It shows that the derived subgroup generated by commutators involving b and A₂ becomes trivial modulo Ap₁, ensuring bounded complexity..

📚 Reference Text:

at most p — 2. This fact will be used repeatedly: after establishing some statement for A we interpret it in AP and then by taking pk-\\\ roots we obtain an additional statement for A. We should point out at this stage that any statement we make will become a triviality should b happen to centralize some AP (as we will know must happen, once we have proved the theorem). Let us repeat here that A is sparse, that r n is the rank of A tv the homocyclic component of exponent p n, and that b p = 1. We will be working inside the split extension G = (A, b), and have to make a large number of observations along the way, which we number consecutively. We shall use repeatedly elementary commutator identities, often without comment. (1) [A u rxb] ^ A p n fl,(,4) = (A p2, A^\ . . . >. This is because inA/A py the subgroup AXAP/Ap is invariant under bA p, and of orderp n; so (A ]AP/AP, bA p) has class at most r x. Since (1) holds for all sparse groups it holds for A p so that (2) [A p29 r2b] ^ A p2 n Q}(Ap) = (A p\ Ap4\ . . . >. On taking/7-th roots in A, (2) gives (3) [A 2,r2b] ^ (A^A^A^Af,...)- When (3) is interpreted in A p, it gives (4) [A'i, r 3b] ë (A p2, AÇ, A$, A p5\ ...). Combining (1) and (2) gives (5) [A l9(rx + r 2)b] ^ A p\ When this is interpreted in A p it gives (6) \A\, (r 2 + r 3)b] â A"\ Combining (3) and (1) gives (7) [A 2, (r, + r 2)b] ^ (A" 2, A"3, A$, ...). When this is interpreted in A p it gives (8) [^, (r 2 + r 3)b\ fk (A p3\ APl A Pl,...). https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press NILPOTENT EXTENSIONS 1037 Now interpreting (7) in A pS " and taking p s 2-th roots, (9) [A s, (rs_, + r s)b] § - We can now proceed with the main steps of the proof. Step I. [A 2, b]p g Ap\ We know from (2) that [A2, r2b]p ^ A p\ If r2 = 0 or 1, we are done. Otherwise r 2 = 2. We now turn to the basic identity (2.6). As b p = 1, we have for every a ^ A 2 [a, bf[a, 2b]V' ..-[a,(p - \)b] p[a,pb] = 1. Commutating this with b r 2 — 2 times, [a, (r 2 - \)bf[a, r2b}^2' . .. [a, (r 2 + p - 3)b] p X [a, (r 2 + p - 2)b] = 1. Since p divides each binomial coefficient, it follows from this and the fact that [A2, r2bf =£ Apl 2 that all but the first and last terms of this identity are in A p

Question 1198 multiple-choice

Adaptive quantum circuits enable dynamic changes in algorithm execution by responding to measurement outcomes, while hypergraph representations offer advanced modeling for their structure. Efficient partitioning of these circuits is essential for optimizing performance on quantum hardware with limited connectivity. Which feature of hypergraph-based partitioning specifically enhances the efficiency of adaptive quantum circuits compared to traditional static methods? 1) The ability to encode classical control constraints, keeping qubits involved in adaptive operations grouped to minimize communication overhead 2) Exclusive use of bipartite graph structures to represent circuit gates and qubits 3) Simplifying all gate interactions to pairwise connections, reducing hyperedge complexity 4) Ignoring measurement outcomes during the partitioning process 5) Assigning each quantum gate to a separate vertex without grouping 6) Applying only static partitioning algorithms without adaptation for hypergraph constraints 7) Restricting partitioning to fixed topologies regardless of circuit adaptivity

✓ Correct Answer:

The correct answer is 1) The ability to encode classical control constraints, keeping qubits involved in adaptive operations grouped to minimize communication overhead.

📚 Reference Text:

Title: Hypergraphic representation for adaptive quantum circuits Year: 2025 Paper ID: 72f7c1bff189463b56a1339f559e7cb0aafdf893 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/72f7c1bff189463b56a1339f559e7cb0aafdf893 Abstract: Adaptive quantum circuits enhance flexibility and efficiency over traditional static circuits by dynamically adjusting their structure and parameters in real-time based on intermediate measurement outcomes. This paper introduces a novel hypergraph representation for adaptive quantum circuits, where groups of gates are considered as participants of hyperedges. By incorporating these gate groups into hyperedges, we create an extended hypergraph that includes constraints usable during the partitioning process. This approach guides the partitioning to maintain groups of ports associated with classical operations, ensuring that the resulting partitions prioritize qubits involved in the same sections of classical operations inherent to the adaptive approach. We present a new hypergraph partitioning algorithm based as an extension of Fiduccia-Mattheyses heuristic, to support hypergraphs created from adaptive quantum circuits. Comparative analysis between static and adaptive methods demonstrates the effectiveness of the proposed hypergraph techniques for adaptive circuits. Experimental results using benchmark quantum circuits validate our theoretical insights, showing improvements in circuit representation for partitioning heuristics. These findings highlight the practical benefits of hypergraph representation in adaptive quantum computing.

Question 1199 multiple-choice

Quantum circuit design plays a crucial role in the efficiency of quantum algorithms, impacting their speed and resource requirements. Innovations in circuits for fundamental operations can lead to substantial improvements in practical quantum computing applications. Which advancement enables both the HHL and Shor's algorithms to achieve better performance through reduced gate count and circuit depth? 1) Implementing error correction codes within the algorithm steps 2) Increasing the number of ancillary qubits used in each operation 3) Parallelizing quantum operations across multiple processors 4) Integrating classical pre-processing before quantum execution 5) Using randomized measurement strategies for output extraction 6) Designing a more efficient quantum circuit for the Quantum Fourier Transform 7) Applying post-selection to filter valid computational outcomes

✓ Correct Answer:

The correct answer is 6) Designing a more efficient quantum circuit for the Quantum Fourier Transform.

📚 Reference Text:

Title: A novel quantum circuit for the quantum Fourier transform Year: 2025 Paper ID: e1efb44e808d9326039ea6f62bf4263e06c4476b Source: semantic-scholar URL: https://www.semanticscholar.org/paper/e1efb44e808d9326039ea6f62bf4263e06c4476b Abstract: The Quantum Fourier Transform (QFT) is a fundamental component of many quantum computing algorithms. In this paper, we present an alternative method for factoring this transformation. Inspired by this approach, we introduce a new quantum circuit for implementing the QFT. We show that this circuit is more efficient than the conventional design. Furthermore, using this circuit, we develop alternative versions of the HHL algorithm and Shor's algorithm, which also demonstrate improved performance compared to their standard implementations.

Question 1200 multiple-choice

Locally maximally entangled (LME) states play a foundational role in quantum information theory, often constructed using group theoretical methods and representation theory. The Bell and GHZ states are canonical examples of such maximally entangled states for two and three quantum subsystems, respectively. Which of the following statements accurately describes a group-theoretic property enabling the construction of a GHZ state for three subsystems using the symmetric group S3? 1) S3 has a one-dimensional trivial representation whose square contains the identity. 2) The three-dimensional regular representation of S3 contains the Bell state as an invariant. 3) The triple tensor product of S3’s two-dimensional irreducible representation contains the trivial representation, guaranteeing an invariant GHZ state. 4) S3 acts transitively only on pairs of subsystems, so cannot be used for GHZ state construction. 5) Only abelian groups are suitable for constructing GHZ states via group representations. 6) The existence of a dual representation is required for constructing GHZ states with S3. 7) GHZ states require the use of SU(2) representations exclusively.

✓ Correct Answer:

The correct answer is 3) The triple tensor product of S3’s two-dimensional irreducible representation contains the trivial representation, guaranteeing an invariant GHZ state..

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Corol- lary 3.3, by ﬁnding a group ˆHthat is a central extension of Hand a representation of ˆH inU(d1)×···×U(dn) that descends to the representation described above under th e maps ˆH→H, (u1,...,un)→u1⊗···⊗un. For example, we can take ˆHto be the subgroup of K=U(d1)× ···×U(dn) generated by the elements ( u1(h),...,un(h)) forh∈Hand the representation to be the deﬁning one. Example 8 below will be of this type. 3.1 Examples Let us now describe various examples of LME states constructed in this way. Example 1: Bell states from dual representations Forn= 2, we can consider any group Hand any irreducible representation Racting on a vector space of dimension dand basiseias ei→Rij[h]ej. (3.3) TakeR∗to be the dual representation, acting on the dual vector space a s e∗ i→e∗ jRji[h−1] (3.4) 14 wheree∗ iare a basis for the dual vector space such that ( e∗ i,ej) =δij. The tensor product of the original vector space Vwith the dual vector space V∗is equivalent to the vector space of linear maps from V→V. The tensor product of RandR∗acts on this space, and always includes a copy of the trivial representation, namely the identity ma p 1≡/summationdisplay iei⊗e∗ i. (3.5) We can check that this is invariant under the combined transformat ions (3.3) and (3.4). In the language of quantum states, the expression (3.5) correspon ds to a normalized state |0/an}b∇acket∇i}ht=1√ d/summationdisplay i|i/an}b∇acket∇i}ht⊗|i/an}b∇acket∇i}ht, (3.6) which is exactly the Bell state (1.2). By choosing H= SU(2), we have examples for every positive integer d. We have seen above that this is the unique LME state for n= 2 up to local unitary transformations. Example 2: the GHZ state from S3 Moving on to tripartite systems, we ﬁrst show how the GHZ state (1 .3) can be obtained from the construction in Corollary 3.3. We need to ﬁnd a group Hwith two-dimensional irreducible representations such that the product of three of th ese contains the trivial rep- resentation. The smallest such group is S3, the permutation group on three elements; this has a unique two-dimensional irreducible representation, and the t ensor product of three of these contains the trivial representation, so this should allow us to construct an LME state in a three-qubit Hilbert space. Though we have already prove n that the GHZ state is the unique such state up to local unitary transformations (sectio n 2.3), it may be useful to demonstrate explicitly that we indeed ﬁnd the GHZ state. We can writ e the two-dimensional representation of S3explicitly via (1) =/parenleftbigg1 0 0 1/parenrightbigg ,(123) =/parenleftbiggω−10 0ω/parenrightbigg ,(132) =/parenleftbiggω0 0ω−1/parenrightbigg , (12) =/parenleftbigg0 1 1 0/parenrightbigg ,(23) =/parenleftbigg0ω ω−10/parenrightbigg ,(13) =/parenleftbigg0ω−1 ω0/parenrightbigg whereω=e2πi 3. It is then straightforward to check that the GHZ state (1.3) is inv ariant underS3acting via the tensor product of three copies of this representat ion: (1)⊗(1)⊗(1),(123)⊗(123)⊗(123),(132)⊗(132)⊗(132), (12)⊗(12)⊗(12),(13)⊗(13)⊗(13),(23)⊗(23)⊗(23). 15 More generally, the GHZ state is invariant under the subgroup of U(2)×U(2)×U(2) con- sisting of elements of the form /parenleftbigg eiα0 0eiβ/parenrightbigg ⊗/parenleftbigg eiθ0 0eiχ/parenrightbigg ⊗/parenleftbigg e−iα−iθ0 0e−iβ−iχ/parenrightbigg (3.7) and /parenleftbigg 0eiα eiβ0/parenrightbigg

Question 1201 multiple-choice

Topological quantum computing utilizes the mathematical properties of braid groups and anyon statistics to achieve fault-tolerant quantum operations. Algebraic frameworks such as the Yang-Baxter equation and the Temperley-Lieb algebra play a crucial role in constructing quantum gates that generate entangled states. Which algebraic construction enables the direct generation of multi-qubit entangled states like GHZ and cluster states from separable basis states, unifying the Hadamard and Bell matrices in quantum gate design? 1) Clifford algebra representations 2) Pauli group operators 3) Hecke algebra matrices 4) Symmetric group elements 5) Dirac operator formalism 6) Braiding operators derived from the Temperley-Lieb algebra 7) Unitary representations of the permutation group

✓ Correct Answer:

The correct answer is 6) Braiding operators derived from the Temperley-Lieb algebra.

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arXiv:1011.6229v1 [quant-ph] 29 Nov 2010Quantum entanglement, unitary braid representation and Temperley-Lieb algebra C.-L. Ho,1,2A.I. Solomon,3,4and C.-H. Oh1 1Department of Physics, and Center for Quantum Technologies National University of Singapore, 117543, Singapore 2Department of Physics, Tamkang University, Tamsui 251, Tai wan, R.O.C.∗ 3Department of Physics and Astronomy, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK 4LPTMC, Universit´ e Pierre et Marie Curie, 75252 Paris Cedex 05, France (Dated: Jun 3, 2010) Important developments in fault-tolerant quantum computa tion using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauﬀman and Lomonaco that a speciﬁc braiding operator from the solution of the Yang-Baxter equation, namely the Bell matri x, is universal implies that in principle all quantum gates can be constructed from braiding operator s together with single qubit gates. In this paper we present a new class of braiding operators fro m the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus un ifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operat ors, these new operators generate directly, from separable basis states, important entangled states s uchas the generalized Greenberger- Horne-Zeilinger states, cluster-like states, and other st ates with varying degrees of entanglement. Introduction. –Recentdevelopmentsinfault-tolerant quantum computation using the braiding of anyons [1], have stimulated interest in applying the theory of braid groupstotheﬁeldsofquantuminformationandquantum computation. In this respect, an interesting result is the realization that a speciﬁc braiding operator is a universal gate for quantum computing in the presence of local uni- tarytransformations[2]. Thisoperatorinvolvesaunitary matrixRthat generates the four maximally entangled Bell states from the standard basis of separable states. This has led to further investigation on the possibility of generating other entangled states by appropriate braid- ing operators [3–5]. In [4], unitary braiding operators were used to realize entanglement swapping and generate the Greenberger-Horne-Zeilinger (GHZ) state [6], as well as the linear cluster states [7]. Further generalizations of thebraidingoperatorstobipartitequantumsystemswith states of arbitrary dimension, i.e., qudits, were obtained by the approach of Yang-Baxterization [8, 9]. The GHZ state was not directly generated by the braiding operator in [4]. The resulting state was trans- formed, by use of a local unitary transformation, to the GHZ state. We argue here that this state does not, in fact, possess the same entanglement properties as the GHZ state. In this note we show how the Bell states, the generalized GHZ states and some cluster-like states may be generated directlyfrom a braiding operator. We adopt a diﬀerent approach, based on the Temperley-Lieb algebra (TLA) [10], to obtain a class of unitary represen- tationsofthe braidgroup, and with it the requiredbraid- ing operator. We ﬁrst obtain an explicit representation of the TLA, and then ﬁnd the braid group representation ∗Permanent addressvia the Jones representation [11]. Braid group and quantum entanglement. – The m-stranded braid group Bmis generated by a set of ele- ments{b1,b2,...,b m−1}with deﬁning relations: bibj=bjbi,|i−j|>1; bibi+1bi=bi+1bibi+1,1≤i

Question 1202 multiple-choice

Quantum algorithms have been widely studied for solving the hidden subgroup problem (HSP), which has broad implications in computational group theory and quantum complexity. Recent advances focus on non-abelian groups, including nilpotent groups of bounded class and prime factors, where group structure is crucial to algorithm design. Which of the following statements accurately describes a key feature of the exact quantum algorithm for the hidden subgroup problem in nilpotent groups of bounded class and bounded prime factors? 1) It requires exponentially many quantum resources in the size of the group. 2) It constructs subgroup states using zero-sum subsequences and operates in polynomial time and resources. 3) It is limited to abelian groups and cannot handle non-abelian group structures. 4) It relies solely on classical subgroup membership testing without quantum techniques. 5) It exclusively uses Kuperberg’s sieve without any subgroup state conversion. 6) It only provides approximate solutions for nilpotent groups of class two. 7) It cannot be reduced to the hidden subgroup membership conversion problem.

✓ Correct Answer:

The correct answer is 2) It constructs subgroup states using zero-sum subsequences and operates in polynomial time and resources..

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as some simpler ones) as subgroup state conversion . This conversion is based on ﬁnding zero sum 123 14 Page 4 of 23 M. Imran, G. Ivanyos subsequences of sufﬁciently long sequences of elements of L. Eventually, in c−1 rounds of iteration, where cis the nilpotency class of G, we compute a copy of the subgroup state corresponding to HG/prime/G/primeinG. (The semi-elementary property ensures the existence of a standard central series of length cwith elementary abelian factors.) Finally, from sufﬁciently many copies of such subgroup states we compute HG/prime/G/primeusing the exact abelian hidden subgroup algorithm of [ 16]. Fortunately, semi- elementary groups occur as factor groups of subgroups of nilpotent groups frequently enough to make a reduction from the HSP to the special case of HSMC possible, see Proposition 1for details. The main result we obtain is the following. Theorem 1 Suppose that G is a nilpotent group of class bounded by a constant and that the prime factors of |G|are also bounded by a constant. We assume that G is a black-box group with unique encoding of elements by /lscript-bit strings. Then there is an exact quantum algorithm that solves the hidden subgroup problem in G using poly(/lscript) operations and poly(log|G|)calls to the subgroup state creating oracle and its inverse. Related results In the exact setting, [ 16] efﬁciently solves the abelian case without the restriction on the prime factors of |G|. There are quite a few related non-exact polynomial-time algorithms. Among them, the result of [ 17], which solves the HSP insolvable groups that have derived series and exponent bounded by constants, is perhaps the closest to Theorem 1. This class of groups covers the groups for which our result is applicable, except those that have exponent divisible by large powers of small primes. Note however, that the case of these groups could be efﬁciently treated by a combination of the reduction of Proposition 1with the algorithm of [ 17]. We remark that the semidirect product group in which the HSP is equivalent to the hidden shift problem over Zn 2kis a nilpotent group of class k. Bonnetain and Naya-Plasencia [ 18] propose a non-exact method whose main ingredient can be considered as a combinationof Kuperberg’s sieve with ﬁnding zero sum subsequences in Z n 2using linear algebra. The case of nilpotency class at most two is efﬁciently treated by the non-exact method of [ 19], without any restriction on the size of the prime factors of |G|.I t is worth mentioning that by technical content, [ 19] can be considered as the closest relative of the present paper. The idea of reducing the HSP to HSMC stems from there, and many ingredients of the reduction appeared in that paper. Also, the key tool of[19], using several coset superpositions and the quantum Fourier transform of a central subgroup can be considered as some (though less transparent) form of subgroup state conversion. In the class two case, however, there is a more powerful tool to cancel outcharacters of the

Question 1203 multiple-choice

In quantum computing, the implementation of quantum gates on multilevel quantum systems (qudits) requires precise control over Hamiltonian dynamics and consideration of both operational fidelity and time constraints. Advanced numerical optimization methods play a critical role in designing control pulses that maximize gate performance. Which statement best explains why minimizing quantum gate implementation time is crucial for high-fidelity operations in experimental quantum computing? 1) It increases the number of basis states available for computation. 2) It allows for the realization of arbitrary global phases in gate operations. 3) It simplifies the energy level structure of the system Hamiltonian. 4) It reduces the exposure of quantum states to environmental decoherence. 5) It ensures gradient-based algorithms always find the global optimum. 6) It eliminates the need for numerical optimal control methods. 7) It guarantees that all control solutions converge to the same minimum time.

✓ Correct Answer:

The correct answer is 4) It reduces the exposure of quantum states to environmental decoherence..

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1 Global phase and minimum time of quantum Fourier transform for qudits represented by quadrupole nuclei V. P. Shauro* and V. E. Zobov Kirensky Institute of Physics, Russian Academy of Sciences, Siberian Branch, Akademgorodok 50, bld. 38, Krasnoyarsk, 6 60036 Russia (Received 22 January 2013) We demonstrate the relation between a global phase of the quantum gate and the layout of energy levels of its effective Hamiltonian required for implementing the gate for minimum time. By an example of the quantum Fourier transform gate for a qudit represented by a quadr upole nucle us with the spin I = 1, the effective Hamiltonians and minimum implementation times for different global phases are found. Using numerical optimal control methods , the problem of the globa l phase in search ing for the optimal pulse shape is considered in detail for the quantum Fourier transform gate at I = 1, 3/2, 2 , and 5/2 . It is shown that at the constrained control time the gradient a lgorithms can converge to the solutions corresponding to different global phases or the same global phase with different minimum times of the gate implementation . PACS number(s): 03.67. -a, 76.60. -k, 02.60.Pn I. INTRODUCTION Implementation of quantum algorithms requires performing basic quantum operations (gates) with maximum fidelity for minimum time [1, 2]. Quantum computations can be performed with the use of not only two -level (qubits) but also multilevel quantum systems (qudits) [3 −5]. The latter have a number of advantages; in particular, a specified size of the computational basis is provided by fewer qudits. In implementation of quantum algorithm s, one must take into account not only the operational complexity ( the number of gat es for algorithm execution [1 ]) but also the time complexity (the algorithm execution time ) [6−11]. The shorter the algorithm execution time , the smaller is the loss caused by the interaction with the environment. The time complexity of quantum logic operations is determined by the quantum system and the control method used. Generally, the existence of minimum (critical ) time cT for implementation of a quantum gate with an allowed error is the fundamental limitation imposed on the speed of quantum operations. The search for effective techniques for controlling quantum system s that implement the gates with maximum fidelity for minimum time is an important problem on the way to the creation of a full-scale quantum computer. In recent years, various numerical methods have become increasingly popular that allow calculat ing control fields for various tasks with relatively * E-mail: rsa@iph.krasn.ru 2 high efficiency . The well -known numerical GRAPE [12] and Krotov [13, 14] algorit hms are based on finding the minimum of a certain objective function al by using information on its gradient. Although the gradient methods yield only a loc al solution, these algorithms are successfully used for controlling quantum systems , since they often converge to the global minimum [15]. Moreover, in the absence of the control field constrain ts, the objective

Question 1204 multiple-choice

Integrated photonic quantum processing units are advancing the implementation of high-dimensional quantum logic using qudits, enabling complex operations and enhanced scalability. Multi-value controlled-unitary (MVCU) gates are central to manipulating entanglement and performing efficient quantum computation in such systems. Which experimental result offers the most direct quantitative evidence for high-fidelity operation of a four-level maximally entangled Bell state on a photonic quantum chip? 1) Quantum state fidelity measurements showing values above 0.95 for four-level Bell states 2) Resistance measurements of thermal-optic phase shifters 3) Interference visibility of Mach-Zehnder interferometers 4) Classical statistical fidelity of Pauli X4 gates 5) Measured truth tables for MVCX gates in different bases 6) Process fidelity of MVCXd gate at 0.952 7) Arbitrary single-qudit projective measurement capability

✓ Correct Answer:

The correct answer is 1) Quantum state fidelity measurements showing values above 0.95 for four-level Bell states.

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state, in which the operations are coherently entangled with the x-register state, thus forming the MVCU entangling gate, where the state-gate entanglement is indicated by the four colourful links; (V) coherent compression of Hilbert space by an indistinguishable erasure of spatial information; (VI) and (VII) arbitrary single -qudit projective measurement in the xand yregisters. Insets: left top, measured resistance of all thermal-optic phase shifters (TOPSs); measured interference visibility of all 2-dimensional Mach-Zehnder Interferometers (MZIs); bottom right, measured classical statistic ﬁdelities ( Fc) for the Pauli X4gate with a mean of 0.988(13) and Fourier F4gate with a mean of 0.967(19). dA microscopy image of the d-QPU chip. It monolithically integrates 451 optical components, including 4 SFWM sources, 116 recon ﬁgurable TOPS, 131 multimode interferometer (MMI) beamsplitters, 4 wavelength-division multiplexing (WDM) ﬁlters, 156 waveguide crossings and 40 grating couplers (GC). The d-QPU chip is wire bounded and can be ﬂexibly controlled by classical electronics, and can be reliably reprogrammed and recon ﬁgured to benchmark a spectrum of different quaternary quantum algorithms.ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-28767-x 4 NATURE COMMUNICATIONS | (2022) 13:1166 | https://doi.org/10.1038/s41467-022-28767-x | www.nature.com/naturecommunications Fig. 3 Characterisation of quaternary multi-value controlled-unitary logic operations. a ,bMeasured density matrices ( ρ) for a four-level maximally entangled Bell state and a fully product state. Column heights (colours) represent absolute values ∣ρ∣(phases Arg( ρ)) of the elements. Quantum state ﬁdelity Fqwas measured to be 0.983(4) and 0.953(3), respectively. c, Measured quantum state ﬁdelities for a complete set of four-level Bell states Ψjii;j, i,j=0,1,2,3. The generalised Bell states are created by operating input states of fi/C12/C12/C11 /C10kj/C12/C12/C11 in the MVCX dgate. Shaded areas atop bars refer to ± 1 σerror bars. The Fqvalues in a–cwere estimated by Monte Carlo the photon counts with photon Poissonian statistics. dReconstructed process matrix ( χ) of the MVCX dgate. It was measured by quantum process tomography with in total 256 quantum state tomographic measurements. We obtained the quantum process ﬁdelity of 0.952, that is de ﬁned as Tr[ χ0χ], where χ0is the ideal matrix. The χmatrix is represented in the standard identity and Pauli basis { I2, X2,Y2,Z2}. Blue and red colours are used to improve the clarity. e–jMeasured truth tables (normalised photon counts) for three MVCU logic gates in two complementary bases {I, II}: e,f, a multi-value controlled- Xd(MVCX d) gate; g,h, a multi-value controlled- Zd(MVCZ d) gate; i,j, a multi-value controlled- Hd(MVCH d) gate. The de ﬁnitions of basis are given as: computational basis ki/C12/C12/C11 ; Fourier basis fi/C12/C12/C11 ; Hadamard basis hi/C12/C12/C11 ; basis li/C12/C12/C11 is another eigenstate of the Hadamard and ai/C12/C12/C11 and bi/C12/C12/C11 are given by rotations, which are provided in Supplementary ( i={0, 1, 2, 3}). Classical statistic ﬁdelities ( Fc1,Fc2) are measured, which are adopted to estimate the lower and upper bound of the complementary classical ﬁdelity: [0.891(2), 0.931(1)] for the MVCX d, [0.912(2), 0.952(1)] for the MVCZ d, and [0.865(1), 0.920(1)] for the MVCH d.I ne–j, the probability distributions are colour coded (key is provided at the right bottom). The values in

Question 1205 multiple-choice

In modern algebra, group rings and their quotients by ideals are studied for their structural properties, often relating to representations and cohomology. The connection between group rings and exterior (Grassmann) algebras can reveal deep results about the freeness and basis of such quotients. Which of the following statements is true regarding the quotient ring Z/T, where T is a certain ideal and A is a group, in relation to the Grassmann (exterior) algebra E over Q? 1) Z/T is isomorphic to a non-free Abelian group with torsion elements. 2) Z/T, with addition, is a free Abelian group whose basis corresponds to wedge products of distinct generators in E. 3) Z/T is a non-Abelian ring with nilpotency class greater than 2. 4) The commutator subgroup of Z/T is trivial, making it a simple group. 5) Z/T cannot be embedded into any algebra over Q with anti-commuting generators. 6) All elements of Z/T are nilpotent under addition. 7) Z/T is generated by elements of the form e_i^2, where e_i are generators of E.

✓ Correct Answer:

The correct answer is 2) Z/T, with addition, is a free Abelian group whose basis corresponds to wedge products of distinct generators in E..

📚 Reference Text:

+ T = -[a h a,]m([a,, a,-]1"1 - 1) + T = -\m\[a i,aj]maai,aj)-\) + T which is equal, by (23), to -|m|([a f, a,] - 1) + T = m([a,, a,] - 1) + 7\ By (26), each element of the form (25) is a linear combination of elements (27) ([a,,, a i2] - 1) • • • ([a /2/_,, a,J - 1) + T, where, by (23), i p ^ /, for all p ^ q (\ Then E is (isomorphic to) the Grassmann (or exterior) algebra on a countably infinite- dimensional vector space over Q with a basis [e, ; | i e N}. It is well-known (and easy to prove) that the set [eH •••e k | k > 0, /, < i 2 < ••• < i k] is a basis of E over Q. Since e] = 0, elements 1 + e t (i € N) are invertable in E and (1 +

Question 1206 multiple-choice

Accurate parameterization of molecular force fields is critical for reliable molecular dynamics simulations in computational chemistry. Modern protocols combine quantum mechanical data and statistical sampling to optimize both intra- and intermolecular interactions. Which protocol is specifically employed to parameterize intermolecular force field terms by fitting to quantum mechanically derived interaction energies of molecular dimers obtained from Monte Carlo simulations? 1) Joyce protocol 2) Pickyroute II 3) RESP fitting 4) OPLS-AA 5) CHARMM-GUI 6) AMBER99 7) MMFF94

✓ Correct Answer:

The correct answer is 2) Pickyroute II.

📚 Reference Text:

previous works.22,32,33,40 i) For each compound, EFFintrais parameterized by means of the Joyceprotocol,22,33 using the optimized geometry, its Hessian matrix and, when re quired, a number of torsional energy scans as reference QM data. Further detail s of the intramolecular parameterization and objective function can be found in the Supporting Informa- tion. ii) Considering the scarce ﬂexibility of most of the investi gated compounds, the param- eterization of EFFinteris performed following Pickyroute II,40i.e.based on rigid MC simulations. Within this framework, the QMD-FF intermol ecular parameters are obtained through an iterative procedure, which consist s in minimizing at each step the squared diﬀerence between the QM interaction energ ies (∆Einter) com- puted over a set of dimer geometries extracted from MC simula tions and the ones 9 computed by means of the QMD-FF intermolecular term ( EFFinter), parameterized in the previous cycle. In other words, a non linear ﬁtting of t hefunctionalobjective function Iinter=/summationtextNgeom k=1[(∆Einter k−EFFinter k)2]e−α∆Einter k /summationtextNgeom k=1e−α∆Einter k(2) is performed, where Ngeomis the number of geometries considered for the target dimer,αa Boltzmann-like weight. The convergence of the iterative p rocedure is monitored by means of the quantity ∆ P, which is deﬁned as ∆P= 1 NgeomsNgeoms/summationdisplay i/bracketleftBig (EFFinter i)c−(EFFinter i)c−1/bracketrightBig2 1 2 (3) where (EFFinter i)cis the intermolecular energy computed with the QMD-FF param - eters obtained in cycle c. Thus ∆Pmeasures the diﬀerence between the intermolec- ular energies obtained with the FF parameters of in two conse cutive cycles. Further details for Pickyparameterization can be found in the original papers32,40and in the Supporting Information. iii) Once both the intra-molecular and the intermolecular c ontributions have been pa- rameterized, they are joined together to form a QMD-FF, suit able for MD simula- tions. iv) Eventually, extended MD simulations are performed with the resulting QMD-FF, on systems composed of a large number of molecules in the NPT en semble. Several macroscopic properties are then computed on the resulting b ulk phases, and the reliability and the accuracy of the employed QMD-FF is valid ated by comparing the simulated properties with those computationally obtai ned adopting literature FFs and with their experimental counterparts. 2.2 QM Calculations All QM calculation are performed with the Gaussian09 package .71Energy optimizations on isolated monomers are carried out for each of the compound s displayed in Figure 1 at 10 DFT level, employing the standard B3LYP functional with the D unning’s correlation con- sistent cc-pv-Dz basis set. The global minimum is obtained b y simultaneously optimizing all internal coordinates, storing the resulting Hessian mat rix, necessary for Joycepa- rameterization. When required, relaxed energy scans are pe rformed by an optimization of all degrees of freedom, but the scanned coordinate. Like the intramolecular term, also the intermolecular ener gy of a pair of monomers is computed at DFT level, but an empirical correction72is applied to the benchmarked functionals, to take dispersion interactions into account , exploring several combinations of functional, dispersion correction and basis set. Additio nal details and the speciﬁc combinations are given in Table

Question 1207 multiple-choice

Quantum error-correcting codes (QECC) are crucial for maintaining the integrity of quantum information in the presence of noise, and new classes of codes continue to be developed to enhance their performance and efficiency. Among these, graph codes and jump codes introduce novel approaches to encoding and decoding quantum data. Which statement accurately describes the equivalence between graph codes and stabilizer codes in quantum error correction? 1) Graph codes and stabilizer codes are both subsets of cyclic codes and thus always share the same code space. 2) Graph codes can only be constructed from unweighted graphs, while stabilizer codes require weighted graphs. 3) Stabilizer codes are a special case of graph codes with restricted adjacency matrices. 4) Only graph codes defined over non-prime finite fields have a stabilizer code equivalent. 5) Every graph code is a stabilizer code, and every stabilizer code can be represented as a graph code. 6) Stabilizer codes use only Pauli X operators, while graph codes use only Pauli Z operators. 7) The equivalence between graph codes and stabilizer codes holds only for classical error-correcting codes.

✓ Correct Answer:

The correct answer is 5) Every graph code is a stabilizer code, and every stabilizer code can be represented as a graph code..

📚 Reference Text:

aspects are still open. The main tasks are to ﬁnd methods for constructing good QECC and efﬁcient algorithms for encodingand decoding, including the correction of the errors. A large family of QECC can be derived from cyclic codes (see [GB99, GGB99, GB00]). Those QECC admit various techniques for encoding and decoding, e. g., based on Fouriertransformations over ﬁnite ﬁelds and quantum shift registers [BG00]. An interesting new classof QECC has been developed in cooperation with the group of G. Alber [ABC +01, AMD03]. Those so-called jump codes exploit side-information about the errors due to the emission of quanta (quantum jumps). The side-information is obtained by continuously monitoringthe quantum system and recording which subsystem emitted, e. g., a photon. An interestingconnection to design theory leads to various constructions of jump codes [BCG +03]. Another new concept of QECC linking various groups in the Schwerpunktprogramm QIV are so-called graph codes which have been introduced by D. Schlingemann and R. F. Werner [SW02]. Similar to the cluster states used in the one-way quantum computer of R. Raussendorf and H. J. Briegel [RB01], the basis states of a graph code are deﬁned via an interaction graph corresponding to the spin–spin coupling Hamiltonian which can be usedfor encoding. Compared to the standard gate model of quantum computation, such a couplingHamiltonian yields a very efﬁcient—intrinsically parallel—algorithm. We have shown that the concepts of graph codes and that of so-called stabilizer codes are equivalent, i. e., any graph code is a stabilizer code and vice versa [GKR02,Sch02,WSR +02]. In the following, we will present the main ideas used in the proof. Starting from an α-dimensional Hilbert space H, a graph code is an αk-dimensional sub- spaceQofH⊗nwhich is spanned by the vectors |x /angbracketright=1 √ αn/summationdisplay y∈Fn pm/parenleftBigk+n/productdisplay j=1j−1/productdisplay i=1χ(zi,zj)Γij/parenrightBig |y/angbracketright. (1.1) 4 1 Algorithms for Quantum Systems — Quantum Algorithms Here the dimension αis a prime power, i. e., α=pmfor some prime number p. The basis states |y/angbracketrightofH⊗nare labeled by vectors y∈Fn pmover the ﬁnite ﬁeld Fpm, and the encoded states |x /angbracketrightare labeled by vectors x∈Fk pm. The coefﬁcients on the right hand side are given by the values of a non-degenerate symmetric bicharacter χon Fpm×Fpm,w h e r e z=(x, y). Finally, the exponents Γijare given by the adjacency matrix Γof a weighted undirected graph with integral weights (corresponding to the coupling strength between the subsystems iand j). Equivalently, the states (1.1) of the graph code Qcan be expressed in the form |x /angbracketright=1 √ αn/summationdisplay y∈Fn pmζ(q(x+y))|y/angbracketright, (1.2) where ζis a non-trivial additive character of Fpandqis a quadratic form on Fk+n pm∼= Fm(k+n) p . This isomorphism shows that it is sufﬁcient to consider graphical quantum codes which are deﬁned over prime ﬁelds. First we show that a graph code deﬁned over the ﬁnite ﬁeld Fpis a stabilizer code, i. e., a joint eigenspace of an abelian subgroup of the error group Ggenerated by Xa:=/summationdisplay y∈Fnp|y+a/angbracketright/angbracketlefty|,andZd:=/summationdisplay y∈Fnpexp(2πi/p)dty|y/angbracketright/angbracketlefty|, fora, d∈Fn p(cf. [KR02a,KR02b]). Starting from (1.2) one can show that the stabilizer of Q is given

Question 1208 multiple-choice

Quantum algorithms often require efficient implementation on real hardware, where device connectivity and circuit complexity play crucial roles in determining experimental success. The transpilation process adapts quantum circuits to fit the constraints of specific quantum devices, affecting both circuit size and fidelity. Which of the following statements best describes the relationship between transpiled circuit size and experimental accuracy on noisy intermediate-scale quantum (NISQ) devices? 1) Larger transpiled circuit size always leads to higher experimental fidelity. 2) Smaller transpiled circuit size guarantees the lowest ℓ1 distance from the ideal outcome. 3) Circuit depth is irrelevant to experimental accuracy when optimizing for gate count. 4) Bidirectional coupling maps eliminate all tradeoffs between compactness and fidelity. 5) Sometimes the smallest transpiled circuit does not yield the most accurate experimental results due to tradeoffs between compactness and fidelity. 6) All NISQ devices achieve maximal accuracy when the circuit uses the fewest gates possible. 7) Transpilation cannot affect the fidelity of quantum experiments.

✓ Correct Answer:

The correct answer is 5) Sometimes the smallest transpiled circuit does not yield the most accurate experimental results due to tradeoffs between compactness and fidelity..

📚 Reference Text:

circuit implementation of quantum walks , Phys. Rev. A 79 (2009), 052335. doi:10.1103/PhysRevA.79.052335. [11] P. Shor, Polynomial-Time Algorithms for Prime Factorization and Di screte Logarithms on a Quantum Computer , SIAM Journal on Computing 26(1997), no. 5, 1484–1509. doi:10.1137/S0097539795293172. [12] M.NielsenandI.Chuang, Quantum Computation and Quantum Information: 10th Anniver sary Edition , Cambridge University Press, New York, 2010. doi:10.1017/CBO9780 511976667. [13] J. Welch, D. Greenbaum, S. Mostame, and A. Aspuru-Guzik, Eﬃcient quantum circuits for diagonal unitaries without ancillas , New Journal of Physics 16(2014), no. 3, 033040. doi:10.1088/1367-2630/16/3/033040. EFFICIENT AND SCALABLE QUANTUM WALK ALGORITHMS 17 [14] A. Barenco, C. Bennett, R. Cleve, D. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, Elementary gates for quantum computation , Phys. Rev. A 52(1995), 3457–3467. doi:10.1103/PhysRevA.52.3457. [15] M. Saeedi and M. Pedram, Linear-depth quantum circuits for n-qubit Toﬀoli gates with no ancilla , Phys. Rev. A87(2013), 062318. doi:10.1103/PhysRevA.87.062318. [16]IBM Quantum Experience , 2019. url:quantum-computing.ibm.com. [17]Qiskit: An Open-source Framework for Quantum Computing , 2019. url:qiskit.org. 18 EFFICIENT AND SCALABLE QUANTUM WALK ALGORITHMS Appendix A.Coupling maps, layouts and samllest transpiled circuit siz es A.1.Coupling maps. These are the IBM quantum computer coupling maps for the ma- chines inthispaper. Thedirectional arrowsshowsource-targetp airforcontrolledoperations. Arrows in both directions mean both qubits can serve as the contro l and target. 01 2 3 4 Figure 9. ibmqx2 coupling map 01 2 3 4 Figure 10. ibmqlondon coupling map 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Figure 11. ibmq16melbourne coupling map EFFICIENT AND SCALABLE QUANTUM WALK ALGORITHMS 19 A.2.Layouts for QFT based shift. Here we show the layout maps for each of the ex- periments with QFT based shift. The qubits and couplings in black are in volved in the respective circuit. 01 2 ibmqx22 0 1 ibmqlondon1 2 0 ibmq16melbourne Figure 12. QFT based shift, 4 sites, 1 step. 01 2 ibmqx22 0 1 ibmqlondon2 0 1 ibmq16melbourne Figure 13. QFT based shift, 4 sites, 2 steps. 02 1 3 ibmqx213 0 2 ibmqlondon1 2 0 3 ibmq16melbourne Figure 14. QFT based shift, 8 sites, 1 step. 20 EFFICIENT AND SCALABLE QUANTUM WALK ALGORITHMS A.3.Smallest transpiled circuit sizes. The smallest circuit sizes that were obtained during simulations are given below for the respective experiments. T hese are not always the same as the circuits that gave the least ℓ1distance from the ideal distribution shown in the main body of the paper. (size , depth) ibmqx2 (11, 7) ibmqlondon (21, 15) ibmq16melbourne (25, 16) Table 4. QFT based shift, 4 sites, 1 step (size , depth) ibmqx2 (18, 11) ibmqlondon (38, 29) ibmq16melbourne (48, 30) Table 5. QFT based shift, 4 sites, 2 steps (size , depth) ibmqx2 (47, 33) ibmqlondon (59, 38) ibmq16melbourne (78, 45) Table 6. QFT based shift, 8 sites, 1 step

Question 1209 multiple-choice

Quantum circuit optimization seeks the most efficient sequence of elementary gates to accomplish a computational task, but the number of possible circuits grows rapidly with circuit length and gate set size. Probabilistic search strategies are increasingly used to address the combinatorial complexity of this process. In a Monte Carlo graph search for quantum circuit optimization, which mechanism ensures that nodes representing more promising circuits are preferentially expanded during the search process? 1) Uniform random sampling of nodes regardless of their quality scores 2) Selecting nodes based exclusively on their chronological order of creation 3) Expanding only nodes that have already led to a solution 4) Assigning each node a selection probability proportional to its quality score, normalized across all nodes 5) Randomly choosing gates without considering node fitness 6) Using edge weights exclusively to determine which nodes to expand 7) Applying exhaustive evaluation to all possible nodes at each step

✓ Correct Answer:

The correct answer is 4) Assigning each node a selection probability proportional to its quality score, normalized across all nodes.

📚 Reference Text:

set of 20 elementary quantum gates can be assembled to build 205combina- tions for quantum circuits of length 5. Thus it is not possible to evaluate all configurations in a feasible time to solve for a specific optimization task. Since the graph model generated by the eval- uation of quantum gates can have cycles, we apply a Monte Carlo search on the graph model with quantum gates as transitions. Thus, given a specific task, every node will receive a qual- ity score which is used to compute a probability for the a selection of this node. Based on the random selection and already explored opera- tions, a new operation is randomly selected to grow the graph. Once a solution is found, an efficient quantum circuit can be generated by computing the shortest path in the graph from the start node to the target node.This strategy is formalised as follows: We have a set of vertices V– associated with quan- tum circuits – of a graph G= (V;E) with V={v1, . . . v n}and we build a probability function p(vi),P ip(vi) = 1, which assigns to each node of the graph a probability for se- lection. Poisson sampling is then exploited as the underlying sampling process. It is assumed that each vertex of the graph is an independent Bernoulli trial. Following standard mathemati- cal conventions, the first-order inclusion proba- bility of the ith element of the graph is denoted by the symbol πi=p(vi). We further asso- ciate to each vertex of the graph an underlying task specific quality score si≥0. (Here larger values of siimply better quality.) Accordingly, we compute the first-order inclusion probability viaπi=siP jsj. This paradigm of Monte Carlo Search [44] and adapted Gibbs sampling [45] is used to iteratively grow a graph containing the effects of ordered quantum operations as trajec- tories in this graph, see Figure 1. Our MCGS procedure is now as follows: Given a set of vertices V={v1, . . . v n}and a probability function p(vi), Poisson sampling is used to select an existing node vifrom the graph. A quantum gate is then randomly se- lected from the set OPof elementary gates ac- cording to a uniform distribution and applied (from the left) to the quantum circuit associ- ated with vi. The result is a new circuit, which is either associated with an existing node or a new one [46]. If the circuit at a node already ex- ists, e.g. vj, a new edge ( vi, vj) is added to the graph, decorated with edge weight given by the applied elementary gate. If the quantum circuit does not exist, the graph is extended by adding a new node vN+1and an edge ( vi, vN+1). Since the probabilities vary with respect to the qual- ity score of the nodes, the graph grows asym- metrically. Once a node is reached which (suffi- ciently accurately) solves the optimization task, a shortest path from v1to the target node gives the shortest available quantum circuit, see also Figure

Question 1210 multiple-choice

In the study of m-edge-coloured graphs, switching operations based on group actions are a powerful tool for understanding graph transformations and symmetries. The computational complexity of deciding when such switchings can achieve certain properties depends strongly on the underlying group structure. Which non-Abelian group is commonly studied for switching operations in m-edge-coloured graphs and is known for its role in permutation problems, with recent dichotomy theorems classifying the complexity of associated colouring and homomorphism decision problems? 1) The cyclic group $C_m$ 2) The symmetric group $S_m$ 3) The Klein four-group $V_4$ 4) The quaternion group $Q_8$ 5) The trivial group $\{e\}$ 6) The abelian group $Z_m$ 7) The direct product group $S_2 \times S_3$

✓ Correct Answer:

The correct answer is 2) The symmetric group $S_m$.

📚 Reference Text:

Title: Switching m-edge-coloured graphs using non-abelian groups Year: 2022 Paper ID: d3c264512dba84184fd702949afc334067d6d930 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d3c264512dba84184fd702949afc334067d6d930 Abstract: Let $G$ be a graph whose edges are each assigned one of the $m$-colours $1, 2, \ldots, m$, and let $\Gamma$ be a subgroup of $S_m$. The operation of switching at a vertex $x$ with respect $\pi \in \Gamma$ permutes the colours of the edges incident with $x$ according to $\pi$. There is a well-developed theory of switching when $\Gamma$ is Abelian. Much less is known for non-Abelian groups. In this paper we consider switching with respect to non-Abelian groups including symmetric, alternating and dihedral groups. We first consider the question of whether there is a sequence of switches using elements of $\Gamma$ that transforms an $m$-edge-coloured graph $G$ to an $m$-edge coloured graph $H$. Necessary and sufficient conditions for the existence of such a sequence are given for each of the groups being considered. We then consider the question of whether an $m$-edge coloured graph can be switched using elements of $\Gamma$ so that the transformed $m$-edge coloured graph has a vertex $k$-colouring, or a homomorphism to a fixed $m$-edge coloured graph $H$. For the groups just mentioned we establish dichotomy theorems for the complexity of these decision problems. These are the first dichotomy theorems to be established for colouring or homomorphism problems and switching with respect to any group other than $S_2$.

Question 1211 multiple-choice

Quantum algorithms have revolutionized computational group theory by offering efficient solutions to problems involving normal subgroups, coset superpositions, and the hidden subgroup problem. These algorithms utilize quantum superposition, Fourier transforms, and oracles to extract group-theoretic information exponentially faster than classical methods. Which critical property must an oracle function possess in a quantum algorithm for efficiently solving the Abelian Hidden Subgroup Problem? 1) It must map each group element to a unique classical bitstring. 2) It must be invariant under all automorphisms of the group. 3) It must be injective when restricted to each subgroup element. 4) It must output uniform random values for each input. 5) It must be constant on cosets of the hidden subgroup and send distinct cosets to orthogonal quantum states. 6) It must commute with the quantum Fourier transform. 7) It must only operate on cyclic groups of prime order.

✓ Correct Answer:

The correct answer is 5) It must be constant on cosets of the hidden subgroup and send distinct cosets to orthogonal quantum states..

📚 Reference Text:

substituting the gene rators in T into the relators, and let N0stand for the normal closure (the smallest normal subgroupc ontaining) ofR0. ThenN=N0sinceN0≤NandG/N0=G/Nby deﬁnition of TandR0. Still some care has to be taken since it is possible that Tgenerates Gonly modulo N, that is it might generate a proper subgroup of G.Therefore some additional elements should be added to R0. LetSbe the generating set for G. Using the constructive membership test for G/N,we express the original generators from SmoduloNwith straight line programs in terms of the elements of T. For each element x∈Swe form the quotient y−1xwhereyis the element obtained by substituting the generators from Tinto the straight line program for xmoduloN. LetS0be the set of all the quotients formed this way. Note that TandS0generate together G. Then one can verify that the normal closure of R0∪S0inGisN. Thus,from R0andS0wecanﬁndgeneratorsfor Nintimepolynomialintheinput size+ ν(G/N) using the normal closure algorithm of [1]. We obtained the de sired result. 4.3 Unique encoding and solvable normal subgroup We conclude this section with some results obtained as combi nation of the ideas presented above with those of Watrous described in [27]. Assume that the enco ding of the elements of Gis unique and a normal solvable subgroup NofGis given by generators. We use the encoding of Gfor that of G/N. The identity test in G/Ncan be implemented by an eﬃcient quantum algorithm for testi ng membership in Ndue to Watrous (Theorem 2). We are also able to produce (sever al copies of) the uniform superposition |N/a\}bracketri}ht=1√ |N|/summationtext x∈N|x/a\}bracketri}hteﬃciently. For solvable subgroups N, we can again apply the result of Watrous (Theorem 2) to produce |N/a\}bracketri}htin polynomial time. We will now show that having suﬃciently many copies of |N/a\}bracketri}htat hand, we can use ideas of Watrous for computing orders of elements of G/Nand even for performing the constructive membership test in Abelian subgroups of G/N. Thus, we will have an eﬃcient quantum implementation of the Beals-Babai algorithms for G/N. We will ﬁrst state a lemma which says that we can eﬃciently so lve the HSP in an Abelian group if we have an appropriate quantum oracle. Lemma 9. LetAbe an Abelian group, and let Xbe a ﬁnite set. Let H≤A, and let f:A→CX (given by an oracle) such that: 7 1. For every g∈A,|f(g)/a\}bracketri}htis a unit vector, 2.fis constant on the left cosets of H,and maps elements from diﬀerent cosets into orthogonal states. Then there exists a polynomial time quantum algorithm for ﬁnd ing the hidden subgroup H. Proof.First we extend naturally ftoG/H: on a coset of H, it takes the value f(h) for an arbitrary member hof the coset. The algorithm is the standard quantum algorith m for the Abelian hidden subgroup problem. We repeat several times the following ste ps to ﬁnd a set of generators for H. •Prepare the initial superposition: |1G/a\}bracketri}ht|0m/a\}bracketri}ht. •Apply the Abelian quantum Fourier transform in Aon the ﬁrst register:/summationtext g∈A|g/a\}bracketri}ht|0m/a\}bracketri}ht. •Callf:/summationtext g∈A|g/a\}bracketri}ht|f(g)/a\}bracketri}ht. •Apply again the Fourier transform in A:/summationtext g∈A/H,h∈H⊥χh(g)|h/a\}bracketri}ht|f(g)/a\}bracketri}ht. •Observe the ﬁrst register. By hypothesis, the states |f(g)/a\}bracketri}htare orthogonal

Question 1212 multiple-choice

In quantum information theory, the fidelity of quantum cloning processes is a key measure of how closely a cloned state matches the original, with mathematical tools used to analyze and visualize these fidelity regions. The constraints and symmetries of quantum states play an important role in determining achievable fidelities. For a universal quantum cloner operating on a Hilbert space of dimension d, what is the value of the ideal fidelity that can be achieved according to standard quantum information theory? 1) d 2) 1 3) 1/d 4) d/2 5) 2/d 6) 1/(d+1) 7) d^2

✓ Correct Answer:

The correct answer is 3) 1/d.

📚 Reference Text:

with irreps V0 (2)(k n1)from formula (9) we obtain following set of equations: F(2) 12=1 18d a2 1(d1)2p 3a1a2(d1) +2p 2a1a3p d1p d+2+3a2 2(d1)2p 6a2a3p d1p d+2+2a2 3(d+2) , F(2) 13=1 18d a2 1(d1) +2a1p 3a2(d1) +p 2a3p d1p d+2 +3a2 2(d1) +2p 6a2a3p d1p d+2+2a2 3(d+2) , F(2) 14=1 9d 2a2 1(d1)2p 2a1a3p d1p d+2+a2 3(d+2) . (55) 15 The similar set of equations we can also obtain for partition (1, 1). Moreover we know that the ﬁdelity from ideal Nis always equal to 1/ d(see Fact 2). In next step we use Mathematica software to generate parametric plots of regions given by formulas of the form (55) together with normalization condition a2 1+a2 2+a2 3=1. Thanks to this we get admissible range of ﬁdelities in every irreducible space labeled by partition ai. Due to Theorem 3 to obtain admissible region of ﬁdelities we have to generate convex hull of allowed regions obtained for every irreducible representation a. To do this we have used Mathematica package ConvexHull3D . One can see that to generate admissible regions for number of clones larger than 3 we need higher-dimensional space to embed convex hull, so we can not represent our results in the graphical form. There is still some way to omit this problem at least partially. Namely we can construct some projection which maps convex hulls from ddimensional space to 3dimensional space, but then of course we lose some information. [1] W. K. Wootters and W. H. ˙Zurek, Nature 299, 802 (1982). [2] D. Dieks, Phys. Lett. A 92, 271 (1982). [3] V . Bu ˘zek and M. Hillery, Phys. Rev. A 54, 1844 (1996). [4] D. Bruß, D. P . DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchiavello, and J. A. Smolin, Phys. Rev. A 57, 2368 (1998). [5] N. Gisin and S. Massar, Phys. Rev. Lett. 79, 2153 (1997). [6] R. F. Werner, Phys. Rev. A 58, 1827 (1998). [7] M. Keyl and R. F. Werner, J. Math. Phys. 40, 3283 (1999). [8] Y.-N. Wang, H.-D. Shi, Z.-X. Xiong, L. Jing, X.-J. Ren, L.-Z. Mu, and H. Fan, Phys. Rev. A 84, 034302 (2011). [9] S. L. Braunstein, V . Bu ˘zek, and M. Hillery, Phys. Rev. A 63, 052313 (2001). [10] N. J. Cerf, J. Mod. Opt. 47, 187 (2000). [11] J. Fiurášek, R. Filip, and N. J. Cerf, Quant. Inform. Comp. 5, 583 (2005). [12] S. Iblisdir, A. Acin, N. J. Cerf, R. Filip, J. Fiurášek, and N. Gisin, Phys. Rev. A 72, 042328 (2005). [13] S. Iblisdir, A. Acin, and N. Gisin, Generalised Asymmetric Quantum Cloning (2005), arXiv:quant-ph/0505152. [14] A. Kay, D. Kaszlikowski, and R. Ramanathan, Phys. Rev. Lett. 103, 050501 (2009). [15] A. Kay, R. Ramanathan, and D. Kaszlikowski, Quant. Inf. Comp. 13, 0880 (2013), arXiv:quant-ph/1208.5574. [16] M. Jiang and S. Yu, J. Math. Phys. 51, 052306 (2010). [17] V . Scarani, S. Iblisdir, N. Gisin, and A. Acin, Rev. Mod. Phys. 77, 1225 (2005). [18] H. Fan, Y.-N. Wang, L. Jing, J.-D. Yue, H.-D.

Question 1213 multiple-choice

Computational spectroscopy of biomolecules often requires specialized methods to accurately model vibrational spectra, especially in systems with strong hydrogen bond networks like those found in membrane proteins. Efficient approaches must balance quantum mechanical accuracy with computational cost, particularly for large and dynamic systems. Which of the following strategies is essential for achieving accurate infrared spectra of internal water molecules in membrane proteins with pentagonal hydrogen bond networks, while maintaining computational efficiency? 1) Using only molecular mechanics for all regions of the system 2) Applying the harmonic approximation to all vibrational modes 3) Restricting quantum mechanical treatment to a single water molecule 4) Omitting structural sampling from molecular dynamics simulations 5) Employing small basis sets without dispersion corrections 6) Dividing the system into group-localized vibrational coordinates, utilizing a sufficiently large quantum mechanical region including the hydrogen bond network and its partners, and incorporating anharmonic effects with molecular dynamics structural sampling 7) Neglecting anharmonicity in hydrogen-bonded vibrational modes

✓ Correct Answer:

The correct answer is 6) Dividing the system into group-localized vibrational coordinates, utilizing a sufficiently large quantum mechanical region including the hydrogen bond network and its partners, and incorporating anharmonic effects with molecular dynamics structural sampling.

📚 Reference Text:

Title: Anharmonic Vibrational Calculations Based on Group-Localized Coordinates: Applications to Internal Water Molecules in Bacteriorhodopsin Year: 2021 Paper ID: da568804eaaf2999e0e46cabfbda5ac73e2bba27 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/da568804eaaf2999e0e46cabfbda5ac73e2bba27 Abstract: An efficient anharmonic vibrational method is developed exploiting the locality of molecular vibration. Vibrational coordinates localized to a group of atoms are employed to divide the potential energy surface (PES) of a system into intra- and inter-group contributions. Then, the vibrational Schrödinger equation is solved based on a PES, in which the inter-group coupling is truncated at the harmonic level while accounting for the intra-group anharmonicity. The method is applied to a pentagonal hydrogen bond network (HBN) composed of internal water molecules and charged residues in a membrane protein, bacteriorhodopsin. The PES is calculated by the quantum mechanics/molecular mechanics (QM/MM) calculation at the level of B3LYP-D3/aug-cc-pVDZ. The infrared (IR) spectrum is computed using a set of coordinates localized to each water molecule and amino acid residue by second-order vibrational quasi-degenerate perturbation theory (VQDPT2). Benchmark calculations show that the proposed method yields the N–D/O–D stretching frequencies with an error of 7 cm–1 at the cost reduced by more than five times. In contrast, the harmonic approximation results in a severe error of 150 cm–1. Furthermore, the size of QM regions is carefully assessed to find that the QM regions should include not only the pentagonal HBN itself but also its HB partners. VQDPT2 calculations starting from transient structures obtained by molecular dynamics simulations have shown that the structural sampling has a significant impact on the calculated IR spectrum. The incorporation of anharmonicity, sufficiently large QM regions, and structural samplings are of essential importance to reproduce the experimental IR spectrum. The computational spectrum paves the way for decoding the IR signal of strong HBNs and helps elucidate their functional roles in biomolecules.

Question 1214 multiple-choice

Quantum control optimization for half-integer spin systems involves tailoring control pulses to implement gates with desired global phases and minimal error. The critical operation time and solution landscape are influenced by mathematical symmetries relating to spin and phase. In quantum gate optimization for half-integer spin systems, which of the following best explains why gates with global phases differing by integer multiples of π can be executed in the same minimal critical time? 1) Group-theoretical properties of half-integer spins cause the operator associated with a π phase shift to be implementable instantaneously, resulting in identical minimal gate times for phase differences of π. 2) The control pulses for gates with different global phases are always identical, eliminating time differences for phase-offset gates. 3) The optimization algorithm always converges to the same solution regardless of global phase, ensuring equal gate times. 4) Only integer-spin systems exhibit phase symmetry in gate critical times, not half-integer spins. 5) Secondary solutions with different critical times are never found for half-integer spin systems. 6) The gate error is independent of both global phase and control time for half-integer spins. 7) Pareto fronts for phase pairs differing by π never coincide in half-integer spin systems.

✓ Correct Answer:

The correct answer is 1) Group-theoretical properties of half-integer spins cause the operator associated with a π phase shift to be implementable instantaneously, resulting in identical minimal gate times for phase differences of π..

📚 Reference Text:

usua lly have significantly different pulse shapes; thus, the formation of the steps in Figs. 2 and 4 is not related to the multiple convergence to the same pulse. The presence of secondary solutions with the same global phase and different critical times is ob served more clearly in the minimization of error 1J (22) for the QFT gate with a fixed global phase , e.g. /8 , at random initial guesses [Fig. 2 (c)]. It can be seen that in this case, only about 10% of the runs converge to the main solutions with gate error 8 110J and two families of secondary solutions with large errors are observed at given control time. (Notice that while 1J is minimizing the solutions correspon ding to other global phases should have large error 11/ 2J for 5 π/8 and 13 π/8 or 11J for 9 π/8. Therefore the probability of convergence to such solutions is extremely low , if any ). 11 The ratio between the number of run s converging to the main and secondary solutions may differ significantly for different phases and control times T. It is inherent in both 1J and 2J scenarios of optimization . Nevertheless, f or the model ( 11) the variation of control time T did not result in a significant decrease in the probability of convergence to the secondary solutions in both scenarios. The results of optimization shown in Fig. 2 were used in establishing the time dependence of the gate error by the PFT method for different global phases at I = 3/2. The Pareto fronts are shown i n Fig. 3. In addition to the main solutions with the shortest critical time for each global phase , several Pareto fronts for the secondary solutions with global phases π/8 and 5π/8 are shown . The shortest time cT = 1.62/q corresponds to global phases π/8 and 9π/8. It can be seen in Fig. 3 that in the case N = 4 the Pareto fronts for global phases π /8 (5π/8) and 9π /8 (13π/8 ) coincide pairwise within the calculation error . This also concerns the secondary solutions for the corresponding global phase s. This c oincidence of the critical time for the gates with the global phase s different by  reflects the general properties of half-integer spins. Indeed, since for half-integer spins the equality 2 ()ziI iii G G G e U e U e e U      is valid and the operator 2ziIe can be realized by a strong field for a negligible time , the minimum time of the gates with differ ent signs should be the same. The same conclusion can be made after consideration of the gates with global phases p and p such as pp   . From expression (2), we have 2 2ppNp p p pN           Since p and p are integer s, this equality is valid only for

Question 1215 multiple-choice

Quantum control theory integrates advanced mathematical frameworks with physical concepts to manipulate and measure quantum systems. Techniques in this field are crucial for implementing reliable operations in quantum computing and information processing. Which mathematical structure is essential for analyzing the controllability of quantum systems by determining which state transitions are possible via available control Hamiltonians? 1) Lie algebras and Lie groups 2) Tensor networks 3) Symplectic manifolds 4) Hilbert-Schmidt operators 5) Fock spaces 6) Gauge theories 7) Von Neumann algebras

✓ Correct Answer:

The correct answer is 1) Lie algebras and Lie groups.

📚 Reference Text:

Title: Introduction to Quantum Control and Dynamics Year: 2007 Paper ID: ad3f9e582c0aa2fec5080f256a04a56008cbd5f2 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/ad3f9e582c0aa2fec5080f256a04a56008cbd5f2 Abstract: QUANTUM MECHANICS States and Operators Observables and Measurement Dynamics of Quantum Systems MODELING OF QUANTUM CONTROL SYSTEMS: EXAMPLES Quantum Theory of Interaction of Particles and Fields Approximations and Modeling: Molecular Systems Spin Dynamics and Control Mathematical Structure of Quantum Control Systems CONTROLLABILITY Lie Algebras and Lie Groups Controllability Test: The Dynamical Lie Algebra Notions of Controllability for the State Pure State Controllability Equivalent State Controllability Equality of Orbits OBSERVABILITY AND STATE DETERMINATION Quantum State Tomography Observability Observability and Methods for State Reconstruction LIE GROUP DECOMPOSITIONS AND CONTROL Decompositions of SU(2) and Control of Two Level Systems Decomposition in Planar Rotations Cartan Decompositions Levi Decomposition Examples of Application of Decompositions to Control OPTIMAL CONTROL OF QUANTUM SYSTEMS Formulation of the Optimal Control Problem The Necessary Conditions of Optimality Example: Optimal Control of a Two Level Quantum System Time Optimal Control of Quantum Systems Numerical Methods for Optimal Control of Quantum Systems MORE TOOLS FOR QUANTUM CONTROL Selective Population Transfer via Frequency Tuning Time Dependent Perturbation Theory Adiabatic Control STIRAP Lyapunov Control of Quantum Systems ANALYSIS OF QUANTUM EVOLUTIONS: ENTANGLEMENT, ENTANGLEMENT MEASURES, AND DYNAMICS Entanglement of Quantum Systems Dynamics of Entanglement Local Equivalence of States APPLICATIONS OF QUANTUM CONTROL AND DYNAMICS Nuclear Magnetic Resonance Experiments Molecular Systems Control Atomic Systems Control: Implementations of Quantum Information Processing with Ion Traps APPENDIX A: POSITIVE AND COMPLETELY POSITIVE MAPS, QUANTUM OPERATIONS, AND GENERALIZED MEASUREMENT THEORY Positive and Completely Positive Maps Quantum Operations and Operator Sum Representation Generalized Measurement Theory APPENDIX B: LAGRANGIAN AND HAMILTONIAN FORMALISM IN CLASSICAL ELECTRODYNAMICS Lagrangian Mechanics Extension of Lagrangian Mechanics to Systems with Infinite Degrees of Freedom Lagrangian and Hamiltonian Mechanics for a System of Interacting Particles and Field APPENDIX C: CARTAN SEMISIMPLICITY CRITERION AND CALCULATION OF THE LEVI DECOMPOSITION The Adjoint Representation Cartan Semisimplicity Criterion Quotient Lie Algebras Calculation of the Levi Subalgebra in the Levi Decomposition Algorithm for the Levi Decomposition APPENDIX D: PROOF OF THE CONTROLLABILITY TEST OF THEOREM 3.2.1 APPENDIX E: THE BAKER-CAMPBELL-HAUSDORFF FORMULA AND SOME EXPONENTIAL FORMULAS APPENDIX F: PROOF OF THEOREM 6.2.1 REFERENCES INDEX Notes and Exercises appear at the end of every chapter.

Question 1216 multiple-choice

Quantum algorithms have demonstrated the ability to efficiently solve algebraic problems in black-box semigroups, impacting cryptographic primitives such as the discrete logarithm problem. These algorithms often blend quantum techniques like Fourier transforms with classical search strategies. In the context of quantum algorithms for black-box semigroups, which sequence of steps enables efficient computation of the discrete logarithm of an element x with respect to an element g? 1) Quantum period-finding to determine the period and index of g, distinguishing whether x is in the tail or cycle, followed by classical binary search to compute the discrete logarithm if x is in the tail 2) Classical exhaustive search over all powers of g, recording when x appears 3) Quantum amplitude amplification to directly locate x without period or index computation 4) Quantum walk algorithms for traversing the semigroup structure to find x 5) Preparation of entangled states encoding all possible powers, followed by measurement to extract x's position 6) Classical computation of all pairwise products in the semigroup to map its structure before searching for x 7) Random sampling of elements and comparison with x until a match is found

✓ Correct Answer:

The correct answer is 1) Quantum period-finding to determine the period and index of g, distinguishing whether x is in the tail or cycle, followed by classical binary search to compute the discrete logarithm if x is in the tail.

📚 Reference Text:

algorithms.) Lemma 2.1. There is an efﬁcient quantum algorithm to determine the index and the period of an element gof a black-box semigroup. Proof. First we ﬁnd the period, as follows. Create the statePM jD1jjijgji=p M for some sufﬁciently large M(it sufﬁces to take M >N2CN). Note that we can computegjefﬁciently even for exponentially large jusing repeated squaring, so this state can be made in polynomial time. Next, we discard the second register. To understand what happens when we do this, suppose we measure the second register. If we obtain an element in the tail of g, then the ﬁrst register is left in a computational basis state, which is useless. However, with probability at least .MtC1/=M1N=M , we obtain an element in the cycle of g, and we are left with anr-periodic state 1p LL1X jD0jx0Cjri 408 A. M. Childs and G. Ivanyos g g2g3gt−1gt=gt+r gt+1 gt+2gt+r−2gt+r−1 Figure 1. The semigroup hgi. for some unknown x02¹t;tC1;:::;tCr1º, whereLis eitherb.Mt/=rc ord.Mt/=re(depending on the value of x0). This is precisely the type of state that appears in Shor’s period-ﬁnding algorithm (see for example [5, Algorithm 5]). After Fourier transforming this state over ZMand measuring, we obtain the out- comek2ZMwith probability Pr.k/D1 LM L1X jD0e2ik.x 0Cjr/=M 2 Dsin2.krL M/ LM sin2.kr M/: A simple calculation (see for example [5, equations (57)–(60)]) shows that the probability of obtaining a closest integer to one of the rinteger multiples of M=r is at least4=2. By efﬁcient classical postprocessing using continued fractions, we can recover rwith constant probability by sampling from such a distribution [19]. Since we are in the cycle of gwith overwhelming probability, the overall procedure succeeds with constant probability (which could be boosted by standard techniques). Given the period of g, we can ﬁnd its index by an efﬁcient classical procedure. Observe that we can efﬁciently decide whether a given element gjis in the tail or the cycle of g: simply compute grby repeated squaring and multiply by gjto computegjCr. IfgjCrDgj, thengjis in the cycle of g; otherwise it is in the tail ofg. Let .gj/´´ 1ifgjCrDgj(i.e.,gjis in the cycle of g), 0otherwise (i.e., gjis in the tail of g): The list. .g/; .g2/;:::; .gN//consists oft1zeros followed by NtC1 ones, so we can ﬁnd tinO.logN/iterations by binary search. Quantum computation of discrete logarithms in semigroups 409 3 Computing discrete logarithms We now show how to efﬁciently compute discrete logarithms in semigroups on a quantum computer. Theorem 3.1. There is an efﬁcient quantum algorithm to compute loggxon input x;g2S(or to determine if no such value exists). Proof. First, we use Lemma 2.1 to compute the index tand the period rofg. Then we determine whether xis in the tail or the cycle of g. As described in the proof of Lemma 2.1, this can be done efﬁciently by determining whether xgrDx. Ifxis in the tail of g, then we compute p, the smallest positive integer such that .xgp/D1. This can be done efﬁciently by using binary search to ﬁnd the ﬁrst 1 in the list. .xg/;

Question 1217 multiple-choice

Quantum walks are quantum analogs of classical random walks and are implemented on quantum computers to investigate quantum algorithm performance. The accuracy of these experiments is influenced by hardware features such as qubit connectivity, gate noise, and circuit characteristics. Which factor most directly enables a quantum device to implement quantum walk circuits with smaller size and depth, resulting in lower deviation from the ideal probability distribution? 1) Higher qubit connectivity allowing more direct qubit interactions 2) Increased number of decoherence events during computation 3) Greater leakage of information from the computational subspace 4) Lower initial state selection precision 5) Reduced number of available qubits on the device 6) Use of circuits with intentionally increased gate noise 7) Absence of transpilation before running circuits

✓ Correct Answer:

The correct answer is 1) Higher qubit connectivity allowing more direct qubit interactions.

📚 Reference Text:

For each experiment, we group together the plots of th e distributions for all the machines, and also the ideal distribution tha t the QW state would have starting from the same state and after the same number of e volutionary steps. A table after each group of plots records the circuit size, the circuit depth, and the ℓ1distance between the ideal distribution and the ones shown. We point out tha t the circuits that give the distributions with the smallest ℓ1distance from the ideal do not always have the smallest transpiled size. We record the smallest circuit sizes in Appendix A Sect ion A.3. The ﬁrst set of plots are for N= 4 sites (n= 2), and 1 step of walk evolution. Recall that the joint velocity-position state is |v,x/an}⌊ra⌋ketri}ht. As a reminder, the joint velocity-position state is |v,x/an}⌊ra⌋ketri}ht. For example, the state |011/an}⌊ra⌋ketri}hthas|v/an}⌊ra⌋ketri}ht=|0/an}⌊ra⌋ketri}ht, and|x/an}⌊ra⌋ketri}ht=|11/an}⌊ra⌋ketri}ht. The initial walk state for this set of experiments is chosen to be |v,x/an}⌊ra⌋ketri}ht=|010/an}⌊ra⌋ketri}ht. The ideal plot shows that the walk has moved both right and the left b y 1 lattice point, having been scattered in both a left and a right moving component. T he measured distribu- tionsshowgoodagreementwiththeexpecteddistributionforIBMQ X2andIBMQX London, but there are other states sampled as well . The distribution from I BM16Melbourne has stronger deviation from the ideal.5The circuit layouts for each case are in Appendix A Section A.2. 4For the terminology in this paper related to the IBM quantum comput ers, the reader might wish to look up IBM quantum computing website [16]. 5We can speculate about the causes and sources of the deviations f rom the ideal. The circuit size and depth, gate noise, leakage, actual sequence of operations and d ecoherence, and other factors, are likely. EFFICIENT AND SCALABLE QUANTUM WALK ALGORITHMS 13 ideal 000 001 010 011 100 101 110 1110.000.150.300.450.60Probabilities 0 0 00.500 00.500 0 0ibmqx2 000 001 010 011 100 101 110 1110.000.150.300.45Probabilities 0.0090.035 0.0090.475 0.0090.447 0.006 0.011 ibmqlondon 000 001 010 011 100 101 110 1110.000.150.300.45Probabilities 0.0310.059 0.0460.440 0.0260.356 0.021 0.021ibmq16melbourne 000 001 010 011 100 101 110 1110.00.10.20.30.4Probabilities 0.0590.090 0.0670.343 0.0340.327 0.046 0.034 Figure 6. QFT based shift, 4 sites, 1 step The table below summarizes the circuit size and depth, and the ℓ1distance of the mea- sured distribution from the ideal distribution. It shows that diﬀere nt architectures transpile to diﬀerent sizes (gate counts) and depths. Comparison with the g ate coupling maps in Ap- pendix A conﬁrms the highest connectivity gate map, which is that of IBMQX2, corresponds to the lowest circuit size and depth, and the least deviation from the ideal distribution. (size , depth) ℓ1distance from ideal ibmqx2 (11, 7) 0.0781 ibmqlondon (26, 18) 0.2031 ibmq16melbourne (33, 21) 0.3301 Table 1. QFT based shift, 4 sites, 1 step The next set of plots and the table are for N= 4 sites ( n= 2), for 2 steps of walk evolution. Initial state of the walk is |v,x/an}⌊ra⌋ketri}ht=|010/an}⌊ra⌋ketri}ht. 14 EFFICIENT AND SCALABLE QUANTUM WALK ALGORITHMS ideal 000 001 010

Question 1218 multiple-choice

In algebraic geometry and number theory, unit points often refer to points on a variety whose coordinates are units in a given ring, and these points are closely related to the structure of multiplicative groups of units. Their properties and normalization techniques are central to the classification and study of algebraic objects over rings. Which of the following statements accurately describes the normalization of unit points in the affine space Ak over a ring O, in connection with the multiplicative unit group U? 1) Unit points must have at least one coordinate equal to zero. 2) The support of a unit point always contains non-unit entries. 3) Unit points can be normalized so that the first coordinate X₀ equals 1, with all coordinates being units. 4) The group of unit points is isomorphic to the additive group of O. 5) Normalizing unit points requires all coordinates to be equal. 6) Only points with non-empty support qualify as unit points. 7) The bijection between unit points and U fails if O is a field.

✓ Correct Answer:

The correct answer is 3) Unit points can be normalized so that the first coordinate X₀ equals 1, with all coordinates being units..

📚 Reference Text:

q^ (d) θa(X) = Σ δjZ1 where i^ = 4^ + as and ord. &7 = Σj tj(2tJ + aj) ord ?j (a) is immediate from the definitions of S(P). We call such points unit points; in the next section we study them carefully. We get (c) and (d) by specifying m to be 1 or 2 in the remarks before Prop. II.l.l. To prove (b) we use: LEMMA II.2.3. Let 0 Φ qeJί and yek* with |ord. y\ < \ord. q. Let a e {0,1}, t e Z and set s = tit + a) ord. q + (2t + a) ord. y. Then: (1) if a = 0, s > 0 (2) i/ α = 1, s> —|ord. y\. For ord. y>0 (respectively ord. y <0) equal- ity occurs if and only if t = — 1 (respectively t = 0). Proof. (1) is trivial. In order to prove (2) note that if ord. y > 0 then s > (2(ί + I)2 - 1) ord. j/, and if ord. y < 0, s > (2ί2 - 1) |ord. y\. LEMMA Π.2.4. Suppose y = (T/J, •—,yg)eGg with |ord. ^| < J ord. g^. S = {j: ord. ^ ^ 0}. Suppose a5 e {0,1}. Tfcew: ord.λa(y)> -Σlord.^|. jes Furthermore, equality holds if αΓ^l) = S and inequality holds if D -fi S. Proof. By (c), λa(y) = Σ bTyz where https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press 82 HORACIO TAPIA-RECILLAS ord. fay1) = Σ^ = Σ tβj + *j) ord. qό + (2tj + a/) ord. ys . So by Lemma II.2.3, ord. (b^1) > — Σljes lord. yό\ giving . Suppose now that a~\l) = S. Then there is precisely one monomial bjy1 in ;iα(#) such that ord. (b^1) = —Σ./es lord 101 (whenever ord. ^ = 0, tά = 0. When ord. ^ > 0, ίj, = —1 and when ord. y3 < 0, t3 = 0). Thus equality holds in . Finally, if a~\T) ~fi S, there is an index j such that ccj = 0 and ord. ^ =£ 0. Then, Sj > 0 > — |ord. ^| and the last assertion follows. Remark (b) is an immediate consequence of Lemma II.2.4 and Theorem II.2.2. (note that iP(λ2a) = λa(y)2 up to multiplication by a non- zero constant independent of a). §IL3 The unit points of Ak Let U denote the multiplicative group of units of the ring O and Uk be the set of unit points of Ak (i.e. points with empty support). In this section we show that φ induces a bijection U9 —> Uk. The injectivity of

Question 1219 multiple-choice

Quantum computation and abstract algebra intersect in the analysis of projectors within finite-dimensional semisimple associative algebras, which play an important role in gauge theory and holographic dualities. Representation theory, especially for symmetric groups, is essential for constructing operators and understanding physical observables in these systems. In the context of finite-dimensional semisimple associative algebras related to symmetric groups, which statement best describes the function of combinatorial bases formed from permutation equivalence classes? 1) They exclusively generate non-unitary operators used in quantum error correction protocols. 2) They serve to label half-BPS operators in conformal field theory without influencing the central elements of the algebra. 3) They underpin the construction of central elements which, upon exponentiation, yield unitary operators suitable for quantum computational tasks. 4) They are used to approximate structure constants in infinite-dimensional Lie algebras. 5) They enable the encoding of classical communication channels in quantum protocols but are not related to algebra decomposition. 6) They determine the physical scaling with the gauge group rank $ N $ in gravity theories without involving projectors. 7) They provide a basis for Clebsch-Gordan coefficients but do not interact with the representation theory of symmetric groups.

✓ Correct Answer:

The correct answer is 3) They underpin the construction of central elements which, upon exponentiation, yield unitary operators suitable for quantum computational tasks..

📚 Reference Text:

representations 23 5.2.3 More general choices of V 24 6 Detection of the Kronecker projector in K(n) 24 7 Detection of the Littlewood-Richardson projector 28 A Representation theory lemmas used in quantum-inspired classical algo- rithms 31 B Solving the power sums Mk 31 C Proof of a property of Kronecker projectors 32 – i – JHEP05(2023)1911 Introduction and outlook With motivations from the holographic AdS/CFT correspondence [ 1–3] and other gauge- string dualities (e.g. [ 4,5]), we consider the quantum computational task of detecting projectors in ﬁnite dimensional associative algebras. These algebras we consider are semi- simple and have a Wedderburn-Artin (WA) decomposition as a direct sum of matrix algebras (see a general exposition of the WA decomposition in, e.g. [ 6]). They are related to symmetric groups Snof all permutations of nobjects. The WA decompositions can be constructed using representation theory data from symmetric groups, such as characters of irreducible representations (irreps), matrix elements of permutations in these irreps and branching or Clebsch-Gordan coeﬃcients. The projectors of interest act as the identity in a ﬁxed matrix block of the WA decomposition and zero elsewhere. They are thus labelled by the representation theory data specifying the matrix blocks. Given their relation to symmetric groups, the algebras also have combinatorial bases related to equivalence classes of permutations. These combinatorial bases are used to construct central elements in the algebras of interest. They are exponentiated to give unitary operators in Hilbert spaces formed by the algebras themselves. Our detection task proceeds with the standard technique of quantum phase estimation (QPE) [ 7], using these unitary operators as quantum gates which are applied to query the projectors. The set-up is usefully viewed in the context of a quantum communication between Bob, who builds the projector and sends it to Alice, who identiﬁes the projector using QPE. Combining results on query complexities from quantum information theory and facts from the representation theory of Sn, we arrive at complexity estimates which are bounded by polynomials in n. The AdS/CFT correspondence gives a conjectured equivalence between string theoretic quantumgravityin AdS 5×S5space-timeandthenon-gravitationalquantumtheoryof N= 4 SYM theory in four dimensions with U(N)gauge group. There are numerous successful tests of the correspondence for interactions of perturbative gravitons (see [ 8] and the review [ 9]). Further interesting tests of AdS/CFT involve the quantum states associated with half-BPS giant gravitons [ 10] and those associated with half-BPS supergravity geometries (LLM geometries) [ 11]. The labelling of half-BPS CFT operators using Young diagrams and the computation of their correlators [ 12,13] plays an important role in the tests involving the interactions of giant graviton branes [ 14–21] and the dynamics of strings in LLM geometries [ 22]. The construction of the CFT operators labelled by Young diagrams and the computation of their correlators for CFT operators of scaling dimension nuses projection operators in the centre of the group algebra C(Sn)of the symmetric group Sn. TheN- dependence of the correlators is controlled by the structure constants of the centre Z(C(Sn)) of the

Question 1220 multiple-choice

Quantum algorithms are often studied for their ability to solve computational problems involving functions and group structures more efficiently than classical algorithms. The hidden shift problem is one such challenge, notable for its connection to Boolean functions and Fourier analysis. Which property governs the efficiency of the quantum algorithm for finding the hidden shift in Boolean functions? 1) The function's degree of non-linearity 2) The minimum influence of the Boolean function 3) The number of variables in the Boolean function 4) The presence of periodic subgroups in the group 5) The cryptographic strength of the function 6) The total number of Fourier coefficients 7) The symmetry of the underlying group structure

✓ Correct Answer:

The correct answer is 2) The minimum influence of the Boolean function.

📚 Reference Text:

Title: Quantum algorithm for the Boolean hidden shift problem Year: 2011 Paper ID: 83503f134ab42ce309685c7d752486da49095394 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/83503f134ab42ce309685c7d752486da49095394 Abstract: The hidden shift problem is a natural place to look for new separations between classical and quantum models of computation. One advantage of this problem is its flexibility, since it can be defined for a whole range of functions and a whole range of underlying groups. In a way, this distinguishes it from the hidden subgroup problem where more stringent requirements about the existence of a periodic subgroup have to be made. And yet, the hidden shift problem proves to be rich enough to capture interesting features of problems of algebraic, geometric, and combinatorial flavor. We present a quantum algorithm to identify the hidden shift for any Boolean function. Using Fourier analysis for Boolean functions we relate the time and query complexity of the algorithm to an intrinsic property of the function, namely its minimum influence. We show that for randomly chosen functions the time complexity of the algorithm is polynomial. Based on this we show an average case exponential separation between classical and quantum time complexity. A perhaps interesting aspect of this work is that, while the extremal case of the Boolean hidden shift problem over so-called bent functions can be reduced to a hidden subgroup problem over an abelian group, the more general case studied here does not seem to allow such a reduction.

Question 1221 multiple-choice

In representation theory and quantum computing, the Fourier transform over finite groups is a crucial tool for decomposing representations into irreducible components. Induced representations, permutation modules, and quantum Fourier transforms all play important roles in leveraging group structure for efficient computation. When block diagonalizing an induced representation from a subgroup H to a group G using quantum algorithms, which of the following steps specifically transforms the basis from |t, v⟩ into a basis labeled by irreducible representations and their multiplicities? 1) Measuring the state in the original group element basis 2) Applying the Hadamard transform to all registers 3) Performing a quantum Fourier transform over the subgroup H only 4) Projecting onto the trivial representation of the subgroup 5) Performing a quantum Fourier transform over the full group G 6) Swapping the transversal and irrep registers 7) Discarding the ancilla register after embedding

✓ Correct Answer:

The correct answer is 5) Performing a quantum Fourier transform over the full group G.

📚 Reference Text:

log(1//epsilon1)). 3.3 Fourier transform over induced representations As mentioned earlier, the usual Fourier transform is a unitary operator, which changes the basis from the basis of group elements {|g/angbracketright|g∈G}to the block diagonal form given in (2.1). In this subsection, we use this transform to construct a Fourier transform for induced representations i.e., a transform that block 10 diagonalizes induced representations. It turns out that the Fourier transform for induced representations allows us to construct the dual Schur transform. Suppose we have an induced representation from HtoGof an irreducible representation σofHi.e., we have↑G Hσ. The computational basis for this space can be written as |t,v/angbracketright, wheretis an element of the transversal and vis a basis vector in the representation space of σ. This induced representation is in general reducible as a representation of Gand can be decomposed as a sum of irreducible representations. The multiplicity of each irreducible representation ρin the decomposition is equal to, using Frobenius reciprocity, the multiplicity of σin the restriction of ρtoH. We now describe how one can perform this transform. The change of basis we want to implement is from the basis labeled |t,v/angbracketrightto a block diagonal basis labeled, say|λ,i,j/angbracketright. Hereλlabels the irreps that appear in the decomposition and iandjare the multiplicity and the irrep space basis vectors. 1. First, we append an ancilla register of size |H|/dim(σ)(which is an integer) to the initial state so that we have a register of size H(excluding the transversal register). We now embed |v/angbracketrightinto this register of sizeHas|σ,u,v/angbracketright, whereulabels the multiplicity space of dimension dim( σ). 2. Next, perform the inverse QFT over Hto get the group basis |h/angbracketright. So including the transversal register, we have|t,h/angbracketright. Now, write this as |g/angbracketright. For the symmetric group, it turns out that this is easy since bothtandhare speci ed in terms of adjacent transpositions. However, if the group basis for certain groups is de ned in a complicated way, then this transform might be non-trivial. 3. Now perform the QFT over Gto obtain the basis |λ,i,j/angbracketright. The label iis the multiplicity index, which runs over the entire dimension of λ. However, for an induced representation, it would run over a smaller set in general. Similarly for the irrep label. If the irrep and the multiplicity index labels can be ordered in such a way that the ones that occur in the decomposition are higher in the ordering, then we can easily return the ancilla register and obtain a clean transform. For the goal of block diagonalization, this ordering is not so important. However, we will see that this can be done for the case of the symmetric group. 3.4 Quantum Fourier transform over permutation modules As explained above, permutation modules are induced representations from the trivial representation of some Young subgroup YTtoSn. The Young subgroup YTcan be regarded as the stabilizer group of some ntupleE= (e1,...,en)of typeTwith entries ei∈[d]. This representation space is spanned by all possible permutations of E. In order to construct the Fourier transform over this space, we rst

Question 1222 multiple-choice

Programmable quantum processors are devices that can implement various quantum channels by applying different program states, with resource requirements determined by the complexity of the channels to be implemented. The conversion of mixed program states to pure program states often affects the necessary dimension of the program register. When purifying mixed program states to pure states for a programmable quantum processor with original program register dimension $ d_P $, what is the first standard upper bound for the required dimension of the enlarged program register? 1) $ d_P + 1 $ 2) $ 2d_P $ 3) $ d_P^2 $ 4) $ d_P^3 $ 5) $ d_P/2 $ 6) $ d_P^2 + 1 $ 7) $ d_P \min\{D, d_P\} $

✓ Correct Answer:

The correct answer is 3) $ d_P^2 $.

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Then we call P∈ CPTP (H1⊗HP,H2), with ﬁnite-dimensional HP, an/epsilon1-programmable quantum processor for a setC⊂CPTP (H1,H2)of channels ( /epsilon1-PQPC), if for every quantum channel T∈C there exists a state πT∈D(HP)such that 1 2/bardblP(·⊗πT)−T(·)/bardbl/diamondmath≤/epsilon1. Accepted in Quantum2021-06-18, click title to verify. Published under CC-BY 4.0. 5 Figure 2: The ﬁgure shows a CPQP UVwith its input, program and output register. To address the Hilbert spaces H1,H2andHP, we refer to the former two as the input and output registers and to the latter as program register. We say that the processor P /epsilon1-implements the classCof channels; for /epsilon1= 0we say thatPexactly implements the class C, and address it as a PQPC. In particular, when C=TUV, we write CPQP UVfor the covariant programmable quantum processor and /epsilon1-CPQPUVfor the approximate version thereof. To allow for potentially mixed states in the program register is natural, since the set TUVis convex, whereas the set of pure states is not. Remark 13. In the literature, where the set of channels Coften consists of isometries or even unitaries, it is customary to impose that the program state πTis pure. While this does not aﬀect the upper bounds on the program dimension (allowing mixed states cannot make it more diﬃcult to program a processor compared to pure states), it is in particular an essential assumption for some lower bounds when unitaries are implemented [27, 47]. On the other hand, it does not aﬀect our lower bounds based on the properties of the Holevo information. It is however always possible to modify a given processor P∈CPTP (H1⊗HP,H2)to a new processorP/prime∈CPTP (H1⊗H/prime P,H2)such that every channel implemented by P, i.e. T/epsilon1=P(·⊗πT), is implemented by P/primeusing a pure program state, i.e. T/epsilon1=P/prime(·⊗|ψT/angbracketright/angbracketleftψT|) for|ψT/angbracketright∈H/prime P. The idea is to purify πTon a larger system, which gives rise to two upper bounds ond/prime P= dimH/prime P. To obtain the ﬁrst, every state πTonHPcan be puriﬁed to a pure state|ψT/angbracketright∈HP⊗HP=:H/prime Pand we can letP/prime:=P◦(idH1⊗HP⊗trHP), showing that d/prime P≤d2 P. The second upper bound is a reﬁnement of the ﬁrst, starting from the observation that we can decompose a mixed program state into a convex combination of pure states, πT=/summationtext ipi|ψTi/angbracketright/angbracketleftψTi|, where each|ψTi/angbracketright/angbracketleftψTi|implements a channel Ti=P(·⊗|ψT/angbracketright/angbracketleftψT|), and hence T/epsilon1=/summationtext ipiTiis a convex combination of channels. The quantum channels CPTP (H1,H2)form a convex set in a space of dimension d2 1(d2 2−1), which can be seen from the Choi-Jamiołkowski isomorphism, hence by Carathéodory’s theorem Tcan be written as a convex combination of not more than D:=d2 1(d2 2−1)+1of theTi, i.e.T/epsilon1=/summationtextD j=1qjTij, withqj≥0summing to 1. Thus, the state π/prime T:=/summationtextD j=1qj|ψTi/angbracketright/angbracketleftψTi|, which has rank ≤r:= min{D,dP}, implements T/epsilon1as well:T/epsilon1=P(·⊗π/prime T). But now we only need an r-dimensional Hilbert space HRto purifyπ/prime T= trHR|ψT/angbracketright/angbracketleftψT|,|ψT/angbracketright∈HP⊗HR. As before, we can let the processor be P/prime:=P◦(idH1⊗HP⊗trHR), showingd/prime P≤dPmin{D,dP}. We conclude by noting that the ﬁrst method is good if dPis already small, whereas the second is preferred if dPis large. The former is the case if we consider unitarily covariant channels (see Example 20). The latter is the case if C=TUVand the

Question 1223 multiple-choice

In modular group algebras FG, where G is a finite p-group and F is a finite field, the normalized unitary subgroup V_* is defined using a classical involution and plays a significant role in algebraic structures. The isomorphism problem investigates whether the structure of V_* is uniquely determined by the group algebra FG for certain classes of groups. For which group classes has it been established that the normalized unitary subgroup V_* is uniquely determined by the structure of the modular group algebra FG? 1) Finite abelian p-groups, 2-groups of maximal class, and non-abelian 2-groups of order at most 16 2) All finite simple groups 3) All finite p-groups with non-trivial center 4) All non-abelian groups of exponent two 5) Symmetric groups and cyclic groups of prime order 6) Finite solvable groups of odd order 7) Only abelian groups of order less than 8

✓ Correct Answer:

The correct answer is 1) Finite abelian p-groups, 2-groups of maximal class, and non-abelian 2-groups of order at most 16.

📚 Reference Text:

Title: Isomorphism problem of Unitary Subgroups of Group Algebras Year: 2019 Paper ID: eb6fce02e8092e1db388b106464397dcdf9dc2db Source: semantic-scholar URL: https://www.semanticscholar.org/paper/eb6fce02e8092e1db388b106464397dcdf9dc2db Abstract: Let V_* be the normalized unitary subgroup of the modular group algebra FG of a finite p-group G over a finite field F with the classical involution *. We investigate the isomorphism problem for the group V_*, that asks when the group V_* is determined by its group algebra FG. We confirm it for classes of finite abelian p-groups, 2-groups of maximal class and non-abelian 2-groups of order at most 16.

Question 1224 multiple-choice

Quantum computers offer novel methods for processing multi-dimensional image data, enabling efficiency gains through quantum encoding and parallel transformations. Quantum amplitude encoding and quantum transforms are key techniques used to achieve exponential speed-ups for image processing tasks. When quantum algorithms process multi-layer images such as RGB color images, what is the primary quantum advantage that enables simultaneous transformation of all image layers without extra quantum cost? 1) Each layer's data is compressed before upload, reducing the total number of required qubits. 2) The pixel values in all layers are stored in classical memory and accessed sequentially by quantum circuits. 3) Quantum gates are applied only to the most significant layer, leaving others unchanged. 4) Each layer is encoded into separate quantum registers and processed in isolation. 5) All layers are encoded with an additional label register, allowing parallel processing via amplitude entanglement and superposition. 6) Image layers are merged into a single grayscale channel before quantum processing. 7) Only the red channel is processed using quantum transforms, with green and blue handled classically.

✓ Correct Answer:

The correct answer is 5) All layers are encoded with an additional label register, allowing parallel processing via amplitude entanglement and superposition..

📚 Reference Text:

can be extended to two, and up to any, dimen- sions. The image data that we want to process requiresthe following amplitude encoding. For two dimensions, values labelled by fx;ygwill be encoded into the ampli- tude of the quantum state jxijyiin the computational basis of registers qxandqy. If this encoding is realized, applying the QCT circuit to both qxandqyregisters sep- arately (and simultaneously) achieves the 2-dimensional Discrete Cosine Transform over the data. Grayscale images. The quantum resources needed to encode and interpolate a gray-scale image with 2n2n pixels using the proposed technique is 2nqubits, plus constant ancilla depending on the uploading method, 2m ancillaryqubitsusedtoenlargetheimageto 2n+m2n+m pixels and 2 ancilla qubits for each dimension the QCT needs to be applied to. An example of an interpo- lated image using this method is displayed on Fig. 2 (left). A 512512image (top left) has been enlarged four times in both axis. This implementation requires 29 + 22 + 4 = 26 qubits, omitting encoding an- cillas, to transform the original image to the enlarged space and has been simulated using the quantum simu- lation library Qibousing images available in the image processing library scikit-image [29]. Multi-layer images. We highlight a genuinely quan- tum speed-up that arises when working with multi-layer data. By properly encoding the multiple layers, we can process all the quantum data with a single call to the QCT interpolation algorithm. Starting from the same encoding technique used for a single image, each layer li of the image will be labelled by state jliiof a new la- bel quantum register ql, withdlog(l)equbits, where lis the total number of layers. That is, the value of pixel fx;ygof layerliwill be encoded in the amplitude of the quantum statejxijyijlii. In this encoding, applying the QCT interpolation al- gorithm in the same way as one would for gray-scale im- ages, acting on quantum registers qxandqyonly, will perform the interpolation to all layers of the image in su- 5 perposition, at no extra quantum cost. Since every pixel value is now entangled with its label, the amplitude in- terference that makes the QFT possible (and eﬃcient) will only act on the amplitudes of pixel states that share the same label state. We showcase an example in Fig. 2 (right), where we interpolate an RGB image using two qubits to keep track of the color channel. Since the pixel dimensions are the same as Fig. 2 (left), the depth of the quantum circuit required for interpolation is the same for both instances. The speed-up provided by acting on all layers in super- position is made more apparent the more layers, or data instancessharingthesameshape, thatareencodedinthe proposed way. Additionally, further quantum advantage can be achieved when implementing quantum transfor- mations to all subsets within an image at the same time. Beyond that, this can be extended to other types of data uploaded in superposition. If the initial cost of upload- ing data using a label register is overcome, any quantum processing using a transformation such as the QFT or

Question 1225 multiple-choice

In quantum information theory, understanding the properties and structure of bipartite quantum states is central to studying entanglement and quantum correlations. Representation theory and information-theoretic concepts are often employed to analyze complex quantum systems and develop measures of entanglement. Which entanglement measure is rigorously defined using an information-theoretic approach, possesses desirable properties such as additivity, and quantifies the minimal entanglement consistent with observed correlations in quantum communication tasks? 1) Logarithmic negativity 2) Squashed entanglement 3) Concurrence 4) Entanglement of formation 5) Relative entropy of entanglement 6) Peres-Horodecki criterion 7) Schmidt rank

✓ Correct Answer:

The correct answer is 2) Squashed entanglement.

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Title: The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography Year: 2006 Paper ID: d6543e5561640c9aa62dc53e3d221cc5094ac2de Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d6543e5561640c9aa62dc53e3d221cc5094ac2de Abstract: This thesis presents a study of the structure of bipartite quantum states. In the first part, the representation theory of the unitary and symmetric groups is used to analyse the spectra of quantum states. In particular, it is shown how to derive a one-to-one relation between the spectra of a bipartite quantum state and its reduced states, and the Kronecker coefficients of the symmetric group. In the second part, the focus lies on the entanglement of bipartite quantum states. Drawing on an analogy between entanglement distillation and secret-key agreement in classical cryptography, a new entanglement measure, `squashed entanglement', is introduced.

Question 1226 multiple-choice

Modular tensor categories arising from quantum groups at roots of unity play a central role in topological quantum computing and the study of 3-manifold invariants. Understanding their modularity and unitarizability involves subtle arithmetic conditions on parameters such as rank, root of unity, and weight labels. Which of the following statements is true regarding the modularity of the category C(so₂r+1, ℓ, q) with ℓ odd? 1) The category is modular for any value of r, provided ℓ is odd and q is a root of unity. 2) The category is modular if and only if q^ℓ = 1 and r is even. 3) Modularity holds only when ℓ is even and q^ℓ = −1. 4) The category is modular whenever ℓ divides 2r+1, regardless of q. 5) Modularity occurs if ℓ is odd and there are no half-integer weight objects. 6) The category is modular if and only if q^ℓ = −1 and r is odd. 7) The modularity depends solely on the unitarizability of the category.

✓ Correct Answer:

The correct answer is 6) The category is modular if and only if q^ℓ = −1 and r is odd..

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that one has modularity for q=ezπi/ℓif and only ifgcd(z,(r+ 1)ℓ) = 1. Moreover, Masbaum and Wenzl [ MW] show that when gcd(ℓ, r+ 1) = 1 the subcategory of C(slr+1, ℓ, q) generated by the simple objects labelled by integer weights is a modular subcategory whose r ank is 1 /(r+ 1) times the rank of the full category. There are a number of other proo fs of this fact, see e.g. [Br] Section 5. Denote this subcategory by Z(Ar), and see Subsection 5.1. 4.3. Type B,ℓodd. The category C(so2r+1, ℓ, q) with ℓodd has been con- sidered to some extent by several authors including Sawin [ S1], [S2] and Le-Turaev [LT]. It is shown in ([ TW2 ], Theorem 9.9) that if ℓis odd, the subcategory of C(so2r+1, ℓ, q) generated by simple objects labelled by integer weights is modular and has rank exactly half of that of C(so2r+1, ℓ, q). Combining the computations in [R1] and the modularity criterion of [ Br] one has: Theorem 4.3.The category C(so2r+1, ℓ, q)withℓodd is modular if and only ifqℓ=−1andris odd. 10 ERIC C. ROWELL Proof. By the modularity criterion we wish to show that there are obs truc- tions to modularity ( i.e.non-trivial objects in the set N, see Theorem 4.1) if and only if the conditions of the theorem are not satisﬁed. By the modularity of the subcategory generated by simple objects labelled by intege r weights, any obstruct- ing object must be labelled by a half-integer weight. In [ R1] the object Xγlabelled by the (half-integer) weight that is furthest from the 0 weig ht in the fundamental alcove is shown to induce an involution of the fundamental al cove (by tensoring withXγ) that preserves q-dimension up to a sign. This implies that Xγis the only potential obstruction to modularity. In [ R1] (Scholium 4.11) the signs of the q-dimensions are analyzed, and the theorem then follows from the explicit compu- tations of Nν γ,λ,dλandθλ(also found in [ R1]) together with Formula (2.2) and the obstruction equation Sγ,λ=dγdλ. The subject of the author’s thesis [ R2] (the results of which can be found in [R1]) is the question of unitarizability of the family of catego riesC(so2r+1, ℓ, q) withℓodd. Using an analysis of the characters of the Grothendieck semirings it is shown that no member of this family of categories is unitar y. In fact, there is a much stronger statement, for which we need the following deﬁ nition: Definition 4.4.A pre-modular category Ois called unitarizable ifOis ten- sor equivalent to a unitary pre-modular category O′. By tensor equivalent we mean there exists a functor preserving the monoidal structure th at is bijective on mor- phisms and such that every object in the target category is is omorphic to an object in the image of the functor. Using a structure theorem of Tuba and Wenzl [ TbW ] it is shown in [ R2] that: Theorem 4.5.Fixqwithq2a primitive ℓth root of unity, ℓodd, and rsatis- fying2(2r+1)< ℓ. Then nobraided

Question 1227 multiple-choice

In quantum computing, group representation theory plays a crucial role in designing algorithms for problems involving symmetry, such as the hidden subgroup problem. Tools like the Clebsch-Gordan transform and Schur’s lemma are used to simplify and analyze quantum states arising from group actions. Which of the following is the primary function of the m-fold Clebsch-Gordan transform in algorithms solving non-abelian hidden subgroup problems? 1) To map each irrep label to its conjugate representation in the group algebra 2) To generate all possible symmetry-adapted states in the left regular representation 3) To decompose the direct product of multiple irreducible representations into a direct sum of irreps, each with specific multiplicities 4) To discard all registers that do not correspond to the trivial representation 5) To apply the quantum Fourier transform to each subgroup element independently 6) To measure multiplicity spaces without changing the representation basis 7) To enforce block diagonalization of the group algebra itself, independent of the state space

✓ Correct Answer:

The correct answer is 3) To decompose the direct product of multiple irreducible representations into a direct sum of irreps, each with specific multiplicities.

📚 Reference Text:

products we can expr ess this as 1 |G|k+1/summationdisplay g∈G/circleplusdisplay µ1,...,µmm/circlemultiplydisplay i=1/bracketleftBigg/summationdisplay hi∈HDµi(ghig−1)⊗Idµi/bracketrightBigg . (47) From this expression we see that we should focus on terms for a ﬁxe d choice of single copy irrep labels, µ1,...,µ m. M=1 |G|k+1/summationdisplay g∈Gm/circlemultiplydisplay i=1/summationdisplay h∈HDµi(ghig−1), (48) which is invariant under the direct product action of the group, /bracketleftBiggm/circlemultiplydisplay i=1Dµi(g)/bracketrightBigg M/bracketleftBiggm/circlemultiplydisplay i′=1Dµi′(g−1)/bracketrightBigg =M. (49) Thus we see that the symmetry of Eq. (43) is going to be related to d irect product represen- tation/circlemultiplytextm i=1Dµi(g). What can we say about the direct product of mirreps represented by/circlemultiplytextm i=1Dµi(g)? Well ﬁst we note that this is, of course, a representation of the group G. As such it will be, in general, reducible. In other words, there exists a basis under whic h m/circlemultiplydisplay i=1Dµi(g) =/circleplusdisplay µInµ⊗Dµ(g). (50) wherenµis the multiplicity of the µth irrep in this decomposition. We will call the transform which enacts the above basis change the m-fold Clebsch-Gordan transform. Notice that this transform can act in a non-separable manner over the diﬀerent sp aces where each irrep lives. It is this transform which we will use below to solve the Heisenberg hidd en subgroup problem. Given the basis change described in Eq. (50), and the fact that RR(g)⊗mcommutes with ρ[H],m, we can use Schur’s lemma to show that ρ[H],mhas a block diagonal form related to this basis change. In particular, the state ρ[H],mwill act trivially on the space where the µ irrep acts in Eq. (50) and will only have non-trivial support over the space arising from the multiplicity of the µth irrep, i.e. the space where identity acts in Eq. (50). To be concrete let us deﬁne the m-fold Clebsch-Gordan transform. It is the transform which takes as input the mirrep labels |µ1/an}bracketri}ht⊗···⊗| µm/an}bracketri}htalong with the spaces upon which these irreps act |vi/an}bracketri}ht⊗···⊗| vm/an}bracketri}htwherevi= 1,...,d µiand transforms this into the basis given by Eq. (50), which is |µ1/an}bracketri}ht ⊗ ··· ⊗ | µm/an}bracketri}ht ⊗ |µ/an}bracketri}ht ⊗ |w/an}bracketri}ht ⊗ |v/an}bracketri}ht, whereµis the irrep label for the subspaces on the RHS of Eq. (50), w= 1,...,n µlabels the multiplicity of these irreps, and v= 1,...,d µlabels the space where the µth irrep acts. Notice that we have not deﬁned an explicit basis for |v/an}bracketri}htand|w/an}bracketri}ht, just as in the QFT over a ﬁnite group, there was a choice in the basis for the |rµ/an}bracketri}htand|lµ/an}bracketri}htbasis. When we encounter the Clebsch-Gordan transform relevan t for the Heisenberg HSCP, we will pick a particular basis. Given the above deﬁnitions we can now describe the a protocol for d istinguishing the hidden subgroup conjugacy states when we are given mcopies of the hidden subgroup state just as we did for the hidden subgroup states: D. Bacon 17 1. Perform QFTs over Gover allmhidden subgroup states. 2. Measure the irrep label for each of these mstates, resulting in outcome µi. 3. Throw away the registers where the left regular representatio n irreps act. 4. Take the mremaining spaces where the right regular representation acts, alo ng with the

Question 1228 multiple-choice

In advanced algebra, the study of group algebras over finite fields often involves analyzing module structures, semidirect products, and varieties defined by group-theoretic identities. Matrix constructions and properties of commutators play a crucial role in classifying these algebraic objects. Which statement accurately describes the group Jf k formed from units in an algebra constructed as a quotient of a matrix algebra over a module involving a finite group G and field F_2? 1) Jf k is generated only by diagonal matrices with entries from F_2. 2) Jf k contains only abelian subgroups and satisfies no nontrivial group-theoretic identities. 3) Jf k is a simple group, with no proper normal subgroups. 4) Jf k is generated solely by group elements g ∈ G, without reference to any special matrix. 5) Jf k is a semidirect product of a subgroup B (generated by a special matrix C and its conjugates) and G, and satisfies a commutator identity w k = 1 in the group algebra F_2G. 6) Jf k is not closed under matrix multiplication and does not form a group. 7) Jf k coincides with the entire matrix algebra over F_2G without any quotient or ideal structure.

✓ Correct Answer:

The correct answer is 5) Jf k is a semidirect product of a subgroup B (generated by a special matrix C and its conjugates) and G, and satisfies a commutator identity w k = 1 in the group algebra F_2G..

📚 Reference Text:

Let ^ be the left 5-submodule of 5 <8>F 2 5 generated by all elements \®g (8 c = 0. Thus,(c+l)E,-<;/«+^* = ^andsoP = (c+l)(E,-gr1)®c+^t as required. This completes the proof of Lemma 3. • https://doi.org/10.1017/S1446788700002056 Published online by Cambridge University Press [11] Solution of a problem of Plotkin and Vovsi 339 Lemma 3 implies that .#* satisfies the identity w k = 1. Indeed, for every hu h 2, h3 e Jf? k, [h u h2, h y] e B k so for some g, e G. If c — [h 4, hs] • • • [h2k+2, h 2k+3], h, e J4? k for all i, then c is of the form where c e G, c = [g 4, gs] • • • [g 2k+2, g 2k+3] for some g,; € G (i = 4,... , 2k + 3). Therefore, by Lemma 3, where P = (? + 1)(£\ ^"') ®c + N k. Note that is a F 2-linear combination (in the group algebra F 2G) of elements of the form d/1,/2] + 1) • • • (l/a-i./z/] + 1) (/ < *), /,• e G (i = 1,...

Question 1229 multiple-choice

Quantum machine learning explores the intersection of quantum computing and neural network architectures, often leveraging symmetry principles from physics and mathematics to enhance modeling capabilities. Representation theory and group dualities play a crucial role in designing algorithms for systems exhibiting complex symmetries. Which mathematical concept enables the systematic decomposition of quantum circuits with SU(d) symmetry into symmetric group (Sn) irreducible representations, facilitating the exploitation of both permutation and unitary symmetries in quantum machine learning? 1) Noether's theorem 2) Galois theory 3) Tensor product decomposition 4) Jordan canonical form 5) Fourier analysis on abelian groups 6) Schur-Weyl duality 7) Stone–von Neumann theorem

✓ Correct Answer:

The correct answer is 6) Schur-Weyl duality.

📚 Reference Text:

Nparameters to solve topological symmetry-protected phases in quan- tum many-body systems, where Nis the system size. One of the key properties of classical CNNs is equivariance, which roughly states that if the input to the neural net- work is shifted, then its activations translate accordingly. There have been several attempts to introduce theoreti- cally sound analogs of convolution and equivariance to quantum circuits, but they have generally been somewhat heuristic. The major diﬃculty is that the translation invari- ance of CNNs lacks a mathematically rigorous quantum counterpart due to the discrete spectrum of spin-based 2691-3399/23/4(2)/020327(18) 020327-1 Published by the American Physical Society HAN ZHENG et al. PRX QUANTUM 4,020327 (2023) quantum circuits. For example, Ref. [ 29] uses the quasilo- cal unitary operators to act vertically across all qubits. In quantum systems there is a discrete set of transla- tions corresponding to permuting the qudits as well as a continuous notion of translation corresponding to spatial rotations by elements of SU (d). Combining these two is the realm of so-called permutational quantum computing (PQC) [ 30]. Therefore, a natural starting point for realiz- ing convolutional neural networks in quantum circuits is to look for permutation equivariance . In one of our related works [ 31], we argued that the natural form of equiv- ariance in quantum circuits is permutation equivariance and we introduce a theoretical framework to incorporate group-theoretical CNNs into the quantum circuits, build- ing on and generalizing the PQC framework to what we call PQC+ [ 31]. In this paper, we further explore PQC+ and its signiﬁ- cance for machine-learning applications. Roughly speak- ing, the PQC+ machine consists of unitary time evo- lutions of k-local SU (d)-symmetric Hamiltonian. As a feature, Schur-Weyl duality between the aforementioned Snand SU (d)actions on qudit systems appears natu- rally and is used throughout the paper. Most importantly, it indicates that any SU (d)-symmetric quantum circuits can be expressed in Snirreducible representations (irreps). Exploiting the power of Snrepresentation theory in quan- tum circuits towards NISQ applications is thus the central theme of the paper. The representation theory of Snhas been found to be a powerful tool in various permutation- equivariant learning tasks, e.g., learning set-valued func- tions [ 32] and learning on graphs [ 33,34]. Most appli- cations of permutation-equivariant neural networks work with a subset of representations of Sn. In contrast, in physi- cal and chemical models where the Hamiltonian exhibits global SU (d)symmetry, such as the Heisenberg model, it is necessary to consider alltheSnirreps (a detailed explanation of this signiﬁcant insight can be found in Sec. V). However, even the best classical fast Fourier transforms (FFTs) over the symmetric group Snrequire at least O(n!n2)operations [ 35,36], which dashes any hope of calculating the Fourier coeﬃcients even for rel- atively small n. Indeed, despite increasing realization of the importance of enforcing SU (2)symmetry, none of the neural-network quantum state (NQS) ansätze are able to respect SU (2)symmetry for all SU (2)irreps, due to the superpolynomial growth of the multiplicities of irreps and

Question 1230 multiple-choice

Quantum Machine Learning (QML) leverages mathematical concepts from representation theory and quantum physics to design neural network architectures that respect symmetry and group actions. The manipulation and transformation of data representations within a QML pipeline are critical for effective equivariant processing and feature extraction. In a QML classification pipeline utilizing Z₂ symmetry, which operation is responsible for reducing the dimensionality of quantum data by tracing out qubits, analogous to pooling in classical neural networks? 1) Applying the Clebsch-Gordan decomposition 2) Implementing the Pauli X operator 3) Encoding classical data into quantum states using an embedding layer 4) Transforming group representations via SWAP operations 5) Processing data with equivariant quantum neural network layers 6) Performing a partial trace to remove qubits and focus on relevant features 7) Defining subrepresentations to decompose quantum architectures

✓ Correct Answer:

The correct answer is 6) Performing a partial trace to remove qubits and focus on relevant features.

📚 Reference Text:

as is usually the case in QML, can be rewritten as ((R S)(g)) (v) =Sg(Ry gv):(25) The kicker of this form is that studying how representa- tions act on linear maps (and thus quantum channels!) boils down to a special case of tensor representations.12 12It should be noted that for some special cases, the dual representa- 22 Figure 9. Change of representations in QML. We consider the task in Fig. 3 of classifying real valued data (x1;x2)2R2. Here, the data labels are symmetric under the action of Z2, whose representation is G1=f11;g, where(x1;x2) = (x2;x1). The ﬁrst step is to embed the data into the state of two qubits via some data-encoding scheme that eﬀectively changes the representation of Z2toG2=f11;SWAPg. Next, one can process the information via some G2-equivariant quantum neural network W(1)such as that in Fig. 8. Further, as we show below we can ﬁnd a map 1: (C2) 2!(C2) 2, given by 1() =1 2(+ SWAPX2X2SWAP) that changes the representation of the group to G3=f11;X 2gsuch that we can process the information with a G3-equivariant quantum neural network W(2). Next, we show a poolingmap 2: (C2) 2!C2(given by the partial trace) which maps two- qubit states to single-qubit states (note that the partial trace can also output mixed quantum states), and which also changes the representation of the symmetry group to G4=f11;Xg. Finally, we can process the information encoded in the single qubit via a G4-equivariant quantum neural network W(3). For instance, the vector space of equivariant maps (for unitary reps S;R), is given by =SgRy g; (26) which is the collection of trivial 1D invariant subspaces of V W, since saying is equivariant is the same as saying (R S)(g) = (see Deﬁnition 12).13In many cases, since tensor representations of common groups are well un- derstood (re: Clebsch-Gordan decomposition), this shift in perspective can be both theoretically and computationally useful. We refer the reader to Fig. 9 where we showcase how diﬀerent representations can play a role in a QML classiﬁ- cation task. Notably, this ﬁgure depicts several key ingre- dients in QML tasks such as embedding layers (mapping classical data to quantum data), quantum neural network layers(to process the information in the quantum states), anddata-pooling layers (where qubits in the system are discarded with the hope of reducing the feature space di- mension while preserving the relevant data features). We note that the ﬁeld of change of representation in GQML is still in its infancy (see [62] for some theory). tionRis isomorphic to the original representation R(this is the case for e.g. SU(2)), but in general this is not true, for instance anySU(d)withd>3. It is however true that Ris irreducible iﬀ Ris, and that (R)=R. 13The trivial representation always breaks into 1D trivial representations—this is almost immediate from the deﬁnition.2. Structure deﬁnitions and basic theory We now have a rich supply of contextualized examples: time to study their structure. To begin, we need an appropriate deﬁnition of a subrep- resentation of a representation. Deﬁnition12. LetR:G!GL(V)be

Question 1231 multiple-choice

Distributed quantum computing enables the execution of quantum algorithms across multiple physically separated quantum nodes by leveraging entanglement, non-local gates, and classical communication. Simulators play a crucial role in modeling the performance and limitations of these systems under realistic hardware constraints. Which of the following features specifically distinguishes the Distributed Quantum Computing Simulator (DQCS) from most existing distributed quantum simulators and platforms? 1) Exclusive support for trapped ion quantum hardware 2) Absence of quantum noise modeling in simulations 3) Use of only static, non-adaptive quantum circuits 4) Limited benchmarking to single-node quantum algorithms 5) Simulation of classical control protocols without quantum gate operations 6) Integrated modules for simulating non-local gates, distributed circuits, and algorithm performance under noise and hardware imperfections 7) Optimization restricted to photonic links between quantum nodes

✓ Correct Answer:

The correct answer is 6) Integrated modules for simulating non-local gates, distributed circuits, and algorithm performance under noise and hardware imperfections.

📚 Reference Text:

different nodes utilizing one ebit/CNOT gate by applying the quantum circuit shown in Fig. 2 [12]. The non-local CNOT gate q0 • Z(q2) q1 q2 •H q3 X(q1) Figure 2: The quantum circuit for the non-local teleportation-based CNOT gate between q0andq3using the Bell pair q1andq2prepared in1√ 2(|00⟩+ |11⟩), followed by two local CNOT gates, measurement on the qubits q1andq2, classical communication and conditional single-qubit quantum gates has been experimentally realized in superconducting qubits [18] and ion qubits [19]. A non-local quantum circuit could be obtained using multiple non-local quantum gates, where the gates are applied sequentially through entangling, measuring, and re-entangling the same qubit with only one extra qubit/node. Recently, distributed Grover’s algorithm has been experimentally realized using 2 dual species trapped ions connected through photonic link, where the nodes were separated by a distance of 2 meters [10]. In this paper, we describe a distributed quantum computing simulator (DQCS) that uses non-local quantum gates between the nodes, creates distributed quan- tum circuits, and evaluates their performance in the presence of noise [12]. We use DQCS to investigate and benchmark the performance of several distributed quantum algorithms [20, 21, 22], demonstrating the applicability of dynamic quantum circuits in DQC that reduce resource overhead and achieve high fi- delity. We illustrate this with a distributed quantum phase estimation (DQPE) example. We study the loading of the probability distribution in DQC, scal- ing with the number of nodes. We use DQCS to run the distributed quantum amplitude estimation (DQAE) algorithm on multiple nodes. The DQCS con- tains modules that provide users various ways to study, run, and evaluate the performance of quantum algorithms in DQC [21]. The DQCS could simulate and optimize the quantum network and circuit layers for distributed quantum algorithms. Several distributed quantum computing [3, 23, 24, 25] and quantum inter- net simulators [26, 27, 28] that have been developed that consider distributed quantum gates, compilers, and classical networks respectively. They consider the control platform for the hardware (Interlin-q) [23], job schedulers [3], op- timization using the quantum message parsing interface (QMPI) [24] for the simulation of DQC, analogous to the high performance computing. They con- sider the noiseless setting that does not include the gate, measurement imper- fections, coupling efficiency, etc. that is crucial to realize DQC in the Noisy intermediate-scale quantum era (NISQ) quantum devices. In addition, they don’t consider other techniques, such as dynamic quantum circuits in the dis- tributed setting, that reduce the number of non-local quantum gates. The key challenge of the DQC is that each non-local two-qubit quantum gate requires two local two-qubit gates, two measurements, and efficient coupling between the nodes. Thus, for a given set of hardware parameters (such as gate fidelity, measurement error rate, and coupling efficiency), it’s crucial to study the per- formance of a quantum algorithm [29] in the DQC in the presence of various imperfections. The goal of the DQCS is to consider the implementation of DQC in the NISQ devices [10], consider the hardware parameters including node

Question 1232 multiple-choice

Quantum walks provide a discrete-time framework to simulate quantum systems, including relativistic fermions described by the Dirac equation. The mathematical properties of spinors and their transformation groups are crucial in faithfully modeling such particles on spatial grids. Why must a quantum walk simulating fermionic behavior on a polar spatial grid extend the range of the polar angle variable from 0 to 4π instead of the conventional 0 to 2π? 1) To allow for periodic boundary conditions in the simulation 2) Because spinors transform under SU(2), requiring a 4π rotation to return to their original state 3) To account for the presence of magnetic monopoles in the system 4) To match the topology of the SO(3) rotation group 5) To increase the resolution of the spatial lattice 6) To simulate bosonic particles with integer spin 7) To model time-reversal symmetry more accurately

✓ Correct Answer:

The correct answer is 2) Because spinors transform under SU(2), requiring a 4π rotation to return to their original state.

📚 Reference Text:

Title: Dirac quantum walks with conserved angular momentum Year: 2020 Paper ID: 6490386ea5ad7257f822cad53dc6588cd5c033cc Source: semantic-scholar URL: https://www.semanticscholar.org/paper/6490386ea5ad7257f822cad53dc6588cd5c033cc Abstract: A quantum walk (QW) simulating the flat (1+2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1 + 2)$$\end{document}D Dirac equation on a spatial polar grid is constructed. Because fermions are represented by spinors, which do not constitute a representation of the rotation group SO(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{SO}}(3)$$\end{document}, but rather of its double cover SU(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{SU}}(2)$$\end{document}, the QW can only be defined globally on an extended spacetime where the polar angle extends from 0 to 4π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4 \pi $$\end{document}. The coupling of the QW with arbitrary electromagnetic fields is also presented. Finally, the cylindrical relativistic Landau levels of the Dirac equation are computed explicitly and simulated by the QW.

Question 1233 multiple-choice

Spin foam models are a framework used in quantum gravity to represent spacetime using networks labeled by group representations. Q-deformation introduces quantum group structures to these models, impacting their mathematical and physical properties. In four-dimensional q-deformed spin foam models, which feature of root of unity representations of Uq(su(2)) is crucial for ensuring the convergence of state sums? 1) They allow for infinite-dimensional representations. 2) They yield finite-dimensional representations that control divergences. 3) They eliminate the need for intertwiners. 4) They make the Lorentz group commutative. 5) They force the deformation parameter to be imaginary. 6) They require braiding to be absent. 7) They guarantee a vanishing cosmological constant.

✓ Correct Answer:

The correct answer is 2) They yield finite-dimensional representations that control divergences..

📚 Reference Text:

Title: Quantum deformation of two four-dimensional spin foam models Year: 2010 Paper ID: cbd94aeba77638934c5f9b64bc5b5dd1d742b76d Source: semantic-scholar URL: https://www.semanticscholar.org/paper/cbd94aeba77638934c5f9b64bc5b5dd1d742b76d Abstract: We construct the q-deformed version of two four-dimensional spin foam models, the Euclidean and Lorentzian versions of the EPRL model. The q-deformed models are based on the representation theory of two copies of Uq(su(2)) at a root of unity and on the quantum Lorentz group with a real deformation parameter. For both models we give a denition of the quantum EPRL intertwiners, study their convergence and braiding properties and construct an amplitude for the four-simplexes. We nd that both of the resulting models are convergent.

Question 1234 multiple-choice

In quantum computing, efficient representations of quantum operations are essential for simulation and algorithm development. Matrix Product Operators (MPOs) and tensor network contractions are widely used to encode and manipulate complex quantum transformations such as the Quantum Fourier Transform (QFT). When compressing the QFT as a Matrix Product Operator using sequential Singular Value Decompositions and truncating small singular values, how does the error bound on the average fidelity loss depend on the bond dimension χ and the number of qubits n? 1) It scales linearly with n and independently of χ. 2) It increases quadratically with both χ and n. 3) It decreases exponentially with n and remains constant with χ. 4) It is proportional to χ⁻² regardless of n. 5) It depends only on the largest singular value of the tensor network. 6) It is independent of both χ and n. 7) It is O(ne^{-χ log(χ/3)/√χ}), meaning the error is independent of n in both average and worst-case scenarios, but requires appropriate scaling of χ for desired accuracy.

✓ Correct Answer:

The correct answer is 7) It is O(ne^{-χ log(χ/3)/√χ}), meaning the error is independent of n in both average and worst-case scenarios, but requires appropriate scaling of χ for desired accuracy..

📚 Reference Text:

T/prime/prime[3]U/prime[2]U[1] V/prime/prime[4] FIG. 3. Contracting a 4-qubit QFTTN into a QFT MPO. In step 1, we contract the ﬁrst two phase MPOs by doing a series of SVDs and we truncate small singular values or keep χof them. Since site 1 is isometric already by contraction of the Htensor and the copy tensor, we stop at site 2, and thus the orthogonality center at the end is at site 2. In step 2, we do a series of SVDs to push the orthogonality center to the last site, and again we truncate singular values or keep χsingular values. In step 3, we repeat the contraction for our current MPO and the next phase MPO. During the steps, we also contract any Htensors encountered. The ﬁnal MPO has the orthogonality center at site 3. for phase MPOs is completely stable (see Appendix Hfor detailed explanations). Therefore, the zip-up algorithm is an optimal algorithm for contracting the QFTTN. In conclusion, by applying the zip-up algorithm over the QFTTN, we can construct a QFT MPO with maxi- mum bond dimension χinO(χ3n2)time. In principle, one could derive a closed-form expression for the QFT MPO, since the QFT has a well-deﬁned formula and the QFT MPO’s site tensors approach some ﬁxed form as n grows (because PDPSSs approach DPSSs in the limit). One potential approach is to use the tensor train (MPS) cross-interpolation [ 43], a technique to approximate the MPS or MPO decomposition when the SVD is incalcu- lable. Therefore, an O(χ3n)or even an O(χ2n)algorithm might exist for constructing the QFT MPO. However, we note that once a QFT MPO is constructed it can be stored in a database and reused later; thus, the construction time needs not to be not considered in algorithms. Addition- ally, our construction scheme has a very small constantassociated with it, and it is fast enough for the num- ber of qubits required for most tasks. Therefore, reducing the construction time of the QFT MPO is less important than reducing the construction time of other parts of the algorithms, such as converting a vector into an MPS. Nev- ertheless, it is interesting to see whether a closed-form expression exists for the QFT MPO. C. Error for compressing the QFT MPO In this subsection, we analyze the error of compressing the QFT MPO. In particular, we show that both the average error and worst-case error are independent of n. The former is easy to benchmark with numerical simulations and the latter guarantees the error of truncating a QFT MPO willnot be ampliﬁed by any input states. We deﬁne Q /prime nto be the truncated MPO representation ofQnwith bond dimension χ. Then a convenient choice of the average error eavis the average squared norm of 040318-11 JIELUN CHEN, STOUDENMIRE, and WHITE PRX QUANTUM 4,040318 (2023) Qn−Q/prime nacting on Haar random states: eav= E ψ∼μH/bracketleftBig/vextenddouble/vextenddouble(Qn−Q/prime n)|ψ/angbracketright/vextenddouble/vextenddouble2/bracketrightBig , (65) which reduces to eav=1 2n/vextenddouble/vextenddoubleQn−Q/prime n/vextenddouble/vextenddouble2 F(66) from Haar integral calculations. We show in Appendix J that eavis bounded by O(ne−χlog(χ/3)/√χ). Therefore, χ needs

Question 1235 multiple-choice

In information theory, entropic vectors derived from the joint distributions of quasi-uniform random variables associated with finite groups provide insight into the informational structure enabled by those groups. Group-theoretic properties play a fundamental role in determining the diversity and representability of these entropic vectors, with implications for network coding strategies. Which of the following statements correctly characterizes the relationship between nilpotent groups and abelian representability for two quasi-uniform random variables (n=2)? 1) All simple non-abelian groups are abelian representable for n=2. 2) Only abelian groups are abelian representable for n=2. 3) A group is abelian representable for n=2 if and only if it is nilpotent. 4) Any group with a cyclic subgroup is abelian representable for n=2. 5) Nilpotent groups are never abelian representable for n=2. 6) Dihedral groups are abelian representable for n=2 regardless of their order. 7) Quasi-dihedral groups are abelian representable for all values of n.

✓ Correct Answer:

The correct answer is 3) A group is abelian representable for n=2 if and only if it is nilpotent..

📚 Reference Text:

Title: On Abelian group representability of finite groups Year: 2012 Paper ID: 67269c1cf75303b9bb767bfb937619360896eaed Source: semantic-scholar URL: https://www.semanticscholar.org/paper/67269c1cf75303b9bb767bfb937619360896eaed Abstract: A set of quasi-uniform random variables $X_1,...,X_n$ may be generated from a finite group $G$ and $n$ of its subgroups, with the corresponding entropic vector depending on the subgroup structure of $G$. It is known that the set of entropic vectors obtained by considering arbitrary finite groups is much richer than the one provided just by abelian groups. In this paper, we start to investigate in more detail different families of non-abelian groups with respect to the entropic vectors they yield. In particular, we address the question of whether a given non-abelian group $G$ and some fixed subgroups $G_1,...,G_n$ end up giving the same entropic vector as some abelian group $A$ with subgroups $A_1,...,A_n$, in which case we say that $(A, A_1,..., A_n)$ represents $(G, G_1, ..., G_n)$. If for any choice of subgroups $G_1,...,G_n$, there exists some abelian group $A$ which represents $G$, we refer to $G$ as being abelian (group) representable for $n$. We completely characterize dihedral, quasi-dihedral and dicyclic groups with respect to their abelian representability, as well as the case when $n=2$, for which we show a group is abelian representable if and only if it is nilpotent. This problem is motivated by understanding non-linear coding strategies for network coding, and network information theory capacity regions.

Question 1236 multiple-choice

Quantum Fourier transformations (QFTs) are essential tools in quantum computing, particularly for algorithms dealing with group-theoretic structures relevant to physics and cryptography. Non-Abelian groups present unique challenges in developing efficient QFTs, with applications spanning hidden subgroup problems and quantum simulations. Which specific non-Abelian group, commonly appearing in models of neutrino mixing and flavor symmetries, has recently been targeted by efficient quantum circuits for its quantum Fourier transformation? 1) $S_4$ 2) $\Delta(27)$ 3) $A_5$ 4) $\mathbb{Z}_N$ 5) $Q_8$ 6) $D_{16}$ 7) $C_3 \times C_3$

✓ Correct Answer:

The correct answer is 2) $\Delta(27)$.

📚 Reference Text:

Title: Highly-efficient quantum Fourier transformations for certain non-Abelian groups Year: 2024 Paper ID: 4c97b959be0561d0ce905b7608d58c1f22b0a366 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/4c97b959be0561d0ce905b7608d58c1f22b0a366 Abstract: Quantum Fourier transformations are an essential component of many quantum algorithms, from prime factoring to quantum simulation. While the standard abelian QFT is well-studied, important variants corresponding to \emph{nonabelian} groups of interest have seen less development. In particular, fast nonabelian Fourier transformations are important components for both quantum simulations of field theories as well as approaches to the nonabelian hidden subgroup problem. In this work, we present fast quantum Fourier transformations for a number of nonabelian groups of interest for high energy physics, $\mathbb{BT}$, $\mathbb{BO}$, $\Delta(27)$, $\Delta(54)$, and $\Sigma(36\times3)$. For each group, we derive explicit quantum circuits and estimate resource scaling for fault-tolerant implementations. Our work shows that the development of a fast Fourier transformation can substantively reduce simulation costs by up to three orders of magnitude for the finite groups that we have investigated.

Question 1237 multiple-choice

Quantum algorithms for non-abelian groups face unique challenges, especially when applied to the Dihedral Hidden Subgroup Problem (DHSP) and its generalizations. The relationship between hidden subgroup problems, coset sampling, and hidden shift problems is central to ongoing research in quantum computation and algorithmic group theory. Which construction provides a group-theoretic framework that generalizes the dihedral group and establishes an equivalence between certain Hidden Subgroup Problems and the hidden shift problem for finite abelian groups? 1) Direct product of two cyclic groups 2) Wreath product of abelian and symmetric groups 3) Free product of abelian groups 4) Central extension by a cyclic group 5) Quotient group of a direct sum 6) Semi-direct product G = A ⋉ C₂ with C₂ acting by inversion on A 7) Abelianization of the dihedral group

✓ Correct Answer:

The correct answer is 6) Semi-direct product G = A ⋉ C₂ with C₂ acting by inversion on A.

📚 Reference Text:

this has success probability: =⎛ ⎝−⎞ ⎠NNP211 2. success (3.22) Theorem 3.23. The standard algorithm for DHSP cannot implement the optimal measurement using one coset sample. Proof. This follows because (3.21 ) and (3.22) are incompatible. □The dihedral hidden subgroup problem  7 3.3 Dihedral coset sampling In the standard HSP algorithm, after the ﬁrst step, we are left with random coset samples as in (3.3 ). In the case of=GDN, the dihedral group of order n, and ⟨⟩ ==HHy xaa, this is explicitly of the form: ∣∣∣⟩ ( ∣ ⟩∣ ⟩ ) (∣ ⟩ ∣ ⟩) (∣ ⟩ ∣ ⟩) (∣ ⟩ ∣ ⟩) (∣ ⟩ ∣ ⟩)∑=+ =+ =+ =⎧ ⎨⎪ ⎩⎪+= +=∈ +− − −Hch c cyx yx yxy x yx y x xy x β xy x β11 2 1 2 1 2 1 2,i f 0 1 2,i f 1hHa βα βα a βα β a α αa α aα α1(3.24) where=cyxβα. Remark 3.25. The second case is reduced to the ﬁrst by the transformation →−αaαif this transformation leaves the distribution of αinvariant. Given samples of the form (∣ ⟩ ∣ ⟩) == +−ψψ x y x1 2,aa ααa α ;(3.26) the DCP is the problem of ﬁnding generators for the hidden subgroup =HHa. The states =ψψaaα;are called the DCP samples for a. For HSP samples produced from the standard algorithm, where αis from the uniform distribution, we may view HSP samples as DCP samples by Remark 3.25 . Remark 3.27. We can encode a DCP sample ψaα;as: (∣ ⟩∣ ⟩ ∣ ⟩∣ ⟩)+−αa α1 201 . Using the fact that =−yxx yαα, this can be encoded (after negating a) as: (∣ ⟩∣ ⟩ ∣ ⟩∣ ⟩)++αa α1 201 , which is another commonly used form used in the literature, especially in the context of the “hidden shift problem ”. 3.4 Generalizations of DCP LetAbe aﬁnite abelian group (written multiplicatively) with identity e. Let →ff A S,:01 be the injective functions to a ﬁnite setSand such that () ( )=fx fx s10 for all∈xA and some ∈sA . The hidden shift problem is the problem of determining the hidden shift ∈sA from evaluations of the functions f0andf1. We refer the reader to previous studies [17 , §5] for a discussion of the hidden shift problem without the condition of injectivity.8 Imin Chen and David Sun Let=⋉GCA 2 , where ⟨⟩=Cτ2 acts from the left on Aby inversion. In the resulting semi-direct product, let ()=yτe,. Then, for elements in the normal subgroup A˜ofGcorresponding to A, i.e., elements of the form ()=αeα ˜ ,, we have that =−−yαy α˜˜1 1, and every element in Gcan be written in the form yα˜orα˜. The HSP for subgroups of the form: ⟨( )⟩ {( ) ( )}==Hτs ee τs,, , , (3.28) is equivalent to the hidden shift problem with shift ∈sA . Indeed, an element in /DoubleCapZ[]G, which is constant on left H-cosets, is a linear combination of sums of the form: ∑=+ ∈+−yα h yα y s α˜˜

Question 1238 multiple-choice

Quantum error correction codes are essential for protecting quantum information against various types of noise, including erasure events where certain subsystems are lost. The analysis of information leakage and fidelity of recovery are key aspects in evaluating the effectiveness of these codes. In the mathematical modeling of quantum erasure channels with multiple possible subsystem losses, which feature most directly enables error correction codes to achieve higher fidelity recovery compared to general error channels? 1) The replacement of lost subsystems with random quantum states 2) The assumption that all errors are independent and identically distributed 3) The use of entanglement between subsystems to prevent information loss 4) The presence of a classical register recording the locations of lost subsystems 5) The restriction to only single-qubit erasures in the system 6) The absence of any information leakage to the environment 7) The use of only deterministic noise maps for system evolution

✓ Correct Answer:

The correct answer is 4) The presence of a classical register recording the locations of lost subsystems.

📚 Reference Text:

erasures at unknown locations, since the latter are necessarily harder to correct against.) For instance, if the ith physical subsystem is lost to the environment with probability qi, then the corresponding noise map is Nð1Þ A→ACð·Þ¼X qijiihijC⊗jϕiihϕijAi⊗trAið·Þ;ð10Þ where we introduce a classical register C, which records which one of the nsystems was lost, and where jϕiiare some fixed states. One can also consider more general erasure scenarios, where any given combination of subsystems can be lost with a given probability. For instance, one might assume that systems A1andA2are simultaneously lost with probability qf1;2g, systems A2andA3are simultaneously lost with probability qf2;3g, and systems A1andA3are lost with probability qf1;3g. More generally, a combination of subsystems, which we label generically by α, can be lost with probability qα; we assume we know exactly which systems are lost. The corresponding general noise map is then NA→ACð·Þ¼X α∈KqαjαihαjC⊗Nα A→Að·Þ;ð11aÞ Nα A→Að·Þ¼j ϕαihϕαjAα⊗trAαð·Þ; ð11bÞ where the register Cencodes the exact locations at which simultaneous erasures occur, where Aαdenotes the physical systems labeled by α(for instance, if α¼f2;3g, then Aα¼A2⊗A3), and where fjϕiαgare fixed states. The sum ranges over a set Kof possible α’s corresponding to erasures which may occur. Technically, Kis any set of subsets of f1;2;…;ng, and Aα¼⊗ i∈αAi. Situations which can be described using this setting include, for instance, any kconsecutive erasures or the erasure of any ksubsystems. C. Characterization via the environment A very useful characterization of the quantities (8)is provided by B´ eny and Oreshkov [30], building upon the decoupling approach to error correction [31,32] . The recoverability of the logical information can be character- ized by studying how much information is leaked to the environment, as represented by a complementary channel dN∘EofN∘E. Recall that a complementary channel ˆFA→C of a quantum channel FA→Bis a channel of the form ˆFA→Cð·Þ¼trB½WA→BCð·ÞW†/C138, where WA→BC is a Stinespring dilation isometry for the map F, i.e., FA→Bð·Þ¼trC½WA→BCð·ÞW†/C138.B ´eny and Oreshkov showCONTINUOUS SYMMETRIES AND APPROXIMATE QUANTUM … PHYS. REV . X 10,041018 (2020) 041018-7 that the fidelity with which one can reverse the action of the encoding and the noise is exactly the fidelity of the total complementary channel to a constant channel: feðN∘EÞ¼max ζFjˆϕiðdN∘E;TζÞ; ð12aÞ fworstðN∘EÞ¼max ζmin jϕiFjϕiðdN∘E;TζÞ;ð12bÞ whereTζð·Þ¼trð·Þζis the constant channel outputting the state ζand where the maximizations range over all quantum states ζon the output system ofdN∘E. Now we determine a complementary channeldN∘Eto the encoding and noise channels. Consider first the single- erasure noise channel (10). A Stinespring dilation of Nð1Þ A→ACon two additional systems C0⊗Eis given as Nð1Þ A→AC¼trC0E½Wð·ÞW†/C138, with WA→ACC0E¼XﬃﬃﬃﬃqipjiiC⊗jiiC0⊗jϕiiAi⊗1Ai→E⊗1AnAi; ð13Þ where 1Ai→Eis an isometric embedding of AiintoEand 1AnAiis the identity operator on all systems Aexcept Ai. Now consider a Stinespring dilation of EL→Aas EL→A¼trF½VL→AFð·ÞV†/C138. Then, we may take dN∘EL→C0EFð·Þ¼trAC½WVð·ÞV†W†/C138 ¼X qijiihijC0⊗trAnAi½Vð·ÞV†/C138;ð14Þ where tr AnAidenotes the partial trace over all systems except Ai(the latter is then embedded in the Esystem). Hence, the complementary channel to the single erasure channel simply gives the erased information to the envi-ronment with the corresponding erasure probability. It is straightforward to see that for the more general

Question 1239 multiple-choice

In quantum information theory, the symmetries of composite systems are described using representation theory of groups like SU(d) and the symmetric group Sₙ. The Schur-Weyl duality provides a framework for understanding how the tensor powers of vector spaces decompose under these joint group actions. Which statement about the decomposition of V⊗ⁿ under the joint action of SU(d) and Sₙ is correct? 1) Irreducible representations are labeled by partitions with at most n rows and d boxes. 2) The symmetric group Sₙ acts trivially on all components of V⊗ⁿ. 3) Only single-row Young diagrams appear for SU(2) tensor decompositions. 4) Each SU(d) irrep may correspond to multiple Sₙ irreps, but never vice versa. 5) The group actions of SU(d) and Sₙ on V⊗ⁿ generally do not commute. 6) The decomposition is indexed by Young diagrams with n boxes and at most d rows, with each SU(d) and Sₙ irrep paired in the sum. 7) The total spin in the SU(2) case is determined by the number of columns in the Young diagram.

✓ Correct Answer:

The correct answer is 6) The decomposition is indexed by Young diagrams with n boxes and at most d rows, with each SU(d) and Sₙ irrep paired in the sum..

📚 Reference Text:

same Young diagrams with n boxes and at most d rows . Theorem (Schur-Weyl duality ):The action of SU(d)and Snon V⊗njointly decompose the space into irreducible representations of both groups in the form V⊗n=/circleplusdisplay λWλ⊗Sλ, where W λand Sλdenote irreps of SU(d), respectively, S n, andλranges over all Young diagrams of size n with at most d rows. Consequently, πSU(d)∼=/circleplusdisplay μWμ⊗1mSU(d),μ,πSn∼=/circleplusdisplay λ1mSn,λ⊗Sλ, where m SU(d),μ=dimSμand m Sn,λ=dimWλ. One can easily verify that πSU(d)andπSncommute (fur- ther properties are described in the Appendix). Consider thesymmetric group algebra C[Sn] consisting of all for- mal ﬁnite sums f=/summationtext iciσi. Its representation is then ˜πSn(f)=/summationtext iciπSn(σi). When there is no ambiguity, we denote by Uσor simply σthe representations πSn(σ). Working from the perspective of Schur-Weyl dual- ity requires, at least theoretically, using the Schur basis rather than the computational basis. A conventional way to build such a basis is conducting sequential coupling and Clebsch-Gordan decompositions of SU (d)representations [30,47,48], which transform common matrix representa- tions of SU (d)into irrep matrix blocks like in Fig. 2(a). Since our focus are ansätze, operators, and quantum cir- cuits with SU (d)symmetry, which commute with πSU(d), and since Schur-Weyl duality and the double commutant theorem (see Appendix) say that they must be established from the group algebra C[Sn], we need to explore Snirreps blocks as in Fig. 2(b) however. We are going to introduce a method to decompose permutation matrices in this picture and explain basic notions in Snrepresentation theory and the Okounkov-Vershik approach, since they are essential to understand the theoretical results in this paper. We ﬁrst consider the so-called permutation module Mμ. In the case of qubits, permutation modules correspond 020327-3 HAN ZHENG et al. PRX QUANTUM 4,020327 (2023) (c) (b) (a) FIG. 2. (a) Decomposition of (C2)6with respect to SU (2)action while (b) is for S6by Schur-Weyl duality. (c) Arrangement of (b), which respects both permutation modules and S6irreps. to sets of Schur-basis elements having diﬀerent readout on total spin components. Fortunately there is an acces- sible way to understand Mμin the tensor-product space V⊗n. To make things simpler, consider the d=2 case of SU(2)−Snduality on (C2)⊗n. Only two-row Young dia- grams λ=(λ1,λ2)appear in this duality and the half of diﬀerence1 2(λ1−λ2)between the lengths of the two rows gives the total spin of the SU (2)-irrep Wλ. The permuta- tion module Mμis isomorphic with the linear span of all computational basis vectors with z-spin components equal to1 2(μ1−μ2). Note that, (C2)⊗n=/circleplustext μMμand each Mμis invari- ant under Snpermutation. Furthermore, Mμcan be further decomposed into Snirreps. For two-row Young diagrams [SU(2)−Snduality], the decomposition is easy: Mμ=/circleplustext λ≥μSλwhere λ,μhave the same size nand we use the dominance order λ/trianglerightequalμifλ1≥μ1.I ns u m m a r y ,w eh a v e (C2)⊗n=/circleplusdisplay μMμ=/circleplusdisplay μ/circleplusdisplay λ/trianglerightequalμSλ∼=/circleplusdisplay λ1mSn,λ⊗Sλ. Isomorphic copies of Sλcome from diﬀerent permuta- tion modules. The largest permutation module contains all distinct Snirreps in (C2)⊗n[see e.g., Fig. 2(c)]. For gen- eral Young diagrams, decompositions of Mμwould have nontrivial multiplicities [ 50,59]. Each Sλcan be decomposed further with respect

Question 1240 multiple-choice

In the study of finite p-groups, subgroup structures, centers, and properties of involutions play key roles in classifying group forms. The analysis of minimal non-abelian subgroups and the behavior of their elements helps determine possible configurations for larger groups. Which of the following statements about a finite non-abelian p-group G with a maximal normal abelian subgroup A is correct under the condition that the exponent of G/A is p? 1) G/A is elementary abelian and its elements all have order p. 2) G/A must contain an element of order greater than p. 3) The center Z always contains a subgroup isomorphic to Q₈. 4) All involutions in G are squares in any subgroup of G. 5) Minimal non-abelian subgroups of G are necessarily abelian. 6) G/A has exponent dividing p². 7) The commutator subgroup of G is trivial.

✓ Correct Answer:

The correct answer is 1) G/A is elementary abelian and its elements all have order p..

📚 Reference Text:

then i¼vaBHis an involution and ib¼va/C01¼ia2 so that iBZðGÞ, contrary to Lemma 1.15. If v2¼b2, then j¼vbBHis an involu- tion and ja¼vba¼ja2so that jBZðGÞ, contrary to Lemma 1.15. It follows that v2¼a2b2, which is a unique involution in Hthat is not a square in H. In that case ðvbÞ2¼v2b2¼ða2b2Þb2¼a2and avb¼ab¼a/C01so that ha;vbiGQ8and Hhvi¼ha;vbi/C2hviGQ8/C2C4.r Lemma 1.17. Our group G has no subgroup isomorphic to Q 8/C2C4/C2C4. Proof. Suppose that K¼Q/C2hci/C2hdicG, where Q¼ha;biGQ8and hciGhdiGC4. We have W1ðKÞ¼ha2;c2;d2iGE8, where ½ac;bd/C138¼½a;b/C138¼a2 so thathac;bdi0¼ha2iand therefore hac;bdiis minimal non-abelian (by Proposi- tion 1.5). But ðacÞ2¼a2c2and ðbdÞ2¼a2d2, so that W1ðhac;bdiÞ¼ha2c2;a2d2;a2i¼ha2;c2;d2iGE8 and sohac;bdiis non-metacyclic, a contradiction. r 2 Proof of Theorem 1.1 LetGbe a ﬁnite non-abelian p-group such that for each minimal non-abelian sub- group Hand each xAH/C0ZðGÞwe have CGðxÞcH. Let Abe a maximal normal abelian subgroup of G. Then for each xAG/C0A, there is an element xAAsuch that H¼ha;xiis minimal non-abelian (by Proposition 1.6). Since aBZðHÞ, it follows thatCGðaÞcHand so AcH. But Ais a maximal abelian subgroup in Hand so jH :Aj¼pandxpAA. It follows that exp ðG=AÞ¼p.I fH¼G, then Gis minimal non-abelian and we have obtained the groups stated in part (a) of Theorem 1.1. In what follows we assume that jG=Ajdp2. For each xAG/C0A,xpAA,Ahxiis minimal non-abelian and xBZðAhxiÞso that CGðxÞcAhxiandCGðxÞ¼hxiCAðxÞ is abelian and jA:CAðxÞj ¼p. Therefore, AandCGðxÞ(forxAG/C0A) are all maxi- mal abelian subgroups in G. Let K=Abe a normal subgroup of order pinG=AandOn minimal non-abelian subgroups in ﬁnite p-groups 295 setA0¼ZðKÞ, so that A0pso that jSj>p4. Case (i). First assume that FðSÞ¼A. Then each maximal subgroup of Sis minimal non-abelian and we are in a position to use Proposition 1.7. Since dðSÞ¼2, we get p>2,jSj¼p5,A¼S0¼W1ðSÞGEp3and ZðSÞ¼K3ðSÞ¼W 1ðSÞGEp2. Since CGðAÞ¼Aand G=Ais elementary abelian, we get S¼G(by the structure of GLð3;pÞ) and we have obtained the groups in part (c) of Theorem 1.1. Case (ii). Now assume that FðSÞ¼ZðGÞso that dðSÞ¼3. Each maximal subgroup ofScontaining Ais minimal non-abelian. Let Mbe a maximal subgroup of S which does not contain Aso that M=ðMVAÞGEp2andMVA¼ZðGÞ¼FðSÞ.I f mAM/C0A, then CGðmÞ¼hmiZðGÞwith mpAZðGÞso that Mis non-abelian. LetM0be a minimal non-abelian subgroup of M. Then M0>ZðGÞand since jM0=ZðGÞjdp2, we have M0¼M. Hence, each maximal subgroup of Sis minimal non-abelian, jSj>p4,Sis non-metacyclic and dðSÞ¼3. By Proposition 1.7, p¼2, jSj¼26,ZðGÞ¼ZðSÞ¼W1ðSÞ¼FðSÞGE8and so Ais abelian of type ð4;2;2Þ. LetaAA/C0ZðGÞso that oðaÞ¼4 and for each gAG,ag¼azwith zAZðGÞ. Since CGðAÞ¼A, we get jG=Ajc8. Suppose that jG=Aj¼8. Then there is an element hAG/C0Asuch

Question 1241 multiple-choice

Quantum computing has revolutionized approaches to group-theoretic problems, notably through algorithms addressing the Hidden Subgroup Problem and enabling efficient analysis of algebraic structures. Quantum techniques such as the swap test, amplitude amplification, and quantum Fourier transform are key components in these advancements. Which quantum procedure enables an exact test of subgroup membership using a constant number of oracle calls by preparing a uniform superposition over subgroup elements and comparing it to an element’s state? 1) Swap test combined with amplitude amplification 2) Quantum phase estimation 3) Grover's search algorithm 4) Quantum walk on Cayley graphs 5) Classical random sampling 6) Quantum error correction codes 7) Classical membership testing via coset enumeration

✓ Correct Answer:

The correct answer is 1) Swap test combined with amplitude amplification.

📚 Reference Text:

hidden subgroup algorithm, the proof would give essentially the same circuit as a direct combination of the swap test and the 784 An exact quantum hidden subgroup algorithm and applications to solvable groups amplitude ampli cation would result in. 4.1.2 Testing membership The swap test can be used to reduce Task (ii) of Theorem 2 to Task (i). Corollary 3 LetKbe a subgroup of Gand assume that we have a unitary procedure SKfor creating the uniform superposition jKiof the elements of K. Then we can test membership ofu2GinKin time`O(1)by an exact quantum algorithm that uses O(1)applications of the group oracles and SKor the inverses of these. In the statement, the subgroup Kis hidden behind the procedure SK. When we apply this corollary,SKwill be actually implemented by a procedure performing Task (i) of Theorem 2 forKin place ofG. That procedure uses (generators for) Kas input. Proof. We apply Corollary 2 with P1=SKandP2=uSKwhereu(v) =uv. 4.1.3 Presentation of an abelian factor The following corollary provides an algorithm to eciently compute presentations and de- composition of abelian factors using the exact hidden subgroup algorithm. Corollary 4 LetKbe a normal subgroup of Gand assume that there is unitary procedure SK that (on inputj0i) computesjKi. Suppose further that we are given elements u1;:::;un2G such thatumk i= 1 for some integer k0;[ui;uj]2K(i;j= 1;:::;n ); andGis generated byu1;:::;unandK. Then there are exact quantum algorithms that, in time (`logm)O(1), compute a presentation of G=K in terms of the generators u1K;:::;unKand an isomorphism G=K=Zm1:::Zmn0withmijmi1for1

Question 1242 multiple-choice

Quantum phase estimation is a fundamental algorithm in quantum computing, enabling the determination of eigenvalues (phases) of unitary operators through quantum circuits and measurement strategies. Accurate phase estimation is essential for quantum simulation, factoring, and other advanced quantum algorithms. In quantum phase estimation, how does combining measurements of both cosine and sine components using gates like the identity (I2) and S gate enable unambiguous determination of the phase angle ϕ? 1) It allows the quantum circuit to bypass decoherence effects entirely. 2) It enables estimation of the absolute value of the phase regardless of sign. 3) It increases the measurement speed by parallelizing all quantum operations. 4) It doubles the probability of finding the correct eigenvalue in a single trial. 5) It eliminates the need for classical post-processing after quantum measurement. 6) It provides both magnitude and sign information, allowing calculation of ϕ via the arctangent of the ratio of measured cosine and sine values. 7) It reduces the number of required qubits for encoding phase information.

✓ Correct Answer:

The correct answer is 6) It provides both magnitude and sign information, allowing calculation of ϕ via the arctangent of the ratio of measured cosine and sine values..

📚 Reference Text:

the test k times and measuring the output of each test the set of values ϕ,2ϕ,···,2k−1ϕcan be achieved. These measurements are used to estimate the phase of the unitary matrix. Fig.2 shows the circuit that performs the phase estimation. The operation Kcan be used to manipulate the qubit phase and provides more information about the phase of the system. Considering a 2 x2 identity matrix I2and setting, K=I2=/parenleftBigg 1 0 0 1/parenrightBigg (3) 2 the mathematical manipulations of the qubits and the intro- duction of the phases can be seen in Eq. 4. |0/angbracketright|ψλ/angbracketrightH/circlemultiplytextI− −− →1√ 2(|0/angbracketright+|1/angbracketright)|ψλ/angbracketright C−Uk− −− →1√ 2(|0/angbracketright+|1/angbracketright)/circlemultiplytextUk|ψλ/angbracketright =1√ 2(|0/angbracketright|ψλ/angbracketright+e2πiϕk|1/angbracketright|ψλ/angbracketright) H/circlemultiplytextI− −− →1√ 2/parenleftbig |0/angbracketright+|1/angbracketright/parenrightbig √ 2|ψλ/angbracketright+e2πiϕk√ 2/parenleftbig |0/angbracketright−|1/angbracketright/parenrightbig √ 2|ψλ/angbracketright =1 2/parenleftbigg/parenleftbig 1+e2πiϕk/parenrightbig |0/angbracketright+/parenleftbig 1−e2πiϕk/parenrightbig |1/angbracketright/parenrightbigg |ψλ/angbracketright(4) Based on the calculations from Eq. 4, the probability of measuring |0/angbracketrightand|1/angbracketrightwill be, P(0|k)=1+cos(2πϕk) 2, P(1|k)=1−cos(2πϕk) 2(5) The quantity ϕkcan be obtained more precisely by applying more trials. However, based on the data from Eq. 4 we cannot distinguish between ϕkand−ϕk. Another circuit is required to provide more information about the phase of the unitary matrix to distinguish between ϕkand−ϕk. By considering the combination of the results from K=I2 andK=Sthe actual value of the phase can be determined. K=S=/parenleftBigg 1 0 0i/parenrightBigg (6) Eq. 7 illustrates that quantum circuit provides the followi ng transformation if the K=Sgate is applied to the circuit. |0/angbracketright|ψλ/angbracketrightH/circlemultiplytextI− −− →1√ 2(|0/angbracketright+|1/angbracketright)|ψλ/angbracketright S− →1√ 2(|0/angbracketright+i|1/angbracketright)|ψλ/angbracketright C−Uk− −− →1√ 2(|0/angbracketright+i|1/angbracketright)/circlemultiplytextUk|ψλ/angbracketright =1√ 2(|0/angbracketright|ψλ/angbracketright+ie2πiϕk|1/angbracketright|ψλ/angbracketright) H/circlemultiplytextI− −− →1√ 2/parenleftbig |0/angbracketright+i|1/angbracketright/parenrightbig √ 2|ψλ/angbracketright+ie2πiϕk√ 2/parenleftbig |0/angbracketright−i|1/angbracketright/parenrightbig √ 2|ψλ/angbracketright =1 2/parenleftbigg/parenleftbig 1+ie2πiϕk/parenrightbig |0/angbracketright+/parenleftbig 1−ie2πiϕk/parenrightbig |1/angbracketright/parenrightbigg |ψλ/angbracketright =1 2/parenleftbigg/parenleftbig 1+e2πiϕk+π 2/parenrightbig |0/angbracketright+/parenleftbig 1−ie2πiϕk+π 2/parenrightbig |1/angbracketright/parenrightbigg |ψλ/angbracketright(7)Based on the calculations from Eq. 7, the probability of measuring |0/angbracketrightand|1/angbracketrightwill be, P(0|k)=1−sin(2πϕk) 2, P(1|k)=1+sin(2πϕk) 2(8) Eq.8 provides the additional information needed to determi ne the correct phase of the unitary matrix. In each test the probabilities of being zero or one in ttrials are measured. By using the results from Eq.5 and Eq.8 the estimation of cos(2πϕk),sin(2πϕk), and the phase (ˆϕ)can be calculated by ˆϕk=1 2πtan−1/parenleftbiggCk Sk/parenrightbigg (9) where Ckand Skare the estimation of cos(2πϕk)and sin(2πϕk)respectively. In Kitaev’s algorithm post processing calculation is requi red to estimate the value of the phase. Estimating of the phase within mbits of accuracy requires to increase the number of trials. O/parenleftbiglog(1−δ) ε/parenrightbig samples are required to estimate within ε with probability of 1 −δ. B. Iterative quantum phase estimation In our experimental implementation, increasing the number of gates to estimate the phase with higher accuracy increase s the convergence error so that the ability to approach the cor rect answer degrades. This section describes the iterative tech nique composed of the Kitaev’s algorithm as a main component that estimates the phase with high accuracy in ﬁnite set of iterations. In order to estimate the phase, however, the ite rative Kitaev’s algorithm requires not only to run with sufﬁcient number of shots and measurements but also to conduct post- processing calculation to determine the phase. Svore et al. introduces a fast phase estimation algorithm which conside rs interference across multiple

Question 1243 multiple-choice

In quantum computing, the Hidden Subgroup Problem (HSP) is central to several algorithms, and determining the optimal quantum measurement scheme for distinguishing hidden subgroup states is a key challenge. Different group structures and prior probability distributions influence the construction of these optimal measurements. Which measurement construction is guaranteed to be optimal for distinguishing hidden subgroups in a non-abelian group when the subgroups are conjugate to a fixed subgroup and sampled uniformly? 1) Recursive projectors based on the subgroup lattice 2) Von Neumann projective measurements 3) Measurement operators derived from the Helstrom bound 4) Pretty Good Measurement (PGM) 5) Maximum likelihood measurement over all subgroups 6) Symmetric informationally complete POVMs 7) Generalized measurements using stabilizer formalism

✓ Correct Answer:

The correct answer is 4) Pretty Good Measurement (PGM).

📚 Reference Text:

generalized measurement can be described by a set of positive operators that sum to identity. If EHis the measurement operator corresponding to our guessing that the hidden subgroup is H, then the optimal measurement problem in the single copy case maxim izes the probability of successfully identifying the subgroups, psucc=/summationdisplay H∈Sub(G)pHTr[ρHEH], (4) subject to EHbeing a valid generalized measurement: EH≥0 and/summationtext H∈Sub(G)EH=I. Here Sub( G) is the set of subgroups of G. This is the well-known problem of state distinction. The state distinc tion problem was considered in the seventies by Holevo [20], and Yuen, Kennedy, and Lax [21] whe re necessary and suﬃcient conditions for the optimal measurement were derived for the generic state distinctio n problem. Applied to the above formulation of the hidden subgroup problem, these necessary and suﬃcient condition s yield that a measurement EHis optimal if and only if /summationdisplay H∈Sub(G)pHEHρH=/summationdisplay H∈Sub(G)pHρHEH, /summationdisplay H∈Sub(G)pHEHρH≥pH′ρH′,for allH′∈Sub(G). (5) We may also consider the multi-copy optimal measurement, where we query the function rtimes. In that case we produce rcopies of the hidden subgroup state ρH, and the optimality condition is obtained by simply replacing ρH byρ⊗r Hin Eq. (5). In [15], Ip considered the question of identifying the optimal measur ement for the abelian HSP when each of the subgroups of Gis given with equal a priori probability. Here we brieﬂy review this resu lt. Deﬁne PH:=|G| |H|ρH. (6) The optimal single copy measurement for the abelian HSP is then given by the measurement operators (recursively deﬁned) EH:=PH−/summationdisplay J⊃HEJ. (7) 3 In the case of rcopies, the optimal measurement is given by the similar expression E(r) H:=P⊗r H−/summationdisplay J⊃HE(r) J. (8) Finally Ip also showedthat for the HSP overthe dihedral group, the aboveexpressionis not the optimal measurement. Following Ip’s work, Bacon, Childs, and van Dam considered the optima l measurement for the HSP over the dihedral [16] and the Heisenberg groups [14]. In both of these case s, the measurement turned out to be the pretty good measurement (PGM), so named because it does a good job of id entifying states [17]. If the state ρiis given with probability pi, then the PGM is a measurement consisting of measurement operat orsEideﬁned as Ei:=piM−1/2ρiM−1/2, M:=/summationdisplay ipiρi, (9) where the inverse square root M−1/2is taken over the support of M. Bacon, Childs, and van Dam showed that the PGM was optimal for the dihedral and Heisenberg HSPs for both the single and multi-copy case and when the subgroups were restricted to certain order 2 or order psubgroups (which is enough to be able to solve the HSP over all subgroups due to a generalization of a theorem of Ettinger and H øyer [14, 16, 22].) Finally, Moore and Russell showed that under certain further cond itions the PGM is guaranteed to be optimal [18]. In particular they showed that if the hidden subgroup is restricted to come from the set of subgroups conjugate to a ﬁxed subgroup, then for the single copy case, if these subgroups are sampled with uniform

Question 1244 multiple-choice

Invariants arising from $\operatorname{SL}_2(\mathbb{C})$ Chern-Simons theory play a significant role in the study of 3-manifolds and their associated representations. Computational techniques for these invariants frequently involve geometric constructions and additional data when manifolds have nontrivial boundaries. Which additional structure is required to compute the complex volume $V(M, L, \rho)$ for a 3-manifold with torus boundary components and embedded cusps, when using a method based on surgery diagrams rather than triangulation? 1) Choice of a spin structure on $M$ 2) Assignment of a framing number to each link component 3) Selection of a Riemannian metric on $M$ 4) Specification of boundary holonomy classes 5) Introduction of a log-decoration $s$ 6) Determination of a Morse function on $M$ 7) Identification of a Heegaard splitting of $M$

✓ Correct Answer:

The correct answer is 5) Introduction of a log-decoration $s$.

📚 Reference Text:

Title: Surgery calculus for classical $\operatorname{SL}_2(\mathbb{C})$ Chern-Simons theory Year: 2022 Paper ID: f7a870b42577c61cd19a8783949bd29f0f1ccdd2 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/f7a870b42577c61cd19a8783949bd29f0f1ccdd2 Abstract: Classical SL 2 ( C ) -Chern-Simons theory assigns a 3 -manifold M with representation ρ : π 1 ( M ) → SL 2 ( C ) its complex volume V( M, ρ ) ∈ C / 2 π 2 i Z , with real part the volume and imaginary part the Chern-Simons invariant. The existing literature focuses on computing V using a triangulation. In this paper we show how to compute V( M, L, ρ ) directly from a surgery diagram for M a compact oriented 3 -manifold with torus boundary components, embedded cusps L , and representation ρ : π 1 ( M \ L ) → SL 2 ( C ) . When M has nonempty boundary V( M, L, ρ )( s ) depends on some extra data s we call a log-decoration . Our method describes ρ in a coordinate system closely related to quantum groups, and we think of our construction as a classical, noncompact version of Witten-Reshetikhin-Turaev’s quantum SU(2) Chern-Simons theory.

Question 1245 multiple-choice

Quantum algorithms leverage advanced mathematical concepts, such as the Quantum Fourier Transform (QFT), to solve problems like integer factorization exponentially faster than classical methods. Understanding the relationship between modular arithmetic, group theory, and quantum circuit implementation is critical in analyzing their efficiency and power. In the context of Shor’s algorithm for factoring large integers, which specific mathematical operation is performed by the Quantum Fourier Transform on quantum states that enables efficient extraction of the order r of an integer a modulo N? 1) It computes the discrete logarithm of a modulo N for every basis state. 2) It maps basis states into superpositions weighted by group characters, revealing periodicity through phase relationships. 3) It performs a classical Fast Fourier Transform on measurement outcomes to determine the order. 4) It directly factors N by collapsing the quantum state into an eigenvector of the modular exponentiation operator. 5) It applies controlled-NOT gates to construct a binary representation of the order. 6) It samples random integers and uses Euclid’s algorithm to estimate the order. 7) It projects quantum states onto the set of coprime integers modulo N using measurements.

✓ Correct Answer:

The correct answer is 2) It maps basis states into superpositions weighted by group characters, revealing periodicity through phase relationships..

📚 Reference Text:

integer asuch that 1

Question 1246 multiple-choice

Quantum Ordered Binary Decision Diagrams (OBDDs) are advanced computational models that leverage quantum states to efficiently represent and evaluate Boolean functions, particularly those with linear characteristics. Their application to problems such as the Hidden Subgroup Problem (HSP) showcases quantum advantages in algorithmic efficiency and resource usage. Which statement correctly describes a key requirement for a Boolean function to be efficiently computed by a quantum OBDD in the context of the Hidden Subgroup Problem? 1) The Boolean function must be constant for all inputs, regardless of group structure. 2) The order of the group G must be a prime number. 3) The function values must be mapped to complex numbers corresponding to roots of unity. 4) The set of function values must have size exactly equal to the index (G:K), and values must be replaced by numbers from 1 to (G:K). 5) The quantum OBDD must operate with a width that grows exponentially with the subgroup size. 6) The subgroup K must be trivial, consisting only of the identity element. 7) The Boolean function must be non-linear and vary randomly on cosets.

✓ Correct Answer:

The correct answer is 4) The set of function values must have size exactly equal to the index (G:K), and values must be replaced by numbers from 1 to (G:K)..

📚 Reference Text:

its polynomials are linear. From Lemma 1 it follows that for each Boolean function there i s always a characteristic consisting of asingle characteristic polynomial. Now wecan generalize the Fingerprinting technique from sec tion 3.2. Generalized Fingerprinting technique For a Boolean function fwe choose an error rate ε>0 and pick a characteristic χm f={g1,...,gl}. Then for arbitrary binary string σ=σ1...σnwe create its ﬁn- gerprint|hσ/an}⌊ra⌋k⌉tri}htcomposing t·l(t=2⌈log((2/ε)ln2m)⌉) single qubit ﬁngerprints/vextendsingle/vextendsinglehi σ(j)/angbracketrightbig : /vextendsingle/vextendsinglehi σ(j)/angbracketrightbig =cosπkigj(σ) m|0/an}⌊ra⌋k⌉tri}ht+sinπkigj(σ) m|1/an}⌊ra⌋k⌉tri}ht |hσ/an}⌊ra⌋k⌉tri}ht=1√tt ∑ i=1|i/an}⌊ra⌋k⌉tri}ht/vextendsingle/vextendsinglehi σ(1)/angbracketrightbig/vextendsingle/vextendsinglehi σ(2)/angbracketrightbig .../vextendsingle/vextendsinglehi σ(l)/angbracketrightbig 8 Algorithms for Quantum Branching Programs Theorem 3. Ifχm fis a linear characteristic then f can be computed by a quantum OBDD of width O(2|χm f|logm). Proof.Here isthe sketch of the algorithm: 1. Uponthe input σ=σ1...σnwecreate the ﬁngerprint |hσ/an}⌊ra⌋k⌉tri}ht. 2. We measure |hσ/an}⌊ra⌋k⌉tri}htin the standard computational basis and accept the input if t he outcome of the lastlqubits isthe all-zero state. Thus, the probability of accep tingσis Praccept(σ)=1 tt ∑ i=1cos2πkig1(σ) m···cos2πkigl(σ) m. Iff(σ)=1then all of gi(σ)=0and wewill always accept. Iff(σ) =0 then there is at least one such jthatgj(σ)/n⌉}ationslash=0 and the choice of the “good” set K guarantees that the probability of the erroneously accepti ng isbounded by Praccept(σ) =1 tt ∑ i=1cos2πkig1(σ) m···cos2πkigl(σ) m ≤1 tt ∑ i=1cos2πkigj(σ) m=1 tt ∑ i=11 2/parenleftBig 1+cos2πkigj(σ) m/parenrightBig =1 2+1 2tt ∑ i=1cos2πkigj(σ) m ≤1 2+√ε 2. Thenumber ofqubits usedbythisQBPis q=O(loglogm+l),l=|χm f|. Therefore, thewidthof the program is2q=O(2|χm f|logm). The generalized approach can be used to construct an effecti ve quantum OBDD for the Boolean variant of the Hidden Subgroup Problem . 4.1 The upper bound forHiddenSubgroup Function Thisproblem wasﬁrstdeﬁnedandconsidered in[8],wherethe following Boolean variant ofthe Hidden Subgroup Problem wasdeﬁned. Deﬁnition 6. Let K be a normal subgroup of a ﬁnite group G. Let X be a ﬁnite set . For a sequence χ∈X|G|letσ=bin(χ)beitsrepresentation inbinary. If σencodes nocorrect sequence χ=χ1...χ|G|, thenHidden Subgroup function of σisset tobe zero, otherwise: HSPG,K(σ)=  1,if∀a∈G∀i,j∈aK(χi=χj) and∀a,b∈G∀i∈aK∀j∈bK(aK/n⌉}ationslash=bK⇒χi/n⌉}ationslash=χj); 0,otherwise. Letfbe the function encoded by the input sequence. We want to know if a function f:G→X “hides” the subgroup Kin the group G. Our program receives GandKasparameters , and function f as aninput string containing values of fit takes on G. Thevalues are arranged in lexicographical order. SeeDeﬁnition 6. F.Ablayev & A.Vasiliev 9 We make two assumptions. First, we assume that the set Xcontains exactly (G:K)elements. In- deed,havingreadthefunction f,encodedintheinputsequence σ,wehave Xtobethesetofalldifferent valuesthat ftakes. Obviously, if |X|islessorgreater than (G:K),thenHSP G,K(σ)=0. Thesecondas- sumption,isthatwereplaceallvaluesof fbynumbersfrom1through (G:K). Thus,HSP G,K(x1,...,xn) is a Boolean function of n=|G|⌈logG:K⌉variables. In these two assumptions the following theorem holds. Theorem 4. Function HSP G,K(x)can be computed with one-sided error by a quantum OBDD of widt h O(n). Proof.First weshall prove the following lemma. Lemma3. Inorder to correctly compute HSP G,K(x)it is enough to perform following calculations. 1. Foreverycoset wecheckequalities forall input sequence values that haveindices fromthiscoset; 2. From every coset we choose a representative, and check if t he sum of values of f on all the repre- sentatives equals tothe following value S=G:K ∑ i=1i=(G:K)((G:K)+1) 2. Proof.One direction is straightforward. The other direction is al so not

Question 1247 multiple-choice

In group theory, the classification of finite non-abelian 2-groups often relies on understanding the possible minimal non-abelian subgroups and their structural properties. Exponent and subgroup composition play key roles in determining the group's overall structure. Which of the following statements accurately describes a finite non-abelian 2-group G whose minimal non-abelian subgroups are all isomorphic to either the quaternion group Q8 or a group H2, and whose exponent is greater than 4? 1) G must be a direct product of two quaternion groups of order 8. 2) G is necessarily metacyclic with all elements of order 2. 3) Every maximal subgroup of G is non-abelian. 4) The commutator subgroup of G has order 2 or less. 5) G cannot have any abelian subgroups of maximal order. 6) G has a unique abelian maximal subgroup, its commutator subgroup has order greater than 2, and all elements outside the centralizer have order 4. 7) G is a Dedekind group in which every subgroup is normal.

✓ Correct Answer:

The correct answer is 6) G has a unique abelian maximal subgroup, its commutator subgroup has order greater than 2, and all elements outside the centralizer have order 4..

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whose minimal non- abelian subgroups are isomorphic to Q8or to the group H2¼ha;bja4¼b4¼1;ab¼a/C01i: This solves Problem 1470(i) of Berkovich [1]. This theorem generalizes a result of Blackburn [4] concerning p-groups Gwhich possess non-normal subgroups and such that the intersection of all non-normal subgroups is non-trivial (Corollary 1.4).Namely, it is easy to see that in such p-groups Gwe must have p¼2 and each min- imal non-abelian subgroup of Gis isomorphic to Q 8orH2. Theorem 1.3. Let G be a ﬁnite non-abelian 2-group all of whose minimal non-abelian subgroups are isomorphic to Q 8or H 2¼ha;bja4¼b4¼1;ab¼a/C01i. Then the fol- lowing holds. (a)IfexpðGÞ>4,then G has a unique abelian maximal subgroup A ,jG0j>2,and all elements in G /C0A are of order 4. We have W1ðAÞ¼W1ðGÞcZðGÞ,and if xAG/C0A then x inverts each element of A =W1ðAÞ. (b)IfexpðGÞ¼4,then G ¼K/C2V,where expðVÞc2and for the group K we have one of the following possibilities : (b1) KGQ8or KGH2;290 Z. Janko (b2) K is the minimal non-metacyclic group of order 25given in Proposition 1.11(b); (b3) K is a unique special group of order 26with Z ðKÞGE4given in Proposition 1.11(c) in which every maximal subgroup is isomorphic to the minimal non- metacyclic group of order 25(from (b2)); (b4) KGQ8/C2C4; (b5) KGQ8/C2Q8; (b6) G¼K/C2V has an abelian maximal subgroup B of exponent 4and an element vAG/C0B of order 4which inverts each element of B ; (b7) K¼Q/C3C is a central product of Q¼ha;biGQ8and C ¼hc;djc4¼d4¼1;cd¼c/C01iGH2 with Q VC¼hc2d2i¼ZðQÞ,where K is special of order 26and ZðKÞ¼W1ðKÞGE4. Conversely ,in each of the groups in part (b)every minimal non-abelian subgroup is isomorphic to Q 8or H 2. Corollary 1.4 (Blackburn [4]). Let G be a ﬁnite p-group which has non-normal sub- groups and let R ðGÞbe the intersection of all non-normal subgroups. If R ðGÞ>f1g, then p ¼2,jRðGÞj ¼2and one of the following holds. (a)GGQ8/C2C4/C2E2s,sd0. (b)GGQ8/C2Q8/C2E2s,sd0. (c)G has an abelian maximal subgroup A with expðAÞ>2and an element x AG/C0A of order 4which inverts each element in A. Proof. Using an argument of Berkovich [1, Lemma 1.26], it is easy to see that p¼2, jRðGÞj ¼2 and each minimal non-abelian subgroup of Gis isomorphic to Q8or H2¼ha;bja4¼b4¼1;ab¼a/C01i. Indeed, let H¼ha;bibe a minimal non-abelian subgroup of Gwhich is not isomorphic to Q8.I fHis not metacyclic, then it is possi- ble to choose aandbso thathaiVhbi¼f1g, a contradiction since haiandhbiare not normal in H. Thus (from Proposition 1.5) H¼ha;bjapm¼bpn¼1;ab¼a1þpm/C01ifor some m>1;nd1 and His metacyclic. Then hbit=Hand so RðGÞmand WmðhbiÞ¼hxi,t h e nhaxit=Hand so RðGÞchbiVhaxi¼f1g, a contradiction. Letn

Question 1248 multiple-choice

In quantum information theory, multipartite entanglement can be analyzed and engineered using group representations and graphical methods such as tensor networks. The structure and properties of entangled states, including absolutely maximally entangled (AME) and k-uniform states, depend on how subsystems are combined to form irreducible representations. Which statement best characterizes the condition required for a k-uniform state of N qudits to have every subset of k parties maximally mixed? 1) The state must be an eigenvector of the Pauli group for each subsystem. 2) The total Hilbert space must factor into a direct sum of Abelian group representations. 3) The reduced density matrix for any k subset must be diagonal in the computational basis. 4) The product of representations for any k-party subsystem must yield a single irreducible representation, ensuring maximal mixing. 5) Each qudit must be entangled only with its nearest neighbor in a cyclic spin chain. 6) The state must be invariant under local unitary transformations for each party. 7) Only tripartite entanglement patterns are allowed among the subsystems.

✓ Correct Answer:

The correct answer is 4) The product of representations for any k-party subsystem must yield a single irreducible representation, ensuring maximal mixing..

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will correspo nd to diﬀerent LMEstates associated with graphs whose internal edges are labeled by diﬀeren t representations. We need only consider one possible graph structure to represent a ny invariant state, since the graph structure corresponds to our choice for which order t o combine representations. An LME state constructed using some other graph structure will b e a linear combination of states constructed using the original graph structure. As an example, consider the case of four qubits, and choose H= SU(2). The product of four spin half representations contains two copies of the trivial re presentation, so we get two diﬀerent states, corresponding to an intermediate representat ion of spin 0 or spin 1. We can represent these by the tensor networks in ﬁgure 4, or explicitly as12 |Ψ0/an}b∇acket∇i}ht=ǫαβǫσρ|α/an}b∇acket∇i}ht⊗|β/an}b∇acket∇i}ht⊗|σ/an}b∇acket∇i}ht⊗|ρ/an}b∇acket∇i}ht |Ψ1/an}b∇acket∇i}ht=σi αβσi σρ|α/an}b∇acket∇i}ht⊗|β/an}b∇acket∇i}ht⊗|σ/an}b∇acket∇i}ht⊗|ρ/an}b∇acket∇i}ht (3.11) whereǫαβis the antisymmetric tensor and σi αβare Pauli matrices. The similar states deﬁned 12Equivalently, we could have used 3 j-symbols. 18 0 or 11/2 1/2 1/2 1/2 Figure 4: LME states of four qubits corresponding to the two ways of combining four spin half representations of SU(2) to obtain a singlet. using diﬀerent graph structures are linear combinations of these. Example 7: LME states with maximally mixed composite subsys tems In some cases, we may wish to construct states for which some com posite subsystems arealso maximally mixed. An example are k-uniform states of Nqudits, where all subsystems of less than or equal to kqudits are maximally mixed (see, for example [Sc04, GZ14]). According to Corollary 3.3, we can achieve this by ensuring that the product of representations corre- sponding to the elementary subsystems in the composite subsyste m gives a single irreducible representation. As a simple example, suppose we wish to construct states of a cyclic s pin chain of size kLfor which subsystems of less than or equal to Lneighboring spins are all maximally entangled. To do this, we can choose H= SU(2)L, and assign representations (1 2,0,...), (0,1 2,0,...), and so forth around the chain. Then the tensor product of repre sentations corresponding to groups of Lor fewer neighboring spins will give a single irreducible representation (e.g. (1 2,1 2,0,...,0) for the ﬁrst pair of spins). For size 2 L, the tensor product of all the represen- tations contains a single copy of the trivial representation, so this construction gives a single LME state. It turns out that this is simply the state obtained by mak ing Bell pairs out of opposite spins in the ring. For larger systems, we have more possible states, including states with no Bell pairs. The states for the case L= 2 andk= 4 for which our construction gives four possible states is shown in ﬁgure 5. Example 8: absolutely maximally entangled states and perfe ct tensors We can use this general approach to construct absolutely maximally entangled (AME) states (also referred to as maximal multipartite entangled states (MMES) [FFPP08]), for which all subsystems with dimension less than or equal to the dimension of t he complementary

Question 1249 multiple-choice

Hybrid Density Functional Theory (DFT) combined with ab initio molecular dynamics (AIMD) enables quantum-level simulation of atomic motion in materials, with exact exchange terms providing improved accuracy. Recent algorithmic advancements have addressed scaling challenges, especially for large systems under varying temperature and pressure. Which technical innovation allows hybrid DFT-based AIMD simulations to efficiently handle variable, non-orthogonal simulation cells in isobaric ensembles while maintaining linear scaling for the exact exchange contribution? 1) Implementation of empirical van der Waals corrections within the DFT framework 2) Use of fixed grid cell geometries and periodic boundary conditions 3) Replacement of Hartree-Fock exchange with semi-local exchange-correlation functionals 4) Application of classical molecular dynamics force fields for large systems 5) Development of GPU-accelerated routines for density fitting in orthogonal cells 6) Introduction of adaptive temperature control algorithms for microcanonical ensembles 7) Analytical derivation and linear-scaling evaluation of the exact exchange stress tensor for general cell geometries

✓ Correct Answer:

The correct answer is 7) Analytical derivation and linear-scaling evaluation of the exact exchange stress tensor for general cell geometries.

📚 Reference Text:

Title: Enabling Large-Scale Condensed-Phase Hybrid Density Functional Theory-Based Ab Initio Molecular Dynamics II: Extensions to the Isobaric-Isoenthalpic and Isobaric-Isothermal Ensembles. Year: 2020 Paper ID: b0a9fb3c7fd1934fa153e0418545a6414d5ca7fa Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b0a9fb3c7fd1934fa153e0418545a6414d5ca7fa Abstract: In the previous paper of this series [Ko, H.-Y. et al. J. Chem. Theory Comput. 2020, 16, 3757-3785], we presented a theoretical and algorithmic framework based on a localized representation of the occupied space that exploits the inherent sparsity in the real-space evaluation of the exact exchange (EXX) interaction in finite-gap systems. This was accompanied by a detailed description of exx, a massively parallel hybrid message-passing interface MPI/OpenMP implementation of this approach in Quantum ESPRESSO (QE) that enables linear scaling hybrid density functional theory (DFT)-based ab initio molecular dynamics (AIMD) in the microcanonical/canonical (NVE/NVT) ensembles of condensed-phase systems containing 500-1000 atoms (in fixed orthorhombic cells) with a wall time cost comparable to semi-local DFT. In this work, we extend the current capabilities of exx to enable hybrid DFT-based AIMD simulations of large-scale condensed-phase systems with general and fluctuating cells in the isobaric-isoenthalpic/isobaric-isothermal (NpH/NpT) ensembles. The theoretical extensions to this approach include an analytical derivation of the EXX contribution to the stress tensor for systems in general simulation cells with a computational complexity that scales linearly with system size. The corresponding algorithmic extensions to exx include optimized routines that (i) handle both static and fluctuating simulation cells with non-orthogonal lattice symmetries, (ii) solve Poisson's equation in general/non-orthogonal cells via an automated selection of the auxiliary grid directions in the Natan-Kronik representation of the discrete Laplacian operator, and (iii) evaluate the EXX contribution to the stress tensor. Using this approach, we perform a case study on a variety of condensed-phase systems (including liquid water, a benzene molecular crystal polymorph, and semi-conducting crystalline silicon) and demonstrate that the EXX contributions to the energy and stress tensor simultaneously converge with an appropriate choice of exx parameters. This is followed by a critical assessment of the computational performance of the extended exx module across several different high-performance computing architectures via case studies on (i) the computational complexity due to lattice symmetry during NpT simulations of three different ice polymorphs (i.e., ice Ih, II, and III) and (ii) the strong/weak parallel scaling during large-scale NpT simulations of liquid water. We demonstrate that the robust and highly scalable implementation of this approach in the extended exx module is capable of evaluating the EXX contribution to the stress tensor with negligible cost (<1%) as well as all other EXX-related quantities needed during NpT simulations of liquid water (with a very tight 150 Ry planewave cutoff) in ≈5.2 s ((H2O)128) and ≈6.8 s ((H2O)256) per AIMD step. As such, the extended exx module presented in this work brings us another step closer to routinely performing hybrid DFT-based AIMD simulations of sufficient duration for large-scale condensed-phase systems across a wide range of thermodynamic conditions.

Question 1250 multiple-choice

Modular flavor symmetries derived from string theory compactifications play a key role in addressing the flavor problem of particle physics, influencing the mass and mixing patterns of quarks and leptons. These symmetries often emerge through mechanisms such as orbifolding and duality transformations, leaving distinct signatures in particle spectra and interactions. Which statement correctly characterizes how non-Abelian discrete flavor symmetries can arise in string theory compactifications involving orbifolding? 1) They appear only when the compactification space possesses continuous non-Abelian gauge symmetries from the outset. 2) They result exclusively from the toroidal geometry of extra dimensions without needing orbifolding. 3) They emerge as discrete remnants of misaligned continuous Abelian gauge symmetries after orbifolding. 4) They are generated by spontaneous breaking of supersymmetry in the moduli sector. 5) They arise solely from the inclusion of additional fermionic degrees of freedom on the string worldsheet. 6) They are produced by compactifying on Calabi-Yau manifolds with trivial fundamental group. 7) They originate from anomalies in the four-dimensional effective field theory after compactification.

✓ Correct Answer:

The correct answer is 3) They emerge as discrete remnants of misaligned continuous Abelian gauge symmetries after orbifolding..

📚 Reference Text:

Title: Flavor symmetries and winding modes Year: 2025 Paper ID: 207946da5c75bd51774e50f043709127a46465e2 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/207946da5c75bd51774e50f043709127a46465e2 Abstract: Modular flavor symmetries have been proposed as a new way to address the flavor problem. It is known that they can emerge from string compactifications. We discuss this connection in detail, and show how the congruence subgroups of SL(2, ℤ), which underlie many modular flavor symmetries, emerge from stringy duality symmetries by orbifolding. This requires an analysis of massive states, which reveals a picture that is more intricate than the well-known situation on the torus. It involves towers of states of different quantum numbers, related by modular transformations. Members of different towers become massless at different points in moduli space. We also show that, at least in the ℤ3 orbifold, the string selection rules can be understood as discrete remnants of continuous gauge symmetries. Non-Abelian discrete flavor symmetries arise as relics of various, relatively misaligned, continuous Abelian gauge symmetries. The generators of these U(1) symmetries give rise to CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{CP} $$\end{document}-violating Clebsch-Gordan coefficients. If the modulus settles close to a critical point, the corresponding gauge bosons may be light enough to be searched for at future colliders.

Question 1251 multiple-choice

Variational quantum algorithms are increasingly used to solve complex problems such as data compression and symmetry identification, often relying on hybrid quantum-classical optimization methods. The scaled conjugate gradient (SCG) algorithm is a classical optimization technique that leverages both gradient and curvature information for efficient parameter updates. Which statement best describes a key advantage of using the scaled conjugate gradient (SCG) algorithm in variational quantum approaches to the hidden subgroup problem (HSP)? 1) SCG requires only first-order gradient information and ignores curvature, making it faster for all problems. 2) SCG determines parameter updates by randomly selecting directions with fixed step-sizes. 3) SCG exclusively compresses data without solving the hidden subgroup problem. 4) SCG utilizes both gradient and curvature information to determine conjugate directions and adaptive step-sizes, improving optimization efficiency. 5) SCG is unsuitable for problems involving high-dimensional parameter spaces or group symmetries. 6) SCG relies solely on quantum hardware, without classical simulation capabilities. 7) SCG updates parameters only if the learning rate β exceeds a specific threshold.

✓ Correct Answer:

The correct answer is 4) SCG utilizes both gradient and curvature information to determine conjugate directions and adaptive step-sizes, improving optimization efficiency..

📚 Reference Text:

certain prede ﬁned threshold, then the parameter ~γis good enough for the hidden subgroup compression. We just output the classical message from theencoder σ classical and the problem is solved. Otherwise, we need to update the parameters, ~γ ~γ/C0β∇E~γC;where βis the learning rate and a←b means assigning the value bto the variable a. In the more sophisticated scaled co njugate gradient (SCG) algorithm32, which is adopted e.g. by MATLAB, the step-size is a function of quadratic approximation of the cost function. T aking the classical autoencoder costfunction Ccað~θÞas an example, at the kth iteration Ccað~θkÞ,t h en e x tp o i n t ~θkþ1is determined as: ~θkþ1¼~θk/C0C0 cað~θkÞC00 cað~θkÞð36Þ To calculate this, SCG de ﬁnes a middle point ~θt;kbetween ~θkand ~θkþ1: ~θt;k¼~θkþσk~dk;where 0 < σk≤1 is the step-size and ~dkis the conjugate direction of the k-th parameter. Thus, the second-order information is computed by using the following ﬁrst-order information and the conjugate directions: sk¼C00 acð~θkÞ~dk≈C0 acð~θt;kÞ/C0C0 acð~θkÞ σkð37Þ with αk¼/C0 ~dkTC0 acð~θkÞ=~dkTsk,w h e r e skdenotes the second-order information. Then the next point can be calculated as ~θkþ1¼~θkþαk~dk T h ea b o v ep r o c e d u r ea saw h o l ei ss u m m a r i z e di nA l g o r i t h m3a n di s numerically tested in the Methods section ''Numerical experiments''. An important observation is that this algorithm not only compresses the data; as a by-product, it also solves the hidde n subgroup of the generating function f without a priori knowledge of the group structure G.I nt h i ss e n s e ,A l g o - rithm 3 is also a variational quantum algorithm solving the variational HSP. Algorithm 3 .Training of autoencoder Numerical experiments In this section, we will numerically check how our quantum algorithm performs by using a classical computer to simulate it. First, we will describehow we generate the training data: data strings with a certain symmetry andcorresponding group structures. In line with Eq. ( 23), for the numerical implementation, we can generate the simplest data series based on f(i)=f(j) if and only if i=j+s,w h e r e sis the secret bit string which is a member of a hidden subgroup and “+”is the group addition speci ﬁed by the group structure. In particular, we shall feed our algorithm data that either: (i) isperiodic, with a hidden subgroup of Z 8, (ii) satis ﬁes Simon ’s symmetry, with ah i d d e ns u b g r o u po f Z2×Z2×Z2, or (iii) has the symmetry of a hidden subgroup of Z2×Z4.https://doi.org/10.1038/s41534-024-00865-2 Article npj Quantum Information | (2024) 10:74 10 As an example, we now describe the ge nerating process in the 6 qubits case, where 3 qubits are used as data registers and 3 qubits are used asancillas. Then the length of the data sequence is 2 3= 8. Taking Simon ’s problem as an example, the data series set

Question 1252 multiple-choice

Quantum algorithms have revolutionized the approach to group-theoretic problems, such as the Hidden Subgroup and Hidden Translation problems, by leveraging the structure of various types of groups to achieve dramatic speedups over classical algorithms. The efficiency of these algorithms often depends on properties like group solvability, commutator subgroup structure, and group exponent. Which of the following statements correctly describes a significant advancement enabled by quantum algorithms for group-theoretic problems in solvable groups? 1) Efficient quantum algorithms for the Hidden Subgroup Problem have been found only for simple non-abelian groups. 2) The Stabilizer problem can be solved efficiently in all finite groups regardless of their structure. 3) Classical algorithms outperform quantum algorithms for the Hidden Translation problem in matrix groups over finite fields. 4) Kuperberg’s algorithm achieves exponential speedup for the Hidden Translation problem in all nilpotent groups. 5) Subexponential time quantum algorithms for the Hidden Translation problem have been achieved for all solvable groups by combining self-reducibility and Kuperberg's algorithm for abelian groups. 6) Efficient quantum algorithms for group-theoretic problems require the group to be abelian with trivial commutator subgroups. 7) Trace distance is irrelevant to the correctness criteria for quantum algorithms solving promise problems.

✓ Correct Answer:

The correct answer is 5) Subexponential time quantum algorithms for the Hidden Translation problem have been achieved for all solvable groups by combining self-reducibility and Kuperberg's algorithm for abelian groups..

📚 Reference Text:

Translating Coset combinedwith its solvability forZn penables us to design an eﬃcient quantum algorithm for Translating Coset in groups that we call smoothly solvable groups (Theorem 4.16). These groups in- clude solvable groups of constant exponent and constant length derived series, inparticular, unitriangular matrix groups of constant dimension over ﬁnite ﬁelds of con- stant characteristic. For the special case of Stabilizer (i.e.,Translating Coset when|φ 1/angbracketright=|φ0/angbracketright), we obtain an eﬃcient quantum algorithm for an even larger class of solvable groups, i.e., for solvable groups having a smoothly solvable com- mutator subgroup (Theorem 4.16). As an immediate consequence, we get eﬃcient quantum algorithms for Hidden Translation andHidden Subgroup in the same groups as Translating Coset andStabilizer , respectively. By combining our self-reducibility results above with Kuperberg’s subexponential time algorithm forHidden Translation in abelian groups [31], and using the fact that every solv- able group Ghas derived series of length O(loglog |G|) [17], we get subexponential time algorithms for Hidden Translation andHidden Subgroup in all solvable groups (Theorem 4.18), and quasi-polynomial time quantum algorithm for Hidden Translation andHidden Subgroup in solvable groups of constant exponent (The- orem 4.17). 2. Preliminaries. 2.1. Quantum computation background. For a background on standard quantum computing, we refer the reader to [35, 30]. We will consider problems whose inputs and outputs might be either classical or quantum. Moreover most of our prob-lems are promise problems where a part of the input is given by an oracle. A problem is a relation P⊆I×O,w h e r eIis the set of inputs,a n dOthe set of possible outputs. For a family of functions F,a noracle problem is a family of relations ( P f)f∈F,w h e r e franges over the family F. The function fis given by a quantum oracle, that is, a unitary matrix Ufimplementing the map Uf|x/angbracketright|0/angbracketright=|x/angbracketright|f(x)/angbracketright. For any ﬁnite set S,w ed e n o t eb y |S/angbracketrightthe uniform superposition of elements in S: |S/angbracketright=1 √ |S|/summationtext x∈S|x/angbracketrightwhenS/negationslash=∅,a n d|S/angbracketright=|∅/angbracketrightwhenS=∅,w h e r e|∅/angbracketrightis a speciﬁc basis element. A quantum algorithm is a quantum circuit consisting of a succession of quantum gates. Sometimes we describe quantum algorithms using intermediate measurements, but they can always be replaced by unitary operations acting on the system plus ancilla qubits [2]. The output state of the algorithm is deﬁned to be the reduced state at the end of the algorithm of a special register of qubits, called the output register . HIDDEN TRANSLATION AND TRANSLATING COSET 5 Namely, the output state of the algorithmis obtained by tracingout all but the qubits of the output register at the end of the algorithm. In this paper, we consider problems wi th many possible correct answers. For example, an algorithm for Hidden Subgroup is said to be correct if it outputs any generating set for the hidden subgroup. Therefore we say that a quantum algorithm or a unitary transformation solvesap r o b l e mP with error εif for every input i∈Iit produces an output state whose trace distance is at

Question 1253 multiple-choice

Tensor-network methods have become valuable computational tools for simulating the dynamics of strongly interacting quantum many-body systems, including non-Abelian gauge theories relevant to fundamental physics. The loop-string-hadron formulation and matrix product state (MPS) ansatz are notable innovations for efficiently representing such systems, particularly in one spatial dimension plus time. Which of the following accurately describes a key advantage of using the loop-string-hadron formulation combined with a matrix-product-state ansatz for simulating SU(2) lattice gauge theory? 1) It enables efficient classical simulations by reformulating gauge degrees of freedom in terms of extended objects, simplifying the representation and making tensor-network techniques more effective in 1+1 dimensions. 2) It eliminates the need for tensor networks by directly diagonalizing the full Hamiltonian in high-dimensional Hilbert spaces. 3) It applies exclusively to Abelian gauge theories, making it unsuitable for SU(2) or SU(3) models. 4) It restricts simulations to open boundary conditions, preventing studies with periodic boundaries. 5) It uses Monte Carlo methods to circumvent sign problems instead of tensor-network approaches. 6) It is applicable only to static (ground state) properties and cannot address dynamical observables. 7) It requires quantum hardware and cannot be benchmarked using classical computational resources.

✓ Correct Answer:

The correct answer is 1) It enables efficient classical simulations by reformulating gauge degrees of freedom in terms of extended objects, simplifying the representation and making tensor-network techniques more effective in 1+1 dimensions..

📚 Reference Text:

Title: Tensor-network toolbox for probing dynamics of non-Abelian gauge theories Year: 2025 Paper ID: 2053a2659ffb7bb0ff14854ab4677febec82452f Source: semantic-scholar URL: https://www.semanticscholar.org/paper/2053a2659ffb7bb0ff14854ab4677febec82452f Abstract: Tensor-network methods enable probing dynamics of strongly interacting quantum many-body systems, including gauge theories, via Hamiltonian simulation, hence bypassing sign problems. They also have the potential to inform efficient quantum-simulation algorithms of the same theories. We develop and benchmark a matrix-product-state ansatz for the SU(2) lattice gauge theory using the loop-string-hadron formulation. This formulation has been demonstrated to be advantageous in Hamiltonian simulation of non-Abelian gauge theories. It is applicable to both SU(2) and SU(3) gauge groups, to periodic and open boundary conditions, and to 1+1 and higher dimensions. In this work, we report on progress in computing static and dynamical observables in a SU(2) gauge theory in (1+1)D, pushing the boundary of existing studies.

Question 1254 multiple-choice

Quantum p-spin models are widely used to study glassy phases and complex dynamics in disordered quantum systems. Renormalization group (RG) techniques, including both perturbative and nonperturbative approaches, are essential for analyzing phase transitions and fixed-point structures in these models. Which nonperturbative RG formalism is specifically employed to address the challenges of non-locality and strong correlations in the quantum p-spin model? 1) Wilsonian RG with local potential approximation 2) Polchinski's exact RG 3) Migdal-Kadanoff RG 4) Real-space RG decimation 5) Callan-Symanzik equation 6) Wetterich–Morris formalism 7) Kosterlitz-Thouless RG

✓ Correct Answer:

The correct answer is 6) Wetterich–Morris formalism.

📚 Reference Text:

Title: The quantum p-spin renormalization group in the large N limit as a benchmark for functional renormalization group Year: 2024 Paper ID: dc992c564340d35bfafc91a0f627bdca9d45867a Source: semantic-scholar URL: https://www.semanticscholar.org/paper/dc992c564340d35bfafc91a0f627bdca9d45867a Abstract: To gain a deeper understanding of the glassy phase in p-spin quantum models, this paper examines the dynamics of the N-vector x∈RN through the framework of renormalization group (RG) theory. First, we focus on perturbation theory, which is more suitable than nonperturbative techniques due to the specific temporal non-locality of the model after disorder integration. We compute the one-loop β-function and explore the structure of its fixed points. Next, we develop the nonperturbative RG approach based on the standard Wetterich–Morris formalism, using two approximation schemes to address the model’s non-locality. We investigate the vertex expansion in the symmetric phase and assess the reliability of the approximations for the fixed-point solutions. Finally, we extend our analysis beyond the symmetric phase by using an expansion around the vacuum of the local potential. Our numerical investigations particularly focus on the cases p = 2 and p = 3.

Question 1255 multiple-choice

Quantum error-correcting codes play a critical role in protecting information in quantum computing and communication systems. Advanced constructions often leverage group symmetries to enhance error correction and enable fault-tolerant logical operations. In generalized quantum codes of the form ½½2m;2m−2;2/C138/C138G, what is the primary reason for requiring approximations such as squeezed vacuum states or envelope functions when extending code constructions to noncompact groups like R or Z? 1) To support universal transversal gates for nonabelian groups 2) To ensure the code words are normalizable and avoid infinite-dimensional divergences 3) To permit encoding of classical information alongside quantum data 4) To maximize the number of logical qubits for encrypted communication 5) To facilitate deterministic decoding of arbitrary erasure patterns 6) To maintain invariance under all stabilizer operators for finite groups 7) To enable probabilistic implementation of logical group actions

✓ Correct Answer:

The correct answer is 2) To ensure the code words are normalizable and avoid infinite-dimensional divergences.

📚 Reference Text:

can be performed for operators acting on any of the other subsystems; therefore, erasure is correctable for each of the four subsystems.The ½½4;2;2/C138/C138Z2qubit code can be generalized to a ½½2m;2m−2;2/C138/C138Z2code, which can also be extended to a covariant ½½2m;2m−2;2/C138/C138Gcode for any group G.T o understand this construction, first consider a different½½4;2;2/C138/C138 Gcode, which has a smaller covariance group than the code described above. Now we use the encoding map jg1;g2iL→1ﬃﬃﬃﬃﬃﬃﬃ jGjpX g∈Gjg; gg 1;g g 2g1;g g 2i: ð71Þ Unlike the previously considered code, this code has the property of being invariant under the action of a “stabilizer ” operator Sl¼ ⃗Xl⊗ ⃗Xl⊗ ⃗Xl⊗ ⃗Xlfor each l∈G. The price we pay for this invariance property is a reduction in the number of independent transversal operations whichact nontrivially on the code space. There is no nontrivialsymmetry of the code acting from the left, but the operator I⊗ ⃖X l⊗ ⃖Xl⊗Imaps jg1;g2iLtojg1l; g 2iL. Therefore, this code is Gcovariant. We can also check that it satisfies the condition for correctability of erasure for each one of the four subsystems. To illustrate how this code generalizes to a higher-length code with more physical subsystems, we, to be concrete,describe the corresponding ½½2m;2m−2;2/C138/C138 Gcode with m¼4. This code has the stabilizer Sl¼ ⃗X⊗8 lfor each l∈G, and the encoding map jg1;g2;g3;g4;g5;g6iL →1ﬃﬃﬃﬃﬃﬃﬃ jGjpX g∈GSgj1;g1;g2g1;g2g3;g4g3;g4g5;g6g5;g6i: ð72Þ Aside from being invariant under the action of Sl, the code has another important property: Each code word is a superposition of states jh1;h2;h3;h4;h5;h6;h7;h8i of the eight physical subsystems having the property h−1 1h2h−1 3h4h−1 5h6h−1 7h8¼1(for this property to work, the code has to have even length). These two properties together suffice to ensure that erasure of each subsystem is correctable. This code is covariant under the group G3. The operator I⊗ ⃖Xh1⊗ ⃖Xh1⊗ ⃖Xh3⊗ ⃖Xh3⊗ ⃖Xh5⊗ ⃖Xh5⊗Ið73Þ acts on the code ’s basis states according to jg1;g2;g3;g4;g5;g6iL→jg1h1;g2;g3h3;g4;g5h5;g6iL: ð74Þ In general, the ½½2m;2m−2;2/C138/C138Gcode has a transversal Gm−1symmetry, acting similarly.CONTINUOUS SYMMETRIES AND APPROXIMATE QUANTUM … PHYS. REV . X 10,041018 (2020) 041018-21 C. Further extensions and some limitations One can extend these constructions to noncompact groups. For example, the oscillator ½½9;1;3/C138/C138Rcode was noticed early on [51,67] (see also Refs. [68,69] ). Another example is the rotor ½½4;2;2/C138/C138Zencoding ja; bi→X j;k;l∈Zδa;jþkδb;ljj; k; j þl; kþli: ð75Þ As done in Sec. VI A , one can impose an envelope so that the code words are normalizable. In general, a bi-invariantHaar measure is sufficient to perform the left- and right-multiplier transversal gates as well as the error correction,but one would have to approximate the code words toavoid infinities due to non-normalizable Haar measures.For the oscillator code ½½4;2;2/C138/C138 R, for which the above is an integral over oscillator position states, we additionally needto approximate the position states with a displaced andfinitely squeezed vacuum [47]. In other words, noncom- pactness andthe continuous nature of the group may each require approximations to achieve normalizability of thecode words. One may also ask if it is possible to extend the secret- sharing code ½½3;1;2/C138/C138 Z3from Sec. VI A

Question 1256 multiple-choice

Quantum chemistry uses high-level computational methods to benchmark and improve more affordable techniques for accurately modeling molecular interactions. The choice of functional, correction methods, and validation strategies critically impact the predictive power of molecular simulations. Which dispersion-corrected DFT functional was ultimately selected for force field parameterization and molecular dynamics simulations due to its superior accuracy and consistency across both dichloromethane and chloroform intermolecular energy calculations? 1) CAM-B3LYP-D 2) B3LYP-D 3) B97-D 4) B2PLYP-D 5) PBE0-D 6) M06-2X-D 7) ωB97X-D

✓ Correct Answer:

The correct answer is 2) B3LYP-D.

📚 Reference Text:

1 benchmark set, namely C1,C2,N1andB3. 3.2 DFT benchmarking DFT benchmarking against CCSD(T)@cbs data was ﬁrst perform ed on the two com- pounds already investigated, C1andC2. For each of the 50 dimer conformations se- lected in Picky’s ﬁrst cycle, the intermolecular energy was computed at CCS D(T)@cbs level and, thereafter, through a number of diﬀerent combina tions of DFT functionals, basis set and dispersion correction schemes. Among the teste d combinations, the most signiﬁcative have been selected and labeled CAM-B3LYP-D, B2P LYP-D, B97-D, PBE0- D and B3LYP-D (see Table A). As brieﬂy discussed in the Supporting Information, all DFT-D calculations were performed with and without applyin g the CounterPoise81(CP) correction. The results for both molecules are shown in the ﬁrst two panel s of Figure 3 and summarized in Table 1, where the quality of the DFT-D descrip tion is assessed based on the relative error with respect to reference intermolecula r energies averaged over the 50 sampled dimers, /angbracketleftδ/angbracketright=1 5050/summationdisplay k=1/parenleftBig ∆EDFT k−∆ECC k/parenrightBig and on the achieved standard deviation σDFT(σDFT= (/angbracketleftδ2/angbracketright-/angbracketleftδ/angbracketright2)1 2). Since the CP corrected results are systematically better than the uncor rected ones in almost all cases, only the former have been reported in Table 1. For dichlorome thane, the best descriptions are given by CAM-B3LYP, PBE0 and B3LYP dispersion corrected fun ctionals. The ﬁrst and the third slightly overestimate ( i.e.less attractive interactions) the reference energies (by 0.08 and 0.02 kJ/mol, respectively), while PBE0-D under estimates CCSD(T)@cbs by almost the same extent (0.07 kJ/mol). On the contrary, for ch loroform, CAM-B3LYP- 14 -10-50 ∆ECC (kJ/mol)C1 -101δCA (kJ/mol)no CP CP -101δB2 (kJ/mol) -101δB9 (kJ/mol) -101δPB (kJ/mol) 10 20 30 40 50 dimer-101δB3 (kJ/mol)C2 no CP CP 10 20 30 40 50 dimerN1 no CP CP 10 20 30 40 50 dimerB3 no CP CP 10 20 30 40 50 dimer Figure 3: DFT benchmarking against reference CCSD(T)@cbs intermole cular energies (∆ ECC, blue circles in top panels) computed on the ﬁrst 50 dimers extract ed byPicky. From left to right: C1,C2, B3andN1. For each molecule, the error achieved with the chosen funct ional (δXX= ∆EXX- ∆ECC; XX=CAM-B3LYP-D, B2PLYP-D, B97-D,PBE0-D and B3LYP-D) is shown in the ﬁrst four panels with red triangles and green squares for CP uncorrected and c orrected results, respectively. D’s/angbracketleftδ/angbracketrightandσDFTare signiﬁcantly high (+0.59 and 0.84 kJ/mol), while the two best IPES descriptions are given by PBE0-D ( /angbracketleftδ/angbracketright= -0.11 kJ/mol) and B3LYP-D ( /angbracketleftδ/angbracketright= +0.11 kJ/mol) functionals. However, B3LYP-D shows less sparse data , yielding a standard deviation of 0.32 kJ/mol against 0.42 kJ/mol achieved with P BE0-D. Since B3LYP-D appears to be the most accurate among the tested functionals, at least for the ﬁrst two investigated halogens, two full param eterizations were carried out with the Pickyprotocol described in Section 2, sampling the QM IPES by mean s of such functional. The resulting B3LYP-D based QMD-FFs, for C1andC2respectively, were thereafter employed in MD simulations. The ﬁnal bulk de nsities achieved from these 15 C1 C2 Functional /angbracketleftδ/angbracketrightσDFT /angbracketleftδ/angbracketrightσDFT (kJ/mol) (kJ/mol) (kJ/mol) (kJ/mol) CAM-B3LYP-D

Question 1257 multiple-choice

Quantum algorithms have revolutionized the ability to solve certain group-theoretic problems, especially those involving periodicity and hidden substructures in Abelian groups. Key algorithmic techniques include quantum Fourier transforms and eigenvalue estimation using phase estimation circuits. Which quantum algorithmic concept provides a unifying framework for order-finding, discrete logarithm, and Abelian stabilizer problems by abstracting them as the task of identifying a subgroup hidden by a function that is constant on cosets and distinct between them? 1) Quantum amplitude amplification 2) Hidden subgroup problem 3) Quantum error correction 4) Grover’s search algorithm 5) Quantum teleportation 6) Quantum random walk 7) Quantum state tomography

✓ Correct Answer:

The correct answer is 2) Hidden subgroup problem.

📚 Reference Text:

arXiv:quant-ph/9903071v1 20 Mar 1999The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer Michele Mosca∗Artur Ekert† May 1998 Abstract A quantum computer can eﬃciently ﬁnd the order of an element i n a group, factors of composite integers, discrete logarithm s, stabilisers in Abelian groups, and hidden orunknown subgroups of Abelian groups. It is already known how to phrase the ﬁrst four problems as the es timation of eigenvalues of certain unitary operators. Here we show ho w the so- lution to the more general Abelian hidden subgroup problem can also be described and analysed as such. We then point out how certain instances of these problems can be solved with only one control qubit, o rﬂying qubits, instead of entire registers of control qubits. 1 Introduction Shor’s approach to factoring [Sh], (by ﬁnding the order of el ements in the multi- plicative group of integers mod N, referred to as Z∗ N) is to extract the period in a superposition by applying a Fourier transform. Another ap proach, based on Kitaev’s technique [Ki], is to estimate an eigenvalue of a ce rtain unitary opera- tor. The diﬀerence between the two analyses is that the ﬁrst o ne considers (or even ’measures’ or ’observes’) the target oroutput register in the standard com- putational basis, while the analysis we detail here conside rs the target register in a basis containing eigenvectors of unitary operators rel ated to the function f. The actual network of quantum gates, as highlighted in [CEM M], is the same for both algorithms; it is helpful to understand both ap proaches. In some cases, which we discuss in Sect. 5, this approach suggests im plementations which do not require a register of control qubits. A more gene ral formulation of the order-ﬁnding problem as well as the discrete logarith m problem, and the Abelian stabiliser problem, is the hidden subgroup problem (or the unknown sub- group problem [Hø]). In the case that Gis presented as the product of a ﬁnite number of cyclic groups (so Gis ﬁnitely generated and Abelian), all of these ∗Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, U.K. and Mathematical Institute, 24-29 St. Giles’, Oxford, OX1 3LB, U.K. †Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, U.K. 1 Figure 1: The function fcan be viewed as the composition of a homomorphism gto a group H, and some 1-to-1 mapping hto the set X. Our hidden subgroup Kwill be the kernel of g, and His isomorphic to G/K. problems are solved by the familiar sequence of a Fourier tra nsform, a function application, and an inverse Fourier transform. In this pape r we describe how this more general problem can also be viewed and analysed as a n estimation of eigenvalues of unitary operators. 2 The Hidden Subgroup Problem Letfbe a function from a ﬁnitely generated group Gto a ﬁnite set Xsuch thatfis constant on the cosets of a subgroup K(of ﬁnite index, since Xis ﬁnite), and distinct on each coset. The hidden subgroup prob

Question 1258 multiple-choice

In representation theory and quantum algebra, the induction process allows constructing representations of a group from those of its subgroups, which plays a pivotal role in the study of structures like the quantum double D. The regular and induced representations have deep connections to group symmetries and invariant subspaces. Which statement correctly describes the relationship between the regular representation of a finite group G and induction from its subgroups? 1) The regular representation is always induced from the largest proper subgroup of G. 2) Any induced representation from a nontrivial subgroup is isomorphic to the regular representation. 3) The regular representation of G is the induced representation from the trivial subgroup {e} to G. 4) Inducing from any subgroup H to G yields an irreducible representation. 5) The regular representation cannot be decomposed into invariant subspaces related to conjugacy classes. 6) The induction process does not apply to Hopf algebras like D. 7) The regular representation is only defined for abelian groups.

✓ Correct Answer:

The correct answer is 3) The regular representation of G is the induced representation from the trivial subgroup {e} to G..

📚 Reference Text:

called induction which plays an important role in the representation theory o fD(G). In this subsection, we describe the construction in general. A s aGmodule, the induced representation can be neatly expressed as a tensor product (over the noncommutati ve ringC[H]) C[G]⊗C[H]V , where⊗C[H]indicates that this tensor product is C[H]-linear, i. e., gh⊗v=g⊗ρ(h)v, forh∈Handv∈V. In order to give an explicit construction in terms of a basis, let us pick a transversal T={t1,t2,...,t k}for HinG(so thatk=|G|/|H|) and ﬁx a basis {|v1/a\}bracketri}ht,...,|vd/a\}bracketri}ht}for the representation space V. The induced representation, which we denote ↑G Hρor simply↑GρwhenHis understood, acts on the vector space C[T]⊗V, whereC[T] ={/summationtext iai|ti/a\}bracketri}ht|ai∈C}is the (Hilbert) space of formal C-linear combinations of elements of T with the inner product /a\}bracketle{tti|tj/a\}bracketri}ht=δij. (The tensor product in this case is the conventional tensor product overC.) In this basis, (↑G Hρ)(g) is given by linearly extending the rule g:|t/a\}bracketri}ht⊗|v/a\}bracketri}ht/ma√st≀→|t′/a\}bracketri}ht⊗h|v/a\}bracketri}ht (3.2) wheret′∈Tandh∈Hare the unique elements for which gt=t′h. There are two facts about induced representations which we u se later. 1. Induction is transitive: if H≤K≤Gthen [↑G K(↑K Hρ)]∼=↑G Hρ. (3.3) 7 2. The regular representation of Gis an induced representation from the trivial representati on of the trivial subgroup{e}. RegG=↑G {e}1. (3.4) Combining the above two facts, we can write RegG=↑G H(↑H {e}1) =↑G HRegH. (3.5) 3.2 Irreducible representations and Fourier analysis over D(G) Arepresentation of a Hopf algebra His a representation of its algebra structure. Speciﬁcally, a representation (ρ,V) of the Hopf algebra His a homomorphism ρ:H→End(V). As above, the algebra D(G) can be given the structure of a representation using left-m ultiplication. The irreducible representations of D(G) have been worked out by Dijkgraaf et al. [ 11] in the physics literature and by Gould [ 17] in the mathematics literature. In the article [ 17], Gould established that the algebra D(G) is semi-simple (when Gis a ﬁnite group), described the irreducible representatio ns of D(G), and developed an associated character theory. In this section, we brieﬂy des cribe his construction of the irreducible representa- tions of D(G), indicate how the regular representation can be broken dow n into irreducible representations, and proceed to construct quantum circuits for the associate d quantum Fourier transform. To simplify the notation in what follows, we adopt Dirac nota tion for the algebras C[G] andCG, treating them as spanned by the orthonormal bases {|g/a\}bracketri}ht|g∈G}and{|h∗/a\}bracketri}ht|h∈G}, respectively. Writing |gh∗/a\}bracketri}htas shorthand for the tensor product |g,h∗/a\}bracketri}ht, the action of D(G) on the regular representation, spanned by the |gh∗/a\}bracketri}ht, is g′(h′)∗|g,h∗/a\}bracketri}ht=δ(h′)g,h|g′g,h∗/a\}bracketri}ht. (3.6) We see that g′(h′)∗takes|g,h∗/a\}bracketri}htto zero unless handh′are conjugates (under the action of g); in the case whereg′(h′)∗does not annihilate |g,h∗/a\}bracketri}ht, it acts by the left Gaction in the ﬁrst coordinate. It follows immediately that each of the subspaces Vh= span({|g,h∗/a\}bracketri}ht,g∈G}) is invariant under the left action of D(G); this provides an initial orthogonal decomposition D(G) =/circleplustext hVh into invariant subspaces. Observe, also, that the action of/summationtext h∈ggh∗onVhis precisely left multiplication bygon the ﬁrst component; this gives Vhthe structure of a G-representation isomorphic to C[G]. As the action of D(G) onVhis

Question 1259 multiple-choice

The study of finite p-groups often involves analyzing their subgroup structure using concepts such as the lower central series, Frattini subgroups, and characteristic matrices. Group invariants like maximal and minimal subgroup indices play a crucial role in classifying these groups. For a finite p-group G with G₃ ≅ Cₚ and trivial Frattini subgroup of the derived subgroup (Φ(G′) = 1), what is the minimal index (Imin) of a subgroup of type A₁ when the parameter m satisfies m ≥ 2? 1) Imin = p 2) Imin = n 3) Imin = p² 4) Imin = m 5) Imin = n + 1 6) Imin = p³ 7) Imin = 2p

✓ Correct Answer:

The correct answer is 3) Imin = p².

📚 Reference Text:

w(G) X w (¯G) Group Remark (a1) (a2)/parenleftbiggw110 0w22/parenrightbigg diag(1,w11)/parenleftbigg1 0 0w11w22/parenrightbigg (E1) t=w11w22 (a1) (b2)/parenleftbiggw11w12 0 0/parenrightbigg diag(w−1 12,w11)/parenleftbigg1 1 0 0/parenrightbigg (E2) (a1) (c2)/parenleftbiggw110 0 0/parenrightbigg diag(1,w11)/parenleftbigg1 0 0 0/parenrightbigg (E3) (b1) (a2)/parenleftbigg0 0 w21w22/parenrightbigg diag(w21,w−1 22)/parenleftbigg0 0 1 1/parenrightbigg (E4) (b1) (b2)/parenleftbigg0w12 w210/parenrightbigg diag(w21,1)/parenleftbigg0w12w21 1 0/parenrightbigg (E5) t=w12w21 (b1) (c2)/parenleftbigg0 0 w210/parenrightbigg diag(w21,1)/parenleftbigg0 0 1 0/parenrightbigg (E6) (c1) (a2)/parenleftbigg0 0 0w22/parenrightbigg diag(1,w−1 22)/parenleftbigg0 0 0 1/parenrightbigg (E7) (c1) (b2)/parenleftbigg0w12 0 0/parenrightbigg diag(w−1 12,1)/parenleftbigg0 1 0 0/parenrightbigg (E8) (c1) (c2)/parenleftbigg0 0 0 0/parenrightbigg /parenleftbigg0 0 0 0/parenrightbigg (E9) Table 3.2: Case 4 in Theorem 3.5 9 groups of Type (E1)–(E9). /square By Theorem 3.2, we have the following theorem. Theorem 3.6. LetGbe a ﬁnite p-group listed in Theorem 3.5. Then the following conclusions hold: (1)Imax= 2for types (A1)–(A3), (C1)–(C5), (D3), (D5), (E3), (E6), (E9) ; (2)Imax=nfor types (B1)–(B3), (D1)–(D2), (D4), (E1), (E4), (E7) ; (3)Imax=mfor types (E2), (E5), (E8) . 4 The case G3∼=CpandΦ(G′) = 1 Suppose that Gis a ﬁnite p-group with G3∼=Cp, Φ(G′) = 1 and G/G3∼=Mp(n,m,1), wheren >1 forp= 2 and n≥m. Let G/G3=/an}bracketle{t¯a,¯b,¯c|¯apn=¯bpm= ¯cp= 1,[¯c,¯a] = [¯b,¯c] = 1/an}bracketri}ht. Then, without loss of generality, we may assume that G=/an}bracketle{ta,b,c/an}bracketri}htsuch that [ a,b] =c. SinceΦ(G′) = 1,cp= 1. Let G3=/an}bracketle{tz/an}bracketri}ht. Sinceapn∈G3, wemayassumethat apn=zw11. Bysimilarreasons, wemay assumethat bpm=zw21, [c,a] =zw12and[c,b] =zw22. Then we get a 2 ×2 matrix over Fp.w(G) = (wij) is called a characteristic matrix of G. Notice that w(G) will be changed if we change the generators a,b. We also call a,ba set of characteristic generators of w(G). Theorem 4.1. LetGbe a ﬁnite p-group such that G3∼=Cp,Φ(G′) = 1andG/G3∼= Mp(n,m,1), wheren >1forp= 2andn≥m. Then the following conclusion hold: (1)Ifm= 1, thenImin= 1; (2)Ifm≥2, thenImin= 2. Proof(1) Since G3=/an}bracketle{t[c,ab],[c,a]/an}bracketri}ht∼=Cp, [c,a]/ne}ationslash= 1 or [c,ab]/ne}ationslash= 1. Hence either /an}bracketle{ta,c/an}bracketri}ht or/an}bracketle{tab,c/an}bracketri}htis anA1-subgroup of index p. ThusImin= 1. (2) Firstly, we claim that Imin>1. Otherwise, Imin= 1. Assume that Dis anA1- subgroup of index p. It is easy to see that D′=G3andd(D/G3) = 2. Since Φ( G)≤D, d(Φ(G)/G3)≤2. On the other hand, Φ( G)/G3=/an}bracketle{t¯ap,¯bp,¯c/an}bracketri}htis of type ( pn−1,pm−1,p), a contradiction. If [c,b] = 1, then [ c,a]/ne}ationslash= 1. In this case, /an}bracketle{tcbp,a/an}bracketri}htis anA1-subgroup of index p2. If [c,b]/ne}ationslash= 1, then, without loss of generality, we may assume that [ c,a] = 1. In this case, /an}bracketle{tb,cap/an}bracketri}htis anA1-subgroup of index p2. To sum up, Imin= 2. /square Theorem 4.2. Suppose that Gis a ﬁnite p-group such that G3∼=Cp,Φ(G′) = 1and G/G3∼=Mp(n,m,1), wheren >1forp= 2andn≥m. Letw(G) = (wij)be a characteristic matrix of G. Then the following conclusions hold: (1)Ifp= 2andm= 1, thenG∈ A3; (2)Ifm >1forp= 2,w22= 0andw12/ne}ationslash= 0, thenG∈ Am+1; (3)Ifm >1forp= 2,w22/ne}ationslash= 0andw12= 0, thenG∈ An+1. 10 ProofWe getImaxby investigating all maximal subgroups of G. LetG3=/an}bracketle{tz/an}bracketri}ht, N=/an}bracketle{tb,ap,c,z/an}bracketri}htandMi=/an}bracketle{tabi,bp,c,z/an}bracketri}ht. Then all maximal subgroups of GareNand Miwhere 0≤i≤p−1. (1) In this case, [ c,b] = [a,b2] = 1. Hence [ c,a]/ne}ationslash= 1. It follows that Mi=/an}bracketle{tc,abi/an}bracketri}ht ∈ A1. Since/an}bracketle{ta2,b/an}bracketri}ht ∈ A1, we have N=/an}bracketle{ta2,b/an}bracketri}ht×/an}bracketle{tc/an}bracketri}ht ∈ A2by Lemma 2.3. Hence G∈ A3. (2) In this case, [ c,b] = 1 and [ c,a]/ne}ationslash=

Question 1260 multiple-choice

Supersymmetric models face significant constraints from cosmology and indirect detection experiments when proposing candidates for dark matter. Moduli fields, their decay channels, and mass hierarchies play central roles in determining viable scenarios for lightest supersymmetric particle (LSP) abundances. Which adjustment to modulus properties can resolve the cosmological tension between string-motivated moduli reheating and the existence of a stable TeV-scale LSP in the MSSM? 1) Increasing the branching ratio of modulus decay into superpartners 2) Lowering the modulus mass below the gravitino mass 3) Enhancing modulus decays to standard model particles exclusively 4) Suppressing the decay rate of the modulus to match that of the gravitino 5) Making the modulus mass much larger than the gravitino mass, reducing branching ratios to superpartners, or increasing modulus decay rates 6) Ensuring moduli couple only to hidden sector gauge bosons 7) Restricting modulus decay channels to only neutralinos

✓ Correct Answer:

The correct answer is 5) Making the modulus mass much larger than the gravitino mass, reducing branching ratios to superpartners, or increasing modulus decay rates.

📚 Reference Text:

and HESS, and the non-thermal wino is ruled out even for very conservative assumptions about the DM pro le in the inner galaxy ( e.g. cored isothermal) [50, 51]. A wino-like LSP can be consistent with these bounds if it is only a subleading component of the total DM density. Using the AMSB relation for M2in terms of m3=2, this forces m'=m3=2&100, signi cantly greater than the generic expectation [51]. The problem is even worse for other neutralino LSP species, since these annihilate less eciently and an even larger value of m' m3=2 is needed to obtain an acceptable relic density. Furthermore, m'>2m3=2also allows the modulus eld to decay to pairs of gravitinos. The width for this decay is typically similar to the total width to SM superpartners [52{55]. For m' m3=2>30 TeV, the gravitinos produced this way will decay to particle-superpartner pairs before nucleosynthesis but after the modulus decays, and recreate the same LSP density problem that forced m' m3=2in the rst place. These results suggest a degree of tension between reheating by string-motivated moduli (withm'm3=2and MPl) and the existence of a stable TeV-scale LSP in the minimal supersymmetric standard model (MSSM). This tension can be resolved if all relevant moduli have properties that are slightly di erent from the na ve expectation; for example m' m3=2 and BR('! 3=2 3=2) 1 [55, 56], an enhanced modulus decay rate with m'm3=2and

Question 1261 multiple-choice

In group theory and algebraic combinatorics, the structure of automorphism groups and regular subgroups is often studied via derivations, homomorphisms, and Cayley digraph representations. The interplay between derivations and functions with cocycle conditions is central to understanding symmetries and classification in these mathematical objects. Which statement best describes the significance of the map θ from Der(W, L) to Hom_H(W, V) when the automorphism group G is 2-closed? 1) θ is always surjective but never injective, regardless of the properties of G. 2) θ is an isomorphism, establishing a bijective correspondence between derivations and compatible functions. 3) θ maps derivations onto a proper subset of Hom_H(W, V) but is neither injective nor surjective. 4) θ is a homomorphism whose kernel consists of all inner derivations only. 5) θ fails to preserve the group structure unless G is abelian. 6) θ provides a one-to-one correspondence only for trivial derivations. 7) θ is injective but not surjective, even if G is 2-closed.

✓ Correct Answer:

The correct answer is 2) θ is an isomorphism, establishing a bijective correspondence between derivations and compatible functions..

📚 Reference Text:

H(w) . Sof( βw) ∈H(w) for any β∈Fp. In particular H(βw) ⊆H(w) for any β∈Fp.I f β/negationslash=0, we have H(w) =H(βw) . (iv) Now let fbe in L0andw1,w2∈W. Consider g=f−w2−f( w 2)(as usual f( w 2)denotes the element of ξ1(U) with image f( w 2)). Now g∈Land g(0)=(f−w2−f( w 2))(0)=f( w 2)−f( w 2)=0, sog∈L0. Moreover g(w 1)= f( w 1+w2)−f( w 2), therefore f( w 1+w2)=g(w 1)+f( w 2)∈H(w 1)+H(w 2)./square We denote by Der (W,L) the group of derivations from WtoLand by Inn (W,L) the group of inner derivations, see [ 8]. Lemma 2 Any regular elementary Abelian subgroup of Gis conjugate to Wξ1(U)if and only if Der(W,L) =Hom(W,ξ 1(U))+Inn(W,L) . Proof Letδbe an element in Der (W,L) . Then H=Xδξ1(U)is a regular elementary Abelian subgroup of G, where Xδ={wwδ|w∈W}. In particular if His conjugate toWξ1(U), then there exists l∈Lsuch that Xδ⊆Wlξ1(U). In other words, for anyw∈Wthere exists a unique cw∈ξ1(U) such that wδ=[w,l]+cw.T h em a p c−:w/mapsto→cwis an homomorphism of Wintoξ1(U). In fact [w1,l]w2+[w2,l]+cw1+cw2=([w1,l]+cw1)w2+[w2,l]+cw2 =(wδ 1)w2+wδ 2 =(w1+w2)δ=[w1+w2,l]+cw1+w2 and so cw1+w2=cw1+cw2. Thus δ=c−+[ −,l]∈Hom(W,ξ 1(U))+Inn(W,L) . 346 J Algebr Comb (2007) 26: 343–355 As any regular elementary Abelian subgroup HofGis of the form Xδξ1(U),f o r someδ∈Der(W,L) , the other side of the implication is totally trivial. /square Assume that the group Gis 2-closed. Every orbital of Gcorresponds to an orbit of the point stabilizer L0,i . e .e v e r yo r b i t a lo f Gcorresponds to a suborbit of the form (w,v)L0={(w,f (w) +v)|f∈L0}=(w,v+H(w)) . The group Gcontains the regular subgroup Wξ1(U)and so every orbital is a Cayley digraph over W×V.I n particular the orbital corresponding to the suborbit (w,v)L0is{((w 1,v1),(w 2,v2))| (w1−w2,v1−v2)∈(w,v+H(w)) }=Cay(W×V,(w,v +H(w))) . Furthermore G=/intersectiontext (w,v) Aut(/Gamma1(w,v)), where /Gamma1(w,v)=Cay(W×V,(w,v +H(w))) . Let us deﬁne the Fp-vector space Hom H(W,V ) ={ψ∈B|ψ(w 1+w2)−ψ(w 1)−ψ(w 2)∈H(w 1)∩H(w 2) for any w1,w2∈W,ψ( 0)=0}, see [6]. Next if δ∈Der(W,L) , we deﬁne θ(δ):W→Vbyθ(δ)(w) =(wδ)(0). Lemma 3 θis an injective homomorphism of Der(W,L) intoHom H(W,V ) .IfGis 2-closed ,thenθis an isomorphism . Proof Clearly θ(δ)( 0)=(0δ)(0)=0. Let w1,w2be elements of W. Setwδ 1=f+g where f∈L0andg∈ξ1(U).W eh a v e θ(δ)(w 1+w2)−θ(δ)(w 1)−θ(δ)(w 2)=(w1+w2)δ(0)−wδ 1(0)−wδ 2(0) =((w 1)δ)w2(0)+wδ 2(0)−wδ 1(0)−wδ 2(0) =wδ 1(−w2)−wδ 1(0)=f(−w2)∈H(w 2). By symmetry (the group Wis commutative) we get that θ(δ)(w 1+w2)−θ(δ)(w 1)−θ(δ)(w 2)∈H(w 1). Therefore θ(δ)∈Hom H(W,V ) . The map θis an homomorphism. If δ1,δ2∈Der(W,L) andw∈W, then θ(δ1+ δ2)(w)=wδ1+δ2(0)=wδ1(0)+wδ2(0)=(θ(δ 1)+θ(δ2))(w) . Moreover θ(0)=0. Finally wαδ=αwδfor any α∈Fpand so θ(αδ) =αθ(δ) . The map θis injective. θ(δ)=0 if and only if wδ(0)=0 for any w∈W.S owδ∈ L0for any w∈W. In particular 0 =(w+w1)δ(0)=((wδ)w1+wδ 1)(0)=wδ(−w1) for any w1,w. Thus wδ=0 for any w. Therefore δ=0. Assume that Gis 2-closed. Let ψbe an element of Hom H(W,V ) . Deﬁne η(w)= ψ(−w)for any w∈W. The function ηlies in the base group B. Furthermore η lies in Hom H(W,V ) , in fact η(0)=ψ(0)=0 and η(w 1+w2)−η(w 1)−η(w 2)= ψ(−w1−w2)−ψ(−w1)−ψ(−w2)∈H(−w1)∩H(−w2)=H(w 1)∩H(w 2).W e claim that ηnormalizes G,i . e .[W,η]⊆L.L e

Question 1262 multiple-choice

In abstract algebra, modules over rings and their submodule structures are fundamental for understanding algebraic systems. Finiteness conditions such as the maximal condition play a crucial role in classifying and analyzing these submodules. Which statement best describes the maximal condition on Q-submodules of a quotient module M_q/M_qT, where Q-submodules are those invariant under a given set of endomorphisms? 1) Every descending chain of Q-submodules eventually stabilizes. 2) Every Q-submodule contains at least one maximal element. 3) The quotient module is simple with respect to Q-submodules. 4) Each Q-submodule is finitely generated. 5) Every Q-submodule is closed under scalar multiplication only. 6) Every ascending chain of Q-submodules eventually stabilizes. 7) The quotient module possesses no proper Q-submodules.

✓ Correct Answer:

The correct answer is 6) Every ascending chain of Q-submodules eventually stabilizes..

📚 Reference Text:

0 for all i u i2, i^, u, we have 7 c ker £ so there is a homomorphism £ : Z(A)/ 7 -> £ such that (a,- + 7)f = 1 + e, (i € N). Since ((fa,. «/J - 1) • • • (K,_,, a l2/] - 1) + r)f = 2leheh • • • ei2l, the set {([a,,, a h] - 1) • • • (K,.,, a, 2J - 1) + T | Z > 0, i, < i 2 < • • • < i 2,} is linearly independent, so it forms a Z-basis for ZA' + 7/ 7. Now to complete the proof of Lemma 9 it remains to note that if T is an ideal of Z(A) generated by elements of Z(A') such that (Z(A') + T/T, +> is a free abelian group and {vj + T \ j e J] is a Z-basis of Z(A') + T/T then the set of all elements al'af-'-Vj + T (jeJ) with rii € Z for all / e N and n t = 0 for almost all / is a basis of Z(A)/ 7 over Z. • Let ^ be a positive integer, N? the set of ordered ^-tuples of elements of N. Suppose that M q is the free right /(A)-module generated by all elements (I'I, i2,... , i q) e hi9. Recall that O is the set of all functions 0 : N -> N such that a < b(f> when a < b. We also write <$> for the corresponding sets of endomorphisms of Z(A) (such that a,0 = ai for all i) and of Z-linear mappings of M q into itself such that ((/,,... , i q)f)4> = (i\4>,... , iq(t>)(f4>), where / e Z(A). A Z(A)-submodule L in Mq is called a <&-submodule if L is closed under all mappings € . The main result of the section is as follows. PROPOSITIONS. For every positive integer q the module M q/MqT satisfies the maximal condition on Q-submodules. PROOF. Recall that Ji is the set of all elements in Z(A)/ 7 which are of the form (24). Define on Jt a linear order denoted by < and a partial order denoted by •<. Let m, m' e Jt', m = mxmj, m' = m\m' 2, where m,, m\ € ^ (i = 1, 2), (29) m, = al'a? • • • + 7, «', = a";a2ni • • • + 7, (30) m 2 = (fa,, a i2] - 1) • • • ([a i2,_,, a, 2J - 1) + 7 (!,<•••< i2l), (31) m^ = ([a,-;, flj;] -!)••• ([a, ;, ,, a^] - 1) + 7 (/;<•••< ^,) https://doi.org/10.1017/S1446788700002056 Published online by Cambridge University Press 348 C. K. Gupta and A. N. Krasil'nikov [20] Define 1, ifn>0; sgn(n) =0, ifn = 0; -1, ifn<0. We write mx < m\ if and only if one of the following conditions (i)-(ii) holds: (i) \n k\ < \n' k\ for some it but

Question 1263 multiple-choice

Quantum computing utilizes quantum mechanical effects to solve certain problems more efficiently than classical computers, with the Quantum Fourier Transform (QFT) serving as a foundational component in many algorithms. Solid-state approaches, such as spin-torque-based qubit manipulation, are under investigation for scalable and robust quantum circuit implementation. Which statement best explains the significance of realizing the Quantum Fourier Transform using spin-torque-based quantum computing architectures? 1) It enables high-fidelity implementation of QFT circuits in solid-state systems, advancing the physical realization of efficient quantum algorithms. 2) It demonstrates the ability to perform quantum error correction solely with spin-torque-based devices. 3) It allows for the direct factoring of large numbers without circuit decomposition. 4) It provides a method to eliminate all sources of quantum noise in solid-state qubits. 5) It makes three-qubit entanglement obsolete for QFT applications. 6) It replaces the need for universal gate sets such as Clifford + T in quantum computation. 7) It proves that spin-torque devices are the only viable platform for quantum key distribution.

✓ Correct Answer:

The correct answer is 1) It enables high-fidelity implementation of QFT circuits in solid-state systems, advancing the physical realization of efficient quantum algorithms..

📚 Reference Text:

Title: Spin-Torque-Based Quantum Fourier Transform Year: 2019 Paper ID: cd58253ad63be31d9fa51a6198cd2d2e451c1caa Source: semantic-scholar URL: https://www.semanticscholar.org/paper/cd58253ad63be31d9fa51a6198cd2d2e451c1caa Abstract: Quantum computing (QC) provides an efficient platform to solve complex problems such as number factoring and searching. The quantum Fourier transform (QFT) is an integral part of quantum algorithms for integer number factoring, phase estimation, discrete algorithms, interchange of position and momentum states, quantum key distribution protocol, multiparty quantum telecommunication, and quantum arithmetic. The theoretical and experimental implementations of QFT on various platforms have been proposed by researchers. Spin-torque-based qubit(s) manipulation is one of the encouraging solid-state device technologies. However, to date, QFT is not realized by spin-torque-based QC architecture. In this article, the spin-torque-based architecture has been modeled with the help of optimized decomposition of quantum circuits for the QFT. Moreover, an optimal-depth Clifford + T gates set-based quantum circuit is utilized to implement the QFT. The performance analysis in terms of fidelity (>99%), magnitude, and phase difference of respective density matrices for different forms of three-qubit QFT provides a novel way of its physical realization to address the complex problems.

Question 1264 multiple-choice

In numerical linear algebra and quantum algorithms, accurately solving linear systems often involves approximating matrix inverses, particularly when dealing with non-Hermitian matrices. Advanced techniques such as Hermitian embedding, polynomial approximations, and operator norm bounds are commonly used to ensure solution robustness and efficient error control. Which of the following statements best characterizes the role of Hermitian embedding in solving linear systems with non-Hermitian matrices? 1) It reduces the condition number of the original matrix to improve convergence speed. 2) It preserves both singular values and condition number, allowing Hermitian-specific approximation techniques to be applied. 3) It guarantees that the matrix inverse can be computed exactly in polynomial time. 4) It transforms the matrix into a diagonal form to simplify spectral analysis. 5) It eliminates the need for truncation thresholds in polynomial approximations. 6) It ensures that all eigenvalues of the matrix become positive. 7) It allows solution estimators to be constructed without considering operator norms.

✓ Correct Answer:

The correct answer is 2) It preserves both singular values and condition number, allowing Hermitian-specific approximation techniques to be applied..

📚 Reference Text:

second term ∥C(g(C)−C−1)b∥2can be upper bounded as follows: ∥C(g(C)−C−1)b∥2≤ ∥C∥2∥(g(C)−C−1)b∥2≤ ∥C∥2∥g(C)−C−1∥2∥b∥2=∥C∥24ν2 ∥C∥2≤4ν2≤4ν, (A9) with the last inequality using the fact that 0 < ν≤1 Hence we note that the solution obtained by our truncation threshold T∈O(K·κClogκC ν) has min {αm}m∈[−T..T ] C TX m=−TαmQm! b−b 2 ≤min x∈CN∥Cx−b∥2+ 4ν (A10) Cis non-Hermitian: Consider the following transformation and embedding matrices: C′=O C C†O b′= b 0 x′= 0 x . (A11) Through the embedding, we can transform our linear system into solving C′x′=b′where C′is now Hermitian. We know that the eigenvalues of C′correspond to the set {±σk(C)}∀k∈[N][1]. Hence the singular values of C′are equal to the set of singular values of C. Thus we have κC′=κCand∥C′∥=∥C∥. Constructing ˆC′=C′ ∥C∥, we see all eigenvalues of ˆC′fall within the domain DκC= [−1,−1 κC]∪[1 κC,1]. By Lemma 14 of [37] and Equation A2, we see that exists g(x) =Pj0 j=0pj ∥C∥j+1xjsuch that ∥g(C′)−C′−1∥ ≤2ν ∥C∥, (A12) To find a decomposition of C′, we first find the decomposition of C†. Using the fact that Qis unitary, thus the adjoint/transpose of QmisQ−m: C†= KX ℓ=−KcℓQℓ!† =KX ℓ=−K¯cℓQ−ℓ=KX ℓ=−K¯c−ℓQℓ, (A13) where ¯ cℓis the conjugate of cℓ. We can then express C′as follows C′=O C C†O = OKP ℓ=−KcℓQℓ KP ℓ=−K¯c−ℓQℓO =KX ℓ=−K 0cℓ ¯c−ℓ0 ⊗Qℓ(A14) One can find that for T∈O(K·κlogκ ν), there exists a set of parameters such that {ˆαm,ˆβm,ˆγm,ˆδm:m∈ [−T..T],ˆαi,ˆβm,ˆγm,ˆδm∈C}such that we can write g(C′) in the following form: g(C′) =j0X j=0pj ∥C∥j+1C′j=j0X j=0pj ∥C∥j+1 KX ℓ=−K 0cℓ ¯c−ℓ0 ⊗Qℓ!j (A15) =TX m=−Tˆβiˆδm ˆαmˆγm ⊗Qm= TP m=−TˆβmQmTP m=−TˆδmQm TP m=−TˆαmQmTP m=−TˆγmQm (A16) Furthermore, observe that the following equality is true: C′g(C′)b′=O C C†O TP m=−TˆβmQmTP m=−TˆδmQm TP m=−TˆαmQmTP m=−TˆγmQm  b 0 (A17) 15 = CTP m=−TˆαmQmCTP m=−TˆγmQm C†TP m=−TˆβmQmC†TP m=−TˆδmQm  b 0 = CTP m=−TˆαmQm b C†TP m=−TˆβmQm b (A18) We now attempt to upper bound the MSE loss obtained from our estimator ˜ x=PT m=−TαmQm: min {αm}m∈[−T..T ] C TX m=−TαmQm! b−b 2 ≤ C TX m=−TˆαmQm! b−b 2 =  CTP m=−TˆαmQm b 0 − b 0 2 (A19) ≤  CTP m=−TˆαmQm b C†TP m=−TˆβmQm b − b 0 2 ≤ ∥C′g(C′)b′−b′∥2, (A20) where the second-to-last inequality can be obtained by simply observing the fact that replacing zero terms in a vector with non-zero terms increases the value of the ℓ2norm, and the last equality is obtained by replacing the term CPT m=−TˆαmQm bC†PT m=−TˆβmQm b⊺ with C′g(C′)b′. Similar to the Hermitian case, we obtain the following: ∥C′g(C′)b′−b′∥2=∥C′C′−1b′−b′∥2+∥C′(g(C′)−C′−1)b′∥2. (A21) Noting that C′−1= O C†−1 C−1O , we can recast the first term as follows: ∥C′C′−1b′−b′∥2= O C C†O O C†−1 C−1O b′−b′ 2 = CC−1O O C†C†−1 b 0 − b 0 2 (A22) = CC−1b 0 − b 0 2 =∥CC−1b−b∥2= min x∈CN∥Cx−b∥2(A23) As∥b′∥= 1, the second term is upper bounded as follows: ∥C′(g(C′)−C′−1)b′∥2≤ ∥C′∥2∥g(C′)−C′−1∥2∥b′∥2=∥C∥24ν2 ∥C∥2≤4ν2≤4ν (A24) Hence we see that for the non-Hermitian case, the solution obtained by our truncation threshold T∈ O(K·κClogκC ν) is still ν-close in terms of the MSE loss. Appendix B: Quantum

Question 1265 multiple-choice

In particle physics, discrete non-abelian symmetry groups are often used to constrain mass matrices, such as Yukawa matrices, leading to specific mixing patterns among particle families. Computational and mathematical techniques help identify suitable groups and construct invariant matrices. Which of the following conditions is necessary for a finite group to be a candidate for a discrete flavor symmetry embedded in U(3), allowing construction of invariant Yukawa matrices based on group representations? 1) The group must be abelian and have only 1-dimensional representations. 2) The order of the group must be a prime number. 3) The group must have both 2- and 3-dimensional irreducible representations, with the order divisible by the dimension of each. 4) The group must have only faithful 1-dimensional representations. 5) The group must be a subgroup of SU(2) with exclusively 2-dimensional representations. 6) The group must be infinite and non-abelian. 7) The group must have no faithful representations in any dimension.

✓ Correct Answer:

The correct answer is 3) The group must have both 2- and 3-dimensional irreducible representations, with the order divisible by the dimension of each..

📚 Reference Text:

UPMNS=Vl† LVν L. (10) II. THE SEARCH FOR DESIRED GROUPS It is clear, that in order to get the form of Yukawa matrices one nee ds to ﬁnd the irreducible representations of the groups which meet our requirem ents. First of all, we want our group to be discrete, non-abelian and to be a subgroup of U(3) (similarlyto A4). Then, itisquiteobviousthat thegroupshouldpossess 2and3dime nsional irreducible representations, which are necessary for construct ion of Eq.(3). To ﬁnd such a group one can make use of GAP: the programme for discrete algebra computation available on the website [7]. One would also need the SmallGroup library [8] andREPSN package [9] which serve as indispensable tools for our purposes. In order to perform the calculation it is recommended to beneﬁt fro m some mathematical theorems: Theorem 2. The order of the ﬁnite group Gis divisible by dimension of its irreducible representation. Theorem 3. A ﬁnite group Gis isomorphic to a ﬁnite subgroup of U(3) if and only if it possesses a faithful (irreducible or reducible) 3-dimensi onal representation. 6 Since our group has got 2 and 3-dimensional irreducible representa tions, following the second theorem, one should ﬁrst verify that the group’s order is d ivided by 6. The proof of the third theorem can be found in many textbooks abo ut ﬁnite groups (see, for example, [10]). On the other hand, the justiﬁcation of th e last statement is quite intuitive. In case of ﬁnite groups, every representation has got it s unitary equivalent (it is possible to present the matrices of the given representation in suc h a basis to make them unitary). Therefore, the faithfulness assures that every eleme ntgofGcan uniquely be expressed as a unitary matrix (see [11] for a more profound analys is of this theorem). III. THE SEARCH FOR YUKAWA MATRICES After ﬁnding the irreducible representations of GF, one has got almost everything to get to know how Yukawa matrices look like. Since the equation restric ting the form of Yukawa matrices has the form of eigenproblems to the eigenvalue 1, its solution is rather straightforward. One can easily compute the eigenspace Wifor each of the generators. The problemarises, whenonewantstodealwiththecommoneigenspace Uoftheeigenspaces Wi. Inordertoperformthiscalculation, onecanfollowverysimplealgorit hm(itisnotnecessarily the most eﬃcient one). To simplify the considerations, let us assume that one has got two n-dimensional spaces SandT. Each space is spanned by its basis vectors: {s1,s2,...,sn} and{t1,t2,..,tn}respectively. Vector /vector abeing the part of the common subspace P=S ∩T can be equivalently expressed in two basis: /vector a=a1/vector s1+a2/vector s2+...+an/vector sn, /vector a=a′ 1/vectort1+a′ 2/vectort2+...+a′ n/vectortn, (11) wherea1,a2,...,an,a′ 1,a′ 2,...,a′ nare some complex coeﬃcients. Denoting the basis vectors of space Svia their components as: /vector s1= s11 s12 . . . s1n , /vector s2= s21 s22 . . . s2n ,..., /vector s n= sn1 sn2 . . . snn  7 and similarly for T, one can rewrite the equality following from Eq.(11) as a set of equat ions: a1s11+a2s21+...+ansn1−a′ 1t11−a′ 2t21−...−a′ ntn1= 0, a1s12+a2s22+...+ansn2−a′ 1t12−a′ 2t22−...−a′ ntn2= 0, . .

Question 1266 multiple-choice

Quantum error-correcting codes are crucial for safeguarding quantum information against errors, especially in systems where physical symmetries and conserved quantities, such as charge or spin, play a role. The effectiveness of such codes can depend on the structure of the physical subsystems and the nature of the charge fluctuations within them. In quantum codes designed to protect conserved quantities, which scenario is most likely to enable the code to approach the theoretical lower bound on worst-case error? 1) Either large local charge fluctuations or a large number of subsystems 2) Restricting charge fluctuations to small values in each subsystem 3) Using only a single subsystem with fixed charge 4) Minimizing the number of subsystems while maximizing charge conservation 5) Employing codes with strictly commuting charge operators and minimal fluctuation 6) Relying solely on classical repetition codes for error correction 7) Decreasing local charge fluctuations below the system's noise threshold

✓ Correct Answer:

The correct answer is 1) Either large local charge fluctuations or a large number of subsystems.

📚 Reference Text:

FAIST et al. PHYS. REV . X 10,041018 (2020) 041018-4 physical subsystems. The term ΔTiin Eq. (2)is a proxy for the magnitude of the charge fluctuations of the code wordstates within each subsystem. We also find that Eq. (2)can be generalized in a number of ways. We can express the limit on code performance interms of other measures besides worst-case entanglementfidelity, such as the entanglement fidelity for a fixed inputstate. We can derive bounds that apply in the case where more than one subsystem is erased or where errors occur on an unknown subsystem rather than a known subsystem.Similarly, the physical charge can have terms acting onseveral subsystems, including overlapping and noncom- muting terms, such as those commonly encountered in many-body physics. We can consider cases where thecharge distribution for a subsystem has infinite range butwith a normalizable tail. We also treat the cases where the code is only approximately covariant with respect to a continuous symmetry, the case of a nonisometric approx-imately charge-conserving code, and the case where thephysical charge operator is not strictly transversal. B. Regimes where our bound is circumvented and criterion for code performance The idea underlying Eq. (2)is that, for erasure correction to work well, one should not be able to learn much aboutthe global value of the charge by performing a localmeasurement on a subsystem. Hence, to be able to correct the errors to good accuracy, the code words need to be able to have either large fluctuations of the local charge values(which requires ΔT i→∞to accommodate these fluctua- tions) or many subsystems ( n→∞) so that the global charge is a sum of many local contributions. In fact, codes can be constructed in either limit for which ϵworst approx- imately matches the scaling in ΔTiandnof the lower bound (2)(see Sec. VI). To study the case of large ΔTi, we consider a normalized variant of the infinite-dimensional covariant code con-structed in Ref. [12]. The infinite-dimensional version encodes one logical rotor [with unbounded Uð1Þcharge] in a code block of three rotors. In the modified version of this code, we truncate the charge of the logical system tof−h;−hþ1;…;þhgand also normalize the physical code words by truncating the charge of the subsystems, eitherby imposing a sharp cutoff on the local charge or by modulating the local charge using a Gaussian envelope of width w. The value of ϵ worstachieved by this code, and our lower bound, both scale like h=w up to a logarithmic factor. Regarding the limit of a large number of subsystems, we observe that a code discussed in Ref. [8]matches the 1=n scaling of our lower bound on ϵworst. Here, the subsystems are qubits, regarded as spin- 1=2particles, and the code space is two-dimensional, spanned by two Dicke states with different values of the total angular momentum Jz along the zaxis. (A Dicke state is a symmetrized super- position of all basis states with a specified Jz). This code iscovariant with respect to z-axis rotations by construction and can be shown to achieve ϵworstscaling

Question 1267 multiple-choice

In group representation theory, the analysis of symmetric groups and their irreducible representations requires understanding combinatorial factors, orthogonality properties, and connections to probability measures such as the Plancherel measure. Techniques involving delta functions, Stirling numbers, and projector manipulations are essential for computations in mathematical physics and quantum theory. Which of the following best describes the mathematical expression for the Plancherel probability weight assigned to an irreducible representation $ R $ of the symmetric group $ S_n $? 1) $\frac{1}{d_R n!}$ 2) $d_R n!$ 3) $(n!)^2 \chi_R(\sigma_1)$ 4) $d_R^2 n!$ 5) $(n!)^2 \chi_R(\sigma_1)\chi_R(\sigma_1)$ 6) $\frac{d_R}{n!}$ 7) $\frac{d_R^2}{n!}$

✓ Correct Answer:

The correct answer is 7) $\frac{d_R^2}{n!}$.

📚 Reference Text:

(n!)2χR(σ1)χR(σ1) =d2 R n!(A.2) Remark. This is independent of the choice of µand is the Plancherel probability weight for the irrep R. Proof of Lemma 2 /summationdisplay τ(Dreg στ(T2))2=/summationdisplay τδ(σ−1T2τ)δ(σ−1T2τ) =δ(σ−1T2 2σ) =δ(T2 2) =n(n−1) 2(A.3) B Solving the power sums Mk The problem of solving the power sum moments Mk=/summationtextN i=1fk ican be sorted by a simple resolution of a linear system that we now present. We equate /summationdisplay f∈FAl(f) =l/summationdisplay k=0cl kMk (B.1) and expand the l.h.s. using equation (3.9) in [ 24] /summationdisplay f∈FAl(f) =N/summationdisplay i=1Al(fi) =N/summationdisplay i=1l!l/summationdisplay r=0/parenleftBigg l r/parenrightBigg/parenleftBigg fi r/parenrightBigg 2r =l/summationdisplay r=02rl! r!/parenleftBigg l r/parenrightBiggN/summationdisplay i=1(fi)r=l/summationdisplay r=02rl! r!/parenleftBigg l r/parenrightBiggN/summationdisplay i=1r/summationdisplay k=0s(r,k)fk i (B.2) – 31 – JHEP05(2023)191where (fi)r=fi(fi−1)...(fi−r+ 1)is the failing factorial, and s(k,r)’s are Stirling numbers of the ﬁrst kind. This expands further as /summationdisplay f∈FAl(f) =l/summationdisplay k=0/bracketleftBiggl/summationdisplay r=ks(r,k)2rl! r!/parenleftBigg l r/parenrightBigg/bracketrightBigg/parenleftBiggN/summationdisplay i=1fk i/parenrightBigg =l/summationdisplay k=0/bracketleftBiggl/summationdisplay r=ks(r,k)2rl! r!/parenleftBigg l r/parenrightBigg/bracketrightBigg Mks(B.3) Thus the coeﬃcient ckis cl k=l/summationdisplay r=ks(r,k)2rl! r!/parenleftBigg l r/parenrightBigg (B.4) Fixingl= 0,1,..., Λ, and having sorted U(l,ρ) = (−1)l(l+ 1)/summationtext f∈FAl(f)/ρ2l+2, we can invert the above triangular system ( B.3) starting from l= 0. C Proof of a property of Kronecker projectors This appendix proves the relation (6.11). Using ˜PR1,R2,R3in(6.8),QR/prime 1,R/prime 2,R/prime 3µµin(6.6), and introducing the notation κR/prime 1R/prime 2=dR/prime 1dR/prime 2 (n!)2, we have ˜PR1,R2,R3QR/prime 1,R/prime 2,R/prime 3µµ =κR/prime 1R/prime 2dR1dR2dR3 (n!)3/summationdisplay τ1,τ2,τ3/summationdisplay σ1,σ2χR1(τ1)χR2(τ2)χR3(τ3) /summationdisplay i1,i2,j1,j2,kDR/prime 1 i1j1(σ1)DR/prime 2 i2j2(σ2)CR/prime 1R/prime 2;R/prime 3,µ i1,i2,kCR/prime 1R/prime 2;R/prime 3,µ j1,j2,kτ3τ1σ1⊗τ3τ2σ2 =κR/prime 1R/prime 2/summationdisplay τ1,τ2,τ3/summationdisplay ˜σ1,˜σ2dR1dR2dR3 (n!)3χR1(τ1)χR2(τ2)χR3(τ3) /summationdisplay i1,i2,j1,j2,kDR/prime 1 i1j1(τ−1 1˜σ1)DR/prime 2 i2j2(τ−1 2˜σ2)CR/prime 1R/prime 2;R/prime 3,µ i1,i2,kCR/prime 1R/prime 2;R/prime 3,µ j1,j2,kτ3˜σ1⊗τ3˜σ2 =δR1R/prime 1δR2R/prime 2κR1R2/summationdisplay τ3/summationdisplay ˜σ1,˜σ2dR3χR3(τ3) n! /summationdisplay i1,i2,j1,j2,kDR1 i1j1(˜σ1)DR2 i2j2(˜σ2)CR1,R2;R3,µ i1,i2,kCR1,R2;R3,µ j1,j2,k(τ3˜σ1⊗τ3˜σ2) =δR1R/prime 1δR2R/prime 2κR1R2/summationdisplay τ3/summationdisplay σ1,σ2dR3χR3(τ3) n! /summationdisplay i1,i2,j1,j2,kDR1 i1j1(τ−1 3σ1)DR2 i2j2(τ−1 3σ2)CR1R2R3;µ i1i2kCR1R2R3;µ j1j2kσ1⊗σ2 =δR1R/prime 1δR2R/prime 2κR1R2/summationdisplay σ1,σ2/summationdisplay i1,i2,j1,j2DR1 i1p1(τ−1 3)DR1 p1j1(σ1)DR2 i2p2(τ−1 3)DR2 p2j2(σ2) ×CR1,R2,R3;µ i1i2kCR1,R2,R3;µ j1j2kσ1⊗σ2 =δR1R/prime 1δR2R/prime 2δR3R/prime 3κR1R2/summationdisplay σ1,σ2DR1 p1j1(σ1)DR2 p2j2(σ2) ×CR1,R2,R3 p1,p2,k/primeDR/prime 3 k/primek(τ3)δνµdR3χR3(τ3) n!CR1R2;R3,µ j1,j2,kσ1⊗σ2 =δR1R/prime 1δR2R/prime 2δR3R/prime 3κR1R2/summationdisplay σ1,σ2DR1 p1j1(σ1)DR2 p2j2(σ2)CR1,R2;R3,µ p1,p2,kCR1,R2;R3,µ j1,j2,kσ1⊗σ2 =δR1,R/prime 1δR2,R/prime 2δR3,R/prime 3QR1,R2,R3µµ (C.1) – 32 – JHEP05(2023)191Open Access. This article is distributed under the terms of the Creative Commons Attribution License ( CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. SCOAP3supports the goals of the International Year of Basic Sciences for Sustainable Development. References [1] J.M. Maldacena, The large N limit of superconformal ﬁeld theories and supergravity ,Adv. Theor. Math. Phys. 2(1998) 231 [hep-th/9711200 ] [INSPIRE]. [2] S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory ,Phys. Lett. B 428(1998) 105 [hep-th/9802109 ] [INSPIRE]. [3] E. Witten, Anti-de Sitter space and holography ,Adv. Theor. Math. Phys. 2(1998) 253 [hep-th/9802150 ] [INSPIRE]. [4]D.J. Gross and W. Taylor, Two-dimensional QCD is a string theory ,Nucl. Phys. B 400(1993) 181 [hep-th/9301068 ] [INSPIRE]. [5] S. Cordes, G.W. Moore and S. Ramgoolam, Lectures on 2-d Yang-Mills theory, equivariant cohomology and topological ﬁeld theories ,Nucl. Phys. B Proc. Suppl. 41(1995) 184 [hep-th/9411210 ] [INSPIRE]. [6] A. Ram, Dissertation, Chapter 1: Representation

Question 1268 multiple-choice

Quantum Chern-Simons theory combines principles of Lie theory, harmonic analysis, and topological field theory to study the quantum geometry of moduli spaces, especially for compact gauge groups on surfaces like the torus. The representation theory of mapping class groups and the construction of Hilbert spaces rely on deep algebraic and geometric structures. In the quantization of Chern-Simons theory for a compact Lie group G on a torus, which mathematical object describes the moduli space of flat connections modulo gauge equivalence? 1) The weight lattice of G 2) The dual of the Lie algebra t 3) The set of roots of G 4) The maximal torus T itself 5) The space of all connections on the torus 6) The quotient (T × T)/W, where T is the maximal torus and W is the Weyl group 7) The group of automorphisms of G

✓ Correct Answer:

The correct answer is 6) The quotient (T × T)/W, where T is the maximal torus and W is the Weyl group.

📚 Reference Text:

in notation from Lie theory, which we brieﬂy rec all in Appendix A, where we further introduce a few quantities we will need in th is paper. In particular we recall that t⊂gis the Lie algebra of TGand that we denote by Z(k)the ﬁnite set of1 k-scaled weights modulo the root lattice of g. In Section 3 we present a generalization of the Weil-Gel’fan d-Zak transform which provides an isometry Z:H(k)→/parenleftbig L2(t)⊗CZ(k)/parenrightbigW. TheWinvariance on the right is with respect to the following acti on ofw∈W w·fγ(θ) =fwγ(wθ) forγ∈ Z(k),θ∈T. (5) For technical reasons, we further conjugate with a discrete Fourier transform FZ(k) (see the deﬁnition from equation (35)). We use the following notation for the corresponding pre-quantum operators action on the square i ntegrable functions ˆρk(S) = (Z◦FZ)−1◦˜S◦(Z◦FZ), ˆρk(T) = (Z◦FZ)−1◦˜T◦(Z◦FZ) (6) and correspondingly for the quantum representation ˆηt(S) = (Z◦FZ)−1◦ηt(S)◦(Z◦FZ), ˆηt(T) = (Z◦FZ)−1◦ηt(T)◦(Z◦FZ). (7) Considering the W-invariant subspace of L2(t)⊗CZ(k)we get the decomposition H(k)≃/parenleftbig L2(t)⊗CZ(k)/parenrightbigW =/parenleftbig L2(F0))⊗C(k)/parenrightbig/circleplusdisplay/parenleftbig L2(F0)⊗C(k)/parenrightbig (8) for which we refer again to appendix A for deﬁnitions and nota tions. Theorem 1. Letjandωbe complex numbers satisfying j4= 1, ω3=in/2j−1. (9) The pre-Quantum representation ˆρksplits into the direct sum of two representa- tions, induced by the decomposition (8) ˆρk= (ˆρ′ k,0⊗ˆρ′′ k,0)⊕(ˆρ′ k,1⊗ˆρ′′ k,1), GENUS ONE COMPLEX QUANTUM CHERN-SIMONS REPRESENATION 5 where, forθ∈F0,α,β∈Ikandα′,β′∈Ik ˆρ′ k,0(S)(f)(θ) =j/integraldisplay F0f(˜θ)/summationdisplay w∈We2πi(wθ,˜θ)kdvolk(˜θ), ˆρ′ k,1(S)(f)(θ) =j/integraldisplay F0f(˜θ)/summationdisplay w∈Wdet(w)e2πi(wθ,˜θ)kdvolk(˜θ), ˆρ′ k,0(T)(f)(θ) = ˆρ′ k,1(T)(f)(θ) =ωe−πi(θ,θ)kf(θ), ˆρ′′ k,0(S)α,β=j−1 /radicalbig |Z(k)|/summationdisplay w∈Wdet(w)e2πi(wα,β)k ˆρ′′ k,1(S)α′,β′=j−1 /radicalbig|Z(k)|/summationdisplay w∈We2πi(wα′,β′)k ˆρ′′ k,0(T)α,β=ω−1eπi(α,α)kδα,β, ˆρ′′ k,1(S)α′,β′=ω−1eπi(α′,α′)kδα′,β′. Letb∈Csuch that |b|= 1,Re(b)>0and (10) is=k1−b2 1+b2, Theorem 2. The quantum representation ˆηtis obtained by conjugation with an unitary operator ˆηt(ϕ) =e−rˆ∆σ◦ˆρk(ϕ)◦erˆ∆σ. It decomposes itself as ˆηt= (ˆη′ t,0⊗ˆρ′′ k,0)⊕(ˆη′ t,1⊗ˆρ′′ k,1). For the particular choice of complex structure σ=ibthe representations on L2(F0) take the following explicit integral form ˆη′ t,0(S)(f)(θ) =jeπ(b−b)(θ,θ)k/integraldisplay F0f(˜θ)/summationdisplay w∈We2πi(wθ,˜θ)ke−π(b−b)(˜θ,˜θ)kdvolk(˜θ), ˆη′ t,1(S)(f)(θ) =jeπ(b−b)(θ,θ)k/integraldisplay F0f(˜θ)/summationdisplay w∈Wdet(w)e2πi(wθ,˜θ)ke−π(b−b)(˜θ,˜θ)kdvolk(˜θ), ˆη′ t,0(T)(f)(θ) =ωi−n 2eπ(b−b)(θ,θ)k/integraldisplay F0f(˜θ)/summationdisplay w∈Weπi(wθ−˜θ,wθ−˜θ)ke−π(b−b)(˜θ,˜θ)kdvolk(˜θ), ˆη′ t,1(T)(f)(θ) =ωi−n 2eπ(b−b)(θ,θ)k/integraldisplay F0f(˜θ)/summationdisplay w∈Wdet(w)eπi(wθ−˜θ,wθ−˜θ)ke−π(b−b)(˜θ,˜θ)kdvolk(˜θ), In Section 4, Lemma 5, we provide the explicit expression for e−rˆ∆σfor generic σ. In Section 5 we provide comparisons to the known result for q uantum Chern– Simons theory with compact gauge group G. 2.Moduli space of flat connections and the pre-quantum line bun dle In this section we brieﬂy recall the construction of the pre- quantum space for Chern-Simons theory. We refer to [Fr1] and [ADW] for more det ails. See Appen- dix A for notations on Lie theory. Given a closed surface Σof genusg= 1, its 6 JØRGEN ELLEGAARD ANDERSEN AND SIMONE MARZIONI fundamental group is abelian, precisely π1(Σ) =Z×Z. LetGbe a compact, simple, connected and simply connected Lie gro up. The moduli space of ﬂat connections is then (11) M= Hom(π1(Σ),G)/G= (T×T)/W, whereTis the maximal torus of GandWis the Weyl group of G. We can lift the description to quotient of its tangent space M= (t⊕t)//parenleftbig (ΛR×ΛR)⋊W/parenrightbig . (12) Since any principal G-bundle on Σis trivial, we can ﬁx a trivialization and so a G-connection A∈Ω(Σ,g)is a one form on Σwith values in the lie algebra g. In particular, we can

Question 1269 multiple-choice

Quantum Phase Estimation (QPE) algorithms are central to many quantum computing applications, but their implementation on current noisy quantum hardware presents significant challenges. Strategies to improve QPE performance are critical for advancing practical quantum computation. Which modification most directly addresses the limitations of Noisy Intermediate-Scale Quantum (NISQ) hardware when implementing Quantum Phase Estimation algorithms? 1) Increasing the number of ancilla qubits in the circuit 2) Replacing controlled gates with single-qubit rotations 3) Utilizing quantum error correction codes exclusively 4) Extending coherence times through hardware cooling 5) Performing more rounds of measurement in Kitaev's algorithm 6) Reducing the number of control gates and phase shift operations 7) Applying post-processing to measurement outcomes

✓ Correct Answer:

The correct answer is 6) Reducing the number of control gates and phase shift operations.

📚 Reference Text:

Title: Experimental Challenges of Implementing Quantum Phase Estimation Algorithms on IBM Quantum Computer Year: 2019 Paper ID: cd1d1ab8500d80825d7c5999501fe5751439360a Source: semantic-scholar URL: https://www.semanticscholar.org/paper/cd1d1ab8500d80825d7c5999501fe5751439360a Abstract: Many researchers have been heavily investigated on quantum phase estimation (QPE) algorithms to find the unknown phase, since QPE is the core building block of the most quantum algorithms such as the Shor's factoring algorithm, quantum sampling algorithms, and finding the eigenvalues of unitary matrices. Kitaev's algorithm and QPE algorithms using inverse Quantum Fourier transform were proposed and widely used by researchers as a key component for their quantum algorithms. In this paper, we explore the experimental challenges of QPE algorithms on Noisy Intermediate-Scale Quantum (NISQ) computers by implementing various QPE algorithms on the state-of-the-art IBM quantum computer. Our experimental results demonstrate that the accuracy of finding the phase using these algorithms are severely constrained by NISQ's physical characteristics such as coherence time and error rates. To mitigate such physical limitations, we propose modified solutions of these algorithms by reducing the number of control gates and phase shift operations. Our experimental results showed that our solutions can significantly increase the accuracy of the finding phase in near-term quantum computers.

Question 1270 multiple-choice

In quantum algorithms for hidden subgroup problems, group theory concepts such as subgroup structure, the center of a group, and the quantum Fourier transform over finite abelian groups play a crucial role in designing efficient procedures. The ability to convert quantum states related to coset spaces is central to these algorithms. Which property of a subgroup L of the center of a group G enables the quantum Fourier transform over L to effectively disentangle coset superpositions in quantum algorithms for group-theoretic problems? 1) L is a maximal subgroup of G 2) L is self-normalizing in G 3) L is cyclic of prime order 4) L contains all commutators of G 5) Every element of L commutes with every element of G 6) L is a simple subgroup 7) L is a subgroup of the derived series of G

✓ Correct Answer:

The correct answer is 5) Every element of L commutes with every element of G.

📚 Reference Text:

⎠. Initially H0={I}and K=G. The commutator subgroup K/prime=G/prime consists of the matrices of the form⎛ ⎝1a13 1 1⎞ ⎠andK/K/prime∼=Z2 9, not elementary. Then Kis replaced by the group of matrices⎛ ⎝13a12a13 13 a23 1⎞ ⎠. (Here and in each step below, the notation aijshould be considered as independent of those used in the previous step(s).) It turns out that this Kis abelian ( K/prime={I}), but not elementary. It gets replaced by the group of matrices of the form⎛ ⎝13a123a13 13 a23 1⎞ ⎠, which is isomorphic toZ3 3and H∩Kconsists of the matrices of the form⎛ ⎝13a12 1 1⎞ ⎠and H0becomes this subgroup. The next Kwill be the normalizer of this H0, which is the group of matrices⎛ ⎝1a12a13 13a23 1⎞ ⎠, whose commutator subgroup K/primeconsists of the matrices of the form⎛ ⎝1a13 1 1⎞ ⎠and K/K/primeis isomorphic to Z3 3,s o K, and hence also K/H0, is semi-elementary. Therefore HSMC will ﬁnd H/H0inK/H0and the procedure terminates. 4 The main conversion LetLbe a subgroup of the center of Gisomorphic to Zn pwhere pis a prime. Then L is a normal subgroup of G. Our aim is to convert a copy of the subgroup state /Xi1G,H to a copy of /Xi1G/L,HL/L. In the light of the second conversion (“pushing”) described in Sect. 2.4, one could do it by converting ﬁrst to a copy of /Xi1G,HL. To this end, it would be desirable to have a procedure that converts the coset superpo- sition|aH/angbracketrightto|aHL/angbracketright. A possible approach would be computing |L/angbracketright=1√|L|/summationtext z∈L|z/angbracketright in a new register, multiplying |aH/angbracketrightwith it to obtain1√|HL|/summationtext z∈L/summationtext x∈H|azx/angbracketright|z/angbracketrightand 123 14 Page 12 of 23 M. Imran, G. Ivanyos then trying to “disentangle” |z/angbracketrightfrom|azx/angbracketright. The quantum Fourier transform of L almost does this job: If we apply it to the second register, we obtain the state 1√|L|/summationdisplay y∈Lω(y,z) √|HL|/summationdisplay z∈L/summationdisplay x∈H|azx/angbracketright|y/angbracketright, where ω=e2πi pand by (,)we denote the standard scalar product of Lmodulo p. For y∈L, let us denote by Pythe linear transformation of CGmapping |x/angbracketrightto 1√|L|/summationtext z∈Lω(y,z)|xz/angbracketright. With this notation, the state we have can be rewritten as 1√|L|/summationdisplay y∈L|Py(aH)/angbracketright|y/angbracketright. Using the assumption that Lis in the center of G, a direct calculation shows that for every x1,x2∈G,w eh a v e |Py(x1x2)/angbracketright=| x1Py(x2)/angbracketright=|(Py(x1)x2)/angbracketright. (1) It is also straightforward to see that for every x∈Gand for every w∈L,w eh a v e |wPy(x)/angbracketright=| Py(x)w/angbracketright=ω−(y,w)|Py(x)/angbracketright. (2) We deﬁne the support of an element |u/angbracketrightof CGas the set of elements appearing with nonzero coefﬁcient in the decomposition of |u/angbracketrightas a linear combination of group elements. Using equality ( 1), one can show that if x1and x2are not in the same left coset of LH, then the states |Py(aH)x−1 1/angbracketrightand|Py(aH)x−1 2/angbracketrightare orthogonal. This is because the support of |Py(aH)x−1 i/angbracketrightis contained in LHx−1 i=(xiLH)−1(i=1,2). On the other hand, if x1H=x2H, then Hx−1 1=Hx−1 2and the two states are equal. By the characterization given in Lemma 1, it follows that for any left coset aH,t h e state1√|G|/summationtext x∈G|x/angbracketright|P0(aHx−1)/angbracketrightis a puriﬁcation of /Xi1G,LH. Of course it is hopeless to enforce y=0i n|Py(aH)x−1 1/angbracketright.

Question 1271 multiple-choice

Quantum photonic devices often rely on precise control of light interference patterns to implement algorithms such as Grover's Search or the Quantum Fourier Transform. Techniques for measuring single-photon outputs are crucial for accurate quantum state reconstruction in such systems. Which of the following experimental approaches is essential for improving the reliability of single-photon measurements in photonic quantum computing platforms that use metasurfaces? 1) Using unheralded photon sources with high spatial resolution cameras 2) Increasing the number of camera pixels to maximize spatial sampling 3) Employing classical light sources for quantum algorithm demonstration 4) Ignoring probabilistic measurement outcomes and taking single-shot readings 5) Implementing heralding techniques to ensure only events with known photon presence are counted 6) Avoiding full-image capture to reduce data volume 7) Relying solely on simulations without experimental verification

✓ Correct Answer:

The correct answer is 5) Implementing heralding techniques to ensure only events with known photon presence are counted.

📚 Reference Text:

SLMsetsthecorrespondingphases,asshownintheﬁrstrowofFigure3(a) .Theoutputstatescorrespondtothecolumnsof the4 ×4identitymatrix,correspondingtothefourlower- right output directions exhibiting signiﬁcant interferencepatterns.Thesimulationandexperimentalinterferencepat-ternsareinagreementandcanbedecoded(inarow-majormanner)toyieldtheexpectedoutputstates. With the previous experience gained from using clas- sical light to conﬁrm the metasurface’s functionality, wenowshiftourfocustodiscussingtheresultsobtainedwithaquantumlightsource.Whileweexpectconsistentresultsby using the two different types of sources, the utilization ofaquantumlightsourcehighlightsakeyaspect:asingle- photoncansimultaneouslyinteractwithallninemetalenses(inasimilarspirittoadoubleYoungslitexperiment),whichshowsthemetasurface’spotentialasalow-powerplatformfor implementing quantum algorithm. It is important tonote that due to the probabilistic nature of single-photondetection, obtaining an accurate output state necessitatessampling numerous measurements of single photons. Thisaligns with the conventional approach of implementingquantum algorithms through a quantum circuit, which involves executing the algorithm multiple times. Repeated measurement plays a vital role in some algorithms, suchas in GS to reduce the risk of false positives and in QFT toachieveapreciseprobabilitydistribution[ 7],[22]. On the other hand, using a quantum light source presentsanadditionalchallengeinextractinginformationdue to the lower number of photons reaching the single-photon camera. To mitigate the impact of environmentalnoiseinthesituationofasmallernumberofsignalphotons,weadopttheheraldingtechnique(asshownin Figure2(d) ). In this case, we use a signiﬁcantly smaller number of pix- els on the camera, around 40 ×40 pixels, to capture the entire interference patterns. This allows us to ensure anadequatenumberofphotonsperpixel,enablingdataacqui-sitionwithinamanageabletimeduration.Forthecompari-son of the results using classical laser and heralded singlephotons, please refer to the Supplementary Materials .I n principle, for our implementation, it is sufficient to placefoursingle-photondetectorstoobtaintheoutputstatepre- cisely at the four exit directions for a two-qubit operation. However,usingasingle-photoncameraallowsustocapturethefullinterferenceimage.Theseadditionaldetectors(nowroughlyaroundeachexitdirection)provideadditionalout-putbases,whicharehelpfulforestablishingamorerobustmeasurement of the original output state by ﬁtting theinterferencepattern,inasimilarspirittoerrorcorrection. R.S.Tanuwijayaetal.:Metasurfaceforprogrammablequantumalgorithms — 933 Figure 4:Experimentalresultsusingaquantumheraldingphotonsource.(aandb)Interferencepatternsafterup-samplingfortheGrover’ssearch (GS)operation(left)andquantumFouriertransform(QFT)operation(right).Eachcolumncorrespondstoadifferentinputstate,andeachrowshows theinterferencepatternsformedindesignateddirections.(candd)Normalizedfittedoutputstateamplitudesextractedfromtheinterferencepatternsfortherespectiveoperations.Thefittingmethodapproximatesthefieldamplitudeatthecenterofthepatternbymodelingtheinterferenc e ofthefourmetalenses. Here,wedevelopaﬁttingprocedureforeachinterference pattern formed at a particular output direction, assumingthat only four waves from the involved metalenses inter-fere with each other. The ﬁtting procedure extracts thecomplex ﬁeld amplitudes exiting each metalens. Based onthisﬁttingprocedure,wecaneffectively performa4 ×up- samplingoftheinterferencepatternimages,whicharepre-sented in Figure 4(a) and (b) for the GS and QFT, respec- tively. Further details regarding the up-sampling processfrom the raw images using the physical ﬁtting model areprovidedinthe SupplementaryMaterials .Theup-sampled interference patterns are displayed in a tabular form witheach column labeling the input state and each row label-ing the output interference pattern at individual outputdirections (output basis), resembling the classical resultsshown in Figure 3by reading out the individual interfer- ence patterns in a row-major manner. At the same time,wehavealsoobtainedthe||| 𝜓(out) i|||(continuousvaluesnow) at each output mode |i⟩during the ﬁtting process. Thesevaluesarepresentedinthesametabularformin Figure4(c) and (d)for the GS and QFT, respectively. In both cases, a comparison is made to the expected results (identitymatrix), resulting in a root-mean-square error (RMSE) of0.148 for all 16 elements in the GS and 0.216 for the QFT.It is worth noting that the larger error observed at thek 0direction in QFT (as shown in Figure 4(d) )c a nb e attributed to the presence of residual beams, which alsoappear in Figure 3(c) , contributing to the larger RMSE in t h eQ F Tc a s e .W en o t et h a tw ec a ns c a l e - u pt h ec u r r e n tapproach by adding more lenses to the metalens array.FigureS3inSupplementaryMaterials showsanexampleof four qubits operation with totally 16 lenses. While furtherscale-up relies much on how accurately the interferencepattern can form from different lenses, we envision thecurrentapproachcanbeimmediatelyusefultominiaturizepart of a large quantum circuit. We also note that in

Question 1272 multiple-choice

In quantum algorithms for the Hidden Subgroup Problem over the Heisenberg group, representation theory and specific transforms are used to analyze and distinguish subgroup states. Understanding the behavior of irreducible representations and their combinations is central to efficient algorithm design in non-abelian groups. When applying the Clebsch-Gordan transform to the direct product of two p-dimensional irreducible representations (σ_{k1} and σ_{k2}) of the Heisenberg group, what occurs if k1 + k2 is not congruent to zero modulo p? 1) The product decomposes into a set of one-dimensional irreps with multiplicity one. 2) The product decomposes into a p-dimensional irrep labeled k′ = k1 + k2, with multiplicity p. 3) The product remains indecomposable and forms an irreducible representation of dimension p^2. 4) The product yields a trivial one-dimensional representation. 5) The decomposition involves a unitary change of basis and gives a structure over the irrep space distinct from multiplicity. 6) The result is a direct sum of p one-dimensional irreps with distinct labels. 7) The product produces no irreducible components and cannot be decomposed further.

✓ Correct Answer:

The correct answer is 2) The product decomposes into a p-dimensional irrep labeled k′ = k1 + k2, with multiplicity p..

📚 Reference Text:

subgroup, we see tha t the trivial and the Ai,j subgroups cannot be distinguished by the above measurement of µ. Indeed, the probability of getting one of the pdimensional irreps in both cases is uniformly1 pand there are p−1 such irreps such that with probability 1 −1 pthe states are not distinguished using just this irrep measurement. We are interested in more than just the probability of the measurem ent of the irrep label. In particular we are most interested in the exact form of the hidden subgroup state produced after we measure the irrep index and throw away the index of the lef t regular representation. When the hidden subgroup is Ai,jand we measure the one dimensional irrep with character χa,bwe obtain, of course, a single one dimensional space with density mat rix just the scalar 1. However when obtain one of the pdimensional irreps, we obtain the state over the p dimensional irrep space, ρk(Ai,j) =1 p/summationdisplay l∈Zpω2−1l(l−1)ik+ljk/summationdisplay r∈Zpωklir|r+l/an}bracketri}ht/an}bracketle{tr|. (58) This latter state is the state which we will be most relevant to our eﬃc ient algorithm for the HSCP over the Heisenberg group. 7 The Clebsch-Gordan Transform over the Heisenberg Group IfwehavetwoirrepsoftheHeisenberggroup, Dµ1andDµ2whataretherulesfordecomposing the direct product of these irreps into irreps? In other words, wh at is the Clebsch-Gordan transform for the 2-fold direct product of two irreps of the Heise nberg group? The ﬁrst case to consider is when both of the irreps are one dimensio nal irreps with characters χa1,b1andχa2,b2. In this case it is easy to see that the direct product of these irreps is a one dimensional irrep with correspondingcharacter χa1+a2,b1+b2where the addition is done over Zp. The second case to consider is when one of the irreps is the one dime nsional irrep with character χa1,b1and the second irrep is a pdimensional irrep with character χk2. 20How a Clebsch-Gordan Transform Helps to Solve ... In this case it is easy to see that χa1,b1((x,y,z))⊗σk2((x,y,z)) =ωk2y+a1x+b1z/summationdisplay r∈Zpωk2zr|r+x/an}bracketri}ht/an}bracketle{tr| (59) Now making the change of basis described by the unitary matrix V=/summationdisplay t∈Zp|t+k−1 2b1/an}bracketri}ht/an}bracketle{tt| /summationdisplay s∈Zpω−a1s|s/an}bracketri}ht/an}bracketle{ts| , (60) toχa1,b1((x,y,z))⊗σk2((x,y,z)) yields Vωk2y+a1x+b2z/summationdisplay r∈Zpωk2zr|r+x/an}bracketri}ht/an}bracketle{tr|V†=σk((x,y,z)). (61) Thusthe directproductofthe onedimensionalirrepwith characte rχa,band thepdimensional irrepσkis apdimensional irrep σk. Further there is no multiplicity in this decomposition. The ﬁnal case to consider is the case where one is forming the direct product of two p dimensional irreps, σk1andσk2. In this case we see that σk1((x,y,z))⊗σk2((x,y,z)) =/summationdisplay r1∈Zpωk1y+k1zr1|r1+x/an}bracketri}ht/an}bracketle{tr1|⊗/summationdisplay r2∈Zpωk2y+k2zr2|r2+x/an}bracketri}ht/an}bracketle{tr2|.(62) Consider ﬁrst the case where k1+k2/ne}ati≀nslash= 0 mod p. Deﬁne the following change of basis W=/summationdisplay a,b∈Zp|a−b/an}bracketri}ht/an}bracketle{ta|⊗|(k1a+k2b)(k1+k2)−1/an}bracketri}ht/an}bracketle{tb|. (63) Applying this transform to σk1((x,y,z))⊗σk2((x,y,z)) results in /summationdisplay r1,r2∈Zpωk′y+(k1r1+k2r2)z|r1−r2/an}bracketri}ht/an}bracketle{tr1−r2|⊗|(k1r2+k2r2)(k′)−1+x/an}bracketri}ht/an}bracketle{t(k1r2+k2r2)(k′)−1|(64) wherek′=k1+k2. Usingu=r1−r2andv= (k1r1+k2r2)(k′)−1this can be reexpressed as ωk′y/summationdisplay u∈Zp|u/an}bracketri}ht/an}bracketle{tu|⊗/summationdisplay v∈Zpωk′vz|v+x/an}bracketri}ht/an}bracketle{tv| (65) which we see is just I⊗σk′((x,y,z)). Thus when k1+k2/ne}ati≀nslash= 0 mod p, the direct product of the twopdimensional irreps labeled by k1andk2is reducible to the irrep labeled by k′=k1+k2 with multiplicity p. Whenk1+k2= 0 mod pwe ﬁnd that σk1((x,y,z))⊗σk2((x,y,z)) =/summationdisplay r1,r2∈Zpωk1z(r1−r2)|r1+x/an}bracketri}ht/an}bracketle{tr1|⊗|r2+x/an}bracketri}ht/an}bracketle{tr2|.(66) Consider the unitary change of basis X=1√p/summationdisplay a,b,c∈Zpω(a+b)c|a−b/an}bracketri}ht/an}bracketle{ta|⊗|c/an}bracketri}ht/an}bracketle{tb|. (67) D. Bacon 21 This transforms the direct product to /summationdisplay u,c∈Zpωk1uz+2xc|u/an}bracketri}ht/an}bracketle{tu|⊗|c/an}bracketri}ht/an}bracketle{tc|, (68) whereu=r1−r2and we have

Question 1273 multiple-choice

Integer commitment schemes play a crucial role in cryptography, especially in constructing zero-knowledge proofs and privacy-preserving protocols. Their efficiency and security depend on properties like hiding, soundness, and support for robust operations over the integers. Which property is notably lacking in single-base commitment schemes of the form c = g^s mod n, thereby limiting their use in zero-knowledge and witness indistinguishable protocols? 1) They are computationally inefficient for large security parameters. 2) They cannot support addition of committed values. 3) They require quadratic communication complexity. 4) They lack the standard hiding property; committing twice to the same value reveals the value. 5) They rely on the discrete logarithm problem instead of the RSA assumption. 6) They cannot be generalized to support integer relations. 7) They are susceptible to chosen-ciphertext attacks.

✓ Correct Answer:

The correct answer is 4) They lack the standard hiding property; committing twice to the same value reveals the value..

📚 Reference Text:

of this is if you want to show that a committed number sis an RSA signature on a given message aw.r.t. public key n,3. What we want to know is that a=s 3+tnfor somet, and this of course must be true over the integers and not just modulo m. Of course, one might be able to solve this by choosing the commitment scheme such that m=n, but this requires that at least you know nat the time the commitment scheme was set up, and also a new instance of the commitment scheme for each n. This is often unreasonable in practice. There are other ways around the problem,see for instance [12], but the protocols are far from optimal, typically one hasto resort to “binary cut-and-choose”, which means communication complexityat least quadratic in the security parameter. Another example of the need forrelations over the integers is the eﬃcient zero-knowledge proofs of Boudot [4] fordemonstrating that a committed number is in a given interval. Here, it is crucialfor eﬃciency that one can prove eﬃciently that committed numbers a,bsatisfy b=a 2over the integers. It should be clear that what we really need here is an integercommitment scheme,thatis,aschemewhere S=Z(oratleastsomelargeﬁniteinterval),and where there is an eﬃcient multiplication protocol that works over the integers. AStatistically-HidingIntegerCommitmentScheme 127 Here, by eﬃcient, we mean constant round protocols requiring only communica- tion linear in the security parameter. 1.2 The Earlier Scheme with Statistically-Hiding Commitment In [14], Okamoto and the second author of this paper presented the ﬁrst eﬃcient integer commitment scheme and also suggested an eﬃcient multiplication pro-tocol. The scheme is based on the strong RSA assumption suggested in [2,14].However, via private communication, we found some gaps in the proof of sound-ness of the associated protocols, one of which we think presents a non-trivial problem which, to the best of our knowledge, has remained open until now.Later in the paper we give a short explanation of the problem in the proof from[14].Weﬁllallthegapshereusingadditionalideaincludingaminormodiﬁcationof the form of a commitment. 1.3 Other Related Works There are several related works inspired by [14] such as [8,3,10,4]. The protocols constructed there generally do not suﬀer from the main problem we mentionedabove.However,thereasonforthisisthattheyuse“commitments”withasinglebase, i.e., a commitment to sis of formc=g smodn. Such a commitment does not satisfy the standard hiding property for commitments. For instance,if a prover commits twice to the same value, this is immediately visible. Thusderived protocols using such commitments are not in general (honest-veriﬁer)zero-knowledge nor witness indistinguishable. Boudot [5] pointed out another problem in the proof of soundness – In this type of protocols (based on a group with a hidden order), the natural protocolfor showing that one knows how to open a commitment ccan in fact only show that the prover can open cor−c(a problem that even [8,3,10,4] cannot avoid). This is not so serious in practice but we suggest a solution to this problem too,by changing the way in which commitments are opened. 1.4 Our Scheme In this paper, we present a commitment scheme that may be seen as a gener- alization of

Question 1274 multiple-choice

Quantum Fourier Transform (QFT) is a key operation in quantum computing, closely related to the classical Fast Fourier Transform (FFT). Optical implementations of QFT exploit waveguide structures and linear optical components to achieve ultrafast signal processing. Which feature of the Quantum Fourier Transform (QFT) circuit makes its output closely resemble the classical FFT with respect to result ordering in multi-qubit systems? 1) The use of multi-level quantum gates for entanglement generation 2) The implementation of direct summation across all input states 3) The production of complex-valued outputs in reverse bit order 4) The exclusive use of single-qubit rotation gates 5) The absence of phase rotation gates in the circuit 6) The transformation of outputs into purely real values 7) The restriction to binary amplitude weighting

✓ Correct Answer:

The correct answer is 3) The production of complex-valued outputs in reverse bit order.

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on a single qubit state to give : 1 210 211 01 0 1 0 1 0 xx xx x x  H (19) Thus it can be seen that the Hadamard transform performs a 2 -point QFT by implementing the basic FFT building block of the Butterfly operation i.e. the sub traction and addition of the two input signals, albeit in a superposition state . Larger count QTFs can thus be implemented by the same basic operation with the additional inclusion of a controlle d phase shifter on the lower leg of the late r stage Butterflies, as detailed when describing the structure of the FFT in Section 3. Thus the QFT can be optically implemented with the waveguide structure described in Section 3 for th e optical implementation of the FFT (the difference being that the input values are in a superposition state rather than taking on explicit values). The FFT decomposition of the QFT using the Hadamard gate as a basic building element together with phase ro tation gates described by matrices of the form12,13:      njn eR 22 .000. ....0 .1000 .0100 .001  (20) This allows the QFT structure for a n-qubit input to be represente d compactly as shown in Figure 6 . Figure 6 . n-qubit QFT quantum gate array structure The decomposition is thus an FFT structure acting on an binary n-qubit input to yield a complex -valued n-qubit output (in reverse bit order). An extension t o a multi -valued QFT with input values weighted by a wavefunction, such as shown by Equation 15 , has been proposed by Muthukrishnan and Stroud11 making the similarity to the cohe rent optical FFT structure clear . An implementation problem arises in the phase resolution required by the final term in the Rn matrices for large n as this scales as 2n. Howeve r, using methods originally proposed by Coppersmith13 and with a later refin ement by Fowler and Hollenberg14 it may be possible to QFT a binary integer of several thousand bits with a phase control no better than π /64 which seems to be technically fe asible. Thus the optical implementation of the QFT uses linear optical components which are essentially identical in form and arrangement to those that can be employed for the coherent optical implementation of a Fourier transform processor. 5. CONCLUSION This paper has described various optical structures that are able to compute the discrete Fourier transform with coherent light, both by direct evaluation of the summation and also by implementation of a structure to implement the basic building block of t he Cooley -Tukey fast Fourier transform algorithm, the Butterfly operation. A n-point discretely sampled coherent wavefront can thus be transformed into n complex valued Fourie r co-efficients in no longer tha n the propagation time of the coherent wavefront through the optical system which , if constructed as a waveguide , may only be tens of millimeters long, implying less than nan osecond

Question 1275 multiple-choice

In the study of group representation theory, understanding how representations decompose and relate to group structure is essential. Concepts such as irreducible representations, regular representations, and induced representations play central roles in both mathematics and physics. For a finite group G, which of the following statements is always true regarding its irreducible representations? 1) The number of irreducible representations equals the order of the group. 2) Every irreducible representation has dimension one. 3) The number of irreducible representations equals the number of conjugacy classes in G. 4) The regular representation decomposes into irreducible representations all with multiplicity one. 5) Induced representations are always irreducible for any subgroup. 6) The trivial representation is not counted among the irreducible representations. 7) The dimensions of all irreducible representations are equal.

✓ Correct Answer:

The correct answer is 3) The number of irreducible representations equals the number of conjugacy classes in G..

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be viewed as a direct sum of these two sub-representations. A representation is called an irreducible representation (irrep) if it does not contain any non-trivial sub-representations. Any representation Vcan be broken up into a direct sum of sub-representations Wand its complement as above. Continuing this process further and breaking up Wand its complement into sub-representations, one can arrive at a decomposition of Vinto irreps: V∼=V1⊕···⊕Vn, where some of the irreps in the decomposition may be equivalent. A special kind of irrep is the trivial irrep which acts on a one-dimensional vector space and takes all the group elements to the identity. Any nite group Ghas a nite number of irreducible representations whose number is equal to the number of conjugacy classes of G. A regular representation of a nite group Gacts on the vector space C[G], wheregacts on any basis vectorhby left multiplication g:h→gh. The regular representation turns out to have the following interesting direct sum decomposition into irreps. C[G]∼=/circleplusdisplay iW⊕di i, (2.1) whereWiis an irrep of G,iruns over all the di erent irreps of Ganddiis the vector space dimension of the irrepWi. The quantum Fourier transform (QFT) over a group Gusually refers to the transform that performs the above block diagonalization. This can be de ned as the following basis transformation. |g/angbracketright→|ρ,i,j/angbracketright, whereρis the label of the irrep, iis the multiplicity space index and jis the irrep space index. From the above decomposition, we can see that the dimension of both the multiplicity space and irrep space are the same and so iandjrun over the same index set labeling the basis vectors. A type of representation that is of particular importance here is the induced representation de ned as follows. Given a subgroup HofGand a representation (ρ,W)ofH, we can construct a representation V ofGas follows. As a vector space, it is the tensor product C[G/H ]⊗W, where C[G/H ]is the space of cosets ofHinG. The action of Gon this basis can be described using a transversal G/H ={t1,t2,...,tm} forHinG, wheremis the number of cosets i.e., m=|G|/|H|. This means that these elements form an orthonormal basis of the vector space C[G/H ]. Let{|w1/angbracketright,...,|wd/angbracketright}be a basis of W. We denote the induced representation by ↑G Hρor↑Gρ(whenHcan be inferred). In this basis, (↑G Hρ)(g)is the following action on basis vectors (which can be linearly extended to other vectors). (↑G Hρ)(g) :|t/angbracketright⊗|wi/angbracketright/mapsto→|t/prime/angbracketright⊗ρ(h)|wi/angbracketright (2.2) wheret/prime∈G/H andh∈Hare the unique elements for which gt=t/primeh. 3 2.2 Irreducible representations of symmetric and unitary groups The symmetric group on nletters is denoted Snand consists of all possible permutations of the nletters. There aren!permutations and every element can be written as a product of transpositions, where a transpo- sition is a swap of two letters. Here we denote a transposition between aandbas(a,b). The representation theory of the symmetric group is discussed in several books (see for example, [12, 23, 36]). The irreducible representations of Snare labeled by Young diagrams, which are diagrams that consist of rows of boxes. A Young diagram corresponds to a partition of n, which is

Question 1276 multiple-choice

Quantum algorithms for hidden subgroup problems play a pivotal role in both computational complexity and cryptography, particularly when considering non-abelian groups and their connections to hard lattice problems. The dihedral hidden subgroup problem (DHSP) exemplifies a central challenge in quantum computing due to its implications for cryptanalysis and post-quantum security. Which of the following statements best describes the consequence if an efficient quantum algorithm for the dihedral hidden subgroup problem (DHSP) were discovered? 1) It would enable factoring large integers in polynomial time using the dihedral group structure. 2) It would allow solving the unique shortest vector problem (SVP) in lattices efficiently, undermining the security of certain lattice-based cryptosystems. 3) It would prove that all non-abelian hidden subgroup problems are solvable in polynomial time. 4) It would provide a polynomial-time algorithm for all instances of the subset sum problem. 5) It would establish that classical algorithms can match quantum query complexity for DHSP. 6) It would eliminate the need for quantum queries in solving abelian HSPs. 7) It would show that post-processing in DHSP requires only logarithmic time.

✓ Correct Answer:

The correct answer is 2) It would allow solving the unique shortest vector problem (SVP) in lattices efficiently, undermining the security of certain lattice-based cryptosystems..

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Inui and Le Gall gave a solution for semidirect product groups of the form Zpk/multicloseright Z pwithpan odd prime [31]. Finally, as mentioned in the introduction, there is also a body of knowl- edge about the query complexity of the hsp. In particular, Ettinger, Høyer, and Knill have shown that O(log|G|) quantum queries of the function f are suﬃcient to determine the hidden subgroup [13]. Unfortunately, the quantum algorithm they present requires time O(|G|). 3.The dihedral hidden subgroup problem The dihedral group of order 2 N, denoted DN, is the group of symmetries of a regular N-sided polygon. This group is generated by two elements r andssatisfying the relations r2=e,sN=e, andrsr=s−1, whereeis the identity element. Here scorresponds to a rotation of the polygon and rcorresponds to a reﬂection. A generic element of the dihedral group can be written as rtskwheret∈Z2ands∈ZN, and group multiplication is given byrt/primesk/primertsk=rt+t/primesk+(−1)tk/prime. (Throughout this article we write ZN to denote Z/NZ.) The dihedral hidden subgroup problem ( dhsp ) was ﬁrst considered by Ettinger and Høyer [15]. They showed that given O(logN) queries to the hidden subgroup oracle for the dihedral group, there exists a quantum algo- rithm whose output contains enough classical information to solve the dhsp , and therefore that the query complexity of the dhsp isO(logN). Unfor- tunately, they were not able to ﬁnd an eﬃcient algorithm to process the output, so this approach has not yet led to an eﬃcient algorithm for the dhsp . A major motivation for attempting to solve the dhsp is a connection to lattice problems discovered by Regev [45]. A d-dimensional lattice is 6 DAVE BACON, ANDREW M. CHILDS, AND WIM VAN DAM the set of all integer linear combinations of dlinearly independent vectors inRdthat form a basis for the lattice. In the shortest vector problem , one attempts to ﬁnd the shortest (nonzero) vector in the lattice given a basis. In particular, in the g(d)unique shortest vector problem , we are promised that the shortest vector is unique and shorter than all other non-parallel vector by a factorg(d). The presumed hardness of certain g(d) unique shortest vector problems is the basis for a cryptosystem proposed by Ajtai and Dwork (in whichg(d) =O(d8)) [1], and a subsequent improvement proposed by Regev (in whichg(d) =O(d1.5)) [46]. Regev showed that an eﬃcient quantum algorithm for the dhsp that works by sampling hidden subgroup states can be used to solve the poly( d) unique shortest vector problem [45], thereby breaking the proposed lattice cryptosystems. Regev also gave a promising path toward solving the dhsp in the form of a connection to the subset sum problem . In the subset sum problem, one is givenknumbers between 0 and N−1, denoted x∈Zk N, and a target t∈ZN, and the goal is to ﬁnd a subset of the knumbers, speciﬁed by a binary vector b∈Zk 2, such that b·x=t, whereb·x:=/summationtextk j=1bjxjmodN. If such a subset exists, then we call ( x,t) a legal subset sum input. Regev has shown that if one can

Question 1277 multiple-choice

In quantum many-body physics, understanding the structure of subsystems is crucial for characterizing entanglement and possible configurations of composite quantum states. The interplay between group actions, tensor spaces, and geometric objects like moment polytopes provides powerful tools for analyzing quantum marginals. Which statement most accurately describes the role of the moment map in the study of the quantum marginal problem for multipartite quantum systems? 1) It defines the probability distribution of measurement outcomes on each subsystem. 2) It identifies the entanglement entropy between subsystems by tracing over their states. 3) It constructs the set of all possible pure state tensors from given marginal spectra. 4) It determines the trace norm used to measure the purity of quantum states. 5) It transforms density matrices into diagonal form under local unitary operations. 6) It translates the possible eigenvalues of reduced density matrices into a geometric structure known as the moment polytope, via group actions on the tensor space. 7) It normalizes the global quantum state to ensure the validity of marginal probabilities.

✓ Correct Answer:

The correct answer is 6) It translates the possible eigenvalues of reduced density matrices into a geometric structure known as the moment polytope, via group actions on the tensor space..

📚 Reference Text:

of the complexity of general moment polytopes. These moment polytopes (and their related problems) are most natural to state from the point of view of quantum systems and their quantum marginals ⁹, so we start by deﬁning them ﬁrst. But before we deﬁne quantum systems some brief notation must be established. LetTenpn0;n1;:::;ndqCn0bCn1b:::bCnddenote thespace of d1dimensional tensors offormatn0n1nd,andletXbeatensorin Tenpn0;n1;:::;ndq. Ifweregard Xasavector, withX:being it’s conjugate transpose, then XXX:is a Hermitian positive semideﬁnite (PSD) operatoron Tenpn0;n1;:::;ndq. Wewilldenoteby kXktrrXs1{2the`2normofX(whenviewed as a vector). With this notation in mind, we then deﬁne a quantum system withd1subsystems as a PSD operator on Tenpn0;n1;:::;ndqwith unit trace /one.superior⁰. Given a quantum system onTenpn0;n1;:::;ndqand a subset It0;1;:::;du, we deﬁne its (quantum) marginals orreduced density matrices bypIqtrIcrs, where trIcdenotes the partial traceovertensorfactors Ict0;:::;duzI. Inthesamewaythat describesthestateoftheentire quantumsystem, pIqcharacterizesthestateofthesubsystemslabeledby I(inananalogousway totheclassicalmarginal ofaprobabilitydistribution). For Itiu, wewritepiq; theseoperators ⁷There is a slight technicality here and the moment map is actually the absolute values squared of the entries of A. ⁸For some of the problems mentioned above, it is non-trivial to phrase them as moment polytopes. ⁹These generalize the classical notion of marginals of a probability distribution on several variables. /one.superior⁰A reader not familiar with the basics of quantum systems may want to skip a couple of paragraphs ahead. 3 areknownasthe one-bodymarginals orone-bodyreduceddensitymatrices of. Eachpiqisuniquely characterized by the property that trrpiqApiqstrrpIn0bIn1b:::bIni1bApiqbIni1b:::bIndqs (1) for all operators ApiqonCni. For a tensor XPTenpn0;n1;:::;ndqand a given subset I t0;1;:::;duof the subsystems, the marginals of Xwith respect to Ihave a particularly simple description: using the standard basis, identify Xwith a matrix MpIq XPLpCnI;CnIcq, where we denote nI:± iPIni. The matrix MpIq Xis known as a ﬂattening, unfolding, or matricization [ Lan12,Hac12] of the tensor X. Then, pIq XMpIq XpMpIq Xq:is its Gram matrix. Given a Hermitian operator onCn(i.e., annnHermitian matrix), we write specpq ps1;:::;snqfor the vector of eigenvalues of , ordered non-increasingly. If is PSD with unit trace then its eigenvalues form a probability distribution, so specpqis an element of Ppnq: tps1;:::;snq:s1¥ ¥sn¥0 :° jsj1u. We also abbreviate Ppn1;:::;ndq:Ppn1q Ppndq. Wewillbeparticularlyinterestedincharacterizingtheeigenvaluesoftheone-body marginals, motivated by the following fundamental problem in quantum mechanics [Kly06]: Problem1.1 (One-body quantum marginal problem) .GivenpPPpn1;:::;ndq, decide if there exists a tensorYPTenp1;n1;:::;ndqsuch that specppiq Yqppiqfor alli1;:::;d. Remark 1.2. Note that the above problem is equivalent to the following, given density matrices pPSD matriceswithunittrace qp1q;:::;pdq,determineifthereexistsatensor(purestate) YPTenp1;n1;:::;ndq suchthatpiq Yiforalli1;:::;d. Sinceaunitarychangeofbasiscomesforfreeoneachsubsystem,only the eigenvalues of p1q;:::;pdqare relevant. Theaboveproblemisextremelyfundamentalfromthepointofviewofquantummany-body physics. It is a special case of the more general quantum marginal problem, which puts constraints on the marginals of multiple systems and is known to be QMA-complete (for growing d) [Liu06]. We note that the normalization to trace one is natural; since trrpiq YskYk2for alli, we can simultaneously rescale all marginals simply by rescaling the tensor. NowwediscusshowProblem1.1canbephrasedasaquestionaboutmomentpolytopes[ NM84, Bri87,Kly06,Res10,VW17,Wal14]. LetGGLpn1q GLpndq, where GLpnqdenotes the group of invertible nn-matrices. Then Gacts onVTenpn0;n1;:::;ndqby pgp1q;:::;gpdqqX:pIn0bgp1qb:::bgpdqqX: As the group acts by rescaling slices of the tensor, we will call any YPGXatensor scaling ofX./one.superior/one.superior What is the moment map in this setting? It turns out that the moment captures exactly the notionofone-bodyquantummarginals. Itismoreconvenienttodeﬁnethemomentmaponthe projectivespace(namelyrestrictourselvestotensorsofunitnorm),sincewedon’tcareaboutthe scalarmultiples. Wewilldenotetheprojectivespacecorrespondingto VbyPpVqandidentifyit

Question 1278 multiple-choice

Quantum Fourier transforms (QFTs) are essential tools in quantum computing, especially for algorithms dealing with group structures. The Heisenberg group Hp, a non-abelian group of order p³, presents unique challenges and opportunities for efficient QFT circuit design. When performing Fourier sampling over the Heisenberg group Hp, why do abelian Fourier methods yield exponentially little information about hidden subgroup parameters such as i? 1) Because the Heisenberg group has no abelian normal subgroups. 2) Because all irreducible representations of Hp are one-dimensional. 3) Because the QFT over Hp cannot be implemented efficiently on a quantum computer. 4) Because the probability distributions after Fourier sampling are always deterministic for Hp. 5) Because, for most outcomes (c ≠ 0), the probabilities are uniform due to quadratic Weil sums, revealing almost no information about i. 6) Because the Heisenberg group is isomorphic to Zp × Zp × Zp. 7) Because 'twiddle factors' in the QFT circuit eliminate information about subgroup parameters.

✓ Correct Answer:

The correct answer is 5) Because, for most outcomes (c ≠ 0), the probabilities are uniform due to quadratic Weil sums, revealing almost no information about i..

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now weare doing ab elian Fourier sampling over anon-abelian group, wehavetoconsider theeffectofapplying Ftoacosetof tAi,j,wheret= (0,0,τ)andτ∈Fp. Note thattAi,j={(µ,µi,/parenleftbigµ 2/parenrightbig i+µj+τ) :µ∈Fp}. Weobtain F|tAi,j/an}bracketri}ht=1 p2/summationdisplay a,b,c∈Fpωcτ /summationdisplay µ∈Fpω(a+bi+cj−ci 2)µ+ciµ2 |a,b,c/an}bracketri}ht. Hence, the probability of observing a particular triple (a,b,c)isp−4|/summationtext µ∈Fpω(a+bi+cj−(ci)/2)µ+ciµ2|2. If c/ne}ationslash= 0, this is a quadratic Weil sum and we can use Fact 1 to conclude t hat the probability of observing (a,b,c)is given by p−3, independent of i,j. The probability of observing (a,b,c),c/ne}ationslash= 0is1−1 p. If c= 0, only terms of the form (−bi,b,0)show up. These terms do give information about i; however, the probability of observing such a term is1 p. Thus, the forgetful abelian method gives exponentially sm all information about i. 2.5 Efﬁcient quantum circuits forthe FourierTransform on Hp The fact that any QFTfor any ﬁnite group is a unitary matrix (when properly normal ized) makes this class of transformations an important source of transforma tions a quantum computer can carry out. The problem of ﬁnding efﬁcient implementations of QFTs interms of quantum circuits wasstudied previously, see [Høy97, Bea97, PRB99, HRTS03, MRR04]. From [MRR04, Theo rem 2] it follows that for any prime ptheQFTfor the Heisenberg group Hpcan by computed in polylog(p)operations. In the following we give an explicit description of an efﬁcient quantum circuit which computes QFTHp. First, note that we are interestedinarealizationonaquantumcomputer whichwork sonqubits. Thismeansthatwehavetoembed thestates and transformations intoaregister of size 2nfor somepositive integer n. Inthefollowing wewill assumethatnisthesmallestintegersuchthat p<2nandwewillidentifythegroupelements (x,y,z)∈ Hp withasubset ofthebinarystrings oflength 3n: ineachofthethreecomponents wechoosethebasisvectors |0/an}bracketri}ht,...,|p−1/an}bracketri}htto represent the respective component of the element (x,y,z). The following proposition shows that a QFTforHpcan be implemented efﬁciently in termsof elementary quantu m gates. Proposition 1 Letpbeprime,let HpbetheHeisenberggroupoforder p3andletIrr(Hp) ={χa,b: (a,b)∈ F2 p} ∪ {ρk:k= 1,...,p−1}denote the irreducible representations of Hp. Then the QFTforHpwith respect to Irr(Hp)can be computed using O(log3p)elementary quantum gates. Proof: First we consider the normal subgroup N∞⊳Hpand compute a Fourier transform for this abelian group. Thisgroupisisomorphictoadirectproduct oftwocyc licgroups, i.e., N∞∼=Zp×Zp. Theelements ofN∞are given by N∞={(0,y,z) :y,z∈Zp}, i.e., we can identify the elements of N∞with those binary strings of length 3nwhich have trivial support on the ﬁrst npositions. Note that the irreducible representations of N∞are given by ψa,bfora,b∈Zp, where ψa,b(0,y,z) := exp(2πi/p(ay+bz)) =ωay+bz p. SinceN∞isnormalthegroup Hpoperates ontheirreducible representations [CR62]. Weden otethisaction by “∗”, i.e., we have a map ∗:Hp×Irr(Hp)→Irr(Hp)which is explicitly given by (x,y,z)∗ψa,b= ψa,b−ax. Next, we choose as a transversal for N∞⊳Hpthe ordered list T= [(x,0,0) :x∈Zp]. We have to be able to efﬁciently implement the images of all irreduci ble representations of Hpevaluated at the 8 elements of T. This is required for the so-called ‘twiddle factors’ in the decomposition of QFTHpalong the subgroup tower {1}⊳N∞⊳Hp. Indeed, we construct a QFTadapted to this subgroup tower, see also [PRB99, MRR04]. We now use the following formula for imp lementing a QFTGwhich holds in the situation where wehave anabelian normal subgroup Nand an abelian factor group G/N: QFTHp=/parenleftbig 11|G/N|⊗QFTN/parenrightbig/parenleftBigg/circleplusdisplay t∈TΦ(t)/parenrightBigg/parenleftBig QFTG/N⊗11|N|/parenrightBig . HereΦdenotes an extension of the decomposition of the regular rep resentation of Ninto irreducibles. Denoting this direct sum by Λ, i.e.,Λ :=/circleplustext

Question 1279 multiple-choice

In combinatorics and coding theory, constructing subsets of vectors with equal sums is a classic problem connected to linear algebra and recursive algorithms. Advanced methods often utilize signed subset sums and structured linear relations among vectors. Which of the following statements most accurately describes the role of a signed subset sum in representing the zero vector within a set of input vectors? 1) It uses only nonnegative integer coefficients to combine vectors whose sum equals zero. 2) It requires that all vectors in the subset are identical to achieve a zero sum. 3) It allows the combination of vectors with arbitrary real coefficients to obtain the zero vector. 4) It represents the zero vector as a linear combination of the input vectors with coefficients restricted to ±1, corresponding to inclusion or subtraction in the signed subset. 5) It finds the shortest possible subset whose sum is zero, without regard to coefficients. 6) It constructs the zero vector using only even-numbered coefficients for each vector. 7) It guarantees that all subsets used are overlapping for maximal coverage.

✓ Correct Answer:

The correct answer is 4) It represents the zero vector as a linear combination of the input vectors with coefficients restricted to ±1, corresponding to inclusion or subtraction in the signed subset..

📚 Reference Text:

Page 15 of 23 14 represented as subsets of the index set, it will not be too misleading to use the term (sub)set for a (sub)sequence. Our strategy will be ﬁnding ppairwise disjoint subsets of input vectors having equal sums. We will achieve this goal by designing a method forﬁnding a non-trivial pair of subsets having equal sum and then, like in [ 21], applying the algorithm recursively to obtain 4, 8, 16, etc. disjoint subsets with equal sum. Note that a pair of disjoint subsets with equal sum can be interpreted as a repre- sentation of the zero vector by a linear combination of the input vectors with nonzero coefﬁcients 1 or −1 only. Based on this, it will be convenient to use the term signed subsets and signed subset sums. A signed subset of a set Sof vectors is formally a function from Sto the set {0,1,−1}. The support of such a signed subset is the set of elements on which the function takes nonzero values. With some sloppiness, we use the term signed subset sum to refer both to the signed subset and to the value of the signed sum. (Technically, a signed subset sum could be a data structure consisting of the description of the signed subset and the value.) We call two or more subset sums disjoint if their supports are pairwise disjoint. Based on the observation that a signedsubset sum of vectors that are results of pairwise disjoint subset sums is again a signed subset sum of the original vectors, one can build signed subset sums hierarchically from smaller disjoint signed subset sums. The trivial subset sum corresponds to the empty set with the zero vector as value. Alinear relation (or just relation for short) among a collection of vectors is for- mally an array of coefﬁcients such that the corresponding linear combination is the zero vector. However, we continue to denote relations by equalities between linear combinations with the zero vector. It is often useful to omit the vectors to whichcoefﬁcient zero is assigned. By taking the signed subsets of the vectors having the same or the opposite coefﬁcient in a linear relation, we obtain a linear relation among pairwise disjoint signed subset sums in which the coefﬁcients are form {1,..., p−1 2} and each coefﬁcient appears at most once. We call such a relation of signed subset sums standard . As an example, let {v1,v2,...,v 10}be a ﬁxed set of vectors satisfying the relation a1v1+a2v2+···+ a10v10=0 for some constants ai∈Zp. Moreover, suppose that the coefﬁcients satisfy the following a1=a2,a3=− a1,anda1≤p−1 2, a4=a6,a5=a7=− a4anda4≤p−1 2, a8=a9=a10anda8>p−1 2. Then we have the standard relation a1(v1+v2−v3)+a4(v4+v6−v5−v7)+(p−a8)(−v8−v9−v10)=0 among the signed subset sums v1+v2−v3,v4+v6−v5−v7and−v8−v9−v10.W e shall build standard linear relations among signed subset sums with smaller and smaller coefﬁcients (among increasingly larger subset sums). The key idea is constructing ﬁrst(p−1)2 4pairwise signed subset sums arranged in a square matrix having a relation 123 14 Page 16 of 23 M. Imran, G. Ivanyos in each row as well

Question 1280 multiple-choice

Localized orbitals are essential in quantum chemistry and materials science for interpreting chemical bonding and enabling scalable electronic structure calculations. Efficient algorithms for generating such orbitals have significant impact on computational cost and interpretability in large systems. Which approach provides a robust, parameter-free method for constructing localized, well-conditioned basis sets for Kohn-Sham orbitals, but is traditionally limited by the computational expense of column pivoted QR factorization? 1) Foster-Boys localization 2) Edmiston-Ruedenberg localization 3) Maximally localized Wannier functions 4) Selected Columns of the Density Matrix (SCDM) method 5) Löwdin orthogonalization 6) Hartree-Fock canonical orbitals 7) Projector Augmented-Wave (PAW) method

✓ Correct Answer:

The correct answer is 4) Selected Columns of the Density Matrix (SCDM) method.

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COMPUTING LOCALIZED REPRESENTATIONS OF THE KOHN-SHAM SUBSPACE VIA RANDOMIZATION AND REFINEMENT ANIL DAMLE, LIN LINy,AND LEXING YINGz Abstract. Localized representation of the Kohn-Sham subspace plays an important role in quantum chemistry and materials science. The recently developed selected columns of the density matrix (SCDM) method [J. Chem. Theory Comput. 11, 1463, 2015] is a simple and robust procedure for nding a localized representation of a set of Kohn-Sham orbitals from an insulating system. The SCDM method allows the direct construction of a well conditioned (or even orthonormal) and localized basis for the Kohn-Sham subspace. The SCDM algorithm avoids the use of an optimization procedure and does not depend on any adjustable parameters. The most computationally expensive step of the SCDM method is a column pivoted QR factorization that identi es the important columns for constructing the localized basis set. In this paper, we develop a two stage approximate column selection strategy to nd the important columns at much lower computational cost. We demonstrate the e ectiveness of this process using the dissociation process of a BH 3NH3molecule, an alkane chain and a supercell with 256 water molecules. Numerical results for the large collection of water molecules show that two stage localization procedure can be more than 30 times faster than the original SCDM algorithm and compares favorably with the popular Wannier90 package. Key words. Localization, Kohn-Sham density functional theory, selected columns of the density matrix, column pivoted QR factorization AMS subject classi cations. 65Z05, 65F30 1. Introduction. Kohn-Sham density functional theory (DFT) [21, 24] is the most widely used electronic structure theory for molecules and systems in condensed phase. The Kohn-Sham orbitals (a.k.a. Kohn-Sham wavefunctions) are eigenfunc- tions of the Kohn-Sham Hamiltonian and are generally delocalized, i.e.each orbital has signi cant magnitude across the entire computational domain. Consequently, the information about atomic structure and chemical bonding, which is often localized in real space, may be dicult to interpret directly from Kohn-Sham orbitals. The con- nection between the delocalized orbitals, and localized ones can be established through alocalization procedure, which has been realized by various numerical methods in the literature [15, 30, 29, 18, 10, 14, 34, 2]. The common goal of these methods is to nd a set of orbitals that are localized in real space and span the Kohn-Sham invariant subspace, de ned as the subspace as that spanned by the Kohn-Sham orbitals. For simplicity, we restrict our discussion to isolated systems and omit the discussion of Brillouin zone sampling in this manuscript. Mathematically, the problem of nding a localized representation of the Kohn- Sham invariant subspace can be formulated as follows. Assume the collection of Kohn-Sham orbitals are discretized in the real space representation as a tall and skinny matrix 2CNnewith orthonormal columns and where N ne. We seek to compute a unitary transformation Q2Cnenesuch that the columns of = Qare localized ,i.e.each column of becomes a sparse vector with spatially localized support Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720 Email: damle@berkeley.edu yDepartment

Question 1281 multiple-choice

Galois rings are algebraic structures that generalize finite fields and play a vital role in coding theory and quantum computing. Quantum algorithms often exploit the properties of these rings to achieve computational speed-ups. Which statement correctly describes a key advantage of implementing the quantum Fourier transform (QFT) over Galois rings in quantum algorithms? 1) It guarantees perfect error correction for all quantum codes. 2) It restricts quantum computation to only cyclic codes. 3) It allows classical algorithms to outperform quantum ones for period finding. 4) It eliminates the need for commutative ring structures. 5) It enables efficient quantum algorithms for uncovering hidden linear structures in finite commutative rings with identity. 6) It confines quantum computation exclusively to Galois fields. 7) It requires exponential-time resources for quantum implementation.

✓ Correct Answer:

The correct answer is 5) It enables efficient quantum algorithms for uncovering hidden linear structures in finite commutative rings with identity..

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Title: Quantum Fourier Transform Over Galois Rings Year: 2009 Paper ID: b36482672368bbe486139d84c039b9e2e3b92d36 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b36482672368bbe486139d84c039b9e2e3b92d36 Abstract: Galois rings are regarded as "building blocks" of a finite commutative ring with identity. There have been many papers on classical error correction codes over Galois rings published. As an important warm-up before exploring quantum algorithms and quantum error correction codes over Galois rings, we study the quantum Fourier transform (QFT) over Galois rings and prove it can be efficiently preformed on a quantum computer. The properties of the QFT over Galois rings lead to the quantum algorithm for hidden linear structures over Galois rings.

Question 1282 multiple-choice

Quantum computation has enabled new approaches to approximating knot invariants, which are mathematical tools for distinguishing knots and links. These developments have revealed deep connections between knot theory, computational complexity, and quantum algorithms. Which computational problem related to knot invariants has been shown to be QCMA-complete, meaning it is as hard as the most difficult problems in the quantum class where proofs are classical but verification is quantum? 1) Exact Jones Polynomial Evaluation 2) Classical Plat Closure Determination 3) HOMFLYPT Polynomial Approximation 4) Quantum Braiding Simulation 5) Increase Jones Plat 6) Tutte Polynomial Calculation 7) #P-hard Bit Extraction of Jones Polynomial

✓ Correct Answer:

The correct answer is 5) Increase Jones Plat.

📚 Reference Text:

ﬁeld theory, that it is possible to achieve universa l quantum computation. In [10], it was shown that topological quantum computers cou ld be eﬃciently simulated by computers based on the standard quantum circuit model, im plying the existence of an eﬃcient algorithm for approximating the Jones polynomial o f certain links obtained from braids. More recently, Aharonov, Jones and Landau [2] gave e xplicit quantum algorithms for approximating the Jones polynomial of either the trace or pl at closure of a braid on a quantum computer. Our ﬁrst contribution in this paper is a family of q uantum circuits implementing the so-called Jones-Wenzl unitary representations of the b raid group, generalizing the circuits for the Jones representation from [2]. We then show how to use our circuits to approximate a family of evaluations of the HOMFLYPT two-variable polyno mial. We also sketch two diﬀerent quantum algorithms for approximating the Jones po lynomial of links obtained via a class of closures generalizing the trace and plat closures . In the sense of [7], our quantum algorithms obtain additive approximations of Jones and HOM FLYPT polynomial evaluations in polynomial-time. In [12, 13], a converse to the above simulation results was pr oved. More speciﬁcally, it was shown there that quantum circuits can be simulated by braids , in such a way that the output probabilities of the quantum circuit are functions of the Jo nes polynomial of the plat closure of some braid of comparable length to that of the original cir cuit. Our third contribution is a simpler proof of their main result. In fact, this work grew fr om our attempts to understand the connections between these works and [2]. Finally, we give applications of the Jones polynomial to qua ntum complexity theory. We begin by restating a result that was proved in [7], showing th at a machine which obtains an additive approximation of the Jones polynomial of the plat c losure of a braid is equally as powerful as a quantum computer. We then introduce two new pro blems, Increase Jones Plat andApproximate Concatenated Jones Plat . The ﬁrst asks if a given braid can be conjugated by another braid from a given class such that th e Jones polynomial of its plat closure is nearly maximal. We prove that this problem is comp lete for the complexity class QCMA, a certain quantum analog of NP. The latter problem asks if a given braid, after being concatenated with itself exponentially many times, has a la rge plat closure. We show that this problem is PSPACE-complete. Finally, we give self-con tained proofs that learning nof the most signiﬁcant bits of the Jones polynomial of the plat c losure of an n-strand braid is a #P-hard problem, while learning its most signiﬁcant bit is PP-hard. This constitutes a simpler, quantum-based proof of the known result that compu ting the Jones polynomial is #P-hard. The original proof of this (see e.g. [39]) relates t he Jones polynomial to the Tutte

Question 1283 multiple-choice

Quantum state discrimination is crucial in quantum information theory, especially when distinguishing between states associated with different symmetries or functions. The query complexity and sample requirements often depend on measurement strategies and the use of entanglement. In the quantum collision problem, what is the minimum number of copies (k) required to reliably distinguish between the Schur(k,d) and Planch(k) quantum states using only individual (local) measurements, and what key resource enables optimal discrimination with fewer copies? 1) k = Ω(log d); entanglement is not necessary 2) k = Ω(d); collective measurements offer no advantage 3) k = Ω(√d); adaptive classical algorithms suffice 4) k = Ω(d²); entangled measurements across many copies are required 5) k = Ω(d³); swap tests outperform entangled strategies 6) k = Ω(1); isometry without extension to unitary is optimal 7) k = Ω(d!); Young diagram combinatorics are irrelevant

✓ Correct Answer:

The correct answer is 4) k = Ω(d²); entangled measurements across many copies are required.

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guishability. Indeed, k= Ω(d2) copies are needed for Ω(1) distinguishability even if we perform the optimal individual measurement on O(1) copies at a time. This follows from the following variant of Lemma 6: 13 Lemma 9. F(Schur(k,d),Planch(k))≥1−k3 12d2(68) F(Schur(k,d),Planch(k))≥1−C−√ k−9 4k2 d2, (69) whereC >1is a universal constant. These bounds are not quite strong enough to match the 1 −O(k2/d2) upper bound on the ﬁdelity from Lemma 6 (which we conjecture is tight), but they still pr ove that achieving the performance of the optimal measurement requires operations th at are entangled across many copies. Proof.We begin with a direct calculation of fk,d:=F(Schur(k,d),Planch(k)) =E λ⊢k/radicaltp/radicalvertex/radicalvertex/radicalbt/productdisplay (i,j)∈λ/parenleftbigg 1+j−i d/parenrightbigg . (70) Consider the contribution from a particular λ, and suppose for now that λ=λT. Then (i,j)∈λ if and only if ( j,i)∈λ, so each factor of 1+( j−i)/dis accompanied by a corresponding factor of 1−(j−i)/d(unlessi=j, in which case this term is 1 and can be ignored). Combining these terms, the entire product can be written as /radicaltp/radicalvertex/radicalvertex/radicalbt/productdisplay (i,j)∈λ/parenleftbigg 1+j−i d/parenrightbigg = /productdisplay (i,j)∈λ/parenleftbigg 1−(j−i)2 d2/parenrightbigg 1 4 . (71) Otherwise, suppose that λ/ne}ati≀nslash=λT. Recall that λandλThave the same probability under the Plancherel distribution. Thus the terms corresponding to λandλTappear in (70) with the same coeﬃcient. Next we use the identity x+y≥2√xy(forx,y≥0) to show /radicaltp/radicalvertex/radicalvertex/radicalbt/productdisplay (i,j)∈λ/parenleftbigg 1+j−i d/parenrightbigg +/radicaltp/radicalvertex/radicalvertex/radicalbt/productdisplay (i,j)∈λT/parenleftbigg 1+j−i d/parenrightbigg ≥2 /productdisplay (i,j)∈λ/parenleftbigg 1−(j−i)2 d2/parenrightbigg 1 4 .(72) Combining (71) and (72), we can lower bound (70) as fk,d≥E λ⊢k /productdisplay (i,j)∈λ/parenleftbigg 1−(j−i)2 d2/parenrightbigg 1 4 ≥1−k3 12d2, (73) where the last inequality uses the fact that the product is minimized f or the diagram λ= (k). This proves (68). The proof of (69) is mostly the same, but also uses bounds on large d eviations of λ1for Plancherel-distributed λ. In particular, equation (1.20) of [7] implies that under Planch( k), Pr(λ1≥3√ k)≤C−√ k(74) for a universal constant C >1 (in fact,C= exp(6cosh−1(3/2)+2√ 5)≈28189). Since ( i,j)∈λ implies that j−i≤λ1−1, we can lower bound the middle expression in (73) to give fk,d≥Pr(λ1≤3√ k)/parenleftbigg 1−9k d2/parenrightbiggk 4 ≥1−C−√ k−9 4k2 d2, (75) which completes the proof. It is interesting to consider the classical analog of applying the swap test to the collision problem. If we compare k/2 sets of 2 objects each, then estimating the number of collisions up to a constant multiple requires k= Ω(d). This is also quadratically worse than the optimal collective estimation procedure on kobjects, but for apparently diﬀerent reasons. 14 4.2 A black box for the quantum collision problem We now deﬁne a version of this problem that allows for adaptive algorit hms by introducing a quantum oracle. The oracle implements the isometry |i/an}bracketri}ht /ma√st≀→ |i/an}bracketri}ht|ψf(i)/an}bracketri}ht, (76) whereB:={|ψ1/an}bracketri}ht,...,|ψd/an}bracketri}ht}is an arbitrary (unknown) orthonormal basis of Cdandfis either a one-to-one function or an r-to-one function. The goal is to determine which is the case using as few queries as possible. In fact, we assume that the isometry is ext ended to a unitary operator Racting on Cd⊗Cdby |i/an}bracketri}ht|y/an}bracketri}ht /ma√st≀→ |i/an}bracketri}htU|y⊕f(i)/an}bracketri}ht, (77) whereU:=/summationtext i|ψi/an}bracketri}ht/an}bracketle{ti|is the unitary matrix eﬀecting a

Question 1284 multiple-choice

Quantum computing leverages unique transformations and representations to process visual information such as images and videos efficiently. Quantum Fourier Transform (QFT) circuits and their multidimensional extensions play a pivotal role in advancing quantum vision systems. Which of the following approaches ensures both theoretical reliability and practical feasibility for quantum image and video processing within a quantum computing framework? 1) Designing QFT circuits without formal correctness proofs and omitting multidimensional extensions 2) Developing complete and formally proven QFT and inverse QFT circuits, extending them to 2D and 3D versions, and demonstrating effectiveness through simulations 3) Utilizing classical Fourier transforms for quantum image analysis without quantum-specific circuit designs 4) Implementing only the inverse QFT for video processing, while neglecting the direct QFT 5) Encoding images in quantum states but restricting processing to one-dimensional algorithms 6) Simulating image processing on classical computers with no quantum circuit involvement 7) Applying quantum vision representation without providing a comprehensive model or framework

✓ Correct Answer:

The correct answer is 2) Developing complete and formally proven QFT and inverse QFT circuits, extending them to 2D and 3D versions, and demonstrating effectiveness through simulations.

📚 Reference Text:

Title: Quantum Fourier Transforms Year: 2021 Paper ID: 11e1a6cc6474a149bcd25089e1e7d28d828b29ac Source: semantic-scholar URL: https://www.semanticscholar.org/paper/11e1a6cc6474a149bcd25089e1e7d28d828b29ac Abstract: Quantum Fourier transform (QFT) plays a key role in many quantum algorithms, but the existing circuits of QFT are incomplete and lacking the proof of correctness. Furthermore, it is difficult to apply QFT to the concrete field of information processing. Thus, this chapter firstly investigates quantum vision representation (QVR) and develops a model of QVR (MQVR). Then, four complete circuits of QFT and inverse QFT (IQFT) are designed. Meanwhile, this chapter proves the correctness of the four complete circuits using formula derivation. Next, 2D QFT and 3D QFT based on QVR are proposed. Experimental results with simulation show the proposed QFTs are valid and useful in processing quantum images and videos. In conclusion, this chapter develops a complete framework of QFT based on QVR and provides a feasible scheme for QFT to be applied in quantum vision information processing.

Question 1285 multiple-choice

In computational quantum algebra, noncommutative Gröbner bases and semidefinite programming (SDP) techniques are utilized to solve optimization problems involving Pauli operators. The structure and enumeration of basis elements play a key role in the efficiency of these algorithms. Which of the following statements accurately describes the enumeration of polynomials in a Gröbner basis for the ideal generated by Pauli operators acting on n qubits? 1) There are exactly 2n polynomials arising from the Pauli commutators. 2) The basis consists of 9n elements, independent of commutators or variable relations. 3) Only the polynomials of the form wj² = 1 contribute to the total count, yielding 3n elements. 4) The Gröbner basis has 3n polynomials of the form wj² = 1, 6n polynomials of the form wjuj ± ivj, and 32·(n choose 2) commutators. 5) Monomial order selection eliminates the need for commutator polynomials in the basis. 6) The number of basis elements is always linear in n, never quadratic. 7) S-polynomials involving more than one index do not reduce to zero and must be excluded from the basis.

✓ Correct Answer:

The correct answer is 4) The Gröbner basis has 3n polynomials of the form wj² = 1, 6n polynomials of the form wjuj ± ivj, and 32·(n choose 2) commutators..

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obner basis for IPauli. It has9 2n(n+ 1) elements. Proof. Firstly, note that the polynomials in Eq. (5.4) belong to IPauli. For instance, xjzj+iyj=−iyj(z2 j−1)−yj(xjyj−izj)zj+ (xjyj−izj)yjzj + (yjxj+izj)yjyj−xj(y2 j−1)zj. We now apply the noncommutative Buchberger algorithm (see, e.g., [ Mor86 ]), to check that the constructed set Bof polynomials is a GB. Firstly, self-obstructions. Given one of the polynomials p∈Bit involves at most two indices i,j. Hence all its nontrivial self-obstructions can also involve only these two indices. But modulo the GB these all reduce to 0 (by Example 5.3). Likewise, given polynomials p1,p2∈B, they involve at most four distinct indices. These same indices are the only ones that can appear in a nontrivial S-polynomial constructed from p1,p2. But, again by Example 5.3, these must reduce to 0 sincep1,p2appear in the GB associated to the chosen four indices. To count the number of elements in this GB, note there are 3 npolynomials of the form w2 j= 1 forw∈{x,y,z}and 1≤j≤n. There are 6 npolynomials of the form wjuj±ivj with{w,u,v}={x,y,z}. Finally, there are 32·/parenleftbign 2/parenrightbigcommutators in B. Adding these numbers we get the desired count. Example 5.5. Changing the monomial order to grlex under which x1

Question 1286 multiple-choice

Quantum Branching Programs (QBPs) extend classical branching program models into the quantum domain and play an important role in understanding the resource requirements and limitations of quantum computation. One key area of research is how the width and qubit requirements for QBPs compare with those of their classical counterparts, especially for functions with high deterministic complexity. For a Boolean function whose minimal deterministic OBDD requires exponential width, what does the established lower bound theorem imply about the minimum width required for a quantum OBDD (QOBDD) computing this function? 1) It allows QOBDDs to achieve constant width regardless of deterministic width. 2) It guarantees that QOBDDs require exponential width, matching the deterministic case. 3) It shows that QOBDD width is always quadratic in the number of variables. 4) It implies QOBDD width is independent of deterministic OBDD width. 5) It establishes that QOBDD width is at least linear in the number of variables. 6) It requires QOBDDs to use more qubits than deterministic OBDDs use bits. 7) It allows QOBDDs with sublinear width for any Boolean function.

✓ Correct Answer:

The correct answer is 5) It establishes that QOBDD width is at least linear in the number of variables..

📚 Reference Text:

the ﬁnal projective measurement). Wedeﬁne acomputation of Qon aninput σ=(σ1,...,σn)∈{0,1}nasfollows: 1. Acomputation of Qstarts from the initial state |ψ0/an}⌊ra⌋k⌉tri}ht; 2. The j-th instruction of Q reads the input symbol σij(the value of x ij) and applies the transition matrixU j=Uj(σij)to thecurrent state |ψ/an}⌊ra⌋k⌉tri}httoobtain thestate |ψ′/an}⌊ra⌋k⌉tri}ht=Uj(σij)|ψ/an}⌊ra⌋k⌉tri}ht; 3. Theﬁnal state is |ψσ/an}⌊ra⌋k⌉tri}ht=/parenleftBigg 1 ∏ j=lUj(σij)/parenrightBigg |ψ0/an}⌊ra⌋k⌉tri}ht. 4. After the l-th (last) step of quantum transformation Q mea sures its conﬁguration |ψσ/an}⌊ra⌋k⌉tri}ht, and the inputσisaccepted withprobability Praccept(σ)=/an}⌊ra⌋k⌉tl⌉{tψσM† accept|Macceptψσ/an}⌊ra⌋k⌉tri}ht=||Maccept|ψσ/an}⌊ra⌋k⌉tri}ht||2 2. Circuit representation. A QBP can be viewed as a quantum circuit aided with an ability t o read classical bits as control variables for unitary operations . That is any quantum circuit is a QBP which does not depend essentially on its classical inputs. xi1• /d23/d22/d21/d20 /d16/d17/d18/d19 ··· xi2 • /d23/d22/d21/d20 /d16/d17/d18/d19··· ...xil ··· • /d23/d22/d21/d20 /d16/d17/d18/d19 |φ1/an}⌊ra⌋k⌉tri}ht U1(1)U1(0)U2(1)U2(0)··· Ul(1)Ul(0)/d78/d77/d13/d13/d13 |φ2/an}⌊ra⌋k⌉tri}ht ··· /d78/d77/d13/d13/d13|ψ0/an}⌊ra⌋k⌉tri}ht .../vextendsingle/vextendsingleφq/angbracketrightbig··· /d78/d77/d13/d13/d13 Herexi1,...,xilis the sequence of (not necessarily distinct) variables den oting classical control bits. Using the common notation single wires carry quantum inform ation and double wires denote classical information and control. F.Ablayev & A.Vasiliev 3 Complexity measures. ThewidthofQisthedimension dof thestatespace Hd,thelengthofQisthe numberlof instructions inthe sequence T. Notethat foraQBPinthecircuit setting another important c omplexity measureexplicitly comesout – a number qof qubits physically needed to implement a corresponding qu antum system with classical control. Fromdeﬁnition it follows that log d≤q. Deﬁnition 2. We call a quantum branching program a q-qubit QBP, if it can be implemented as a classically-controlled quantum system based on qqubits. Acceptance criteria. A QBPQ computes the Boolean function f with one-sided error if there exists anε∈(0,1)(called an error) such that for all σ∈f−1(1)the probability of Qaccepting σis 1 and for allσ∈f−1(0)the probability of Qerroneously accepting σisless than ε. Read-once branchingprograms. Deﬁnition 3. We call a QBPQ a quantum OBDD(QOBDD) or read-once QBPif each variable x ∈ {x1,...,xn}occurs inthe sequence T of transformations of Qat most once. For the rest of the paper we’re only interested in QOBDDs, i.e . the length of all programs would be n(the number of input variables). GeneralizedLowerBound. ThefollowinggenerallowerboundonthewidthofQOBDDswasp roven in [3]. Theorem 1. Let f(x1,...,xn)be a Boolean function computed by a quantum read-once branch ing pro- gram Q. Then width(Q)=Ω(logwidth(P)) where Pisa deterministic OBDDof minimal width computing f (x1,...,xn). That is, the width of a quantum OBDD cannot be asymptotically less than logarithm of the width of the minimal deterministic OBDDcomputing the same functi on. And since the deterministic width of many “natural” functions isexponential [13],weobtain the linear lower bound for these functions. 3 Algorithmsfor QBPsBasedon Fingerprinting Generally [10], ﬁngerprinting – is a technique that allows to present objects (words over so me ﬁnite alphabet) bytheir ﬁngerprints , whicharesigniﬁcantlysmallerthantheoriginals. Itisus edinrandomized and quantum algorithms to test equalityof some objects (binary strings) with one-sided error by sim ply comparing their ﬁngerprints. In this paper we develop a variant of the ﬁngerprinting techn ique adapted for quantum branching programs. At the heart of the method is the representation of Boolean functions by polynomials of special type, which wecall characteristic . 3.1 Characteristic PolynomialsforQuantum Fingerprintin

Question 1287 multiple-choice

In quantum computing, distinguishing between different quantum states often relies on the probabilistic outcomes of measurements in randomly chosen bases, with group representation theory providing the mathematical framework for analyzing these situations. The statistical distance between probability distributions resulting from such measurements is crucial for the effectiveness of quantum algorithms in group-theoretic problems. Which of the following best describes the scaling behavior of the lower bound on the l1 (total variation) distance between probability distributions resulting from quantum measurements of totally mixed states associated with different group-theoretic structures, when measured in random orthonormal bases? 1) The lower bound decreases exponentially with the size of the group 2) The lower bound scales inversely with the square of the dimension of the representation 3) The lower bound is proportional to the inverse square root of a relevant parameter, such as the dimension (e.g., Ω(1/√r₁)) 4) The lower bound increases linearly with the number of "good" coordinates 5) The lower bound remains constant regardless of the group or representation size 6) The lower bound scales logarithmically with the order of the group 7) The lower bound is proportional to the sum of the norms of the random vectors

✓ Correct Answer:

The correct answer is 3) The lower bound is proportional to the inverse square root of a relevant parameter, such as the dimension (e.g., Ω(1/√r₁)).

📚 Reference Text:

ors of all vectors inW′are/radicalbig (1±ǫ)d, whereǫ=O(δ).E0occurs with probability at least 1−2exp(−α))over the 17 choice of the Gaussians. Let E1denote the event that |/an}bracketle{tw′i|w′j/an}bracketri}ht|<2δfor allw′i,w′j∈W′,i/ne}ationslash=j.E1 occurs withprobability atleast 1−2exp(−α)over thechoice oftheGaussians. Callacoordinate jgoodif/summationtextp i=1|α′i j|2>p+√p. LetE2denote the event that more thandθ 2coordinatesjare good.E2occurs with probability at least 1−exp(dθ/8)over the choice of theGaussians. Now suppose that all three events E0,E1,E2occur. LetS′be the analogous probability vector deﬁned withrespect to W′instead ofW. Then,/bardblS−S′/bardbl1=O(p−1/2). For agoodcoordinatej /vextendsingle/vextendsingle/vextendsingle/vextendsingleS′ j−1 d/vextendsingle/vextendsingle/vextendsingle/vextendsingle=1 (1+ǫ)d−1 d+1√p(1+ǫ)d =−ǫ (1+ǫ)d+1√p(1+ǫ)d =−O/parenleftbigg−1 dp/parenrightbigg +Ω/parenleftbigg1 d√p/parenrightbigg = Ω/parenleftbigg1 d√p/parenrightbigg . Thethird step above follows from the fact that ǫ=O(δ) =O(p−1). Hence, /bardblS′−U/bardbl1≥/summationdisplay j:jgood|S′ j−T′ j|>dθ 2·Ω/parenleftbigg1 d√p/parenrightbigg = Ω/parenleftbigg1√p/parenrightbigg . Finally, /bardblS−U/bardbl1≥ /bardblS′−U/bardbl1−/bardblS−S′/bardbl1= Ω(p−1/2)−O(p−1) = Ω(p−1/2). Theconﬁdence bound is Pr[E0∧E1∧E2]>1−4exp(−α)−exp(dθ/8) = 1−exp(−Ω(α)), sinceδ=O(1). Thiscompletes the proof of the lemma. Wearenow in aposition toﬁnally prove Theorem 1. Proof:(of Theorem 1) Letρbe an irreducible dρ-dimensional unitary representation of G. We follow the notation of Deﬁnition 1 for ρ. LetV′′ 2denote the orthogonal complement of V′ 1inV′ 1+V′ 2. Letσi denote the totally mixed state in Viandσ′′ 2denote the totally mixed state in V′′ 2+ (V1∩V2). By Proposi- tion 2,/bardblσ2−σ′′ 2/bardbltr<2δ1/2(1−δ2)−1/4. LetBρbe a random orthonormal basis for ρ. LetPi:=PG,Bρ Hi denote the probability distributions on the vectors of Bρgot by Fourier sampling the states |Hi/an}bracketri}htrespec- tively, conditioned on ρbeing observed. Then Piis the probability distribution got by measuring σiin the basis Bρ. LetP′′ 2denote the probability distribution got by measuring σ′′ 2in the basis Bρ. Then, /bardblP2−P′′ 2/bardbl1<2δ1/2(1−δ2)−1/4. Deﬁner1,2= rank(Π 1,2). Notethatri=r′ i+r1,2. Suppose case 1 of Deﬁnition 1 applies. Let Wbe a random orthonormal set of r′ 1+r′ 2+r1,2vectors inCdρ. Deﬁne probability distributions S,Twith respect to Was in Lemma2. Bysymmetry, P1=Sand P′′ 2=T. Note thatr′ 1+r′ 2+r1,2≤r1+r2andr′ 1+r′ 2+r1,2≤dρ

Question 1288 multiple-choice

Quantum mechanics on group manifolds, such as U, provides powerful methods for studying representation theory and its applications in gauge and string theories. Techniques like bosonization and large N expansions are central to solving problems in collective field theory and partition functions of quantum field theories. Which concept specifically facilitates the calculation of tensor product multiplicities, such as Littlewood-Richardson coefficients, within the framework of group manifold quantum mechanics? 1) Eguchi-Kawai reduction 2) Partition function computation 3) Large N limit of free fermions 4) Representation theory on U 5) Bosonization into collective fields 6) Leading order O(N^0) expansions 7) Zero magnetic field limit analysis

✓ Correct Answer:

The correct answer is 4) Representation theory on U.

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Title: Conformal Field Theory Techniques for Large N Group Theory Year: 1993 Paper ID: 12fe61f148bc3bd10d8f569dfdafc73d5da366b4 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/12fe61f148bc3bd10d8f569dfdafc73d5da366b4 Abstract: We show how to use quantum mechanics on the group manifold U(N) as a tool for problems in U(N) representation theory. The quantum mechanics reduces to free fermions on the circle, which in the large N limit become relativistic. The theory can be bosonized giving the Das-Jevicki-Sakita collective field theory. The formalism is particularly suited to problems involving tensor product multiplicity (Littlewood-Richardson) coefficients. As examples, we discuss the partition function of two-dimensional Yang-Mills theory on the sphere, and the zero magnetic field limit of D-dimensional Eguchi-Kawai Yang-Mills theory. We give the leading O(N^0) solution of the latter theory, using a method which allows computing corrections. Largely (but not completely) superseded by hep-th/9311130.

Question 1289 multiple-choice

In topological quantum computation, invariants of knots and links can be efficiently approximated using quantum algorithms that exploit the structure of group representations and entanglement. The plat closure process is especially relevant for modeling the braiding and fusion of particles in these systems. Which property of a particle is required for the quantum algorithm that approximates plat closures to yield a link invariant via the amplitude ⟨α|τΛ(θ)|α⟩? 1) The particle must be its own antiparticle, corresponding to a self-conjugate representation (Λ̄ = Λ). 2) The particle must transform under a non-abelian representation of the symmetry group. 3) The particle must be associated with a reducible representation. 4) The particle must have a fixed electric charge. 5) The particle must belong to a representation with dimension greater than one. 6) The particle must have an associated braiding operator that is diagonal in the computational basis. 7) The particle must be described by a representation that is not induced from a subgroup.

✓ Correct Answer:

The correct answer is 1) The particle must be its own antiparticle, corresponding to a self-conjugate representation (Λ̄ = Λ)..

📚 Reference Text:

which ¯Λ = Λ, i. e., Λ is its own conjugate. For adjacent pairs Λ⊗2, deﬁne the state |Φ/a\}bracketri}ht=1√dΛ/summationdisplay v|v,v/a\}bracketri}ht, (4.3) 13 where the sum runs over all the elements of an orthonormal bas is ofVΛanddΛis its dimension. This maximally entangled state is the copy of the trivial represe ntation in the tensor product. Now, deﬁne the state |α/a\}bracketri}ht=|Φ/a\}bracketri}ht⊗n(4.4) and, given a braid θ∈B2nthat realizes the desired link under the Plat closure, let th e representation of θ in the above tensor power be τΛ(θ). The plat closure yields a link invariant: PlΛ(θ) =/a\}bracketle{tα|τΛ(θ)|α/a\}bracketri}ht. (4.5) Physically, one can view the plat closure as follows: partic le-antiparticle pairs are created, their world lines are braided and then they are annihilated. The plat closure i s the amplitude of obtaining the vacuum at the end of this process. One also requires that the particl e be its own antiparticle since we need the representation to be its own conjugate. 4.2 Quantum algorithms for link invariants In this section, we present quantum algorithms to additivel y approximate the trace and plat closures of braids in some irrep Λ of D(G). We use the QFT over D(G) to accomplish this: critically, the Rmatrices can be implemented eﬃciently in the regular, combinatorial representation so we may use the QFT over D(G) to implement it inside the irreps. Then we take the trace ove r the irrep of choice. We describe this in more detail below. First, we need the deﬁnition of additive a pproximation (see Bordewich et al. [ 9]). Deﬁnition 1. Given any function f:D→Cand a normalization u:Z+→R+, an additive approxi- mation for the pair (f,u)is a probabilistic algorithm which given any x∈Dandǫ>0produces an output g(x), such that Pr[|f(x)−g(x)|>ǫu(|x|)]<1/4, in time polynomial in |x|andǫ−1. In order to approximate the trace closure of some braid θwritten as a word in the generators σi, we ﬁrst construct the representation of θinD(G)⊗nby implementing the Rmatrices in D(G). In the case of ﬁnite group doubles, the Rmatrix is a unitary operator and, in fact, a permutation oper ator in the standard basis. Recall that it is deﬁned as R=/summationdisplay gg⊗g∗; thus its action on a basis element |g1h∗ 1/a\}bracketri}ht⊗|g2h∗ 2/a\}bracketri}htis given by R|g1h∗ 1,g2h∗ 2/a\}bracketri}ht=/summationdisplay g|gg1h∗ 1,g2(gg2h2)∗/a\}bracketri}ht=|hg−1 2 2g1h∗ 1,g2h∗ 2/a\}bracketri}ht. Therefore, the action of σ1=TRis σ1|g1h∗ 1,g2h∗ 2/a\}bracketri}ht=|g2h∗ 2,hg−1 2 2g1h∗ 1/a\}bracketri}ht. This operator is a left multiplication by a group element fol lowed by a swap which can be eﬃciently imple- mented. In order to take the trace, we must be able to produce random ba sis vectors in the irrep Λ embedded inVD(G). The basis vectors inside the irrep ( h,ρ) can be constructed using the fact that it is an induced representation. If {w1,...,w d}are a basis for Vρ, then{t1⊗w1,...t k⊗wd}is a basis for Vρ,h, where ti∈Thas before. Assuming that we can embed the vectors |wi/a\}bracketri}htin the vector space VZ(h), then the vectors |h∗,tj,wi/a\}bracketri}htare basis vectors of the irrep ( h,ρ) embedded in VD(G). Now, using the Hadamard test, we can ﬁnd the inner

Question 1290 multiple-choice

Quantum groups are algebraic structures that generalize symmetries in mathematics and physics, with representation theory playing a crucial role in understanding their properties. "Easy" quantum groups have a special feature where combinatorial methods, particularly those involving partitions, can be used to analyze their representations. Which aspect of "easy" quantum groups enables their fusion rules in representation theory to be directly described using combinatorics of set partitions? 1) The use of noncommutative differential operators 2) The characterization of intertwiner spaces via partition combinatorics 3) The existence of a universal enveloping algebra 4) The classification through Lie algebra root systems 5) The presence of a Cartan subalgebra 6) The implementation of quantum cohomology techniques 7) The reliance on categorical duality principles

✓ Correct Answer:

The correct answer is 2) The characterization of intertwiner spaces via partition combinatorics.

📚 Reference Text:

Title: On the representation theory of partition (easy) quantum groups Year: 2013 Paper ID: 65658c902fad95ddcba28d01d3ab25e9aa9a9a16 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/65658c902fad95ddcba28d01d3ab25e9aa9a9a16 Abstract: Compact matrix quantum groups are strongly determined by their intertwiner spaces, due to a result by S.L. Woronowicz. In the case of easy quantum groups, the intertwiner spaces are given by the combinatorics of partitions, see the inital work of T. Banica and R. Speicher. The philosophy is that all quantum algebraic properties of these objects should be visible in their combinatorial data. We show that this is the case for their fusion rules (i.e. for their representation theory). As a byproduct, we obtain a unified approach to the fusion rules of the quantum permutation group $S_N^+$, the free orthogonal quantum group $O_N^+$ as well as the hyperoctahedral quantum group $H_N^+$.

Question 1291 multiple-choice

In finite group theory, p-groups are fundamental objects whose structure is analyzed using concepts such as commutator subgroups, Frattini subgroups, and maximal subgroups. The classification of non-abelian p-groups often involves determining properties of their subgroups and how these relate to group actions and generators. Which property is characteristic of a finite non-abelian p-group that is an A₁-group with rank 2 and commutator subgroup of order p? 1) All maximal subgroups are non-abelian 2) The group's center is trivial 3) The Frattini subgroup is equal to the center 4) The group has rank 3 5) Its commutator subgroup is abelian but of order greater than p 6) The group has no abelian maximal subgroups 7) The quotient by the Frattini subgroup is simple

✓ Correct Answer:

The correct answer is 3) The Frattini subgroup is equal to the center.

📚 Reference Text:

We need the following lemmas. Lemma 2.1. ([9, Lemma 2.2]) Suppose that Gis a ﬁnite non-abelian p-group. Then the following conditions are equivalent: (1)Gis anA1-group; (2)d(G) = 2and|G′|=p; (3)d(G) = 2andΦ(G) =Z(G). Lemma 2.2. ([1, Theorem 2.7]) Suppose that Ghas at least two A1-subgroups of index p. The the following conclusions hold: (1)Ifd(G) = 3, thenΦ(G)≤Z(G); (2)Ifd(G) = 2, thenc(G)≤3,Φ(G′)G3≤C2 pandG/Φ(G′)G3∈ A1. Lemma 2.3. ([1, Corollary 2.4]) (1) LetMbe anAt-group, and Abe an abelian group of order pk. ThenG=M×Ais anAt+k-group. (2)LetMbe anAt-group with |M′|=p,G=M∗A, whereAis abelian with order pk+1andM∩A=M′. ThenGis anAt+k-group. Lemma 2.4. ([5, Lemma 2.1]) Suppose that pis odd,{1,η}is a transversal for (F∗ p)2 inF∗ p. Then the following matrices form a transversal for the congr uence classes of invertible matrices of order 2overFp: 2 (1)/parenleftbigg0 1 −1 0/parenrightbigg ,(2)/parenleftbiggν1 −1 0/parenrightbigg ,(3)/parenleftbigg1 0 0ν/parenrightbigg ,(4)/parenleftbigg1 1 −1r/parenrightbigg , whereν= 1orη,r= 1,2,...,p−2. Lemma 2.5. ([1, Lemma 4.3]) The following matrices form a transversal for the con- gruence classes of invertible matrices of order 2overF2. (1)/parenleftbigg1 0 0 1/parenrightbigg ,(2)/parenleftbigg0 1 1 0/parenrightbigg ,(3)/parenleftbigg1 0 1 1/parenrightbigg . Lemma 2.6. ([5, Lemma 2.3]) Suppose that pis a prime (p= 2is possible ). For odd p, {1,η}is a transversal for (F∗ p)2inF∗ p. Then the following matrices form a transversal for the congruence classes of non-invertible matrices of or der2overFp: (1)/parenleftbigg0 1 0 0/parenrightbigg ,(2)/parenleftbigg0 0 0ν/parenrightbigg ,(3)/parenleftbigg0 0 0 0/parenrightbigg , whereν= 1orη. 3 The case G3≤Φ(G′)∼=Cp Suppose that Gis a ﬁnite p-group with G3≤Φ(G′)∼=CpandG/Φ(G′)∼=Mp(n,m,1), wheren >1 forp= 2 and n≥m. Let G/Φ(G′) =/an}bracketle{t¯a,¯b,¯c|¯apn=¯bpm= ¯cp= 1,[¯c,¯a] = [¯b,¯c] = 1/an}bracketri}ht. Then, without loss of generality, we may assume that G=/an}bracketle{ta,b,c/an}bracketri}htsuch that [ a,b] =c. SinceG3≤Φ(G′)∼=Cp,cis of order p2and Φ(G′) =/an}bracketle{tcp/an}bracketri}ht. Sinceapn∈Φ(G′), we may assume that apn=cw11p. By similar reasons, we may assume that bpm=cw21p, [c,a] =cw12pand [c,b] =cw22p. Then we get a 2 ×2 matrix over Fp.w(G) = (wij) is called a characteristic matrix of G. Notice that w(G) will be changed if we change the generators a,b. We also call a,ba set of characteristic generators of w(G). In this paper, w(G) is always a characteristic matrix of G, anda,bis always a set of characteristic generators of w(G). Theorem 3.1. LetGbe a ﬁnite p-group such that G3≤Φ(G′)∼=CpandG/Φ(G′)∼= Mp(n,m,1)wheren >1forp= 2andn≥m. Then the following conclusions hold: (1)Ifm= 1, thenp= 2,[c,b] =c2andGhas a unique A1-subgroup of index p; (2)Imin= 2form≥2. Proof(1) Ifm= 1, since 1 = [ a,bp] =cp[c,b](p 2)andcp/ne}ationslash= 1, then p= 2 andc2= [c,b]. Since [c,ab][c,a] =c2, we have [ c,a] =c2or [c,ab] =c2. Without loss of generality, we may assume that [ c,a] =c2. Hence [ c,ab] = 1. Notice that the maximal subgroups of G are/an}bracketle{ta,Φ(G)/an}bracketri}ht=/an}bracketle{tc,a/an}bracketri}ht,/an}bracketle{tb,Φ(G)/an}bracketri}ht=/an}bracketle{tc,b,a2/an}bracketri}htand/an}bracketle{tab,Φ(G)/an}bracketri}ht=/an}bracketle{tc,ab/an}bracketri}ht. Since/an}bracketle{tc,a/an}bracketri}ht ∈ A1, /an}bracketle{tc,b,a2/an}bracketri}ht /ne}ationslash∈ A1and/an}bracketle{tc,ab/an}bracketri}htis abelian, /an}bracketle{tc,a/an}bracketri}htis the unique A1-subgroup of index p. (2) Ifm≥2, then we claim that Imin>1. Otherwise, we may assume that Dis an A1-subgroup of index p. It is easy to see that D′= Φ(G′) andd(D/Φ(G′)) = 2. Since 3 Φ(G)≤D,d(Φ(G)/Φ(G′))≤2. On the other hand, Φ( G)/Φ(G′) =/an}bracketle{t¯ap,¯bp,¯c/an}bracketri}htis of type (pn−1,pm−1,p), a contradiction. Hence Imin>1. Since|G3| ≤p,

Question 1292 multiple-choice

Supersymmetric models of particle physics can be extended with hidden sectors and mechanisms for asymmetric dark matter, involving new gauge symmetries and particle content. The arrangement of mass spectra and symmetry properties is crucial for ensuring cosmological viability and compatibility with experimental constraints. In a hidden sector model with a spontaneously broken U(1)x symmetry and vector-like chiral superfields Y and Yc, which configuration ensures that only the dark matter fermion and the lightest scalar are stable, thus suppressing indirect detection signals from late-time annihilation? 1) Assigning identical U(1)x charges to both Y and Yc fields 2) Choosing scalar soft masses so all scalars remain stable 3) Setting the hidden gaugino mass to be much larger than Y’s mass parameter 4) Arranging the hidden Higgs vevs to preserve the U(1)x gauge symmetry 5) Allowing R-parity to be violated in the hidden sector 6) Selecting flavor symmetry breaking patterns that destabilize all hidden sector particles 7) Choosing the mass spectrum so only the DM fermion and the lightest scalar are stable

✓ Correct Answer:

The correct answer is 7) Choosing the mass spectrum so only the DM fermion and the lightest scalar are stable.

📚 Reference Text:

All three features require non-trivial additional structure in the underlying mechanisms of supersymmetry breaking or mediation [108, 109]. V. VARIATION #2: ASYMMETRIC HIDDEN U(1) As a second extension of the MSSM, we investigate a theory of hidden asymmetric dark matter (ADM) [110{113]. In the ADM framework, the DM particle has a distinct antiparticle, and its abundance is set mainly by a particle-antiparticle asymmetry in analogy to baryons, and this tends to suppress indirect detection signals from late-time annihilation if very little anti-DM is present [114{117]. The ADM theory we consider is nearly identical to the hidden U(1)xtheory studied in Section IV, but with an additional pair of vector-like hidden chiral super elds YandYcwithU(1)xchargesxY=1. We assume that a small asymmetry in the Ydensity is generated during moduli reheating, in addition to the much larger symmetric density, and we compute the resulting relic densities and experimental signals. A. Mass Spectrum and Decays The superpotential in the hidden sector is same as that considered in Sec. IV up to a new mass term for the YandYcmultiplets, WYYYc: (34) We also include the new soft supersymmetry breaking terms Lsoftm2 eYjeYj2+m2 eYcjeYcj2(bYeYeYc+ h.c.): (35) As in Sec. IV, we x the hidden gaugino mass to its AMSB value with bx= 2(1 + 1), accounting for the new super elds. We also do not impose minimal AMSB values for the 19 scalar soft terms, but take them (as well as 0andY) to be of similar size to the gaugino soft mass. Finally, we arrange parameters so that the hidden Higgs scalars develop expectation values and spontaneously break the U(1)x. The mass spectrum of the hidden sector follows the minimal model considered in Sec. IV, but now a new Dirac fermion of mass m =Yand two complex scalars 1;2. The scalar mass matrix in the ( eY;eYc) basis is M2 eY= jYj2+m2 eY~Db Y bYjYj2+m2 eYc+~D;! ; (36) where ~D=g2 x2cos 2+xYgxg0v2cos 2 =2. Takingm2 eY=m2 eYcfor convenience, the mass eigenvalues are m2 1;2=jYj2+m2 eYq ~2 D+jbYj2: (37) In what follows we will refer to the lighter scalar 1as . This theory preserves both the usual R-parity as well as a non-anomalous global U(1) avour symmetry among the YandYcmultiplets, and can support multiple stable states. The number of stable particles depends on the mass spectrum. To allow for dominantly asymmetric dark matter, we will focus on spectra with m x 1>m +msuch that the decay x 1! + is possible, and the only stable hidden states are and . If this channel is not kinematically allowed, the x 1neutralino will also be stable and can induce overly large gamma ray signals as in the previous section. We also choose soft masses such that mx< m ; m to allow both states to annihilate eciently into hidden vectors. With this mass ordering, the lightest hidden states will be the vector Xand the hidden Higgs hx 1. Both will decay to the SM in the same way as in the minimal model of Sec. IV. The lightest

Question 1293 multiple-choice

The Sum-of-Squares (SoS) hierarchy is a central tool in optimization and computational complexity, providing increasingly tight relaxations for NP-hard problems. Its quantum analog, the quantum Lasserre hierarchy, extends these techniques to systems involving non-commutative operators, such as those found in quantum constraint satisfaction problems. Which statement best describes the role of pseudo-expectations within the Sum-of-Squares (SoS) hierarchy? 1) They specify the exact assignment of variables in Boolean satisfiability problems. 2) They are probability distributions over feasible solutions to polynomial optimization problems. 3) They are dual solutions in semidefinite programs that enforce global integrality. 4) They are linear functionals on polynomials of bounded degree that satisfy positivity and approximate expectations as the degree increases. 5) They are non-convex relaxations used to directly solve NP-hard problems in polynomial time. 6) They represent quantum states in the relaxation of local Hamiltonian problems. 7) They are rounding algorithms that convert relaxed solutions to exact solutions.

✓ Correct Answer:

The correct answer is 4) They are linear functionals on polynomials of bounded degree that satisfy positivity and approximate expectations as the degree increases..

📚 Reference Text:

Level dof this hierarchy involves optimizing over “pseudo-expectations” – linear functionals µ which are defined on polynomials of degree at most 2din thezivariables and which satisfy Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 4 µ(1) = 1 andµ(f2)≥0for any polynomial fwith degree at most d.1The set of pseudo- expectations contains the set of “actual” expectation values which can be obtained from probability distributions assigning values to the variables zi, and converges to this set as d→∞ . For this reason, the SoS hierarchy gives a converging series of upper bounds on the true value of the Max Cut problem. Additionally, for any constant d, the leveld SoS upper bound can be found in polynomial time, though in practice this optimization quickly becomes infeasible as dgrows large. The best-known approximation algorithm (i.e., constructive lower bound algorithm) for Max Cut is due to Goemans and Williamson and can be viewed as an algorithm which “rounds” the pseudo-expectation produced by a level 1 Lasserre relaxation to an actual expectation [GW95]. In general, Lasserre relaxations in conjunction with rounding have been powerful techniques for upper bounding and approximating hard constraint satisfac- tion problems (CSPs) [BRS11]. In fact, assuming the Unique Games Conjecture (UGC) of Khot [Kho02], Raghavendra [Rag08, Rag09] showed that it is NP-hard to outperform a canonical approximation algorithm based on rounding from the first level of the SoS hierarchy for any classical constraint satisfaction problem . Quantum Lasserre hierarchy The success of SDP relaxations for classical CSPs has motivated their use in approximations to local Hamiltonian problems [BH13, BGKT19, HO22] and in particular Quantum Max Cut [GP19, PT21, PT22, Kin23, KPS18]. Formu- lating these quantum relaxations is non-trivial and builds on results in non-commutative polynomial optimization [HM04] and its application to problems in quantum information [DLTW08, NPA08]. The quantum Lasserre relaxation for Quantum Max Cut [GP19, PT22] was defined with respect to the n-qubit Pauli algebra. A solution to the level- drelaxation assigns pseudo-expectation values to operators corresponding to Pauli polynomials, i.e., polyno- mials of degree at most 2din the non-commuting variables {Xi,Yi,Zi}fori= 1,...,n . In analogy with the classical situation, the pseudo-expectation of the identity is fixed to be 1 and, for any Pauli polynomial Pwith degree at most d, the pseudo-expectation of its square must be non-negative. The relaxation can be viewed as relaxing from operator algebraic states (positive linear functionals) on the n-qubit Pauli algebra to pseudo-states that are positive only with respect to squares of degree- dpolynomials. In contrast to the classical case, the relationship between the quantum Lasserre Hi- erarchy and approximation algorithms for local Hamiltonian problems is much less well understood. In [GP19], Gharibian and Parekh give an approximation algorithm for Quan- tum Max Cut based on a rounding from the first level of the quantum Lasserre hierarchy. Later works then gave algorithms which rounded from the second level of the hierarchy and lead to better approximation ratios [PT21, PT22]. In [Lee22], a modified Lasserre relax- ation was introduced, which considered only

Question 1294 multiple-choice

In quantum algorithms for algebraic problems, the matrix sum problem over the finite field Zp is fundamental, especially for cases with two equations and two variables. The structure of solutions often depends on properties of polynomials and quadratic residues within Zp. When solving the matrix sum problem for k=2 over Zp, what is the expected proportion of cases in which the discriminant Δ is a nonzero square (i.e., a quadratic residue), thus yielding two distinct solutions for the variables? 1) Nearly all cases, approaching 100% as p increases 2) Approximately one-third of cases 3) Exactly 1/p of all cases 4) Only when p is a prime congruent to 1 mod 4 5) One quarter of all possible cases 6) About half the cases, since roughly half of elements in Zp are quadratic residues 7) Proportion depends on explicit values of x1, x2, y1, y2, w, v

✓ Correct Answer:

The correct answer is 6) About half the cases, since roughly half of elements in Zp are quadratic residues.

📚 Reference Text:

Therefore the matrix sum problem can be written k/summationdisplay j=1/parenleftbiggbjsbj(1−bj) 0bj/parenrightbigg/parenleftbiggxj yj/parenrightbigg =/parenleftbiggw v/parenrightbigg . (37) Ifk= 1, then it is not hard to see that the probability of having a s olution isO(1/p), i.e., exponentially small in log p. However, if we take k= 2, then there are as many variables as (scalar) equations, and we ﬁnd the matrix sum problem b1x1+sb1(1−b1)y1+b2x2+sb2(1−b2)y2=w (38) b1y1+b2y2=v (39) with uniformly random x1,x2,y1,y2,w,v∈Zp. The equations (38–39) can be solved as follows. Deﬁne ∆ := (2wy1+vy1−v2−2vx1)(y1+y2)y2+(vy2+x1y2−x2y1)2. (40) If ∆ is a nonzero square in Zpandy1,y2,y1+y2/ne}ati≀nslash= 0, then there are two solutions for ( b1,b2): b1=vy1+x2y1−x1y2±√ ∆ y1(y1+y2)andb2=vy2+x1y2−x2y1∓√ ∆ y2(y1+y2). (41) It is straightforward to check that ( b1,b2) is indeed a solution. If ∆ = 0 then these two solutions are the same, and there is only one solution. Finally, if ∆ is n onsquare, then there are no solutions. To calculate the success probability of the pgm, we must determine the probability that y1,y2,y1+y2/ne}ati≀nslash= 0 and ∆ is a square in Zp. First, note that since y1,y2are uniformly random, we havey1,y2,(y1+y2)/ne}ati≀nslash= 0 with probability ( p−1)(p−2)/p2= 1−O(1/p). Now rewrite ∆ as ∆ = 2wy1y2(y1+y2)+/bracketleftbig v(y1−2x1−v)(y1+y2)y2+(vy2+x1y2−x2y1)2/bracketrightbig .(42) Assuming 2 y1y2(y1+y2)/ne}ati≀nslash= 0, ∆ depends linearly on w. Hence, if we ﬁx x1,x2,y1,y2,vand choose wuniformly at random from Zp, then ∆ will also be uniformly random in Zp. Of the possible values of ∆ ∈Zp, we have (p−1)/2 cases with√ ∆/ne}ati≀nslash= 0, (p−1)/2 cases with ∆ not a square, and one case with ∆ = 0. Therefore, under the assumption y1,y2,y1+y2/ne}ati≀nslash= 0, we have Pr(ηx,y w,v= 0) =1 2−1 2p(43) Pr(ηx,y w,v= 1) =1 p(44) Pr(ηx,y w,v= 2) =1 2−1 2p. (45) In particular, we see that Pr( ηx,y w,v= 2) =1 2−O(1/p). Therefore, by Lemma 2, Pr(success) ≥ 1−O(1/p). Since the above discussion gives an explicit solution of the matrix sum problem, and since arithmetic in the ﬁnite ﬁeld Zpcan be performed in time poly(log p), it is straightforward to eﬃciently implement the quantum sampling transformation ( 26).2Combined with the fact that the success probability of the pgmis large, this shows that the hidden subgroup problem in the Heisenberg group can be solved eﬃciently. 2For example, given the ability to compute a list of solutions b1,...,b ηwithηsmall, the following simple trick can be used to eﬃciently create the uniform superposition. Crea te the labeled superposition/summationtext j∈Zη|j,bj/angbracketright/√η, Fourier transform the ﬁrst register (over Zη), and measure the ﬁrst register. When the outcome is 0, which occurs with probability 1 /η, the desired state is obtained. Finally, use O(η) repetitions to boost the success probability close to 1, and implement the measurement unitarily. 12 6.2 The general case More generally, consider the group Zr p⋊ϕZpfor any constant rand for any automorphism ϕ, i.e., for any matrix µsatisfyingµp=I. It is easy to see that matrices related by a similarity transformation correspond to isomorphic groups. Thus, wit hout loss of generality, we can assume thatµis in Jordan canonical form, or in other words, that it is zero everywhere except

Question 1295 multiple-choice

Quantum computing introduces new approaches for processing and interpolating classical data, utilizing specialized quantum algorithms and transforms. These techniques are essential for efficient signal reconstruction, image processing, and overcoming bottlenecks in data uploading. Which quantum algorithm enables efficient interpolation of band-limited signals by leveraging the principles of the Nyquist-Shannon sampling theorem, and is characterized by polynomial-time complexity relative to qubit count? 1) Quantum Phase Estimation 2) Quantum Principal Component Analysis 3) Quantum Fourier Transform 4) Quantum Grover Search 5) Quantum Error Correction 6) Quantum Amplitude Amplification 7) Quantum Support Vector Machine

✓ Correct Answer:

The correct answer is 3) Quantum Fourier Transform.

📚 Reference Text:

Eﬃcient quantum interpolation of natural data Sergi Ramos-Calderer1, 2 1Quantum Research Centre, Technology Innovation Institute, Abu Dhabi, UAE. 2Departament de Física Quàntica i Astrofísica and Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain. (Dated: July 19, 2022) We present eﬃcient methods to interpolate data with a quantum computer that complement up- loading techniques and quantum post-processing. The quantum algorithms are supported by the eﬃcient Quantum Fourier Transform (QFT) and classical signal and imaging processing techniques, and open the door of quantum advantage to relevant families of data. We showcase a QFT inter- polation method, a Quantum Cosine Transform (QCT) interpolation geared towards natural data, and we improve upon them by utilizing a quantum circuit’s capabilities of processing data in su- perposition. A novel circuit for the QCT is presented. We demonstrate the methods on probability distributions and quantum encoded images, and discuss the precision of the resulting interpolations. I. INTRODUCTION Uploading a large amount of classical data into a quan- tumstateremainsabottleneckforquantumcomputation applications. Quantum states indeed oﬀer a large Hilbert space to encode classical data, but uploading one element at a time makes a quantum approach ineﬃcient, no mat- ter its promises regarding processing power on the up- loaded state. New strategies are needed to overcome the initial threshold of uploading data into a quantum state. To this end, we present diﬀerent methods that inter- polate smooth probability distributions and natural data over a larger space that can alleviate the data uploading eﬀort, dramaticallyinsomecases. Thismethodsbuildon classical resampling techniques that employ the Fourier Transform to interpolate band-limited signals. The orig- inal amplitude encoded distribution is ﬁrst Fourier trans- formed, then complemented with vanishing high frequen- cies and, ﬁnally, an inverse Fourier transform over the larger space delivers the interpolated probability distri- bution. To be precise, the relevant fact that makes this approachusefulwheninterpolatingclassicaldistributions in a quantum computer is that the Quantum Fourier Transform(QFT)iseﬃcient, thatis, itonlyneedsapoly- nomial number of operations as a function of the number of qubits involved. The QFT interpolation scheme was adapted with suc- cess in the context of Tensor Networks in Ref. [1], and introduced to enhance the result of solutions of partial diﬀerential equations in a quantum computer in Ref. [2], wheretheaccuracyoftheinterpolationisstudied. Weex- pand upon this concept with more diverse eﬃcient quan- tum transformations and further utilize superposition for more eﬃcient interpolation algorithms. These interpolation methods can also be understood as a discrete-to-discrete instance of the Nyquist-Shannon sampling theorem. The sampling theorem states that all the information of a signal with ﬁnite band-width can be captured by samples obtained at a ﬁnite rate, known as Nyquistrate. Weincorporatetheseideaswhendiscussing the accuracy of the proposed resampling techniques. Theuse of Fourier-like transforms is ubiquitous when recon- structing continuous signals from samples. The QFT can introduce eﬃcient counterparts to techniques used in classical signal processing, and we present some ex- amples in the context of data interpolation and image resampling. Classical image processing of natural data has been drawn towards the Discrete Cosine Transforms (DCT) [3]

Question 1296 multiple-choice

Invariant theory studies the behavior of fixed fields under the action of finite groups, and Noether’s problem explores whether these fixed fields are rational over a base field. The rationality of such fields has significant implications for field extensions and Galois theory, especially in the context of p-groups and roots of unity. For a non-abelian p-group G of order pⁿ that contains a cyclic subgroup of index p, which condition on the field K ensures that the fixed field K is rational over K? 1) K is algebraically closed 2) K contains a primitive p²-th root of unity 3) K has characteristic p 4) K contains all nth roots of unity for any n 5) K contains a primitive p-th root of unity 6) K contains a primitive pⁿ⁻²-th root of unity 7) K is a finite field of order pⁿ

✓ Correct Answer:

The correct answer is 6) K contains a primitive pⁿ⁻²-th root of unity.

📚 Reference Text:

Title: Noether’s Problem for Some p-Groups Year: 2007 Paper ID: 0f66c3cb0ec2a901c5e196b6dc545fe4f7a5ed9d Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0f66c3cb0ec2a901c5e196b6dc545fe4f7a5ed9d Abstract: Let K be any field and G be a finite group. Let G act on the rational function field K(x g : g ∈ G) by K-automorphisms defined by g · x h = x gh for any g, h ∈ G. Noether’s problem asks whether the fixed field K(G) = K(x g : g ∈ G) G is rational (=purely transcendental) over K. We will prove that if G is a non-abelian p-group of order p n containing a cyclic subgroup of index p and K is any field containing a primitive p n−2 -th root of unity, then K(G) is rational over K. As a corollary, if G is a non-abelian p-group of order p 3 and K is a field containing a primitive p-th root of unity, then K(G) is rational.

Question 1297 multiple-choice

Quantum algorithms for the hidden subgroup problem (HSP) play a crucial role in computational group theory and cryptography, leveraging mathematical structures like Hermite normal form and Fourier transforms to achieve efficiency. Understanding how parameters such as group order and smoothness impact complexity and implementation is key to advancing solutions in various quantum models. Which of the following statements accurately describes the quantum query complexity for solving the hidden subgroup problem in finite abelian groups of rank n and how the overhead changes when m is prime versus composite? 1) The quantum query complexity is Θ(n); the overhead is constant for prime m and O(log^2 m) for composite m. 2) The quantum query complexity is Θ(log n); the overhead is O(n^2) for prime m and constant for composite m. 3) The quantum query complexity is Θ(√n); the overhead is O(log m) for prime m and O(m^2) for composite m. 4) The quantum query complexity is Θ(m); the overhead is O(n) for prime m and O(log m) for composite m. 5) The quantum query complexity is Θ(n^2); the overhead is constant for both prime and composite m. 6) The quantum query complexity is Θ(log^2 n); the overhead is O(log n) for prime m and O(n log m) for composite m. 7) The quantum query complexity is Θ(1); the overhead is O(m) for prime m and O(log^2 n) for composite m.

✓ Correct Answer:

The correct answer is 1) The quantum query complexity is Θ(n); the overhead is constant for prime m and O(log^2 m) for composite m..

📚 Reference Text:

provided by the oracle Uf. A unique description of the output is helpful for avoiding that it gets entangled with the garbage. The Hermite normal form appears to be a good choice, as it will also be useful in some of our applications of the hidden subgroup algorithm. For a constant m, e.g., the product of the rst 100 primes, the theorem gives an ecient exact solution of the HSP in the strictest model (i) (equivalent to quantum Turing machines). For a smooth m, e.g., when it is input in unary, or it is a product of primes of size polynomial in nandk, the theorem could be used in model (ii), that is in the context of uniform circuits based on an in nite set of gates. For general m, through the implementation of Mosca and Zalka [10], one obtains a result in the strongest model (iii). For m= 21, input by in unary, or more generally, when mcan be computed in deterministic polynomial time from [log m] (counted in unary), application of [10] again results in an ecient hidden subgroup algorithm in the (in nitely based) uniform circuit model (ii). Koiran, Nesme and Portier showed in [20] that the bounded-error quantum query com- plexity of the hidden subgroup problem in a nite abelian group of rank n(such as the group Zn mk) is (n). Thus, regarding the number of queries, the overhead of our exact method for Zn mis constant factor if mis a prime number and O(log2m) in the general case. For Zn mk the overhead is O(klog2m) (orO(k) whenmis prime). In the classical setting, Nayak [21] proposed a simple deterministic polynomial time method that solves the hidden subgroup problem in an abelian group GusingO(p jGj) queries. The method has an extension to the non-commutative case that uses O(p jGjlogjGj) queries. Note that for G=Zn 2, the classical randomized query complexity of the HSP is shown to be (p jGj) by Simon [3], so in this group Nayak's algorithm has a constant overhead only. M. Imran and G. Ivanyos 775 Applications. Watrous [19] gave a polynomial time quantum algorithm for computing the order of a solvable black-box group with unique encoding of elements. Black-box groups are used to obtain general algorithms for various "real" groups such as the multiplicative groups modulo given integers, permutation groups and matrix groups over nite elds. A matrix group over the q-element eld Fqis just a subgroup of the group GLn(Fq) of invertible nby nmatrices over Fq. Elements of a black-box group are represented by binary strings of a certain length `and the group operations are given by oracles and as input, a generating set for the group is given. See Section 2 for more details. We use the exact Fourier transform of Zmand the hidden subgroup algorithm of Zn m(for various values of n) to devise an exact version of Watrous's algorithm and to solve some other problems for abelian and solvable black-box groups with unique encoding of elements whose order is

Question 1298 multiple-choice

In the study of algebraic groups and Lie theory, graph automorphisms significantly influence the structure and classification of subgroups, including centralizers and maximal commuting sets. These concepts are foundational for understanding the symmetry and subgroup lattices in groups of Lie type. In the context of algebraic groups with graph automorphisms, which centralizer is identified as a reductive group of dimension 36 and specifically of type C4? 1) The centralizer of an involution in SL4 2) The maximal torus in E6 3) The centralizer of a regular unipotent element in D4 4) The centralizer of a graph automorphism in A4 5) The centralizer of a diagonal automorphism in SL8 6) The centralizer Cy(rt^a) arising from the action of a graph automorphism on root subgroups 7) The centralizer of a Coxeter element in F4

✓ Correct Answer:

The correct answer is 6) The centralizer Cy(rt^a) arising from the action of a graph automorphism on root subgroups.

📚 Reference Text:

T-root subgroup. If CYT = a then one checks from the known action of the graph automorphism r that Va~ Cy(rt") if and only if to:11 centralizes Va. A direct check of roots shows that such root subgroups Vo: span D. Now suppose that CYr *a. One checks from the Chevalley commutator relations that in each case ( U('(, v(XT) = u('( x u()(T• and hence Cy( rt".) n ( Ua, Va,) is 1-dimensional. There are 24 pairs of roots interchanged by r, and hence dim L( Cy( rta,))?: 24 +dim L(D) = 36. On the other hand, the same arguments at the level of the Lie algebra show that dim(L(Y) n C( rta.)) = 36. Thus Cy( rtaJ is a reductive group of dimension 36, and Dis a maximal commuting set of fundamental subgroups SL2 (this is already true in Y). It follows that C y( rt" ) = C4, as claimed. Now let Y = A2n-I• so Y = SL2n-Here it will be convenient to replace r by r' = hb, where b is the inverse-transpose map and h is the n x n matrix ( O;.n + i-;) (that is, the matrix with entries 1 on the opposite diagonal and entries 0 elsewhere). Then r' E Yr. Now r' normalizes the lower triangular group and also the diagonal group T. An easy computation shows that any element of r' T is T-conjugate to an element of the form r't, where t = diag(l, ... , 1, Cn+t• •. ., c2") for some C; E FP. If we also assume that r't corresponds to an involution in Y < r), then (r't)2EZ(Y). It then follows that t=diag(l, .. ., 1,c, .. ., c) with c= ±1. Then Cy(r't) is Dn or Cn according as c is 1 or -1. LEMMA 2.8. There are local subgroups U3(2) X Gi(2) in 2£6(2), and Li(3) x ~(3) in £7(3). These subgroups are unique up to G0-conjugacy. Proof. First consider L = £7(3). The ad joint algebraic group G contains three conjugacy classes of involutions, with centralizers A 1 D6, A1< wi) and T1 £6 ( w2) (see [17]). Let a be an involution in G0 with Ca(a) = Ti£6(w2). Here w2 induces a graph automorphism on the £6• By Proposition 2. 7, we may pick an involution b E Cda) such that Ca(a, b )0 = h Write F =Cc( a, b )0. Then I;~ Ca(F). Since C0(b) contains F, b must be conjugate to a, so Ca(b) = T;E~ ( w;). Then F ~ E~ and so Cc(F) contains (T1, T;). We claim that C0(F) is reductive. For otherwise, FCa(F) lies in a parabolic subgroup P of G. Since P contains F == ~, P must be an E6-parabolic with F fixing a 1-space of the unipotent radical. But F also fixes a 1-space of the unipotent radical of the opposite parabolic, contradict ing the fact that C0(F)0.;; P. This proves the claim. Also Cc(F)0 n C0(a) = T1, so

Question 1299 multiple-choice

Quantum algorithms for hidden subgroup problems are foundational in quantum computing, with significant implications for cryptography and computational theory. The efficiency of these algorithms often depends on trade-offs between quantum memory usage and running time. Which algorithm achieves polynomial quantum space for solving the dihedral hidden subgroup problem by increasing time complexity, representing a trade-off compared to earlier approaches? 1) Ettinger-Hoyer algorithm 2) Shor's factoring algorithm 3) Graph isomorphism quantum algorithm 4) Simon's algorithm 5) Grover's search algorithm 6) Kuperberg's original algorithm 7) Regev's modification of Kuperberg's algorithm

✓ Correct Answer:

The correct answer is 7) Regev's modification of Kuperberg's algorithm.

📚 Reference Text:

problem. Ettinger and Hoyer [15] developed a quantum algorithm that produces output sufﬁcient to solve the dihedral hidden subgroup problem. However, in this case exponential time is required to reconstruct the hidden subgroup from the output. The fastest known quantum algorithm for the dihedral hidden subgroup problem was intro- duced in 2003 by Kuperberg [16] . This approach uses time and query complexity of 2O(plogN), where Ncorresponds to the parameter of underlying group DN. Unfortunately, Kuperberg’s al- gorithm also requires 2O(plogN)quantum space. Another subexponential-time algorithm is due to Regev [17], whose modiﬁcation of Kuperberg’s approach resulted in a polynomial-space algorithm at a price of a slow-down in time complexity. In this dissertation, a quantum algorithm that solves the hidden subgroup problem for the generalized quaternion group is developed. 5 CHAPTER 2 A QUANTUM MODEL OF COMPUTATION 2.1 The Quantum Mechanical Framework Quantum mechanics emerged as a mathematical model for physics to address problems encountered by classical theories in their attempts to explain the results of certain experiments. The following four postulates deﬁne the content of quantum mechanics. Postulate 1 (State Space). The state space of a system is a Hilbert space, and the system is com- pletely described by its state vector, a unit vector in the system’s state space. A single qubit constitutes a two-level system whose state is described by a vector in a two- dimensional Hilbert space. Any two vectors with the same global phase are equivalent. Postulate 2 (State Evolution). The evolution of a closed system is described by a unitary trans- formation. That is, for any evolution of the closed system there exists a unitary operator U such that if the system is in the state jy1iat time t1, then its state at time t2is jy2i=Ujy1i: (2.1) A closed quantum system evolves in accordance with the Schrödinger equation i¯h¶ ¶tjy(t1)i=Hjy(t1)i; (2.2) where ¯his the reduced Planck’s constant, and His the Hamiltonian, an operator that corresponds to the total energy of the system. The eigenvalues of Hare the possible energies of the system. 6 For a time-independent Hamiltonian, the state of the system at a later time t2is given by jy(t2)i=eiHt ¯hjyi(t1): (2.3) Thus, the time evolution of the system is governed by the unitary operator U=eiHt ¯h: (2.4) The deﬁning relation for any unitary operator Uis UU†=U†U=I; (2.5) where U†denotes the adjoint of U. Since a quantum computer is an n-qubit quantum system, the time evolution of its wave function obeys the Schrödinger equation and is described by a unitary operator. In quantum com- putation, unitary operators are implemented as unitary gates. Postulate 3 (State Combination). The state space of a composite system is the tensor product of the state spaces of its component systems. Given nsystems in the states jy1i;jy2i;:::;jyni, the state of their combined system is jy1i jy2i ::: jyni=jy1y2:::yni: (2.6) IfH1andH2are Hilbert spaces of dimensions N1andN2, respectively, then their tensor product H=H1 H2 (2.7) has dimension dim(H) =N1N2: (2.8) 7 IfH1andH2are endowed with bases juiiandjvii, respectively, then the basis for His given

Question 1300 multiple-choice

Quantum circuits designed for floating-point arithmetic must ensure accurate representation of special cases such as underflow, overflow, and sub-normal numbers. The use of specialized qubits and gate logic is essential for handling these conditions during operations like squaring. In a quantum circuit performing squaring with 3 exponent qubits (NE=3), which input exponent value requires a controlled-NOT gate to flip an indicator qubit signifying truncation of the result to zero? 1) ∣111⟩ 2) ∣000⟩ 3) ∣011⟩ 4) ∣010⟩ 5) ∣101⟩ 6) ∣100⟩ 7) ∣110⟩

✓ Correct Answer:

The correct answer is 2) ∣000⟩.

📚 Reference Text:

underflow/overflow protection (N M¼3and N E¼3). Figure 5. Quantum circuit used to obtain exponent for squaring operation, including sub-normal numbers and under/ overflow protection (N M¼3and N E¼3). 12Quantum Computing and Communications For certain normalized input numbers the squaring operation leads to outputs truncated to 0 or to the non-zero sub-normal numbers discussed in Section 5.3. The quantum circuits discussed so far need to be modified in a number of ways to deal with this possible sub-normal output. Figure 4 illustrates the required changes for NM¼3 and NE¼3. Two additional qubits are needed. Qubit ∣isubi¼∣0iis used as indication that result is a sub-normal number. Qubit ∣icuti¼∣0iis similarly used to define cases with output truncated to 0. Both qubits are initialized to ∣1i. Then, before the mantissa multiplication step takes place, a first modification is intro-duced, shown on the left-hand side of Figure 4 . For N E¼3, only inputs with exponent ∣000iwill need truncating to 0, as shown in the first 4-qubit controlled- NOT gate flipping ∣icutito∣0i. For NE¼3, inputs with exponent ∣001iare guaranteed to lead to sub-normal output (or 0), and for these cases ∣isubiis set to ∣0i, using the second 4-qubit controlled-NOT gate with ∣isubias target. The mantissa-multiplication step shown in Figure 2 remains unchanged (i.e. qubits ∣isubiand∣icutiare not used). The next required modification relates to the ‘copy- ing’of the result of the mantissa multiplication to output register ∣imr1∣imr0iand the application of increments to the output exponent. The additional logic needed is Figure 6. Quantum circuit used to set output mantissa for squaring operation, including sub-normal numbers and underflow/overflow protection (N M¼4and N E¼3). 13Quantum Algorithms for Nonlinear Equations in Fluid Mechanics DOI: http:/ /dx.doi.org/10. 5772/intechopen.9 5023 shown on the right-hand side of Figure 3 . First, for ∣imp5i¼∣1i, setting ∣ier0i¼∣1i becomes conditional of both ∣isubi¼∣1iand∣icuti¼∣1i. The next two 4-qubit gates are used to guarantee that correct output with exponent ∣111iis created for inputs with exponents ∣101iand∣110i. The remaining gate operations perform the ‘copy- ing’of the mantissa squared into ∣imr1∣imr0itaking into account the possible sub- normal output (cases with ∣isubi¼∣0i). The steps for ∣isubi¼∣1iare the same as in the corresponding circuit for squaring without the sub-normal number modifica- tions. A further set of circuit modifications to deal with sub-normal numbers is required in the quantum circuit used to obtain the output exponent. Figure 5 shows the additional operations required relative to the original quantum circuit shown inFigure 3 . Three additional CNOT operations are introduced just before performing theQFT3. For ∣isubi¼∣0iand∣icuti¼∣0ithe initialization of ∣imp 1∣imp0iis modified so that the subsequent steps will produce the correct result for the expo- nent. The three CNOT operations also appear in the ‘uncompute ’stage at the right- hand side of the circuit. Further changes comprise two 4-qubit controlled-NOToperations on ∣ier2iand∣ier1irequired to create ∣111iexponents for inputs with exponent ∣110i. For a fixed value of N Eit is important to note that the additional complexity introduced by increasing NMis limited. In fact, the quantum circuit shown on the left-hand side of Figure 4 does

Question 1301 multiple-choice

Quantum error correction codes often utilize auxiliary qubits, known as parity qubits, to enforce constraints and improve fault tolerance. The encoding and decoding of such codes requires precise gate operations and resource management. In a quantum code where parity qubits are dynamically added and removed to impose k-body constraints, which operation is primarily used to initialize and impose the parity constraints among data and parity qubits? 1) Application of CNOT gates controlled by data qubits 2) Hadamard gates followed by measurement in the X basis 3) SWAP gates between parity and data qubits 4) Phase gates applied sequentially to all qubits 5) Controlled-Z gates acting on pairs of data qubits 6) Toffoli gates between data and parity qubits 7) Measurement of data qubits in the Z basis without gate operations

✓ Correct Answer:

The correct answer is 1) Application of CNOT gates controlled by data qubits.

📚 Reference Text:

an encoding and decoding strategy to add or remove an arbitrary number of parity qubits to or from the code. Suppose we start with the logical qubits, each of them encoded in a data qubit. We can now add aphysical qubit ðijÞcorresponding to the parity of qubits ( i) and ( j), by initializing this qubit in the state j0iand then imposing the parity on it with two CNOT gates controlled by qubits ( i) and ( j), as shown in Fig. 3(b). This procedure adds an additional parity qubit to the code and guaranteesthat the corresponding constraint C¼σðiÞ zσðjÞ zσðijÞ z is satisfied. Following this procedure, it is possible to add parity qubits and constraints by applying this strategy iteratively. Instead of directly obtaining the parity information from the data qubits, one can also use the parity qubits included in local constraints, as for example the plaquettes shown in Fig. 2: By definition, the parity of all but one qubit of a constraint is always encoded in the state of the remaining qubit, i.e., σðijÞ zσðjkÞ zσðklÞ zσðliÞ zjψi¼j ψi ⇒σðijÞ zjψi¼σðjkÞ zσðklÞ zσðliÞ zjψi: This transition and the encoding circuits are shown in Figs. 3(c)–3(d). Conversely, applying the same gate sequence to a parity- encoded set of qubits removes the targeted qubit from the code and projects it to the state of the corresponding constraint. If the constraint was fulfilled, this is the state j0i. For a layout as depicted in Fig. 2with nlogical qubits, this procedure allows encoding and decoding circuits with an overall circuit depth of nþ1. The exact circuit is given in the Supplemental Material [51]. The described procedure does not only offer a way to build up or collapse the all-to-all connected LHZ scheme, but also holds the possibility to switch between different variants of the parity mapping and adapt the number of parity qubits to the algorithmic requirements during com- putation. For example, if some interactions are not needed for a certain algorithm, it is not always necessary to have the corresponding parity qubits in the code. A reduction in the number of parity qubits results in shorter logical lines and thus fewer physical gates. In addition to the two-body parities as introduced in Eq. (1), it is also possible to encode(c)(a) (d)(b) FIG. 3. Encoding and decoding circuits to add or remove a qubit and the corresponding constraint to or from the code.(a) Qubit ðijÞis directly encoded via the adjacent data qubits ( i) and ( j) in a three-body constraint, with the circuit (b). (c) Qubit ðijÞis encoded using other parity qubits in a four-body constraint, with the circuit (d). A qubit without a label indicates a qubitwithout any parity information, being in the state j0i.PHYSICAL REVIEW LETTERS 129, 180503 (2022) 180503-3 higher-order k-body parity qubits in the same manner, when using suitable layouts (see Ref. [47]for more details). As an example, a three-body parity qubit can be used toenable a nontrivial interaction between three logical qubits. Universal gate set. —Single-qubit

Question 1302 multiple-choice

In the study of topological quantum field theories (TQFTs) derived from quantum groups at roots of unity, understanding the structure of the Drinfeld double and its center is essential for analyzing mapping class group representations and algebraic invariants. The quadratic Casimir element in Uq(sl2) plays a pivotal role in decomposing representations and constructing central idempotents. Which of the following statements correctly describes the center of the Drinfeld double D(Bq) for the quantum sl2-Borel algebra Bq at a primitive l-th root of unity? 1) The center is isomorphic to the group algebra C[Z/l] alone. 2) The center is generated exclusively by the quadratic Casimir element X of Uq(sl2). 3) The center is isomorphic to C[Z/l]⊗V, where V is the center of Uq(sl2). 4) The center coincides with the set of all nilpotent elements in D(Bq). 5) The center consists solely of scalar multiples of the identity in D(Bq). 6) The center is generated by the cointegral and R-matrix elements. 7) The center is trivial for all values of l greater than 2.

✓ Correct Answer:

The correct answer is 3) The center is isomorphic to C[Z/l]⊗V, where V is the center of Uq(sl2)..

📚 Reference Text:

we ﬁnd that the projectiv e phasecof the functor Φ in (1.10) for a universal TQFT over a double D(A) is given by: c=ν−3. 3.The Relation of Universal and Semisimple TQFT’s: An Example In this section we shall analyze the proposed representation of th e mapping class group D of the punctured torus explicitly in the example of the double of the q uantum-sl2-Borel algebraBq. 1.) The Algebra D(Bq): Letqbe a primitive l-th root of unity where l= 2m+ 1, m∈Z≥1. We denote by Bqthe Hopf algebra with generators e, k±1and relations: kek−1=qe, kl= 1 el= 0 ∆(k) =k⊗k∆(e) =e⊗1 +k2⊗e S(e) =−k−2e S(k) =k−1ǫ(e) = 0ǫ(k) = 1.(3.75) 26 As PBW-basis for Bqwe choose enkjwithn= 0,...,l−1 andj∈Z/l. The leftcointegral ofBqis given by x= (l−1/summationdisplay j=0kj)el−1(3.76) and the left integralwith normalization m(x) = 1 is m(enkj) =q2δj,2δn,(l−1). (3.77) Themoduliof these integrals are easily found to be a=k2and α(k) =q, α(e) = 0 (3.78) so that ω=q2. (3.79) Since we assumed lto be odd we can choose as generators of the dual algebra B∗ qthe modulus αand the linear form fdeﬁned by < f,enkj>=δn,1. The following relations together with those in (3.75) can be used as a deﬁnition for the doub leD(Bq) containing BqandB∗ qwith opposite comultiplication: αfα−1=q2f αl= 1 fl= 0 αeα−1=q−2e kfk−1=q−1f ef−fe=α−k2 ∆(α) =α⊗α∆(f) =f⊗α+1⊗f(3.80) We shall sometimes refer to the Z- gradation ofD(Bq) which is deﬁned on the generators bygr(e) = +1,gr(k) = 0,gr(f) =−1, andgr(α) = 0. The universal R-matrix of this algebra is R= (l−1/summationdisplay n=0q−n(n−1) 2 [n]!en⊗fn)(1 l/summationdisplay i,j∈Z/lq−ijkj⊗αi) (3.81) Here [n]! = [n][n−1]...[1] with [ n] =qn−q−n q−q−1. If we compute the expressions in (2.23) and (2.24) for the integrals in (3.76) and (3.77) we obtain the following isom orphisms between BqandB∗ q: βl(kjen) =q−(l−1−n)(l−2−n) 2 [l−1−n]!fl−1−n1 l/summationdisplay i∈Z/lqi(j−2)αi+1. (3.82) and βl(αifn) = (−1)n[n]!q−n(n+3) 2−2iel−1−n/summationdisplay j∈Z/lqj(i−1)kj. (3.83) 27 As an associative algebra D(Bq) is isomorphic to the product C[Z/l]⊗Uq(sl2), where the central group algebra C[Z/l] is generated by z:=α−mk (3.84) The generators of the Uq(sl2) factor are deﬁned by E:=z−1e, F:=−f, K:=αmk (3.85) and obey the relations KEK−1=q2E, KFK−1=q−2F, (3.86) EF−FE=K−K−1. (3.87) 2.) The Center of D(Bq): Thanks to the above decomposition the center of D(Bq) is given by C[Z/l]⊗VwhereVis the center of Uq(sl2). In order to give a description of V it is convenient to introduce the projections πj(K) =1 l/summationdisplay i∈Z/lq2ijKij∈Z/l (3.88) on the eigenspaces of Kwith eigenvalue q−2j. Furthermore we introduce the projections Tj=l−1−j/summationdisplay s=j+1πs(K)j= 0,...,m−1. (3.89) The standard quadratic Casimir of Uq(sl2) is given by: X=EF+qK−1+q−1K q−q−1. (3.90) The trivially graded part UoofUq(sl2) (gr(E)=1,gr(F)=-1,gr(K)=0) is a free module over the ringC[K] with basis{Xj}j=0,...,l−1and the minimal equation for Xis: l−1/productdisplay j=0(X−b(j)) = 0, (3.91) where the roots b(j) =b(l−1−j) :=q(2j+1)+q−(2j+1) q−q−1(3.92) 28 are of order two for j= 0,...,(m−1) and of order one for j=m. Using the polynomials φj(X) =/productdisplay 0≤s≤(l−1):b(s)/ne}ationslash=b(j)(X−b(s))j= 0,...,m (3.93) of order ( l−2) forj < mand of order ( l−1) forj=mwe can deﬁne the idempotents and nilpotents associated to X: Pj=1 φj(b(j))φj(X)−φ′ j(b(j)) φj(b(j))2(X−b(j))φj(X)j= 0,...,m Nj=1

Question 1303 multiple-choice

In quantum information theory, the Schur transform leverages group representation theory to efficiently decompose quantum systems associated with symmetric and unitary group symmetries. Understanding how representations combine and decompose is foundational for quantum algorithms and applications such as data compression and state manipulation. Which coefficients determine how irreducible representations of the unitary group combine in Clebsch-Gordan decompositions and also describe the decomposition of symmetric group representations via Young subgroups? 1) Littlewood-Richardson coefficients 2) Stirling numbers 3) Kronecker coefficients 4) Gelfand-Tsetlin patterns 5) Weyl coefficients 6) Hook-length formulas 7) Schur polynomials

✓ Correct Answer:

The correct answer is 1) Littlewood-Richardson coefficients.

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the unitary group. Then they use this circuit to construct the Schur transform by applying it iteratively. This gives us a clue as to how to construct the dual transform if we look at the so called Littlewood- Richardson (LR) coecients. The LR coecients give the number of times an irrep of the unitary group appears in the CG decomposition of two unitary group irreps. The same coecients describe the decompo- sition of induced representations of the symmetric group from certain Young subgroups. This suggests that we need to investigate permutation modules of the symmetric group since they are exactly such induced representations. We can now see that block diagonalizing these induced representations gives us the dual Schur transform. We will not directly work with LR coecients or the related Kostka numbers, but embed them in a larger space. This dual version is polynomial in logd,nandlog(1//epsilon1), where as the one in [5] is polynomial in d,nandlog(1//epsilon1). However, as mentioned above, in Harrow's thesis [18], a short description is given of how to make the algorithm from [5] polynomial in logd,nandlog(1//epsilon1). A classical algorithm to compute the CG coecients for SU(d) is given in [1]. This paper is organized as follows. In Section 2, we explain the necessary background in representation theory, namely the notions of induced representations, symmetric and unitary group representations and the structure of permutation modules of the symmetric group. In Section 3, we describe the construction of quantum Fourier transforms over induced representations. This construction was also used in [31] for Fourier transforms over quantum doubles over nite groups. Then, we apply this QFT for permutation modules which are essentially induced representations. In Section 4, we construct the dual algorithm for the Schur transform and show that the construction is O(poly (n,logd,log 1//epsilon1))elementary operations. Finally, in Section 5, we present some conclusions and other potential applications of our subroutines. 2 2 Background in representation theory 2.1 Basics of induced representations In this section, we brie y describe the representation theoretic concepts such as irreducible representa- tions (irreps, for short), regular representations and induced representations. Induced representations are important in this article since the dual Schur transform is essentially a block diagonalization of induced representations. These concepts for nite groups are described in several texts such as [37, 12]. Here we follow the development in [12]. A representation of a nite group on a nite dimensional complex Hilbert space Vis a homomorphism from the group Gto the unitary group on the vector space U(V)i.e., a representation is ρ:G→U(V). For every nite or compact group, any representation on Vcan be made unitary i.e., ρ(g)is a unitary matrix for allg∈G. Very often, the space which the representation ρmaps to, is identi ed with the representation. Two representations ρandρ/primeof a group Gacting on the same vector space Vare considered equivalent if there exists a unitary Usuch thatUρ(g)U†=ρ/prime(g)for allg∈G. A subspace WofVis called a sub- representation if ρ(g)preservesWfor allg∈G. In this case, the orthogonal complement of WinVis also a sub-representation and Vcan

Question 1304 multiple-choice

Quantum algorithms pose significant challenges to the security of post-quantum cryptographic schemes, especially those based on hard lattice problems. Accurate estimation of algorithmic costs is crucial for evaluating the practical resilience of schemes like Kyber, Saber, and TFHE against future quantum attacks. Which aspect most critically limits the practical accuracy of quantum attack cost estimates on lattice-based cryptography when using the quantum query model? 1) The assumption that lattice reduction can always be performed in linear time 2) Neglecting the impact of discrete Gaussian secret distributions on attack difficulty 3) Ignoring the polynomial factors in cost formulas by focusing solely on main exponents 4) Overestimating the efficiency of classical memory access in quantum search algorithms 5) Relying on the unrealistic availability of Quantum Random Access Classical Memory (QRACM) 6) Assigning constant cost to all matrix-vector multiplications regardless of modulus size 7) Failing to account for the complexity of cryptographic key generation procedures

✓ Correct Answer:

The correct answer is 5) Relying on the unrealistic availability of Quantum Random Access Classical Memory (QRACM).

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be done in polynomial time for all reasonable choices of χssuch as the discrete Gaussian. Finally, the ﬁnal search (Step 28), given by Lemma 3, takes an expe cted time Gqc(χkenums) and makes an expected Gqc(χkenums) calls to O. Therefore the overall cost of this step is O(Gqc(χkenums)·(3)) =O/parenleftig√ Dpkfft/2Gqc(χkenums)poly(logn)/parenrightig .(4) To conclude the proof, just note that the total cost of the algor ith, up to polynomial factors is (1)+(4). ⊓ ⊔ 6 Application Inthissectionwegive roughestimatesfortheimpactofouralgorithmonthecost of solving lattice parameters from the literature. In particular, we consider NIST PQC Round 3 candidate Saber [DKR+20] and NIST PQC to-be-standardised candidateKyber[SAB+20] andsomeTFHEparameters[CGGI20].4Such amea- surement is complicated by two major obstacles. –The cost given in Theorem 7 is the sum of two costs: lattice reduction and a quantum search.Roughlyspeaking,loweringthe ﬁrstsummand incr easesthe second and vice versa. In other words, the ﬁnal cost is obtained b y balancing the two summands. For the ﬁrst summand cost estimates in various cost models are available. In particular, estimates in quantum circuit mode ls are available [AGPS20]. Thus, to give precise cost estimates we require qu antum circuit costs for the oracles in Algorithm 3. It is clear that such cost s would be substantial when compared with e.g. [AGPS20]. In the latter, the costed circuit is essentially an XOR followed by an adder. Here, we have to imple - ment matrix vector products mod qwhich will cost signiﬁcantly more. We consider designing and costing quantum circuits for elementary ope rations such as linear algebra mod qbeyond the scope of this paper. For this reason, we cost our algorithm in the quantum query model only, both for the oracle inside lattice reduction and our quantum search. In this model, all or acle queries are assigned unit cost. As just outlined, this is unrealistic bu t gives a “best case” estimate from the perspective of an attacker. Simila rly, in the 4These schemes do not use discrete Gaussians as secret distri butions. However, we simply ignore this diﬀerence here. 23 same spirit as the “Core-SVP” model, we cost algorithms by simply dro p- ping theO() around cost estimates, which means our estimates may over- or underestimate the true cost; however, all evidence so far poin ts towards underestimating the costs. –A second major obstacle is that our algorithm critically relies on QRACM , a possibly unrealistic resource as already pointed out in e.g. [AGPS20]. T hus, even armed with a quantum circuit for our oracle, we would have to as sume a QRACM oracle (for which we, following previous work [AGPS20], assign unit cost for querying). This would not permit us to draw conclusions about realistic costs of solving instances of lattice problems. Table 2. Dual attack cost estimates. All costs are logarithms to base two. Scheme CC CN C0 GE19 QN Q0 This work This work (QN) (Q0) Kyber 512 139.2 134.4 115.4 139.5 124.4 102.7 119.3 99.7 Kyber 768 196.1 190.6 173.7 191.9 175.3 154.6 168.3 150.0 Kyber 1024 262.4 256.1 241.8 252.0 234.5 215.0

Question 1305 multiple-choice

In computational group theory and quantum algorithms, the hidden subgroup problem (HSP) in nonabelian groups such as the affine group Ap requires sophisticated techniques beyond classical abelian methods. Representation theory and Fourier analysis play crucial roles in distinguishing group substructures and solving these problems efficiently. Which of the following statements most accurately explains why nonabelian Fourier transforms are necessary for solving the hidden subgroup problem in the affine group Ap? 1) Nonabelian Fourier transforms are required because abelian groups lack any subgroup structure. 2) Abelian Fourier transforms cannot distinguish between conjugate subgroups in Ap, whereas nonabelian Fourier analysis leverages high-dimensional representations to preserve subgroup distinctions. 3) Nonabelian Fourier transforms are only needed when the group order is a prime number. 4) Random bases always outperform adapted bases in nonabelian Fourier analysis for Ap. 5) The abelian Fourier transform is only ineffective if the group is cyclic. 6) Hidden shift problems never require nonabelian methods for solution. 7) In nonabelian groups, all subgroups are automatically trivial under Fourier analysis.

✓ Correct Answer:

The correct answer is 2) Abelian Fourier transforms cannot distinguish between conjugate subgroups in Ap, whereas nonabelian Fourier analysis leverages high-dimensional representations to preserve subgroup distinctions..

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cuts across these blocks, mixing diﬀerent“frequencies” ρ kand canceling out the useful information. To be more precise, a random basis is not adapted to the subgroup structure of Ap; it does not “know” that ρdecomposes into a direct sum of the ρkwhen restricted to Nq. Second, it is worth noting that for the values of qfor which we have an algorithm for full (as opposed to information-theoretic) reconstruction, namely q=p/polylog( p), a random basis works as well, since the /lscript1distance δbecomes 1 /polylog( p). Based on the strong evidence from representation theory that some bases are much betterfor computation than others, we conjecture that, for some families of groups, adaptedbases allow full reconstruction while random bases do not; but this remains an openquestion. Third, while we focused above on distinguishing conjugate subgroups from each other, in fact our proof shows that if q

Question 1306 multiple-choice

Post-quantum cryptography seeks secure algorithms that can withstand quantum attacks, with lattice-based digital signature schemes playing a crucial role in protecting resource-constrained IoT devices. Hardware/software co-design can significantly accelerate these cryptographic operations for practical deployment. Which optimization technique is specifically designed to reduce memory accesses and latency in the Number Theoretic Transform (NTT) on SIMD architectures for NTRU-based signature schemes? 1) Pipelined instruction scheduling 2) Layer merging 3) Hash-based signature aggregation 4) Asynchronous memory access 5) Key encapsulation merging 6) Differential power analysis 7) Polynomial ring switching

✓ Correct Answer:

The correct answer is 2) Layer merging.

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Title: PQNTRU: Acceleration of NTRU-Based Schemes via Customized Post-Quantum Processor Year: 2025 Paper ID: b877a6969536f14b9f72e57a76e7c420fa6e1af9 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b877a6969536f14b9f72e57a76e7c420fa6e1af9 Abstract: Post-quantum cryptography (PQC) has rapidly evolved in response to the emergence of quantum computers, with the US National Institute of Standards and Technology (NIST) selecting four finalist algorithms for PQC standardization in 2022, including the Falcon digital signature scheme. Hawk is currently the only lattice-based candidate in NIST Round 2 additional signatures. Falcon and Hawk are based on the NTRU lattice, offering compact signatures, fast generation, and verification suitable for deployment on resource-constrained Internet-of-Things (IoT) devices. Despite the popularity of ML-DSA and ML-KEM, research on NTRU-based schemes has been limited due to their complex algorithms and operations. Falcon and Hawk's performance remains constrained by the lack of parallel execution in crucial operations like the Number Theoretic Transform (NTT) and Fast Fourier Transform (FFT), with data dependency being a significant bottleneck. This paper enhances NTRU-based schemes Falcon and Hawk through hardware/software co-design on a customized Single-Instruction-Multiple-Data (SIMD) processor, proposing new SIMD hardware units and instructions to expedite these schemes along with software optimizations to boost performance. Our NTT optimization includes a novel layer merging technique for SIMD architecture to reduce memory accesses, and the use of modular algorithms (Signed Montgomery and Improved Plantard) targets various modulus data widths to enhance performance. We explore applying layer merging to accelerate fixed-point FFT at the SIMD instruction level and devise a dual-issue parser to streamline assembly code organization to maximize dual-issue utilization. A System-on-chip (SoC) architecture is devised to improve the practical application of the processor in real-world scenarios. Evaluation on 28 $nm$nm technology and field programmable gate array (FPGA) platform shows that our design and optimizations can increase the performance of Hawk signature generation and verification by over 7$\times$×.

Question 1307 multiple-choice

In topological group theory, pseudocompactness is an important property that relates to the boundedness of continuous functions and interacts intricately with the algebraic structure and cardinality of infinite Abelian groups. The classification of Abelian torsion groups that admit pseudocompact group topologies often hinges on set-theoretic concepts such as admissibility and specific group constructions. Which of the following statements is true regarding the existence of a pseudocompact group topology on Abelian groups constructed as direct sums over infinite cardinals? 1) Any direct sum of finite Abelian groups over a countable cardinal always admits a pseudocompact group topology. 2) Direct products of finite Abelian groups over any infinite cardinal always admit a pseudocompact group topology. 3) A direct sum of a finite Abelian group over an infinite admissible cardinal can admit a pseudocompact group topology. 4) The socle of any p-group must be trivial for the group to admit a pseudocompact topology. 5) Cofinal admissibility conditions are unnecessary in the classification of pseudocompact torsion Abelian groups. 6) Pseudocompactness of a group topology implies the group must be compact. 7) Any Abelian torsion group constructed from inadmissible cardinals admits a pseudocompact topology.

✓ Correct Answer:

The correct answer is 3) A direct sum of a finite Abelian group over an infinite admissible cardinal can admit a pseudocompact group topology..

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and H ^@ξθΑΡξ^@2κΡξ (with each Ρξ = F). Since \A\ < T and \A\ < y < 2K, there are disjoint subsets B and C of 2* such that A^ B and | | = |C| = y. The group 0$ec^ contains an isomorph S of @i=@ξ€ΒΓξ® S satisfies G' « G and //c 0i6XF5 s G' e 0i<2,cF5. Since N = \p~^(H) is G^-dense in FK, the same is true of tp~1(G/) (an isomorph of G). Now 1.1 and 1.4(b) complete the proof. D 3.12. Corollary. Let y be an infinite admissible cardinal, let Fbe afinite Abelian group with \F\ > l, and let G = 0yF0 @i1. That G e^ (a) is clear from the isomorphism G = 02,cF Faof 3.1. En route to the classification of those Abelian torsion groups which admit a pseudocompact group topology, we first consider/7-groups. We treat the general case fully in 3.19; the following example, included s an aid to the reader, is intended to illustrate the principles and the difficulties encountered in the general case. 3.14. Theorem. Let y be an inadmissible cardinal, let p e P, and define G0 = 0y Z (p) ® 02r Z (p2) , and G, = 02y Z (p) Θ 0y Z (p2) . Then (a) G0 admits a pseudocompact group topology, and (b) G! does not admit a pseudocompact group topology. Proof. (a) is clear from what has preceded. (For example: 2y = 2log(2v), so 3.11(ii) ensures G0 e ^(a) for every a such that 2y < α < 22V.) (b) The image of Gi under the map χ -» px is the group 0yZ(p), whose cardinality is inadmissible. Since the continuous image of a pseudocompact space is pseudocompact, evidently the group G! has no pseudocompact group topology. α Bereitgestellt von | Technische Informationsbibliothek Hannover Angemeldet Heruntergeladen am | 27.03.18 14:50 332 W. W. Comfort, D. Remus 3.15. Definition. A (possibly finite) sequence γΐ9 γ2, ..., yk ... of cardinal numbers satisfies the cofinal admissibility condition if for every k with yk inadmissible there is i>k such that y{ > yk and y{ is admissible. The following algebraic result summarizes what we need to know of the structure of Abelian/?-groups of bounded order. (As usual, the socle of a/?-group is the kernel of the homomorphism χ ->px.) The first part of 3.16 is Standard

Question 1308 multiple-choice

Quantum Teichmuller theory extends classical Teichmuller theory using quantum algebraic methods, often involving projective representations of mapping class groups of surfaces such as the torus. The modular group, denoted $SL_2(\mathbb{Z})$, plays a central role in the symmetries of the torus and in the study of noncommutative tori. Which statement accurately describes the mapping class group of the torus and its connection to the modular group? 1) It consists of all $2\times2$ integer matrices with determinant 1, forming the group $SL_2(\mathbb{Z})$. 2) It comprises only diagonal matrices over real numbers representing scaling symmetries. 3) It is isomorphic to the symmetric group $S_3$ of permutations. 4) It consists of upper triangular matrices with integer entries and determinant zero. 5) It is equivalent to $GL_2(\mathbb{R})$, the group of invertible $2\times2$ real matrices. 6) It includes all $2\times2$ integer matrices with determinant not equal to zero. 7) It is the set of permutation matrices of size $2\times2$ over integers.

✓ Correct Answer:

The correct answer is 1) It consists of all $2\times2$ integer matrices with determinant 1, forming the group $SL_2(\mathbb{Z})$..

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Title: A Projective Representation of the Modular Group. Year: 2020 Paper ID: 5823b11c7cfc41d22626188bfc31380bb275c5a6 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/5823b11c7cfc41d22626188bfc31380bb275c5a6 Abstract: Quantum Teichmuller theory assigns invariants to three-manifolds via projective representations of mapping class groups derived from the representation of a noncommutative torus. Here, we focus on a representation of the simplest non-commutative torus which remains fixed by all elements of the mapping class group of the torus, $SL_2(\mathbb{Z})$. Also known as the modular group. We use this representation to associate a matrix to each element of $SL_2(\mathbb{Z})$; we then compute the trace and determinant of the associated matrix.

Question 1309 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) play a fundamental role in computational group theory and cryptography, enabling efficient solutions to problems previously considered infeasible for classical computers. Recent advances have extended these algorithms to broader classes of groups and emphasized the importance of exactness and error management in quantum computation. Which property specifically enables the presented exact quantum algorithm for the Hidden Subgroup Problem to operate efficiently within nilpotent groups of bounded class when the group order consists solely of small prime factors? 1) The existence of a unique normal subgroup in all nilpotent groups 2) The presence of a trivial center in groups with small prime order 3) The availability of fast Fourier transforms over non-Abelian groups 4) The use of commutative subgroup decomposition only 5) The reliance on bounded-error quantum measurements 6) The structural restriction provided by bounded nilpotency class and small prime factors, allowing effective reduction to simpler quotient groups 7) The requirement for infinite group presentations in matrix groups

✓ Correct Answer:

The correct answer is 6) The structural restriction provided by bounded nilpotency class and small prime factors, allowing effective reduction to simpler quotient groups.

📚 Reference Text:

(and hence the HSP in the related semidirect product groups) appears to be quite popular in post-quantumcryptography, see, e.g., [ 8] by Castryck and V ander Meeren and [ 9] by Alagic and Russell. In [ 10], Bae and Lee propose a polynomial time solution to a continuous version of the hidden shift problem. A quantum procedure is exact if it returns a correct output (after a ﬁnal measurement) with probability one. Besides that exact quantum algorithms can be considered ascounterparts of deterministic classical methods, their measurement-free versions can serve as ingredients of larger unitary procedures. A further possible practical advantage is that, as they do not introduce further errors, using them can help control the errorof a procedure working an input with errors and also containing other, non-exact ingredients. The method of [ 1] has an exact version, so it is natural to ask that in which classes of groups can the HSP be solved by an exact quantum algorithm in polynomial time. Brassard and Hoyer [ 11] presented a polynomial time exact method that works in Z n 2. In [12], Cai and Qiu proposed a simpler efﬁcient exact method for Simon’s problem (a special, though arguably the hardest instance of the HSP in Zn 2). Efﬁcient exact algorithms with optimal query complexity for the HSP in Zn 2appeared independently in [13] by Bonnetain and in [ 14] by Wu et al. Mosca and Zalka in [ 15] proposed an efﬁcient exact solution of the discrete logarithm problem in cyclic groups of known order. An exact quantum algorithm for the HSP in Zn mkfor general mwas presented recently in [ 16], settling the case of abelian groups under the assumption that a multiple of the prime factors of the order of the group is known. In this paper we present an approach to solving the hidden subgroup problem in nilpotent groups that have nilpotency class O(1). Our main result is a polynomial time exact quantum algorithm for the HSP in such groups only having prime factors also of size O(1)in their order. An example for such groups is the group of the 4 nby 4 n block unitriangular matrices over the integers modulo 10dwith block size n. These are the 4 nby 4nmatrices divided into 16 rectangular blocks of size nwith integer entries, considered modulo 10d, whose 4 diagonal blocks are equal to the nbynidentity matrix, whereas the 6 blocks below the diagonal have zero entries. These matrices can be represented by 6 dn2decimal digits describing the entries of the blocks above the 123 Zero sum subsequences and hidden subgroups Page 3 of 23 14 diagonal. The order of the group is 106dn2, and it is nilpotent of class 3. To obtain sufﬁciently general results, we assume that the group Gis given as a black-box group with unique encoding. The main strategy of our algorithm is essentially a reduction to instances of the hidden subgroup problem in quotient groups of subgroups of G.W e choose an input model

Question 1310 multiple-choice

In theoretical computer science, the classification of Boolean operations and their computational complexity is closely linked to algebraic structures known as clones, with Post's Lattice providing a comprehensive organization of these clones. Quantum algorithms such as Simon's have demonstrated significant speedups for certain problems that are hard classically, especially in the context of operations defined on two-element domains. Which property of Boolean clones determines whether the Hidden Kernel Problem (HKP) admits a polynomial-time quantum solution but not a classical one? 1) The presence of the XOR function within the clone 2) The inclusion of all constant functions in the clone 3) The clone being closed under negation 4) The existence of monotone functions in the clone 5) The algebraic structure of the clone as classified by Post's Lattice 6) The clone containing only unary operations 7) The absence of idempotent operations in the clone

✓ Correct Answer:

The correct answer is 5) The algebraic structure of the clone as classified by Post's Lattice.

📚 Reference Text:

follows from the fact that every homo- morphismofDnisalsoahomomorphismof BnwhenClo (B) ⊆ Clo(D)(⇔B/√re⌋edesequalD). The observation implies that HKP (Bn) is equivalent to HKP (Dn)whenever Clo (B)=Clo(D). In- stead of considering all possible 2-element algebras B, we therefore consider only those algebras with distinct clones ofterm operations. The task of describing all clones of operations on a 2- elementdomain(alsoknownasbooleanclones),wasfamously undertakenin1941byEmilPost[ 16].Orderedbyinclusion, the set of all boolean clones is known as Post’s Lattice , and hasaparticularlyregularstructure(seeFigure 1).Theoper- ationsgiveninFigure 1areintermsofthefamiliarbooleansφ(n)=|0/a\}⌊ra⌋ketri}htH H ... |0/a\}⌊ra⌋ketri}ht |0/a\}⌊ra⌋ketri}ht ... |0/a\}⌊ra⌋ketri}htH /hatwideφH... ...n m Figure2. Thecircuitsφ(n)ofDeﬁnition 3.2. operations,withthepossibleexceptionofthe majorityfunc- tion,deﬁned maj(x,/y.alt,z)≔/parenleftbigx∧/y.alt/parenrightbig∨(x∧z)∨(/y.alt∧z). Animportant propertyof thisoperation is that x=maj(/y.alt,x,x)=maj(x,/y.alt,x)=maj(x,x,/y.alt). We end this subsection by deﬁning the quantum circuit whichwillyieldapolynomial-timequantumsolutiontoHKP (Bn) for many of the algebras we consider. This circuit is essen- tially thesame asSimon’s original circuit [ 19]. Deﬁnition 3.2. Letφ:{0,1}n→ {0,1}m, and deﬁne the 2n+m-dimensionalunitarytransformation /hatwideφbyitsactionon basis vectors, /hatwideφ/parenleftbig|x/a\}⌊ra⌋ketri}ht ⊗ |/y.alt/a\}⌊ra⌋ketri}ht/parenrightbig≔|x/a\}⌊ra⌋ketri}ht ⊗ |/y.alt+φ(x)/a\}⌊ra⌋ketri}ht, wherex∈ {0,1}n,/y.alt∈ {0,1}m, and “+” is componentwise addition modulo2.Thequantumcircuit sφ(n)isthe circuit sφ(n)≔/parenleftbigH⊗n⊗Im/parenrightbig/hatwideφ/parenleftbigH⊗n⊗Im/parenrightbig ofsizen+m,wheremeasurement isperformedon theﬁrst nqubits. The circuit sφ(n)is given graphically in Figure 2. Notethatwecanalwaystake m≤n,sothecircuitisofsize Θ(n). 4 The HKP and Post’sLattice: Algorithms In this section we present positive results for solving the HiddenKernelProblemforPost’sLatticeinpolynomialtime usingclassicalandquantumalgorithms.Themainquantum result is Theorem 4.1and the main classical result is Corol- lary4.2,stated below. Theorem4.1. LetBbea2-elementalgebrasuch thatoneof 1.MPT∞ 0/√re⌋edesequalB, 2.MPT∞ 1/√re⌋edesequalB, or 3.AP /√re⌋edesequalB holds and letA=Bn. The quantum circuit sφ(n)of Deﬁni- tion3.2solvesHKP(A)withprobability 1−1/τinΘ(n+lg(τ)) iterations. Corollary4.2. LetBbea2-elementalgebrasuchthatoneof LICS’20,July8–11,2020,Saarbrücken,Germany Ma/t_thew Moore and TaylorWalenczyk MPT∞ 0MPT∞ 1⊤ A AP0 ≈Z2 APDM :classical and quantum poly-time solutions exist:only quantum poly-time solution (super-polynomial speedup) Figure 3. The fragment of Post’s Lattice for which the HKP is solvable in polynomial-time using a quantum algo- rithm (whole diagram). Bold lines indicate clones exhibit- ingasuper-polynomialspeedupwith aquantumalgorithm (i.e. no classical polynomial-time algorithm exists, see The - orem5.1),whileplain linesindicate clonesforwhichaclas- sical polynomial-timealgorithmexists. 1.MPT∞ 0/√re⌋edesequalB, 2.MPT∞ 1/√re⌋edesequalB, or 3.DM /√re⌋edesequalB holds and letA=Bn. ThenHKP(A)is solvable classically in O(n)time. Thebestwaytosummarizethesetworesultsisintermsof Post’sLattice,seeFigure 3.Inparticular,powersofalgebras whoseclonesarecontainedintheintervalbetween APand Ahave HKP solvable in polynomial-timeusing a quantum algorithm, but not a classical one (see Theorem 5.1). Com- bined with the results of the next section, this provides a complete quantum and classical classiﬁcation of the algo- rithmic complexity of the HKP for powers of 2-element al- gebras. Theremainderofthesectionisdevoted toproving Theo- rem4.1andCorollary 4.2.Weproceedbyprovingaseriesof lemmas,beginningﬁrst with two lemmasleading upto the famousresult of Simon[ 19],whichwe phrase andprove in termsofthe Hidden Kernel Problem. Lemma 4.3. LetD≤/parenleftbigZn 2/parenrightbig2. Forall/afii10069.ital∈/parenleftbigZn 2/parenrightbig2, /summationdisplay.1 d∈D(−1)/afii10069.ital·d=/braceleftBigg |D|if/afii10069.ital·d=0for alld∈D, 0otherwise, where/afii10069.ital·dis the dot product modulo 2 of /afii10069.italandd(regarded asZ2-vectors).Proof.Deﬁne sets D0=/braceleftbig d∈D|/afii10069.ital·d=0/bracerightbig andD1=/braceleftbig d∈D|/afii10069.ital·d=1/bracerightbig . Observethatthesesetsaredisjointand D=D0∪D1,sothey partitionD.Thus,/summationdisplay.1 d∈D(−1)/afii10069.ital·d=|D0| − |D1|. Itisnothardtoseethat D0isasubgroupofD.IfD1=∅,then thesummationisequalto |D|and/afii10069.ital·d=0foralld∈D.Let us assume therefore that D1/nequal∅and letd∈D1. It follows thatd+D0⊆D1andd+D1⊆D0. HenceD1=d+D0 and soD1is a coset of D0. Therefore |D1|=|D0|and the summation equals0. /square Lemma 4.4. Letθbe a congruence of Zn 2and letX⊆θ. The setXgeneratesθwith probability 1−1/τ, where|X| ∈ Θ(n+lg(τ)). Speciﬁcally, if |X|=2nthen the probability of generation is 1−2−n. Proof.Letψbeanarbitrarycongruenceof Zn 2.Wehave/afii10069.italψ h ifandonlyif (/afii10069.ital+h)ψ0,sothecongruenceclassof ψcontain- ing 0∈Zn 2uniquely characterizes ψ. Call this congruence classHψ, Hψ=/braceleftbig /afii10069.ital∈Zn 2|/afii10069.italψ0/bracerightbig . It’snothardtoseethat HψisasubgroupofZn 2.Inthisman- nersubgroups(normalsubgroups,ingeneral)correspondto congruences. LetX0={x+/y.alt| (x,/y.alt) ∈X}and note that X0is con- tainedinthecongruenceclassof0in θ.Fromtheparagraph above, we have Cg (X)/nequalθif

Question 1311 multiple-choice

In symplectic geometry and mathematical physics, reduction by symmetry is a powerful method for simplifying phase spaces in systems with group actions. The Kirillov Poisson structure plays a central role in the study of reduced phase spaces related to Lie groups and their Lie algebras. When a Lie group G acts freely and properly on its cotangent bundle T*G by left translation, which of the following statements correctly describes the resulting quotient space T*G/G and its associated Poisson structure? 1) T*G/G is isomorphic to the space of left-invariant vector fields on G with a canonical symplectic structure. 2) T*G/G is a trivial fiber bundle over G with the standard Poisson bracket from T*G. 3) T*G/G is isomorphic to the dual of the Lie algebra g*, inheriting the Kirillov Poisson structure induced from T*G. 4) T*G/G is a contractible space with no nontrivial Poisson structure. 5) T*G/G is naturally identified with the Lie algebra g itself, carrying the Lie bracket as a Poisson structure. 6) T*G/G corresponds to the set of G-orbits in T*G, but lacks any induced Poisson structure. 7) T*G/G is isomorphic to the space of G-invariant functions on T*G, with a trivial Poisson bracket.

✓ Correct Answer:

The correct answer is 3) T*G/G is isomorphic to the dual of the Lie algebra g*, inheriting the Kirillov Poisson structure induced from T*G..

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(i.e. Φ∗ gH=Hwhere as usual Φ∗ gH=H◦Φg), then (Φ g◦γ)(t) is again a solution. If we do not want to consider solutions that can be transformed int o each other in this way to be essentially diﬀerent we are led to consider the space M/G as the ’true’ phase space of the system. The only observables (i.e. functions on M) that one is concerned with are those which are themselves G-invariant and therefo re descend to observables onM/G. Denote the set of smooth G-invariant functions on MbyO, and let then φ,ψ∈O. Using the fact that Φ gpreserves the Poisson structure for all g∈Gand thatφandψareG-invariant we can easily deduce that {φ,ψ}is also an element of O, i.e.Ois a Poisson subalgebra of C∞(M). Letπ:M→M/G be the canonical projection. The pullback map π∗:C∞(M/G)→Ois in fact an isomorphism. It 5 assigns to a function ˆφonM/G the function ˆφ◦πonMwhich is constant along the G-orbits. What one wants is now to deﬁne a Poisson structure {.,.}∗onC∞(M/G) such that the Poisson algebras ( C∞(M/G),{.,.}∗) and (O,{.,.}) are isomorphic. This would mean that all ( G-invariant) information of the original phase space is tran sferred to the Poisson algebra ( C∞(M/G),{.,.}∗). Therefore deﬁne {ˆφ,ˆψ}∗= (π∗)−1{π∗ˆφ,π∗ˆψ} (2.2) for all ˆφ,ˆψ∈C∞(M/G). Obviously π∗is a Poisson algebra isomorphism between the two Poisson algebras. So starting from a G-invariant theory we have arrived at a formulation in which all essential information is contain ed and all redundancy has been eliminated. Consider now a G-invariant theory in which the group manifold itself is the c onﬁg- uration space (for example a particle moving on Gwith a Hamiltonian that is invariant underG). The phase space of such a system is of course T∗G, the cotangent bundle of G. The left action of the group on itself, denoted by Lg, induces a left G−actionL∗ gon T∗Gwhich is in fact a Poisson action w.r.t. the canonical Poisso n structure on T∗G (T∗Gcarries a natural Poisson structure because it is a cotangen t bundle). Therefore we can do what we did before and extract the G-invariant piece of the theory by going to the quotient manifold T∗G/G (2.3) Obviously this space is nothing but the space of left invaria nt one forms which means that it is isomorphic to g∗, the dual of the Lie algebra of G. The canonical projection π:T∗G→g∗(2.4) is therefore simply the pullback to the ﬁber over the identit y element of Gby means of left translation. As before the Poisson structure on T∗Ginduces a Poisson structure ong∗. This Poisson structure on g∗is called the ’Kirillov’ Poisson structure. It is possible to obtain (by explicit calculation) a more con crete formula for the Kirillov Poisson structure [15]. It reads {F,G}(ξ) =ξ([gradξF,gradξG]) (2.5) whereFandGare smooth functions on g∗,ξ∈g∗and gradξF∈gis determined by d dǫF(ξ+ǫη)|ǫ=0=η(gradξF) (2.6) for allη∈g∗. In this paper we will always take Gto be a simple group which means that the Cartan-Killing form ( .,.) on its Lie algebra is non-degenerate. This allows us 6 to identify the Lie algebra gwith its dual,

Question 1312 multiple-choice

In higher representation theory and quantum algebra, unitary 2-representations and their invariants play a pivotal role in the study of topological quantum field theories and symmetry. Fusion categories and their pivotal structures are essential for defining categorical traces and constructing topological invariants. Which statement accurately describes a key relationship between the category of transformations of the identity on the 2-category of unitary 2-representations of a finite group and equivariant vector bundles? 1) It is equivalent to the category of all vector bundles over the group without equivariance. 2) It is equivalent to the category of conjugation equivariant vector bundles over the group, equipped with the fusion tensor product. 3) It is equivalent to the category of commutative algebras graded by the group. 4) It is equivalent to the category of symmetric monoidal functors from the group to vector spaces. 5) It is equivalent to the category of projective group representations. 6) It is equivalent to the category of pointed sets with a group action. 7) It is equivalent to the category of Galois extensions of the ground field.

✓ Correct Answer:

The correct answer is 2) It is equivalent to the category of conjugation equivariant vector bundles over the group, equipped with the fusion tensor product..

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Title: On unitary 2-representations of finite groups and topological quantum field theory Year: 2009 Paper ID: 08c0445f73d8e0b250fa02e64ea0f7d0245c40b9 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/08c0445f73d8e0b250fa02e64ea0f7d0245c40b9 Abstract: This thesis contains various results on unitary 2-representations of finite groups and their 2-characters, as well as on pivotal structures for fusion categories. The motivation is extended topological quantum field theory (TQFT), where the 2-category of unitary 2-representations of a finite group is thought of as the `2-category assigned to the point' in the untwisted finite group model. The first result is that the braided monoidal category of transformations of the identity on the 2-category of unitary 2-representations of a finite group computes as the category of conjugation equivariant vector bundles over the group equipped with the fusion tensor product. This result is consistent with the extended TQFT hypotheses of Baez and Dolan, since it establishes that the category assigned to the circle can be obtained as the `higher trace of the identity' of the 2-category assigned to the point. The second result is about 2-characters of 2-representations, a concept which has been introduced independently by Ganter and Kapranov. It is shown that the 2-character of a unitary 2-representation can be made functorial with respect to morphisms of 2-representations, and that in fact the 2-character is a unitarily fully faithful functor from the complexified Grothendieck category of unitary 2-representations to the category of unitary equivariant vector bundles over the group. The final result is about pivotal structures on fusion categories, with a view towards a conjecture made by Etingof, Nikshych and Ostrik. It is shown that a pivotal structure on a fusion category cannot exist unless certain involutions on the hom-sets are plus or minus the identity map, in which case a pivotal structure is the same thing as a twisted monoidal natural transformation of the identity functor on the category. Moreover the pivotal structure can be made spherical if and only if these signs can be removed.

Question 1313 multiple-choice

Quantum algorithms have shown significant promise in addressing the Hidden Subgroup Problem (HSP), a central challenge in computational group theory and quantum computation. The nonabelian case of HSP is especially important due to its connection with hard problems like graph isomorphism and its role in advancing quantum algorithmic techniques. Which group family is known to admit efficient quantum algorithms for the nonabelian Hidden Subgroup Problem? 1) Wreath products such as $ \mathbb{Z}_2^k \wr \mathbb{Z}_2 $ 2) Arbitrary symmetric groups $ S_n $ 3) All abelian groups of unbounded exponent 4) Nonabelian simple groups 5) Infinite nonabelian groups 6) Groups with nontrivial center only 7) Nilpotent groups of unbounded class

✓ Correct Answer:

The correct answer is 1) Wreath products such as $ \mathbb{Z}_2^k \wr \mathbb{Z}_2 $.

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yield an eﬃcient quantum algorithm for graph automorphism (see, e.g., Jozsa [16] for a review). This was the impetus behind the development ofthe ﬁrst nonabelian quantum Fourier transform [2] and is, in part, the reason that thenonabelian HSP has remained such an active area of research in quantum algorithms. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 940 MOORE, ROCKMORE, RUSSELL, AND SCHULMAN In general, we will say that the HSP for a family of groups Ghas a Fourier sampling algorithm if a procedure similar to that outlined above works. Speciﬁcally, the algorithm prepares a coset state as deﬁned above, |cH/angbracketright=1/radicalbig |H|/summationdisplay h∈H|ch/angbracketright, over a random coset cHof the hidden subgroup H, computes the (quantum) Fourier transform of this state, and measures the result. After a polynomial number of suchtrials, a polynomial amount of classical computation, and, perhaps, a polynomialnumber of classical queries to the function hto conﬁrm the result, the algorithm produces a set of generators for the subgroup Hwith high probability. When Gis abelian, measuring a state’s Fourier transform has a clear meaning: one observes the frequency χwith probability equal to the squared magnitude of the transform at that frequency. In the case where Gis anonabelian group, however, in order to deﬁne a full measurement it is necessary to select bases for each repre-sentation of G. (We explain this in more detail below.) The subject of this article is the relationship between this choice of basis and the information gleaned from themeasurement: are some bases more useful for computation than others? Since we are typically interested in exponentially large groups, we will take the size of our input to be n= log |G|. Throughout, “polynomial” means polynomial in nand thus polylogarithmic in |G|. 1.1. Nonabelian hidden subgroup problems. Although a number of inter- esting results have been obtained on the nonabelian HSP, the groups for which eﬃcientsolutions are known remain woefully few. On the positive side, Roetteler and Beth[22] give an algorithm for the wreath product Z k 2≀Z2. Ivanyos, Magniez, and Santha [15] extend this to the more general case of semidirect products K/multicloseleftZk 2, where Kis of polynomial size, and also give an algorithm for groups whose commutator subgroup isof polynomial size. Friedl, Ivanyos, Magniez, Santha, and Sen [8] solve a problem theycall hidden translation and thus generalize this further to what they call “smoothlysolvable” groups: these are solvable groups whose derived series is of constant lengthand whose abelian factors are each the direct product of an abelian group of boundedexponent and one of polynomial size. (See also section 8.) In another vein, Ettinger and Høyer [6] show that the HSP is solvable for the dihedral groups in an information-theoretic sense; namely, a polynomial number of quantum queries to the function oracle gives enough information to reconstruct thesubgroup, but the best known reconstruction algorithm takes exponential time. Moregenerally, Ettinger, Høyer, and Knill [7] show that for arbitrary groups the HSP can be solved information-theoretically with a ﬁnite number of quantum queries. However,their algorithm

Question 1314 multiple-choice

In algebraic geometry, group schemes provide a framework for studying groups whose elements are geometric objects, such as points of schemes, with group operations given by morphisms. Abelian varieties are a special class of projective group schemes central to number theory and geometry. Which property is essential for a scheme A to be a commutative group scheme over a field k, as established via morphisms m (multiplication), -1A (inverse), and OA (identity)? 1) The scheme A must have a nontrivial automorphism group. 2) Every morphism from A to another scheme must be an isomorphism. 3) The morphisms m, -1A, and OA must satisfy the group axioms diagrammatically, including associativity, commutativity, unit, and inverse. 4) The underlying topological space of A must be contractible. 5) A must be integral over its coordinate ring. 6) The scheme A must be affine and connected. 7) All graded extensions of the coordinate ring of A must be finite.

✓ Correct Answer:

The correct answer is 3) The morphisms m, -1A, and OA must satisfy the group axioms diagrammatically, including associativity, commutativity, unit, and inverse..

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By Lemma 1.2.1, there is a 0eRv such that θ(x) Φ 0. Thus we have an evaluation homomorphism Spec. (Rθ). This gives us a fc-valued point Px of A. Px depends only on the class of x modulo Γ, and we have defined a function: https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press ABELIAN VARIETIES 75 φ:Gg/Γ->Ak The following standard facts will be needed later on. LEMMA 1.3.1. Let NczM be graded rings with M integral over N. Then the open sets Spec. (Mn), n e Nif i > 0, cover Proj. (M) and the maps Spec. (Mn) -> Spec. (Nn) piece together to give a morphism Proj. (M) ->Proj.(N). LEMMA 1.3.2. Let M and N be graded algebras and φ19φ2' Proj. (M) —•Proj. (N) morphisms. Suppose further that φι and 0, and M is a domain. Then φι — ψ2. We are now ready to interpret the results of the last two sections geometrically. THEOREM 1.3.3. Let β\R' -*Rf be the map θ{X, Y) -> Θ{XY,XY~ι). Then: (1) Rf is integral over β(R;). (2) A' = Proj. (R') is the scheme theoretic product A x A of A with itself over k. (3) β induces a morphism ^5*:AχA->AχA. (4) The map Ak x Ak-> Ak x Ak induced by β* takes (Px,Py) to \*xy> * xy-d Proof. Assertions (1) and (2) come from Propositions 1.1.5 and 1.1.4. Lemma 1.3.1 and (1) give a morphism Af -»A' induced by β. Since Af identifies with A x A we get the morphism β* of (3), and (4) follows from the definition of β. With the notations above let: 1) m: A x A > A be the morphism A x A > A x A -^> A where πx is projection on the first factor. 2) — 1A: A -»A be the morphism induced by the automorphism θiX) -* ΘiX-1) of R. 3) OΛ: A > A be the morphism A > Spec, (k) > A where e is the λ -valued point Pih...tV. THEOREM 1.3.4. With the operations defined above A is a commuta- https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press 76 HORACIO TAPIA-RECILLAS tive group scheme over k. The map φ: Gg/Γ —> Ak constructed at the beginning of this section is a group homomorphism. Proof (In outline). To show that A is a commutative group scheme we must verify the commutativity of certain diagrams expressing the associative and commutative law, and the existence of a unit and in- verse. For example, for associativity we must show that the morphisms m o (id^ x m) and m o (m x idJ from A x A x A -> A are the same. To do this we choose affine open subsets U and V on A x A x A and A such that m o (id x m) and m o (m x id) take U

Question 1315 multiple-choice

Quantum annealing offers a novel approach to simulating lattice gauge theories by encoding physical systems on discrete computational architectures. Non-Abelian gauge groups, such as dihedral groups, introduce additional complexity through their non-commuting symmetry operations. Which method enables simulation of time evolution in lattice gauge theories by encoding the entire history of a quantum system in an extended Hilbert space? 1) Trotter-Suzuki decomposition 2) Quantum Monte Carlo sampling 3) Tensor network renormalization 4) Variational quantum eigensolver 5) Operator splitting technique 6) Adiabatic quantum state transfer 7) Feynman clock formalism

✓ Correct Answer:

The correct answer is 7) Feynman clock formalism.

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Title: Dihedral lattice gauge theories on a quantum annealer Year: 2022 Paper ID: 0bee4a3cbc9b825f89a4b81c02d64b664fdd2904 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0bee4a3cbc9b825f89a4b81c02d64b664fdd2904 Abstract: We study lattice gauge theory with discrete, non-Abelian gauge groups. We extend the formalism of previous studies on D-Wave’s quantum annealer as a computing platform to finite, simply reducible gauge groups. As an example, we use the dihedral group D n $D_{n}$ with n = 3 , 4 $n=3,4$ on a two plaquette ladder for which we provide proof-of-principle calculations of the ground-state and employ the known time evolution formalism with Feynman clock states.

Question 1316 multiple-choice

In supersymmetric theories of particle physics, the nature of the lightest supersymmetric particle (LSP) and its interactions are key to understanding dark matter production in the early universe. Non-thermal mechanisms, such as moduli reheating, can significantly affect the relic abundance of candidate particles. Which strategy is proposed to resolve the overproduction of neutral MSSM LSPs in standard moduli reheating scenarios? 1) Increasing the reheating temperature far above the modulus mass scale 2) Extending the MSSM by adding a hidden U(1) sector 3) Enhancing bino mixing to further suppress annihilation rates 4) Relying solely on thermal freeze-out processes 5) Using sneutrino LSPs exclusively as dark matter candidates 6) Decreasing the decay coefficient c to minimize particle production 7) Tuning the branching ratio to gravitinos to unity

✓ Correct Answer:

The correct answer is 2) Extending the MSSM by adding a hidden U(1) sector.

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with very low relic density along the m 0 1= 500 GeV contour corresponding to an s-channelA0pseudoscalar resonance. In general, for m'm3=2, we nd that a Higgsino-like LSP also tends to produce too much dark matter when it is created in moduli reheating. As for the wino, this can be 7We also fail to nd any such points for c= 100 andm'= 2m3=2. 8The details of our indirect detection analysis will be presented in the next section. 10 m3/2 [100 T e V] µ/M 2mϕ/m3/2= 100 NFW Einasto CoredRC= 0.1 kp c Con tr.γ= 1.2 12345678910 0.1 1-1-0.500.511.52 log10Ω/tildewideN1/Ωcdm1000700 400 100 LEP Excl.TRH T fo; to the right TRH< T foand the production is non-thermal. The remaining lines correspond to bounds from ID for di erent galactic DM distributions, and the area below and to the right of these lines is excluded. avoided for larger values of m'=m3=2, as demonstrated by Fig. 1, although one must still ensure that the very heavy modulus does not decay signi cantly to gravitinos. These results can be extended to an arbitrary MSSM neutralino LSP. In general, mixing with a bino will further suppress the annihilation cross section, leading to an overproduction of dark matter for m'm3=2. The only loophole we can see is a very strong enhancement of the annihilation from a resonance or coannihilation [79]. This requires a very close mass degeneracy either between 2 m and the mass of the resonant state, or between m and the coannihilating state, with coannihilation further suppressed at low reheating temperatures. The only other viable LSP candidates in the MSSM are the sneutrinos. These annihilate about as eciently as a Higgsino-like LSP [80], and therefore also tend to be overproduced. The MSSM sneutrinos also have a very large scattering cross section with nuclei, and bounds from direct detection permit them to be only a small fraction of the total DM density [81]. Having expanded slightly on the ndings of Ref. [51], we conclude that a neutral MSSM LSP is typically overproduced in moduli reheating unless m' m3=2(with tiny BR 3=2), N 1, or the decay coecient c 100 is very large. None of these features appears to be generic in string compacti cations. We call this the moduli-induced MSSM LSP problem . For this reason, we turn next to extensions of the MSSM with more general LSP candidates that can potentially avoid this problem. 11 IV. VARIATION #1: HIDDEN U(1) The rst extension of the MSSM that we consider consists of a hidden U(1)xvector multipletXand a pair of hidden chiral

Question 1317 multiple-choice

Matrix groups over modular rings play an important role in algebraic geometry and cryptography, particularly in the study of elliptic curves and their automorphisms. Understanding the structure and decomposition of matrices in GL₂(Z/NZ) is crucial for analyzing group actions and implementing efficient cryptographic algorithms. Which statement accurately describes the stabilizer of a point of order N on an elliptic curve with respect to matrix groups over Z/NZ? 1) It is always the group of lower triangular matrices in GL₂(Z/NZ). 2) It is always the cyclic subgroup generated by a permutation matrix. 3) It is equivalent to the group of diagonal matrices for all N. 4) It consists solely of the identity matrix for composite N. 5) It is a conjugate of the group of upper triangular matrices, known as a Borel subgroup. 6) It is the set of matrices with trace zero in GL₂(Z/NZ). 7) It is the entire GL₂(Z/NZ) group for prime N.

✓ Correct Answer:

The correct answer is 5) It is a conjugate of the group of upper triangular matrices, known as a Borel subgroup..

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algebra s.Journal of Symbolic Computation , 9(3):355–373, 1990. [RS06] Alexander Rostovtsev and Anton Stolbunov. Public-key cry ptosys- tem based on isogenies. IACR Cryptology ePrint Archive , 2006:145, 2006. [Sho97] Peter W. Shor. Polynomial-time algorithms for prime factoriz ation anddiscretelogarithmsonaquantumcomputer. SIAM J. Comput. , 26(5):1484–1509, 1997. [Sil09] Joseph H Silverman. The arithmetic of elliptic curves , volume 106. Springer Science & Business Media, 2009. [The22] The Sage Developers. SageMath, the Sage Mathematics Software System (Version 9.4) , 2022.https://www.sagemath.org . [Voi18] John Voight. Quaternion algebras. preprint, 13:23–24, 2018. [Wat69] William C Waterhouse. Abelian varieties over ﬁnite ﬁelds. In An- nales scientiﬁques de l’ ´Ecole Normale Sup´ erieure , volume 2, pages 521–560, 1969. [Wes21] Benjamin Wesolowski. The supersingular isogeny path and en do- morphism ring problems areequivalent. CryptologyePrint Archive, Report 2021/919, 2021. https://ia.cr/2021/919 . 32 M. Chen, M. Imran, G.Ivanyos, P.Kutas, A.Leroux, C. Petit A Proof of Theorem 5.10 for arbitrary N The main diﬃculty in the general case is that there is no PLUdecomposition in GL2(Z/NZ). A simple example is the case where N= 6 and the matrix we need to decompose is/parenleftbigg 2 3 3 2/parenrightbigg . This happens because the ﬁrst column of an invertible matrix (in this c ase with determinant 1) may contain no invertible element. However, if one takes a matrix whose top left corner is an invertible e lement, then it can be written as a product of a lower and an upper triangular matrix (which then can be further decomposed as in Theorem 5.10). Let/parenleftbigga b c d/parenrightbigg be an arbitrary matrix in GL 2(Z/NZ). Since it is invertible there existsk∈Z/NZsuch thata+kcis invertible and such a kcan be computed eﬃciently (using Chinese remaindering). This implies that/parenleftbigg1k 0 1/parenrightbigg/parenleftbigga b c d/parenrightbigg has an invertible top left corner (and is invertible as a matrix) hence can be w ritten asLDU. This implies that/parenleftbigga b c d/parenrightbigg can be written as U1LDU2whereU1,U2are upperunitriangularmatrices.Finally,forgeneral Nthegroupoflowerandupper triangularmatricesareno longercyclicbut onecancompute genera torsofZ/NZ an then everything else from the proof carries over. B Elementary proof that the stabilizers are Borel subgroups Proposition B.1. LetXbe a point of order NinEand let us denote the stabilizer of E/XbyGX. ThenGXis a conjugate of the group of upper triangular matrices. Proof.LetP,Qbe some basis of E[N]. ThenX=αP+βQand there exists integersx,ysuch thatαx+βy≡1 (modN), otherwise Xwould have lower order. One has the following simple identity: /parenleftbiggα−y β x/parenrightbigg ·(1,0)T= (α,β)T. Now the condition that σ(A) =λAimplies that /parenleftbigg a b c d/parenrightbigg ·/parenleftbigg α−y β x/parenrightbigg ·(1,0)T=λ/parenleftbigg α−y β x/parenrightbigg ·(1,0)T. Now one has that /parenleftbigg α−y β x/parenrightbigg−1 ·/parenleftbigg ab c d/parenrightbigg ·/parenleftbigg α−y β x/parenrightbigg ·(1,0)T=λ·(1,0)T. Title Suppressed Due to Excessive Length 33 Since the middle matrix runs through conjugation is an automorphism one has to look at the problem of ﬁnding the matrices/parenleftbigg xy z u/parenrightbigg for which/parenleftbigg xy z u/parenrightbigg ·(1,0)T= λ(1,0)T.From comparing coeﬃcients one gets that z= 0 and all those matrices also satisfy this condition. This provesthat GAis just the the conjugate of upper triangular matrices by the matrix/parenleftbiggα−y β x/parenrightbigg

Question 1318 multiple-choice

Quantum error correction is essential for mitigating noise in quantum computers, with Clifford circuits playing a pivotal role due to their special properties. Advanced optimization algorithms and symmetry exploitation are increasingly used to improve the efficiency and scalability of error correction schemes in these systems. Which optimization strategy most effectively reduces the computational complexity when searching for optimal verification sequences in Clifford circuit error correction by leveraging symmetry to ignore equivalent solutions? 1) Randomized search with adaptive thresholding 2) Genetic algorithms with dynamic mutation rates 3) Gradient descent with regularization 4) Exhaustive search over all possible sequences 5) Greedy search combined with resource state selection 6) Simulated annealing with sequence pruning 7) Identification of automorphisms in the search space

✓ Correct Answer:

The correct answer is 7) Identification of automorphisms in the search space.

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Title: Optimized Clifford Noise Reduction: Theory, Simulations and Experiments Year: 2025 Paper ID: 23c397e67454fe0347aeffaf4e96a8e54df4fc4c Source: semantic-scholar URL: https://www.semanticscholar.org/paper/23c397e67454fe0347aeffaf4e96a8e54df4fc4c Abstract: We propose several optimizations of the CliNR partial error correction scheme which implements Clifford circuits by consuming a resource state. Errors are corrected by measuring a sequence of Pauli operators that we refer to as the verification sequence. We first propose a global optimization algorithm searching for a verification sequence resulting in a low logical error rate using tabu search. Then, we introduce a proxy for the logical error rate which is easier to evaluate and we design a two-step optimization algorithm. First, a verification sequence minimizing the proxy is computed, then this sequence is refined by reintroducing the logical error rate. Finally, we identify a large group of automorphisms of the search space which preserve the proxy and we use this symmetry to reduce the size of the search space. This results in a 168 × (respectively 20,160 ×) reduction of the size of the search space for the optimization of verification sequences with three (respectively four) Pauli operators. Our numerical simulations for 20-qubit Clifford circuits with size 400 under the ion chain model show that our optimization algorithms improve the performance of CliNR by 25% and that the two-step optimization achieves the same results as the global optimization with 64% fewer evaluations of the logical error rate. Finally, we perform experiments on a 36-qubit trapped ion quantum computer, without mid-circuit measurements, showing that the CZNR variant of CliNR is at breakeven.

Question 1319 multiple-choice

Quantum optimal control utilizes numerical optimization techniques to design control pulses that implement quantum gates with high fidelity and minimal duration. In such approaches, understanding the trade-offs between gate error and control time is crucial for achieving time-optimal gate implementations in physical systems. Which of the following best describes the primary function of the Pareto Front Tracking (PFT) method in quantum control optimization for gate implementation? 1) It randomly generates control pulses to maximize the diversity of initial conditions. 2) It efficiently maps the trade-off curve between gate error and control time by using optimized pulses at one control time as starting points for neighboring times. 3) It restricts the control field amplitude to maintain hardware feasibility during optimization. 4) It analytically determines the global phase associated with quantum gates in spin-1 systems. 5) It increases the number of time steps to stabilize numerical calculations during optimization. 6) It guarantees convergence to the global minimum for any initial pulse guess. 7) It eliminates the influence of the Hamiltonian form on achievable control pulse solutions.

✓ Correct Answer:

The correct answer is 2) It efficiently maps the trade-off curve between gate error and control time by using optimized pulses at one control time as starting points for neighboring times..

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operator sU [26]: () ()k klit k l k l si t i tkl sk l k l kli t I e if U eeut i t I ifit                            where  are the eigenvalues of sU and spin projection operator I describes the interaction with the control field in Hamiltonian (11) along the respective axis ,xy . Since the gradient meth od converges to a local minimum , which can be differ ent from the global one, the calculations must be rep eated for many times with different guesses of initial 9 pulse (0)u in (24) . In our calculations f or S = 500 , the initial pulse shape was formed by setting random amplitude s in the range [ −3, 3] at each 50th point of the time interval [0, T] with the further spline interpolati on at the rest points. After that, during the optimization run , the control field amplitude is free to change without any constraints . The run is stopped when the gate error is ( ) 810kJ or ( 1) ( ) 6 ( 1)10k k kJ J J   . To determine the exact critical time for the different global phases of the gate, we calculate d the dependence of the gate error on control time T (Pareto front) by a Pareto front tracking (PFT) method [11]. In this technique, the o ptimal pulse obtained for control time T is taken as an initial pulse for the optimization run for time TT . We set 0.01T . The number of time steps at passing from T to TT was fixed; i.e., step length t was varied. Such a run-to-run variation of t does not significant ly affect the calculation accuracy at large S; in our viewpoint, however, it makes the calculation more stable as compare d with the regime when t is fixed and S is varied . The PFT technique makes it possible to significantly reduce the calculation cost and calculate the Pareto fronts for certain families of solutions with high efficien cy. B. Numerical results Figure 1 shows the Pareto fronts for the QFT gate on the spin I = 1 obtained by minimizing error 1J (22). For the three possible values of the global phase , the dependences are similar, but shifted by the time scale . As cri tical time s cT, we take the values for which 5 110J . The shortest time 1.83/cTq corresponds to the global phase 5 / 6 and the longest time 4.44 /cTq is obtained for /6 , which is qualitatively cons istent with the results given in Table I. The quantitative difference is related to the fact that , considering the Hamiltonian in form (15) , we significantly limited the class of possibl e solutions. To obtain an exact analytical solution, the explicitly

Question 1320 multiple-choice

The Hidden Subgroup Problem (HSP) is central to quantum computing, especially in the study of non-abelian groups such as the finite Heisenberg group $ H_p $. Quantum algorithms and group-theoretic properties play a crucial role in determining the efficiency of HSP solutions for these groups. Which property of the non-central order $ p $ subgroups of the finite Heisenberg group $ H_p $ most directly explains why standard abelian Fourier sampling fails to efficiently distinguish the hidden subgroup in quantum algorithms? 1) They are direct products of cyclic groups and hence abelian. 2) Their Fourier transforms yield vectors forming mutually unbiased bases. 3) They are all normal subgroups within $ H_p $. 4) They coincide with the center and commutator subgroups. 5) They have polynomially many conjugacy classes. 6) Their group multiplication is commutative. 7) They can be efficiently generated by two elements.

✓ Correct Answer:

The correct answer is 2) Their Fourier transforms yield vectors forming mutually unbiased bases..

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to identify H. Weillustrate the power of therandom strong method via a 2 concrete example of the HSP over ﬁnite Heisenberg groups Hpof orderp3, wherep≥3is a prime. Hpis deﬁned as the following set of upper triangular matrices: Hp:=   1x z 0 1y 0 0 1 :x,y,z∈Fp  . (1) Aconvenient encodingfortheelementsof Hpistowrite (x,y,z),wherex,y,z∈Fpmatchthecomponents inequation (1). Thecomposition of twoelements isthen give n by (x1,y1,z1)◦(x2,y2,z2) = (x1+x2,y1+y2,z1+z2+x1y2), and the inverse of an element is given by (x,y,z)−1= (−x,−y,xy−z). It is easy to see that the classical randomised query complexity of theHSPon Hpisθ(p). Thegeneric quantum algorithm of Ettinger, Høyer andKnill [EHK04]achieves O(logp)querycomplexity, but attheexpense of pO(logp)quantum operations. An algorithm with 2θ(√logp)quantum operations can be obtained by combining the ideas of [Kup03] and [FIM+03]. However,thequery complexity of thisalgorithm isalso 2θ(√logp). Itseemsnon-trivial todesign a quantum algorithm with (logp)O(1)query complexity and total running time pO(1). In the following paragraphs, we indicate how various existing methods for no n-abelian HSP fail to achieve this goal. After that, we show how the random strong method attains this goal, illustrating the power of random bases in Fourier sampling. Itcanbeshownthat Hpisasemidirect product oftheform Zp⋉(Zp×Zp),wherethenormal subgroup is given byN∞:={(0,y,z) :y,z∈Fp}and the complement by A0,0:={(x,0,0) :x∈Fp}. The commutator subgroup of Hpis given by [Hp,Hp] ={(0,0,z) :z∈Fp}, which is also the centre ζ(Hp). Thecommutatorsubgroupisisomorphicto Zp;henceitisabelianbutnot smoothlyabelian (anabeliangroup Gissaidtobesmoothlyabelian [FIM+03]ifitisthedirect product ofasubgroup ofbounded expone nt and asubgroup of size (log|G|)O(1)). TheBaer subgroup turns out to be κ(Hp) =ζ(Hp). IfA≤ Hp,|A|=p, then|κ(Hp)A| ≤p2; therefore for such an A,[G:κ(Hp)A]≥p. In fact, we will see later that there are (p2+p+1)orderpsubgroups of Hp. Thus, the methods of [GSVV04, Gav04, IMS03, FIM+03] are not applicable in order to solve the HSP for Hpefﬁciently. For more details about the Heisenberg group, se e Section 2. The chief obstacle to ﬁnding hidden subgroups in Hparises from the order psubgroups of Hpother than its centre. There are (p2+p)such orderpsubgroups; we shall call them Ai,j,i∈Fp∪{∞},j∈Fp. Theforgetful abelian method (i.e. Fourier sampling over the abelian group Zp×(Zp×Zp)instead of the non-abelian group Hp∼=Zp⋉(Zp×Zp)), weak Fourier sampling, strong Fourier sampling in the natural representation basis of Hp(i.e. the representation basis adapted to the distinguised subgroup tower {1}⊳N∞⊳Hp)aswellasstrongFouriersamplinginthe Zp-Fouriertransformofthenaturalrepresentation basis give exponentially small information about the index iofAi,j. For more details, see Section 2.4. For now, we give an intuitive description of the main difﬁculty p osed by these subgroups. Suppose the hidden subgroup is Ai,jfor somei∈Fp∪ {∞},j∈Fp. With exponentially high probability, Fourier sampling overHpgivesusarepresentation uniformlyatrandomfromoneofthe (p−1)irreduciblerepresentations ρk of degreepfork= 1,...,p−1ofHp. Suppose onesuch representation ρkshowsup. Thestateessentially collapses toavector |ψk,i,j/an}bracketri}ht ∈Cp,i.e.,(Hp,Ai,j)isaGelfandpair for all i,j(see also[MR05]for Gelfand pairs in the context of the HSP).The vectors |ψk,i,j/an}bracketri}hthave the property that |/an}bracketle{tψk,i,j|ψk,i′,j′/an}bracketri}ht|=/braceleftBigg1√p:i/ne}ationslash=i′,for allj,j′, δj,j′:i=i′, 3 i.e., they form a set of (p+1)mutually unbiased bases [WF89] of Cp. The main difﬁculty is that it is not clearapriorithatthereisanyorthonormal basisthatcanpa irwisedistinguish betweenthese (p2+p)vectors with inverse polynomial probability. Note that the so-call edhidden conjugate problem [MRRS04] is easy to solve information-theoretically for Hp; the conjugacy classes of the order psubgroups are deﬁned by i and the above property says that {|ψk,i,j/an}bracketri}ht}jis an orthonormal

Question 1321 multiple-choice

In quantum information theory, the classification of multipartite entangled states often involves analyzing the dimensions of Hilbert spaces and the action of symmetry groups such as local unitaries. Understanding when locally maximally entangled (LME) states exist, and the structure of their moduli spaces, requires careful dimension counting and algebraic constraints. Suppose a multipartite quantum system has subsystems of dimensions d₁, d₂,.., dₙ, with the largest subsystem dimension denoted dn. Under what condition is the moduli space of locally maximally entangled states modulo local unitary transformations (SLME/K) guaranteed to be empty? 1) When dn is strictly greater than the product of the dimensions of the other subsystems 2) When dn equals the sum of the dimensions of the other subsystems 3) When dn is strictly less than the product of the other subsystem dimensions 4) When dn is equal to half the product of the other subsystem dimensions 5) When dn is strictly less than half the product of the other subsystem dimensions 6) When all subsystem dimensions are equal 7) When dn is exactly equal to the product of the other subsystem dimensions

✓ Correct Answer:

The correct answer is 1) When dn is strictly greater than the product of the dimensions of the other subsystems.

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of gi). We deﬁne Rto be the number of these equations that are independent (i.e. the rank of the rectangular matrix that deﬁnes t he linear equations) for a generic point ψ- this gives the dimension of the generic orbit in H. This may be computed by evaluating the rank in the case where the coeﬃcients ψare left as undetermined variables. Then dimS= dimG−R (5.4) 19For a discussion of the relationship between the dimension of the GIT quotient and the dimension of the stabilizer for a generic state in the quantum information literature, see, e.g., [MOA13]. 32 and dim(P(H)//G) = dim(P(H))−R. (5.5) Implementing this method using Maple, we have calculated the dimensio n in many speciﬁc examples and found that the results agree in all cases with the pred ictions of Theorems 1.1 and 1.2. 5.1 Proof of theorems governing existence and dimension of SLME We are now ready to outline a proof of the key results Theorems 1.1 a nd 1.2 for the existence and dimension of SLME/K. We refer to the companion paper [BRV] for the complete proof. Here, the aim is to prove as much as possible with minimal mathematical background, and give an overview of the remaining steps. Boththeoremsfollowbyanalyzingthefollowingrecursiveresultfort hedimensionof SLME/K: Theorem 5.1. Consider a multipart quantum system described by Hilbert sp aceH=H1⊗ ···⊗Hnwith subsystems Hiof dimension di, withd1/lessorequalslantd2/lessorequalslant···/lessorequalslantdn. LetSLME/Kbe the space of locally maximally entangled states up to identiﬁca tion by local unitary transforma- tionsK= SU(d1)×···×SU(dn). Then 1. Ifdn>d1···dn−1, thenSLME/Kis empty. 2. Ifdn=d1···dn−1, thenSLME/Kis a single point. 3. Ifdn≤1 2d1···dn−1, thenSLME/Kis non-empty. It is a single point in the case where n= 3andd1=d2=d3= 2, has dimension d−3in the case where n= 3andd1= 2, d2=d3=dand has dimension d1···dn−d2 1−···−d2 n+n−1in all other cases. 4. If1 2d1···dn−1d1···dn−1 In this case, we violate the necessary conditions (2.6), so there ca nnot be any LME states, and the GIT quotient must be empty. 33 •Case 2:dn=d1···dn−1 In this case, the Hilbert space is a tensor product of two subsyste ms of dimension dn=d1···dn−1. Since the density matrices for these subsystems have the same s pec- trum, having the elementary dn-dimensional subsystem maximally mixed implies that thecompositesubsystem composedoftheﬁrst n−1elementarysubsystems isalsomax- imally mixed. This implies that each density matrix for the ﬁrst n−1 subsystems is maximally mixed, so our state is LME if and only if the two complementary dndimen- sional subsystems are maximally mixed. Describing the state as a dn×dnmatrixψiI, the condition that both subsystems are maximally mixed is that ψ†ψ=ψψ†=1 dn1 1; this holds if and only if ψiI=1√dnUiIwhereUis a unitary matrix. By an SU( dn) transformation on the dn-dimensional elementary subsystem and an overall phase ro- tation, we can bring the state to the form ψiI=1√dnδiI. This is just

Question 1322 multiple-choice

Quantum algorithms have significantly impacted the study of hidden shift problems (HSP) in the context of group theory, particularly for nonabelian groups relevant to cryptography. The structure and subgroup composition of affine groups over finite fields play a crucial role in efficient algorithm design for these problems. Which subgroup of the affine group Ap over Z_p consists specifically of affine functions with slope 1 and forms a normal subgroup? 1) Hq, the nonnormal subgroup of size q 2) H, the nonnormal subgroup representing lines through the origin 3) N, the subgroup isomorphic to Z_p consisting of translations (additive shifts) 4) Hb, the conjugate subgroup stabilizing point b 5) Nq, the normal subgroup of lines with slopes as powers of a 6) Z∗_p, the multiplicative group of units modulo p 7) Hq, the subgroup representing lines intersecting the diagonal at (b, b)

✓ Correct Answer:

The correct answer is 3) N, the subgroup isomorphic to Z_p consisting of translations (additive shifts).

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(multiplicative) cosets of a known multiplicative subgroup of Z∗ p. These functions have been studied in some depth for their pseudorandom properties, and several instances have been suggestedas cryptographically strong pseudorandom generators. By associating f swith its Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 944 MOORE, ROCKMORE, RUSSELL, AND SCHULMAN isotropy subgroup, and using our reconstruction algorithm to ﬁnd that subgroup, we give an eﬃcient quantum algorithm for the hidden shift problem in the case wheref(x) is a function of x’s multiplicative order mod rfor some r= polylog( p). This generalizes the work of van Dam, Hallgren, and Ip [4], who give an algorithm forhidden shift problems in the case where fis precisely a multiplicative character. Finally, in section 8 we show that the set of groups for which the HSP can be solved in polynomial time has the following closure property: if H={H n}is a family of groups for which we can eﬃciently solve the HSP and K={Kn}is a family of groups for which |Kn|= polylog( |Hn|), we can also eﬃciently solve the HSP for the family {Gn}, where each Gnis any extension of KnbyHn. This subsumes the results of [13] on Hamiltonian groups, and also those of [15] on groups with commutatorsubgroups of polynomial size. We note that subsequent to this work, Bacon, Childs, and van Dam [1] found additional algorithms for the HSP in the aﬃne groups. Their approach uses the“pretty good measurement,” which they showed is optimal for certain cases of theHSP (see also [21]). Their work extends to a number of other group families, such asthe Heisenberg groups. 2. The aﬃne and q-hedral groups. LetA pbe the aﬃne group , consisting of ordered pairs ( a, b)∈Z∗ p×Zp, where pis prime, under the multiplication rule (a1,b1)·(a2,b2)=( a1a2,b1+a1b2).Apcan be viewed as the set of aﬃne functions f(a,b):Zp→Zpgiven by f(a,b):x/mapsto→ax+bwhere multiplication in Apis given by function composition. Structurally, Apis a semidirect product Z∗ p/multicloseleftZ p∼=Zp−1/multicloseleftZ p. Its subgroups can be described as follows: •LetN∼=Zpbe the normal subgroup of size pconsisting of elements of the form (1 ,b). Geometrically, this is the set of aﬃne functions with slope 1. •LetH∼=Z∗ p∼=Zp−1be the nonnormal subgroup of size p−1 consisting of the elements of the form ( a,0). Geometrically, this is the set of lines passing through the origin. •For each b∈Zp, the conjugate subgroup Hb=( 1,b)·H·(1,−b) consists of elements of the form ( a,(1−a)b). In the action on Zp,Hbis the stabilizer of b; geometrically, Hbis the set of lines intersecting the diagonal at ( b, b). •Ifa∈Z∗ phas order q, where qdivides p−1, let Nq∼=Zq/multicloseleftZ pbe the normal subgroup consisting of all elements of the form ( at,b). Geometrically, Nqis the set of lines whose slope is a power of a. •Similarly, if a∈Zphas order q, let Hqbe the nonnormal subgroup Hq= /angbracketleft(a,0)/angbracketrightof size q. Then Hqconsists of the elements of the form ( at,0), and its conjugates Hb q=( 1,b)·Hq·(1,−b) consist of the elements of the form (at,(1−at)b). Geometrically, these are the subsets of

Question 1323 multiple-choice

Quantum algorithms for group-theoretic problems often rely on properties of group structure, especially when dealing with non-abelian groups formed as semidirect products. Efficient solutions to the hidden subgroup problem (HSP) in such groups can depend on subgroup normality and group solvability. In the context of solving the hidden subgroup problem for a group G = A ⋊ϕ Z_p, where A is abelian and p is prime, which of the following conditions most directly enables the efficient application of quantum algorithms to identify the hidden subgroup? 1) The automorphism ϕ is trivial for all elements of A 2) The subgroup H1 (the intersection of the hidden subgroup with A) is normal in G 3) The group G is simple and non-solvable 4) The prime p divides the order of A 5) The subgroup generated by the generator of Z_p acts freely on A 6) The hidden subgroup contains the center of G 7) The automorphism ϕ acts as the identity on the generator of Z_p

✓ Correct Answer:

The correct answer is 2) The subgroup H1 (the intersection of the hidden subgroup with A) is normal in G.

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state hiding the subgroup H. In the standard approach, one attempts to determine H using samples of hidden subgroup state ρH. 2.2 Semidirect product groups Thesemidirect productof two groups A,Bis deﬁnedin terms of a homomorphism ϕ:B→Aut(A), whereAut( A) denotes theautomorphismgroupof A. Thesemidirect productgroup A⋊ϕBconsists of the elements ( a,b) witha∈Aandb∈B. With group operations in AandBwritten additively, the group operation in A⋊ϕBis deﬁned as ( a,b)(a′,b′) = (a+ϕ(b)(a′),b+b′). It is not hard to show that group inversion satisﬁes ( a,b)−1= (ϕ(−b)(−a),−b). We consider the hidden subgroup problem for semidirect prod uct groups G=A⋊ϕZp, where Ais an abelian group and pis prime. In this case, since ϕis a homomorphism of a cyclic group, it is determined entirely by ϕ(1) (in particular, ϕ(p) is the identity map); hence, with a slight abuse of notation, we let ϕ:=ϕ(1) (an automorphism of A) henceforth, and deﬁne ϕb:= (ϕ◦···◦ϕ) (b times), giving ϕ(b) =ϕb. We will be especially interested in cyclic subgroups of such a semidirect product group. For any a∈A, we have /an}bracketle{t(a,1)/an}bracketri}ht={(0,0),(a,1),(a+ϕ(a),2),(a+ϕ(a) +ϕ2(a),3),...}. For convenience, we introduce the function Φ(b):A→Adeﬁned by Φ(b)(a) :=b−1/summationdisplay i=0ϕi(a) (3) such that the elements of /an}bracketle{t(a,1)/an}bracketri}htare (a,1)b= (Φ(b)(a),bmodp) forb∈N. 3 3 Reduction to cyclic subgroups To simplify the hidden subgroup problem over semidirect pro duct groups, we can reduce the prob- lem to that of ﬁnding either the trivial hidden subgroup or a h idden cyclic subgroup of order p. This reduction generalizes a result of Ettinger and Høyer, w ho reduced the dihedral hspto the problem of ﬁnding a trivial hidden subgroup or a hidden reﬂec tion (an order 2 subgroup). Lemma 1 (cf. Theorem 2.3 of [8]). To ﬁnd an eﬃcient algorithm for the hspoverA⋊Zpwith pprime, it suﬃces to ﬁnd an eﬃcient algorithm for the hspoverA2⋊Zpfor anyA2≤Awith the promise that either His trivial or H=/an}bracketle{t(d,1)/an}bracketri}htfor somed∈A2with|H|=p. Proof.The proof proceeds along the lines of the Ettinger-Høyer red uction. Their idea is to factor out the part of the hidden subgroup that lies within A. For generalized dihedral groups ( p= 2, ϕ(a) =−a), their reduction goes through unchanged, as noted in [10]. However, in some cases there is an additional complication coming from the fact tha t theA-part of the hidden subgroup may not be normal in G. Fortunately, we can deal with this case separately. Letf:G→Shide a subgroup H. LetG1:=A×{0}, and letH1:=H∩G1=:A1×{0}. Since frestricted to the abelian group G1hides the subgroup H1, we can eﬃciently ﬁnd generators for H1by solving an abelian hidden subgroup problem. We would like to factor out H1, so we check whether it is normal in G. Forg= (a,b)∈Gand h1= (h,0)∈H1, we havegh1g−1= (a,b)(h,0)(ϕ−b(−a),−b) = (ϕb(h),0). Therefore H1/triangleleftequalGif and only if ϕ(H1) =H1, whereϕ(H1) :={(ϕ(h),0) :h∈A1}. IfH1/ne}ati≀nslash=H, thenϕ(H1) =H1: since pis prime, there must be some ( d,1)∈H, and for any h∈A1, (d,1)(h,0)(d,1)−1= (ϕ(h),0)∈H, henceϕ(h)∈A1. IfH1=H, it could be that ϕ(H1)/ne}ati≀nslash=H1. However, given a generating set for H1, we can check whether H1/triangleleftequalG(for example, we can use the results of [31]; note that Gis solvable since its commutator subgroup is abelian1). If we

Question 1324 multiple-choice

In quantum computing and representation theory, the interplay between group symmetries and basis constructions is fundamental for designing efficient algorithms. Understanding how operators and basis labeling schemes relate across unitary and symmetric groups is crucial for manipulating quantum states. Which statement accurately describes the relationship between Young-Jucys-Murphy (YJM) elements and SU(d) irreducible representation (irrep) basis vectors in the context of Schur-Weyl duality? 1) YJM elements are block-diagonal in the SU(d) irrep basis, providing partial eigenvalue information only. 2) YJM elements are strictly diagonal in the SU(d) irrep basis, with eigenvalues corresponding to tableau content vectors. 3) YJM elements act as ladder operators in the SU(d) irrep basis, shifting spin quantum numbers. 4) YJM elements do not commute with Casimir operators and cannot be simultaneously diagonalized. 5) YJM elements are strictly diagonal only in the computational basis, not in the SU(d) irrep basis. 6) YJM elements correspond to Clebsch-Gordon coefficients in the SU(d) irrep basis. 7) YJM elements label multiplicity spaces in the decomposition of SU(d) irreps but are not diagonal operators.

✓ Correct Answer:

The correct answer is 2) YJM elements are strictly diagonal in the SU(d) irrep basis, with eigenvalues corresponding to tableau content vectors..

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algebraic. Thus, the gate action drawing from the group algebra C[Sn] is basis independent. In par- ticular, it can be implemented directly in the computational basis without computing the Fourier coeﬃcients—this is a key observation that underpins the superexponential quan-tum speedup. (b) This identiﬁcation allows us to apply both classical tools from SU (d)representation theory as well as the Okounkov-Vershik approach to Schur basis no matter how it is established. As an example, we show inSec. IV B an eﬃcient algorithm to generate Schur-basis states required for optimization and learning tasks. (c) A detailed examination on Schur basis enables us covert all the previous results about S n-CQA to, what we call, SU(d)-CQA with Snsymmetry. For two-row Young diagrams, this conjecture was shown to be correct in Ref. [ 74], where the author studied the question by1 2-spin eigenfunctions instead of YJM ele- ments. The general case for SU (d)−Snduality still holds and can be proven in a surprisingly easy way using YJM elements and the Okounkov-Vershik approach. We present details in the Appendix. Lemma 3: Under SU(d)−Snduality, sequentially cou- pled Casimir operators commute with YJM elements. As a brief illustration of this result, let us consider thesequentially coupled total spin basis |j1,...,jn;m/angbracketright of SU (2).T h e spin component m andspin labels j kare determined by spin operator Sn zas the summation of all half Pauli- Zmatrices1 2Ziat each isite and sequen- tial coupled Casimir operators J2 k=(Sk x)2+(Sk y)2+(Sk z)2, respectively, [we abuse our language for simplicity as true eigenvalues of J2 karejk(jk+1)]. Since they commute with YJM elements, J2 kXi|j1,...,jn,m/angbracketright=XiJ2 k|j1,...,jn;m/angbracketright.I t is well known from linear algebra that commutative oper- ators can be simultaneously diagonalized and we elaborate this fact with the following theorem. Theorem 4: YJM elements are strictly diagonal under theSU(d)irrep basis built by sequential Clebsch-Gordon decompositions. Conversely, sequentially coupled Casimir operators are strictly diagonal under the Young basis decomposed by branching rule. We now illustrate by examples the correspondence between spin labels and content vectors for the simplest SU(2)−Snduality, then we go to the general case. Let |j1,...,jn;m/angbracketrightbe any SU (2)-irrep basis vector. Theorem 4 says that it is also a Young basis element, thus we can talk about its eigenvalues (content vector) (αT(1),...,αT(n)) with respect to the YJM elements (recall that Tdenotes a denotes a standard Young tableau, or equivalently the corresponding GZ path from the Bratteli diagram like that from Figs. 3and4). An equivalence between two label- ing schemes means the spin label J={j1,...,jn}uniquely determines the content vector αT=(αT(1),...,αT(n))and vice versa. SU(2)case: for brevity, let us denote SU (2)-irrep basis vectors by |J;m/angbracketright. It is possible to ﬁnd two basis ele- ments |J;m/angbracketright,|J/prime;m/angbracketrightwith the same spin component mbut diﬀerent spin labels. This is due to the fact that (C2)⊗n would decompose into copies of isomorphic SU (2)irreps [Fig. 2(a)]. On the other hand, a Young basis element is then denoted by |αT;μ/angbracketrightwhere μ, as explained in Sec. II, 020327-8 SPEEDING UP LEARNING QUANTUM... PRX QUANTUM 4,020327 (2023) comes with choosing the

Question 1325 multiple-choice

The Quantum Fourier Transform (QFT) is a key element in quantum computing, critical for algorithms such as Shor’s factoring and phase estimation. Advanced implementations seek to overcome hardware limitations by using high-dimensional encodings and novel circuit elements. Which approach enables a fully quantum, high-dimensional implementation of the Quantum Fourier Transform by transferring multi-qubit states to resonators and leveraging cross-Kerr interaction for state preparation? 1) Using classical feedback with qubit recycling in standard quantum circuits 2) Employing additional ancilla qubits for each precision increment 3) Mapping multi-qubit states to a single resonator via perfect state-transfer and utilizing a second oscillator with cross-Kerr interaction and projective measurement 4) Applying Grover's algorithm for Hilbert space expansion 5) Implementing QFT exclusively with superconducting qubits and no resonators 6) Performing QFT by iterative gate optimization on noisy intermediate-scale quantum devices 7) Relying on photonic quantum circuits without state-transfer or nonlinear interactions

✓ Correct Answer:

The correct answer is 3) Mapping multi-qubit states to a single resonator via perfect state-transfer and utilizing a second oscillator with cross-Kerr interaction and projective measurement.

📚 Reference Text:

Title: Quantum Fourier Transform in Oscillating Modes Year: 2019 Paper ID: dfcd40097a4cd830412d2c75313cb66b7f199da9 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/dfcd40097a4cd830412d2c75313cb66b7f199da9 Abstract: Quantum Fourier transform (QFT) is a key ingredient of many quantum algorithms. In typical applications such as phase estimation, a considerable number of ancilla qubits and gates are used to form a Hilbert space large enough for high-precision results. Qubit recycling reduces the number of ancilla qubits to one, but it is only applicable to semi-classical QFT and requires repeated measurements and feedforward within the coherence time of the qubits. In this work, we explore a novel approach based on resonators that forms a high-dimensional Hilbert space for the realization of QFT. By employing the perfect state-transfer method, we map an unknown multi-qubit state to a single resonator, and obtain the QFT state in the second oscillator through cross-Kerr interaction and projective measurement. A quantitive analysis shows that our method allows for high-dimensional and fully-quantum QFT employing the state-of-the-art superconducting quantum circuits. This paves the way for implementing various QFT related quantum algorithms.

Question 1326 multiple-choice

Quantum algorithms have revolutionized computational approaches to algebraic number theory, enabling efficient solutions to problems that are classically considered hard. One important application is the computation of the unit group of a number field, which has significant implications for both mathematics and cryptography. Which innovation is essential for efficiently solving the hidden subgroup problem over continuous groups such as ℝⁿ in the context of quantum algorithms for number fields? 1) Utilizing elliptic curve methods for group structure identification 2) Applying Grover's search for subset sum problems 3) Employing Shor’s algorithm for integer factorization 4) Representing real-valued lattices using Gaussian-weighted superpositions and suitable encoding 5) Adopting classical LLL lattice reduction without quantum enhancement 6) Implementing brute-force enumeration of units 7) Using block cipher techniques for group masking

✓ Correct Answer:

The correct answer is 4) Representing real-valued lattices using Gaussian-weighted superpositions and suitable encoding.

📚 Reference Text:

A quantum algorithm for computing the unit group of an arbitrary degree number ﬁeld Kirsten Eisenträger∗ Department of Mathematics The Pennsylvania State University eisentra@math.psu.edu and Harvard UniversitySean Hallgren† Dept. of Computer Science and Engineering The Pennsylvania State University hallgren@cse.psu.eduAlexei Kitaev Kavli Institute for Theoretical Physics University of California, Santa Barbara kitaev@kitp.ucsb.edu and California Institute of Technology Fang Song Department of Combinatorics & Optimization and Institute for Quantum Computing University of Waterloo fang.song@uwaterloo.ca ABSTRACT Computing the group of units in a ﬁeld of algebraic numbers is one of the central tasks of computational algebraic num- ber theory. It is believed to be hard classically, which is of interest for cryptography. In the quantum setting, eﬃcient algorithms were previously known for ﬁelds of constant de- gree. We give a quantum algorithm that is polynomial in the degree of the ﬁeld and the logarithm of its discriminant. This is achieved by combining three new results. The ﬁrst is a classical algorithm for computing a basis for certain ideal lattices with doubly exponentially large generators. The sec- ond shows that a Gaussian-weighted superposition of lattice points, with an appropriate encoding, can be used to provide a unique representation of a real-valued lattice. The third is an extension of the hidden subgroup problem to continuous groups and a quantum algorithm for solving the HSP over ∗Partially supported by National Science Foundation grant DMS-1056703 and by the National Security Agency (NSA) under Army Research Oﬃce (ARO) contract number W911NF-12-1-0522. Part of this work was done while the ﬁrst author was visiting Harvard University and MIT. †Partially supported by National Science Foundation awards CCF-0747274 and CCF-1218721, and by the National Se- curity Agency (NSA) under Army Research Oﬃce (ARO) contract number W911NF-12-1-0522. Part of this work was done while visiting MIT. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for proﬁt or commercial advantage and that copies bear this notice and the full citation on the ﬁrst page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior speciﬁc permission and/or a fee. STOC ’14, May 31 - June 03 2014, New York, NY , USA Copyright is held by the owner/author(s). Publication rights licensed to ACM. ACM 978-1-4503-2710-7/14/05$15.00. http://dx.doi.org/10.1145/2591796.2591860.the group Rn. Categories and Subject Descriptors F.2.2 [Analysis of Algorithms and Problem Complex- ity]: Numerical Algorithms and Problems General Terms Algorithms, Theory Keywords Quantum Algorithms, Unit Group, Computational Alge- braic Number Theory 1. INTRODUCTION The problems where quantum algorithms have exponen- tial speedups over the best known classical algorithm have mostly been of number theoretic origin. Shor found quan- tum algorithms for factoring and discrete log [Sho97] and Hallgren found a quantum algorithm for solving Pell’s equa- tion [Hal07]. These algorithms were further generalized to ﬁnding the unit group of a number ﬁeld and related prob- lems [Hal05, SV05]. The running time is measured in terms of the

Question 1327 multiple-choice

In quantum computing, the progression of algorithms is often analyzed using mathematical principles that describe changes in information and probability distributions. One such principle is majorization, which has implications for algorithm efficiency and the theoretical foundations of quantum information. Which statement best describes the Majorization Principle as applied to optimal quantum algorithms? 1) It asserts that quantum algorithms must be irreversible to achieve speedup over classical methods. 2) It states that outcome probabilities remain uniformly distributed throughout the computation. 3) It requires that the entanglement between qubits decreases with each computational step. 4) It posits that the probability distributions of outcomes become increasingly ordered toward the correct solution at each step. 5) It only applies to classical algorithms using randomized processes. 6) It demands that all quantum algorithms use bosonic particles for information encoding. 7) It specifies that majorization is exclusive to the quantum fast Fourier transform algorithm.

✓ Correct Answer:

The correct answer is 4) It posits that the probability distributions of outcomes become increasingly ordered toward the correct solution at each step..

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Title: Observation of Majorization Principle for quantum algorithms via 3-D integrated photonic circuits Year: 2016 Paper ID: 4642c21acce169d6bf6a7afb6e176ff7af4437c0 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/4642c21acce169d6bf6a7afb6e176ff7af4437c0 Abstract: The Majorization Principle is a fundamental statement governing the dynamics of information processing in optimal and efficient quantum algorithms. While quantum computation can be modeled to be reversible, due to the unitary evolution undergone by the system, these quantum algorithms are conjectured to obey a quantum arrow of time dictated by the Majorization Principle: the probability distribution associated to the outcomes gets ordered step-by-step until achieving the result of the computation. Here we report on the experimental observation of the effects of the Majorization Principle for two quantum algorithms, namely the quantum fast Fourier transform and a recently introduced validation protocol for the certification of genuine many-boson interference. The demonstration has been performed by employing integrated 3-D photonic circuits fabricated via femtosecond laser writing technique, which allows to monitor unambiguously the effects of majorization along the execution of the algorithms. The measured observables provide a strong indication that the Majorization Principle holds true for this wide class of quantum algorithms, thus paving the way for a general tool to design new optimal algorithms with a quantum speedup.

Question 1328 multiple-choice

In quantum error correction, stabilizer codes use algebraic structures to protect quantum information against local disturbances. The toric code is a foundational example that utilizes lattice topology and specific operator constructions to achieve fault tolerance. Which property ensures that the toric code can correct any error affecting up to k qubits, provided k does not exceed ⌊(L-1)/2⌋ for a lattice of size L? 1) The existence of a unique ground state for every region of the lattice 2) The use of non-Hermitian operators to define errors 3) The suppression of fundamental group notation in code definitions 4) Encoding information in noncontractible loops only 5) The presence of edge operators acting independently of the code projector 6) The satisfaction of the Knill-Laflamme conditions via stabilizer projector commutation relations 7) The ability of Pauli Y operators to create contractible loops

✓ Correct Answer:

The correct answer is 6) The satisfaction of the Knill-Laflamme conditions via stabilizer projector commutation relations.

📚 Reference Text:

in deﬁning the fundamental group up to isomorphism, so we suppress it from the notation from now on. Accepted in Quantum2020-09-17, click title to verify. Published under CC-BY 4.0. 8 Figure 1: An example arrangement is shown of the regions A⊂B. Region Aconsists of the subset of edges in the red shaded rectangle, and likewise for Bin blue. Each region includes the edges in its rectangular boundary, as shown by the shading. 3.1 The toric code is a QECC: a warm-up In this section, we warm up by proving that the toric code is a QECC, which was shown in [13]. The toric code is a special case of Kitaev’s models, so this warm-up shows a weaker result than Theorem 3.1, but we will improve upon both of these points in the next section. We proceed by showing that the toric code obeys the Knill-Laﬂamme conditions, which state that a set of errors E={Ei}is correctable by an error correcting code represented by a projector Ponto the code subspace if and only if PE† iEjP=αijP (27) whereαijform the entries of a Hermitian matrix (see Theorem 2.1). The projection operator for the toric code is given by P=/productdisplay v∈V1 +Xv 2/productdisplay p∈F1 +Zp 2. (28) Now, consider a general error on k∈Nqubits. Since tensor products of Pauli operators span all possible operators, it is suﬃcient to consider the errors E(k)≡/braceleftBigg/circlemultiplydisplay e∈Skσe:Sk⊆E,|Sk|≤k,σe∈{X,Y,Z}/bracerightBigg . (29) We claim that if k>⌊L−1 2⌋, whereLis the size of the lattice, then E(k)is not correctable. To see this, ﬁrst note k>⌊L−1 2⌋impliesk≥⌈L 2⌉. Thus, we can form the operator E† iEjthat is a tensor product ofXalong a noncontractible loop by choosing appropriate Ei,Ej. Then,E† iEjtransforms two orthogonal states in the codespace into each other and therefore PE† iEjP/negationslash∝P. Now suppose k≤⌊L−1 2⌋. We ﬁrst compute the commutation relations (X⊗I⊗···⊗I)1 +Xv 2=1 +Xv 2(X⊗I⊗···⊗I) (X⊗I⊗···⊗I)1 +Zp 2=1 + (−1)i(p)Zp 2(X⊗I⊗···⊗I),(30) whereIis the identity operator on C[Z2]andi(p)is an indicator for whether the ﬁrst edge is on the boundary of p. Similarly, (Z⊗I⊗···⊗I)1 +Xv 2=1 + (−1)i(v)Xv 2(Z⊗I⊗···⊗I) (Z⊗I⊗···⊗I)1 +Zp 2=1 +Zp 2(Z⊗I⊗···⊗I).(31) Now, we can represent, up to a phase, E† iEjas a product of Pauli’s of the form I⊗···⊗I⊗σ⊗I⊗ ···⊗I, whereσ∈{X,Z}. We then commute Pacross each of the factors. We ﬁrst consider the edges on which a Pauli Zis acted on. Then, unless every vertex is incident to an even number of them, there will exist a vertex vfor whichc(v) = 1, which would imply PE† iEjP= 0. Otherwise, the edges form loops. Since there are at most 2k≤L−1edges acted on, the loops must be contractible. A similar argument holds for Pauli Xwhere there we work in the dual lattice. We conclude that PE† iEjP= 0orE† iEjis, up to a phase, a product of Xv,Zp, which act trivially on the ground space. Hence the Knill-Laﬂamme condition is satisﬁed. Accepted in Quantum2020-09-17, click title to verify. Published under CC-BY 4.0. 9 3.2 States with locally zero energy density are locally indistinguishable We now give a proof of Theorem 3.1. Consider

Question 1329 multiple-choice

In quantum computing, techniques such as block-encoding and singular value transformation are used to efficiently manipulate matrices and their eigenvalues within quantum circuits. These methods enable the application of complex functions to matrix singular values using polynomial approximations. Which of the following statements correctly describes the outcome of applying a polynomial singular value transformation to a block-encoded matrix using efficient quantum algorithms? 1) The eigenvalues of the matrix are replaced with their exponential values. 2) The matrix is projected onto a subspace determined by its largest singular value. 3) The singular values are eliminated, leaving only the matrix's phase information. 4) The singular values are mapped through the chosen polynomial, resulting in a block-encoding of the polynomial applied to the matrix with controlled approximation error. 5) The matrix is converted into a diagonal matrix with entries equal to the polynomial's coefficients. 6) The singular values are sorted in descending order and stored in a quantum register. 7) The matrix is transformed into a unitary that acts only on its ancilla qubits.

✓ Correct Answer:

The correct answer is 4) The singular values are mapped through the chosen polynomial, resulting in a block-encoding of the polynomial applied to the matrix with controlled approximation error..

📚 Reference Text:

4.0. 18 We begin by rewriting the above expression a bit. Recall that ∆kis an integer from 0 to2k+1−1encoding the kless signiﬁcant bits of λj. If we subtract ∆k/2nfromλj, we obtain a multiple of 1/2n−kplus something <1/2nwhich we ﬂoor away. Then, bit k(λj) indicates if this is an even or an odd multiple. That means we can write: bitk(λj) =parity/parenleftbigg ﬂoor/parenleftbigg 2n−k/parenleftbigg λj−∆k 2n/parenrightbigg/parenrightbigg/parenrightbigg (83) Sinceλj−∆k 2nis a multiple of 1/2n−kplus something <1/2n, we equivalently have that 2n−k/parenleftBig λj−∆k 2n/parenrightBig is an integer plus something less than 1/2k. For such values, we can write the parity(ﬂoor( x)) function in terms of a squared cosine that has been ‘ampliﬁed’: parity (ﬂoor(x)) =amp/parenleftbigg cos2/parenleftbiggπ 2[x+φ]/parenrightbigg/parenrightbigg (84) amp(x) =/braceleftBigg 1ifx>1/2 0ifx<1/2(85) where the shift φcenters the extrema of the cosine in the intervals where xoccurs. There- fore: bitk(λj) =amp/parenleftbigg cos2/parenleftbiggπ 2/bracketleftbigg 2n−k/parenleftbigg λj−∆k 2n/parenrightbigg +φ/bracketrightbigg/parenrightbigg/parenrightbigg (86) =amp/parenleftbigg cos2/parenleftbigg π/bracketleftbigg 2n−k−1/parenleftbigg λj−∆k 2n/parenrightbigg +φ/2/bracketrightbigg/parenrightbigg/parenrightbigg (87) =amp/parenleftBig cos2/parenleftBig πλ(k) j/parenrightBig/parenrightBig (88) Where we have deﬁned: λ(k) j:= 2n−k−1/parenleftbigg λj−∆k 2n/parenrightbigg +φk (89) for somek-dependent choice of phase φk=φ/2. By applying a phase shift conditioned on the |∆k/angbracketrightregister, and then iterating it 2n−k−1 times, we can construct a block-encoding of the unitary with eigenvalues λ(k) j. Our goal is now to transform this unitary as follows: /summationdisplay je2πiλ(k) j|ψj/angbracketright/angbracketleftψj| →/summationdisplay j/bracketleftBig amp/parenleftBig cos2/parenleftBig πλ(k) j/parenrightBig/parenrightBig/bracketrightBig |ψj/angbracketright/angbracketleftψj| (90) To obtain a cosine use linear combinations of unitaries [BCK15, CKS15] to take a take a linear combination with the identity: /summationdisplay je2πiλ(k) j+ 1 2|ψj/angbracketright/angbracketleftψj|=/summationdisplay jcos/parenleftBig πλ(k) j/parenrightBig/bracketleftbigg eiπλ(k) j|ψj/angbracketright/bracketrightbigg /angbracketleftψj| (91) =/summationdisplay j/vextendsingle/vextendsingle/vextendsinglecos/parenleftBig πλ(k) j/parenrightBig/vextendsingle/vextendsingle/vextendsingle·/bracketleftbigg ±eiπλ(k) j|ψj/angbracketright/bracketrightbigg /angbracketleftψj| (92) where on the previous line ±indicates sign/parenleftBig cos/parenleftBig πλ(k) j/parenrightBig/parenrightBig . This way the above is a sin- gular value decomposition of the block-encoded matrix. So all that is left to do is to approximately transform the singular values aof the matrix above by: a→amp(a2) (93) We will accomplish this using singular value transformation. Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 19 Lemma 10. Singular value transformation. SayAis a square matrix with singular value decomposition A=/summationtext iai|ψl i/angbracketright/angbracketleftψr i|, andp(x)is a degree- deven polynomial with real coeﬃcients satisfying |p(x)|≤1for|x|≤1, and suppose Ahas a block-encoding UA. Then, for anyδ >0, there exists a O(poly(d,log(1/δ)))time classical algorithm that produces a circuit description of a block-encoding of the matrix: ˜p(A) :=/summationdisplay i˜p(ai)|ψr i/angbracketright/angbracketleftψr i| (94) where ˜p(x)is a polynomial satisfying |˜p(x)−p(x)|≤δfor|x|≤1. This block-encoding makesdqueries toUAorU† A(not controlled). If the block-encoding of Ahasmancillae, then that of ˜p(A)hasm+ 1ancillae. In the special case when m= 1and the unitary implementing the block-encoding has the form UA=/summationtext iVi⊗|ψl i/angbracketright/angbracketleftψr i|where theViare qubit reﬂections, then it is possible to make the block-encoding of ˜p(A)also just have one ancilla. Proof.This is a slightly simpliﬁed version of the main result of [GSLW18]. The m > 1 case involves an application of Corollary 18 which demands an extra control qubit that is also postselected. For an accessible review on the subject see [MRTC21]. We brieﬂy elaborate on the m= 1special case when UA=/summationtext iVi⊗|ψl i/angbracketright/angbracketleftψr

Question 1330 multiple-choice

Quantum groups, particularly q-deformed analogues of classical compact groups, play a significant role in modern representation theory and mathematical physics. Their character theory is deeply intertwined with quantum algebra and probabilistic methods in infinite-dimensional analysis. Which of the following statements correctly describes the role of the Drinfeld–Jimbo quantized universal enveloping algebras in the study of q-deformed classical compact groups of types B, C, and D? 1) They provide commutative subgroups used to define random walks on group duals. 2) They classify all finite-dimensional irreducible representations of classical unitary groups. 3) They serve as quantum analogues of universal enveloping algebras whose representations form the foundation for the character theory of q-deformed groups. 4) They establish spectral decompositions for symplectic and orthogonal groups without reference to quantum deformation. 5) They generalize the notion of group actions on probability spaces in classical stochastic processes. 6) They construct Markov chains on finite sets arising from non-quantized Lie algebras. 7) They restrict the definition of quantized characters to only type A classical groups.

✓ Correct Answer:

The correct answer is 3) They serve as quantum analogues of universal enveloping algebras whose representations form the foundation for the character theory of q-deformed groups..

📚 Reference Text:

Title: Characters of infinite-dimensional quantum classical groups: BCD cases Year: 2021 Paper ID: 776ef716ac51fb40dd8ce197930f5b4c71526683 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/776ef716ac51fb40dd8ce197930f5b4c71526683 Abstract: We study the character theory of inductive limits of [Formula: see text]-deformed classical compact groups. In particular, we clarify the relationship between the representation theory of Drinfeld–Jimbo quantized universal enveloping algebras and our previous work on the quantized characters. We also apply the character theory to construct Markov semigroups on unitary duals of [Formula: see text], [Formula: see text], and their inductive limits.

Question 1331 multiple-choice

In quantum information theory, rigorous analysis often involves approximating infinite or complicated sets of quantum states by finite sets using tools from convex geometry and functional analysis. δ-nets and volume bounds are standard techniques for discretizing high-dimensional spaces in this context. What is the explicit cardinality bound for a δ-net in the one-norm covering the set of quantum states on a d-dimensional Hilbert space as given by volume estimates from quantum information theory? 1) (d/δ)^{d^2} 2) (4/δ)^{d^2} 3) (2/δ)^{d^2} 4) (2/δ)^{2d^2} 5) (12/δ)^{d^2} 6) (8/δ)^{2d} 7) (1/δ)^{4d}

✓ Correct Answer:

The correct answer is 4) (2/δ)^{2d^2}.

📚 Reference Text:

of S′∪ {1 d /BD}as a real vector space Wthat contains all the sets S,S′,S′′. All operators in a∈Wobey tr{a}= 1. Obviously, this vector space does contain matrices that are not in S(Cd) - they have negative eigenvalues. We want to construct a ﬁnite ap proximation S′′′ toS′′such that S′′⊂conv(S′′′)⊂ S(Cd). OnW, we may still use the one-norm /bardbl·/bardblas a norm. The intersection B:= rebdS(Cd)∩WsatisﬁesS′′⊂ conv(B), and it is thus our goal to construct an approximation satisfying S′′⊂conv(S′′′)⊂conv(B). OnW, aδ-net in 1-norm of cardinality no more than N(δ)≤(2/δ)2d2can be found via the volume bound from [35] (Lemma 2.6) in its application to quantum states (see e.g. [6], Lemma 5.1 and note that quantum states are just special instances of quantum channels) . We additionally need to show that the 14 blow-up ( S′′)εis contained in conv( B) for small enough ε >0. This is done by taking any normalized vectorx∈Cdand computing, for some a∈Wsatisfying /bardbla−Nδ(ρs)/bardbl ≤ε, /an}bracketle{tx,ax/an}bracketri}ht=/an}bracketle{tx,(a−Nδ(ρs))x/an}bracketri}ht+/an}bracketle{tx,Nδ(ρs)x/an}bracketri}ht (110) ≥ /an}bracketle{tx,(a−Nδ(ρs))x/an}bracketri}ht+δ/2 (111) ≥δ/2−ε, (112) and forε≤δ/2 this is larger than or equal to zero. So, applying Nδguarantees us that ( Nδ(S′))δ/2⊂ conv(B)⊂ S(Cd). It can be read oﬀ from [43], Theorems 3.1.6 and 1.8.5 and above cardina lity bounds for δ-nets that |S′′′| ≤(12/δ)2d2can be chosen. Now replace δbyδn= 12n−1/(4d2), then|S′′′| ≤√n. Thus, tr{N†⊗n δn(Pn)ρsn} ≥1−2−n(αε2+d2+√n nlog(2n))∀ρsn∈S⊗n. (113) At the same time, looking back to our equations (39) to (47), we see that the following is valid: tr{N†⊗n δ(Pn)σ⊗n}= tr{PnN⊗n δ(σ⊗n)} (114) ≤2−n(infρ∈SD(ρ/bardblNδ(σ))−Θ(n,ε,d,Ndeltan(σ)). (115) It still holds lim n→∞Θ(n,ε,d,Ndeltan(σ)) = 0, but by monotonicity of the relative entropy it is not yet clear that this is in general the optimal exponent. By assumptio n, infρ∈SD(ρ/bardblσ)<∞. Then also infρ∈SD(ρ/bardblNδn(σ))<∞so that for every ρ∈Sand forδn< td, we also have D(ρ/bardblNδn(σ)) =D(ρ/bardblσ)−tr{ρ(log(Nδn(σ))−log(σ))} (116) ≥D(ρ/bardblσ)−d·max ilog(ti (1−δn)ti+δn/2) (117) ≥D(ρ/bardblσ)−d·log(t1 (1−δn)t1+δn/2) (118) ≥D(ρ/bardblσ)−d·log(t1 (1−δn)t1) (119) =D(ρ/bardblσ)+d·log(1−δn). (120) Clearly, this proves tr{N†⊗n δn(Pn)σ⊗n} ≤2−n(infρ∈conv(S)D(ρ/bardblσ)+d·log(1−δn)−Θ(n,d,ε,Nδn(σ)). (121) Proof of the lower bound on the error exponent in Theorem 4 for inﬁnite|S|. Luckily, in this converse part of the proof we do not have to approx imateSfrom the outside, an approximation from the inside is enough. We will need the Fannes-Aud enaert inequality that yields the following bound: |D(ρ/bardblσ)−D(ρ′/bardblσ)| ≤τlog(d)+h(τ)+τlog(1/td) =:c(τ), (122) whereτ:=1 2/bardblρ−ρ′/bardbl. Then, asbefore, weapproximateaninﬁnite set conv( S)by asequenceconv( Sn)n∈N of convex sets, each conv( Sn) having ﬁnitely many extremal points {ρx}(n)}x∈Xn, where|Xn| ≤√nand for eachρ∈conv(S) andn∈Nthere isρ′∈conv(Sn) such that /bardblρ−ρ′/bardbl ≤12n−1/4d2. Then, by playing everything back to the ﬁnite case and using the scaling of |Xn|we get tr{Aσ⊗n} ≥2−n(infρ∈conv(S)D(ρ/bardblσ)+c(12n−1/4d2)+Γ(n,ε,d,σ,√n)). (123) This proves our claim, since lim n→∞Γ′(n,ε,d,√n) = 0 for all ε∈(0,1). 15 Proof of our no-go-result. Let a permutation-invariant operator A∈ B((Cd)⊗n) satisfy the inequalities 0≤A≤ /BDand 1−ε≤tr{Aρ⊗n} ∀ρ∈ S(Cd) (124) for some ε >0. Then due to Schur-Weyl duality, /BD−Acan be written /BD−A=/summationtext λAλwith each Aλ satisfying Pλcλ≥Aλ≥0 for some 1 ≥cλ≥0, where Pλis the projection onto the isotypical subspace corresponding to λ. It follows that if we average an Aλover the unitary group, we get for every ρthat tr{/integraldisplay dUU⊗nAλU†⊗nρ⊗n} ≤/integraldisplay dUtr{U⊗n( /BD−A)U†⊗nρ⊗n} ≤/integraldisplay

Question 1332 multiple-choice

Quantum algorithms have been shown to provide exponential speedups for certain problems defined over discrete groups. Extending these results to continuous settings, such as real vector spaces, presents unique challenges and opportunities for practical applications. Which statement accurately describes a key innovation in quantum algorithms for the continuous hidden shift problem on $\mathbb{R}^n$? 1) They only work for discrete groups and cannot be generalized to continuous spaces. 2) They require discretizing the continuous space before applying quantum techniques. 3) They introduce ε-random linear disequations and employ continuous oracle functions, enabling a polynomial-time quantum solution in dimension $n$. 4) They rely on classical algorithms that outperform quantum approaches for continuous problems. 5) They demonstrate that quantum speedups are limited to low-dimensional cases in continuous spaces. 6) They necessitate the use of non-linear oracle functions for efficient computation. 7) They are restricted to finite domains due to the limitations of quantum hardware.

✓ Correct Answer:

The correct answer is 3) They introduce ε-random linear disequations and employ continuous oracle functions, enabling a polynomial-time quantum solution in dimension $n$..

📚 Reference Text:

Title: Quantum algorithm based on the ε-random linear disequations for the continuous hidden shift problem Year: 2019 Paper ID: 2c023a645e935cf24b21df50adadcc2b963fa08c Source: semantic-scholar URL: https://www.semanticscholar.org/paper/2c023a645e935cf24b21df50adadcc2b963fa08c Abstract: There have been several research works on the hidden shift problem, quantum algorithms for the problem, and their applications. However, all the results have focused on discrete groups with discrete oracle functions. In this paper, we define the continuous hidden shift problem on $\mathbb{R}^n$ with a continuous oracle function as an extension of the hidden shift problem, and also define the $\varepsilon$-random linear disequations which is a generalization of the random linear disequations. By employing the newly defined concepts, we show that there exists a quantum computational algorithm which solves this problem in time polynomial in $n$.

Question 1333 multiple-choice

Optimizing quantum circuits is essential for practical quantum computing, particularly when implementing complex multi-qubit gates such as CNOT, CCZ, and Toffoli. Techniques like parity qubit encoding and gate decomposition can substantially reduce resource requirements and circuit depth. When constructing a logical Toffoli (CCNOT) gate using parity qubits and efficient gate decomposition, which approach enables this gate to be realized using only two-qubit operations and single-qubit rotations, thereby minimizing resource overhead? 1) Implementing Toffoli directly via three simultaneous CNOT gates without encoding 2) Using only sequential application of Hadamard gates to all qubits 3) Mapping the Toffoli gate onto measurement-based quantum computation 4) Decomposing the Toffoli gate into Hadamard and CCZ gates, then reducing CCZ to two-qubit operations with parity encoding 5) Utilizing error correction codes to replace Toffoli gates with logical Pauli operations 6) Applying only single-qubit phase rotations to all logical qubits 7) Encoding the Toffoli as a cascade of SWAP gates between parity qubits

✓ Correct Answer:

The correct answer is 4) Decomposing the Toffoli gate into Hadamard and CCZ gates, then reducing CCZ to two-qubit operations with parity encoding.

📚 Reference Text:

+/floorleftBig max/parenleftBig/vextendsingle/vextendsingle/vextendsinglen 2−c/vextendsingle/vextendsingle/vextendsingle,/vextendsingle/vextendsingle/vextendsinglen 2−t/vextendsingle/vextendsingle/vextendsingle/parenrightBig/floorrightBig +k/bracketrightBig +3, (11) where candtrepresent the index of the control and the target qubit, respectively, and k=/braceleftbigg 1i f |n 2−c|=|n 2−t| 0 otherwise .(12) The circuit depth dCNOT corresponds to the depth of two ˜Rx gates minus the depth saved due to gate canceling, as depicted in Fig. 2. The number of required (physical) CNOT gates is 2(n−1+|c−t|), which is less than the CNOT count for two logical Hadamard gates, also due to canceling gates. Further-more, seven single-qubit rotations are needed to construct alogical CNOT gate (six for the Hadamard parts and three for the CZpart, where the Rzrotations on the data qubit can be merged). C. Intrinsic higher-order interactions It is possible to encode the cumulative parity of multiple logical qubits ( qi) in a single parity qubit, with operator cor- respondences ˜σ(q1) z˜σ(q2) z···˜σ(qn) z|ψ/angbracketright=σ(q1q2···qn) z |ψ/angbracketright (13) for|ψ/angbracketright∈HCF. As an example, consider the three-body parity qubit ( ijk). With this qubit, a logical three-body interaction of the form exp/parenleftbig iφ˜σ(i) z˜σ(j) z˜σ(k) z/parenrightbig (14) is directly accessible via the physical single-qubit operation exp/parenleftbig iφσ(ijk) z/parenrightbig . (15) Note that the placement of parity qubits σ(q1q2···qn) z involved in more than two logical lines typically requires a tailored qubitlayout [ 29]. This approach is in particular useful for solving combinatorial optimization problems with higher-order inter-actions (see Sec. III D ). D. Derived higher-order interactions Alternatively, we can combine parity qubits with actual physical two-qubit gates to obtain logical multiqubit gates.Consider, for example, a physical controlled phase gate CP(i,(jk)) φbetween a parity qubit ( jk) and a data qubit ( i). For a computational basis state |si/angbracketright|sj/angbracketright|sk/angbracketright,s{i,j,k}∈{0,1}, this gate applies a phase if and only if sj/negationslash=skandsi=1. Flipping the parity qubit ( jk), this can be used to construct a logical 2-controlled phase gate (and in particular a CCZgate forφ=π)a s CC˜P(i,j,k) φ=C˜P(i,j) φ/2C˜P(i,k) φ/2σ(jk) xCP(i,(jk)) φ/2σ(jk) xP(i) −φ/2. (16) Here P(i) −φ/2is a single-qubit phase gate and up to a global phase equivalent to a single-qubit rotation R(i) z(−φ/2). The action of the operators in Eq. ( 16) on all computational basis states is given in Table I.TABLE I. Phases acquired by application of the constituent oper- ators for the CC˜P(i,j,k) φ gate on a computational basis state |si/angbracketright|sj/angbracketright|sk/angbracketright. For all basis states, the phases of the gate sequence sum up to the required phase. Physical operations, which are applied directly to the parity or data qubits, are written without a tilde, while a tilde denoteseffective logical operations. sisjsk CC˜P(i,j,k) φC˜P(i,j) φ/2C˜P(i,k) φ/2σ(jk) xCP(i,(jk)) φ/2σ(jk) xP(i) −φ/2 000 0 0 0 0 0 001 0 0 0 0 0 010 0 0 0 0 0 011 0 0 0 0 0100 0 0 0 φ/2 −φ/2 101 0 0 φ/20 −φ/2 110 0 φ/20 0 −φ/2 111 φφ / 2φ/2 φ/2 −φ/2 The Toffoli gate, an important constituent of many quan- tum circuits [ 2,36–38], can be decomposed as CCNOT(i,j,k)=H(k)CCZ(i,j,k)H(k)(17) and, together with Eq. ( 16), reduced to two-qubit operators in the parity

Question 1334 multiple-choice

In quantum information theory, the tensor product of Hilbert spaces plays a crucial role in describing composite systems. Understanding the structure of symmetric and anti-symmetric subspaces within these tensor products is fundamental for analyzing properties of bosonic and fermionic states. For a Hilbert space of dimension d, which of the following statements correctly describes the dimensions of the symmetric and anti-symmetric subspaces for k = 2, and their relation to the full tensor product space (ℂ^d)⊗2? 1) The symmetric subspace has dimension d^2, the anti-symmetric subspace has dimension zero, and their sum equals d^2. 2) The symmetric subspace has dimension (d+1 choose 2), the anti-symmetric subspace has dimension (d choose 2), and their sum equals d^2. 3) The symmetric subspace has dimension (d choose 2), the anti-symmetric subspace has dimension (d+1 choose 2), and their sum equals 2d^2. 4) The symmetric subspace has dimension d, the anti-symmetric subspace has dimension d^2−d, and their sum equals d^2. 5) The symmetric and anti-symmetric subspaces both have dimension d, and their sum equals 2d. 6) The symmetric subspace has dimension 2d, the anti-symmetric subspace has dimension d^2−2d, and their sum equals d^2. 7) The symmetric subspace has dimension d^2−1, the anti-symmetric subspace has dimension 1, and their sum equals d^2.

✓ Correct Answer:

The correct answer is 2) The symmetric subspace has dimension (d+1 choose 2), the anti-symmetric subspace has dimension (d choose 2), and their sum equals d^2..

📚 Reference Text:

unordered subset of kelements from a set of d elements, which is/parenleftbigd k/parenrightbig . We will now present a proposition that establishes a relationship between the symmetric and anti-symmetric subspaces. Proposition 21. We haveP(d,k)† asymP(d,k) sym = 0. In particular P(d,k) asym andP(d,k) sym are orthogonal with respect to the Hilbert-Schmidt inner product. Proof. We have: P(d,k)† asymP(d,k) sym =P(d,k) asymP(d,k) sym =1 k!/summationdisplay π∈Sksgn (π)Vd(π)P(d,k) sym =/parenleftigg 1 k!/summationdisplay π∈Sksgn (π)/parenrightigg P(d,k) sym = 0,(89) where we used that Vd(π)P(d,k) sym =P(d,k) sym and that1 k!/summationtext π∈Sksgn (π) = 0 since: 1 k!/summationdisplay π∈Sksgn (π) =1 k!/summationdisplay π∈Sksgn (τπ) = sgn (τ)1 k!/summationdisplay π∈Sksgn (π) =−1 k!/summationdisplay π∈Sksgn (π), (90) whereτis any permutation. Note that, for k= 2 (andd > 1), we have that, dim/parenleftbig Sym2(Cd)/parenrightbig + dim/parenleftbig ASym2(Cd)/parenrightbig =/parenleftbigd+1 2/parenrightbig +/parenleftbigd 2/parenrightbig =d2, this implies that (Cd)⊗2= Sym2(Cd)⊕ASym2(Cd). SinceP(d,2) sym andP(d,2) asym are both linear combinations of permutations, they commute with U⊗2 for all unitaries U. Consequently, they are elements of a basis for the commutant Comm(U(d),k= 2)and they are orthogonal to each other. Furthermore, the commutant can have at most dimension 12 two, since it is spanned by the identity and Flip operators (Example 14). Thus, P(d,k) sym andP(d,k) asym form an orthogonal basis for the commutant. Since the moment operator is the orthogonal projector onto the commutant (Theorem 7), we can derive the formula for the second-order moment operator (already derived in Corollary 13), usingP(d,k) sym andP(d,k) asym as the basis. We express the moment operator as: E U∼µH/bracketleftbig U⊗2OU†⊗2/bracketrightbig =/angbracketleftigP(d,2) sym/vextenddouble/vextenddouble/vextenddoubleP(d,2) sym/vextenddouble/vextenddouble/vextenddouble 2,O/angbracketrightig HSP(d,2) sym/vextenddouble/vextenddouble/vextenddoubleP(d,2) sym/vextenddouble/vextenddouble/vextenddouble 2+/angbracketleftigP(d,2) asym/vextenddouble/vextenddouble/vextenddoubleP(d,2) asym/vextenddouble/vextenddouble/vextenddouble 2,O/angbracketrightig HSP(d,2) asym/vextenddouble/vextenddouble/vextenddoubleP(d,2) asym/vextenddouble/vextenddouble/vextenddouble 2(91) =Tr/parenleftig P(d,2) symO/parenrightig Tr/parenleftig P(d,2) sym/parenrightigP(d,2) sym +Tr/parenleftig P(d,2) asymO/parenrightig Tr/parenleftig P(d,2) asym/parenrightigP(d,2) asym (92) =Tr(O) + Tr( FO) d(d+ 1)/parenleftbiggI+F 2/parenrightbigg +Tr(O)−Tr(FO) d(d−1)/parenleftbiggI−F 2/parenrightbigg (93) =Tr(O)−d−1Tr(FO) d2−1I+Tr(OF)−d−1Tr(O) d2−1F, (94) where in the first equality we used the fact that P(d,2) sym andP(d,2) asym are an orthonormal basis for the commutant, we utilized the fact that P(d,2) sym andP(d,2) asym are orthogonal projectors (meaning that they square to themselves and are Hermitian), in the third equality we substituted P(d,2) sym =1 2(I+F) andP(d,2) asym =1 2(I−F)along with their respective traces to arrive at the desired formula. Now we show a theorem that states that for all quantum states |ϕ⟩in ad-dimensional Hilbert space, the moment operator of |ϕ⟩⟨ϕ|⊗kis a uniform linear combination of permutations, which can be written in terms of the projector on the symmetric subspace P(d,k) sym. Theorem 22. Letd,k∈N. For all|ϕ⟩∈Cd, the moment operator is a uniform linear combination of permutations: E U∼µH/bracketleftig U⊗k|ϕ⟩⟨ϕ|⊗kU†⊗k/bracketrightig =P(d,k) sym Tr/parenleftig P(d,k) sym/parenrightig, (95) withP(d,k) sym =1 k!/summationtext π∈SkVd(π)andTr/parenleftig P(d,k) sym/parenrightig = dim/parenleftbig Symk(Cd)/parenrightbig =/parenleftbigk+d−1 k/parenrightbig . Proof. For allσ∈Sk, we left-multiply both sides of Eq. (30) by Vd(σ−1) and obtain: Vd(σ−1)M(k) µH/parenleftig (|ϕ⟩⟨ϕ|)⊗k/parenrightig =/summationdisplay π∈SkcπVd(σ−1)Vd(π), (96) whereM(k) µH/parenleftig (|ϕ⟩⟨ϕ|)⊗k/parenrightig :=E U∼µH/bracketleftig/parenleftbig U|ϕ⟩⟨ϕ|U†/parenrightbig⊗k/bracketrightig . Using the fact that Vd(σ−1) commutes with U⊗kfor all unitaries Uand thatVd(σ−1)|ϕ⟩⊗k=|ϕ⟩⊗k, the LHS is: Vd(σ−1)M(k) µH/parenleftig (|ϕ⟩⟨ϕ|)⊗k/parenrightig =M(k) µH/parenleftig (|ϕ⟩⟨ϕ|)⊗k/parenrightig . (97) The RHS is: /summationdisplay π∈SkcπVd(σ−1)Vd(π) =/summationdisplay

Question 1335 multiple-choice

Quantum computing hardware platforms, such as neutral atoms and superconducting circuits, require specialized circuit compilation techniques to efficiently implement algorithms like QFT and QAOA. Optimization and adaptation to native gate sets are crucial for practical quantum algorithm execution. Which compiler is specifically used to translate and optimize quantum circuits for hardware constraints such as qubit connectivity and native gate sets in both neutral atom and superconducting circuit platforms? 1) Qiskit 2) Cirq 3) pytket 4) Q# 5) Forest 6) QuTiP 7) Pennylane

✓ Correct Answer:

The correct answer is 3) pytket.

📚 Reference Text:

Author Accepted Manuscript version arising from this submission. This project was fundedwithin the QuantERA II Programme that has received fund-ing from the European Union’s Horizon 2020 researchand innovation programme under Grant Agreement No.101017733. W.L. and D.B. conceived the project. D.B. and C.D. developed the models for neutral atoms and D.B. forsuperconducting circuits. D.B. performed all numerical sim-ulations. All authors analyzed and discussed the data andcontributed to the writing of the manuscript. The authors declare no competing interests. APPENDIX A: QUANTUM CIRCUITS FOR QFT AND QAOA 1. Standard gate model The quantum Fourier transform (QFT) is a key ingre- dient in Shor’s algorithm for integer factorization [ 1] and thus a prototypical application for quantum computers. Whilea typical circuit representation of a QFT contains exclu-sively Hadamard gates, H, and controlled-phase gates, R n= diag{1,1,1,exp{2πi/2n}}, a more efﬁcient representation with fewer gates and lower circuit depths can be constructedusing SW AP gates [ 45]. Figure 6(a) illustrates the circuit for a QFT with N=4 qubits. The quantum approximate optimization algorithm (QAOA) aims at ﬁnding approximate solution tocombinatorial optimization problems [ 4]. For instance, let us consider the task to ﬁnd the ground state of the Nqubit spin glass Hamiltonian H z=N/summationdisplay n,m=1 n

Question 1336 multiple-choice

Reconstructing configuration interaction (CI) expansions from complex wave functions is important in quantum chemistry, especially when dealing with strongly correlated electron systems and large active spaces. Recent innovations combine matrix-product state representations with algorithmic techniques to efficiently identify significant electronic configurations. Which optimization strategy employs quantum information theory to enhance the crossover and mutation steps in a genetic algorithm for searching large Hilbert spaces for important CI configurations? 1) Simulated annealing with local orbital rotations 2) Genetic algorithm with quantum information theory-guided crossover and mutation 3) Direct diagonalization of the Hamiltonian matrix 4) Hartree-Fock-based excitation generation 5) Cluster expansion using coupled-cluster theory 6) Stochastic selection via Monte Carlo sampling 7) Tensor decomposition using singular value thresholding

✓ Correct Answer:

The correct answer is 2) Genetic algorithm with quantum information theory-guided crossover and mutation.

📚 Reference Text:

Title: Efficient Reconstruction of CAS-CI-Type Wave Functions for a DMRG State Using Quantum Information Theory and a Genetic Algorithm. Year: 2017 Paper ID: ee47288cc3f38eb77596777fb4e5031e27e820f3 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/ee47288cc3f38eb77596777fb4e5031e27e820f3 Abstract: We improve the methodology to construct a complete active space-configuration interaction (CAS-CI) expansion for density-matrix renormalization group (DMRG) wave functions using a matrix-product state representation, inspired by the sampling-reconstructed CAS [SR-CAS; Boguslawski , K. ; J. Chem. Phys. 2011 , 134 , 224101 ] algorithm. In our scheme, the genetic algorithm, in which the "crossover" and "mutation" processes can be optimized based on quantum information theory, is employed when reconstructing a CAS-CI-type wave function in the Hilbert space. Analysis of results for ground and excited state wave functions of conjugated molecules, transition metal compounds, and a lanthanide complex illustrate that our scheme is very efficient for searching the most important CI expansions in large active spaces.

Question 1337 multiple-choice

The affine group Ap over a prime field Zp is a fundamental algebraic structure in group theory and cryptography, consisting of ordered pairs with a specific multiplication rule and rich subgroup architecture. Quantum algorithms for the hidden shift and hidden subgroup problems often exploit the interplay between these subgroups and group extensions. Which subgroup of the affine group Ap over Zp consists precisely of all elements of the form (1, b), representing lines with slope 1 under the group’s action? 1) The subgroup Hq consisting of elements (at, 0) 2) The nonnormal subgroup H ≅ Z∗p 3) The normal subgroup N ≅ Zp 4) The conjugate subgroup Hb consisting of elements (a, (1−a)b) 5) The affine group Ap itself 6) The normal subgroup Nq ≅ Zq ⋉ Zp 7) The commutator subgroup of Ap

✓ Correct Answer:

The correct answer is 3) The normal subgroup N ≅ Zp.

📚 Reference Text:

(multiplicative) cosets of a known multiplicative subgroup of Z∗ p. These functions have been studied in some depth for their pseudorandom properties, and several instances have been suggestedas cryptographically strong pseudorandom generators. By associating f swith its Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 944 MOORE, ROCKMORE, RUSSELL, AND SCHULMAN isotropy subgroup, and using our reconstruction algorithm to ﬁnd that subgroup, we give an eﬃcient quantum algorithm for the hidden shift problem in the case wheref(x) is a function of x’s multiplicative order mod rfor some r= polylog( p). This generalizes the work of van Dam, Hallgren, and Ip [4], who give an algorithm forhidden shift problems in the case where fis precisely a multiplicative character. Finally, in section 8 we show that the set of groups for which the HSP can be solved in polynomial time has the following closure property: if H={H n}is a family of groups for which we can eﬃciently solve the HSP and K={Kn}is a family of groups for which |Kn|= polylog( |Hn|), we can also eﬃciently solve the HSP for the family {Gn}, where each Gnis any extension of KnbyHn. This subsumes the results of [13] on Hamiltonian groups, and also those of [15] on groups with commutatorsubgroups of polynomial size. We note that subsequent to this work, Bacon, Childs, and van Dam [1] found additional algorithms for the HSP in the aﬃne groups. Their approach uses the“pretty good measurement,” which they showed is optimal for certain cases of theHSP (see also [21]). Their work extends to a number of other group families, such asthe Heisenberg groups. 2. The aﬃne and q-hedral groups. LetA pbe the aﬃne group , consisting of ordered pairs ( a, b)∈Z∗ p×Zp, where pis prime, under the multiplication rule (a1,b1)·(a2,b2)=( a1a2,b1+a1b2).Apcan be viewed as the set of aﬃne functions f(a,b):Zp→Zpgiven by f(a,b):x/mapsto→ax+bwhere multiplication in Apis given by function composition. Structurally, Apis a semidirect product Z∗ p/multicloseleftZ p∼=Zp−1/multicloseleftZ p. Its subgroups can be described as follows: •LetN∼=Zpbe the normal subgroup of size pconsisting of elements of the form (1 ,b). Geometrically, this is the set of aﬃne functions with slope 1. •LetH∼=Z∗ p∼=Zp−1be the nonnormal subgroup of size p−1 consisting of the elements of the form ( a,0). Geometrically, this is the set of lines passing through the origin. •For each b∈Zp, the conjugate subgroup Hb=( 1,b)·H·(1,−b) consists of elements of the form ( a,(1−a)b). In the action on Zp,Hbis the stabilizer of b; geometrically, Hbis the set of lines intersecting the diagonal at ( b, b). •Ifa∈Z∗ phas order q, where qdivides p−1, let Nq∼=Zq/multicloseleftZ pbe the normal subgroup consisting of all elements of the form ( at,b). Geometrically, Nqis the set of lines whose slope is a power of a. •Similarly, if a∈Zphas order q, let Hqbe the nonnormal subgroup Hq= /angbracketleft(a,0)/angbracketrightof size q. Then Hqconsists of the elements of the form ( at,0), and its conjugates Hb q=( 1,b)·Hq·(1,−b) consist of the elements of the form (at,(1−at)b). Geometrically, these are the subsets of

Question 1338 multiple-choice

In quantum information theory, the representation theory of symmetry groups such as SU(d) plays a key role in determining how quantum codes can be constructed and which operations they support. The scaling of representation dimensions and subsystem sizes fundamentally limits fault-tolerant quantum computation. For a quantum code covariant with the full unitary group on the logical space and storing multiple logical qubits with high accuracy, what is the implication of the mathematical bounds on local subsystem dimensions with respect to error correction and transversal gate implementation? 1) All physical subsystems can remain small regardless of the desired accuracy and logical qubit count. 2) At least one physical subsystem must be very large, especially as error correction becomes more stringent or more logical qubits are encoded. 3) Subsystem dimensions are completely independent of code accuracy or gate requirements. 4) The bounds guarantee universal transversal gates without any restrictions on subsystem size. 5) Fault-tolerant codes can always be implemented with minimal physical resources. 6) Logical qubit count only affects the number of subsystems, not their dimension. 7) Increasing the number of logical qubits only requires linear scaling of subsystem dimensions.

✓ Correct Answer:

The correct answer is 2) At least one physical subsystem must be very large, especially as error correction becomes more stringent or more logical qubits are encoded..

📚 Reference Text:

one generator Jzis diagonal in the canonical basis, and the integer λ¼λ1¼2jdetermines the highest spin attainable in that irrep. For the fundamentalrepresentation λ¼ð1;0ÞofSUð3Þ, the two simultaneously diagonalizable generators are the two Gell-Mann matricesthat are diagonal in the canonical basis. Since the entries in λare decreasing, the largest eigenvalue that any generator could have is λ 1; i.e., kTðiÞ λk∞≤λ1. It turns out that the irrep that minimizes the dimension out of all irreps with fixed λ1 is the completely symmetric irrep ðλ1;0;0;…;0Þ. The dimension of this irrep is the dimension of the symmetric subspace on λ1number of d-dimensional systems, which is a polynomial of degree d−1inλ1. Therefore, in order to fit in a system of dimension di, the largest possible λ1is of the order of Oðd1=ðd−1Þ i Þ. Now, any general representation can be decomposed into irreps, and a generator Tis simply T¼⨁Tλ, where Tλis the corresponding generator for each irrep. We then have kTk∞¼max λkTλk∞. So, if a representation fits in the system dimension di, then it cannot contain any irrep λwith λ1larger than Oðd1=ðd−1Þ i Þ. For a code that is covariant with respect to the full unitary group on the logical space, we have d¼dL, and picking a simple standard generator for our earlier bound (26),w e obtain the following theorem. Theorem 6. (Approximate Eastin-Knill theorem for quantum computation). Consider a SUðdLÞ-covariant code as in Fig. 1. Then, the local subsystem dimensions must obey max ilndi≥ln/C18dL−1þ⌈½2nϵworstðN∘EÞ/C138−1⌉ dL−1/C19 ð54aÞ ≥lnðdL−1Þ 2nϵworst−lnð1þð2nϵworstÞ−1Þ 2nϵworst: ð54bÞ A similar bound can be obtained for the figure of merit ϵe, based on the (fixed-input) entanglement fidelity, by making in Eq. (54a) the replacement ϵworst→dLϵe=2. In other words, any code that (a) stores a large amount of quantum information and (b) admits universal transversal gates has severe restrictions on its ability to recover fromerasure errors. Ifϵ worst is kept constant and for dL→∞, then we can ignore the second term in Eq. (54b) , and we see that there must be an ifor which ln ðdiÞgrows linearly in ln ðdLÞwith a proportionality factor 1=ð2nϵworstÞ. We can imagine that each physical subsystem of the physical system is com-posed of m i∼lndiqubits lumped together. Then if, for instance, we wish to achieve a precision of ϵworst∼10−3, then we must have the scaling ln ðdiÞ≳ð500=nÞlnðdLÞ, or, equivalently, mi¼log2ðdiÞ≳ð500=nÞlog2ðdLÞ. Con- cretely, for ten logical qubits ( dL¼210) encoded into n¼10subsystems, Eq. (54b) tells us that at least one physical subsystem needs to be of a respectable dimension log 2ðdiÞ≥216qubits; i.e., at least one of the unitary tensor factors in the physical transversal unitary must act jointly onmi≥216qubits lumped together into a single physical subsystem. In order to consider the regime of extremely high accuracy, we can recast the bound (54a) asCONTINUOUS SYMMETRIES AND APPROXIMATE QUANTUM … PHYS. REV . X 10,041018 (2020) 041018-17 max ilndi≥ðdL−1Þln/C181 2ϵworstndL/C19 ð55Þ (see proof in Supplemental Material, Sec. E. 2 [37]). Suppose we wish to accurately resolve individual logical basis states of a highly mixed logical state. The logicalinformation might, for instance, be

Question 1339 multiple-choice

In quantum field theory, the beta function determines how the coupling constants evolve with changes in energy scale, which is crucial for predicting behaviors in gauge theories such as QCD and QED. High-order loop corrections, such as five-loop calculations, provide refined theoretical predictions and rely on advanced computational methods to manage the complexity of Feynman diagrams and divergences. Which statement is true regarding the five-loop corrections to the beta function in Quantum Chromodynamics (QCD) within the $\overline{\mathrm{MS}}$ renormalization scheme? 1) The five-loop corrections are significantly smaller than the four-loop corrections, indicating a well-behaved perturbative expansion. 2) The five-loop corrections are larger than all lower-order corrections, suggesting instability in the perturbative series. 3) The five-loop corrections eliminate the need for renormalization in QCD. 4) The five-loop corrections show non-perturbative effects dominate at all energy scales. 5) The five-loop corrections reverse the sign of the beta function, making QCD coupling constant increase with energy. 6) The five-loop corrections are only relevant in Abelian gauge theories like QED. 7) The five-loop corrections cause the breakdown of asymptotic freedom in QCD.

✓ Correct Answer:

The correct answer is 1) The five-loop corrections are significantly smaller than the four-loop corrections, indicating a well-behaved perturbative expansion..

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Title: The five-loop beta function of Yang-Mills theory with fermions Year: 2017 Paper ID: e809beecbae48ee2b355e46db0adf89bb9cd1c62 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/e809beecbae48ee2b355e46db0adf89bb9cd1c62 Abstract: A bstractWe have computed the five-loop corrections to the scale dependence of the renormalized coupling constant for Quantum Chromodynamics (QCD), its generalization to non-Abelian gauge theories with a simple compact Lie group, and for Quantum Electrodynamics (QED). Our analytical result, obtained using the background field method, infrared rearrangement via a new diagram-by-diagram implementation of the R* operation and the Forcer program for massless four-loop propagators, confirms the QCD and QED results obtained by only one group before. The numerical size of the five-loop corrections is briefly discussed in the standard MS¯$$ \overline{\mathrm{MS}} $$ scheme for QCD with nf flavours and for pure SU(N) Yang-Mills theory. Their effect in QCD is much smaller than the four-loop contributions, even at rather low scales.

Question 1340 multiple-choice

In lattice-based cryptanalysis, techniques such as modulus switching and Fourier analysis are used to efficiently distinguish candidate secrets in attacks on schemes like Learning With Errors (LWE). Normalization and careful parameter selection are critical for the statistical reliability of such attacks. Which parameter is specifically introduced to reduce the candidate search space from q^kfft to p^kfft, thereby increasing the efficiency of dual lattice attacks? 1) BKZ block size 2) Modulus switching target p 3) Cutoff parameter C 4) Normalization constant α 5) Sample size D 6) Scaling factor ψ(sfft) 7) Variance of the error distribution

✓ Correct Answer:

The correct answer is 2) Modulus switching target p.

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AT latqIklat/parenrightbigg , whereαis a constant equal toσe σsand is used for normalisation in the case that s,ehave diﬀerent distributions. We ﬁnd Dshort vectors of the form/parenleftbigg α·xj yj,lat/parenrightbigg in the column space of Busing some short vectors sampling procedure (see Al- gorithm 1). Then, given a list LofDsuch vectors let yj,ﬀt:=xT j·Aﬀtand yj,enum:=xT j·Aenum. We can then deﬁne the function FL(˜senum,˜sﬀt) = ℜ/parenleftBigg 1 ψ(˜sﬀt)/summationdisplay jexp/parenleftBigg/parenleftBigg/floorleftbiggp q·yj,ﬀt/ceilingrightbiggT ·˜sﬀt+p q·yT j,enum·˜senum−p q·xT j·b/parenrightBigg ·2iπ p/parenrightBigg/parenrightBigg 10 Table 1. Dual attack parameters. parameters explanation n,m,χs,χeLWE parameters as in Deﬁnition 1 β0,β1 BKZ block size β0and sieving dimension β1 p modulus switching target modulus µ the target success probability 0 <µ<1 σ2 s,σ2 e variances of χsandχerespectively senum,sﬀt,slatcomponents of scovered by exhaustive search, FFT and lattice reduction ˜senum,˜sﬀt guesses for senummodqandsﬀtmodp Nenum(senum) number of guesses with ˜senumwith probabilities larger than the probability of senumwhen drawing sfromχs. kenum,kﬀt,klatdimension of senum,sﬀtorslatrespectively Aenum,Aﬀt,AlatA·s=Aenum·senum+Aﬀt·sﬀt+Alat·slat α scaling/normalisation factor α:=σe/σs L,D list/number of short vectors returned by sieving oracle D:=|L| ψ(sﬀt) = exp(2πicq′ p/summationtext tst) whereq′=q/gcd(p,q),cq′:= 0 when q′is odd andcq′:= 1/2q′whenq′is even C FFT cutoﬀ value for scoring function Φ deﬁned in Section 2.3 φfp(µ),φfn(µ) =Φ−1/parenleftBig 1−µ 2·Nenum(senum)·pkfft/parenrightBig ,Φ−1/parenleftbig 1−µ 2/parenrightbig Deq exponential factor coming from the clean Fourier coeﬃcient of the error in the dual attack equations for required samples Dround exponential factor coming from the rounding when modulus switching for required samples Darg≈1/2, improvement factor coming from considering the complex argument of the Fourier coeﬃcient rather than only the magni - tude. Dfpfn a polynomial factor controlling false positives and negati ves 11 for all˜senum∈Zkenumqand˜sﬀt∈Zkfftp. First, note that ˜sﬀtis modp≪qrather thanqmeaning that only up to pkfftcandidates have to be considered rather thanqkfft. Applying modulus switching here is the key innovation of [MAT22], but since its details do not matter for us here, we refer the reader to [MAT22] for details. Second, here ψ(˜sﬀt) is a complex factor of norm 1 deﬁned in [MAT22, p. 25, proof of Lemma 5.4] and easily computable (cf. Table 1). The f unctionFL essentially performs an FFT on values drawn from a certain distribut ion. Via an analysis that we do not reproduce, one can show that the fun ction FLabove has the following properties with respect to some cutoﬀ para meterC (computed below): –If˜senum/n⌉}a⊔ionslash=senumthenFL(˜senum,˜sﬀt)C –There might be ˜sﬀt/n⌉}a⊔ionslash=sﬀtsuch thatFL(senum,˜sﬀt)>C. The ﬁrst point corresponds to a wrong guess. In this case, values on which the FFT is performed follow a uniform distribution and the expected v alue of FL(˜senum,˜sﬀt) is 0. The second point corresponds to the correct guess. In this case, values on which the FFT is performed essentially follow a normal distri- bution with nonzero mean and therefore the expected value of FL(senum,sﬀt) is nonzero. By carefully choosing the value of C, and taking suﬃciently many samples in the list, we can ensure that these properties hold with high proba- bility. The third point follows from the fact that [MAT22] performs a m odulo switching operation that can introduce some errors and makes the analysis of

Question 1341 multiple-choice

In the study of finite p-groups, particularly 2-groups, the structure of maximal subgroups and centralizers plays a crucial role in group classification. Groups such as the quaternion group, elementary abelian groups, and metacyclic groups are central examples in this area. Which property must hold for all non-central elements in a finite P-group as classified by the structure of their centralizers? 1) Every centralizer contains the Frattini subgroup as a proper subset. 2) Every centralizer is cyclic or of minimal order. 3) Every centralizer is a direct product of two non-cyclic subgroups. 4) Every centralizer is equal to the whole group. 5) Every centralizer is abelian and non-cyclic. 6) Every centralizer is isomorphic to the quaternion group. 7) Every centralizer has index 2 in the group.

✓ Correct Answer:

The correct answer is 2) Every centralizer is cyclic or of minimal order..

📚 Reference Text:

1/a\}bracketri}ht; (5)/a\}bracketle{ta,b,c/vextendsingle/vextendsinglea2n−3=b22= 1,c2=a2b2,[a,b] =b2,[c,a] =a2n−4,[c,b] = 1/a\}bracketri}ht; (6) Q8×C2; (7)/a\}bracketle{ta,b,c/vextendsingle/vextendsinglea4=c4= 1,a2=b2,[a,b] =a2,[c,a] =c2,[c,b] = 1/a\}bracketri}ht; FINITE p-GROUPS 375 (8)/a\}bracketle{ta,b,c,d/vextendsingle/vextendsinglea4=b4= 1,c2=a2b2,b2=d2,[a,b] =b2,[a,c] =a2,[a,d] = 1,[b,c] = 1,[b,d] =a2,[c,d] =c2/a\}bracketri}ht. Proof.Ifn≤5, then, by classiﬁcation of 2-groups of order ≤25, the conclusion holds. In following assume n≥6 andGis aP-group. By induction hypothesis, each maximal subgroup of Gis abelian or isomor- phic to one of the groups (1)–(5), (7) and (8). If Ghas a maximal subgroup which is isomorphic to one of the groups (2)–(5), (7) and (8), then Gis isomor- phic to one of the groups (2)–(5) and (8) by Lemma 5.1, Lemmas 5.3, 5.4, 5.5, 5.6 and 5.7. Assume every maximal subgroup of Gis abelian or metacyclic minimal non- abelian. By Lemma 3.1, every maximal subgroup of Gis metacyclic. If Gis not metacylcic, then Gis minimal non-metacyclic. It follows that |G| ≤25by Lemma 2.5. This contradicts |G| ≥26. ThusGis metacyclic. IfGis minimal non-abelian, then we get the group (1). IfGis not minimal nonabelian, then Gis a metacyclic A2-group. By Lemma 2.2, |G′|= 4. Assume G=/a\}bracketle{ta,b/a\}bracketri}ht, whereG′ Z(G). Thus |CG(x)| ≤16. It follows that CG(x) is abelian and hence d(CG(x))≤2. Thus there exists y∈Gsuch that CG(x) =/a\}bracketle{tx,y/a\}bracketri}ht. It follows that CG(x)//a\}bracketle{tx/a\}bracketri}ht=/a\}bracketle{t¯y/a\}bracketri}htis cyclic. /square Acknowledgments. We cordially thank the referee for valuable comments. All these helped us to improve the whole paper substantially. 376 L. ZHANG, J. WANG, AND H. QU References [1] Y. Berkovich, Groups of Prime Power Order, Volume 1 , Walter de Gruyter, Berlin, 2008. [2] Y. Berkovich and Z. Janko, Structure of ﬁnite p-groups with given subgroups , Contemp. Math. 402, 13–93, Amer. Math. Soc., Providence, RI, 2006. [3] N. Blackburn, Generalizations of certain elementary theorems on p-groups, Proc. London Math. Soc. (3) 11(1961), 1–22. [4]

Question 1342 multiple-choice

Quantum information processing leverages quantum algorithms and gate operations to solve complex problems more efficiently than classical computing. Qudit systems, which generalize qubits to d-level quantum states, enable novel circuit designs and computational strategies in quantum computing. In the context of implementing the Quantum Fourier Transform (QFT) on an n-qudit, d-level quantum system, which gate plays a crucial role in creating equal superpositions over all d computational basis states? 1) The standard controlled-NOT gate 2) The Toffoli gate 3) The generalized Hadamard gate (Hd) 4) The swap gate 5) The generalized phase gate (Rd k) 6) The measurement gate 7) The Pauli-X gate

✓ Correct Answer:

The correct answer is 3) The generalized Hadamard gate (Hd).

📚 Reference Text:

of |ψ2〉. By measuring the ﬁrstN quantum state of |ψ3〉we can obtain the natural number string we want that is offset up to a constant g(a1),g(a2),g(a3),... ,g(aN) (92) using a single query of the oracle function. The Bernstein-Vazirani algorithm clearly demonstrates the power of quantum computing. It outperforms the best classicalalgorithm in terms of speed by a factor of N[95]. The qudit generalizations of the Bernstein-Vazirani algorithm helps uscomprehend the potential of the qudit systems. 3.2 Qudit Algorithms for the Hidden Abelian Subgroup Problems Many of the widely used quantum algorithms such as the discrete Fourier transform, the phase estimation and the factoring ﬁt into the framework of the hidden subgroup problem (HSP). In thissection, we review the qudit generalization of these algorithms.The qudit Fourier transform is discussed in Section 3.2.1 and its application, the PEA is reviewed in Section 3.2.2 . A direct application of these algorithms, Shor ’s factoring algorithm performed with qutrits and in metaplectic quantumarchitectures is also introduced Section 3.2.2 . 3.2.1 Quantum Fourier Transform With Qudits The quantum Fourier transform algorithm (QFT) is realizable on a qubit system [ 125]. QFT, as the heart of many quantum algorithms, can also be performed in a qudit system [ 145, 165]. In an N-dimensional system represented with n d-dimensional qudits, the QFT, F(d,N), where N/equalsdn, transforms the computational basis {|0〉,|1〉,... ,|n−1〉} (93) FIGURE 13 | Illustration of a qudit multipod linkage: the top is in the original basis and the bottom is in the Morris-Shore basis. Δis a common detuning between a common (ancilla) state and other qudit states, Ωk represents the single-photon Rabi frequencies. State |b〉is a superposition of the qudit states weighted by the couplings Ωk;|un〉are the states that are not in the dynamics. Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504 13Wang et al. Qudits and High-Dimensional Quantum Computing into a new basis set [ 26] F(d,N)|j〉/equals1⎥radicaltpext⎥radicaltpext N√⎥summationdisplayN−1 k/equals0e2πijk/N|k〉. (94) For convenience, we write an integer jin a base- dform. If j>1 then j/equalsj1j2/jn/equalsj1dn−1+jn−2 2+/+jnd0(95) and, if j<1, then j/equals0.j1j2/jn/equalsj1d−1+j2d−2+/+jnd−n. (96) The QFT acting on a state |j〉can be derived and rewritten in a product form as |j〉/equals|j1j2/jn〉11 dn/2⎥summationdisplaydn−1 k/equals0e2πijk/dn|k〉 /equals1 dn/2⎥summationdisplayd−1 k1/equals0/⎥summationdisplayd−1 kn/equals0e2πij⎥parenleftBig⎥summationtextn l/equals1kld−l⎥parenrightBig|k1k2/kn〉 /equals1 dn/2⎥summationdisplayd−1 k1/equals0/⎥summationdisplayd−1 kn/equals0⊗n l/equals1e2πijkld−l|kl〉 /equals1 dn/2⊗n l/equals1⎡⎢⎢⎣⎥summationdisplayd−1 kl/equals0e2πijkld−l|kl〉⎤⎥⎥⎦. This process can be realized with the quantum circuit shown in Figure 10 , and the fully expanded expression of the product form is shown on the right side of the ﬁgure. The generalized Hadamard gate Hdin the ﬁgure is de ﬁned as Hd:/equalsF(d,d) which effects the transform Hd⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglejn〉/equals|0〉+e2πi0.jn|1〉+/+e2(d−1)πi0.jn|d−1〉. (97) The matrix representation of Hdis ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝11 / 1 1e 2πi0.1/ e2πi0.(d−1) «« 1 « 1e2(d−1)πi0.1/ e2(d−1)πi0.(d−1)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (98) In the circuit the Rd kgate is a phase gate that has the expression Rd k/equals⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝10 / 0 0e 2πi/dk/ 0 «« 1 « 00 / e2πi(d−1)/dk⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (99) The black dots in the circuit are multi-value-controlled gates that apply Rd kto the target qudit jtimes for a control qudit in state |j〉. In order to complete the Fourier

Question 1343 multiple-choice

In finite group theory, the study of p-groups—groups whose order is a power of a prime—relies heavily on understanding subgroup structures, centers, and properties such as maximal class and commutator relations. These elements are crucial for classifying and distinguishing types of p-groups. Which of the following statements is true regarding finite p-groups of maximal class? 1) The center of a p-group of maximal class is always of order p squared. 2) Every maximal subgroup of a p-group of maximal class is abelian. 3) The Frattini subgroup of a p-group of maximal class is equal to the whole group. 4) If a maximal subgroup is of maximal class, then the whole group must be of maximal class. 5) The commutator subgroup of a p-group of maximal class is always trivial. 6) All p-groups of maximal class are cyclic. 7) Metacyclic subgroups cannot exist in p-groups of maximal class.

✓ Correct Answer:

The correct answer is 4) If a maximal subgroup is of maximal class, then the whole group must be of maximal class..

📚 Reference Text:

discuss case by case. IfGis the group (1) in Lemma 2.6, then, by Lemma 3.2, Gis the group (1). IfGis the group (2) in Lemma 2.6, then Z(G) is a cyclic subgroup of index p2. Letg∈G\Z(G). Then CG(g)< G. Obviously, CG(g)≥ /a\}bracketle{tg/a\}bracketri}htZ(G). Thus CG(g) =/a\}bracketle{tg/a\}bracketri}htZ(G). Hence CG(g)//a\}bracketle{tg/a\}bracketri}ht∼=Z(G)/Z(G)∩/a\}bracketle{tg/a\}bracketri}htis cyclic. That is, G is aP-group. This is the group (4). IfGis the group (3) in Lemma 2.6, then, by Lemma 2.9, G1is abelian or metacyclic non-abelian. Since |G| ≥35,|G1| ≥34. It follows that there exists a subgroup H/triangleleftequalGsuch that |H|= 34withd(H)≥2 andHis abelian or metacyclic non-abelian. By Lemma 3.1, d(H) = 2. By Lemma 2.8, ℧1(H)≇ C9. HenceH∼=C2 9or M3(2,2). Thus ℧1(H)≤Z(H) and℧1(H)∼=C2 3. Since Gis of maximal class, |Z(G)|= 3. So there exists x∈℧1(H)\Z(G). SinceH is not cyclic, H//a\}bracketle{tx/a\}bracketri}htis not cyclic. It follows from H≤CG(x) thatCG(x)//a\}bracketle{tx/a\}bracketri}ht is not cyclic. That means Gis not aP-group. IfGisthegroup(4)in Lemma2.6,then, bycalculations, weget Z(G) =/a\}bracketle{tap/a\}bracketri}ht is a subgroup of index p3and/a\}bracketle{ta,y/a\}bracketri}htis a abelian maximal subgroup. Since |G′|=p2,Ghas a unique abelian maximal subgroup by Lemma 2.4. Let g∈G\Z(G). Then /a\}bracketle{tap,g/a\}bracketri}ht ≤CG(g)< G. IfCG(g) =/a\}bracketle{tap,g/a\}bracketri}ht, thenCG(g)//a\}bracketle{tg/a\}bracketri}ht is cyclic. If CG(g)>/a\}bracketle{tap,g/a\}bracketri}ht, thenCG(g)⋖G. Since/a\}bracketle{tap,g/a\}bracketri}ht ≤Z(CG(g)) and |CG(g)//a\}bracketle{tap,g/a\}bracketri}ht|=p,CG(g) is abelian. Thus CG(g) =/a\}bracketle{ta,y/a\}bracketri}htandg=aiyj, where (i,p) = 1 or ( j,p) = 1. It follows that CG(g)//a\}bracketle{tg/a\}bracketri}htis cyclic. Hence Gis a P-group. This is the group (5). /square 5.P-groups of order even Lemma 5.1. AssumeGis aP-group of order 2nandn≥5,Mis a maximal subgroup of G. IfMis of maximal class, then Gis of maximal class. Proof.Otherwise, by Lemma 2.7, G/G′∼=C3 p. It follows that ℧1(G) =G′=M′=℧1(M) FINITE p-GROUPS 371 and hence exp( G) = exp(M).SinceMis of maximal class, Mhas a maximal subgroup Hwhich is cyclic by the classiﬁcation of 2-groups of maximal class. LetK= Ω2(H). ThenK∼=C4andKcharHcharM/triangleleftequalG. HenceK/triangleleftequalG. By N/Ctheorem, G/CG(K)/lessorsimilarAut(K)∼=C2. SinceMisofmaximalclass, |Z(M)|= 2. Hence K/not⋖essequa⋖Z(M). ThusG/CG(K)∼= C2. SinceGis not of maximal class, CG(K) is not cyclic. Since n≥5, K≤℧1(M′)≤Φ(CG(K)). It follows that CG(K)/Kis not cyclic. This contradicts that Gis aP-group. /square Lemma 5.2. AssumeGis aP-group of order 2nandn≥6,Mis a maximal subgroup of GandM=/a\}bracketle{ta,b,c/vextendsingle/vextendsinglea2n−4=b22= 1,c2=a2b2,[a,b] =b2,[c,a] = [c,b] = 1/a\}bracketri}ht. Then (1)Z(M) =/a\}bracketle{ta2,c/a\}bracketri}ht=/a\}bracketle{tb2,c/a\}bracketri}ht∼=C2×C2n−4andΩ1(M) =/a\}bracketle{ta2n−5,b2/a\}bracketri}ht∼=C2 2; (2) Φ(G) =℧1(G)≤Z(M)≤Z(G); (3)G′≤Ω1(Z(M)) = Ω 1(M); (4)M= Ωn−4(G), in particular, o(x) = 2n−3for allx∈G\M. Proof.(1) Obviously. (2) Firstly, we prove Z(M)≤Z(G). Ifa2/∈Z(G), thenCG(a2)//a\}bracketle{ta2/a\}bracketri}htis cyclic since Gis aP-group. Since CG(a2)≥M,M//a\}bracketle{ta2/a\}bracketri}htis cyclic. This is a contradiction. Thus a2∈Z(G). Similarly, c∈Z(G). Secondly, we prove x2∈Z(M) for all x∈G. Since x2∈M, it suﬃces to prove x2∈Z(G). If not, then, since CG(x2)//a\}bracketle{tx2/a\}bracketri}ht ≥Z(M)/a\}bracketle{tx/a\}bracketri}ht//a\}bracketle{tx2/a\}bracketri}htandG is aP-group,Z(M)/a\}bracketle{tx/a\}bracketri}ht//a\}bracketle{tx2/a\}bracketri}htis cyclic. Since /a\}bracketle{tx2/a\}bracketri}ht ≤Φ(Z(M)/a\}bracketle{tx/a\}bracketri}ht),Z(M)/a\}bracketle{tx/a\}bracketri}htis cyclic. In particular, Z(M) is cyclic. This is a contradiction. (3) By (2), G/Z(G) is elementary abelian. By Lemma 2.3, G′is elementary abelian. In particular, G′≤Ω1(Z(M)) = Ω 1(M). (4)Ifnot, thenΩ n−4(G) =G. By(2)and(3), Gis4-abelian. Since n−4≥2, exp(G) = exp(Ω n−4(G)) = 2n−4. Hence exp( ℧1(G)) = 2n−5. It follows that Φ(G) =℧1(G)≤Ωn−5(Z(M)) =/a\}bracketle{tb2,c2/a\}bracketri}ht= Φ(M). Thusd(G) =d(M)+1 = 4. Takex∈G\M. Then/a\}bracketle{ta,b,c,x/a\}bracketri}ht=G. Since G′≤Ω1(M) =/a\}bracketle{ta2n−5,b2/a\}bracketri}ht, [a,x]∈ /a\}bracketle{ta2n−5/a\}bracketri}htor [a,bx]∈ /a\}bracketle{ta2n−5/a\}bracketri}ht. Without loss of generality, we can assume [a,x]∈ /a\}bracketle{ta2n−5/a\}bracketri}ht. If [a,x] = 1, then CG(a)/(/a\}bracketle{ta/a\}bracketri}htΦ(G))≥ /a\}bracketle{t¯c,¯x/a\}bracketri}ht∼=C2 2. HenceCG(a)//a\}bracketle{ta/a\}bracketri}htis not cyclic. This is a contradiction. So [ a,x]

Question 1344 multiple-choice

Programmable quantum processors are devices that implement quantum channels based on the input program state, with their operational capabilities determined by mathematical structures such as commutants and convex sets. The connection between group symmetries, channel geometry, and processor design is central to quantum information theory. Which condition guarantees that a set of quantum channels admits an exact measure-and-prepare programmable quantum processor with a finite number of pure program states? 1) The channels are all trace-preserving but not necessarily unital. 2) The commutant associated with the group symmetry is non-abelian. 3) The set of channels forms a continuous convex manifold with infinitely many extreme points. 4) The Choi-Jamiołkowski states corresponding to the channels are all mixed states. 5) The program register is measured using a non-projective POVM only. 6) The set of Choi-Jamiołkowski states is isomorphic to a polytope with finitely many extreme points. 7) The channels are covariant under SU(H1) but the commutant is non-abelian.

✓ Correct Answer:

The correct answer is 6) The set of Choi-Jamiołkowski states is isomorphic to a polytope with finitely many extreme points..

📚 Reference Text:

that channels implemented by a processor that is covariant with respect to the special unitary group SU(H1)are unital using a similar argument. We will now consider how to construct covariant programmable quantum processors in the case whereKis abelian. 3.1 Covariant programmable quantum processors from extreme points In this section, we consider a special class of CPQP UV, the measure-and-prepare CPQP UV. Wewillshowthat TUVcanbeimplementedbyameasure-and-prepareCPQP UVifandonly ifthecommutant Kisabelian. Beforewestatethemainresult,wepresentalemmarequired foritsproofwhichisacorrespondencebetweenthecommutantandtheaﬃliatedstatespace which we require in order to show our ﬁrst main result for exact programmability. Lemma 16. LetUbe an irrep of a compact group G, and letVbe another representation ofG. Then, the following are equivalent: i)Kis abelian. ii)JUVis isomorphic to a simplex. iii)JUVis isomorphic to a polytope. Accepted in Quantum2021-06-18, click title to verify. Published under CC-BY 4.0. 8 Proof.“i)⇒ii)” This statement is mentioned in Ref. [40] without proof. Let B∈Kand letKbe an abelian matrix algebra. This implies that nk= 1for allk∈{1,...,K}. Furthermore, let Kbe the number of irreps appearing in the direct sum decomposition of ¯U⊗V. We obtain B=K/circleplusdisplay k=11bk⊗xk wherexk∈C. Moreover, B≥0if and only if xk≥0for allk∈{1,...,K}andtr(B) = 1 if and only if/summationtext kbkxk= 1. According to Lemma 15, extreme points of K∩D (H1⊗H 2) are also extreme points of JUV. The extreme points of K∩D (H1⊗H 2)are of the form xi= 1/bifor somei∈{1,...,K}andxl= 0for alll/negationslash=i. We identify the Kextreme points ofJUVwithKpoints in RK. Therefore,JUVis isomorphic to a (K−1)-simplex. “iii)⇒i)” We prove this statement by contraposition, i.e. we show if Kis non-abelian, JUVis not isomorphic to a polytope. If Kis non-abelian, there is a k∈{1,...,K}such that the corresponding block is of dimension nk>1. Let us consider elements of the form Bϕ=/parenleftbigg1 bk|ϕk/angbracketright/angbracketleftϕk|⊗1bk/parenrightbigg ⊕0, where|ϕk/angbracketright/angbracketleftϕk|∈D (H/prime k). The normalization1 bkyields tr(Bϕ) = 1and furthermore, Bϕ≥ 0, i.e.Bϕ∈D(H1⊗H 2). These elements are extreme points of JUVby Lemma 15. Thus, there are inﬁnitely many extreme points of JUV. Hence, the set cannot be isomorphic to a polytope which has ﬁnitely many extreme points by deﬁnition. The last implication “ii) ⇒iii)” is obvious. Let us now state the deﬁnition of the class of processors we consider in this section. Deﬁnition 17 (Measure-and-prepare PQPC).LetH1andH2be separable Hilbert spaces. Then, we call a PQP CPwith ﬁnite-dimensional HPa measure-and-prepare PQP Cfor a set of quantum channels Cif there exists a K∈N, a POVM{Ei}i∈{1,...,K}⊂D(HP)and a set of quantum channels Tk∈C,k∈{1,...,K}, such that P(A⊗B) =K/summationdisplay k=1tr[EkB]Tk(A)∀A∈B(H1), B∈B(HP). Proposition 18. LetC⊂ CPTP(H1,H2)be a convex set of quantum channels. Then, there is a measure-and-prepare PQP Cif and only ifJC, the corresponding set of Choi- Jamiołkowski states, is isomorphic to a polytope. Moreover, if such a measure-and-prepare PQPCexists,dPcan be chosen to be the number of extreme points and the program states can be chosen pure. Proof.We ﬁx a quantum channel T∈Cwith its corresponding Choi-Jamiołkowski state cT∈JC. SinceJCis isomorphic to a polytope, it is spanned by Kextreme points cTkand can therefore be written as convex combination of these cT=K/summationdisplay k=1xkcTk, wherexk∈[0,1]and/summationtextK k=1xk= 1. By the Choi-Jamiołkowski isomorphism, there is a channelTk∈TUVcorresponding to each of the extreme points cTkofJC, i.e.Tcan be linearly decomposed T(·) =K/summationdisplay k=1xkTk(·)

Question 1345 multiple-choice

Quantum algorithms for hidden subgroup problems utilize the structure of finite Abelian groups to efficiently extract information about hidden subgroups. Techniques such as group decomposition and quantum Fourier transforms are central to these algorithms. Which property of finite Abelian groups allows quantum algorithms to reduce hidden subgroup problems to simpler cases involving groups of prime power order? 1) The existence of non-Abelian subgroups in every finite Abelian group 2) The partitioning of group elements into cosets of equal size 3) The ability to decompose any finite Abelian group into a direct product of cyclic groups 4) The requirement that all group elements have order two 5) The invariance of group operations under arbitrary permutations 6) The necessity for the group order to be a power of two 7) The restriction that only trivial subgroups exist in finite Abelian groups

✓ Correct Answer:

The correct answer is 3) The ability to decompose any finite Abelian group into a direct product of cyclic groups.

📚 Reference Text:

in a basis containing the eigen vectors of the unitary operators which we apply to it (as done in [Ki] and [CEMM], for example), and this gives us (5), from which it is easy to see and analyse the e ﬀect of the inverse Fourier transform even when it does not perfectly match the s ize of G. 4.3 Finding Hidden Subgroups As discussed in Sect. 2, any ﬁnite Abelian group Gis the product of cyclic groups. In light of the order-ﬁnding algorithm, which also p ermits us to factor, we can assume that the group Gis represented as a product of cyclic groups of prime power order. Further, for any product of two groups GpandGqwhose orders are coprime, any subgroup KofGp×Gqmust be equal to Kp×Kq from some subgroups KpandKqofGpandGqrespectively. We can therefore consider our function fseparately on GpandGqand determine KpandKq separately. Thus we can further restrict ourselves to group sGof prime power order. This not only simpliﬁes any analysis, it could reduce the size of quantum control registers necessary in any implementation of these algorithms. Let us thus assume that G=Zpm1×Zpm2×· · ·× Zpmlfor some prime pand positive integers m1≤m2≤ · · · ≤ ml=m. The ‘promise’ is that fis constant on cosets of a subgroup K, and distinct on each coset. The hidden subgroup Kis{k= (k1, k2, . . ., k l)|f(x) =f(x+k) for all x∈G}. In practice, this will usually be a consequence of the nature of f, as in the case of discrete logarithms where f(x1, x2) =ax1bx2, or whenever fis constructed as h◦gfor 8 some homomorphism gfromGto some ﬁnite group H, and a 1-to-1 mapping hfromHto the set X. LetUfbe an operator which maps |x/an}brack⌉tri}ht |0/an}brack⌉tri}htto|x/an}brack⌉tri}ht |f(x)/an}brack⌉tri}ht. Deﬁne e1= (1,0, . . .,0),e2= (0,1,0, . . .,0), and so on. Let us also consider an oper- ator related to Uf,Uf(xej), which maps |x/an}brack⌉tri}ht |f(y)/an}brack⌉tri}htto|x/an}brack⌉tri}htUf(xej)|f(y)/an}brack⌉tri}ht= |x/an}brack⌉tri}ht |f(y+xej)/an}brack⌉tri}ht. In the case of Simon’s Problem, the operator Uf(x(0,1,0)) maps |1/an}brack⌉tri}ht |f(y1, y2, y3)/an}brack⌉tri}htto|1/an}brack⌉tri}htUf(0,1,0)|f(y1, y2, y3))/an}brack⌉tri}ht=|1/an}brack⌉tri}ht |f(y1, y2+ 1, y3)/an}brack⌉tri}ht and does nothing to |0/an}brack⌉tri}ht |f(y1, y2, y3)/an}brack⌉tri}ht. For each t= (t1, t2, . . ., t l), 0≤tj< pmj, satisfying l/summationdisplay j=1pm−mjhjtj= 0 mod pmfor all h∈K (10) deﬁne |Ψt/an}brack⌉tri}ht=/summationdisplay a∈G/Kexp −2πi pml/summationdisplay j=1pm−mjtjaj |f(a)/an}brack⌉tri}ht. (11) We are summing over a set of representatives of the cosets of Kmodulo G, and by condition (10) on t, this sum is well-deﬁned. Let Tdenote the set of t satisfying (10), which corresponds to the group of characte rs ofG/K. The |Ψt/an}brack⌉tri}ht are eigenvectors of each Uf(xej), with respective eigenvalues exp(2 πixt j/pmj) . By determining these eigenvalues, for j= 1,2, . . ., l , we will determine t. If we had|Ψt/an}brack⌉tri}htin an auxiliary register, we could estimate tj/pmjusing Uf(xej)by the technique of the previous section. If we use F−1 pmjwe would determine tjexactly, or we could use the simpler F−1 2k, for some k >log2(pmj), and obtain tjwith high probability. For simplicity, we will use F−1 pmj. In practice we could use F−1 2k for a large enough kso

Question 1346 multiple-choice

In additive combinatorics and computational number theory, finding zero-sum signed subsets within vectors over finite fields has important consequences for algorithm design and cryptography. Efficient algorithms often exploit recursive constructions and disjoint subset structures to ensure tractable solutions. Which of the following best describes the significance of constructing pairwise disjoint signed subsets when seeking a zero-sum solution among vectors in Zn_p? 1) It guarantees independence among subset sums, enabling deterministic polynomial-time algorithms for non-trivial zero-sum solutions. 2) It ensures that all subset sums are duplicates, allowing for faster hash collision detection. 3) It forces the subset sums to always be equal, simplifying combinatorial proofs. 4) It maximizes the probability of subset sums being non-empty, regardless of vector dimension. 5) It restricts the algorithm to only exponential-time solutions due to increased complexity. 6) It eliminates the need for recursive bounding of parameters in the construction. 7) It provides a direct reduction to the classic subset sum problem without structural modifications.

✓ Correct Answer:

The correct answer is 1) It guarantees independence among subset sums, enabling deterministic polynomial-time algorithms for non-trivial zero-sum solutions..

📚 Reference Text:

vectors of dimension ⌈d/2⌉n. We apply procedure Ato ﬁnd pairwise disjoint signed subsets M1,..., Mdsuch that/summationtextd i=1iu/prime i=0, where u/prime iis the signed sum of the uks corresponding to the signed subset Mi. Now for each 1≤i≤d,u/prime iis the concatenation of vectors wij(j=⌊d/2⌋+ 1,..., d). Here, for 1≤i,j≤d,wijstands for the signed subset sum obtained by joining the signed subset sums ukjaccording to the signed subset Mi. The signed subset sums wijare pairwise disjoint, not all of them are empty and they satisfy the relations d/summationdisplay i=1iwij=0(j=⌊d/2⌋+ 1,..., d) and d/summationdisplay j=1jwij=0(i=1,..., d). 123 14 Page 18 of 23 M. Imran, G. Ivanyos We subtract the sum of the last ⌈d/2⌉(“horizontal”) relations of the second kind from the sum the ⌈d/2⌉(“vertical”) relations of the ﬁrst kind and obtain the relation ⌊d/2⌋/summationdisplay i=1d/summationdisplay j=⌊d/2⌋+1iwij−d/summationdisplay i=⌊d/2⌋+1⌊d/2⌋/summationdisplay j=1jwij+d/summationdisplay i,j=⌊d/2⌋+1(i−j)wij=0. Notice that for ⌊d/2⌋+ 1≤i,j≤d,w eh a v e |i−j|≤⌊ d/2⌋. Therefore, by ﬂipping signs where appropriate and then joining the signed subsets with equal coefﬁ- cients, we obtain pairwise disjoint subset sums w/prime 1,...,w/prime ⌊d/2⌋with/summationtext⌊d/2⌋ i=1iw/prime i=0. The subset sums w/prime ican all be empty only if each wijis empty when i/negationslash=jand at least one of iand jis greater than ⌊d/2⌋. Assume that this is the case. Then, if there is an index i>⌊d/2⌋such that wiiis non-empty then wiimust be itself a non-trivial zero subset sum and gives a one-term solution. Otherwise not all wijare empty for i,j≤⌊d/2⌋and/summationtext⌊d/2⌋ i,j=1iwij=0. /intersectionsq/unionsq Iterated application of the method of Lemma 3gives the following result. Proposition 4 Given S ±(p,n)=pO(plogp)nO(p)vectors from Zn p, a non-trivial signed subset sum representing the zero vector can be found in deterministic timepoly(S ±(p,n)). Proof Put d0=p−1 2,h0(n)=n+1 and deﬁne di=⌊ di−1/2⌋and hi(n)= hi−1(n)hi−1(⌈di−1/2⌉n)recursively for i=1,...,⌊logd0⌋. As among any h0(n)= n+1 vectors from Zn pa non-trivial linear relation can be found in time poly (nlogp), recursive applications of Lemma 3give that among h⌊logd0⌋(n)vectors a single non- trivial signed subset sum (that is, a linear relation with nonzero coefﬁcients ±1 only) can be found in time poly (h⌊logd0⌋(n)logp). We show by induction that hi(n)≤⎛ ⎝i−1/productdisplay j=0⌈dj/2⌉⎞ ⎠2i−1 (n+1)2i. (3) Fori=0, both sides are equal to n+1. Assume that the inequality holds for 0 ≤i< ⌊logd0⌋. Then we also have hi(⌈di/2⌉n)≤⎛ ⎝i−1/productdisplay j=0⌈dj/2⌉⎞ ⎠2i−1 (⌈di/2⌉n+1)2i. Using ⌈di/2⌉n+1≤⌈di/2⌉(n+1), we obtain hi(⌈di/2⌉n)≤⌈di/2⌉2i−1⎛ ⎝i/productdisplay j=0⌈dj/2⌉⎞ ⎠2i−1 (n+1)2i. (4) 123 Zero sum subsequences and hidden subgroups Page 19 of 23 14 Multiplying inequalities ( 3) and ( 4) and using hi+1(n)=hi(n)hi(⌈di/2⌉n), we obtain hi+1(n)≤⎛ ⎝i/productdisplay j=0⌈dj/2⌉⎞ ⎠2i (n+1)2i+1, which is inequality ( 3)f o r i+1 in place of i.U s i n g dj≤d0/2j≤d0=p−1 2, inequalities ( 3)f o r i=⌊logd0⌋give h⌊logd0⌋≤⎛ ⎜⎜⎝/ceilingleftBig logp−1 2/ceilingrightBig −1/productdisplay j=0/ceilingleftbiggp−1 4/ceilingrightbigg⎞ ⎟⎟⎠2/ceilingleftBig logp−1 2/ceilingrightBig −1 (n+1)2/ceilingleftBig logp−1 2/ceilingrightBig =/ceilingleftbiggp−1 4/ceilingrightbigg/ceilingleftBigp−1 4/ceilingrightBig/ceilingleftBig logp−1 2/ceilingrightBig (n+1)/ceilingleftBigp−1 2/ceilingrightBig . Therefore, we have h⌊logd0⌋(n)=pO(plogp)nO(p). /intersectionsq/unionsq We interpret a non-empty zero sum signed subset as a non-trivial collision between two disjoint subset sums. (Non-trivial means that at most one of the subsets can be empty.) We use the short-term collision for such a pair. We have

Question 1347 multiple-choice

In advanced cryptography, group-theoretic structures such as non-abelian groups and semidirect products are employed to strengthen encryption protocols. Achieving high security standards like IND-CCA is crucial for defending against adaptive chosen ciphertext attacks. Which property distinguishes the use of semidirect product groups in encryption protocols that achieve full IND-CCA security compared to earlier approaches based on non-abelian groups? 1) Semidirect product groups always require abelian subgroup assumptions for security. 2) Semidirect product groups are limited to IND-rCCA security in practice. 3) Protocols based on semidirect products depend on complex group hashing schemes. 4) Semidirect product groups are less flexible in real-world cryptographic design. 5) Semidirect product groups allow protocols to achieve full IND-CCA security without relying on computational assumptions about abelian subgroups. 6) Only non-abelian group protocols can be used for post-quantum cryptography. 7) Semidirect product protocols demand stricter conditions than non-abelian group protocols.

✓ Correct Answer:

The correct answer is 5) Semidirect product groups allow protocols to achieve full IND-CCA security without relying on computational assumptions about abelian subgroups..

📚 Reference Text:

Title: A construction of encryption protocols over some semidirect products Year: 2023 Paper ID: d53dba7bca669a4c3b85adcb9134c79e39bc50ea Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d53dba7bca669a4c3b85adcb9134c79e39bc50ea Abstract: Abstract In CANDARW ’18, Isobe et al. proposed a secure encryption protocol on non-abelian groups based on the Anshel–Anshel–Goldfeld key exchange protocol. There have remained two weak points on the protocol: one is that the protocol is indistinguishable against adaptive chosen ciphertext attack (IND-CCA) in a slightly restricted sense, what they call IND-rCCA secure, and the other is that the conditions imposed on groups and hashing schemes are too strict to make the protocol practical. In this article, we propose an IND-CCA secure protocol that resolves those problems. The key idea is to employ some specific semidirect product as platform groups, so that we can achieve the exact IND-CCA security from concise conditions on groups and hashing schemes. Our protocol is not dependent on any computational assumptions on abelian subgroups.

Question 1348 multiple-choice

Quantum Max Cut (QMC) is a quantum variant of the classical Max Cut problem, where optimization is performed over quantum states and techniques such as representation theory and semidefinite programming (SDP) are used to compute solutions for specific graph families. Understanding which methods yield exact solutions for QMC is crucial for algorithm development in quantum optimization. For which classes of graphs do both representation-theoretic methods and semidefinite programming (SDP) relaxations yield exact solutions for the maximum eigenvalue of the Quantum Max Cut Hamiltonian? 1) Weighted star graphs with arbitrary edge weights 2) Complete k-partite graphs, including cliques and crown graphs 3) Double-star graphs with uniform edge weights 4) Bipartite graphs with an odd number of vertices 5) Tree graphs with uniform edge weights 6) Path graphs with weighted edges 7) Cyclic graphs with non-uniform edge weights

✓ Correct Answer:

The correct answer is 2) Complete k-partite graphs, including cliques and crown graphs.

📚 Reference Text:

highest eigenvalue of Quantum Max Cut can be computed exactly, but with diverging aims. We provide a method to compute (in exact arithmetic) the maximum eigenvalue of the QMC Hamiltonian on a certain family of graphs with uniform edge-weights, extending previously known results on exact solutions to Quantum Max Cut [ LM62 ]. This result builds on the characterization of the QMC Hamiltonian in terms of irreducible representations of the symmetric group via Schur-Weyl duality, and is largely independent of our construction of the swap relaxations. [TRZ+], on the other hand, is concerned with understanding instances where the SDP relaxations exactly solve the Quantum Max Cut problem as this can improve the analysis of approximation algorithms. Specifically, [ TRZ+] investigates whether either the first swap relaxation or the second quantum Lasserre relaxation over Paulis are exact or not on a number of graphs. Interestingly, there are cases where the SDP hierarchies considered by [TRZ+] are proven to be inexact (see Section 4 of [ TRZ+]) but for which exact arithmetic solutions are known. An example of this is the QMC Hamiltonian with uniform edge-weights on a clique with an odd number of vertices, which have exact solutions via representation Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 87 theory (see, for example, Section 2.5 of this paper). One special class of graphs solved by our methods are complete k-partite graphs for any constant k, special cases of which include the clique and the crown graph. These latter graphs are shown to be exactly computable by the SDPs in [ TRZ+]. In other cases, such as a uniformly weighted double-star graph and a star graph with positive weights, the SDP relaxations are shown to be numerically exact but our representation theoretic method is unable to provide an exact solution. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 88

Question 1349 multiple-choice

The hidden subgroup problem (HSP) is a foundational challenge in quantum computing, with significant implications for cryptography and computational complexity. Quantum algorithms for HSP are especially important when dealing with non-abelian groups constructed as semidirect products of cyclic groups. Which group structure admits an efficient polynomial-time quantum algorithm for the hidden subgroup problem even when elements are encoded in a black-box manner? 1) Z_pr ⋊ Z_p, where p is an odd prime 2) Z_q ⋊ Z_q, where q is an even prime 3) Z_pr ⋊ Z_r, where r divides p 4) Z_pr ⋊ Z_2, with p and r arbitrary primes 5) Z_p ⋊ Z_pr, with p and r distinct odd primes 6) Z_pr ⋊ Z_q, with q not dividing p-1 7) Z_pq ⋊ Z_p, with q and p distinct primes

✓ Correct Answer:

The correct answer is 1) Z_pr ⋊ Z_p, where p is an odd prime.

📚 Reference Text:

Title: Efficient quantum algorithms for the hidden subgroup problem over semi-direct product groups Year: 2004 Paper ID: c2080522eb34a9130533dd6af69814ad69408ae7 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c2080522eb34a9130533dd6af69814ad69408ae7 Abstract: In this paper, we consider the hidden subgroup problem (HSP) over the class of semidirect product groups Zpr ⋊ Zq, for p and q prime. We first present a classification of these groups in five classes. Then, we describe a polynomial-time quantum algorithm solving the HSP over all the groups of one of these classes: the groups of the form Zpr ⋊ Zp, where p is an odd prime. Our algorithm works even in the most general case where the group is presented as a black-box group with not necessarily unique encoding. Finally, we extend this result and present an efficient algorithm solving the HSP over the groups Zprm ⋊ Zp.

Question 1350 multiple-choice

Quantum cloning machines are theoretical devices that attempt to duplicate unknown quantum states, but their performance is fundamentally restricted by quantum mechanics. The geometry of allowed clone fidelities and the symmetries of the underlying transformations are key to understanding the limits of quantum information processing. In the case of universal quantum cloning from one input to three outputs (1→3 UQCM), which property characterizes the symmetry of the convex hull of achievable clone fidelities? 1) Invariance under permutation of two clones only 2) Reflection symmetry across the fidelity axis 3) Translational symmetry along the fidelity plane 4) Rotational symmetry by 2π/3 5) Mirror symmetry about the origin 6) Absence of any symmetry 7) Invariance under scaling of fidelities

✓ Correct Answer:

The correct answer is 4) Rotational symmetry by 2π/3.

📚 Reference Text:

ﬁxed and have values determined by the chosen point. apart from the above-mentioned reconstruction of states from the convex hull, we can try to ﬁnd, for example, all allowed quantum states which satisfy some required condition for relations between ﬁdelities F1k. For example, for 1 !3 9 universal cloning machines we can demand the following constraint F12+F13=2F14, (19) where we take maximization over F12. Such a reconstruction was presented in our previous paper regarding admissible region of ﬁdelities for the qubit case [19]. Finally, having these states we can reconstruct a cloning machine which returns clones with ﬁdelities fi, corresponding to ﬁdelities F1igiven by the chosen point. We have also interesting interpretation of the most bottom part of our plots as optimal anti-clones. First of all one can notice that our convex hulls are invariant with respect to rotations around straight line F12=F13=F14by the angle b=2p/3 in the case 1!3UQCM and they are symmetric with respect to the straight line F12=F13 in the case 1!2UQCM . The most bottom point is determined by the intersection between symmetry line and convex hull and it corresponds to a minimum value of ﬁdelities which are equal in these cases. In the future, it would be interesting how to obtain optimal clones starting from our method. Numerically, it is not that hard, one just needs to add a cut to the general region to end with optimal region of ﬁdelities. Analytically the answer does not seem to be so trivial, but we still hope that the employed group theoretical techniques are interesting and may provide some new insight into the inner structure of the optimal universal asymmetric quantum cloners. Finally, let us note that to solve a M!N(M

Question 1351 multiple-choice

In quantum information theory and representation theory, the computation of averages over the unitary group using the Haar measure often involves sophisticated techniques related to permutation operators, traces, and group algebras. Understanding the interplay between moment operators, permutation symmetry, and their underlying mathematical structures is essential for analyzing random quantum processes. Which statement most accurately describes how the coefficients in the expansion of the moment operator over permutation operators are determined? 1) By maximizing the determinant of the permutation matrix associated with each operator 2) By taking the inverse of the trace of each permutation operator acting on the space 3) By diagonalizing the unitary operators in the tensor product space 4) By counting the fixed points in the action of each permutation on the basis 5) By computing the eigenvalues of the Gram matrix of permutation products 6) By solving a linear system involving traces of permutation matrices, using the Gram matrix and its pseudo-inverse, resulting in Weingarten coefficients 7) By evaluating the commutator of each permutation operator with the moment operator

✓ Correct Answer:

The correct answer is 6) By solving a linear system involving traces of permutation matrices, using the Gram matrix and its pseudo-inverse, resulting in Weingarten coefficients.

📚 Reference Text:

approach that allows us to evaluate the moment operator and, consequently, compute integrals over the Haar measure. The following theorem presents a recipe for accomplishing this task. Theorem 10. (Computing moments) Let O∈L/parenleftbig (Cd)⊗k/parenrightbig . The moment operator can then be expressed as a linear combination of permutation operators: E U∼µH/bracketleftbig U⊗kOU†⊗k/bracketrightbig =/summationdisplay π∈Skcπ(O)Vd(π), (30) where the coefficients cπ(O)can be determined by solving the following linear system of k!equations: Tr/parenleftig V† d(σ)O/parenrightig =/summationdisplay π∈Skcπ(O) Tr/parenleftig V† d(σ)Vd(π)/parenrightig for allσ∈Sk. (31) This system always has at least one solution. 5 Proof. Equation (30) follows from Lemma 6.2, which states that M(k) µH(O) =E U∼µH/bracketleftbig U⊗kOU†⊗k/bracketrightbig ∈ Comm(U(d),k), and from Schur-Weyl duality (Theorem 9). To obtain the linear system of equations, we begin by multiplying both sides of Eq. (30) by V† d(σ) and taking the trace for all σ∈Sk. This yields: /summationdisplay π∈Skcπ(O) Tr/parenleftig V† d(σ)Vd(π)/parenrightig = Tr/parenleftig V† d(σ)M(k) µH(O)/parenrightig (32) = Tr/parenleftig M(k) µH/parenleftig V† d(σ)O/parenrightig/parenrightig (33) = Tr/parenleftig V† d(σ)O/parenrightig , (34) where in the first equality we used Eq. (30), in the second equality we used that V† d(σ) commutes withU⊗k, and in the last equality we used the fact that the moment operator is trace preserving (Lemma 6.1). Since M(k) µH(O)∈span (Vd(π) :π∈Sk), a solution to this linear system of equations always exists. The previous theorem provides an explicit expression for the coefficients of the moment operator in the permutation basis as cπ(O) =/summationtext σ∈Sk(G+)π,σTr/parenleftbig V† σO/parenrightbig . Here, Gis the Gram matrix, i.e. that matrix with coefficients Gπ,σ= Tr/parenleftig V† d(π)Vd(σ)/parenrightig , and G+is its pseudo-inverse. This result allows us to express the moment operator in terms of the so-called Weingarten coefficients Wg(π−1σ,d):= (G+)π,σ[51, 55–57]2, as follows: E U∼µH/bracketleftbig U⊗kOU†⊗k/bracketrightbig =/summationdisplay π,σ∈SkWg(π−1σ,d) Tr/parenleftig V† d(σ)O/parenrightig Vd(π). (35) The Weingarten coefficients Wg(π−1σ,d)can be written in terms of characters of the symmetric group. However, we will not explore this aspect here. For reference, see [51, 55, 59, 60]. The Gram matrix Ghas a simple expression in terms of the cycle structure of the permutations, given by: Gπ,σ= Tr/parenleftig V† d(π)Vd(σ)/parenrightig = Tr/parenleftbig Vd(π−1)Vd(σ)/parenrightbig = Tr/parenleftbig Vd(π−1σ)/parenrightbig =d#cycles (π−1σ), (36) where the last equality follows from the fact that Tr(Vd(π)) =/summationtext i1,...,ik∈[d]k/angbracketleftbig i1/vextendsingle/vextendsingleiπ−1(1)/angbracketrightbig ···/angbracketleftbig ik/vextendsingle/vextendsingleiπ−1(k)/angbracketrightbig and observing that this sum has d#cycles (π)nonvanishing terms and they are all equal to 1. This fact is also evident in the tensor networks notation that we will introduce in section 6. Proposition 11. Forπ∈Sk, the permutation matrices Vd(π)are linearly independent if k≤d, but linearly dependent if k>d . Proof. Let us consider first the case where k≤d. We assume that there exist complex coefficients απ∈Cfor allπ∈Sksuch that the following equation holds: /summationdisplay π∈SkαπVd(π) = 0. (37) Since,k≤d, we can choose kdistinct elements i1,...,ik∈[d] and apply to the right of both sides of the previous equation the state |i1⟩⊗···⊗|ik⟩, obtaining: /summationdisplay π∈Skαπ/vextendsingle/vextendsingleiπ−1(1)/angbracketrightbig ⊗···⊗/vextendsingle/vextendsingleiπ−1(k)/angbracketrightbig = 0. (38) For allσ∈Sk, we multiply to the left by/angbracketleftbig iσ−1(1)/vextendsingle/vextendsingle⊗···⊗/angbracketleftbig iσ−1(k)/vextendsingle/vextendsingleand deduce ασ= 0. Thus, we have shown linear independence. If k>d , then consider the operator:

Question 1352 multiple-choice

In representation theory and quantum algebra, W-algebras can be constructed from simple Lie algebras using geometric and algebraic methods, such as Fock realizations and principal bundles. These constructions often involve oscillator algebras, matrix representations, and the action of symmetry groups. Which statement most accurately describes the role of the character χ of a Borel subgroup in constructing finite-dimensional representations of a simple Lie algebra via line bundles? 1) It defines the commutation relations of the oscillator algebra. 2) It specifies the dimension of the Hilbert space for the Fock module. 3) It determines the decomposition of representations into irreducible components. 4) It ensures the irreducibility of the Fock realization. 5) It is used to construct the associated complex line bundle whose holomorphic sections carry a representation of the Lie algebra. 6) It selects the set of positive roots for the Lie algebra. 7) It generates a matrix representation of the universal enveloping algebra.

✓ Correct Answer:

The correct answer is 5) It is used to construct the associated complex line bundle whose holomorphic sections carry a representation of the Lie algebra..

📚 Reference Text:

the ﬁnite quantum algebra W(2) 3. So if we have a repre- sentation of sl2⊕u(1) in terms of n×nmatrices, we immediately get a representation of¯W(2) 3in terms of n×nmatrices. In the appendix we give explicit formulas for the quantum Miu ra transformations associated to the ﬁnite Walgebras that can be obtained from sl4. 6.1. Fock realizations of quantum finite Walgebras Using the quantum Miura transformation we can turn any Fock r ealization of g0 into a Fock realization of the corresponding ﬁnite Walgebra. In this section we brieﬂy discuss how we can obtain Fock realizations for simple Lie al gebras. We will then show in the example of W(2) 3that its ﬁnite dimensional representations of can be realiz ed as submodules of certain Fock modules. As usual deﬁne an oscillator algebra Oto be an algebra generated by operators 48 {ai,a† i}n i=1with the following commutation relations [ai,aj] = [a† i,a† j] = 0 ; [ai,a† j] =δij (6.2) In physics terminology this algebra describes a collection ofnuncoupled harmonic oscillators. The Hilbert space of such a system is isomorphi c to the space C[z1,...,z n] of polynomials in nvariables, and (the quantum operators) aianda† jcan be realized as ai=∂ ∂zianda† i=zi (6.3) This is called the ’Fock realization’ of the (quantum) harmo nic oscillator algebra. Since any polynomial Pcan be reduced to a multiple of 1 by successive application of {ai} we can construct any polynomial Qout ofPby the action of the universal enveloping algebra of the oscillator algebra. This means that the Fock r ealization is irreducible. A Fock realization of a Lie algebra gis an injective homomorphism from gto the universal enveloping algebra of the oscillator algebra. As we shall need this later we now discuss how to obtain a Fock r ealization for an arbitrary simple Lie algebra g(see for details [30]). LetGbe a simple Lie group with Lie algebra g. Furthermore let B−be its lower Borel subgroup and N+the group associated to the positive roots. Consider the cos et space B−\G={B−g|g∈G} (6.4) The triple ( G,B −\G,B −) is a principal bundle with total space G, basespace B−\G and structure group B−. Let nowχ:B−→Cbe a 1-dimensional representation (a character) of the Lie group B−, then we can construct the associated complex line bundle LχoverB−\G Lχ=G×C/∼ (6.5) where (b.g,χ(b−1)z)∼(g,z) for allb∈B−. Denote the set of holomorphic global sections of this bundle by Γ χ. The group Gacts from the right on Lχas follows [(g,z)]·g′= [(gg′,z)] (6.6) and this induces of course an action of G(and therefore of g) on Γ χ. Now letHandb−be the Cartan subalgebra and negative Borel subalgebra of g respectively. The derivative of χin the unit element φχ(b) =d dtχ(etx)|t=0x∈g (6.7) 49 is a one dimensional representation of b−. However this implies that 0 = [ φχ(h),φχ(e−α)] = −/a\}bracketle{tα,h/a\}bracketri}htφχ(eα) whereh∈Hande−αis a rootvector of g, which means that φχ(e−α) = 0 for all −α∈∆+. Therefore φχdetermines really an element Λ χofH∗. It is also clear that all elements of H∗can be obtained in this way. It

Question 1353 multiple-choice

The loop Hecke algebra extends the classical Hecke algebra to the setting of the loop braid group, providing new structures relevant in algebraic combinatorics and representation theory. Its basis construction involves combinatorial and algebraic techniques, with connections to quantum superalgebras and dualities. Which combinatorial object is used to enumerate the basis elements of the loop Hecke algebra, thereby confirming a conjecture about its dimension? 1) Motzkin paths 2) Young tableaux 3) Integer partitions 4) Parking functions 5) Dyck paths 6) Noncrossing matchings 7) Ferrers diagrams

✓ Correct Answer:

The correct answer is 5) Dyck paths.

📚 Reference Text:

Title: A basis and Schur-Weyl duality for the loop Hecke algebra Year: 2025 Paper ID: 3e72696ed9bf7acb46bc05cde9141682c6fb3ebe Source: semantic-scholar URL: https://www.semanticscholar.org/paper/3e72696ed9bf7acb46bc05cde9141682c6fb3ebe Abstract: The loop Hecke algebra is a generalization of the Hecke algebra to the loop braid group, introduced by Damiani, Martin and Rowell. We give a new presentation of the loop Hecke algebra provided a mild condition on the parameter and give a basis. We use higher linear rewriting theory to show linear independence and the combinatorics of Dyck paths to compute the cardinality of the basis. This yields a conjecture of Damiani-Martin-Rowel. We also give a representation theoretic interpretation of the loop Hecke algebra in terms of (non-semisimple) Schur-Weyl duality involving the negative half of quantum $\mathfrak{gl}_{1|1}$.

Question 1354 multiple-choice

Computational quantum chemistry employs advanced methods to model the spatial and electronic properties of atoms and molecules, especially in systems with three electrons such as lithium. Techniques like Hartree-Fock variants and spin-coupled approaches are central for accurately predicting atomic behavior and chemical reaction pathways. In computational studies of three-electron atoms, which method specifically allows for more flexible treatment of electron spin coupling, leading to improved calculations of spin-dependent properties such as Q in certain lithium states? 1) Restricted Hartree-Fock (RHF) 2) Density Functional Theory (DFT) 3) Green's Function (GF) approach 4) Coupled Cluster Singles and Doubles (CCSD) 5) Spin-Orbital Generalized Interaction (SOGI) method 6) Configuration Interaction Singles (CIS) 7) Complete Active Space Self-Consistent Field (CASSCF)

✓ Correct Answer:

The correct answer is 5) Spin-Orbital Generalized Interaction (SOGI) method.

📚 Reference Text:

appropriate for Li+ rather than solvmg for them self-consistently for Li. Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jspQUANTUM THEORY OF MANY-ELECTRON SYSTEMS. V 1083 dependence of the orbitals. The point at 60° agrees with independent GF calculations. However, the wave functions near Z=60° do not connect smoothly with the GF function indicating an instability in the GF function. It is interesting to note that for all four states of Li the UHF value of Q(O) is quite close to the SOGI value (see Table Ilc). In summary, these calculations indicate that the G1 description of the spatial properties of three-electron atoms is quite good. That is, to a very good approxi- \ \ / I I I I / / ----------- ' ------- (b) ' ' ' \ \ I I I ---' ' ' I I FIG. 8. (a) Orbital 1 of the 3A2• state of square H4• The edge of the figure is 7.0 a.u.long; shortest distance between protons is 2.54 a.u. The lowest contour is 0.01 and the interval between contours is 0.024; the highest contour is 0.322. Based on the Gaussian basis set (see text). (b) Orbital 3 of the 3 A2• state of square H •. Scale as in (a). Lowest contour 0.01, highest contour 0.41, interval 0.04. ~3 ffiE2 ffiE3 ~4 2 4 3 5 2 5 2 5 5 4 4 3 FIG. 9. The standard Young tableaux for the shape [3, 2]. mation, the two core electrons are singlet paired (as in G1); only for spin-dependent properties is it impor tant to allow for a more general coupling of the electron spins. When this more general coupling is introduced, we obtain a very good value for Q(O) for the 2 2S state and perhaps the 2 2P state of Li. 2. The Ha System Physically it is clear that, for three-electron atoms, Z should be near zero since Z should be determined primarily by the core electrons (for a two-electron ion the ground state is a singlet which corresponds to Z=O, the addition of valence electrons should not change this appreciably since they are much further from the nucleus) . In addition, for a system such as HeR at typical molecular distances, we expect Z to be primarily determined by the pair of orbitals on the He and hence to be near zero. However, for a system of three hydrogen atoms the optimum pairing should be a function of nuclear configuration and should be significantly different from zero for some configurations. We have carried out SOGI calculations for several points on the reaction path of H2+D---7H+HD with the primary objectives being to determine (1) how well the SOGI method will predict potential energy surfaces for chemical reactions, (2) how the SOGI orbitals change as the system moves along the mini mum energy reaction path, and (3) how spatial sym metry restrictions enter into our descriptions of a chem ical reaction. Basis Sets and Configurations:

Question 1355 multiple-choice

Compact quantum groups are mathematical structures that generalize classical groups using noncommutative geometry and play a central role in quantum algebra and theoretical physics. Their representation theory, including concepts like inductive limits and tensor product constructions, helps understand symmetries in infinite-dimensional quantum systems. Which of the following statements best captures the representation-theoretic significance of transformations involved in analyzing q-central probability measures on paths in the q-Gelfand–Tsetlin graph? 1) They provide a way to classify finite-dimensional irreducible representations of classical Lie groups. 2) They define the fusion rules for tensor products in finite quantum groups. 3) They determine the spectrum of the Laplacian on associated quantum homogeneous spaces. 4) They generate the automorphism group of the q-Gelfand–Tsetlin graph. 5) They specify the commutation relations between generators of the quantum group. 6) They give an explicit representation-theoretic interpretation linking probabilistic distributions on graph paths to asymptotic character formulas. 7) They construct universal C*-algebras associated with quantum group duals.

✓ Correct Answer:

The correct answer is 6) They give an explicit representation-theoretic interpretation linking probabilistic distributions on graph paths to asymptotic character formulas..

📚 Reference Text:

Title: Inductive limits of compact quantum groups and their unitary representations Year: 2019 Paper ID: dc37234287cf528758f80db17ecc7c3b97df3135 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/dc37234287cf528758f80db17ecc7c3b97df3135 Abstract: We introduce inductive limits of compact quantum groups and their unitary representation theories. Those give a more explicit representation-theoretic meaning to our previous study of quantization of characters and central probability measures in the asymptotic representation theory. We also study tensor product representations of the infinite-dimensional quantum unitary group. As a by-product, we give an explicit representation-theoretic interpretation to certain transformations that play an important role in analyzing q-central probability measures on the paths in the q-Gelfand–Tsetlin graph.

Question 1356 multiple-choice

In the representation theory of the symmetric group Sn, central elements of the group algebra and projectors onto irreducible representations play a key role in understanding symmetries and character theory. The algebraic structure involves conjugacy classes, partitions, and the behavior of normalized characters under group operations. When a central element Tμ of the group algebra C(Sn), associated with a partition μ, acts by multiplication on a projector PR onto an irreducible representation labeled by Young diagram R, what is the resulting action? 1) The projector PR is annihilated, resulting in zero. 2) The projector PR is scaled by the normalized character of Tμ in representation R. 3) The projector PR is transformed into a central element associated with a different partition. 4) The projector PR is mapped to a sum of all projectors corresponding to irreducible representations. 5) The projector PR remains unchanged, acting as the identity. 6) The projector PR is replaced by a conjugacy class sum over Sn. 7) The projector PR induces a permutation of partitions in the center.

✓ Correct Answer:

The correct answer is 2) The projector PR is scaled by the normalized character of Tμ in representation R..

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sections. Appendix A gives the proofs of two lemmas of quantum inspired classical algorithms used in the main text. Appendix B describes a change of basis in the space of Casimirs of U(N) relevant to section 4. Appendix C proves a property of projectors relevant to section 6. Note.As this paper was approaching completion, we became aware of [ 40] and an associated talk at the Perimeter Institute on the “Quantum Complexity of Kronecker coeﬃcients”. This has some overlaps with section 6 in the use of QPE and projectors to address the computational complexity of Kronecker coeﬃcients. 2 Background We review some background which will be used in this paper. We start with a review of key properties of the group algebra C(Sn)of the symmetric group Snand its centre Z(C(Sn))which is the subspace that commutes with all elements of C(Sn). This is based – 3 – JHEP05(2023)191on standard textbooks in group theory e.g. [ 41] and the relevant formulae are also collected in appendices of recent physics papers such as [ 32,38]. This is followed by a review of Quantum Phase Estimation (QPE) [ 7] algorithm from quantum computation. 2.1 The group algebra C(Sn)and its centreZ(C(Sn)) LetC(Sn)be the group algebra of the symmetric group Sn, i.e. the group algebra formed by linear combinations/summationtext σ∈Snλσσ,λσ∈C. LetZ(C(Sn))be the center of C(Sn). There is a basis ofZ(C(Sn))formed by sums over group elements in conjugacy classes. Conjugacy classes inSnare labelled by partitions µofn(denotedµ/turnstileleftn). For a conjugacy class Cµ⊂Snwe have a central element Tµ=/summationdisplay σ∈Cµσ, (2.1) obeyingγTµγ−1=Tµ, for anyγ∈Sn. We are interested, in particular, in conjugacy classes µ= [k,1n−k]deﬁned by a single cycle of length kand all remaining cycles of length 1. These central elements have the form T2=1 2/summationdisplay 1≤i1/negationslash=i2≤n(i1i2) = (12) + (13) + ... T3=1 3/summationdisplay 1≤i/negationslash=j/negationslash=k≤n( (ijk) + (ikj) ) = (123) + (132) + (124) + (142) + ... ... Tk=1 k/summationdisplay 1/negationslash=i1/negationslash=i2···/negationslash=ik(i1i2···ik) (2.2) where the cycles of length 1are not shown. We will refer to the Tkas cycle central elements inZ(C(Sn)). The product of two general central elements for µ,ν/turnstileleftnyields TµTν=/summationdisplay λCλ µνTλ (2.3) where the integer structure constants Cλ µνcan be organized as a matrix Cµwith matrix elements: (Cµ)λ ν=Cλ µν. (2.4) It is known that the eigenvalues of (Cµ)λ νare the normalized characters /hatwideχR(Tµ) :=χR(Tµ) dR in the irrep labelled by the Young diagram R/turnstileleftnof dimension dR. Another basis for Z(C(Sn))is given by the projectors PRlabelled by irreducible representations R(corresponding to Young diagrams with nboxes, i.e.R/turnstileleftn): PR=dR n!/summationdisplay σ∈SnχR(σ)σ (2.5) The multiplication of the conjugacy class elements with the projectors gives TµPR=/hatwideχR(Tµ)PR (2.6) – 4 – JHEP05(2023)191where/hatwideχR(Tµ)is the normalized character obtained by dividing the trace of the matrix DR(Tµ)representing Tµin the irrep Rby the dimension dR /hatwideχR(Tµ) =χR(Tµ) dR=tr (DR(Tµ)) dR(2.7) Therefore, the linear operator of multiplication by TµinZ(C(Sn), with matrix elements (Cµ)λ ν, has eigenvectors PRassociated with the eigenvalue /hatwideχR(Tµ). We will consider χmax µ= max R/turnstileleftn/hatwideχR(Tµ) (2.8) as a normalization factor such that the magnitude of eigenvalue of Tµ/χmax µassociated with PRis bounded by 1,

Question 1357 multiple-choice

Quantum algorithms often leverage group theory and Fourier analysis to solve hidden subgroup problems, which are foundational for problems like factoring and discrete logarithms. Strong Fourier sampling plays a crucial role in efficiently reconstructing hidden subgroups, especially in nonabelian groups. Which key advantage does strong Fourier sampling provide over weaker methods when solving hidden subgroup problems in nonabelian groups such as q-hedral and affine groups? 1) It always allows solution without considering representation details. 2) It reveals row and column indices in group representations, enabling full reconstruction of hidden subgroups. 3) It eliminates the need for quantum resources in the computation. 4) It restricts measurements to random basis, losing structure-adapted information. 5) It is only effective for abelian groups and fails for nonabelian groups. 6) It ignores nonabelian structure, relying solely on abelian components. 7) It provides no improvement over "forgetful" abelian approaches in information-theoretic reconstruction.

✓ Correct Answer:

The correct answer is 2) It reveals row and column indices in group representations, enabling full reconstruction of hidden subgroups..

📚 Reference Text:

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. C OMPUT . c/circlecopyrt2007 Society for Industrial and Applied Mathematics Vol. 37, No. 3, pp. 938–958 THE POWER OF STRONG FOURIER SAMPLING: QUANTUM ALGORITHMS FOR AFFINE GROUPS AND HIDDEN SHIFTS∗ CRISTOPHER MOORE†, DANIEL ROCKMORE‡, ALEXANDER RUSSELL§,AND LEONARD J. SCHULMAN¶ Abstract. Many quantum algorithms, including Shor’s celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem , in which an unknown subgroup Hof a group Gmust be determined from a quantum state ψoverGthat is uniformly supported on a left coset of H. These hidden subgroup problems are typically solved by Fourier sampling : the quantum Fourier transform of ψis computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identiﬁed: the weak standard method , where only representation names are measured, and the strong standard method , where full measurement (i.e., the row and column of the representation, in a suitably chosen basis,as well as its name) occurs. It has remained open whether the strong standard method is indeedstronger, that is, whether there are hidden subgroups that can be reconstructed via the strongmethod but notby the weak, or any other known, method. In this article, we settle this question in the aﬃrmative. We show that hidden subgroups Hof the q-hedral groups, i.e., semidirect products Z q/multicloseleftZ p, where q|(p−1), and in particular the aﬃne groups Ap, can be information-theoretically reconstructed using the strong standard method. Moreover, if |H|=p/polylog( p), these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. Wecompare our algorithms to two weaker methods that have been discussed in the literature—the“forgetful” abelian method, and measurement in a random basis—and show that both of these areweaker than the strong standard method. Thus, at least for some families of groups, it is crucial to usethe full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group’s subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivatedhidden shift problems , generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over whichthe hidden subgroup problem can be solved eﬃciently. Key words. quantum computation, hidden subgroup problem, Fourier analysis, group repre- sentations AMS subject classiﬁcations. 81P68, 20C35 DOI. 10.1137/S0097539705447177 1. The hidden subgroup problem. One of the principal quantum algorithmic paradigms is the use of the abelian Fourier transform to discover a function’s hiddenperiodicities. In the examples relevant to quantum computing, an oracle function f deﬁned on an abelian group Ghas “hidden periodicity” if there is a “hidden” subgroup HofGso that fis precisely invariant under translation by Hor, equivalently, fis constant on the cosets of Hand takes distinct values on distinct cosets. The hidden ∗Received by the editors March 9, 2005; accepted for publication (in

Question 1358 multiple-choice

In signal processing, efficient algorithms for computing the Discrete Fourier Transform (DFT) rely on specialized matrix structures and permutation techniques. One key optimization is the recursive factorization of DFT matrices for inputs of length that is a power of two. Which statement accurately describes the role of the bit-reversal permutation matrix $ P_n $ in the radix-2 factorization of a $ 2^n \times 2^n $ DFT matrix? 1) $ P_n $ is a diagonal matrix whose entries are primitive roots of unity used for scaling. 2) $ P_n $ permutes matrix columns according to the order of their magnitude. 3) $ P_n $ partitions the DFT matrix into non-recursive blocks for direct computation. 4) $ P_n $ is a lower-triangular matrix that sorts indices by ascending order. 5) $ P_n $ reverses the binary representation of row indices, is involutory and unitary, and enables recursive block partitioning for efficient FFT computation. 6) $ P_n $ multiplies each row of the DFT matrix by a distinct complex number. 7) $ P_n $ is used to eliminate complex conjugate symmetry in the output spectrum.

✓ Correct Answer:

The correct answer is 5) $ P_n $ reverses the binary representation of row indices, is involutory and unitary, and enables recursive block partitioning for efficient FFT computation..

📚 Reference Text:

2 0 6 4 2 0 7 6 5 4 3 2 1 000 001 010 011 100 101 110 111, where in the matrix on the right the exponents are given modul o8. The binary representation of the row indices are shown on the right side of the matrix. Permuting the rows s uch that the binary representation of the row indices reverse in order results in the matrix ˜F′ 8: ˜F′ 8= 0 0 0 0 0 4 0 4 0 2 4 6 0 6 4 20 0 0 0 0 4 0 4 0 2 4 6 0 6 4 2 0 1 2 3 0 5 2 7 0 3 6 1 0 7 6 54 5 6 7 4 1 6 3 4 7 2 5 4 3 2 1 000 100 010 110 001 101 011 111=/bracketleftbigg˜F′ 4˜F′ 4 ˜F′ 4+˜Ω4˜F′ 4+˜Ω4+4/bracketrightbigg , 4 with ˜F′ 4= 0 0 0 0 0 2 0 2 0 1 2 3 0 3 2 1 ,˜Ω4= 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 . The left side of the above expression shows a block partition ing of˜F′ 8in terms of the bit-reversed exponent DFT matrix of half the dimension, ˜F′ 4. Here we again applied the modulo 8equivalence and the property that the entries in˜F′ 4must be doubled when used in ˜F′ 8becauseω4=ω2 8. This partitioning can be written in terms of the permuted DFT matrices as: F′ 8=1√ 2/bracketleftbiggF′ 4F′ 4 F′ 4Ω4−F′ 4Ω4/bracketrightbigg ,with Ω4= diag(1,ω1 8,ω2 8,ω3 8), where the minus sign in the (2,2)-block comes from ω4 8=−1. The generalization of the permutation and partition to a DFT matrix of dimension 2nforn>2 requires the deﬁnition of Ω 2n, which we give below. Deﬁnition 8. DeﬁneΩnas the following 2n×2ndiagonal matrix: Ωn= Ω2n:= ω0 2n+1 ω1 2n+1 ... ω2n−1 2n+1 , whereω2n+1:= exp(−2πı 2n+1). The block partitioning of the 4 ×4 DFT matrix that we demonstrated in Example 1 can be generali zed for any 2n×2nDFT matrix as stated in the following theorem. Theorem 1 (See [18, 15]) .LetF′ n=PnFnbe the2n×2nbit-reversed DFT matrix, where Pnis the bit reversal permutation matrix acting on nbits. Then F′ nadmits the following factorization: F′ n=1√ 2/bracketleftbiggF′ n−1 F′ n−1 F′ n−1Ωn−1−F′ n−1Ωn−1/bracketrightbigg , =/bracketleftbiggF′ n−1 F′ n−1/bracketrightbigg/bracketleftbiggIn−1 Ωn−1/bracketrightbigg/parenleftbigg1√ 2/bracketleftbiggIn−1In−1 In−1−In−1/bracketrightbigg/parenrightbigg , = (I2⊗F′ n−1)(In−1⊕Ωn−1)(H⊗In−1),(7) whereΩn−1is given by Deﬁnition 8. The subdivision of the permuted DFT matrix in a 2 ×2 block matrix is also called a radix-2 splitting in the literature. The bit reversal permutation Pnsatisﬁes the following identity Pn(v1⊗···⊗vn) =vn⊗···⊗v1. (8) wherevk∈C2fork= 1,···,n. The matrix Pnis involutary, because P2 n=In, i.e., reversing the bits twice gives the original ordering. It is also unitary because it is a perm utation matrix. Theorem 1 can be applied repeatedly to obtain the complete ra dix-2 factorization of the DFT matrix. This is formalized in the following theorem. Theorem 2. The2n×2nDFT matrix Fncan be factored as: Fn=PnF′ n=PnA(0) nA(1) n···A(n−1) n,

Question 1359 multiple-choice

Quantum algorithms leverage group-theoretic problems such as the Hidden Subgroup Problem (HSP) to achieve computational speedups for tasks in cryptography and complexity theory. A variety of solution approaches, including coset sampling and Fourier analysis, are applied depending on the underlying group structure. Which technique enables quantum algorithms to extract information about a hidden subgroup by preparing a superposition over cosets and performing measurements, often using Fourier analysis? 1) Grover search 2) Phase estimation 3) Coset sampling 4) Quantum amplitude amplification 5) Quantum walk 6) Classical Markov chain sampling 7) Adiabatic evolution

✓ Correct Answer:

The correct answer is 3) Coset sampling.

📚 Reference Text:

Title: The Hidden Subgroup Problem Year: 2010 Paper ID: 2a2c9f2e021778dfdede7563d82131b3c898e436 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/2a2c9f2e021778dfdede7563d82131b3c898e436 Abstract: We give an overview of the Hidden Subgroup Problem (HSP) as of July 2010, including new results discovered since the survey of arXiv:quant-ph/0411037v1. We recall how the problem provides a framework for efficient quantum algorithms and present the standard methods based on coset sampling. We study the Dihedral and Symmetric HSPs and how they relate to hard problems on lattices and graphs. Finally, we conclude with the known solutions and techniques, describe connections with efficient algorithms as well as miscellaneous variants of HSP. We also bring various contributions to the topic. We show that in theory, we can solve HSP over a given group inductively: the base case is solving HSP over its simple factor groups and the inductive step is building efficient oracles over a normal subgroup N and over the factor group G/N. We apply this analysis to the Dedekindian HSP to get an alternative abelian HSP algorithm based on a change of the underlying group. We also propose a quotient reduction by the normal group obtained via Weak Fourier Sampling. We compute the exact expression of Strong Fourier Sampling over the dihedral group, showing how the previous reduction is natural and matches the standard one. We also give a reduction of rigid graph isomorphism problem to HSP over the alternating group. For this group and other simple groups, we propose maximal subgroup reduction as a possible approach. We also analyse Regev's algorithm for the poly(n)-uniqueSVP, prove how the degree of the polynomial is related to the oracle complexity used and we suggest several variants.

Question 1360 multiple-choice

In quantum information theory, the generation and classification of entangled states are pivotal for implementing quantum protocols and error correction. The interplay between braid group representations and unitary matrices provides a framework for constructing multi-qubit entangled states. Which property uniquely allows the Bell matrix to generate n-qubit GHZ-like entangled states via braid group representations, making it universal under local unitary transformations? 1) It preserves the separability of basis states under tensor product operations. 2) It commutes with all single-qubit Pauli operators. 3) It produces a symmetric superposition of all computational basis states. 4) It diagonalizes the Yang-Baxter equation for arbitrary dimensions. 5) It ensures that measurement outcomes are always identical for all entangled qubits. 6) It leaves the entanglement unchanged after applying arbitrary permutations of qubits. 7) It satisfies the Yang-Baxter equation, enabling superpositions of basis states and their bit-flipped conjugates, thereby generating entangled states equivalent to GHZ states up to local unitary transformations.

✓ Correct Answer:

The correct answer is 7) It satisfies the Yang-Baxter equation, enabling superpositions of basis states and their bit-flipped conjugates, thereby generating entangled states equivalent to GHZ states up to local unitary transformations..

📚 Reference Text:

for applications of the braid group in quantum computation, one requires its unitary representations. For an m-qubit system the usual 2m×2munitary representation of Bmemployed in the literature is bi=I⊗...⊗I⊗R⊗I⊗...⊗I(i= 1...m−1)(2) whereIis the 2×2 unit matrix and Ris a 4×4 unitary matrix that acts on both the i-th and (i+1)-th qubits; that is, occupying the ( i,i+1) position. The ﬁrst of the two braid group relations in (1) is automatically satisﬁed by the form (2). To fulﬁll the second relation, Rmust satisfy (R⊗I)(I⊗R)(R⊗I) = (I⊗R)(R⊗I)(I⊗R).(3) This relation is sometimes called the (algebraic) Yang- Baxter equation. One of the simplest solutions of (3) that produces entanglement of states is the matrix R=1√ 2 1 0 0 −1 0 1−1 0 0 1 1 0 1 0 0 1 . (4) When acting on the standard basis {|00/an}bracketri}ht,|01/an}bracketri}ht,|10/an}bracketri}ht,|11/an}bracketri}ht}, Rgenerates the four maximally entangled Bell states 2 (|00/an}bracketri}ht ± |11/an}bracketri}ht)/√ 2 and (|01/an}bracketri}ht ± |10/an}bracketri}ht)/√ 2. Here we adopt the convention |0/an}bracketri}ht= (1,0)tand|1/an}bracketri}ht= (0,1)t, wheret denotes the transpose. Following [5] and [8], we shall callRthe Bell matrix1. In the presence of local unitary transformations, Ris a universal gate [2]. The representation (2) can also be used to generate maximally entangled n-qubit states which are equivalent, up to local unitary transformation, to the GHZ states [4]. To see this, let us take the n= 3 qubit case, and consider the action of b1b2on the separable state |000/an}bracketri}ht: |ψ/an}bracketri}ht=b1b2|000/an}bracketri}ht=1 2(|000/an}bracketri}ht+|011/an}bracketri}ht+|101/an}bracketri}ht+|110/an}bracketri}ht).(5) |ψ/an}bracketri}htis related to the GHZ state |GHZ/an}bracketri}ht= (|000/an}bracketri}ht+ |111/an}bracketri}ht)/√ 2 by a local unitary transformation as |ψ/an}bracketri}ht=H⊗H⊗H|GHZ/an}bracketri}ht, H=1√ 2/parenleftbigg 1 1 1−1/parenrightbigg (6) whereHis the Hadamard matrix (or gate). That the state |ψ/an}bracketri}htis said to be equivalent to the GHZ state is based on the fact that local unitary transforma- tions donot alter the degreeofentanglement2. Neverthe- less, it is evident that they have very diﬀerent entangle- ment properties. For instance, after making a measure- ment on any one of the three qubits, the other two qubits of the GHZ state become separable, whereas those of |ψ/an}bracketri}ht are still in one of the maximally entangled Bell states! It would be more desirable if one could generate the GHZ statesdirectlyfrom the braiding operators without re- course to any local unitary transformation. A common feature of the Bell states and the GHZ states is that they have the form of the superposition of a separable product state |a1a2...ak···an/an}bracketri}ht ≡ |a1/an}bracketri}ht|a2/an}bracketri}ht···|an/an}bracketri}htwith its conju- gate state |¯a1¯a2...¯ak···¯an/an}bracketri}ht, which has all ak’s changed from 0 to 1, and 1 to 0, i.e., ¯ ak= 1,0 ifak= 0,1, respectively. Thus the state |00/an}bracketri}htis conjugate to |11/an}bracketri}ht, |001/an}bracketri}htis conjugate to |110/an}bracketri}ht, etc. As pointed out after Eq.(4), the Bell matrix essentially superimposes each two-qubit basis state on its conjugate, as does the Hadamard matrix in the one-qubit case. We wish to generalize the Hadamard and Bell matri- ces to higher dimensions (i.e., to nqubits), so that they generate generalized GHZ states from separable states directly. We want these matrices to be representatives of certain braiding operators of the braid group. Hence the maintaskistoﬁndanappropriateunitaryrepresentation of

Question 1361 multiple-choice

Quantum algorithms like the Quantum Fourier Transform (QFT) are sensitive to various types of errors when implemented on real quantum hardware. Understanding error characterization methods is essential for accurate quantum computation and efficient error correction strategies. Which method for systematic error characterization in QFT leverages the separable nature of output states, requires only 3n measurement projections, and is limited to algorithms with product-state outputs? 1) Density matrix reconstruction using 4n measurements 2) Separable state analysis using 3n projections 3) Pauli twirling error mitigation 4) Randomized benchmarking 5) Quantum process tomography 6) Error propagation via Kraus operators 7) Gate-set characterization via cross-entropy benchmarking

✓ Correct Answer:

The correct answer is 2) Separable state analysis using 3n projections.

📚 Reference Text:

Zin each of the qubits, and found the one qubit reduced matrices. To get the pure state with most alignment, no optimization is needed, because for one qubit matrices the principal eigen- vector jÁii(the eigenvector with maximal eigenvalue) is the state with most alignment, getting jÁi=jÁi1 jÁi2. Rev.Mex.Fis. 68031402 CHARACTERIZINGERRORSFOR QUANTUMFOURIER TRANSFORM ON IBM QUANTUM 5 FIGURE5.FidelitiesEq.( 17)forthemethodusing 3nprojections. F1in gray bar, F2in white bar,and F3in black bar. TABLEV.Fidelities, Eq. ( 17) of second method, using 3nprojec- tions,for ibmqsantiago. Inputstate F2 F3 j00i 0.88471 0.98792 j01i 0.90541 0.99226 j10i 0.88857 0.99460 j11i 0.89133 0.99426 Finallythedesiredseparableunitaryrotationforeachba- sisstate U=U1U2isfound,andsimilarlytotheprevious method a Depolarizing Chanel is applied on the theoretical statejÃi, and the resulting state ½Dch=²(jÃ)iis compared with the transformation of the experimental state under U: ½0=U½expUy. The ﬁdelity between ½Dchand½0is shown for the four input states in TableV and shownin Fig. 5. 5. Conclusions In this paper we have managed to study the performance of the Quantum Fourier Transform algorithm in a real quantum computer. The loss of information due decoherence effects, could be corrected by adding redundancy as in quantum cor-rectingcodes,butingeneralcouldbecomputationallyexpen- sive. We propose two methods to model systematic errors: the ﬁrstmethodusestheisotropicindexthatseparatesthelossof information, by decoherence, from the systematic error, that potentially could be correctable determining a unitary trans- formation that could correct any input state. The method needs to reconstruct the density matrix from 4nmeasure- ments, and uses the alignment to determine the unitary cor- rection matrix for each input basis state. Meanwhile, in the secondmethoditisonlynecessarytomeasure 3nprojections, taking advantage of the fact that the theoretical output of the QFTalgorithm of each basis state, is separable in all qubits. The ﬁrst method, although very expensive from a computa- tionalpointofview,ismoregeneralandcanbeusedtomodel systematic errors of any algorithm, the second can be only usedforalgorithmswithseparableoutputstatesasthe QFT. EventhoughthetheoreticaloutputstateoftheQFTalgorithm is separable, experimental errors can introduce some entan- glement. Nonetheless, the experimental result is expected to be reasonably close to being separable for minor errors. As canbeseeninFigs.4and5,theresultsareverysimilarwhich seems to support this assumption. As can be seen in these same ﬁgures, in actual IBM Q computers, the unitary errors are much less relevant than the decoherence errors. Nevertheless, this method could be used as calibration via software, using the Ucmatrix, once itisdeterminedexperimentally,regardlessothermethodsare needed to correct decoherence error. Eventhoughthisanalysisshowscorrectingsystematicer- rors at the current state of the art IBM Q computers is of minor importance, this models could be used to visualize (Fig. 5) the weight of the different types of errors that af- fect quantum computation and could prove especially useful in educational contexts. Acknowledgements We acknowledge the use of IBM Quantum Services for this work. The views expressed are those of the authors, and do not reﬂect the ofﬁcial policy or position of IBM or the IBM Quantum team. 1.A. Fonseca de Oliveira, E. Buksman, I. Cohn and J. Garc´ıa Lopez de Lacalle, Characterizing error propaga- tion in quantum circuits: the Isotropic Index, Quantum Inf. Process. 16(2017) 48, https://doi.org/10.1007/ s11128-016-1507-5 . 2.E. Magesan, D. Puzzuoli, C. E. Granade and D. G. Cory, Mod- eling quantum noise for efﬁcient testing of fault-tolerant cir-

Question 1362 multiple-choice

Quantum algorithms often rely on specific measurement strategies and efficient circuit implementations to solve computational problems such as the generalized hidden shift problem. Foundational results in quantum information theory enable practical realization of abstract measurement schemes on quantum computers. Which statement accurately describes how Neumark's theorem enables the implementation of the Pretty Good Measurement (PGM) in quantum algorithms for the generalized hidden shift problem? 1) It allows the PGM to be realized as a classical reversible computation. 2) It ensures that any measurement operator can be decomposed into tensor products of Pauli matrices. 3) It restricts the PGM to only rank-one projective measurements in the original Hilbert space. 4) It requires the construction of entangled ancilla states for all measurements. 5) It permits the PGM to be efficiently simulated with purely classical resources. 6) It guarantees that any POVM, including the PGM, can be implemented as a unitary operation followed by a projective measurement in an extended Hilbert space. 7) It mandates that the measurement operators for the PGM be diagonal in the computational basis.

✓ Correct Answer:

The correct answer is 6) It guarantees that any POVM, including the PGM, can be implemented as a unitary operation followed by a projective measurement in an extended Hilbert space..

📚 Reference Text:

k≫logN/logM, and a constant number ofsolutionsfor k≈logN/logM. Thisintuitioncanbeformalizedasfollows: Lemma 2.For the generalized hidden shift problem with M=⌊N1/k⌋ withk≥3andNsuﬃciently large, Pr(1≤ηx w≤4)is lower bounded by a constant. A proof is given in the appendix. Together, Lemmas 1 and 2 show that the pgmhas at least a constant probability of successfully identifying the hidden shift. In fact, it turns out that the pgmis thepovmthat maximizes the probability of successfully determining sgiven the states (9). For more details, we refer the reader to Section 4 of [2] and Section 4.4 of [3]. Finally, to give an eﬃcient algorithm based on the pgm, we must show how the measurement can be implemented eﬃciently on a quantu m com- puter. Such an implementation can be achieved using Neumark ’s theorem QUANTUM ALGORITHM FOR A GENERALIZED HIDDEN SHIFT PROBLEM 7 [23], which states that any povmcan be realized by a unitary transforma- tionUon the system together with an ancilla followed by a measurem ent in the computational basis. For a povmconsisting of Nrank one operators Fj=|fj/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tfj|in aD-dimensional Hilbert space, Uhas the block form U=/parenleftbigg V X Y Z/parenrightbigg (20) where the rows of the N×DmatrixVare theD-vectors /an}⌊ra⌋k⌉tl⌉{tfj|, i.e.,V=/summationtextN j=1|j/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tfj|. (In particular, if all the vectors |fj/an}⌊ra⌋k⌉tri}hthave unit length, as will be the case below, X= 0.) Recall from (13) that the povmoperators for the pgmon hidden sub- group states can be written Ej=/summationdisplay x∈Zk NEx j⊗|x/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tx| (21) where Ex j:=|ex j/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tex j| (22) with |ex j/an}⌊ra⌋k⌉tri}ht:=1√ Nk/summationdisplay w∈ZNωwj|Sx w/an}⌊ra⌋k⌉tri}ht. (23) Inotherwords,each Ejisblockdiagonal, withblockslabeledby x∈Zk N, and where each block is rank one. Thus, the measurement can be imp lemented in a straightforward way by ﬁrst measuring the block label xand then performingthe povm{Ex j}j∈ZNconditional on the ﬁrstmeasurement result. To implement the povm{Ex j}j∈ZNusing Neumark’s theorem, we would like to implement the unitary transformation Uxwith the upper left sub- matrix Vx=1√ N/summationdisplay j,w∈ZNω−wj|j/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tSx w|. (24) It is convenient to perform a Fourier transform (over ZN) on the left (i.e., on the index j), giving a unitary operator ˜Uxwith upper left submatrix ˜Vx=1 N/summationdisplay j,w,v∈ZNω(w−v)j|w/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tSx v| (25) =/summationdisplay w∈ZN|w/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tSx w|. (26) Therefore, the pgmcan be implemented eﬃciently if we can eﬃciently per- form a unitary transformation satisfying |w,x/an}⌊ra⌋k⌉tri}ht /ma√sto→ |Sx w,x/an}⌊ra⌋k⌉tri}ht (27) for all matrix sum instances ( x,w) withηx w>0. We refer to (27) as quan- tum sampling of solutions to the matrix sum problem. If we can eﬃciently quantum sample from matrix sum solutions, then by running th e circuit in 8 ANDREW M. CHILDS AND WIM VAN DAM reverse, we can eﬃciently implement ˜Ux, and hence the desired measure- ment. By applying these unitary transformations directly to the s tate (9), we can obtain a description of the algorithm without reference to generalized measurement. Performing the inverse of the quantum samplin g transforma- tion (27) followed by a Fourier transform, we obtain the stat e ρ′:=1 Nk/summationdisplay x∈Zk N|ρ′ x,x/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tρ′ x,x| (28) where |ρ′ x/an}⌊ra⌋k⌉tri}ht:=1√ MkN/summationdisplay j,w∈ZNωw(s−j)/radicalbig ηxw|j/an}⌊ra⌋k⌉tri}ht. (29) Roughly speaking, if the distribution of ηx wis close to uniform, then the sum overwin (29)

Question 1363 multiple-choice

Quantum phase and energy estimation algorithms are crucial for extracting spectral information from quantum systems, and various computational frameworks optimize their resource efficiency and error management. Singular value transformation (SVT) has emerged as a powerful alternative to traditional methods like the Quantum Fourier Transform (QFT) for these tasks. Which of the following is a key advantage of using singular value transformation (SVT) over Quantum Fourier Transform (QFT) in phase estimation algorithms for black-box Hamiltonian simulation? 1) SVT always produces exact energy eigenvalues without error. 2) SVT eliminates the need for all ancilla qubits in quantum circuits. 3) SVT achieves optimal asymptotic performance in black-box settings with fewer ancillary resources. 4) SVT replaces eigenstate input requirements with arbitrary input states. 5) SVT inherently avoids all ancilla entanglement, ensuring perfect error isolation. 6) SVT requires median amplification sorting networks for reliable output. 7) SVT and QFT produce identical circuit depth for all systems.

✓ Correct Answer:

The correct answer is 3) SVT achieves optimal asymptotic performance in black-box settings with fewer ancillary resources..

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the |ψj/angbracketright, which is why this algorithm only works if the input is an eigenstate. We then use iterative estimators for the remaining bits to compute the rest of the estimate. This time their outputs will be deterministic, but there is no point in doing uncomputation since the superposition has already collapsed. Instead, we do the same thing as for the k= 0estimator: compute the next bit and some garbage, and discard the garbage. The ﬁnal answer is either ﬂoor (2nλj)or ﬂoor (2nλj)−1for the same reason as in Corollary 13. 3 Performance Comparison Above, we have presented a modular framework for phase and energy estimation using the key ingredients in Theorem 12 and Theorem 15 respectively. These results already demon- strate several advantages over textbook phase estimation as presented in Proposition 5. First, they eliminate the QFT and do not require a sorting network to perform median ampliﬁcation. Instead, they rely on just a single tool: singular value transformation. Second, they require far fewer ancillae. Our improved phase estimation algorithm requires no ancillae at all, and is merely ‘with phases’ so arguably does not even need uncomputation for some applications. On the other hand, textbook phase estimation requiresO((n+ log(α−1)) log(δ−1))ancillae in order to implement median ampliﬁcation. Given a block-encoding of a Hamiltonian with aancillae, then our energy estimation algorithm requires a+n+ 3ancillae. But in order to even compare Proposition 5 to The- orem 15 we need a method to perform energy estimation using textbook phase estimation. This is achieved through Hamiltonian simulation. Which method of Hamiltonian simulation is best depends on the particular physical system involved. Hamiltonian simulation using the Trotter approximation can perform exceedingly well in many situations [SHC20]. However, in our analysis we must be agnostic to the particular Hamiltonian in question, and furthermore need a uniﬁed method for comparing the performance. Hamiltonian simulation via singular value transformation [LC17, GSLW18], lets us compare Proposition 5 and Theorem 15 on the same footing. After all, this method features the best known asymptotic performance in terms of the simulation time [Childs&20] in a black-box setting. Singular value transformation constructs an approximate block-encoding of eiHtwith ancillae. Since eiHtisunitary, intheidealcasetheancillaestartinthe |0/angbracketrightstateandarealso guaranteed to be mapped back to the |0/angbracketrightstate. But in the approximate case the ancillae are still a little entangled with the remaining registers, so they become an additional source of error. Certainly the ancillae cannot be re-used to perform singular value transformation again, because then the errors pile up with each use. Thus, we are in a similar situation to Lemma 3, where we must discard some qubits and take into account the error. Therefore,wemustdosomeadditionalworkbeyondtheHamiltoniansimulationmethod presented in [GSLW18], to turn the approximate block-encoding of a unitary into a chan- nel that approximates the unitary in diamond norm. The main trick for this proof is to Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 35 consider postselection of the ancilla qubits onto the |0/angbracketrightstate. Then the error splits into two parts: the error of the channel when the

Question 1364 multiple-choice

Quantum computing offers new approaches to problems that are intractable for classical computers, such as large integer factorization. Key algorithms like Shor’s rely on specialized quantum operations that set them apart from classical methods. Which of the following features enables quantum computers to implement the quantum Fourier transform in a way that can provide exponential speedup over classical algorithms for integer factorization? 1) Use of classical random number generators for input states 2) Application of irreversible logic gates in computation 3) Encoding information solely in binary electrical signals 4) Use of parallel electronic circuits without quantum effects 5) Reliance on serial processing of classical bits 6) Manipulation of wavefront superposition states within quantum registers 7) Direct classical simulation of quantum entanglement

✓ Correct Answer:

The correct answer is 6) Manipulation of wavefront superposition states within quantum registers.

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processing times . However, for some problems , such as the factorisation of large integers, even this very high processing speed is not enough as these pro blems have solution times which scale exponentially with n. This has led to interest over the last twenty years in the development of algorithms and hardware implementations suitable for quantum computing . The quantum Fouri er transform has turned out to form a central component in some of the most important quantum algorithms , most notabl y the Shor al gorithm for integer factoris ation. Optical hardware components and configurati ons proposed to implement the quantum Fourier transform have been shown to be closely related to those proposed for the coherent optical implementation of the discrete Fourier transform using the matrix decompositions employed in the fast Fourier transform algorithm , the essential difference being the wavefront superposition state of the input and output registers in the quantum Fourier transform . This would, however, make possible the exponential increase in proces sing power required to solve such problems as large integer factorisation. REFERENCES 1. Goodman J.W., Introduction to Fourier Optics, McGraw -Hill, Second Edition, (1996). 2. Vander Lugt A., “ Signal detection by complex spatial filtering”, IEEE Trans. Inform. Theory, Vol. IT -10, pp. 139-145, (1964) 3. Shor P. W., “Polynomial -time algorithms for prim e factorization and discrete logarithms on a quantum computer”, SIAM J. Comput., Vol. 26(5), pp. 1484 -1509, (1997). 4. Cooley J. W., Tukey, J. W., “An algorithm for the machine computation of complex Fourier series”, Maths. Comput. Vol. 19, pp. 297 -301, (1965 ). 5. Brigham E. O., The Fast Fourier Transform, Prentice -Hall Inc, First Edition, (1974). 6. Liu, M -K., Principles and Applications of Optical Communications, Irwin, (1996). 7. Das, P. K., Optical Signal Processing Fundamnetals, Springer -Verlag, (1991). 8. Siegman A., E., “Fibre Fourier optics”, Opt. Lett., Vol. 26(16), pp. 1215 -1217, (2001). 9. Braunstein S. L., “ Factoring numbers ”, www -users.cs.york.ac.uk/schmue/comp/comp.html, on -line February 2013. 10. Barak R., Ben -Aryeh Y., ”Quantum fast Fourier transform and quantu m computation by linear optics”, J. Opt. Soc. Am. B, Vol. 24(2), pp. 231 -240, (2007). 11. Muthukrishan A ., Stroud C. R., “Quantum fast Fourier transform using multilevel atoms”, Journal of Modern Optics, Vol. 49, pp. 2115 -2127, (2002). 12. Ekert A., Jozsa R., “Sho r’s quantum algorithm for factorizing numbers” , Rev. Mod. Phys., Vol. 68, pp. 733 -753, (1996). 13. Coppersmith D., “An approximate Fourier transform useful in quantum factoring”, IBM Research Report No. RC19642, T.J. Watson Research Center, Yorktown Heights, New York, (1994) (unpublished). 14. Fowler, A. G., Hollenberg L. C. L., “Scalability of Shor’s algorithm with limited set of rotation gates”, Physical Review A, Vol. 70, 032329 -1:7, (2004).

Question 1365 multiple-choice

Quantum algorithms often address computational problems by transforming them into group-theoretic questions involving hidden subgroups. The ability to efficiently reconstruct these subgroups depends on the methods used for extracting information from quantum states, especially in nonabelian groups. Which method is strictly necessary for reconstructing hidden subgroups in certain nonabelian groups such as q-hedral and affine groups, where weaker approaches fail? 1) Weak Fourier sampling using only representation names 2) Measurement in a random basis 3) Strong Fourier sampling capturing full representation data 4) Classical subgroup enumeration algorithms 5) Abelian Fourier sampling methods 6) Ignoring group structure and using brute-force search 7) Approximate measurement of coset labels

✓ Correct Answer:

The correct answer is 3) Strong Fourier sampling capturing full representation data.

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Title: The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts Year: 2005 Paper ID: 0ffce04825147d96afbdbbb84869f4fb9c0fd1a3 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0ffce04825147d96afbdbbb84869f4fb9c0fd1a3 Abstract: Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup $H$ of a group $G$ must be determined from a quantum state $\psi$ over $G$ that is uniformly supported on a left coset of $H$. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of $\psi$ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups $H$ of the $q$-hedral groups, i.e., semidirect products ${\mathbb Z}_q \ltimes {\mathbb Z}_p$, where $q \mid (p-1)$, and in particular the affine groups $A_p$, can be information-theoretically reconstructed using the strong standard method. Moreover, if $|H| = p/ {\rm polylog}(p)$, these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the “forgetful” abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems, generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently.

Question 1366 multiple-choice

Quantum information theory investigates how information is transmitted, degraded, and preserved in quantum systems, often using mathematical structures such as groups and operators. Understanding how quantum channels behave under environmental noise is essential for developing reliable quantum technologies. Which approach leverages representation theory to transfer properties from classical Markov kernels on compact groups to quantum channels, enabling improved estimates of capacities and decoherence times? 1) Using tensor product decompositions to model entanglement entropy 2) Applying classical probability measures to quantum observables directly 3) Employing group transference techniques via representation theory 4) Calculating channel capacities solely through spectral analysis 5) Utilizing Monte Carlo simulations of quantum dynamics 6) Applying only commutative functional inequalities to operator algebras 7) Modeling noise exclusively with Markov chains lacking group symmetry

✓ Correct Answer:

The correct answer is 3) Employing group transference techniques via representation theory.

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1 Group transference techniques for the estimation of the decoherence times and capacities of quantum Markov semigroups. Ivan Bardet∗, Marius Junge†, Nichcolas Laracuente†, Cambyse Rouzé‡, Daniel Stilck França§ ∗Institut National de Recherche en Informatique et en Automatique, Paris, France †Department of Mathematics University of Illinois at Urbana-Champaign Champaign, IL, USA ‡Faculty of Mathematics, Technische Universität München, Munich, Germany §Centre for the Mathematics of Quantum Theory, University of Copenhagen, Copenhagen, Denmark Abstract —Capacities of quantum channels and decoherence times both quantify the extent to which quantum information can withstand degradation by interactions with its environment. However, calculating capacities directly is known to be intractable in general. Much recent work has focused on upper bounding certain capacities in terms of more tractable quantities such as speciﬁc norms from operator theory. In the meantime, there has also been substantial recent progress on estimating decoherence times with techniques from analysis and geometry, even though many hard questions remain open. In this article, we introduce a class of continuous-time quantum channels that we called transferred channels , which are built through representation theory from a classical Markov kernel deﬁned on a compact group. In particular, we study two subclasses of such kernels: Hörmander systems on compact Lie-groups and Markov chains on ﬁnite groups. Examples of transferred channels include the depolarizing channel, the dephasing channel, and collective decoherence channels acting on dqubits. Some of the estimates presented are new, such as those for channels that randomly swap subsystems. We then extend tools developed in earlier work by Gao, Junge and LaRacuente to transfer estimates of the classical Markov kernel to the transferred channels and study in this way different non-commutative functional inequalities. The main contribution of this article is the application of this transference principle to the estimation of decoherence time, of private and quantum capacities, of entanglement-assisted classical capacities as well as estimation of entanglement breaking times, deﬁned as the ﬁrst time for which the channel becomes entanglement breaking. Moreover, our estimates hold for non-ergodic channels such as the collective decoherence channels, an important scenario that has been overlooked so far because of a lack of techniques. I. I NTRODUCTION IN quantum mechanics, the evolution of a global, closed system leads to a unitary evolution of states. However, any realistic quantum system undergoes dissipative dynamics, due to its unavoidable interaction with its surrounding environment. Understanding how this noise limits the usefulness of these systems forvarious information processing tasks is of central importance to the development of quantum technologies. The dynamics of open systems are modeled by completely positive trace preserving maps. In the Markovian approximation, continuous time evolutions are then modeled by quantum Markov semigroups (Tt)t≥0of such maps. Given a concrete quantum Markov semigroup, it is then important to identify short time versus long time behaviour of the evolution. For example, it is important to know how long entanglement can be preserved. This remains a challenging problem. Even for classical systems, precise decoherence time estimates are very delicate, see [1], [2]. The aim of this

Question 1367 multiple-choice

Lattice reduction algorithms such as LLL play a central role in computational algebraic number theory, particularly when generalized to rings of integers in number fields. The properties of norm-Euclidean rings are crucial for ensuring efficient division algorithms and adapting classical lattice techniques. Which property is necessary for a number field's ring of integers to guarantee the existence of a division algorithm analogous to the Euclidean algorithm, thereby enabling the direct extension of the LLL lattice reduction algorithm? 1) The ring of integers must be a principal ideal domain. 2) The ring of integers must be norm-Euclidean, i.e., there is division with remainder using the norm function. 3) The number field must have class number one. 4) The number field must be totally real. 5) The ring of integers must be finitely generated as a module over Z. 6) The field must be Galois over Q. 7) The discriminant of the field must be square-free.

✓ Correct Answer:

The correct answer is 2) The ring of integers must be norm-Euclidean, i.e., there is division with remainder using the norm function..

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divisible by 4or a primep≡3 (mod 4) , one can take d= 4ord=pin the statement of the lemma. Norm-Euclidean rings OK.Recall that Z[i] =Z[ζ4]andOQ(i√ 3)=Z[ζ3]are norm-Euclidean: let k= 3or4, in order to divide a∈Z[ζk]byb∈Z[ζk]one rounds each coordinate of a/b∈Q[ζk]and denotes the result by q, and then one setsr=a−bq; ifNK/Q(r)< NK/Q(b)for alla,bwe say that Z[ζK]is norm-Euclidean. Napias [42] showed that the LLL algorithm e xtends naturally to the norm-Euclidean rings Z[i]andZ[ζ3]. 14 Razvan Barbulescu and Adrien Poulalion Deﬁnition 310 (norm-Euclidean ring, Prop 1.2.5 of [14], Def 2 of [32]) The Euclidean minimum of a number ﬁeld Kis MK:= max x∈Kmin y∈OKNK/Q(x−y). In particular, when Kis imaginary quadratic, MK= max x∈Cminy∈OK|x−y|2. IfMK<1we say that Kis norm-Euclidean. Example 2 (Prop 1.2.4 of [14],Prop 3 of [32]). The following ﬁelds are norm- Euclidean: –Q(ζk)fork∈ {1,3,4,5,7,8,9,11,12,15,16,20,24}; –Q(i√ d)ford∈ {1,2,3,7,11}. Deﬁnition 311 (OK-LLL-reduced basis) Let K=Q(i√ d)be an imaginary quadratic ﬁeld. We identify Kwith its embedding in Csuch that√ −dis mapped to a complex number of positive imaginary part. We consider the dot product /an}⌊ra⌋k⌉tl⌉{t·,·/an}⌊ra⌋k⌉tri}htofCmand we set /⌊ard⌊lx/⌊ard⌊l=/radicalbig /an}⌊ra⌋k⌉tl⌉{tx,x/an}⌊ra⌋k⌉tri}ht, which is a norm. Letb1,b2,...,b mbe independent in Cm. We say that L:=⊕m j=1Z[ζk]bj is aZ[ζk]-lattice and {bj}is a basis. Let {b∗ 1=b1,b∗ 2,...,b∗ m}be the Gram- Schmidt orthogonalization of {bj}with respect to the dot product. We set µi,j= /an}⌊ra⌋k⌉tl⌉{tbi,b∗ j/an}⌊ra⌋k⌉tri}ht//vextenddouble/vextenddoubleb∗ j/vextenddouble/vextenddouble2. We say that {bj}isOK-LLL-reduced with respect to MK<δ<1 if the following two conditions hold: 1./⌊ard⌊lµi,j/⌊ard⌊l ≤MKfor1≤j

Question 1368 multiple-choice

In quantum information theory, locally maximally mixed (LME) states are multipartite states where each subsystem appears completely random when observed individually. The existence and characterization of LME states depend sensitively on the dimensions of the subsystems and underlying group symmetries. For a three-partite quantum system with one qubit (dimension 2) and two other subsystems of dimensions B and C, which of the following statements correctly describes the possible existence and structure of LME states when B = C? 1) LME states exist only if B = C = 2, and the space of such states is always discrete. 2) LME states exist for any B, but only if C ≥ 2B, and are unique up to local unitary transformations for all B. 3) LME states exist for any B = C, and are described by B unit vectors in R³ summing to zero, modulo SO(3) rotations; for B ≥ 3 the space is of real dimension 2(B−3). 4) LME states exist only if B and C are coprime integers, with no continuous family of solutions. 5) LME states for B = C do not admit a geometric characterization via vectors in R³. 6) LME states exist only for B = C = 3, and are not invariant under SO(3) symmetry. 7) For B = C ≥ 4, the LME states are always unique without any continuous degrees of freedom.

✓ Correct Answer:

The correct answer is 3) LME states exist for any B = C, and are described by B unit vectors in R³ summing to zero, modulo SO(3) rotations; for B ≥ 3 the space is of real dimension 2(B−3)..

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construction: n= 3,2 =d1≤d2≤d3 As a more interesting example, consider the case with n= 3 where the ﬁrst subsystem is a qubit (i.e. d1= 2). We will assume without loss of generality that ( d1,d2,d3) = (2,B,C) with 2≤B≤C. In this case, the necessary conditions (2.6) require that B≤C≤2B (2.8) for the existence of locally maximally mixed states. This case is again sim ple enough that we can explicitly solve the algebraic equations on the coeﬃcients ψi1i2i3obtained by demanding that each subsystem is maximally mixed. Our detailed analysis is presen ted in appendix A. The result of this analysis is that LME states exist if and only if C=Bor (B,C) = 11 (NK,(N+ 1)K). This also follows from the general result Theorem 1.1, as we explain in example 5.6 below. ForB=C, we can describe all possible LME states in terms of a set {/vector ni}ofBunit vectors inR3adding to zero. We associate to /vector nithe qubit state |/vector ni/an}b∇acket∇i}htfor whichS/vector ni|/vector ni/an}b∇acket∇i}ht= (/planck⋊ver2pi1/2)|/vector ni/an}b∇acket∇i}ht, whereS/vector ni=/vector n·/vector σis the spin operator associated with direction /vector n. Then the state 1√ BB/summationdisplay i=1|ni/an}b∇acket∇i}ht⊗|i/an}b∇acket∇i}ht⊗|i/an}b∇acket∇i}ht (2.9) is LME, and all LME states for ( d1,d2,d3) = (2,B,B) can be obtained from states of this form by local unitary transformations. Transformations which sim ultaneously rotate all the vectors in R3also correspond to local unitary transformations, so the space o f LME states is equivalent to the space of Bunit vectors in R3adding to zero with equivalence under SO(3). This gives a unique state up to local unitary transformations for th e casesB= 2 andB= 3, and a space of real dimension 2( B−3) forB≥3. For (B,C) = (NK,(N+ 1)K), the analysis in appendix Ashows that there is a unique LME state up to local unitary transformations. In the case K= 1, we can write this state explicitly as |Ψ(2,N,N+1)/an}b∇acket∇i}ht=1√ N+1N/summationdisplay b=1/braceleftBigg/radicalbigg N+1−b N|0/an}b∇acket∇i}ht⊗|b/an}b∇acket∇i}ht⊗|b/an}b∇acket∇i}ht+/radicalbigg b N|1/an}b∇acket∇i}ht⊗|b/an}b∇acket∇i}ht⊗|b+1/an}b∇acket∇i}ht/bracerightBigg . (2.10) For general K, we can write the LME state as a tensor product |Ψ(2,N,(N+1))/an}b∇acket∇i}ht⊗|Ψ(K,K)/an}b∇acket∇i}ht (2.11) where|Ψ(K,K)/an}b∇acket∇i}htis the Bell state |Ψ(K,K)/an}b∇acket∇i}ht=1√ KK/summationdisplay i=1|i/an}b∇acket∇i}ht⊗|i/an}b∇acket∇i}ht. (2.12) The tensor product state lives in a Hilbert space (H2⊗HN⊗HN+1)⊗(HK⊗HK) =H2⊗(HN⊗HK)⊗(HN+1⊗HK) =H2⊗HNK⊗H(N+1)K (2.13) as desired, where subscripts indicate dimensions. 3 Locally maximally entangled states from representa- tion theory In this section, we describe a special class of LME states that can b e constructed using data coming from the representation theory of arbitrary ﬁnite and com pact groups. We will see 12 that most of the examples in the previous section can be understoo d in this way and that this method allows construction of explicit LME states in many other c ases (or possibly even for all cases ( d1,...,dn) where LME states exist; see section 3.3). We begin with Theorem 3.1. Let|Ψ/an}b∇acket∇i}htbe a pure state in a quantum system with Hilbert space H=HA⊗H¯A, whereAand¯Alabel a subsystem and its complementary subsystem. Let Hbe a subgroup of U(dA)×U(d¯A)that acts irreducibly on HAsuch that |Ψ/an}b∇acket∇i}htis invariant under Hup to a phase (i.e.h|Ψ/an}b∇acket∇i}ht=eiφ(h)|Ψ/an}b∇acket∇i}htfor allh∈H). Then the reduced density matrix ρAis maximally mixed. Proof.If|Ψ/an}b∇acket∇i}htis invariant under Hup to

Question 1369 multiple-choice

Quantum algorithms are transforming the landscape of cryptography by providing efficient solutions to mathematical problems that are hard for classical computers. Isogeny-based cryptosystems, which rely on the difficulty of certain algebraic problems, are especially impacted by these advances. Which cryptographic problem underpinning the security of the pSIDH cryptosystem can be solved in quantum polynomial time when N is prime, thereby rendering pSIDH insecure against quantum attacks? 1) Factoring large composite integers 2) Computing discrete logarithms in elliptic curve groups over composite order 3) Finding shortest vectors in lattice-based cryptosystems 4) Determining isomorphisms between supersingular elliptic curves 5) Evaluating the security of code-based cryptosystems 6) Solving hidden subgroup problems in abelian groups 7) Principal Quaternion Lifting Problem (PQLP)

✓ Correct Answer:

The correct answer is 7) Principal Quaternion Lifting Problem (PQLP).

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5.7 in Algorithm 4. Algorithm 4: QuaternionDecomposition (N,A,B,C,D) Input:N,A,B,C,D as in Problem 5.7. Output: x1,x2,x3∈Rsuch that A+Bj=λx1j(C+Dj)x2j(C+Dj)x3jmodNO0whereλis some integer that is coprime to N. 1t0←root of/parenleftBig n(A)/parenleftbig n(C)−pn(D)/parenrightbig +2pReR(A¯BC¯D)/parenrightBig −4qpImR(A¯BC¯D)t+ /parenleftBig qn(A)/parenleftbig n(C)−pn(D)/parenrightbig −2qpReR(A¯BC¯D)/parenrightBig t2= 0 mod N. 2x3←1+t0i 3x1,x2←EquivNormConjugationProduct (x3) 4Return x1,x2,x3. 5.4 Quantum algorithm for the PQLP As discussed earlier, Theorem 5.1 implies that we can solve the IsERP in quan- tum polynomial time. However, Theorem 5.1 only guarantees a rando mized polynomial-time algorithm, and it might be advantageous to avoid that inside a quantum algorithm. So instead of lifting elements inside the quantum a lgorithm we liftO(logN) elements ﬁrst and then utilize them to make the lifting proce- dure inside the quantum algorithm deterministic (and free of any heu ristic after the precomputation has succeeded). Theorem 5.10. There is an algorithm that solves the PQLP in quantum poly- nomial time. 28 M. Chen, M. Imran, G.Ivanyos, P.Kutas, A.Leroux, C. Petit Proof.We provide a proof for the case where Nis a prime number. The proof for general Nis in Appendix A. For any matrix Min GL2(Z/NZ),Mcan be written asPL·D·UwherePis a permutation matrix, Lis a lower unitriangular (it is lower triangular with 1-s in the diagonal), Dis diagonal, and Uis upper unitriangular(itisuppertriangularwith1-sinthediagonal).Thisdeco mposition can be found in polynomial time using Gaussian elimination. Now one has t o decompose L,DandUseparately.Anylowerunitriangularmatrixcanbewritten as a power of A=/parenleftbigg 1 0 g1/parenrightbigg wheregis a generator of ( Z/NZ)∗. Similarly, every upper unitriangular matrix can be written as a power of B=/parenleftbigg1g 0 1/parenrightbigg . Any diagonal matrix can be written as CkDlwhere C=/parenleftbigg g0 0 1/parenrightbigg ,D=/parenleftbigg 1 0 0g/parenrightbigg . This shows that every element in GL 2(Z/NZ) can be written as PAaBbCcDd. Thus instead of lifting elements inside the main quantum algorithm using Theo- rem 5.1 one can precompute lifts of P,A,B,C,D and then decompose a matrix M∈GL2(Z/NZ) asAaBbCcDdto obtain a lift of M. This decomposition requires several instances of the discrete loga rithm prob- lem inZ/NZwhich can be computed in quantum polynomial time. The only issue is that if one computes a powersmooth lift of A(orB,CorD) thenAa is not going to be powersmooth if ais large. To circumvent this issue one also computes lifts of A2k,B2k,C2kandD2kfor everykbetween 1 and log2(N). Furthermore, one computes lifts which are coprime (this can be ens ured easily asStrongApproximation can lift an element in Z[i]jto any number that is bigger thanpO(1)). This wayPAaBbCcDdwill also be powersmooth as it is the product of 4logN+1 powersmooth numbers. Lifting 4log N+1 numbers can be done in classical polynomial time using Theorem 5.1. ⊓ ⊔ 6 Impact on isogeny-based cryptography The most important application of Theorem 1.4is that it breakspSIDH in quan- tum polynomial time as Nis a prime number in pSIDH. Another application is on SCALLOP [ DFFK+23]. Even though Theorem 1.4 does not break SCAL- LOP, it shows that its security reduces to the problem of evaluating the secret prime degree isogeny (up to a scalar). In [ DFFK+23] it is already discussed that one can deduce some

Question 1370 multiple-choice

In the theory of abelian varieties and their moduli spaces, theta functions play a pivotal role in constructing canonical maps that embed or identify these varieties via explicit coordinates. The properties of such maps, especially injectivity and bijection, are often established using analytic and algebraic techniques. Which method is used to prove the bijection property of the map ψ: Ug → Ug, defined by x ↦ (θ₁(x)/θ₀(x),.., θg(x)/θ₀(x)), in the context of moduli of abelian varieties? 1) Galois descent argument 2) Use of universal covering spaces 3) Contraction mapping principle relying on completeness 4) Lefschetz hyperplane theorem 5) Study of tangent spaces at fixed points 6) Application of Hilbert's Nullstellensatz 7) Counting rational points over finite fields

✓ Correct Answer:

The correct answer is 3) Contraction mapping principle relying on completeness.

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of the canonical map φ: Gg/Γ ->Ak to Ug is a bijection of Ug with Uk. Proof. If x e U°, it follows from remark (b) of § II.2. that φ(x) e Uk. In order to prove bijectivity, it is enough to show the following: (1) ψ: Ug -> Ug x -> (Θ1(x)/Θo(x), , θg(x)/θ0(x)) is 1-1 and onto. (2) Two unit points with the same values of Xlf , Xg must be equal. We proceed to prove (1) and (2). We may normalize the θt so that θo — l+ , and θό = Xj + . Then -ψ is "close to the identity" so (1) is intuitively clear. To give a rigorous proof, suppose u = (uu , ug) e C/α. Let T:Ug ->Ug be the map #—>ίc—ψ(x) + u. It suffices to show that T has a unique fixed point. Let r = min. (ord. q^). If x, y e Ug set ord. (x — y)= min. ord. (xj — ^). We know that 0OC2O = Σ C/^7 where iό = 4^ and ord. C, = Σ 2^ord. ^. So if / ^ (0, , 0), ord. C7 > r. It follows that if x,yeUg: https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press ABELIAN VARIETIES 83 (a) ord. (θo(x) - ΘJiy)) > °rd. (x — y) + r . Let θf(X) = θj(X) — XjθoiX). A similar calculation gives: (b) ord. (flfix) - θf(y)) > ord. (x - y) + r . Now the difference between the j'th coordinate of T(x) and of T(y) is θf(x)/θo(x) - θf(y)/θo(y). Using (a), (b) and the fact that θo(x) and θo(y) are units, we see that this has ord. > ord. (x — y) + r. So T is a contrac- tion mapping. Since k is complete, so is U9, and T has a unique fixed point. To prove (2) note that for any a, (^"^Πί-i ^d2)θa is an element of R8g which only contains terms X1 with 0 < ij < 4. So we may write: where Fa is a homogeneous polynomial of degree 4g with coefficients in k. Lift Fa to a homogeneous Fa with coefficients in 0. Then θlg-ι(\[gi=10$θa and Fα(^0, #i, , 0^) differ by an element of JίR%gt<ΰ. Since iϋ2,
, χg(P)) + cga(χβ(P)) where fa, ga have coefficients in Φ. Suppose now that P and Q are unit points with Z/P) = XS(Q). Then X£P) and Xa(Q) are in 0 and each Xj(P) is a unit. and an easy induction show that XJJP) = Xa(Q) mod Cn for all n. So Zα(P) = Xa(Q) and P = Q. THEOREM II.3.2. Ak is injective. Proof. Suppose ψ{x) = ^(1) = P. Modifying a; by an element of

Question 1371 multiple-choice

Quantum algorithms have revolutionized approaches to integer factorization, posing significant implications for cryptography. Shor’s algorithm, in particular, leverages quantum parallelism and the Quantum Fourier Transform to factor large numbers exponentially faster than classical methods. Which step in Shor’s algorithm crucially depends on finding an even order r of the function f(a) = x^a mod N, with x coprime to N, in order to successfully compute non-trivial factors of N? 1) Initial selection of the modulus N 2) Application of window functions to suppress spectral leakage 3) Implementation of binary phase control in quantum hardware 4) Use of the general number field sieve for classical preprocessing 5) Period-finding via quantum computation using the Quantum Fourier Transform 6) Application of greatest common divisor (GCD) calculations before order determination 7) Repeating the algorithm until a suitable coprime x is found

✓ Correct Answer:

The correct answer is 5) Period-finding via quantum computation using the Quantum Fourier Transform.

📚 Reference Text:

transform, requirin g only binary phase control of th e wavefunction rather than the fine phase control implied by an exact QFT implementation. However, in many examples of period findin g, non -periodic matching leads to spectr al leakage effects which will disrupt operation of the quantum Hadamard transform, a lthough it may be possible to improve this with a window function . 2. THE SHOR ALGORITHM FOR LARGE INTEGER FACTORIZATION The factorization of large integer numbers into their prime fac tors is a difficult problem, computations scaling heavily with n, the bit length of the integer . The ease of forming a large number as the product of two smaller factors, but in contrast the considerable difficulty in determining these from the composite number if they are unknown , forms the basis of commercial public key en cryption systems such as RSA3. Thus efficient factorization algorithms would allow defeating such encryption schemes and so be of great technical importance, as well as having more theoretical applications to discovering new large prime numbers in pure mathematics. The most efficient classical algorithm is the general number field sieve which scales , for an n-bit number as approximately O(exp ((log n)1/3 (log log n)2/3)). In contrast , the Shor algorithm4, by employing the parallelism inherent to quantum computation, scales as O(n2(log n(log log n)), i.e. exponentially faster than the best known classical algorithm. However, t he physical implementation of a quantum computer capable of implementing the Shor algorithm has proved very difficult due to the necessity to maintain the required superposition states through the many quantum logic gates required to implement the algorith m for realistically useful integer sizes. To date, the largest number factorized using a quantum computer implementing Sh or’s algorithm is the number 21, which employed optical techniques for its hardware realization .5 Recently, proposals have been made for the scaling of integrated ion trap based quantum logic gates and the resources estimated to scale such a device to factor the long bit length numbers employed in cryptographic systems.6 The Shor algorithm4,7 factorises an integer N by finding the period, r, of a function f(a) = xa mod N, where x must be in the range 1

Question 1372 multiple-choice

In quantum computing, qudits generalize the concept of qubits to systems with d levels, allowing more complex algebraic structures and gate operations. The classification and explicit construction of quantum gates for qudits are essential for developing advanced quantum algorithms and fault-tolerant quantum architectures. Which statement correctly characterizes the Clifford hierarchy for qudits, specifically regarding the group structure of its levels? 1) All levels of the Clifford hierarchy for qudits form mathematical groups regardless of dimensionality. 2) Only the first level of the Clifford hierarchy for qudits is not a group; higher levels always are. 3) The Clifford hierarchy for qudits contains no group structures at any level due to modular arithmetic complications. 4) Levels C1 and C2 of the Clifford hierarchy for qudits are always abelian groups. 5) For d=2, the Clifford hierarchy levels above C2 always remain groups; for d>2, they do not. 6) Every level of the Clifford hierarchy for qudits is a group if the gates are diagonal in the computational basis. 7) The Pauli group (C1) and Clifford group (C2) for qudits are mathematical groups, while higher levels (Ck≥3) are generally not groups except for certain diagonal subsets.

✓ Correct Answer:

The correct answer is 7) The Pauli group (C1) and Clifford group (C2) for qudits are mathematical groups, while higher levels (Ck≥3) are generally not groups except for certain diagonal subsets..

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(or the generalized Pauli group) for a single qudit as [ 11, 67, 124, 130]. G/equals⎥braceleftbiggτcDχ→⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingleχ→∈Z2 d,c∈Zd⎥bracerightbigg(Zd/equals{0,1,... ,d−1}),(21) where χ→is a two-vector with elements from Zd. With these preliminary concepts de ﬁned in Eqs. 18-21, we now de ﬁne the Clifford group as the following: the set of the operators that mapsthe Weyl-Heisenberg group onto itself under conjugation is calledtheClifford group [124, 157 ]; C/equals⎥braceleftbigC∈U(d)⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingleCGC †/equalsG⎥bracerightbig. (22) A recursively de ﬁned set of gates, the so-called Clifford hierarchy, was introduced by Gottesman and Chuang as Ck+1/equals⎥braceleftbigU⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingleUC1U†4Ck⎥bracerightbig, (23) forC1the Pauli group [ 66]. The sets Ck≥3do not form groups, although the diagonal subsets of C3, which is our focus here, do form a group. The following derivations follow those in Ref. 71. The explicit formula for building a Clifford unitary gate with F/equals⎥parenleftBiggαβ cδ⎥parenrightBigg∈SL(2,Zd),χ→/equals⎥parenleftBiggx z⎥parenrightBigg∈Z2 d (24) is C⎥parenleftbigF⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingleχ→⎥parenrightbig/equalsD(x|z)VF, (25) VF/equals⎧⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩1⎥radicaltpext⎥radicaltpext d√⎥summationdisplayd−1 j,k/equals0τβ−1(αk2−2jk+δj2)|j〉〈k|,β≠0 ⎥summationdisplayd−1 k/equals0ταck2|αk〉〈k|, β/equals0.(26) The special case β/equals0 is particularly relevant to the later derivation, anddet⎛⎝⎥summationdisplayd−1 k/equals0ταck2|k〉〈k|⎞⎠/equalsταc 6(2d−1)(d−1)d, /equals⎧⎨ ⎩τ2αc,d/equals3, 1, ∀d>3,(27) can be shown. In the d/equals3 case, we use C ⎥parenleftbigg⎥bracketleftbigg10 c1⎥bracketrightbigg⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥bracketleftbiggx z⎥bracketrightbigg⎥parenrightbigg∈SU⎥parenleftbigp⎥parenrightbig∀p>3 (28) and det⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝C ⎥parenleftBigg⎥bracketleftBigg10 c1⎥bracketrightbigg⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥bracketleftbiggx z⎥bracketrightbigg⎥parenrightbigg⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠/equalsτ 2cforp/equals3. (29) With all the mathematical de ﬁnitions at hand, we are ready to give an explicit form of the qudit π/8 gate. We choose the qudit gateUυto be diagonal in the computational basis and claim that, ford>3,Uυhas the form Uυ/equalsU(υ0,υ1,...)/equals⎥summationdisplayd−1 k/equals0ωυk|k〉〈k|(υk∈Zd). (30) A straightforward application of Eqs 20 and 30yields UυD(x|z)U† υ/equalsD(x|z)⎥summationdisplay kωυk+1−υk|k〉〈k|. (31) AsUυis to be a member of C3, the right hand side of Eq. 31 must be a Clifford gate. We ignore the trivial case UυD(0|z)U† υ/equalsD(0|z) and focus on the case UυD(1|0)U† υin order to derive an explicit expression for Uυ. We de ﬁnec′,z′,ε′∈Zdsuch that UυD(1|0)U† υ/equalsωε′C ⎥parenleftbigg⎥bracketleftbigg10 c′1⎥bracketrightbigg⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥bracketleftbigg1 z′⎥bracketrightbigg⎥parenrightbigg(32) From Eqs 26 and31we see that the right-hand side of Eq. 32 is the most general form, and we note that U∈SU(d)implies ωc′U∈SU(d). We rewrite the left-hand side of Eq. 32 using Eq. 31and right-hand side using Eq. 26 and obtain D(1|0)⎥summationdisplay kωυk+1−υk|k〉〈k|/equalsωε′D(1|z′)⎥summationdisplayd−1 k/equals0τc′k2|k〉〈k|. (33) After canceling common factors of D(1|0), an identity between two diagonal matrices remains such that ωυk+1−υk/equalsωε′τz′ωkz′τc′k2(∀k∈Zd), (34) or, equivalently, using Eq. 20 , υk+1−υk/equalsε′+2−1z′+kz′+2−1c′k2. (35) From here, we derive the recursive relation υk+1/equalsυk+k⎥parenleftbig2−1c′k+z′⎥parenrightbig+2−1z′+ε′. (36) We solve for the υkwith a boundary condition υ0/equals0, Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504 5Wang et al. Qudits and High-Dimensional Quantum Computing υk/equals1 12k⎥braceleftbigc′+k⎥bracketleftbig6z′+(2k−3)c′⎥bracketrightbig⎥bracerightbig+kε′, (37) where all factors are evaluated modulo d. For example, with d/equals5, theﬁfth root of unity Eq. 14 isω/equalse2πi/5and choosing z’/equals1,c’/equals 4 and ε’/equals0, we obtain υ/equals(υ0,υ1,υ2,υ3,υ4)/equals(0,3,4,2,1) (38) so that Uυ/equals⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ω000 0 0 0ω−200 0 00 ω−100 00 0 ω20 00 00 ω1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(39) The diagonal elements of U υare powers of ωthat sum to zero modulo dand, consequently, det (Uυ)/equals1. For the d/equals3 case, because of Eq. 27 extra work is needed for solving a matrix equation similar to Eq. 32 .W e ﬁrst introduce a global phase factor eiϕsuch that det⎛⎝eiϕ⎥summationdisplayd−1 k/equals0τck2|k〉〈k|⎞⎠/equals10ϕ/equals4πc/9. (40) The ninth root of unity Eq. 14

Question 1373 multiple-choice

Quantum resampling algorithms are designed to manipulate data encoded in quantum registers by adjusting sampling rates, leveraging quantum operations for efficiency. A key tool in such algorithms is the quantum Fourier transform, which enables modifications to the frequency representation of quantum states. In quantum data processing, what is the primary function of the quantum Fourier transform when performing resampling operations on data stored in quantum registers? 1) Modifying the number of high-frequency encoding qubits for efficient upsampling and downsampling 2) Measuring quantum states to collapse their amplitudes to classical values 3) Generating entanglement between qubits for secure communication 4) Implementing error correction codes to protect quantum data 5) Initializing registers to the zero state before data encoding 6) Physically moving qubits between different quantum processors 7) Performing classical discrete Fourier transform calculations on measurement results

✓ Correct Answer:

The correct answer is 1) Modifying the number of high-frequency encoding qubits for efficient upsampling and downsampling.

📚 Reference Text:

Title: Quantum frequency resampling Year: 2025 Paper ID: 56a000cff5bed1b9e9d0008941144c6b65fec639 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/56a000cff5bed1b9e9d0008941144c6b65fec639 Abstract: In signal processing, resampling algorithms can modify the number of resources encoding a collection of data points. Downsampling reduces the cost of storage and communication, while upsampling interpolates new data from limited one, e.g., when resizing a digital image. We present a toolset of quantum algorithms to resample data encoded in the probabilities of a quantum register, using the quantum Fourier transform to adjust the number of high-frequency encoding qubits. We discuss advantage over classical resampling algorithms.

Question 1374 multiple-choice

Quantum chemical calculations and molecular docking are key techniques in the computational evaluation of metal complexes for potential anticancer activity. In silico ADMET profiling and toxicity classification are essential for assessing drug-likeness and safety. Which of the following statements correctly describes the significance of a vanadium complex exhibiting a high binding constant (−10.1 kcal/mol) and a low inhibition constant (0.0378 μMol) in molecular docking studies against breast cancer cell proteins? 1) It indicates poor fit of the complex in the protein’s active site and low therapeutic potential. 2) It suggests strong and specific binding to the target proteins, indicating high inhibitory potential. 3) It implies rapid metabolic degradation and unsuitable drug-likeness. 4) It means the complex is highly toxic and not safe for further development. 5) It reflects a large HOMO-LUMO gap, denoting low chemical reactivity. 6) It demonstrates insufficient absorption and distribution in biological systems. 7) It shows unreliable docking results due to high RMSD values.

✓ Correct Answer:

The correct answer is 2) It suggests strong and specific binding to the target proteins, indicating high inhibitory potential..

📚 Reference Text:

Title: Integrated quantum chemical calculations, predictive toxicity assessment, absorption, distribution, metabolism, excretion and toxicity profiling and molecular docking analysis to unveil the therapeutic potential of non‐oxovanadium(IV) and organotin(IV) complexes targeting breast cancer cells Year: 2024 Paper ID: 1e1a8cd096dd89dd28d59c9de06df9e27d2f5054 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/1e1a8cd096dd89dd28d59c9de06df9e27d2f5054 Abstract: In this work, theoretical calculations of o‐phenylphenol‐based non‐oxovanadium(IV) and organotin(IV) complexes, previously prepared and reported by our group, have been carried out by density functional theory (DFT). Density functional theory quantum chemical computations were used to explore the structural and spectroscopic characteristics of the complexes in this study. The inhibitory nature of complexes were revealed via molecular docking research, which were performed against selected breast cancer cell proteins, 5NWH and 3HB5. The optimization and stability of complexes 1–6, were conducted using optimized DFT/B3LYP/6–311++G (d, p) level. Simulated computations of the molecular electrostatic potential surface were also performed to analyze the reactive behavior of the non‐oxovanadium(IV) and organotin(IV) complexes. The stability and molecular reactivity of the molecules were computed using the HOMO‐LUMO energies, energy gap, chemical potential (μ), electronegativity (χ), hardness (η), and softness (S) values. In silico analysis through molecular docking, ADMET properties and toxicity evaluation was used to assess its anticancer activity, drug‐likeness property and toxicity. The binding constant value, evaluated from molecular docking, was found to be very promising, −10.1 kcal mol−1 observed for vanadium complex 5 and the complexes were found to exhibit inhibition constant as low as 0.0378 μMol. Root‐mean‐square deviation (RMSD) has been carried out to validate molecular docking studies, which have been found to be below 2.0 Å for the complexes, indicating successful docking of the ligand‐protein complex by the program. The complexes, evaluated for their toxicity behavior in terms of Lethal Dose, based on Globally Harmonized System (GHS), have been found to be chemical safe falling under the category III and V and hence can find use as future metallo‐based drugs.

Question 1375 multiple-choice

In quantum computing, the Hidden Subgroup Problem (HSP) is a foundational challenge that generalizes several important quantum algorithms and has significant implications for cryptography. Efficient quantum algorithms for HSP vary widely depending on the group structure and measurement strategies employed. Which of the following statements is TRUE regarding efficient quantum algorithms for the Hidden Subgroup Problem on groups of the form $ \mathbb{Z}_r^p \rtimes \mathbb{Z}_p $? 1) The first efficient quantum algorithm for these groups utilizes entangled measurements across multiple copies of hidden subgroup states. 2) All efficient quantum algorithms for these groups rely exclusively on single-copy measurements. 3) Entangled measurements cannot achieve polynomial-time performance for these groups. 4) Kuperberg’s algorithm provides a polynomial-time solution for these groups without entangled measurements. 5) The matrix sum problem is unrelated to quantum algorithms for the HSP in these groups. 6) Efficient algorithms for these groups require only classical post-processing after quantum state preparation. 7) Hidden subgroup states for these groups cannot be used to distinguish subgroup structures efficiently.

✓ Correct Answer:

The correct answer is 1) The first efficient quantum algorithm for these groups utilizes entangled measurements across multiple copies of hidden subgroup states..

📚 Reference Text:

the hidden subgr oup problem is polynomial, and in particular, that only polynomially many copies of the hid den subgroup states are suﬃcient to solve the problem in general [9]. However, measurements tha t operate on a single hidden subgroup state at a time are in general notsuﬃcient—in particular, they are insuﬃcient for the symmet - ric group [22]. Kuperberg’s algorithm for the for the dihedr alhspdepends essentially on using entangled measurements on multiple copies of the hidden sub group states [20], but unfortunately does not run in polynomial time. As far as we know, our algorit hm for the hspover groups of the form Zr p⋊ Zpis the ﬁrst eﬃcient quantum algorithm to use entangled measu rements across multiple copies (speciﬁcally, rcopies) of the hidden subgroup states. This result provides hope that the distinguishability of polynomially many copies of hidd en subgroup states may lead to further eﬃcient algorithms through the implementation of entangle d measurements. The remainder of the article is organized as follows. In Sect ion 2.1 we review the hidden subgroup problem and the standard approach to solving it wit h a quantum computer, and in Section 2.2 we review some relevant facts about semidirect p roduct groups. Then, in Section 3, we give the reduction to cyclic subgroups. In Section 4, we pr esent the pretty good measurement approach in detail, describing the relationship between hi dden subgroup states and the matrix sum problem, computing the success probability of the pgm, proving its optimality, and explaining how 2 to implement it on a quantum computer. We apply the approach t o speciﬁc groups in Sections 5 and 6, and we conclude in Section 7 with a discussion of the res ults and some open problems. 2 Deﬁnitions 2.1 Hidden subgroup problem LetGbe a ﬁnite group. We say that a function f:G→S(whereSis a ﬁnite set) hides the subgroupH≤Giffis constant and distinct on left cosets of HinG. The hidden subgroup problem is the following: given the ability to query the func tionf, ﬁnd a generating set for H. To approach the hspwith a quantum computer, we must have the ability to query fin superpo- sition. In particular, we are provided with a quantum oracle Ufacting asUf:|g,y/an}bracketri}ht /ma√st≀→ |g,y⊕f(g)/an}bracketri}ht for allg∈Gandy∈S, where⊕denotes the bitwise exclusive or operation, and the element s ofS are represented by bit strings of length poly(log |G|). An eﬃcient quantum algorithm for the hsp is an algorithm using this black box that ﬁnds a generating se t forHin time poly(log |G|). The standard approach to solving the hspon a quantum computer is as follows. Create a superposition over all elements of the group and then query t he function in superposition, giving Uf:1/radicalbig |G|/summationdisplay g∈G|g,0/an}bracketri}ht /ma√st≀→1/radicalbig |G|/summationdisplay g∈G|g,f(g)/an}bracketri}ht. (1) Next, discard the second register, leaving the ﬁrst registe r in a mixed state whose form depends on the hidden subgroup H, ρH:=|H| |G|/summationdisplay g∈K|gH/an}bracketri}ht/an}bracketle{tgH|with the coset states |gH/an}bracketri}ht:=1/radicalbig |H|/summationdisplay h∈H|gh/an}bracketri}ht, (2) whereK⊂Gis a complete set of left coset representatives of HinG. We callρHthehidden subgroup

Question 1376 multiple-choice

Quantum algorithms leverage unique features of quantum mechanics to solve problems in group theory and number theory that are difficult for classical computers. The hidden subgroup problem (HSP) serves as a unifying framework for understanding many quantum algorithms, with connections to eigenvalue estimation techniques. Which statement most accurately describes the relationship between the Abelian hidden subgroup problem and quantum eigenvalue estimation? 1) Abelian HSP is unrelated to eigenvalue estimation and relies solely on quantum Fourier transform. 2) Abelian HSP can only be solved using large quantum registers of control qubits. 3) Eigenvalue estimation is used exclusively for non-Abelian HSP, not Abelian HSP. 4) Abelian HSP requires factoring integers prior to applying eigenvalue estimation. 5) The connection between Abelian HSP and eigenvalue estimation is limited to classical algorithms. 6) Abelian HSP cannot be unified with other quantum algorithms through eigenvalue estimation. 7) Abelian HSP can be analyzed and solved using quantum eigenvalue estimation, providing a unified framework for various quantum algorithms.

✓ Correct Answer:

The correct answer is 7) Abelian HSP can be analyzed and solved using quantum eigenvalue estimation, providing a unified framework for various quantum algorithms..

📚 Reference Text:

Title: The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer Year: 1998 Paper ID: c4e61b8d75991fbcd774e462a8373effecb3216c Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c4e61b8d75991fbcd774e462a8373effecb3216c Abstract: A quantum computer can efficiently find the order of an element in a group, factors of composite integers, discrete logarithms, stabilisers in Abelian groups, and hidden or unknown subgroups of Abelian groups. It is already known how to phrase the first four problems as the estimation of eigenvalues of certain unitary operators. Here we show how the solution to the more general Abelian hidden subgroup problem can also be described and analysed as such. We then point out how certain instances of these problems can be solved with only one control qubit, or flying qubits, instead of entire registers of control qubits.

Question 1377 multiple-choice

Fusion algebras generalize the structure of representation rings for quantum groups, capturing the combinatorial rules for fusing irreducible representations. Asymptotic invariants such as uniform Følner and Kazhdan constants provide quantitative measures of amenability and rigidity in these algebraic systems. Which of the following statements about the exponential growth rate in fusion algebras associated with quantum groups is correct? 1) It remains constant for all choices of the deformation parameter q in quantum SO_q(3) but varies in SU_q(2). 2) It is minimized for fusion algebras corresponding to abelian classical groups. 3) It vanishes identically for fusion algebras of free unitary quantum groups. 4) It is always equal to the Kazhdan constant for semisimple compact Lie groups. 5) It depends only on the dimension of the underlying quantum group and not on its fusion rules. 6) It can be determined for all q-deformations of semisimple, simply connected, compact Lie groups and for all free unitary quantum groups. 7) It is undefined for discrete duals of quantum groups due to the lack of irreducible corepresentations.

✓ Correct Answer:

The correct answer is 6) It can be determined for all q-deformations of semisimple, simply connected, compact Lie groups and for all free unitary quantum groups..

📚 Reference Text:

Title: Asymptotic invariants for fusion algebras associated with compact quantum groups Year: 2025 Paper ID: 40cdf3f1221b29d0d636fa41f1c1197e6a26c6ae Source: semantic-scholar URL: https://www.semanticscholar.org/paper/40cdf3f1221b29d0d636fa41f1c1197e6a26c6ae Abstract: We introduce and study certain asymptotic invariants associated with fusion algebras (equipped with a dimension function), which arise naturally in the representation theory of compact quantum groups. Our invariants generalise the analogous concepts studied for classical discrete groups. Specifically we introduce uniform F\o lner constants and the uniform Kazhdan constant for a regular representation of a fusion algebra, and establish a relationship between these, amenability, and the exponential growth rate considered earlier by Banica and Vergnioux. Further we compute the invariants for fusion algebras associated with % discrete duals of quantum $SU_q(2)$ and $SO_q(3)$ and determine the uniform exponential growth rate for the fusion algebras of all $q$-deformations of semisimple, simply connected, compact Lie groups and for all free unitary quantum groups.

Question 1378 multiple-choice

In computational group theory, efficiently solving problems over finite abelian groups often relies on the construction of small generating sets. The structure and decomposition of these groups play a critical role in algorithmic applications, such as the Hidden Subgroup Problem. For a finite abelian group of order n, what is the maximal size bound for each set in a generating pair that guarantees coverage of the entire group and enables a deterministic algorithm to solve the Hidden Subgroup Problem with at most 4√n queries? 1) 2√n 2) n 3) log₂n 4) √n + 1 5) n/2 6) 4n 7) n²

✓ Correct Answer:

The correct answer is 1) 2√n.

📚 Reference Text:

pair for G 2. Then(S1×T1),(S2×T2)is a generating pair for G. Proof. Any element g∈G1may be expressed as g1+g2, where g1∈S1and g2∈S2. Similarly, any element h∈G2may be expressed as h1+h2, where h1∈T1and h2∈T2. Then(g,h) = (g1,h1)+( g2,h2)∈(S1×T1)+( S2×T2). The construction for a general ﬁnite abelian group combines t he above building blocks. Theorem 3.5. For any ﬁnite abelian group G with order n, there is a generatin g pair S 1,S2for G such that|S1|and|S2|are both at most 2√n. Proof. By the fundamental theorem of abelian groups, any non-trivi al ﬁnite abelian group Gmay be expressed as a direct product of cyclic groups of prime pow er order: G∼= Zpk1 1×Zpk2 2×···× Zpkl l, (3.1) where the integers piare primes, not necessarily distinct, and the integers ki≥1 for all i∈[l]. We prove the lemma by strong induction on l, the number of cyclic groups in a decomposition of G. Ifl=1, the statement follows from Lemma 3.2. Suppose the statement holds for all non-trivial ﬁnite abelia n groups which have a decomposi- tion as above with at most mcyclic groups, for some m≥1. Suppose l:=m+1, and consider a group Gwith order nand a decomposition with lcyclic groups as in eq. (3.1). Suppose for some i∈[l], the integer kiis even. Let rbe such an index. We let G1:=Zpkrrand G2 the direct product of the remaining cyclic groups so that G∼=G1×G2. By Lemma 3.2, there is a generating pair S1,S2forG1such that|S1|=|S2|=√n1, where n1:=prkr. By the induction hypothesis, there is a generating pair T1,T2forG2such that|T1|,|T2|≤2√n2, where n2:=|G2|. By Lemma 3.4, we get a generating pair (S1×T1),(S2×T2)forGsuch that|S1×T1|,|S2×T2|≤ 2√n1n2=2√n. Suppose for all i∈[l], the integers kiare odd. Suppose for some i,j∈[l],i/\⌉}atio\slash=j, we have pi= pj. Let r,sbe such a pair of indices. We let G1:=Zpkrr×Zpkssand G2the direct product of the 5 remaining cyclic groups so that G∼=G1×G2. By Lemma 3.3, there is a generating pair S1,S2 forG1such that|S1|=|S2|=√n1, where n1:=prkrpsks. By the induction hypothesis, there is a generating pair T1,T2forG2such that|T1|,|T2|≤2√n2, where n2:=|G2|. By Lemma 3.4, we get a generating pair (S1×T1),(S2×T2)forGsuch that|S1×T1|,|S2×T2|≤2√n1n2=2√n. Suppose for all i∈[l], the integers kiare odd, and the primes piare all distinct. Then, by the Chinese Remainder Theorem, we have G∼=Zn. The statement now follows from Lemma 3.2, by noting that for n≥2 we have⌈√n⌉≤2√nand⌊√n⌋+1≤2√n. The algorithm for the abelian Hidden Subgroup Problem follo ws directly from the existence of a suitable generating pair for the underlying group. Corollary 3.6. There is a deterministic algorithm with query complexity at m ost4√n that solves the Hidden Subgroup Problem over an abelian group G with order n. Moreover, if the input function hides the non-trivial subgroup H, the algorithm ﬁnds all the elements o f H. Proof. By Theorem 3.5, there is a generating pair S1,S2for the group Gsuch that|S1|,|S2|≤2√n. We query the oracle function fat−xfor all x∈S1and at all elements y∈S2. If the function is injective on the set of points queried, we say fis injective. Otherwise, we say that fis not injective, and output{x+y:x∈S1,y∈S2,f(−x) = f(y)}. If the function is injective, the

Question 1379 multiple-choice

In quantum information science, hybrid systems combining qubits and qutrits are explored to enhance computational power and error resilience. Nuclear Magnetic Resonance (NMR) experiments enable the simulation of such systems by engineering spin interactions and energy level splittings. When emulating a qubit-qubit-qutrit (QQT) system using a four-qubit NMR setup, which modification to the chemical shifts of spins 3 and 4 correctly recreates the qutrit’s quadrupolar splitting? 1) Set both ω3 and ω4 equal to the base frequency Ω3 2) Add J-coupling terms to ω3 but not to ω4 3) Subtract DQ from both ω3 and ω4 4) Set ω3 = Ω3 + DQ and ω4 = Ω3 − DQ 5) Use Larmor frequencies only without quadrupolar adjustments 6) Set ω3 = ω4 regardless of DQ value 7) Add J′ij coupling strengths to both ω3 and ω4

✓ Correct Answer:

The correct answer is 4) Set ω3 = Ω3 + DQ and ω4 = Ω3 − DQ.

📚 Reference Text:

spin states +1, 0 and −1 respectively. Each qubit un- dergoes six single spin ﬂips shown in blue (spin 1) and brown (spin 2) respectively. Qutrit transitions are shown in purp le. their Larmor frequencies i.e. the free precession of the qubits(qutrit) in a static magnetic ﬁeld. DQrepresents the eﬀective quadrupolarsplitting ofthe qutrit levels and the corresponding term in the Hamiltonian accounts for thestaticﬁrst-orderquadrupolarinteraction. Jijsdenote the strength of the scalar coupling interactions between ithandjthqubits, and J′ ijrepresents an interaction term involving the qutrit. The QQT Hamiltonian in Eqn. (6) can be emulated by fourqubitssuchthattwoofthequbitsmimicathree-level system. We use spins 3 and 4 of our four-qubit system to mimic a qutrit, such that the qutrit subspace containing levels 1,2 is emulated by the third qubit and the qutrit subspace with levels 2,3 is emulated by the fourth qubit, with the corresponding chemical shifts: ω3= Ω3+DQ andω4= Ω3−DQ. The Hamiltonian of a four-qubit QQQQ system (as-suming a speciﬁc coupling pattern J13=J14=J′ 13, J23=J24=J′ 23andJ34= 0) is given by HQQQQ=ω1I1z+ω2I2z+ω3I3z+ω4I4z+J12I1zI2z +J′ 23I2z(I3z+I4z)+J′ 13I1z(I3z+I4z) (7) whereIizrepresents the z-component of the magnetiza- tion vector of the ithspin. The NMR spectrum and mul- tiplet pattern of HQQQQwith the assumed Jcoupling values resembles the ﬁrst orderNMR spectrum of a QQT system (wherein each qubit multiplet contains six transi- tions and the qutrit multiplet contains eight transitions). Figure 1 shows the energy level diagramof the four-qubit QQT emulator (energy levels numbered from 1-16) and an actual QQT system (energy levels numbered from 1- 12). We use a 5-Fluorotryptophan molecule to emulate the QQT system, with one19F and three1H nuclei repre- senting the four qubits. The molecular structure and NMR parameters are depicted in Figure 2(a) and the thermal equilibrium 1D NMR spectrum is shown in Fig- ure 2(b). The T 1and T 2relaxation times for all four qubits range between 1.17 - 3.96 s and 0.82 - 2.28 s re- spectively. The line intensities in the coupling pattern of 5-Fluorotryptophan diﬀers slightly from that of the ideal QQT emulator, however hybrid qudit gates imple- mentation can be emulated on this system by tailoring theJ−evolution intervals to obtain the desired angles of rotation. The multiplet pattern and single quantum NMR transitions of the QQT emulator are shown in Fig- ure 3; labels below each spectral line refer to the corre- sponding transitions as per Figure 1(a). The mapping between the transitions (labeled Q1, Q2, T) of the QQT system and the four-qubit QQT emulator (labeled q1, q2, q3, q4) is given in Table 1. For an ideal QQT emula- tor, the lines of the ithqubit occurring at ωi±Ji3∓Ji4 overlapwith each other, while in the 5-Fluorotryptophan molecule, Ji3/ne}ationslash=Ji4(i∈[1,2]) and the lines occurring at ωi±Ji3∓Ji4are separated by 2 |Ji3−Ji4|. However this molecule can emulate a QQT system whenever the intensities of the transitions of the ithspin occurring at ωi±Ji3∓Ji4are equal. B. Experimental implementation of the QFT The NMR pulse sequence for the experimental imple- mentation of a hybrid qudit QFT on the four-qubit QQT NMRemulatorisshowninFigure4. Ahybridcontrolled- rotation gate RH j,k(where ‘j’ is

Question 1380 multiple-choice

Finite group representation theory studies the ways in which groups act on vector spaces through linear transformations, a topic deeply connected to quantum mechanics via operator methods. Techniques involving commuting operators have been used to streamline key computations and provide clearer physical interpretation. Which of the following advantages is specifically associated with using eigenfunctions of commuting operators in finite group representation theory? 1) It eliminates the need for character tables in all cases. 2) It guarantees the existence of a unique basis for each group representation. 3) It replaces linear algebraic methods with purely combinatorial techniques. 4) It enables representations only for infinite groups. 5) It restricts the application to abelian groups. 6) It allows systematic labeling of irreducible representations and calculation of characters, bases, matrix elements, and Clebsch-Gordan coefficients. 7) It requires solving non-commuting operator eigenvalue problems.

✓ Correct Answer:

The correct answer is 6) It allows systematic labeling of irreducible representations and calculation of characters, bases, matrix elements, and Clebsch-Gordan coefficients..

📚 Reference Text:

Title: PHYSICAL METHODS OF GROUP REPRESENTATION THEORY (I) A NEW APPROACH TO THE THEORY OF FINITE GROUP REPRESENTATIONS Year: 1977 Paper ID: 14fc2bba6e2f8c7c5a086f9bd46da901a0f55f8d Source: semantic-scholar URL: https://www.semanticscholar.org/paper/14fc2bba6e2f8c7c5a086f9bd46da901a0f55f8d Abstract: This paper advocates a new approach to the theory of finite group representa-tions by applying exclusively the method of commuting operators in quantum me-chanics. The basic problems of group representation theory such as the labeling of irreducible representations, the finding of characters, irreducible bases and matrix ele-ments and the CG coefficients et al are all simplified to the solving of the eigenfunc-tions of a certain complete set of commuting operators. This method has the advan-tage of being concise in theory and easily manageable in practice.

Question 1381 multiple-choice

Quantum error correction in many-body systems often exploits conserved quantities and system symmetries to achieve robustness against specific types of errors. Codes constructed from symmetric states like Dicke states can be tailored to protect logical information against erasures and local disturbances in models such as the Heisenberg or Ising spin chains. In a one-dimensional translation-invariant Heisenberg spin chain, which of the following code constructions most effectively protects logical qubits from errors that change the total magnetization by up to 2d, and why? 1) Using product states of spins with arbitrary magnetization values 2) Encoding logical information in Dicke states with magnetization values separated by steps of 2d+1 3) Choosing code words as eigenstates of the spin chain with consecutive magnetization values 4) Superposing states with random magnetization spacings to create code words 5) Constructing code words from non-symmetric superpositions with identical magnetization 6) Encoding logical information in states with fixed positions of up-spins, regardless of total magnetization 7) Using superpositions of basis states with magnetization values differing by 1

✓ Correct Answer:

The correct answer is 2) Encoding logical information in Dicke states with magnetization values separated by steps of 2d+1.

📚 Reference Text:

if a system A iis provided to the environment, the latter cannot tell whether the global state isjEiorjE0i, and, hence, the two energy levels form a two- dimensional code space that is approximately error cor-recting against erasures at known locations. For example, consider the code developed in Appendix D in Ref. [8], in the context of a 1D translation-invariant Heisenberg spin chain. Here, we consider as the relevant charge the total magnetization M¼Pσ i Zof the spin chain. The code words jhnmiin Appendix D in Ref. [8]are Dicke states with respect to total magnetization —i.e., they are a superposition of canonical n-spin basis states that all have some fixed magnetization m: jhnmi¼/C18n n=2þm=2/C19−1=2X s∶P sj¼mjsin: ð48Þ The code is covariant with respect to total magnetization by construction, by defining the magnetization charge operator in the abstract logical system to correspond to the magnetization of the corresponding code word. The valuesmare spaced out by steps of 2dþ1, thus ensuring that any errors which change the magnetization by at most 2d cannot cause logical bit flips. This trick —using a suffi- ciently large spacing between code words so that they arenot mapped into each other by errors —has analogs in Calderbank-Shor-Steane codes, related multiqubit codes [59], and bosonic error correction [60].H o w e v e r ,t os h o w that such errors are indeed correctable, one still has to make sure that expectation values of errors with each code word do not depend on the code word in the large- nlimit. This code ’s approximation parameter as an approximate error-correcting code against the erasure of a constantnumber of sites scales as ϵ worstðN∘EÞ¼Oð1=nÞð 49Þ (cf. Supplemental Material, Sec. D. 3 [37]). On the other hand, our bound (26) also displays the same scaling:ϵworstðN∘EÞ¼Ωð1=nÞ: ð50Þ In consequence, this code has an approximation parameter that displays the same scaling as our bound, meaning thatour bound is approximately tight in the regime n→∞. D. Thermodynamic codes with an Ising Hamiltonian A noteworthy feature of Theorem 2 is that the physical observable T Acan include terms fTαgwhich act on overlapping subsystems. In particular, then, TAcould be the Hamiltonian of a many-body quantum system in whichthe subsystems fA ighave nontrivial interactions. Several examples of approximate quantum error- correcting codes in interacting many-body systems arealready discussed in Ref. [8]. Here, we present a simple illustrative example showcasing the application of ourgeneral theorem. Consider a simple 1D quantum Isingspin chain at zero field, described by the Hamiltonian H Ising¼Xn i¼2σZ i−1σZ i; ð51Þ onnsites and with an open boundary condition. Energy eigenstates are diagonal in the Zbasis, where the total energy is the number of domain walls —that is, the number of links with σZ i−1σZ iequal to −1. We can define a code in the same spirit as the Dicke-state code described above, where the code word jmiis now a uniform superposition of all states with exactly mdomain walls. Again, we need to choose a restricted set of values for min order to ensure good

Question 1382 multiple-choice

In finite group theory, the construction and structure of subgroups within exceptional groups of Lie type, such as E₈(4), rely on detailed analysis of extensions, root subgroups, and actions of Cartan subgroups. Understanding the interplay of these components is central to the classification of maximal subgroups and the study of group representations. Which statement accurately describes the positive root subgroups within the Sylow 5-subgroup U of the split extension 5³.SL₃(5) inside E₈(4)? 1) There are exactly nine distinct positive root subgroups, each normalized by a Sylow 2-subgroup of SL₃(5). 2) The positive root subgroups are all non-abelian and act transitively on the Cartan subgroup T. 3) Only three T-invariant subgroups of U of order 5 exist, corresponding to the negative root groups. 4) There are six distinct T-invariant subgroups of U of order 5, called positive root subgroups, each determined by its centralizer in T. 5) The positive root subgroups form a cyclic group of order 30 under multiplication. 6) Every positive root subgroup is centralized by the entire Cartan subgroup T. 7) The number of positive root subgroups equals the dimension of the Cartan subgroup T.

✓ Correct Answer:

The correct answer is 4) There are six distinct T-invariant subgroups of U of order 5, called positive root subgroups, each determined by its centralizer in T..

📚 Reference Text:

in E8(4). The subgroup L4(5) will be constructed as the group generated by two conjugates of the local subgroup Na(E) = 53.SL3(5) constructed in Lemma 2.10. Write P= Na(E). We require a preliminary lemma concerning the group P. LEMMA 5.2. P is a split extension of E by SL3(5). Proof Let e EE#, and consider Cp(e). This has the form F.SLi(5), where, in the notation of the proof of Lemma 2.10, F= (a11 b1, a2, b2) ==51+4, an extraspecial group. By the construction of E (Lemma 2.10), CG(e) = X1X2 with each X;==SL 5. Since Cp(e) normalizes each X;, we conclude that Cp(e)/(e) acts completely reducibly on FI ( e). Thus FI ( e) has precisely six proper non-trivial Cp(e )/ (e)-invariant subgroups. Those lifting to extraspecial groups 51+2 come in pairs (orthogonal complements under the action of Sp4(5) on F). Since E is abelian, C p( e) therefore normalizes a complement EI ( e) to EI ( e) with E abelian. As SL2(5) has trivial first cohomology on the natural 2-dimensional module, we have E = (e) x S with S invariant under SL2(5). It now follows that a Sylow 5-subgroup of P splits over E, and hence P splits, as required. Proof of Theorem 5.1. In view of Lemma 5.2 we may write P =ED, with D == SL3(5). Let T == 24 x 24 be a Cartan subgroup of D, and let Ube a Sylow 5-subgroup of P such that T ~Np( U) and U = E( Un D ). Consider the six T-composition factors of U. Each has centralizer 24 in T, and a direct check shows that these six subgroups of Tare distinct. It follows that there are precisely six T-invariant subgroups of U of order 5. We call these positive root subgroups; thus a positive root subgroup is determined by its centralizer in T. The group D contains three other T-invariant subgroups of order 5. These are the usual root groups in SL3(5) corresponding to negative roots. Each has centralizer in T equal to that of some positive root group, and we call such a subgroup the opposite root group. Label root subgroups as follows: E = Uo:Ua+f3Uo:+/3+r• with (e) = Uo:+/3+r Write D = ( U±/3• U±r), where notation is chosen so that S = UrUf3+r (here Sis the group constructed in the proof of Lemma 5.2). The usual commutator relations hold among the root groups in P, this group being isomorphic to a maximal parabolic subgroup of L4(5). 46 A. M. COHEN, M. W. LIEBECK, J. SAXL AND G. M. SEITZ Now E=UrUf3+rUix+f3+r (where Eis as in Lemma 5.2). Consider E(U±1i):;;; C0(e) =X1X2=SL5°SL5. The group (U±/3) is diagonally embedded_ here (that is, it intersects each X; trivially), and it normalizes the intersection of E with each X;. But F intersects each X; in an extraspecial 51 +z, and the action of ( U ±fl) on E / ( e) is irreducible. Consequently E n X; = ( e) for i = 1,

Question 1383 multiple-choice

Quantum algorithms play a critical role in computational number theory and cryptography, especially for problems involving algebraic structures in number fields. Cyclotomic fields, generated by roots of unity, often enable more efficient algorithms due to their rich symmetries and algebraic properties. Which development allows the quantum algorithm for computing unit groups in cyclotomic fields to be reformulated as a hidden subgroup problem (HSP) rather than a continuous hidden subgroup problem (CHSP), thereby reducing the required number of qubits to approximately O(m^2) instead of O(m^5)? 1) The generalization of the Buchmann-Pohst algorithm to arbitrary number fields 2) The use of quantum error-correcting codes in the algorithm’s implementation 3) The application of Shor’s algorithm for discrete logarithms in finite fields 4) The integration of Galois automorphisms without additional conjectures 5) The switch from classical to quantum gates for lattice reductions 6) The employment of Hallgren’s algorithm for real quadratic fields 7) A conjecture about the size of class groups for certain cyclotomic fields

✓ Correct Answer:

The correct answer is 7) A conjecture about the size of class groups for certain cyclotomic fields.

📚 Reference Text:

arXiv:2303.03978v1 [cs.CR] 7 Mar 2023The special case of cyclotomic ﬁelds in quantum algorithms for unit groups Razvan Barbulescu1⋆and Adrien Poulalion2 1Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400, Talence, France razvan.barbulescu@u-bordeaux.fr 2Alice & Bob, Corps des Mines, France adrien.poulalion@mines.org Abstract. Unit group computations are a cryptographic primitive for which one has a fast quantum algorithm, but the required numb er of qubits is ˜O(m5). In this work we propose a modiﬁcation of the algorithm for which the number of qubits is ˜O(m2)in the case of cyclotomic ﬁelds. Moreover, under a recent conjecture on the size of the class g roup of Q(ζm+ζ−1 m), the quantum algorithms is much simpler because it is a hidden subgroup problem (HSP) algorithm rather than its err or estima- tion counterpart: continuous hidden subgroup problem (CHS P). We also discuss the (minor) speed-up obtained when exploiting Galo is automor- phisms thnaks to the Buchmann-Pohst algorithm over OK-lattices. 1 Introduction The diﬃculty to compute class groups and its cardinality, th e class group, plays an important role in cryptography. Notably, they are at the f oundation of a time commitments protocol [49], a scheme of homomorphic encrypt ion [15] and a ver- iﬁable delay function [50]. Note that the particular cases p lay an important role as the former two examples use quadratic ﬁelds and for the lat ter the cyclotomic case can be the fastest. From a perspective of theoretical computer science, these s chemes are broken because there exists a quantum algorithm of polynomial time complexity [10]. However, from a cryptographic point of view it is required to obtain a more precise estimation of the amount of qubits and of quantum gat es, as it was done for the other primitives of public key cryptography[7,4,40 ,6,44,51,29,45,40]. Let us recall the chronology of the works addressing this que stion. Note that both in the classical and quantum algorithms class group and unit group are very similar and they are sometimes computed simultaneousl y in order to have a halting condition, as in the classical algorithm of Buchma nn and McCurley. In 1994, Shor designed a quantum algorithm to factor integer s and, respectively, solve the discrete log problem in cyclic groups [48]. Kitaev [33] reformulated the algorithms by reducing both problems to ﬁnding the set of per iods of functions ⋆The ﬁrst author has been funded by the Hybrid quantum initiat ive (HQI) of the France 2030 program. 2 Razvan Barbulescu and Adrien Poulalion deﬁned over Zrwith ﬁniterand taking values in a ﬁnite set; this is the hidden subgroup problem (HSP). Note that the parameter raﬀects the time and space complexity only in a polynomial manner. In 2002, Hallgren [28]3reduced the computation of a fundamental unit of a real quadratic ﬁeld to ﬁnding the set of periods of a function deﬁned over R. More generally, if a function deﬁned over Rmhas a lattice (discrete subgroup) of periods, then ﬁnding the set of periods is the continuous hid den subgroup prob- lem (CHSP). As for

Question 1384 multiple-choice

Quantum algorithms for group-theoretic problems leverage techniques such as superposition creation, group action simulation, and efficient encodings to solve computationally complex tasks. These methods are particularly effective when applied to solvable groups due to their recursive structure. Which quantum algorithmic procedure enables the creation of a uniform superposition over the orbit of a quantum state under a group action and can be efficiently implemented using the Translating Coset Superposition algorithm as a subroutine? 1) Quantum Fourier Transform 2) Quantum Walks 3) Hidden Subgroup Sampling 4) Quantum Phase Estimation 5) Coset State Preparation 6) Orbit Superposition Algorithm 7) Abelian Group Decomposition

✓ Correct Answer:

The correct answer is 6) Orbit Superposition Algorithm.

📚 Reference Text:

unitary hides a subgroup HofZr×Zl1×···×Zjgenerated by a generator of typeu=( 1,a1,a2,...,a j), where xN=ga1 1ga2 2...gaj jN. Therefore encN(xN)= enc(ga1 1ga2 2...gaj j). The subgroup H, and therefore the generator uof this particular form, can be found in quantum time poly( /lscript) since this is the solution of Hidden Subgroup for abelian groups extended to functions having quantum ranges [26]. Finally, condition (2) is easily satisﬁed. Indeed, by the compatibility of our en- coding, group operations over G/Ncan be simulated by one call to the group oracle forG. Then the result enc(x), for some x∈G, has to be converted to encN(xN) using the above procedure. 4.2. Orbit superposition. Inthissection, weshowthatcomputingtheuniform superposition of the orbit of a given state is reducible to instances of Translating 18 FRIEDL, IVANYOS, MAGNIEZ, SANTHA, AND SEN Coset Superposition . In the following deﬁnition, we denote by |G·ϕ/angbracketrightthe state 1 √ |G(|φ/angbracketright)|/summationtext |φ/prime/angbracketright∈G(|φ/angbracketright)|φ/prime/angbracketright⊗s,w h e r e|ϕ/angbracketright=|φ/angbracketright⊗s. Definition 4.9. Let|φ/angbracketright∈Γ.L e tsbe a positive integer and let |ϕ/angbracketright=|φ/angbracketright⊗s. Orbit Superposition (G,α,s)is the partial unitary transformation that maps state |ϕ/angbracketright|ϕ/angbracketright|G/angbracketrightto|G·ϕ/angbracketright|ϕ/angbracketright|1G/angbracketright. Then the following algorithm implements Orbit Superposition . OS(G,α,s ) Input: |φ/angbracketright⊗2s|G/angbracketright 1. Apply the group element in 3rd register to the first sregisters: /summationdisplay x∈G|x·φ/angbracketright⊗s|φ/angbracketright⊗s|x/angbracketright=/summationdisplay |ϕ/prime/angbracketright∈G(|ϕ/angbracketright)|ϕ/prime/angbracketright|ϕ/angbracketright|xG|φ/angbracketright/angbracketright, where|ϕ/angbracketright=|φ/angbracketright⊗s 2. Perform the inverse of Translating Coset Superposition (G,α,s ) (which maps |x·φ/angbracketright⊗s|φ/angbracketright⊗s|1G/angbracketrightto|x·φ/angbracketright⊗s|φ/angbracketright⊗s|xG|φ/angbracketright/angbracketright): |G·ϕ/angbracketright|ϕ/angbracketright|1G/angbracketright. Theorem 4.10. LetGbe a black-box group with unique encoding of length /lscript, and letαbe a group action on Γ.L e t|φ/angbracketright∈Γ.L e tsbe a positive integer and |ϕ/angbracketright= |φ/angbracketright⊗s.OS(G,α,s)implements Orbit Superposition (G,α,s)using as a black box one implementation of Translating Coset Superposition (G,α,s)and quantum timepoly(/lscripts)for the remaining computation. 4.3. Translating coset self-reducibility in solvable groups. The purposeof this section is to prove Theorem 4.11 stating the reducibility of Translating Coset in some solvable group GtoTranslating Coset in proper normal subgroups and factors of Gunder some conditions. Given a group action αofGon a ﬁnite set Γ of mutually orthogonal quantum states, we deﬁne for every proper normal subgroup NGthe group action αNofG/Non{|N·φ/angbracketright:|φ/angbracketright∈Γ}byαN(xN,|N·φ/angbracketright)= |x·(N·φ)/angbracketrightfor every x∈Gand|φ/angbracketright∈Γ. Note that this action is independent of the chosen coset representative x; it depends only on the coset xN. For a group action like αson Γsthe group action ( αs)Nwill act on states such as|N·ϕ/angbracketright=1 √ |N(|φ/angbracketright)|/summationtext |φ/prime/angbracketright∈N(|φ/angbracketright)|φ/prime/angbracketright⊗s,w h e r e|φ/angbracketright∈Γa n d|ϕ/angbracketright=|φ/angbracketright⊗s. Note that the use of our notion of compatible encodings allows us to treat the oracle for αas an oracle for αN. In the followingalgorithm, we implicitly use the encoding encofGfor its elements z∈G, and a compatible encoding encNforG/N(given by Theorem 4.8) for its cosets zN∈G/N. Last, for a subset S⊆G, the notation S/Nis the following subset of G/N:S/N={xN:x∈S}. In particular, for any subgroup H≤G,w eh a v e uHN/N ={uhN:hN∈HN/N}. TCS(G,N,α,s (t+1 ) ) Hypothesis :NGwith compatible encoding for G/N Input: |φ0/angbracketright⊗s(t+1)|φ1/angbracketright⊗s(t+1)|1G/angbracketright Ancilla: |N/angbracketright⊗2t|0/angbracketright 1. Perform ttimes OS (N,α,s )on blocks |φ0/angbracketright⊗s|N/angbracketright and then ttimes on blocks |φ1/angbracketright⊗s|N/angbracketright: |N·ϕ0/angbracketright⊗t|φ0/angbracketright⊗s|N·ϕ1/angbracketright⊗t|φ1/angbracketright⊗s|1G/angbracketright|1G/angbracketright⊗2t|0/angbracketright, where|ϕ0/angbracketright=|φ0/angbracketright⊗sand|ϕ1/angbracketright=|φ1/angbracketright⊗s. HIDDEN TRANSLATION AND TRANSLATING COSET 19 2. XOR the compatible encoding of 1G/N into the ancilla register |0/angbracketright: |N·ϕ0/angbracketright⊗t|φ0/angbracketright⊗s|N·ϕ1/angbracketright⊗t|φ1/angbracketright⊗s|1G/angbracketright|1G/angbracketright⊗2t|1G/N/angbracketright. 3. Perform Translating Coset Superposition (G/N, (αs)N,t)on

Question 1385 multiple-choice

In the representation theory of semisimple algebras associated with symmetric groups, the structure and decomposition of ideals are crucial for understanding the classification of irreducible modules. Matrix ideals often arise from irreducible representations of symmetric groups and are connected to key concepts like induced representations and diagonalization. Which statement correctly describes the dimension of the induced irreducible representation $ F^A_a $ of the algebra $ A^0_n(d) $ associated with an irreducible representation $ j_a $ of $ S(n-2) $, when $ \det Q(a) \neq 0 $? 1) It equals $ (\dim j_a)^2 $. 2) It is given by $ n^2 \dim j_a $. 3) It is $ (n-1)\dim j_a $. 4) It equals the rank of the matrix $ Q(a) $. 5) It is $ n \cdot \dim j_a $. 6) It is always 1 regardless of $ j_a $. 7) It equals $ \dim y_n $ for each $ j_a $.

✓ Correct Answer:

The correct answer is 3) It is $ (n-1)\dim j_a $..

📚 Reference Text:

algebra A0 n(d)is the following Theorem 10. The algebraA0 n(d)is a direct sum of two ideals A0 n(d) =MN (24) and the idealsMandNhas different structures. a) The idealMis of the form M=M aU(a), (25) where U (a)are ideals of the algebra A0 n(d)characterized by the irreducible representations jaof the group S (n2), such thatja2Vd[S(n2)]and U(a) =spanCfuab ij(a):a,b=1, ..., n1, i,j=1, ..., wag (26) 11 with uab ij(a)upq kl(b) =dabQbp ik(a)uaq il(a). (27) The ideals U (a)are matrix ideals such that U(a)'M(rank Q(a),C), (28) in particular when detQ(a)6=0we have U(a)'M((n1)dimja,C). (29) b) The idealNhas the following structure N'M nM(dimyn,C), (30) where the matrix ideals M (dimyn,C)are generated by irreducible representations ynof the group S (n1)that are included in the representation Vd[S(n1)]i.e.ynare such that dh(n). The matrix ideals contained in the ideals MandNcontains all minimal left ideals i.e. all irreducible represen- tations of the algebra A0 n(d). The next theorems describes all these representations. The structure of the irreducible representations of the algebra A0 n(d), included in the ideal M, is completely determined by irreducible representations jaof the group S(n2), therefore we will denote them Fa A. Theorem 11. The irreducible representations Fa Aof the algebraA0 n(d)contained in the ideal U (a)M (see Theorem 10) are indexed by the irreducible representations jaof the group S (n2), such that ja2Vd[S(n2)]and ifffn jn:yn2 indS(n1) S(n2)(ja),jn=1, ..., dim yngis the reduced basis of the induced representation indS(n1) S(n2)(ja), then the vectors ffn jn:ln6=0gfrom the basis of the irreducible representation of the algebra A0 n(d)and the natural generators of A0 n(d)act on it in the following way V0(an)fn jn(a) =å r,jrå kq lrz†(a)ra jrkz(a)an kjnp lnfr jr(a), (31) where the summation is over rsuch that lr6=0.Due to the condition ja2Vd[S(n2)]the eigenvalues lnof Q(a)are non-negative. The unitary matrix Z (a) = ( z(a)an kjn)has the form z(a)an kjn=dimyn q Nn jn(n1)!å s2S(n1)yn jnjn(s1)das(q)ja kr[(an1)s(qn1)], (32) with Nn jn=dimyn (n1)!å s2S(n1)yn jnjn(s1)dqs(q)ja rr[(qn1)s(qn1)], (33) where the indices q =1, ..,n1,r=1, .., dim jaare ﬁxed and such that Nn jn>0. For more details see [21]. Whenever sn2S(n1)we have V(sn)fn jn(a) =å r,jryn injn(sn)fn in(a). (34) In particular when detQ(a)6=0,(i.e. when all ln6=0) then the representation Fa Ais the induced representation indS(n1) S(n2)(ja)(in the reduced form) for the subalgebra Vd[S(n1)]A0 n(d). In this case the dimension of the irreducible representation is equal to dimFa A= (n1)dimja=dim(indS(n1) S(n2)(ja)). (35) 12 When detQ(a) = 0,(i.e. when one, up to the multiplicity, eigenvalue lqof Q(a)is equal to 0),then the irreducible repre- sentation ofA0 n(d)is deﬁned on a subspace fyn jn:ln6=lqgof the representation space indS(n1) S(n2)(ja)and the representation has dimension is equal to dimFa A=dim(S(n1) S(n2)(ja))dimyq=rank Q(a). (36) This case takes the place when d=iai1 (37) for some aiin the partition a= (a1, ..,ai, ..,ak)characterizing the irreducible representation ja, under condition that n= (a1, ..,ai+1, ..,ak)characterizes the representation ynof S(n1). The ideal U (a)is a direct sum of dimFa Aof irreducible representations Fa A. In particular matrices z(a)an kjndiagonalize matrix Q(a)ab kl, i.e. we have following Proposition 12. å akå blz†(a)ra jrkQ(a)ab klz(a)bm ljm=drmdjrjmlm

Question 1386 multiple-choice

Nuclear magnetic resonance (NMR) quantum computing utilizes pulse sequences to control qubit states and interactions, enabling the implementation of quantum algorithms such as the quantum Fourier transform (QFT). Accurate manipulation and measurement techniques are crucial for assessing operational fidelity and error rates in these systems. In a three-qubit NMR quantum computer performing the quantum Fourier transform, which method is primarily used to selectively refocus unwanted qubit-qubit couplings during algorithm execution? 1) Application of a specific sequence of (π) pulses to invert spin states 2) Continuous application of strong magnetic fields to suppress all couplings 3) Use of composite pulses with randomized phases 4) Implementation of error-correcting codes before each gate operation 5) Measurement-based feedback to dynamically adjust pulse timing 6) Cooling the sample to near absolute zero to eliminate interactions 7) Employing gradient fields to spatially separate qubit resonances

✓ Correct Answer:

The correct answer is 1) Application of a specific sequence of (π) pulses to invert spin states.

📚 Reference Text:

1 8J23/parenrightig −(π)2 x−(π)1 −x−(π)2 −x− (π)3 −x−(π)2 x−/parenleftbigπ 2/parenrightbig3 y−(π)2 −x.(11) This sequence includes a number of ( π) pulses to refocus couplings during the intervals they should be inactive. The pulse sequence takes advantage of knowl- edge of the starting state of the system at the beginning and end of the program by replacing Hadamard transforms withπ 2pulses. In the mid- dle of the sequence the full Hadamard was indeed used. Figure 1 shows selected theoretical and exper- imental spectra following the quantum Fourier transform of the state I1 z+I2 z+I3 zon the three qubit NMR quantum computer. The ﬁdelity of the QFT calculated using the measure F=1 2+1 2Tr(ρtheoryρexp)/radicalig Tr(ρ2 theory)/radicalig Tr(ρ2exp)(12) is 87%. Here ρis the density matrix minus the part that is proportional to the identity (in NMR, this is called the ‘reduced’ density ma- trix; it should not be confused with the reduced density matrix got by partially tracing the den- sity matrix for a composite quantum system over 3 3.38 3.4 3.42 x 104−10123x 108 8.3 8.32 8.34 8.36 8.38 x 104−10123x 108 9.68 9.69 9.7 9.71 x 104−10123x 108200 300 400 500−100−50050100150200A 700 800 900 1000 1100−100−50050100150200B 2450 2500 2550 2600 2650−100−50050100150200C Figure 1: The three carbon-13 atoms of alanine after performance of the QFT. The top spectra are theoretical while the bottom are experimen- tal. Peaks in NMR spectra show the diﬀerence in energy level of single spin ﬂips. Each spin has four peaks since its energy level is dependent on whether the other two spins are up (along the magnet) or down. This shows up clearly in spin Bwhich has resolved J-couples to both of the other spins. The J-coupling between the Aand Cspins is very small and, therefore, the four peaks are not totally resolved. These peaks tell the magnitude of only some of the terms of the density matrix.some of its subsystems). This measure reﬂects both imperfections in the applied pulses and de- lays, as well as decoherence. To a ﬁrst approx- imation, decoherence during the course of the QFT attenuates the entire density matrix. This is shown in ﬁgure 2. Therefore, we can approxi- mately separate the errors caused by experimen- tal imperfections by renormalizing ρexpto its at- tenuated average. Using this the ﬁdelity of the operations themselves is above 98% over the 6 gates in (11). The ﬁdelity of 87% corresponds to an error rate of 97.7% over the six gates which, while high, does not attain the error rate of 10−4 required for robust quantum computation [22]. These errors arise primarily from spatial inho- mogeneities in the radio frequency ﬁelds which we believe can be improved. In conclusion, using NMR, the QFT has been implemented on a three bit quantum system and the ﬁdelity with which we can transform an ini- tially diagonal state has been measured. Al- though the ﬁdelity does not reach that required for fault tolerant computing, it is easily high enough to permit studies on small quantum sys- tems including quantum

Question 1387 multiple-choice

In group theory, the extension problem for countable abelian p-groups involves understanding how such groups can be combined with nilpotent p-groups, with particular attention to the decomposition into divisible and reduced parts. The rank of the divisible part plays a crucial role in simplifying the classification of possible group extensions. For an abelian p-group A decomposed as the direct sum of a divisible part D and a reduced part R, and a non-divisible nilpotent p-group B, under which condition does the extension property for A and B become equivalent to the extension property for R and B? 1) When the reduced part R is finite 2) When the nilpotency class of B is 1 3) When B is a divisible group 4) When the rank of D is at least p 5) When A is cyclic 6) When the rank of D is at most p - 2 7) When R has no elements of order p

✓ Correct Answer:

The correct answer is 6) When the rank of D is at most p - 2.

📚 Reference Text:

shows that [Aj, (rs + p - 2)b\ ^ #*+ 1 for; â s - 1. In proving (23) it remains to consider the term [Aps, (rs + p - 2)b\. But now, from (22), we get that [Ap, (rs + p - 2)b] g < [Ap2, (p - 2)6], ...,[A"s-\Ap ~ 2)b],[A p$p - 2)b] >. From our induction hypothesis in (20) together with (18) we see that [Ap, (rs+p- 2)b] g A"k + \ This completes the proof of (23). Thus to finally complete the proof of (20) and so of the theorem, we only need the following familiar type of lemma: LEMMA. If[A ps, mb] îâ Ap when m ^ 2, then [Aps, (m - $b] ^ A pk+\ Proof. For every a in A s, [a, (m - \)b] p[a, mbY l} ...[a,(rn+p- 3)b] p [a, (m + p - 2)b] = 1. As usual the hypotheses of the lemma give that all factors other than the first and last are in A p , and we have to show the same for the last. Taking p-th roots in the hypothesis, we get [As,mb] ^ {A vA\,APl...) thus [As,(m +p- 2)b]^ ([A U(P ~ 2)b], [Ap2, (p - 2)H [A? 1, (p ~ 2)b]). From (24), (19) and (18) we see that [As, (m + p - 2)b] ^ Ap" + '. Thus [As, (m - \)b] p S Apk+l and the lemma is proved. Combined with (23), this establishes that [As, bf ë Apk + \ This completes the proof that [A, b] p = 1, and we want finally to show https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press 1044 J. BUCKLEY AND J. WIEGOLD that G has class c â rx + 2. Since [A, b] has exponent /?, we have that then [A, 2b] ^ [v4j, b] since ^ is central, and indeed [A,(r x + 2)fc] ^ [A l9(rx + 1)0]. But [A x, rxb] ^ ^4 P from (1), so it is central, and we have that [A,(r x +2)b] ^[A x,(rx + \)b] = 1. So G is of class at most r x + 2, as claimed. Note that the case r x = 0 causes no problems, for then v4j = 1 and [^, b] = A p so that [A, b] is central. 5. Extensions of /^-groups: general countable A, Before considering countable />-groups in full generality we show that we can confine attention to reduced groups. THEOREM 5.1. Let A be an abelian p-group with divisible part D, let A = D X R and suppose that B is a non-divisible nilpotent p-group. Then Jf(A, B) if and only if the rank r of D is at most p - 2 andJf(R, B). Proof KJf{A, £), then^XD, B) and^CR, B). Since B is non-divisible, it has C p as image and so^(D, Cp). The result of [4] used earlier, together with the fact that D is central in every extension of D by C shows that

Question 1388 multiple-choice

Quantum Hidden Subgroup (QHS) algorithms are central to quantum computing, especially for tackling problems involving group-theoretic structures. Their extension to non-abelian groups presents significant theoretical and practical challenges. Which aspect of algorithm design is considered the primary obstacle in constructing new non-abelian Quantum Hidden Subgroup algorithms, particularly in relation to encoding quantum states and extracting hidden subgroup information? 1) Efficient implementation of the quantum Fourier transform for all group types 2) Choosing the correct transversal for coset representation 3) Minimizing the number of qubits required for group element encoding 4) Ensuring the oracle function is bijective 5) Guaranteeing that measurement outcomes are always unique 6) Using only abelian group structures in the initialization 7) Restricting queries to classical functions over groups

✓ Correct Answer:

The correct answer is 2) Choosing the correct transversal for coset representation.

📚 Reference Text:

arXiv:quant-ph/0403229v1 31 Mar 2004Quantum Hidden Subgroup Algorithms: The Devil Is in the Details Samuel J. Lomonaco, Jr.aand Louis H. Kauﬀmanb aDepartment of Computer Science and Electrical Engineering, Unive rsity of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, Maryland 21250,USA bDepartment of Mathematics, Statistics, and Computer Science, U niversity of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607-7045 ,USA ABSTRACT We conjecture that one of the main obstacles to creating new non- abelian quantum hidden subgroup algorithms is the correct choice of a transversal. 1. INTRODUCTION New quantum algorithms are crucially needed to demonstrate that f uture quantum computers will be more than highly specialized machines, but instead highly versatile general purp ose devices. Unfortunately, the current choice of quantum algorithms available for future quantum computin g devices is surprisingly meager. With some minor (but important) exceptions, there are only three available cla sses of quantum algorithms, namely: 1) Quantum hidden subgroup (QHS) algorithms, i.e., Shor/Simon-like a lgorithms, 2) Amplitude ampliﬁcation algorithms, i.e., Grover-like algorithms, and 3) Quantum algorithms that simulate quantum systems on a quantum computer. In this paper, we focus on the development of new non-abelian QHS a lgorithms because of their tantalizing promise of exponential speedup over existing classical algorithms. For example, should there be a major breakthrough in non-abelian QHS algorithm development, then ther e is the enticing possibility that it might be possible to develop a polynomial-time QHS algorithm for the graph isomo rphism problem. In spite of the eﬀorts of many researchers, the number of new no n-abelian QHS algorithms is very small and scattered. Why is it so diﬃcult to generalize QHS algorithms to non-ab elian groups? Before we can even guess at an answer to this question, we ﬁrst ne ed to deﬁne what we mean by a quantum hidden subgroup algorithm. Further author information: S.J.L., Jr. E-mail: Lomonaco@ umbc.edu: L.H.K. E-mail: kauﬀman@uic.edu 2. THE QUANTUM HIDDEN SUBGROUP ALGORITHM PARADIGM We begin by ﬁrst deﬁning the classical hidden subgroup problem: Hidden Subgroup Problem. Given a group Gand a map f:G−→H(not necessarily a morphism) of Ginto a group H, is it possible to ﬁnd an invariant subgroup KofGsuch that the map fcan be factored into the composition f=ι◦γof two maps γ:G−→G/Kandι:G/K−→H, whereγdenotes the natural epimorphism, and where ιdenotes an injection. The subgroup Kis called a hidden subgroup . The quantum analog of the classical hidden problem is deﬁned as follow s: Generic Quantum Hidden Subgroup Problem. LetHGandHHbe Hilbert spaces with respective orthonormal bases {|g/an}bracketri}ht:|g∈G}and{|h/an}bracketri}ht:h∈H}. Assume that f:G−→His given as a unitary transformation Uf:HG⊗HH−→ H G⊗HH |g/an}bracketri}ht|h/an}bracketri}ht /mapsto−→ | g/an}bracketri}ht/vextendsingle/vextendsinglef(g)h−1/angbracketrightbig Determine the hidden subgroup of fby making as few queries as possible to the blackbox Uf. We can now in turn deﬁne what we mean by a QHS algorithm that solves t he above QHS problem: Thegeneric quantum hidden subgroup (QHS) algorithm is essentially (ignoring many subtleties) the following: •Step O. Initialize two registers |Left-Reg>inHGand|Right-Reg >inHHto produce the initial state |ψ0/an}bracketri}ht=|0/an}bracketri}ht|1/an}bracketri}ht, assuming additive notation for G, and multiplicative notation for H. •Step 1.

Question 1389 multiple-choice

In finite group theory, 2-groups are groups whose order is a power of 2 and their subgroup structure reveals much about their properties. Certain 2-groups exhibit complex configurations of maximal subgroups and specific commutator and Frattini subgroup behavior. Which of the following statements correctly describes the commutator subgroup G′ of a 2-group G with order 2^m+3 (m ≥ 2), minimal number of generators d = 2, a unique abelian maximal subgroup, and quotient G/Z isomorphic to the dihedral group of order 8? 1) G′ is trivial and equal to the Frattini subgroup Φ 2) G′ is cyclic of order 2 and coincides with the center Z 3) G′ is isomorphic to the Klein four-group V4 4) G′ is isomorphic to the cyclic group C4 5) G′ is nonabelian and has order 8 6) G′ is equal to G and is noncyclic 7) G′ is isomorphic to the direct product C2 × C2

✓ Correct Answer:

The correct answer is 4) G′ is isomorphic to the cyclic group C4.

📚 Reference Text:

[G, G′] =/an}bracketle{tz/an}bracketri}ht∼=C2,E=/an}bracketle{tv, x/an}bracketri}ht∼=D8or Q8, E/triangleleftequalG,G=E/an}bracketle{ta/an}bracketri}ht,G/E ∼=C2m,Φ(G) =G′/an}bracketle{ta2/an}bracketri}htis abelian, H=E/an}bracketle{ta2/an}bracketri}ht, Z(G) =/an}bracketle{ta2, z/an}bracketri}ht,A=/an}bracketle{tax, v/an}bracketri}htis an abelian maximal subgroup of G,M=/an}bracketle{tv, a/an}bracketri}ht is metacyclic minimal nonabelian of order 2m+2and|G|= 2m+3,m≥2. Proof. IfGhas more than one abelian maximal subgroup, then all three maximal subgroups of Gare abelian, a contradiction. Hence Ais a unique abelian maximal subgroup of G. It follows that Γ 1={A, M, H }, where Mis 66 Z. BO ˇZIKOV AND Z. JANKO minimal nonabelian. The subgroup A∩M= Φ(G) is abelian, Φ( M)<Φ(G) and|Φ(G) : Φ(M)|= 2. Also, Φ( M) = Z( M)≤Z(G) so that for an element m∈M−A, CΦ(G)(m) = Z( M). If C A(m)>Z(M), then G=M∗Cwith C= C G(M) and M∩C= Z(M), contrary to d( G) = 2. It follows that CA(m) = Z( M) = Z( G) and so |G: Z(G)|= 8. From |G|= 2|Z(G)||G′| (Proposition 1.3) follows that |G′|= 4. For each x∈G−A, CA(x) = Z( M) and so x2∈Z(M) = Z( G) which implies that xinverts A/Z(M). IfA/Z(M)∼= E4, then Φ( G) =℧1(G)≤Z(M), a contradiction. Hence A/Z(M)∼=C4 and since m∈M−Ainverts A/Z(M), we have G/Z(M)∼=D8. Taking a∈A−Φ(G), then /an}bracketle{ta/an}bracketri}htcovers A/Z(M) and so v= [a, m]∈Φ(G)−Z(M). We get 1 = [a, m2] = [a, m][a, m]m=vvmand so vm=v−1 which implies that o( v) = 4. Indeed, if o( v) = 2, then [ v, m] = 1 which contradicts the fact that C Φ(G)(m) = Z( M). We get G′=/an}bracketle{tv/an}bracketri}ht∼=C4and [v, m] =v2implies that M′=/an}bracketle{tv2/an}bracketri}ht. Since v2is a square in M, it follows thatMis metacyclic (Proposition 1.1). In particular, |Ω1(Φ(G))| ≤4 which together with A/Z(M)∼=C4gives Ω 1(A)≤Φ(G) and so Ais also metacyclic. HereH= Φ(G)/an}bracketle{tam/an}bracketri}htis our third maximal subgroup of G. From C Φ(G)(am) = Z(M) follows that His nonabelian with Z( H) = Z( M) and so Proposition 1.3 gives |H′|=2. If d( H) = 2, then Proposition 1.1 gives that Hwould be minimal nonabelian, a contradiction. Hence d( H)≥3 and so His the only maximal subgroup of Gwhich is nonmetacyclic. In fact d( H) = 3 since Φ( G) is metacyclic. We are in a position to use Theorem 87.12 in [2] forn= 2 since G′∼=C4. This gives the generators and relations described in our th eorem, where we have used the notation from Theorem 87.12 in [2]. Theorem 2.2.LetGbe a2-group with d(G) = 2which has exactly one maximal subgroup Hwhich is neither abelian nor minimal nonabelian. If the other two maximal subgroups H1andH2are minimal nonabelian with H′ 1=H′ 2, then one of the following holds: (a)Gis one of the groups given in Theorem 87.10 in [2]. (b)Gis the group of order 25given in Theorem 87.14 in [2]. (c) G=/an}bracketle{th, x|h2n= 1, n≥2,[h, x] =s, s2= 1,[s, h] =z, z2= [z, h] = [z, x] = [x, s] = 1, x2∈ /an}bracketle{tz/an}bracketri}ht/an}bracketri}ht, where |G|= 2n+3,G′=/an}bracketle{tz, s/an}bracketri}ht∼=E4,K3(G) = [G, G′] =/an}bracketle{tz/an}bracketri}ht∼=C2, Φ(G) =G′/an}bracketle{th2/an}bracketri}htis abelian and maximal subgroups of GareH1= /an}bracketle{tG′, h/an}bracketri}ht,H2=/an}bracketle{tG′, xh/an}bracketri}ht(both are nonmetacyclic minimal nonabelian) andH=/an}bracketle{tx, s, h2/an}bracketri}htwithd(H) = 3andH′ 1=H′ 2=H′=/an}bracketle{tz/an}bracketri}ht. Proof. HereA=H1∩H2= Φ(G) is a maximal normal abelian subgroup ofG. Set H′ 1=H′ 2=/an}bracketle{tz/an}bracketri}ht

Question 1390 multiple-choice

Symbolic Swap Algebra is an algebraic structure designed to abstract the behavior of swap operations among elements, closely connected to the representation theory of the symmetric group S_n and applications in quantum information theory. Its definition relies on generators, specific relations, and connections to group algebras and irreducible representations. Which property characterizes the ideal Iswap_n in the Symbolic Swap Algebra Aswap_n with respect to irreducible representations of the symmetric group S_n? 1) It contains all polynomials invariant under the action of S_n. 2) It is generated by swap variables corresponding to odd permutations only. 3) It restricts the dimension of the algebra to be at most n-1. 4) Its elements vanish under all irreducible representations associated with Young tableaux containing three or more rows. 5) It consists of elements commuting with all swap variables s_ij. 6) Its elements vanish under all irreducible representations corresponding to Young tableaux with at most two rows. 7) It is generated by the squares of the swap variables minus the identity.

✓ Correct Answer:

The correct answer is 6) Its elements vanish under all irreducible representations corresponding to Young tableaux with at most two rows..

📚 Reference Text:

in Remark 3.2. Equivalently, we could have defined the Symbolic Swap Algebra as a quotient of the free algebra. Consider the free algebra C⟨sij|1≤i

Question 1391 multiple-choice

In algebraic geometry, graded rings associated with theta functions on abelian varieties exhibit rich algebraic and geometric structures, including integrality and finite generation properties. The interplay between group actions, graded components, and matrix identities is central to understanding these rings. Which statement accurately describes a consequence of the graded ring of theta functions being finitely generated over a field k? 1) The ring has dimension exactly equal to the rank of the underlying abelian variety. 2) Every graded component is a free module over k of finite rank. 3) The associated projective variety possesses well-behaved geometric properties, such as the existence of a projective embedding. 4) All elements of the ring are invertible. 5) The ring is necessarily semisimple. 6) The ring becomes non-Noetherian in positive characteristic. 7) The graded structure is preserved only when k has characteristic zero.

✓ Correct Answer:

The correct answer is 3) The associated projective variety possesses well-behaved geometric properties, such as the existence of a projective embedding..

📚 Reference Text:

a degree 3 graded endomorphism η of R". ηoη = a3 and R" is integral over η(R"). Proof. Similar to that of Proposition 1.1.5 and based on the identities: {XYZ)(XZ~ιT){XY-ιT-1) = X3 (XYZXX-ΎTXYZ-'T-1) = Y3 (XYZ)(X-ιZT~ι)(Y-ιZT) = Z% (XZ-ιT)(X-ιYT)(J-ιZT) = Γ3 . Remark. The proof of Proposition 1.1.5 essentially rests on the fact that AoA1 = 2/ where A is the matrix L _JV Similarly, Prop- osition 1.1.7 uses the fact that BoBt = 3/ where B1 1 1 01 10-11 1-1 0-1 0 1-1-1 §1.2. Finite generation of R In this section we show that the graded ring of theta functions is https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press ABELIAN VARIETIES 71 a finitely generated algebra over k. In the course of the proof stronger results are obtained, namely: (1) if char, k Φ 2, R2 generates R2m for large m. (2) if char, k Φ 3, Rz generates R3m for large m. LEMMA 1.2.1. The elements of Rλ have no common zero in Gg. Proof. Let qj = sίu by extending k we may assume qs — b) with bj e fc*. If / — (i19 , ig), ίj = 2m^ + r^ , let Cz = ft 6;J Π ^\T and set ^(Z) = Σ C7ZJ. Since ord. Cτ = |- ^ ij ord. g^ + 2r>s ΐris ord. j/rs and the matrix (ord. s/tj) is positive definite, φe^. Clearly we have: Thus and φ(VjX) - bjιXjWX) j = 1,2, . , g . Let 0(X, Γ) = ^(ZD^CZΓ-1). The proof of Proposition 1.1.5 shows that Θ e R[. So Θ is in the image of Rx ® Rx. Now suppose all the ele- ments of Rλ vanish at some point x e Gg. Then Θ(x9 Y) = 0, so 0. Then the elements of Rm which are power series in Xf have no common zero in Gg. Proof. Let x be any element of Gg. By Lemma 1.2.1 there is a θ e Riis/tj) such that θ(xm) Φ 0. Then by the remark preceding Proposi- tion 1.1.2. θ(Xm) eRm(j^ij) and does not vanish at x. Let n > 0. We assume for now that char, k does not divide n and the group Un of %-th roots of unity is contained in k. Recall that Sm https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press 72 HORACIO TAPIA-RECILLAS For m > 0 and u = (ul9 ---,ug)eUZ let Rm,u denote the set of ele- ments θe^ satisfying the following functional relation: PROPOSITION 1.2.3. Let n, Un and Sm be as above. Then each Rm>u is a subspace of Sm of dimension (2m)g and Sm = Θ(w) Rm,Uy (^) e Ugn. Proof. Let Tt: & -> & be the operators of Theorem 1.1.3. If θ e RmtU then Tt(θ) = thθ and T&θ) = β. By the proof of Theorem 1.1.3, Sm is the subspace of

Question 1392 multiple-choice

In the study of finite p-groups, classification often relies on subgroup structure, commutator relations, and algebraic invariants such as characteristic matrices. Understanding how these matrices determine isomorphism classes is crucial in computational and theoretical group theory. Which condition is sufficient to guarantee that two finite p-groups G and Ḡ, with G/Φ(G′) ≅ Mp(n,m,1) and G³ ≤ Φ(G′) ≅ Cₚ, are isomorphic? 1) Their orders are equal and both have cyclic centers. 2) There exists a subgroup of index p² in both groups. 3) The lower central series of both groups is identical. 4) There exists an invertible matrix X over Fₚ transforming the characteristic matrix of G to that of Ḡ via explicit equations. 5) Both groups have abelian maximal subgroups. 6) Both groups have the same number of generators. 7) Their commutator subgroups are isomorphic as abelian groups.

✓ Correct Answer:

The correct answer is 4) There exists an invertible matrix X over Fₚ transforming the characteristic matrix of G to that of Ḡ via explicit equations..

📚 Reference Text:

without loss of generality, we may assume that [ c,a] = 1 or [ c,b] = 1. If [c,a] = 1, then /an}bracketle{tap,b/an}bracketri}htis anA1-subgroup of index p2. If [c,b] = 1, then /an}bracketle{ta,bp/an}bracketri}htis an A1-subgroup of index p2. To sum up, Imin= 2. /square Theorem 3.2. Suppose that Gis a ﬁnite p-group such that G3≤Φ(G′)∼=Cpand G/Φ(G′)∼=Mp(n,m,1)wheren >1forp= 2andn≥m. Letw(G) = (wij)be a characteristic matrix of G. Then the following conclusions hold: (1)Ifw22=w12= 0, thenG∈ A3; (2)Ifw22= 0andw12/ne}ationslash= 0, thenG∈ Am+1; (3)Ifw22/ne}ationslash= 0andw12= 0, thenG∈ An+1. ProofWe getImaxby investigating all maximal subgroups of G. LetN=/an}bracketle{tb,ap,c/an}bracketri}ht andMi=/an}bracketle{tabi,bp,c/an}bracketri}ht. Then all maximal subgroups of GareNandMiwhere 0≤i≤ p−1. Note that m >1 forp >2 by Theorem 3.1 (1). (1) In this case, [ c,a] = [c,b] = 1. By calculation, Mi=/an}bracketle{tbp,abi/an}bracketri}ht ∗ /an}bracketle{tc/an}bracketri}htandN= /an}bracketle{tb,ap/an}bracketri}ht∗/an}bracketle{tc/an}bracketri}htwhere/an}bracketle{tbp,abi/an}bracketri}ht ∈ A1and/an}bracketle{tb,ap/an}bracketri}ht ∈ A1. By Lemma 2.3, Mi∈ A2andN∈ A2. HenceG∈ A3. (2) In this case, [ c,b] = 1. By Theorem 3.1 (1), m >1. By calculation, Mi= /an}bracketle{tc,abi/an}bracketri}ht ∗ /an}bracketle{tcbw12p/an}bracketri}htwhere/an}bracketle{tc,abi/an}bracketri}ht ∈ A 1. By Lemma 2.3, Mi∈ Am. Ifp= 2, then N=/an}bracketle{tc,b,a2/an}bracketri}htis abelian. If p >2, thenN=/an}bracketle{tb,ap/an}bracketri}ht∗/an}bracketle{tc/an}bracketri}ht ∈ A2. To sum up, G∈ Am+1. (3) In this case, [ c,b]/ne}ationslash= 1 and [ c,a] = 1. By calculation, N=/an}bracketle{tc,b/an}bracketri}ht ∗ /an}bracketle{tca−w22p/an}bracketri}ht where/an}bracketle{tc,b/an}bracketri}ht ∈ A1. By Lemma 2.3, N∈ An. Ifp= 2, then M0=/an}bracketle{tc,a,b2/an}bracketri}htis abelian andM1=/an}bracketle{tc,ab/an}bracketri}ht ∗/an}bracketle{tb2/an}bracketri}ht ∈ Am. Ifp >2, thenM0=/an}bracketle{tc,a,bp/an}bracketri}ht=/an}bracketle{ta,bp/an}bracketri}ht ∗ /an}bracketle{tc/an}bracketri}ht ∈ A2and Mi=/an}bracketle{tc,abi/an}bracketri}ht∗/an}bracketle{tcbiw22p/an}bracketri}ht ∈ Amfori= 1,2,...,p−1. To sum up, G∈ An+1. /square Theorem 3.3. Suppose that Gand¯Gare ﬁnite p-groups such that G3≤Φ(G′)∼=Cp andG/Φ(G′)∼=Mp(n,m,1), wherep >2andn≥m≥2. Let two characteristic matrices of Gand¯Gbew(G) = (wij)andw(¯G) = (¯wij)respectively. Then G∼=¯G if and only if there exists X=/parenleftbiggx11x12 x21pn−mx22/parenrightbigg , an invertible matrix over Fp, such that /parenleftbigg¯w11 ¯w21/parenrightbigg =|X|−1/parenleftbiggx11x12pn−m x21x22/parenrightbigg/parenleftbiggw11 w21/parenrightbigg and /parenleftbigg¯w12 ¯w22/parenrightbigg =X/parenleftbiggw12 w22/parenrightbigg . ProofSuppose that a,band ¯a,¯bare two set of characteristic generators of w(G) and w(¯G) respectively. Let θbe an isomorphism from ¯GontoG. We have Φ( ¯G)θ= Φ(G) and Ωm(¯G)θ= Ωm(G) since these four groups are characteristic in Gor¯G. So we may let ¯aθ=ax11bx12φ1,¯bθ=ax21pn−mbx22φ2 4 whereφ1∈Φ(G),φ2∈Φ(G)∩Ωm(G) andX:=/parenleftbiggx11x12 x21pn−mx22/parenrightbigg is an invertible matrix over Fp. By calculations, ¯cθ= [¯a,¯b]θ= [¯aθ,¯bθ]≡[ax11bx12,ax21pn−mbx22]≡c|X|(mod Φ( G′)). Transforming ¯ c¯w11p= ¯apnbyθ, we have c|X|¯w11p=ax11pnbx12pn. Hence |X|¯w11= (x11,x12pn−m)/parenleftbiggw11 w21/parenrightbigg (3.1) Transforming ¯ c¯w21p=¯bpmbyθ, we have c|X|¯w21p=ax21pnbx22pm. Hence |X|¯w21= (x21,x22)/parenleftbiggw11 w21/parenrightbigg (3.2) By Equation 3.1 and 3.2, /parenleftbigg¯w11 ¯w21/parenrightbigg =|X|−1/parenleftbiggx11x12pn−m x21x22/parenrightbigg/parenleftbiggw11 w21/parenrightbigg (3.3) Transforming ¯ c¯w12p= [¯c,¯a] byθ, we have c|X|¯w12p= [c|X|,ax11bx12]. It follows that c¯w12p= [c,a]x11[c,b]x12=cw12x11p+w22x12p. Hence ¯w12= (x11,x12)/parenleftbiggw12 w22/parenrightbigg (3.4) Transforming ¯ c¯w22p= [¯c,¯b] byθ, we have c|X|¯w22p= [c|X|,ax21pn−mbx22]. It follows that c¯w22p= [c,a]x21pn−m[c,b]x22=cw12x21pn−m+1+w22x22p. Hence ¯w22= (x21pn−m,x22)/parenleftbiggw12 w22/parenrightbigg (3.5) By Equation 3.4 and 3.5, /parenleftbigg¯w12 ¯w22/parenrightbigg =/parenleftbiggx11x12 x21pn−mx22/parenrightbigg/parenleftbiggw12 w22/parenrightbigg (3.6) Conversely, if there exists an invertible matrix X=/parenleftbiggx11x12 x21pn−mx22/parenrightbigg overFp such that Equation 3.3 and 3.6, then, by using the above argum ent, it is easy to check that the map θ: ¯a/mapsto→ax11bx12,¯b/mapsto→ax21pn−mbx22is an isomorphism from ¯GontoG./square Ifp= 2,m≥2 andn≥3, then we also have Equation 3.3 and 3.6. If p= 2, m= 1 and n≥3, thenb2∈Z(G). It follows that 1 = [ a,b2] =c2[c,b]. Hence [c,b] =c2. Thatis, w22= 1. Inthiscase,

Question 1393 multiple-choice

Quantum algorithms for group-theoretic problems often rely on efficient techniques for order-finding and solving hidden subgroup problems, especially when hardware resources are limited. Advances in state preparation, control qubit reduction, and semi-classical computation are critical for practical implementations on near-term quantum devices. Which optimization allows the estimation of k/r and (km mod r)/r using only log r qubits instead of the traditional 2 log₂r qubits in quantum order-finding algorithms? 1) Knowing the order r in advance 2) Doubling the number of target register qubits 3) Applying repeated quantum Fourier transforms 4) Using non-homomorphic function mappings 5) Performing eigenvalue estimation before order-finding 6) Ignoring measurement results until after all rotations 7) Increasing the coherence time of all qubits

✓ Correct Answer:

The correct answer is 1) Knowing the order r in advance.

📚 Reference Text:

rby applying the order-ﬁnding algorithm if necessary. By usi ng Uaxwith one control register we can approximate k/r, and by using Ubxwith another control register we can approximate ( kmmodr)/rand then extract mmodulo r/gcd(r, k). Note that since we know r, we only need log rbits of precision when estimating k/rand (kmmodr)/r, instead of 2 log2rwhen using continued fractions. Note further that, knowing r, it may be possible to actually place |Ψk/an}brack⌉tri}htinto the target register (by direct construction or otherwi se) for some known k, and thus only require onecontrol register with over log2rqubits to estimate ( kmmodr)/r. One way of doing this is to keep the target register after we have applied the order-ﬁnding algorithm and observ ed an estimate of k/rin the control register. At this point, the target register i s almost entirely in the state |Ψk/an}brack⌉tri}ht, and we could now just estimate the eigenvalue of Ubxon this eigenstate, which we know will be ( kmmodr)/r. 5.2 One Control Bit Consider the case that we have an eﬃcient computational mean s of mapping |f(y)/an}brack⌉tri}htto|f(y+x)/an}brack⌉tri}htfor any x. If we consider fto be of the form h◦gfor a homomorphism g, we are requiring that his the identity or some other function with enough structure that we can eﬃciently map h(g(y)) to h(g(y+x)) = h(g(y)+g(x)). In this case we can eﬃciently solve the hidden subgroup pr oblem with only one control bit or a sequence of ﬂying qubits [THLMK]. We illustrate this method for the problem of ﬁnding the order of an element ain a group H. Figure 2 shows the relationship between F−1 2nand the controlled multipli- cations by powers of ain the order-ﬁnding algorithm. As already pointed out in [GN], the measurements could be performed before the cont rolled rotations. The quantum controlled rotations could then be replaced wit h ‘semi-classically’ 11 Figure 2: We start with ( |0/an}brack⌉tri}ht+|1/an}brack⌉tri}ht)(|0/an}brack⌉tri}ht+|1/an}brack⌉tri}ht)(|0/an}brack⌉tri}ht+|1/an}brack⌉tri}ht)|Ψk/an}brack⌉tri}ht=/summationtext7 x=0|x/an}brack⌉tri}ht |Ψk/an}brack⌉tri}ht. The controlled multiplications create the state/summationtext7 x=0exp(2πik/r)|x/an}brack⌉tri}ht |Ψk/an}brack⌉tri}ht. The remaining gates create the state/vextendsingle/vextendsingle/tildewidestk/r/angbracketrightbig (apart from reversing the order of the qubits) which we then observe. The H-gates correspond to Hadamard trans- forms, and the Rj-gates correspond to a controlled phase shift of exp(2 πi/2j) on state |1/an}brack⌉tri}ht. Figure 3: Here we employ a ‘semi-classical’ version of F−1 23. We could mea- sure each qubit before it is used as a control, perform the con trolled rotations ‘semi-classically’, and the probability of observing each possible output state |x1/an}brack⌉tri}ht |x2/an}brack⌉tri}ht |x3/an}brack⌉tri}htis the same as in Fig. 2. 12 controlled rotations of the subsequent qubits (that is, the control bit is measured and, if the outcome is 1, the rotation is done quantumly). Thi s brings us to Fig. 3, where we observe further that all the operations on the ﬁrs t qubit could be performed before we even prepare the second qubit. All the op erations could be done sequentially, starting from the ﬁrst qubit, the resu lts of measuring the previous qubits determining how to prepare the next qubit be fore measurement. This means we could in

Question 1394 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) are fundamental in areas such as cryptography and computational group theory, with efficiency often hinging on group structure and algorithmic techniques. Advanced methods like strong Fourier sampling with adapted bases have expanded the scope of solvable HSPs, but significant challenges remain for more complex nonabelian groups. Which statement accurately describes a key limitation encountered when applying efficient quantum algorithms to the Hidden Subgroup Problem in certain groups such as the dihedral group? 1) The subgroup generators cannot be obtained even when the hiding set is polynomially sized. 2) The strong standard method fails to find any coset representatives for abelian groups. 3) Finding coset representatives becomes infeasible if the size of the query set grows superpolynomially with the input. 4) Quantum algorithms always solve HSP in matrix groups regardless of query set size. 5) Efficient solutions depend on the absence of level sets in the hiding function. 6) Adapted bases are only applicable to abelian groups and not nonabelian ones. 7) Gauss sum bounds prevent the use of Fourier sampling in HSP quantum algorithms.

✓ Correct Answer:

The correct answer is 3) Finding coset representatives becomes infeasible if the size of the query set grows superpolynomially with the input..

📚 Reference Text:

generators for L. We can reconstruct Sin classical polynomial time simply by querying the function hon all of K. Then L∩Kis the set of all ksuch that f(k)=f(1), and we construct Sby adding elements of L∩Kto it one at a time until they generate all of L∩K. To identify LH, as in [8] we deﬁne a new function f/primeonHconsisting of the unordered collection of the values of fon the corresponding left coset of K: f/prime(h)={f(g)|g∈t(h)·K}. Each query to f/primeconsists of |K|= poly( n) queries to f. The level sets of f/primeare clearly the cosets of LH, and so we reconstruct LHby solving the HSP on H. This yields a setTof generators for LH. It remains to ﬁnd a representative η(h)i nL∩(t(h)·K) for each h∈T.W e simply query f(g) for all g∈t(h)·Kand set η(h)t oa n y gsuch that f(g)=f(1). Since |T|=O(log|H|) = poly( n), this can be done in polynomial time, completing the proof. Unfortunately, we cannot iterate this construction more than a constant number of times, since doing so would require a superpolynomial number of queries to ffor each query of f/prime.I fKhas superpolynomial size, it is not clear how to obtain η(h), even when Hhas only two elements. Indeed, this is precisely the diﬃculty with the dihedral group. 9. Conclusion and directions for further work. We have shown that the “strong standard method,” applied with adapted bases, solves certain nonabelianHSPs in quantum polynomial time that cannot be solved using measurements inrandom bases or “forgetful” abelian approaches. While we are still very far from an algorithm for HSP in the symmetric group S n or for Graph Automorphism, a global understanding of the power of strong Fourier sampling remains an important goal. Perhaps the next class of groups to try beyondthe aﬃne and q-hedral groups are matrix groups such as PSL 2(p), whose maximal Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. THE POWER OF STRONG FOURIER SAMPLING 957 subgroups are isomorphic to Ap, and which include one of the inﬁnite families of ﬁnite simple groups. Appendix A. Notes on exponential sums. The basic Gauss sum bounds the inner products of additive and multiplicative characters of Fp, the ﬁnite ﬁeld of prime cardinality p. Deﬁnitive treatments appear in [19, section 5] and [17]. Considering Fpas an additive group with pelements, we have padditive characters χs:Fp→C, fors∈Fp, given by χs:z/mapsto→ωsz p, where, as above, ωp=e2πi/pis a primitive pth root of unity. Likewise considering the elements of F∗ p=Fp\{0}as a multiplicative group, we have p−1 characters ψt:F∗ p→C, for t∈F∗ p, given by ψt:gz/mapsto→ωtz p−1, where ωp−1=e2πi/(p−1)is a primitive ( p−1)th root of unity and gis a multiplicative generator for the (cyclic) group F∗ p. With this notation the basic Gauss sum is the following.Theorem 8. Letχ sbe an additive character and ψta multiplicative character ofFp.I fs/negationslash=0andt/negationslash=1, then /vextendsingle/vextendsingle/vextendsingle/summationdisplay z∈F∗ pχs(z)ψt(z)/vextendsingle/vextendsingle/vextendsingle=√ p. Otherwise /summationdisplay z∈F∗ pχs(z)ψt(z)=⎧ ⎪⎨ ⎪⎩p−1ifs=0,t=1, −1ifs=0,t/negationslash=1, 0 ifs/negationslash=0,t=1. See [19, section 5.11] for a proof. This basic result has been spectacularly generalized. In the body

Question 1395 multiple-choice

Quantum computing systems often face significant challenges due to noise, which can degrade the performance of critical algorithms such as Grover's algorithm and the quantum Fourier transform. Understanding how noise affects algorithmic accuracy is essential for designing scalable quantum hardware. Which of the following statements best describes the advantage of deriving analytical formulas for noisy multiqubit quantum circuits over relying solely on Monte Carlo statistical modeling? 1) Analytical formulas eliminate all sources of noise in quantum circuits. 2) Analytical formulas always produce exact results regardless of noise level. 3) Analytical formulas provide direct predictions of algorithmic accuracy as a function of qubit number and noise amplitude, supporting efficient design and scaling decisions. 4) Monte Carlo statistical modeling is unnecessary when analytical formulas are available. 5) Analytical formulas can only be applied to single-qubit quantum systems. 6) Analytical formulas require physical implementation for validation. 7) Analytical formulas are less reliable than results from random simulations.

✓ Correct Answer:

The correct answer is 3) Analytical formulas provide direct predictions of algorithmic accuracy as a function of qubit number and noise amplitude, supporting efficient design and scaling decisions..

📚 Reference Text:

Title: The influence of quantum noise on the Grover algorithm and quantum Fourier transform: quantum operations theory approach Year: 2017 Paper ID: d325fde466637206253e48e0de7bf56b6061e7dd Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d325fde466637206253e48e0de7bf56b6061e7dd Abstract: The method of noisy multiqubit quantum circuits modeling is proposed. The analytical formulas for the dependence of quantum algorithms accuracy on qubits count and noise level are obtained for Grover algorithm and quantum Fourier transform. It is shown that the proposed approach is very much in line with results obtained by Monte Carlo statistical modeling method. The developed theory makes it possible to predict the influence of quantum noise on the accuracy of the forward-looking multiqubit quantum systems under development.

Question 1396 multiple-choice

Variational quantum algorithms are pivotal in simulating quantum spin systems, especially on Noisy Intermediate-Scale Quantum (NISQ) devices. Enforcing symmetry properties within neural quantum state ansätze can significantly impact computational efficiency and physical accuracy. Which specific symmetry, when enforced in variational quantum algorithms for spin systems, leads to a polynomial reduction in the Hilbert space and ensures physically meaningful results? 1) SU(2) symmetry 2) C4 rotational symmetry 3) Marshall sign rule 4) Time-reversal symmetry 5) U(1) symmetry 6) Z2 parity symmetry 7) Translation symmetry

✓ Correct Answer:

The correct answer is 1) SU(2) symmetry.

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space. In theNISQ application, especially between 10 to 50 qubits, we have much better scaling, see Fig. 9. Numerous eﬀorts in applying NQS variational archi- tecture to represent the complicated sign structure in the frustrated regime essentially use the energy as the only criterion for assessing its accuracy. This would result in the optimized low-energy variational states in frustrated regime still obeying the Marshall sign rules even though the true ground state is likely to deviate from it signif- icantly [ 58], or breaks the SU (2)symmetry [ 57]. The preservation of spatial symmetry has been the core topic of discussion in the literature, with proposed C4equivariant CNN. However, on the two-dimensional model Heisen- berg model, the spatial symmetry consideration can only reduce the search space redundancy by a constant fac- tor, thus scaling very poorly at even intermediate n.B y reinforcing SU (2)symmetry, we achieve a polynomial reduction of Hilbert space and ensure the result to be physi- cally reasonable, hence oﬀering a second criterion to assess the variational ansätze. The number of qubits scaling linearly with the number of qubits naturally circumvent the issue of having gener- alization property, a crucial property for the NQS ansätze to function [ 92]. In fact, in a related work of us [ 31], we showed that, making use of the representation theory of the symmetric group, this leads to the superexponential quan- tum speedup. To this end, it is unlikely that any classically trained ansätze are capable to reinforce the global SU(2) symmetry of the system. A. Numerical simulation We provide numerical simulations to showcase the eﬀectiveness of the Sn-CQA ansätze, using a JAX auto- matic diﬀerentiation framework [ 93]. The implementation of the Sn-CQA ansätze utilizes the classical Fourier-space 11 9 101011 2 1 59 74 7 1253 2 81 6 34 6 812 FIG. 10. Square and kagome lattices. 020327-13 HAN ZHENG et al. PRX QUANTUM 4,020327 (2023) FIG. 11. For the 3 ×4 lattice, in either case, the Sn-CQA ansätze are able to converge to the ground state at least 10−4precision with explicitly reinforced SU(2) symmetry. This can be seen from the expectation of Sn-CQA never falls below the exact groundstate, while nonsymmetry respecting algorithms inevitably do. The numerical results are subject to room for further development, for instance, with better gradient-descent algorithms such as to utilize the Hessian, since we have only 200–500 learnable parameters to optimize. Therefore, we expect the performance and convergence rate of S n-CQA ansätze to further increase with perhaps more reﬁned tuning. activation by working in the Snirreducible representation subspace where the ground-state energy lies. This would impact the stability of the numerical simulations, which imply the best-suited models are with 8–16 spins. This bot- tleneck in computational resource, as shown in Sec. III, presents no issue for a potential larger-scale implementa- tion on quantum computers. The benchmarked examples with RBM and group-equivariant convolutional neural net- work (GCNN) [ 53] are drawn from NetKet [ 94] tutorial [95], which form the baseline

Question 1397 multiple-choice

In the study of quantum group symmetries, compact quantum groups are encoded algebraically by unitarizable Hopf *-algebras, which play a central role in noncommutative geometry and operator algebras. The concept of quantum torsors generalizes classical torsors to these noncommutative settings, facilitating deep connections between representation theory and algebraic extensions. What mathematical equivalence underpins the relationship between unitary comodules of a unitarizable Hopf *-algebra and quantum torsor analogues, involving *-fibre functors and A-*-Galois extensions with a positive Haar measure? 1) An equivalence of categories between *-fibre functors on unitary A-comodules and A-*-Galois extensions possessing a positive Haar measure 2) An isomorphism of groups between the automorphism group of A and the group of quantum torsors 3) A duality between the representation category of A and its module category 4) A homological correspondence between projective comodules and injective modules over A 5) A bijection between classical torsors and quantum torsors under the action of A 6) An equivalence of fibered categories over the category of vector spaces 7) An adjunction between A-*-Galois extensions and the category of unitary modules

✓ Correct Answer:

The correct answer is 1) An equivalence of categories between *-fibre functors on unitary A-comodules and A-*-Galois extensions possessing a positive Haar measure.

📚 Reference Text:

Title: Galois extension for a compact quantum group Year: 1999 Paper ID: 4534c2cc6772a4d1f80b2e06393179dcbbc3a278 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/4534c2cc6772a4d1f80b2e06393179dcbbc3a278 Abstract: The aim of this paper is to introduce and to investigate the analogues of torsors for compact quantum groups and to study their role in representation theory. Let A be a unitarizable Hopf *-algebra: we show that there is a category equivalence between *-fibre functors on unitary A-comodules and A-*-Galois extension endowed with a positive Haar measure.

Question 1398 multiple-choice

Quantum process tomography is a key technique for analyzing errors in quantum gate implementations, especially in systems such as nuclear magnetic resonance (NMR) quantum computing. In these experiments, radio-frequency (RF) control sequences are designed to improve the fidelity of complex operations like the Quantum Fourier Transform (QFT). Which experimental strategy led to a significant improvement in gate fidelity during Quantum Fourier Transform implementation on a spin system? 1) Using RF control sequences that compensate for RF field inhomogeneity 2) Increasing the temperature of the sample to reduce decoherence 3) Applying only single-qubit rotations without entangling operations 4) Omitting refocusing of the internal spin Hamiltonian 5) Measuring only pure input states with high initial correlation 6) Extending the QFT operation time beyond 30 milliseconds 7) Eliminating all hardware noise through post-processing algorithms

✓ Correct Answer:

The correct answer is 1) Using RF control sequences that compensate for RF field inhomogeneity.

📚 Reference Text:

alanine, save for an average overall rate o f attenuation. Of course, QPT could be performed on a simpler gate, as in Ref. [15] where it was u sed to derive a relaxation superoperator for free precession, but the goal he re is to use QPT to learn about the coherent and incoherent errors committed during implem entation of a complex unitary transformation, the QFT. Therefore, this simple eﬀective relaxation superoperator was assumed to accelerate the simulations of the overall QFT (no re laxation was assumed during the pulse design simulations, since they are not intended to co rrect for such eﬀects). VI. AN OVERVIEW OF THE EXPERIMENTAL RESULTS CompleteQPTwasperformedtwiceusingdiﬀerentsetsofstronglym odulatingRFcontrol sequences to implement single spin rotations (Section IV). The ﬁrst such iteration was done with the sequences described in Ref. [3], which refocussed the evolu tion under the alanine molecule’s internal spin Hamiltonian ( Hsin Eqn. 19) during the sequence. The second iteration used control sequences that not only refocussed the in ternal Hamiltonian, but also compensated for inhomogeneity in the RF ﬁeld itself [10]. Thus, the se cond set of control sequences was expected to produce much smaller incoherent erro rs than the ﬁrst, while the coherent and decoherent errors were expected to be rough ly the same for both. The improvement obtained with the RF-compensated set is illustrated in F ig. 6, which plots the correlations and attenuated correlations obtained for each o f the 64 product operator basis states used as inputs for the two iterations against one anot her. Further evidence for substantial improvements with the RF-compensated control sequences may be obtained from the Kraus operator plots and statistics in Fig. 7 (see below for their interpretation). Having demonstrated this clear-cut improvement, all subsequent analysis will be given only for the results obtained wiht the compensated control pulses. The correlation provides an estimate of the accuracy of the exper imental QFT implemen- tation, but without considering the loss of magnetization due to dec oherence or incoherence. The average of the input state correlations over all 64 basis state s was 0.96, while the aver- age correlation following application of the QFT was 0 .82. A minimization search, however, found pure states which, after operating on them with the experim ental superoperator, had a correlation with the same pure state following the theoretical QFT as low as 0 .45. The average attenuated correlation (or gate ﬁdelity [3]), which also ta kes into account the loss of 17 magnetization, was 0 .64, indicating that about 22% of the magnetization was lost over the ca. 30 ms needed to implement the QFT (see Section IV). These numb ers are in-line with expectations based on other recent applications of the RF-compe nsated control sequences [4]. About half of this magnetization loss was expected due to intrinsic decoherence, and since approximately the same amount of decoherence occurred du ring the input states’ read- out as during the output states’, the remainder is probably due to r esidual uncompensated incoherence and/or

Question 1399 multiple-choice

Qudits are quantum systems with more than two levels, offering opportunities to increase computational power and efficiency in quantum computing compared to traditional qubits. Topological quantum computing seeks to harness exotic particles and topological effects to improve error resilience and gate operations. Which physical effect is specifically utilized to implement non-Clifford gates necessary for universal quantum computation in parafermion-based topological qudit systems? 1) Quantum Zeno effect 2) Aharonov-Casher effect 3) Landau-Zener tunneling 4) Stark effect 5) Fano resonance 6) Josephson effect 7) Doppler shift

✓ Correct Answer:

The correct answer is 2) Aharonov-Casher effect.

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and avoided level crossings [ 160, 161 ]. Another suggestion for exploiting qudit dynamics concerns using a degenerate two-level system with theadditional freedom perhaps improving the energy gap and thusincreasing success probability [ 156]. A dearth of studies have taken place to date into qudit-based adiabatic quantum computing. The one proposal thus farconcerns a quantum adiabatic algorithm for factorization ontwo qudits [ 166]. Speci ﬁcally, they consider two qudits of possibly different dimensions, thus necessitating a hybrid two-qudit gate [ 39]. They propose a time-dependent effective Hamiltonian to realize this two-qudit gate and its realization as radio-frequency magnetic ﬁeld pulses. For this model, they simulate factorization of each of the numbers 35, 21, and 15 fortwo quadrupole nuclei with spins 3/2 and 1, respectively,corresponding to qudit dimensions of 4 and 3, respectively. 4.3 Topological Quantum Computing With Qudits Topological quantum computing offers advantages over other forms of quantum computing by reducing quantum errorcorrection overheads by exploiting topological protection. Somework has been done on topological quantum computing with quditsby proposing quantum computing with parafermions [ 49, 74 ]. Majorana fermions are expected to exhibit non-abelian statistics, which makes these exotic particles, or theirquasiparticle analogue, sought after for anyonic quantumcomputing [ 90]. Majorana fermions can be generalized to Z d parafermions, which also exhibit non-abelian statistics and reduce to standard Majorana fermions for d/equals2. One advantage Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504 16Wang et al. Qudits and High-Dimensional Quantum Computing ofd>2 is that parafermion braiding is an entangling operation. Importantly, encoding a qudit of dimension din the four- parafermion fusion space enables all single-qudit Clifford gatesto be generated modulo phase terms [ 74]. Clifford gates do not provide a universal set of gates for quantum computing. A non-Clifford gate can be achieved forparafermions encoded into parafermion zero modes by exploitingthe Aharonov-Casher effect, physically implemented by move ahalf-ﬂuxon around the parafermionic zero modes. Combining this non-Clifford gate with the Clifford gates achieved byparafermion braiding yields a universal gate set of non-abelianquantum computing with qudits [ 49]. 5 IMPLEMENTATIONS OF QUDITS AND ALGORITHMS The qubit circuit and qubit algorithm have been implemented on various physical systems such as defects in solids [ 27, 81, 120 ]; quantum dots [ 104, 127 ]; photons [ 113, 132 ]; super conducting systems [ 29, 31 ]; trapped ions [ 14, 15 ]; magnetic [ 7, 18, 32, 148 ] and non-magnetic molecules [ 30, 152 ]. For each physical representation of the qubit, only two levels of states are usedto store and process quantum information. However, manyquantum properties of these physical systems have more than two levels, such as the frequency of the photon [ 106]; energy levels of the trapped ions [ 91]; spin states of the nuclear magnetic resonance systems [ 48] and the spin state of the molecular magnetic magnets [ 115]. Therefore, these systems have the potential to represent qudit systems. In this section, we brie ﬂy review several physical platforms that have been used

Question 1400 multiple-choice

In quantum computing, braiding operators from braid group representations are used to generate entangled multi-qubit states such as GHZ and Bell states. These operators play a critical role in constructing universal quantum gates and robust entanglement for quantum information tasks. Which operator, when applied to a two-qubit separable basis state, produces all four Bell states and is also sufficient for universal quantum computation when combined with local unitary gates? 1) The Bell matrix (B(2,1)) 2) The Swap gate 3) The Pauli-X gate 4) The Toffoli gate 5) The Fredkin gate 6) The Hadamard gate 7) The Phase gate

✓ Correct Answer:

The correct answer is 1) The Bell matrix (B(2,1)).

📚 Reference Text:

k-th qubit |ak/an}bracketri}htis |0/an}bracketri}htor|1/an}bracketri}ht. The resulting states are separable in the ﬁrst (k−1) qubits, but entangled in the other ( n−k+ 1) qubits. In particular, for k= 1, the operator B(n,1) entangles the state |a1a2...ak···an/an}bracketri}htwith its conjugate state|¯a1¯a2...¯ak···¯an/an}bracketri}ht, thus giving the generalized GHZ states. We see that these states can indeed be obtained from separable basis states by the braiding operator. We now give a few examples of the braiding operator B(n,1) fork= 1 andn= 1,2,3. From now on we chooseθ=π/8. This gives d=−√ 2, anda,b=±1/√ 2. Without loss of generality, we take a=b= 1/√ 2. The four matrix elements in (14) are da2=dab=−1/√ 2 and db2+A−2=A4dab=−i/√ 2. Explicitly, B(n,1) has the form B(n,1) =/parenleftigg −1√ 20 0−i√ 2/parenrightigg ⊗n j=2I(17) +/parenleftigg 0i√ 2 −1√ 20/parenrightigg ⊗n j=2σ1. 4 Forn= 1,B(1,1) =−1√ 2/parenleftbigg 1−i 1i/parenrightbigg is, up to global phases, equivalent to the Hadamard gate. For n= 2: B(2,1) =−1√ 2 1 0 0 −i 0 1−i0 0 1i0 1 0 0i . (18) This is equivalent to the Bell matrix up to global phases, and it gives all four Bell states from the separable stan- dard basis. For example, when acting on the states |00/an}bracketri}htand|10/an}bracketri}ht, it gives −(|00/an}bracketri}ht+|11/an}bracketri}ht)/√ 2 and−i(|10/an}bracketri}ht − |01/an}bracketri}ht)/√ 2, respectively. Note, however, the diﬀerence between the appearance of this matrix in our approach, and the Bell matrix Rin (4). There the Bell matrix Ris the solution of the alge- braic Yang-Baxter equation (3), and is the basic building block of the braid generators biin (2). In our approach the matrix (18) is obtained from the product of the ma- trices representing the braid generators b1andb2, i.e., it represents the braid b1b2. In a sense, we have factorized R. It was mentioned in the Introduction that the main impetus to using braid group representations in quan- tum computing is that the Bell matrix is a universal gate [2]. SinceB(2,1) is equivalent to Rin generating the Bell states, it should also be a universal gate. To prove that, it suﬃces to show, following [2], that the universal CNOT gate can be generated from B(2,1) and local uni- tary transformations. This is indeed the case, as we have CNOT = (α⊗β)B(2,1)(γ⊗δ), where α=1√ 2/parenleftbigg 1i 1−i/parenrightbigg , β=1√ 2/parenleftbigg 1−i i−1/parenrightbigg , γ=1√ 2/parenleftbigg −1i 1i/parenrightbigg , δ=/parenleftbigg 1 0 0−1/parenrightbigg .(19) Generalized GHZ states for larger ncan be obtained accordingly. Cluster-like states. – As mentioned in the Introduc- tion, the GHZ state is rather fragile in its entangle- ment, as it becomes separable after one of its qubits is measured. Multi-qubit systems which possess more robust entanglement can in fact be generated using B(n,k). As an example, we consider the result ofapplying a braiding operator B(n,k) on a generalized GHZ state |φ/an}bracketri}htgenerated from |00...00/an}bracketri}htwith the braid- ing operator B−1(n,1) =B†(n,1). We have |Φ/an}bracketri}ht= B−1(n,1)|00...00/an}bracketri}ht= (|00...00/an}bracketri}ht+i|11...11/an}bracketri}ht)/√ 2. Upon applying B(n,k) to|φ/an}bracketri}ht, we get B(n,k)|Φ/an}bracketri}ht=1 2(|00···00/an}bracketri}htk−1|00···00/an}bracketri}htn−k+1 +|00···00/an}bracketri}htk−1|11···11/an}bracketri}htn−k+1 +|11···11/an}bracketri}htk−1|00···00/an}bracketri}htn−k+1(20) − |11···11/an}bracketri}htk−1|11···11/an}bracketri}htn−k+1). Here |00···00/an}bracketri}htk−1≡ | 0/an}bracketri}ht1|0/an}bracketri}ht2···|0/an}bracketri}htk−1, |00···00/an}bracketri}htn−k+1≡ |0/an}bracketri}htk|0/an}bracketri}htk+1···|0/an}bracketri}htn, etc. This state is an entangled state for n≥2 andk>1. Unlike the GHZ

Question 1401 multiple-choice

Quantum computing has led to the development of algorithms that outperform classical methods for certain mathematical and cryptographic problems. Understanding the types of quantum algorithms and the foundational problems they address is crucial for grasping advances in computational theory and cryptography. Which quantum algorithm fundamentally threatens cryptographic systems like RSA and Diffie-Hellman by providing a polynomial-time solution to integer factorization and discrete logarithms, tasks that are considered hard for classical computers? 1) Grover’s algorithm 2) Shor’s algorithm 3) Bernstein-Vazirani algorithm 4) Deutsch-Jozsa algorithm 5) Hallgren’s algorithm 6) Simon’s algorithm 7) Ajtai-Dwork algorithm

✓ Correct Answer:

The correct answer is 2) Shor’s algorithm.

📚 Reference Text:

an equivalent but more convenient model, the quantum circuit. These ideas led to the search for further applications of a quantum computer, initially resulting in solutions to somewhat contrived problems, e.g., the Deutsch-Jozsa algorithm [6], the Simon algorithm [7], and the Bernstein-Varzani algorithm [8]. Nonetheless, these early results demonstrated the existence of speciﬁc problems that can be handled dramatically faster by a quantum computer. A breakthrough that elevated quantum computation from a primarily academic exercise to a matter of great practical signiﬁcance came in 1994 when Shor [9] introduced polynomial- time quantum algorithms for factoring numbers and computing the discrete logarithm. The widely used public key cryptography systems RSA and Difﬁe-Hellman rely on the fact that there are no polynomial-time classical algorithms for these problems. When implemented on a quantum computer, Shor’s algorithms will break these encryption schemes in a matter of seconds. Another masterpiece of quantum-computational software, Grover’s algorithm [10], was introduced in 1996, offering a solution for unstructured search problems with quadratic speedup over classical search algorithms. Recently Hallgren [11] developed a quantum algorithm for Pell’s equation with exponential speedup over any existing classical algorithm. Quantum algorithms that outperform classical algorithms can be separated into three cate- gories [12]: 1. Algorithms that are based on the quantum Fourier transform (e.g., Shor’s algorithm). 3 2. Quantum search algorithms. 3. Quantum simulations. The algorithm developed in this dissertation ﬁts into the ﬁrst category. 1.3 The Hidden Subgroup Problem in Quantum Computing 1.3.1 General Formulation and Special Cases The hidden subgroup problem can be stated for any group Gas follows. Deﬁnition 1.3.1. Letf:G!Xmap a group Gto a ﬁnite set X. Let Hbe a subgroup of Gsuch that for any g1;g22G,f(g1) =f(g2)if and only if g1H=g2H. Determine the generating set of H. Almost all known quantum algorithms that run super-polynomially faster than their clas- sical counterparts also solve special cases of the hidden subgroup problem. Some examples are listed in Table 1.1. Table 1.1: Special Cases of The Hidden Subgroup Problem Problem Group Set Hidden Subgroup Function G X H f Deutsch Z2f0;1gf0g f(0)6=f(1) Z2 f(0) =f(1) Simon Zn 2f0;1gnf0;sg,s2f0;1gnf(sx) =f(s) Period Z any rZ f(x+r) =f(x) Finding r2Z Order Z ak rZ f(x) =ax Finding k2Zr r2Z f(x+r) =f(x) ar=1 Discrete ZrZr ak f(t;ts)g f(x1;x2) =akx1+x2 logarithm k2Zr t2Zr f(x1;x2) =f(x1+t;x2ts) ar=1 s2Zr 4 1.3.2 The Dihedral Hidden Subgroup Problem Deﬁnition 1.3.2. The dihedral hidden subgroup problem is the hidden subgroup problem with G=DN, where DNis the dihedral group of order 2 N DN= x;yjxN=y2=yxyx=1 (1.4) The set of all integral linear combinations of a set of nlinearly independent vectors in Rn is called a lattice. The problem of ﬁnding the shortest vector in a lattice is referred to as the unique shortest vector problem. The Ajtai-Dwork public key cryptosystem [13] is based on the hardness of this problem. It has been shown [14] that if there exists a solution to the dihedral hidden subgroup problem that samples cosets, then there exists a quantum algorithm that solves the poly(n)-unique shortest vector

Question 1402 multiple-choice

Quantum circuit recompilation often involves transforming circuits with complex entanglement and non-trivial optimization landscapes. The efficiency of algorithmic strategies in this domain depends heavily on the interplay between adaptivity and classical post-processing methods. Which of the following best explains why adaptive quantum algorithms can achieve exponential computational advantage over non-adaptive classical methods in certain circuit recompilation tasks? 1) Adaptive algorithms can always avoid local minima in any optimization landscape. 2) Non-adaptive methods benefit from volume-law entanglement, making them more efficient. 3) Adaptive algorithms exploit feedback from quantum hardware to efficiently navigate unimodal, non-separable loss landscapes, revealing hidden circuit structures. 4) Non-adaptive classical post-processing is equally efficient as adaptive quantum algorithms for highly entangled circuits. 5) Adaptive methods require exponentially more data from quantum hardware than non-adaptive methods. 6) Non-adaptive techniques utilize high-magic circuits to outperform adaptive quantum algorithms. 7) Adaptive algorithms are only advantageous for separable optimization problems.

✓ Correct Answer:

The correct answer is 3) Adaptive algorithms exploit feedback from quantum hardware to efficiently navigate unimodal, non-separable loss landscapes, revealing hidden circuit structures..

📚 Reference Text:

Title: Advantage for Discrete Variational Quantum Algorithms in Circuit Recompilation Year: 2025 Paper ID: be231ee5cfc2b78948aa090fa3ff866752a06421 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/be231ee5cfc2b78948aa090fa3ff866752a06421 Abstract: The relative power of quantum algorithms, using an adaptive access to quantum devices, versus classical post-processing methods that rely only on an initial quantum data set, remains the subject of active debate. Here, we present evidence for an exponential separation between adaptive and non-adaptive strategies in a quantum circuit recompilation task. Our construction features compilation problems with loss landscapes for discrete optimization that are unimodal yet non-separable, a structure known in classical optimization to confer exponential advantages to adaptive search. Numerical experiments show that optimization can efficiently uncover hidden circuit structure operating in the regime of volume-law entanglement and high-magic, while non-adaptive approaches are seemingly limited to exhaustive search requiring exponential resources. These results indicate that adaptive access to quantum hardware provides a fundamental advantage.

Question 1403 multiple-choice

In discrete mathematics and probabilistic combinatorics, Azuma’s inequality is often used to bound the probability that a martingale sequence deviates significantly from its expected value. Such techniques are crucial in analyzing randomized algorithms and constructions involving periodic phenomena, such as those found in Boolean function representations. Which property of the random variables Xi = cos(2πkig/m) is essential for verifying that the sequence Y0 = 0, Y1, Y2,.. forms a martingale suitable for Azuma’s inequality? 1) The variance of each Xi is always one 2) Each Xi is an independent random variable 3) The sum ∑i=1^k Xi is always positive 4) The conditional expectation E[Yk+1|Y1,..,Yk] equals Yk 5) Each Xi is uniformly distributed on [−1,1] 6) The sequence Yk is strictly increasing 7) The sequence Yk is deterministic

✓ Correct Answer:

The correct answer is 4) The conditional expectation E[Yk+1|Y1,..,Yk] equals Yk.

📚 Reference Text:

binary decision diagrams .SIAM Monographs on Discrete MathematicsandApplications . SIAM Press. F.Ablayev & A.Vasiliev 11 A Proofof Lemma2 Proof.Using Azuma’s inequality (see, e.g., [10]) we prove that a ra ndom choice of the set Kis “good” with positive probability . Let1≤g≤m−1andletKbethesetof tparametersselected uniformlyatrandomfrom {0,...,m− 1}. Wedeﬁnerandom variables Xi=cos2πkig mandYk=∑k i=1Xi. Wewanttoprovethat Azuma’sinequal- ity is applicable to the sequence Y0=0,Y1,Y2,Y3, ..., i.e. it is a martingale with bounded differences. First, weneed toprove that E[Yk]<∞. From the deﬁnition of Xiit follows that E[Xi]=1 mm−1 ∑ j=0cos2πjg m Consider the following weighted sum of mthroots of unity 1 mm−1 ∑ j=0exp/parenleftbigg2πjg mi/parenrightbigg =1 m·exp(2πigm/m)−1 exp(2πig/m)−1=0, sincegisnot amultiple of m. E[Xi]isexactly the real part of the previous sum and thus isequal t o 0. Consequently, E[Yk]=∑k i=1E[Xi]=0<∞. Second, we need to show that the conditional expected value o f the next observation, given all the past observations, isequal to thelast observation. E[Yk+1|Y1,...,Yk]=1 mm−1 ∑ j=0/parenleftbigg Yk+cos2πjg m/parenrightbigg =Yk+1 mm−1 ∑ j=0cos2πjg m=Yk Since|Yk+1−Yk|=|Xk+1|≤1fork≥0weapply Azuma’s inequality toobtain Pr(|Yt−Y0|≥λ)=Pr/parenleftBigg |t ∑ i=1Xi|≥λ/parenrightBigg ≤2exp/parenleftbigg −λ2 2t/parenrightbigg Therefore, weinduce that the probability of Kbeing not “good” for 1 ≤g≤m−1isat most Pr/parenleftBigg |t ∑ i=1Xi|≥√ εt/parenrightBigg ≤2exp/parenleftBig −εt 2/parenrightBig ≤1 m fort=⌈(2/ε)ln2m⌉. Hence the probability that constructed set is not “good” for at least one 1 ≤g≤m−1 is at most (m−1)/m<1. Therefore, there exists aset which is “good” for all 1 ≤g≤m−1. Thisset will also be “good” for all g/n⌉}ationslash=0 modmbecause cos2πk(g+jm) m=cos2πkg m.

Question 1404 multiple-choice

Quantum algorithms often leverage group-theoretic structure to efficiently solve problems that are classically hard. The hidden subgroup problem (HSP) on finite abelian groups is a central example with applications to factoring and discrete logarithms. Which statement best describes the role of the quantum Fourier transform (QFT) in solving the abelian hidden subgroup problem (HSP)? 1) The QFT directly reveals the stabilizer subgroup by measuring the function values. 2) The QFT maps cosets of the hidden subgroup to orthogonal subspaces labeled by group elements. 3) The QFT projects the quantum state onto the basis labeled by individual group elements, unveiling the subgroup structure. 4) The QFT transforms the superposition over coset elements into a superposition over group characters, enabling extraction of information about the hidden subgroup. 5) The QFT is used to create a uniform superposition over all elements of the group prior to function evaluation. 6) The QFT guarantees exponential speedup for all group-theoretic problems, regardless of group structure. 7) The QFT is only applicable to cyclic groups and cannot be generalized to arbitrary finite abelian groups.

✓ Correct Answer:

The correct answer is 4) The QFT transforms the superposition over coset elements into a superposition over group characters, enabling extraction of information about the hidden subgroup..

📚 Reference Text:

[7]. 5 The abelian hidden subgroup problem Given the above developments it is exciting to observe that t he concept of periodicity and the construction of the Fourier transform may be generalized to apply to anyﬁnite group G. Our discussion so far pertains simply to the special cases of the additive group of integers modulo N(for factoring) and the product group Zp−1× Zp−1(for evaluating discrete logarithms). The generalized viewpoint will also provide considerable i nsight into the workings of the Fourier transform. We will now outline the essential ideas i nvolved restricting attention in this section to the case of ﬁnite abelian groups. LetGbe any ﬁnite abelian group. Let f:G→Xbe a function on the group (taking values in some set X) and consider K={k∈G:f(k+g) =f(g) for allg∈G} (14) (Note that we write the group operation in additive notation ).Kis necessarily a subgroup ofGcalled the stabilizer or symmetry group of f. It characterizes the periodicity of fwith respect to the group operation of G. For factoring where GwasZN,Kwas the cyclic subgroup of all multiples of r. The condition (14) is equivalent to saying that fis constant on the cosets of KinG. (Recall that the cosets are subsets of Gof the form g+K={g+k:k∈K}and they partition all of Ginto disjoint parts of equal size |K|). 8 Given a device that computes f, our aim is to suitably determine the “hidden subgroup” Ke.g. we may ask for a set of generators for Kor for an algorithm that outputs a ran- domly chosen element of K. More precisely we wish to obtain this information in time O(poly(log |G|)) where |G|is the size of the group and the evaluation of fon an input counts as one computational step. (Note that we may easily determin eKin timeO(poly( |G|)) by simply evaluating and examining all the values of f). We begin as in our examples by constructing the state |f/an}bracketri}ht=1/radicalbig |G|/summationdisplay g∈G|g/an}bracketri}ht|f(g)/an}bracketri}ht and read the second register. Assuming that fis suitably non-degenerate – in the sense that f(g1) =f(g2) iﬀg1−g2∈Ki.e. thatfis one-to-one within each period – we will obtain in the ﬁrst register |ψ(g0)/an}bracketri}ht=1/radicalbig |K|/summationdisplay k∈K|g0+k/an}bracketri}ht (15) corresponding to seeing f(g0) in the second register and g0has been chosen at random. In eq. (15) we have an equal superposition of labels correspond ing to a randomly chosen coset ofKinG. NowGis the disjoint union of all the cosets so that if we read the la bel in eq. (15) we will see a random element of a random coset, i.e. a labe l chosen equiprobably from all ofG, yielding no information at all about K. The general construction of a “Fourier transform on G” will provide a way of eliminating g0from the labels (just as in the case of ZN) and the resulting state will then provide direct information about K. Let Hbe a Hilbert space with a basis {|g/an}bracketri}ht:g∈G}labeled by the elements of G. Each group element g1∈Ggives rise to a unitary “shifting” operator U(g1) onHdeﬁned by U(g1)|g/an}bracketri}ht=|g+g1/an}bracketri}ht for allg For any coset g0+Klet us write |g0+K/an}bracketri}htfor the

Question 1405 multiple-choice

In noncommutative algebra and quantum group theory, Clifford-Hopf algebras and their spectra play a significant role in understanding algebraic structures and their representations, especially in relation to quantum deformations and braid group actions. The properties of the prime and maximal spectra, the structure of the quantum deformation matrix, and the construction of ideal quotients are central to these topics. Which condition ensures that the quantum deformation matrix qij satisfies the braid group relations in a quantum system, allowing the quantum 2-cocycle to yield a unitary representation of the braid group Bn? 1) qij must be a symmetric matrix with all entries equal to 1 2) qij must be an involution, such as a Hadamard matrix 3) qij must be diagonalizable over the real numbers 4) qij must be nilpotent with order two 5) qij must have distinct eigenvalues 6) qij must be strictly lower triangular 7) qij must be positive definite and Hermitian

✓ Correct Answer:

The correct answer is 2) qij must be an involution, such as a Hadamard matrix.

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an antipode because, a s we have seen in (24), it is the prime and not the maximal spectra of Ywhich is important: the inverse image of the maximal spectra does not have to be maxima l. This is a known result for the Cliﬀord-Hopf algebra [17], for example, which can be se en explicitly via the biconvolution deﬁned by Cl(η,ξ), where the algebra is deﬁned via Cl(V,η∈V∗⊗V∗) and the coalgebra is deﬁned via Cl(V,ξ∈V⊗V). In this case, it is only for a special choice of biconvolution, namely when the biconvolution is antipodal, that Cl(η,ξ) is isomorphic to the Cliﬀord-Hopf algebra. The key point for us is that creating a homogeneous coordinate ring of the form of (23) for the polynomial ring in (6) does not mean that the twist in (25 ) is mapped to the same set of homogenous coordinates. Therefore, the general r esult for a ﬁnite temperature quantum system is that although the path weights in (3) form a one- dimensional unitary representation of the fundamental group of the system (deﬁne d viaxi∈Xαandxj∈Xβ), this representation is not generally faithful, because xi∗xj=χ(α,β)qijxjxi/ne}ationslash=χ(β,α)qjixixj=xj∗xi (28) 10 ie. the general quantum deformed coordinates in (24) do not form an associative group multiplication rule in conjunction with the 2-cocyle. Nonetheless, eve n though the represen- tation is unfaithful, the above 2-cocyle does form a one-dimensiona l unitary representation of the Braid group Bnif thexisatisfy the following relations [5] xi∗xi+1∗xi=xi+1∗xi∗xi+1, xi∗xj=xj∗xi|i−j|>1 (29) Central to this deﬁnition is notion of the general closure of the bra idxwhich is deﬁned for the two braids xi,xj∈Bnand the nonnegative integers αandβ, satisfying 2 α+β=n, by χ(α,β)(xi,xj)(x). Again, we have used the same notation for the braid closure and 2 -cocyle to make their connection explicit. A simple way to satisfy (28) and the second relation in (29) is to make the quantum deformation matrix qijan involution, for example, by chosing qijto be a Hadamard matrix. If qis chosen in this way then the |i−j|>1 condition is automatically satisﬁed from our previous choice of qas an alternating bicharacter ( qii= 1). Similarly, we can satisfy the ﬁrst relation in (29) by using just the ass ociativity properties of the alternating bicharacter of the quantum deformation matrix and 2-cocyles (xi∗xj)∗xk=χ(α,β)χ(β,α)xixjxk=qjkqikxkxixj=qji,kxkxixj (30) The most limiting assumption for deﬁning a general quantum system, therefore, is to chooseqasaninvolution. Morevover, itisonlynecessary tomakethisvery sp eciﬁc choice for the quantum deformation if we are dealing with the prime spectra rat her than the maximal spectra, where we do not have the inclusion Xα֒→Xβ. Hence, for a ﬁnite temperature quantum system if we do want to probe this general inclusion, via χ(α,β)(xi,xj)(x), we need to identify the ideal quotient of the homogeneous generators in (2 6) and (27) in order to deﬁne the more general resolution of (28), rather than the spec iﬁc (faithful) one where qis an involution. Given the two ideals IandJ= (x1,x2) this is deﬁned via I:J= (I: (x1))∩(I: (x2)) =/parenleftbig x−1 1(I ∩(x1))/parenrightbig ∩/parenleftbig x−1 2(I ∩(x2))/parenrightbig (31) Hence, the intersection

Question 1406 multiple-choice

Quantum algorithms exploit superposition, entanglement, and algebraic structures to solve problems more efficiently than classical approaches. Strategies such as Grover search, collision-finding, and rank estimation highlight the interplay between quantum query complexity and problem structure. When both input and output bases of a function are unknown, which statement most accurately describes the performance of quantum algorithms for finding collisions or hidden structure? 1) Quantum algorithms can efficiently distinguish r-to-one from one-to-one functions using Grover search alone. 2) Fourier-Schur sampling enables exponential speedup regardless of basis knowledge. 3) No algorithm offers better performance than random queries followed by brute-force search. 4) Schur duality guarantees optimal quantum speedup in the absence of known bases. 5) Isometric oracles allow for polynomial speedup with weak basis information. 6) Black box rank estimation is always more efficient than collision-finding under unknown bases. 7) Grover's algorithm provides quadratic speedup even when all bases are unknown.

✓ Correct Answer:

The correct answer is 3) No algorithm offers better performance than random queries followed by brute-force search..

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Ambainis. The algorithm ﬁrst queries Ron (d/r)1/3basis states of the form |j/an}bracketri}ht⊗|0/an}bracketri}htfor uniformly randomj, and then measuresthe second register, producing a table Lofuniformly random functionvalues f(i), whicharealldistinctwithhighprobability. ThenweperformaGrove r search for a state U|i/an}bracketri}htsuch thatf(i)∈L. There are two components to the Groversearch. First, we need to identify our target space. Let Π :=/summationtext i∈f−1(L)U|i/an}bracketri}ht/an}bracketle{ti|U†. Note that trΠ is either (d/r)1/3orr(d/r)1/3, and that we can perform the reﬂection 1 −2Π using only two queries (one of Rand one of R†). This implementation can be done by brute force in timeO(r(d/r)1/3), or by Grover search in time O(/radicalbig r(d/r)1/3)—either way, regardless of whether fis one-to-one or r-to-one, and using only two queries. Next, we need a starting state |ψ/an}bracketri}htthat we can eﬃciently map to and from the |0/an}bracketri}htstate, so that we can both initialize it and perform the reﬂection 1 −2|ψ/an}bracketri}ht/an}bracketle{tψ|. One such |ψ/an}bracketri}htis a state that is maximally entangled between the ﬁrst register and an ancilla register of equal dimension. Then we have /an}bracketle{tψ|(I⊗Π)|ψ/an}bracketri}ht= trΠ/d. The algorithm now alternates ℓtimes between the operations 1 −2|ψ/an}bracketri}ht/an}bracketle{tψ|and 1−2Π. We chooseℓso that iffisr-to-one then the probability of ﬁnding a solution is nearly 1, while iffis one-to-one then the probability is far from one. Thus, we will be ab le to distinguish the two cases with constant advantage. Following [12], we set ℓ:=/floorleftBigg 4/π/radicalbigg sin−1/parenleftBig (r/d)2 3/parenrightBig/floorrightBigg = Θ((d/r)1/3). (90) The probability of ﬁnding a solution is ≥1−O(r1/3/d2/3) iffisr-to-one and is ≤ 1/r+O(r1/3/d2/3) iffis one-to-one. Ifinsteadboth the input andoutput basesareunknown, then wek nowofnoalgorithmbet- ter than making O(d) queries with random inputs and using the algorithm of Section 4.1. •No access to R†:Suppose that instead of the full unitary oracle (77), Ris simply the isometry (76), mapping |i/an}bracketri}htto|i/an}bracketri}ht|ψf(i)/an}bracketri}ht. In this case, the best algorithm we know queries the oracleO(logd) times on each of√ dinputs (so that there is a collision among those inputs with reasonably high probability), and then searches the res ulting table (either by brute force or by Groversearch; in either case, making no furthe r queries) using Lemma 10 to perform comparisons. We conjecture that this is optimal (up to the log factor), but we know of no lower bound better than the Ω( d1/3) bound from the classical problem. •Black box rank estimation: Suppose again that Ris an isometry, but now mapping |i/an}bracketri}ht to|i/an}bracketri}ht|ψi/an}bracketri}ht, and we are only promised that either {|ψi/an}bracketri}ht}forms a basis for Cdor that 17 1 d/summationtext i|ψi/an}bracketri}ht/an}bracketle{tψi|is proportional to a projector onto a ( d/r)-dimensional subspace. Here the best algorithm we know is to query RonO(d) inputs and use the rank estimation proce- dure from Section 4.1. We conjecture that this is optimal, but again h ave only the lower bound of Ω( d1/3). 5 Discussion In this article, we have shown that weak Fourier-Schur sampling typ ically provides insuﬃcient information to solve the hidden subgroup problem. Nevertheless, it remains possible that Schur duality could be a useful tool for the hsp. Just as weak Fourier sampling can always be used to reﬁne the state space into smaller

Question 1407 multiple-choice

The hidden subgroup problem (HSP) is a central challenge in quantum computing, underpinning several groundbreaking quantum algorithms. It involves determining a hidden subgroup within a finite group using a function that is constant and distinct on each left coset. Which of the following conditions is necessary for efficient quantum algorithms to solve the hidden subgroup problem in arbitrary finite abelian groups? 1) The group must have a nontrivial center 2) A classical Fourier transform over the group must be available 3) A quantum Fourier transform over the group must be available 4) The function must be injective on all group elements 5) The subgroup must be normal only for non-abelian groups 6) The function must be surjective onto its codomain 7) The group must be a direct product of cyclic groups

✓ Correct Answer:

The correct answer is 3) A quantum Fourier transform over the group must be available.

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of poly(log |G|) bits. Consider a function f:G →SwhereSis some ﬁnite set whose elements can also be eﬃciently represented as strings of poly(log |G|) bits. In the hsp, we are given such a function and promised that it is constant and distinct on left cosets of some subgroup H ≤ G . In other words, f(g1) =f(g2) if and only if g1andg2 are in the same left coset of H. The hidden subgroup problem is, given the ability to query the function f, to produce a generating set for the subgroup H. In the quantum version of the hspwe are given a unitary operator Uf that computes the function f. Explicitly, this quantum oracle acts as (1) Uf:|g,y/angbracketright /mapsto→ |g,y⊕f(g)/angbracketright for allg∈ Gandy∈S, where ⊕is the bitwise exclusive or operation. If we input the basis state |g,0/angbracketrightinto this oracle, it simply evaluates the function: Uf|g,0/angbracketright=|g,f(g)/angbracketright. Our goal is to use this black box to ﬁnd generators of the hidden subgroup in a time polynomial in log |G|. In the standard approach to solving the hidden subgroup problem with a quantum computer (used by all known quantum algorithms for the hsp), one inputs a superposition over all group elements into the ﬁrst register and |0/angbracketrightinto the second register, giving (2) Uf:1/radicalbig |G|/summationdisplay g∈G|g,0/angbracketright /mapsto→1/radicalbig |G|/summationdisplay g∈G|g,f(g)/angbracketright. Suppose we now discard the second register. Due to the promise on f, the state of the ﬁrst register is then a mixed state whose form depends on the hidden subgroup H, ρH:=|H| |G|/summationdisplay g∈K|gH/angbracketright/angbracketleftgH| (3) where K ⊂ G is a complete set of left coset representatives of HinG(of size |K|=|G|/|H|), and where we have deﬁned the coset states |gH/angbracketright:=1/radicalbig |H|/summationdisplay h∈H|gh/angbracketright. (4) We will call ρHthehidden subgroup state corresponding to the subgroup H. Early quantum algorithms, including Deutsch’s algorithm [9], the Deutsch- Jozsa algorithm [8], the Bernstein-Vazirani algorithm [3], Simon’s algorithm OPTIMAL MEASUREMENTS FOR THE DIHEDRAL HSP 5 [50], and Shor’s algorithm [49], all solve examples of the abelian hspbut were not originally described in this language. The formulation in terms of a hidden subgroup was presented by Boneh and Lipton [4], who also noted the connection between the hspover the symmetric group and the graph isomorphism problem. Thehsp over arbitrary ﬁnite abelian groups has an eﬃcient quantum algorithm [4, 23, 35, 49, 50]. Hallgren, Russell, and Ta-Shma proved that thehsphas an eﬃcient quantum algorithm whenever the subgroup His promised to be normal and there is an eﬃcient quantum Fourier transform over the group G[24]. Grigni, Schulman, Vazirani, and Vazirani showed that thehspover “almost abelian” groups has an eﬃcient quantum solution [21], and this result was extended by Gavinsky to “near-Hamiltonian” groups [20]. P¨ uschel, R¨ otteler, and Beth gave an eﬃcient quantum algorithm for thehspover the wreath product Zn 2≀Z2[47], and Friedl et al. showed how to solve the hspover a semidirect product Zn pk/multicloserightZ2for a ﬁxed prime power pk[17]. Moore, Rockmore, Russell, and Schulman gave an eﬃcient quantum algorithm for the hspover certain semidirect product groups, the q-hedral groups [39], and

Question 1408 multiple-choice

In the representation theory of symmetric groups and quantum algorithms, state labeling and basis transformations play a crucial role in exploiting group symmetries for computational efficiency. The interplay between Young subgroups, irreducible representations, and tableaux underlies many advanced algorithms in quantum computation and algebraic combinatorics. When a quantum algorithm constructs a state as a sum over transversals of a Young subgroup YT in the symmetric group Sn and performs Group Projection Estimation (GPE) followed by measurement of the irrep of YT, which of the following statements best describes the resulting basis for the multiplicity space? 1) It remains labeled by standard Young tableaux (SYTs) without modification. 2) It is projected onto a basis labeled by character values of irreducible representations. 3) It transforms to a basis labeled by elements of the symmetric group. 4) It rotates from the SYT basis to a normalized version λ(YT)|T⟩, which corresponds one-to-one with semi-standard Young tableaux (SSYTs) when states are non-zero. 5) It collapses to a single vector indexed by the partition λ. 6) It becomes labeled by the conjugacy classes of the subgroup YT. 7) It is mapped to a basis indexed by tensor products of group representations.

✓ Correct Answer:

The correct answer is 4) It rotates from the SYT basis to a normalized version λ(YT)|T⟩, which corresponds one-to-one with semi-standard Young tableaux (SSYTs) when states are non-zero..

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for states of the form/summationtext tat|t/angbracketright, performing GPE and measuring the irrep of YTwould always give the trivial irrep. 2. We show that when the trivial irrep of YTappears in the measurement, the computational basis of the multiplicity space is rotated from SYTs to (the normalized version of) λ(YT)|T/angbracketright, whereTis a SYT. 3. Then we show that the states λ(YT)|T/angbracketrightare in one-to-one correspondence to SSYTs whenever the states are non-zero. Part (i) To show that we always get the trivial irrep of YTfor states of the form/summationtext tat|t/angbracketright, wheretis the transversal ofYTinSn, we track the state through the steps of the algorithm. 1. Suppose we had the state/summationtext tat|t/angbracketrightA, wheretis an element of the transversal of YTinSn. We take an ancilla register Bconsisting of log|YT|qubits to get/summationtext tat|t/angbracketrightA|0/angbracketrightB. 13 2. Perform inverse QFT over YTon theBregister to obtain an equal superposition over group elements ofYT. This allows us to view the registers AandBtogether as the group basis of Sn. This step takes O(polylog|YT|) time since the QFT over YTcan be done eﬃciently for Young subgroups (as they are direct products of symmetric groups). We now have the state/summationtext t,hbt|t/angbracketrightA|h/angbracketrightB. Herehruns over all the elements of YTandbt=at//radicalbig |YT|. 3. Perform GPE, which consists of the following steps. This takes O(polylog |YT|+ poly(n)) time since controlledRoperations can be done in poly ntime and there are two QFTs over YT. •Take a register Cof log|YT|qubits initialized to |0/angbracketrightand obtain an equal superposition over YTin it. This gives us the state/summationdisplay t,h1,h2ct|t/angbracketrightA|h1/angbracketrightB|h2/angbracketrightC, (3.3) whereh1andh2run over all elements of YTandct=bt//radicalbig |YT|. •Conditioned on the register C, perform a controlled right multiplication on the group basis of Snin registersABi.e., perform/summationdisplay h∈YT|h/angbracketright/angbracketlefth|C⊗R(h)AB, (3.4) whereRis the right multiplication in Sn. This gives the state /summationdisplay t,h1,h2ctR(h2)AB(|t/angbracketrightA|h1/angbracketrightB)|h2/angbracketrightC. (3.5) This state can be rewritten as /summationdisplay t,h1,h3ct(|t/angbracketrightA|h3/angbracketrightB)|h−1 1h3/angbracketrightC, (3.6) where we have replaced h1h2ash3. •Perform QFT( YT) onC. This gives us the following state /summationdisplay µ,k,l,h 1,h3,t/radicalbig dµ |YT|[µ(h−1 1h3)]k,lct|t,h 3/angbracketrightAB|µ,k,l/angbracketrightC, (3.7) whereµruns over all the irreps of YTandkandlrun over its dimension. The sum over h1forcesµ to be the trivial irrep of YT. Thus, for states of the form/summationtext tat|t/angbracketright, we will always get the trivial irrep when we measure µ. Part (ii) If we had done a QFT over Snand not performed GPE before that, we would have had the basis of the multiplicity space labeled by SYTs. Here we show that, after performing GPE and when µ(the irrep label ofYT) is trivial, the basis of SYTs is rotated to λ(YT)|T/angbracketright, whereTis a SYT of shape λ. Starting with a state of the form |th1/angbracketrightand performing GPE would give us the state /summationdisplay µ,k,l,h 2/radicalbig dµ |YT|[µ(h2)]k,lR(h2)|t,h 1/angbracketrightAB|µ,k,l/angbracketrightC, (3.8) Next we perform a QFT over Snon this state to get the following state /summationdisplay h2,λ,µ,T 1,T2,k,l/radicalbig dµdλ |YT|n![λ(th1)]T1,T2[µ(h2)]k,l(|λ/angbracketright⊗λ(h2)|T1/angbracketright⊗|T2/angbracketright)AB|µ,k,l/angbracketrightC, (3.9) 14 whereT1andT2are SYTs. Since the right regular representation acts only on the ﬁrst register, we have the λ(h2) acting only on|T1/angbracketright. Whenµis trivial the basis state |T1/angbracketright, which corresponds to a SYT gets taken to λ(YT)|T1/angbracketright. Part (iii) We show

Question 1409 multiple-choice

Quantum Chern-Simons theory is a topological quantum field theory that associates quantum invariants to 3-manifolds and knots using various gauge groups. The representation of mapping class groups on quantum Hilbert spaces is fundamental for understanding equivalence under surface symmetries in this framework. Which mathematical tool is employed to construct explicit quantum representations of the genus one mapping class group for complex gauge groups in quantum Chern-Simons theory? 1) Fourier-Mukai transform 2) Schur-Weyl duality 3) Generalized Weil-Gel’fand-Zak transform 4) Hecke algebra 5) Langlands correspondence 6) Atiyah-Singer index theorem 7) Seiberg-Witten invariants

✓ Correct Answer:

The correct answer is 3) Generalized Weil-Gel’fand-Zak transform.

📚 Reference Text:

arXiv:1608.06872v2 [math.QA] 23 Dec 2016THE GENUS ONE COMPLEX QUANTUM CHERN-SIMONS REPRESENTATION OF THE MAPPING CLASS GROUP JØRGEN ELLEGAARD ANDERSEN AND SIMONE MARZIONI Abstract. In this paper we compute explicitly, following Witten’s pre scrip- tion, the quantum representation of the mapping class group in genus one for complex quantum Chern-Simons theory associated to any simp le and simply connected complex gauge group GC. We use a generalization of the Weil- Gel’fand-Zak transform to exhibit an explicit expression f or the representation. 1.Introduction In his seminal paper [Wi1], Witten introduce the study of qua ntum Chern- Simons theory with non-abelian gauge group G. For the compact gauge groups G=SU(N), the so called Witten-Reshetikhin-Turaev TQFT’s was ﬁrst c on- structed by Reshetikhin and Turaev for G=SU(N)[RT1, RT2, T] and subse- quently by Blanchet, Habegger, Masbaum and Vogel in [BHMV1, BHMV2, B1] us- ing skein theory. Recently Ueno jointly with the ﬁrst author of this paper showed in a series of four papers [AU1, AU2, AU3, AU4] that the Witten-R eshetikhin-Turaev TQFT is the same as the TQFT coming from Conformal Field Theor y [TUY, BK] as was also proposed to be the case in Witten’s original paper [Wi1]. Witten further suggested in the same paper that the geometric quantization of the moduli space of ﬂat connections should be related to this theory and he dev eloped this approach further in his joint paper with Axelrod and Della Pietra [ADW ]. Following shortly after, Hitchin gave a rigorous account of this work in [H]. Fo r a purely diﬀerential geometric account of the construction of this connection se e [A6, AG1, AGL]. By combining the work of Laszlo [La1] with the above mention wor k of Ueno and the ﬁrst author, it has now been conﬁrmed that one can use the geom etric quantiza- tion of the moduli space of ﬂat connections as an alternative construction of the Witten-Reshetikhin-Turaev TQFT. This has been exploited i n, among others, the works [A3, AMU, A4, AHi, A5, A7, AHJMMc]. In the paper [Wi2] Witten also proposed a way to construct the mapping class groups representations of quantum Chern-Simons theory for complex gauge groups. This theory received much less attention, but the time now se ems ripe to fully develop this theory. Recently Kashaev has, jointly with the ﬁrst author of this paper, rigorously constructed the n= 2theory in [AK1, AK2, AK3]. First the level k= 1theory was examined in detail in [AK1, AK2] and recently has t he picture for all levels become clear and in fact, a very general scheme has been proposed in [AK3] for construction of TQFT’s from Pontryagin self-du al Locally compact groups. It is further veriﬁed in that paper that the scheme fo r the LCA R×Zk yields quantum Chern-Simons theory for PSL(2,C)at levelk. See also [AS] for Supported in part by the center of excellence grant “Center f or quantum geometry of Moduli Spaces" from the Danish National Research Foundation (DNRF 95). 1 2 JØRGEN ELLEGAARD ANDERSEN AND SIMONE MARZIONI

Question 1410 multiple-choice

In quantum information theory, universal quantum cloning machines (UQCMs) aim to create approximate copies of an unknown quantum state, subject to fundamental quantum constraints such as the no-cloning theorem and the monogamy of entanglement. Understanding the mathematical framework behind the limits of cloning fidelity is crucial for quantum cryptography and communication. Which mathematical tool is specifically used to characterize the admissible region of cloning fidelities by decomposing the possible overlaps of pure states with respect to the commutant of the product group in universal quantum cloning? 1) Hilbert-Schmidt norm 2) Lagrangian multipliers 3) Irreducible representations in group representation theory 4) Bell inequalities 5) Quantum error correction codes 6) Tensor network states 7) Bloch sphere parametrization

✓ Correct Answer:

The correct answer is 3) Irreducible representations in group representation theory.

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Group representation approach to 1!Nuniversal quantum cloning machines Michał Studzi´ nski1,2, Piotr ´Cwikli´ nski1,2, Michał Horodecki1,2and Marek Mozrzymas3 1Institute of Theoretical Physics and Astrophysics, University of Gda´ nsk, 80-952 Gda´ nsk, Poland 2National Quantum Information Centre of Gda´ nsk, 81-824 Sopot, Poland 3Institute for Theoretical Physics, University of Wrocław, 50-204 Wrocław, Poland (Dated: October 29, 2018) In this work, we revisit the problem of ﬁnding an admissible region of ﬁdelities obtained after an application of an arbitrary 1 !Nuniversal quantum cloner which has been recently solved in [A. Kay et al., Quant. Inf. Comput 13, 880 (2013)] from the side of cloning machines. Using group- theory formalism, we show that the allowed region for ﬁdelities can be alternatively expressed in terms of overlaps of pure states with recently found irreducible representations of the commutant U U . . . U U, which gives the characterization of the allowed region where states being cloned are ﬁgure of merit. Additionally, it is sufﬁcient to take pure states with real coefﬁcients only, which makes calculations simpler. To obtain the allowed region, we make a convex hull of possible ranges of ﬁdelities related to a given irrep. Subsequently, two cases: 1 !2 and 1!3 cloners, are studied for different dimensions of states as illustrative examples. PACS numbers: 03.67.-a, 03.65.Fd, 03.67.Hk Keywords: quantum cloning; asymmetric quantum cloning; qudit; representation theory I. INTRODUCTION A basic feature of entanglement is that contrary to classical correlations, it is monogamous. For example, if there is maximal entanglement between two parties, then no other party can be entangled with those parties. More generally, if Ais entangled with Band C, then the entanglement must be considerably weaker. This phenomenon gives rise to the fact that quantum information cannot be copied, in contrast with information from the ’classical world’. In other words, one is not able to copy perfectly an arbitrary quantum state. In terms of monogamy, if one wants to prepare some number of copies of the initially unknown quantum state, ﬁdelities of cloning cannot be all equal to 1, there is a trade-off. This basic feature is known as ’no-cloning theorem’ and was recognized by Wootters and ˙Zurek [1], and Dieks [2]. On the other hand, copying is possible, but the quality of the copy can be very bad sometimes. That’s why, the goal of ﬁnding the ultimate bounds for the quality of copying is an important task. A big effort has been made to solve it, starting from the work of Hillery and Bu ˘zek [3]. In general, the subject was studied intensively, both for symmetric (all ﬁdelities are equal) Universal Quantum Cloning Machines ( UQCM ) [4–8], and asymmetric (unequal ﬁdelities) UQCM [8–16]. See also [17, 18] for reviews. Nevertheless, for a long time there was a ’gap’ in studies of quantum cloning - there was no general results on an admissible region of ﬁdelities for universal asymmetric 1!Nquantum cloning machines. The problem has been solved just recently in a series of papers [14, 15] from the point

Question 1411 multiple-choice

Quantum computing architectures increasingly leverage high-dimensional systems such as qudits, which offer advantages over traditional qubit-based platforms. These systems enable the representation and manipulation of more complex quantum states within fewer physical particles. Which of the following is a direct computational advantage of using qudit-based quantum processors with dimension d over qubit-based systems? 1) Increased energy consumption due to higher state density 2) Necessity for longer coherence times to maintain entanglement 3) Ability to represent arbitrary unitary matrices with fewer physical systems, offering a scaling advantage of (log₂d)² 4) Reduced universality in algorithm implementation compared to qubits 5) Lower fidelities in state preparation and measurement 6) Limitation to only two-level quantum algorithms 7) Requirement for strictly superconducting circuit platforms

✓ Correct Answer:

The correct answer is 3) Ability to represent arbitrary unitary matrices with fewer physical systems, offering a scaling advantage of (log₂d)².

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obtained from QTM transitions driven by externalramping magnetic ﬁeld. The transitions between the⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle+ 1 2〉↔⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle−1 2〉states and⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle+3 2〉↔⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsingle−3 2〉is achieved via applying appropriate resonate frequencies ]12and ]23. Relaxation and coherence times are important aspects to be analyzed for theTbPc 2system, and this process is accomplished by imaging the initialized nuclear spin trajectory in real-time. Statistical analysis of the nuclear spin coherence time makes use of the spin ˆa€“lattice relaxation times by ﬁtting the data for an exponential form (y/equalsexp(−t/T1))and yields T1≈17 s for mI/equals ±1 2and T1≈34sfor mI/equals±3 2with ﬁdelities of F⎥parenleftbigmI/equals±1 2⎥parenrightbig≈93% and F⎥parenleftbigmI/equals±3 2⎥parenrightbig≈87% accordingly [ 115]. The TbPc 2SMM can be used to execute Grover ’sa l g o r i t h m ,w h e r e the alternation of the mIstate contained in the TbPc 2molecular qubit are treated by resonance frequencies [ 63, 99 ].TABLE 1 | Normalized photon counts and comparison of the true phase ϕand the experimentally estimated phase ϕ′for each eigenstate of ⎥hatwideU1(Eq. 110 ) and⎥hatwideU2 (Eq. 111 )[106]. ⎥hatwideU1 Eigenstate |0t〉 |1t〉 |2t〉 E0 0.9948 ±0.0004 0 .0101 ±0.0004 0 .0122 ±0.0005 E1 0.0023 ±0.0002 0 .9805 ±0.0009 0 .0120 ±0.0005 E2 0.0029 ±0.0002 0 .0094 ±0.0004 0 .9758 ±0.0010 True Phase ,ϕ 02 π/34 π/3 Est. Phase ,~ϕ 1.972π 0.612π 1.394π Error ,|ϕ−~ϕ| 2π1.4% 2 .7% 3 .0% ⎥hatwideU2 Eigenstate |0t〉 |1t〉 |2t〉 E0 0.878±0.002 0 .316±0.003 0 .143±0.002 E1 0.032±0.001 0 .530±0.003 0 .318±0.003 E2 0.090±0.002 0 .154±0.002 0 .539±0.003 True Phase ,ϕ 00 .3511 π 1.045π Est. Phase ,~ϕ 1.859π 0.377π 1.045π Error ,|ϕ−~ϕ| 2π7.1% 1 .3% 0 .0% FIGURE 14 | Electronic level structure of the trapped ion. The carrier of the quantum information is the qutrit states |0〉,|1〉,a n d |2〉.|0’〉is an auxiliary level used for the conditional two-qutrit gate. Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504 19Wang et al. Qudits and High-Dimensional Quantum Computing 6 SUMMARY AND FUTURE OUTLOOK OF QUDIT SYSTEM 6.1 Summary of the Advantages of Qudit Systems Compared to Qubit Systems Throughout the article we discuss and review many aspects of the qudit systems such as qudit gates, qudit algorithms, alternativecomputation models and implementations. Most gates andalgorithms based on qudits have some advantages over those for qubits, such as shorter computational time, lower requirement of resources, higher availability, and the ability to solve morecomplex problems. The qudit system, with its high-dimensionalnature, can provide more degrees of freedom and largercomputational space. This section summarizes the advantagesof the qudit system compared to the qubit system. Qudit gates have the advantage of a larger working Hilbert space which reduces the number of qudits needed to represent anarbitrary unitary matrix. In our discussion of universality inSection 2.1.2 , the qudit method proposed by Muthukrishnan and Stroud ’sh a sa (log 2d)2s c a l i n ga d v a n t a g eo v e rt h eq u b i t case. Furthermore, Luo and Wang show that with their proposed universal computation scheme [

Question 1412 multiple-choice

In advanced mathematical physics and algebraic geometry, toric varieties and Clifford algebras play a key role in modeling symmetries and quantum systems. The construction of varieties often relies on the interplay of lattices, cones, and graded coordinate rings. Which statement correctly describes the structure of the toric variety associated with D-dimensional light cone coordinates in terms of its coordinate ring and symmetry properties? 1) It is defined as the spectrum of the homogeneous coordinate ring C[x0, x1,.., xD]/I, with each variable corresponding to a compact direction in Euclidean space. 2) The toric variety is determined by the dual lattice Λ and is homeomorphic to the real torus (S^1)^D, representing a compactified space. 3) Its affine charts are glued using cones that correspond to the irreducible representations of the exterior algebra, ensuring antisymmetry in all coordinates. 4) The construction uses non-graded coordinate rings, focusing solely on symmetric polynomial functions without regard to parity. 5) The associated variety is defined by the stabilizer subgroup of the Clifford algebra, reflecting only spatial symmetries. 6) It is given by TS = spec C[ˇΛ] ≃ spec C[x±_i]_{i=1}^D = (C×)^D, where the coordinate ring encodes the algebraic torus structure and multiplicative symmetries for each coordinate direction. 7) The toric variety is described by a non-homogeneous coordinate ring and is only applicable to even-dimensional spinor representations.

✓ Correct Answer:

The correct answer is 6) It is given by TS = spec C[ˇΛ] ≃ spec C[x±_i]_{i=1}^D = (C×)^D, where the coordinate ring encodes the algebraic torus structure and multiplicative symmetries for each coordinate direction..

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are given by Ei=ηiiand the irreducible representations of ClHare therefore isomorphic to those of the Cliﬀord algebra Cl. Similarly, for odd Dthe irreducible representations of ClHare related to those of Cl, but deﬁned in one dimension higher. We are therefore led to the conc lusion that in order to treat the general algebra of the polynomial ring in (5), applicable to a quantum system whereb→b′in (2), we must weaken the construction of the above antipode suc h that it is graded, via some form of quantum group deformation: general ﬁn ite-temperature quantum systems cannot be solved exactly by stabilizer or normalizer constr uctions of the Cliﬀord algebra. IV. QUANTUM DEFORMATION OF A TORIC VARIETY A. Toric Variety Our approach will now be to identify the variety that is associated wit h the ideal quotient (I:J) that we are seeking. Our ﬁrst step will be to picture the parity sym metry operator of (12) as being equivalent to introducing light cone coordinates, with t he Lorentzian signature 7 (D,1), for the Cliﬀord ideal IClin (11) ds2=−dt2+δijdxidxj→ds2=−2dx+dx−+δijdxidxj (19) wherei,j= 1,...,D−1,x+=1√ 2(t+x) andx−=1√ 2(t−x), and to identify the variety associated with this coordinate system. Formally, the variety that is associated with this coordinate system is deﬁned for a general polynomial ring (deﬁned in theD+1 variables xi) via the homogeneous coordinate ring X[19][20]. This can be written as the quotient ring X=C[x0,x1,x2,...,xD]/I (20) whereIis a homogeneous ideal, and it is important to note that by specifying t hat the above ideal is homogenous an implicit form of grading is assumed, which allows us to focus (initially) onthe parity invariant case. To deﬁne this ideal and grading further, we let Λ be a lattice of rank DandˇΛ its dual lattice, where Λ R= Λ⊗ZRis called a regular k-dimensional cone deﬁned as a convex subset σ⊂ΛR[19] via σ=R≥0/an}bracketle{txi/an}bracketri}htk i=1=/braceleftBiggk/summationdisplay i=1aixi|ai∈R≥0/bracerightBigg (21) where{x1,...,xk}is some subset of Λ which can be extended into a basis. The set of all cones can then be combined to form the following set, known as a com plete regular fan s(D) ={σI}I⊂{1,...,D+1}, σI=R≥0/an}bracketle{txj/an}bracketri}htj∈I (22) Finally, to form the projective space associated with the D-dimensional light cone coor- dinates for (16) we can identify the aﬃne varieties associated with e achσand can then glue these together [20], since these varieties Aσ= specC[ˇσ] form the natural inclusion Aσ֒→Aσ′<σ, where ˇσ={ˇλ∈ˇΛ| /an}bracketle{tˇλ,v/an}bracketri}ht ≥0,∀v∈σ}deﬁnes the dual cone. It follows that the variety describing the light cone coordinates in (16), which is kno wn as a toric variety, is deﬁned via TS= specC[ˇΛ]≃specC[x± i]D i=1= (C×)D(23) whereC[x± i]D i=1isthe homogeneous coordinateringfor thealgebraictorus( C×)D. Therefore, although we have made progress towards deﬁning a suitable ideal qu otient we still need to introduce further grading in order to construct the polynomial rin g in (6), since it follows from (20) that the polynomial ring in (6) is only the homogeneous coo rdinate ring of the above projective space not of the space associated with the exte rior algebra. 8 B. Quantum Deformation The quantum deformation of the above toric

Question 1413 multiple-choice

Quantum algorithms have revolutionized computational group theory, enabling efficient solutions to problems involving group orders, element orders, and membership testing in various group types. These advances are particularly significant in the study of matrix groups and black-box groups with accessible oracles. Which of the following statements accurately describes a scenario where quantum algorithms can efficiently solve constructive membership testing in finite groups? 1) When the group is cyclic and elements are encoded non-uniquely 2) When the group is non-Abelian and no oracle for element order is available 3) When the group is a finite black-box group with oracles for element order and constructive membership in elementary Abelian subgroups 4) When the group is infinite and lacks a straight line program representation 5) When the group elements have ambiguous encodings and subgroup membership is undecidable 6) When the group is Abelian but the generators are unknown 7) When the group has only composite order elements and lacks an oracle for membership

✓ Correct Answer:

The correct answer is 3) When the group is a finite black-box group with oracles for element order and constructive membership in elementary Abelian subgroups.

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+ ν(G) or toZp wherepis a prime dividing |G|. Of course, if Gis solvable one can insist that the representations of all the cyclic factors be of the second kind. It turns out that for some of the tasks in the hypotheses of The orem 4 there are eﬃcient quantum algorithms. By Shor’s results [25], the oracle for c omputing discrete logarithms can be implemented by a polynomial time quantum algorithm. Also, a superset of the primes dividing |G|can be obtained in polynomial time by quantum algorithms in t he most natural cases. For example, if Gis a matrix group over a ﬁnite ﬁeld, say G≤GL(n,q) then such a superset can be obtained by factoring the number ( qn−1)(qn−q)···(qn−qn−1), the order of the group GL(n,q). The same method works even for factors of matrix groups over ﬁ nite ﬁelds. If Gis (a factor of) a ﬁnite matrix group of characteristic zero, then the situati on is even better because in that case the prime divisors of Gare of polynomial size. But in any case, one can note that the s uperset of the primes dividing the order of Gis only used in Theorem 4 to compute (and factorize) the order s of elements of Gas well as those of matrices over ﬁnite ﬁelds of size at most |G|. This latter task can also be achieved by a quantum algorithm in polynomial time. In addition, we remark that the algorithm for testing member ship can be understood in a stronger, constructive sense, (see Section 5.3 in [4]), which is the proper generali zation of the constructive membership test in the Abelian case. For this w e need the notion of a straight line program on a set of generators. This is a sequence of expressions e1,...,eswhere each eiis either of the form xi:=hwherehis a member of the generating set or of the form xi=xjx−1 kwhere 0< j,k < i . It turns out that for elements gofGone can also require that a straight line program expressing gin terms of the generators be returned. Therefore, one can im mediately derive from Theorem 4 the following result. Corollary 5. LetGbe a ﬁnite black-box group with not necessarily unique encod ing. Assume that the following are given: 5 (a) an oracle for computing the orders of elements of G, (b) an oracle for the constructive membership tests in eleme ntary Abelian subgroups of G. Then the following tasks can be solved by quantum algorithms of running time polynomial in the input size +ν(G): (i) constructive membership test in G, (ii)–(v) as in Theorem 4. 4 Quantum implementations Inthissection wewilldiscussseveralcaseswhentheremain ingtasksinthehypothesesofCorollary5 can also be eﬃciently implemented by quantum algorithms. 4.1 Unique encoding If we have auniqueencodingfor theelements of theblack-box groupGthen we can useShor’sorder ﬁnding method. As we will show, in that case there is also an eﬃ cient quantum algorithm for the constructive membership test in elementary (and non-eleme ntary) Abelian subgroups. Therefore we will get the following result. Theorem 6. Assume that Gis a

Question 1414 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) often incorporate concepts from group theory, number theory, and quantum circuit design to efficiently identify hidden structures within groups. The Heisenberg Hidden Subgroup Conjugacy Problem (HSCP) leverages such techniques to achieve efficient computation and high success probability. In the quantum algorithm for the Heisenberg Hidden Subgroup Conjugacy Problem, which technique is used to efficiently implement the unitary transformation U2 that outputs a square root modulo a prime p, controlled by a quantum bit in superposition? 1) Using Grover's algorithm to search for roots 2) Applying the Quantum Fourier Transform over Z_p directly 3) Utilizing phase estimation to approximate roots 4) Employing classical brute-force search with quantum parallelism 5) Decomposing the unitary via the Clebsch-Gordan transform 6) Running a classical square root algorithm in reverse and applying a Hadamard transform to the control qubit 7) Using Shor's algorithm for factoring to extract roots

✓ Correct Answer:

The correct answer is 6) Running a classical square root algorithm in reverse and applying a Hadamard transform to the control qubit.

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2)1 p/bracketrightbigg2 =1 2+O(1 p2) (79) Thus we see that with probability approximately1 2a measurement will result in the outcome |a/an}bracketri}ht. Since we know k1andk2, we can easily invert a=ik1k2 k1+k2and ﬁnd i, the label of the hidden subgroup conjugacy. Thus, assuming that we can eﬃciently implement U2we have shown how to eﬃciently solve the HSCP for the Heisenberg. How do we implement U2eﬃciently? This can be done via an idea explained in the original Heisenberg hidden subgroup algorit hm of Bacon, Childs, and van Dam[14]. In particular we note that there exists an eﬃcient d eterministic classical algorithm which can compute the two square roots of a number modu lop. In particular this algorithm can work with a control bit labeling which of the two square r oots is returned: V|x/an}bracketri}ht|b/an}bracketri}ht=|(−1)b√x/an}bracketri}ht|b/an}bracketri}htwhere√xis a canonical square root. Then consider running this algorithm backwards, with an input on the control bit being |b/an}bracketri}ht=1√ 2(|0/an}bracketri}ht+|1/an}bracketri}ht). After this operation, performa Hadamardon the controlqubit and measure this register. Ifthe outcome is|0/an}bracketri}ht, the applied transform on the non-control register will take1√ 2(|√x/an}bracketri}ht+|−√x/an}bracketri}ht) to|x/an}bracketri}htas desired. This will occur with probability1 2. Thus we ﬁnd that we can eﬃciently implement U2with probability1 2such that our full algorithm succeeds with probability1 4+O(1 p2). We have thus shown that the Clebsch-Gordan transform plus some quantum post process- ing can be used to eﬃciently solve the Heisenberg HSCP and hence solv e the full Heisenberg HSP. 8.1 Comparison to the PGM Heisenberg HSP Algorithm Above we have shown how to arriveat an eﬃcient quantum algorithm f or the Heisenberg HSP by focusing on the Heisenberg HSCP and using the extra symmetry o f this state to motivate D. Bacon 25 the use of a Clebsch-Gordan transform to solve the problem. How d oes this compare with the algorithm derived by Bacon, Childs, and van Dam[14]? We will see be low that we have, in eﬀect, derived a portion of this algorithm. Here we will brieﬂy recap the hidden subgroup algorithm of [14] for th e Heisenberg group which was derived using the PGM formalism. The ﬁrst step begins with t he standard prepa- ration of the hidden subgroup state. One then performs a quantu m Fourier transform over Z2 pon the registers corresponding to the yandzvalues of the group element ( x,y,z), and measuring these registers, obtaining with uniform probability over Zpa value for yandz. After this one is left with the state, |i,j,y,z/an}bracketri}ht=1√p/summationdisplay b∈Zpωbjy+2−1b(b−1)iy+bjz|b/an}bracketri}ht, (80) for random uniform y,z∈Zp. When we have two copies of this state, we obtain four uniform random variables, y1,z1,y2,z2∈Zpalong with the state |i,j,y1,z1/an}bracketri}ht⊗|i,j,y2,z2/an}bracketri}ht=1 p/summationdisplay b1,b2∈Zpωj(b1y1+b2y2)+2−1i(b1(b1−1)y1+b2(b2−1)y2)+j(b1z1+b2z2)|b1,b2/an}bracketri}ht (81) If we rewrite this state in terms of the variables u=b1y1+b2y2 v= 2−1b1(b1−1)y1+2−1b2(b2−1)y2+b1z1+b2z2 (82) the two copy state can be expressed as 1 p/summationdisplay b1,b2∈Zpωuj+vi|b1,b2/an}bracketri}ht (83) Onethennotesthatifonecouldenactatransformationwhichtake stheanequalsuperposition of the two solutions to the quadratic equations deﬁned in Eq. (82) t o their respective values ofuandv, then we could transform the above state into 1 p/summationdisplay u,v∈Zp|u,vsolve the quadratic equationωuj+vi|u,v/an}bracketri}ht (84) from which a QFT over Z2 pwill immediately

Question 1415 multiple-choice

In quantum information theory, unitary k-designs and state k-designs are important mathematical constructs that facilitate the simulation of random quantum processes by finite sets of unitaries or quantum states. The efficiency and quality of these designs are closely linked to the dimensions of associated Hilbert and symmetric subspaces. For an n-qubit quantum system with Hilbert space dimension d = 2ⁿ, which of the following statements is true regarding the minimum cardinality required for a set S to form a unitary k-design? 1) The cardinality must be at least k! for any value of n. 2) The cardinality must be at least 2ⁿ for any value of k. 3) The cardinality must be at least d² for any value of k. 4) The cardinality must be at least dᵏk! for any value of n. 5) The cardinality must be at least d²k k!, which grows exponentially with n. 6) The cardinality must be at least n²k, independent of d. 7) The cardinality must be at least d/k!, which decreases with increasing k.

✓ Correct Answer:

The correct answer is 5) The cardinality must be at least d²k k!, which grows exponentially with n..

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that the cardinality of the set Smust satisfy |S|≥d2k k!. Consequently, if we are considering an n-qubit system where the Hilbert space dimension isd= 2n, the cardinality of Smust grow at least exponentially with the number of qubits. For a more comprehensive analysis of lower bounds of k-design, we recommend referring to the following references [39, 59, 65, 66]. The following proposition provides equivalent definitions of unitary k-design: Proposition 35 (Equivalent definitions of unitary k-design.) .Letνbe a probability distribution over a set of unitaries S⊆U. Then,νis a unitary k-design if and only if: 1. E U∼ν/bracketleftbig U⊗kOU†⊗k/bracketrightbig =E U∼µH/bracketleftbig U⊗kOU†⊗k/bracketrightbig for allO∈L/parenleftbig (Cd)⊗k/parenrightbig . 2. E V∼ν/bracketleftbig V⊗k⊗V∗⊗k/bracketrightbig =E U∼µH/bracketleftbig U⊗k⊗U∗⊗k/bracketrightbig . 3.F(k) ν= dim (Comm(U( d),k)). 4. E V∼ν[p(V) ] = E U∼µH[p(U) ]for all polynomials p(U)homogeneous of degree kin the matrix elements of Uand homogeneous of degree kin the matrix elements of U∗. Proof. The equivalence between 1 .and 2.is shown in Observation 28 and the one between 2 . and 3.is shown in Proposition 34. To show that 2 .implies 4., we observe that any homogeneous polynomial p(U) of degree kin the elements of Uand in the elements of U∗can be written asp(U) = Tr/bracketleftbig AU⊗k⊗U∗⊗k/bracketrightbig for a matrix A∈L/parenleftbig (Cd)⊗2k/parenrightbig with entries the coefficients of the polynomial. Similarly, to show that 4 .implies 2., we note that, for all matrices A∈L/parenleftbig (Cd)⊗2k/parenrightbig , pA(U):= Tr/bracketleftbig AU⊗k⊗U∗⊗k/bracketrightbig is a homogeneous polynomial of degree kin both the elements of U andU∗. Therefore we have Tr/parenleftig AE V∼ν/bracketleftbig V⊗k⊗V∗⊗k/bracketrightbig/parenrightig = Tr/parenleftbigg AE U∼µH/bracketleftbig U⊗k⊗U∗⊗k/bracketrightbig/parenrightbigg for allA∈L/parenleftbig (Cd)⊗2k/parenrightbig , (176) which implies the point 2. 24 Moreover, if νis a uniform distribution over a set of unitaries Swhich forms a group, then ifdim (Comm( S,k)) = dim (Comm(U( d),k)), then, by Proposition 32 and Proposition 34, νis a k-unitary design. Now we turn to the definition of k-designs for distributions over sets of states, which are known asstatek-designs or sphericalk-designs [67]. Definition 36 (State k-design) .Letηbe a probability distribution over a set of states S⊆Cd. The distribution ηis said to be a state k-design (or also spherical k-designs) if and only if: E |ψ⟩∼η/bracketleftig |ψ⟩⟨ψ|⊗k/bracketrightig =E |ψ⟩∼µH/bracketleftig |ψ⟩⟨ψ|⊗k/bracketrightig . (177) A unitaryk-designνinduces ak-state design for the probability distribution over states U|ψ0⟩, whereUis drawn from νand|ψ0⟩is fixed. This can be seen by setting O= (|ψ0⟩⟨ψ0|)⊗kin Definition 27. Similarly to the unitary k-design case, we can define the so-called state frame potential . Definition 37 (State frame potential) .The state frame potential is defined as: F(k,η) state :=E |ψ⟩,|ϕ⟩∼η/bracketleftig |⟨ψ|ϕ⟩|2k/bracketrightig . (178) We now state a proposition that relates the state frame potential to k-designs: Proposition 38 (State frame potential design) .Letηbe a probability distribution over a set of statesS⊆Cd. Then: F(k,η) state≥/parenleftbig dim/parenleftbig Symk(Cd)/parenrightbig/parenrightbig−1, (179) where dim/parenleftbig Symk(Cd)/parenrightbig =/parenleftbigd+k−1 k/parenrightbig . Equality holds if and only if ηis ak-state design. Proof. We denote A(k) µH:=E |ψ⟩∼µH/bracketleftig |ψ⟩⟨ψ|⊗k/bracketrightig and A(k) η:=E |ψ⟩∼η/bracketleftig |ψ⟩⟨ψ|⊗k/bracketrightig . We have: /vextenddouble/vextenddouble/vextenddoubleA(k) η−A(k) µH/vextenddouble/vextenddouble/vextenddouble2 2= Tr/bracketleftbigg/parenleftig A(k) η−A(k) µH/parenrightig2/bracketrightbigg (180) = Tr/bracketleftbigg/parenleftig A(k) η/parenrightig2/bracketrightbigg + Tr/bracketleftbigg/parenleftig A(k)

Question 1416 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) are central to advances in computational complexity and cryptography, especially as quantum hardware evolves in the NISQ era. Efficient resource usage and query complexity are key factors distinguishing quantum and classical approaches. Which of the following statements accurately describes the qubit requirements per oracle for distributed quantum algorithms versus centralized quantum algorithms when solving the generalized Simon’s problem? 1) Distributed algorithms use n+m qubits per oracle, while centralized algorithms use n−t+m qubits per oracle. 2) Both distributed and centralized algorithms use the same number of qubits per oracle, specifically n−t+m. 3) Distributed algorithms require more qubits per oracle than centralized algorithms due to increased communication overhead. 4) Centralized algorithms use n−t qubits per oracle, whereas distributed algorithms use n+m qubits per oracle. 5) Distributed algorithms use n−t+m qubits per oracle, while centralized algorithms use n+m qubits per oracle. 6) Both types of algorithms require n−t qubits per oracle for optimal circuit depth. 7) Centralized algorithms use n qubits per oracle and distributed algorithms use m qubits per oracle.

✓ Correct Answer:

The correct answer is 5) Distributed algorithms use n−t+m qubits per oracle, while centralized algorithms use n+m qubits per oracle..

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S⊥ l. Subsequently, we employ the classical Gaussian elimination meth od to obtain Sl. Once we have determined Sl, we can utilize Algorithm 5to identify the hidden subgroupS. In the following, we demonstrate the correctness of Algorithm 5. Denote by S∗=2kl−1/uniondisplay j=0/braceleftBigg/parenleftBiggkl/summationdisplay i=1γiei/parenrightBigg v/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglev∈V∗ j/bracerightBigg , (33) whereγi∈{0,1},j=/summationtextkl i=12i−1γi. To prove the correctness of Algorithm 5, we prove that S=S∗. First, we prove that S∗⊆S. Lets∗=/parenleftBig/summationtextkl i=1γiei/parenrightBig vbeanyelementbelongingto S∗.Sincef(0n−tvi) =f(ei0t), by the deﬁnition of the generalized Simon’s problem, we have f/parenleftBigg 0n−t/parenleftBiggkl/summationdisplay i=1γivi/parenrightBigg/parenrightBigg =f/parenleftBigg/parenleftBiggkl/summationdisplay i=1γiei/parenrightBigg 0t/parenrightBigg . (34) 13 Moreover, for any v∈Vj, there is f/parenleftbig 0n−tv/parenrightbig =f/parenleftBigg 0n−t/parenleftBiggkl/summationdisplay i=1γivi/parenrightBigg/parenrightBigg . (35) Therefore, we have f(0n−tv) =f/parenleftBig/parenleftBig/summationtextkl i=1γiei/parenrightBig 0t/parenrightBig . According to the deﬁnition of the generalized Simon’s problem, we have s∗=/parenleftBig/summationtextkl i=1γiei/parenrightBig v∈S. Thus,S∗⊆S. Then, we prove that S⊆S∗. Lets=slsrbe any element belonging to S, wheresl∈Sl,sr∈Sr. Since{ei|ei∈{0,1}n−t,1≤i≤kl}is the basis of Sl, we also have sl∈/braceleftBiggkl/summationdisplay i=1γiei/vextendsingle/vextendsingle/vextendsingleei∈{0,1}n−t,γi∈{0,1}/bracerightBigg . (36) By the deﬁnition of the generalized Simon’s problem, we have f(0n−tsr) =f(sl0t). Furthermore, there exists γi∈{0,1}such that f(0n−tsr) =f(sl0t) =f/parenleftBigg/parenleftBiggkl/summationdisplay i=1γiei/parenrightBigg 0t/parenrightBigg =f/parenleftBigg 0n−t/parenleftBiggkl/summationdisplay i=1γivi/parenrightBigg/parenrightBigg .(37) From the deﬁnition of V∗ j, we havesr∈V∗ j. Thus,s∈S∗. Consequently, S⊆S∗. 7 Comparisons with other algorithms First, we compare Algorithm 1with the distributed classical randomized algorithm for solving the generalized Simon’s problem. Based on the previous analysi s, Algorithm 1needsO(n−t−kl) queries for ﬁnding the group Sl. However, in order to ﬁnd the groupSl, the distributed classical randomized algorithm needs to query oracl es Ω/parenleftBig max/braceleftBig kl,√ 2n−t−kl/bracerightBig/parenrightBig times. Table 1 Comparison of Algorithm 1with distributed classical randomized algorithm. Algorithms Query complexity Algorithm 1 O(n−t−kl) Distributed classical randomized algorithmΩ/parenleftBig max/braceleftBig kl,√ 2n−t−kl/bracerightBig/parenrightBig 14 Therefore, Algorithm 1oﬀers exponential acceleration compared to the distributed classical randomized algorithm. Second, we compare Algorithm 4with the distributed classical deterministic algo- rithm for solving the generalized Simon’s problem. According to the p revious analysis, Algorithm 4needsO(n−t) queries for ﬁnding the group Sl. However, in order to ﬁnd the groupSl, the distributed classical deterministic algorithm needs to que ry oracles Ω/parenleftBig max/braceleftBig kl,√ 2n−t−kl/bracerightBig/parenrightBig times. Table 2 Comparison of Algorithm 4with distributed classical deterministic algorithm. Algorithms Query complexity Algorithm 4 O(n−t) Distributed classical deterministic algorithmΩ/parenleftBig max/braceleftBig kl,√ 2n−t−kl/bracerightBig/parenrightBig Hence, Algorithm 4provides the advantage of exponential acceleration compared to the distributed classical deterministic algorithm. Third, we compare Algorithm 1and Algorithm 4with the centralized quantum algorithm for solving the generalized Simon’s problem. In Algorithm 1and Algorithm 4, the number of actual functioning qubits for each oracle is only n−t+m. However, in the centralized quantum algorithm, the number of actual functioning q ubits for each oracle isn+m. Table 3 Comparison of Algorithm 1and Algorithm 4with the centralized quantum algorithm for solving the generalized Simon’s prob lem. AlgorithmsThe number of actual functioning qubits for each oracle Algorithm 1 n−t+m Algorithm 4 n−t+m The centralized quantum algorithm for solving the generalized Simon’s problemn+m Consequently, Algorithm 1and Algorithm 4enable the reduction of circuit depth and the physical implementation of algorithms in the NISQ era. Finally, we compare Algorithm 2and Algorithm 5with the best previously proposed distributed quantum algorithm

Question 1417 multiple-choice

Quantum optical protocols often utilize photons, quantum dots, and a sequence of specialized operations to implement logic gates and entanglement in quantum information processing. Techniques such as spin-photon interactions, feed-forward control, and frequency detuning are central to these procedures. In a protocol where photon B's quantum state is manipulated via spin filters and reflection operators, which specific condition must be satisfied for the reflection operator R1 to effectively govern the interaction between quantum dot 1 and the photons? 1) The photon frequency equals the cavity resonance (ω = ωc) 2) The photon frequency is detuned by κ from the cavity resonance (ω − ωc = κ) 3) The photon frequency is detuned by half the cavity linewidth from the cavity resonance (ω − ωc = κ/2) 4) The cavity resonance is twice the photon frequency (ωc = 2ω) 5) The reflection operator is applied only when the spin filter is inactive 6) The photon frequency is less than the cavity resonance by twice the cavity linewidth (ω − ωc = -2κ) 7) The reflection operator requires both photons A and B to be entangled prior to interaction

✓ Correct Answer:

The correct answer is 3) The photon frequency is detuned by half the cavity linewidth from the cavity resonance (ω − ωc = κ/2).

📚 Reference Text:

and photons (A and B) is applied, and the procedure of switches (S1~S6) is performed in accordance with the time table shown in Fig. 3. After the state, |ϕ〉1, passes through SF on path 2 of photon B, the state is transformed into xR Rx LR xR Lx LL ixR Rx LR xR Lx LL1 2() 2() , (10)1QD1,SF 21 B1 A1 2B2 A1 3B2 A1 4B1 A1 e1 1B2 A1 2B1 A1 3B1 A1 4B2 A1 e1ϕϕ|⇒ |= −| |− || +| |+ || ⊗|− −| |+ || +| |+ || ⊗|+⟩⟩ ⟩⟩ ⟩⟩ ⟩⟩ ⟩⟩ ⟩ ⟩⟩ ⟩⟩ ⟩⟩ ⟩⟩ ⟩ where the reflection operator, R1ˆ (ω − ωc = κ/2), in Eq. 8 applies to the above interaction of QD1 and photons. According to the measurement result regarding QD1, if the electron spin state is in −e1, feed-forward (S1 and PF-R) is performed on photons A and B; otherwise, e1+, it is not, as described in Fig. 3. Thus, the output state, |ϕ〉O:1, of QD1 gate is expressed as ⟩⟩ ⟩⟩ ⟩⟩ ⟩⟩ ⟩⟩ ϕϕ|⇒ |= || +| |+ || +| |.−xR Rx LR xR Lx LL (11) 2feed forwardornot O:11 B2 A1 2B1 A1 3B1 A1 4B2 A1 [Rk operator and BS]. After the Rk operator on path 2 and BS is applied to photon B, as described in Fig. 3, the output state, |ϕ〉O:1, of QD1 gate is transformed into the state, |ϕ〉3, as follows: ⟩⟩ ⟩⟩ ⟩⟩ ⟩⟩ ⟩⟩ ⟩⟩ ⟩⟩ ⟩⟩ ⟩⟩ϕϕ|⇒ |= || +| |+ || +| | +− || +| |+ || −| |.π π (12)xR Rx LR xR Lx eL L xR Rx LR xR Lx eL L1 2() 1 2()i iO:1Rk,BS 31 B1 A1 2B1 A1 3B1 A1 42/2 B1 A1 1B2 A1 2B2 A1 3B2 A1 42/2 B2 A1k k Subsequently, this state, |ϕ 〉3, is injected into the QD2 gate in order to merge the split paths (1 and 2) of photon B. [QD2 gate using reflection operator Rˆ2 (ω−ωc = 0)]. After the operation of the interaction, R2ˆ, between QD2 and photon B with the preparation of electron spin 2 state as e2+ , the state, |ϕ 〉3, is transformed into xR Rx LR xR Lx eL L xR Rx LR xR Lx eL L1 2() 1 2() , (13)i i3QD2 41 B1 A1 2B1 A1 3B1 A1 42/2 B1 A1 e2 1B2 A1 2B2 A1 3B2 A1 42/2 B2 A1 e2k k⟩⟩ ⟩⟩ ⟩⟩ ⟩⟩ ⟩⟩ ⟩ ⟩⟩ ⟩⟩ ⟩⟩ ⟩⟩ ⟩ϕϕ|⇒ |= −| |+ || −| |+ || ⊗|− +− || +| |+ || −| | ⊗|+π π where the reflection operator, R2ˆ (ω−ωc = 0), in Eq. 8 applies to the above interaction of QD2 and photon B. When measuring an electron spin 2 state in QD2, if the electron spin state is in e2− , feed-forward (PF-R) is performed to photon A; otherwise, +e1, feed-forward (S2, PF-R, and PF-L) is performed on photons A and B, as described in Fig. 3. Finally,

Question 1418 multiple-choice

Non-perturbative studies of quantum field theories often employ lattice discretization and advanced computational frameworks to explore the dynamics of gauge groups relevant for particle physics. Symplectic gauge groups such as Sp(2N) offer rich symmetry structures and are utilized in constructing models beyond the Standard Model. In investigations of Sp(4) gauge theory using lattice methods, which representation of fermions is particularly notable for exploring novel dynamical and symmetry-breaking patterns? 1) Fundamental representation 2) Adjoint representation 3) Two-index antisymmetric representation 4) Singlet representation 5) Three-index symmetric representation 6) Vector representation 7) Two-index symmetric representation

✓ Correct Answer:

The correct answer is 3) Two-index antisymmetric representation.

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Title: Lattice studies of Sp(2N) gauge theories using GRID Year: 2023 Paper ID: 218eb8980eb0a997e3e826d447014115e063b9e3 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/218eb8980eb0a997e3e826d447014115e063b9e3 Abstract: Four-dimensional gauge theories based on symplectic Lie groups provide elegant realisations of the microscopic origin of several new physics models. Numerical studies pursued on the lattice provide quantitative information necessary for phenomenological applications. To this purpose, we implemented Sp(2N) gauge theories using Monte Carlo techniques within Grid, a performant framework designed for the numerical study of quantum field theories on the lattice. We show the first results obtained using this library, focusing on the case-study provided by the Sp(4) theory coupled to Nas = 4 Wilson-Dirac fermions transforming in the 2-index antisymmetric representation. In particular, we discuss preliminary tests of the algorithm and we test some of its main functionalities.

Question 1419 multiple-choice

Quantum chemistry offers several computational approaches for predicting the properties of transition metal diatomic molecules, which often exhibit challenging electronic structures due to strong electron correlation and multiple spin states. Evaluating the accuracy of these methods is crucial for reliable modeling in inorganic and materials chemistry. Which computational method demonstrated superior agreement with experimental bond dissociation energies for 3d transition metal diatomics, achieving a mean absolute error of approximately 1.4 kcal/mol and maximum errors around 3 kcal/mol? 1) B97 density functional theory 2) Conventional Hartree-Fock theory 3) CCSD coupled cluster singles, doubles, and perturbative triples 4) Phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) with correlated sampling 5) Multiconfiguration self-consistent field (MCSCF) theory 6) Density fitting MP2 (DF-MP2) 7) Semiempirical PM6 method

✓ Correct Answer:

The correct answer is 4) Phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) with correlated sampling.

📚 Reference Text:

Title: On Achieving High Accuracy in Quantum Chemical Calculations of 3 d Transition Metal-Containing Systems: A Comparison of Auxiliary-Field Quantum Monte Carlo with Coupled Cluster, Density Functional Theory, and Experiment for Diatomic Molecules. Year: 2019 Paper ID: ba1e127f794007438990d4d63d66c5336320d3a9 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/ba1e127f794007438990d4d63d66c5336320d3a9 Abstract: The bond dissociation energies of a set of 44 3 d transition metal-containing diatomics are computed with phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) utilizing a correlated sampling technique. We investigate molecules with H, N, O, F, Cl, and S ligands, including those in the 3dMLBE20 database first compiled by Truhlar and co-workers with calculated and experimental values that have since been revised by various groups. In order to make a direct comparison of the accuracy of our ph-AFQMC calculations with previously published results from 10 DFT functionals, CCSD(T), and icMR-CCSD(T), we establish an objective selection protocol which utilizes the most recent experimental results except for a few cases with well-specified discrepancies. With the remaining set of 41 molecules, we find that ph-AFQMC gives robust agreement with experiment superior to that of all other methods, with a mean absolute error (MAE) of 1.4(4) kcal/mol and maximum error of 3(3) kcal/mol (parentheses account for reported experimental uncertainties and the statistical errors of our ph-AFQMC calculations). In comparison, CCSD(T) and B97, the best performing DFT functional considered here, have MAEs of 2.8 and 3.7 kcal/mol, respectively, and maximum errors in excess of 17 kcal/mol (for the CoS diatomic). While a larger and more diverse data set would be required to demonstrate that ph-AFQMC is truly a benchmark method for transition metal systems, our results indicate that the method has tremendous potential, exhibiting unprecedented consistency and accuracy compared to other approximate quantum chemical approaches.

Question 1420 multiple-choice

Quantum dot-cavity systems are essential in quantum optics for enabling controlled photon emission and strong light-matter interaction. The efficiency of such systems is influenced by parameters like coupling strength, cavity and side-leakage rates, and noise contributions. In a quantum dot-cavity system operating in the strong coupling regime with g/κ = 2.4, γ/κ = 0.1, and low side-leakage (κs/κ = 0.01), which outcome is most expected for reflectance, noise, and leakage rates? 1) High reflectance (≈1), very low noise and leakage rates, minimal vacuum noise and sideband leakage impact 2) Low reflectance, high noise and leakage rates, significant loss to side modes 3) High reflectance, high noise rates, substantial decoherence from vacuum fluctuations 4) Moderate reflectance, negligible noise, but substantial leakage into side modes 5) Low reflectance, low noise, but high phase shift variability 6) Moderate reflectance, high noise, minimal effect from side-leakage 7) High noise and leakage rates, low reflectance, strong sensitivity to detuning

✓ Correct Answer:

The correct answer is 1) High reflectance (≈1), very low noise and leakage rates, minimal vacuum noise and sideband leakage impact.

📚 Reference Text:

inter - action of the QD-cavity, and contrast it with the case presented in Eq. 5 (the ideal case: ω = b Rb()outi nˆˆ ). Therefore, the coefficients of noise, N(ω), and leakage, S(ω), can be calculated with c Xω ω=− , as follows: ωγκ ωω γω ωκ κ ωκκ ωω γ ωω γω ωκ κ= −+ −+ ++ =−− + −+ −+ ++Ng ii g Si ii g() [( )/ 2][( )/ 2/ 2], ()[( )/ 2] [( )/ 2][( )/ 2/ 2], (18)cc s sc cc s2 2 where the reflection coefficient, R (ω), is obtained from Eq. 5. Then, we can acquire the noise rate-phase shift (|nh| - φnh) [|n0| - φn0], as well as the leakage rate-phase shift (|sh| - φsh) [|s0| - φs0] from the coefficients of noise, ≡ϕNn ei hhnh ≡ϕNn e [ ]i 00n0, and leakage, ≡ϕS sei hhsh Ss e [ ]i 00s0 ≡ϕ, corresponding to the (hot: g ≠ 0) [cold: g = 0] cavity. The graphs shown in Fig. 5 represent the values of the reflectances (|rh|, |r0|), noise rates (|nh|, |n0|), leakage rates (|sh|, |s0|), and phase shifts (φrh, φr0, φnh, φn0, φsh, φs0) for the detuning frequency as 2(ω − ωc)/κ in terms of the differences in the side-leakage rates (κs/κ = 0.01 and 1.00) with g /κ = 2.4, γ/κ = 0.1, and c Xω ω=− . As illus- trated in Fig. 5, if we take the small side-leakage, κs/κ = 0.01, with g/κ = 2.4 and γ/κ = 0.1, the values of the reflec- tances, noise rates, leakage rates, and phase shifts can be respectively acquired as | rh| ≈ 1 (≈1), |r0| ≈ 1 (≈1), |nh| ≈ 0.1 (≈ 0.1), |n0| = 0 (= 0), |sh| ≈ 0 (≈ 0), |s0| ≈ 0.1 (≈ 0.2), and φrh ≈ 0 (≈ 0), φr0 ≈ −π/2 (= π), φnh ≈ 0 (≈ 0), φn0 = 0 ( = 0), φsh ≈ 11π /20 (= π), φs0 ≈ −3π/4 (= π) in the adjusted frequencies as ω − ωc = κ/2 (ω − ωc = 0). Based on these values, we can confirm that the affections of the vacuum noise, N (ω), and leaky modes, S (ω) (side- band leakage and absorption) will be small (ignored), due to the extremely small values of noise rates (|nh|, |n0|) and leakage rates (|sh|, |s0|) for g /κ = 2.4 and γ/κ = 0.1, respectively, with κs/κ = 0.01 (small side-leakage rate)40,42,53–56. In addition, for the quantifying efficiency and performance of the QD-cavity system, we should modify the reflection operator, R()ˆω in Eq. 7, into the practical reflection operator, ωˆR( )p, including to the affections of the vacuum noise, N(ω), and leaky modes, S(ω) (sideband leakage and absorption), as follows: re ne se RR LL re se RR LLR( )[ () () () ] () [( )( )] () , (19)ii i iiph() h() h() 0() 0()rh nh sh r0 s0ωω ωω ωω=| |+ || +| | ×| 〉〈|⊗|↓〉〈↓|+| 〉〈|⊗|↑〉〈↑| +|

Question 1421 multiple-choice

In quantum chemistry, the optimization of many-electron wavefunctions often requires careful consideration of both spatial orbitals and electron spin coupling. Orthogonal transformations are commonly used to rotate basis sets, enabling a variational search for the most physically accurate state of the system. Which statement accurately describes the role of an orthogonal matrix L in optimizing the wavefunction for a multi-electron quantum system with respect to spin coupling? 1) L is used to diagonalize the electron-nuclear potential energy operator exclusively. 2) L projects the wavefunction onto a subspace containing only singlet spin states. 3) L imposes normalization constraints on the spatial orbitals during optimization. 4) L rotates the spin basis, allowing for the variational optimization of spin functions so that the wavefunction remains an eigenfunction of S² and satisfies the Pauli exclusion principle. 5) L transforms spatial coordinates to maximize overlap between orbitals of different atoms. 6) L minimizes the total charge density of the system via unitary transformation. 7) L ensures that the wavefunction contains only triplet configurations by fixing indices.

✓ Correct Answer:

The correct answer is 4) L rotates the spin basis, allowing for the variational optimization of spin functions so that the wavefunction remains an eigenfunction of S² and satisfies the Pauli exclusion principle..

📚 Reference Text:

spin function to be one of the arbitrarily chosen basis vectors in the !~'-dimensional space of spin functions. This arbitrariness can be removed by allow ing for a general rotation of the coordinate axes in our space of spin functions. We will require that this rota tion give the best single spin function possible. We now derive the necessary equations for finding this optimum spin function. The rotation is most easily obtained by applying an Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jspQUANTUM THEORY OF MANY-ELECTRON SYSTEMS. V 1075 orthogonal transformation to the representative mat rices, U11(r), to obtain U~x is an eigenfunction of S2 and satisfies Pauli's principle. Similarly from the argu ments of Sec. II of Paper I we have that E = ( G~x I H I ~if>x) I ( G~x I G~x) =(if> I HI Ou~'Lif>)(x I wi/'Lx)/( (if> I Ou~)(x I wJ/'Lx)) =(if> I HI Ou~'Lif>)/ (if> I Ou~). (15) At this point we will restrict if> to be a product of one electron spatial orbitals and require that the energy be stationary not only against first-order vatiations in each orbital but also against first-order variations in L. That is, we require that the spatial orbitals and the spin coupling scheme be optimal simultaneously. The energy expression can be rewritten as E= L LliLli(if> I HI 0;/'if>)/L LtkLu(if> I Okz11if>). ij kl (16) We must minimize the energy subject to the constraints that Lk L1k2 = 1 and that (cf>m I cf>m) = 1 for all m. There fore we introduce Lagrange multipliers and minimize the expression I= ( LLt;Ltj(if> I HI 0;/if>)/L LtmLtn(if> I Omn"if>))- L fk{c/>k I c/>k)-X L Ltn2• (17a) ij mn k n This leads to a set of N +!~" coupled equations for the best if> and L. TheN equations for the best if> are integra differential equations and we will call them the spatial SOGI equations. They can be written so that the only permutational operator which appears is the transformed orthogonal unit 011~'L. The f~' equations for the best L are nonlinear algebraic equations which involve if> only in

Question 1422 multiple-choice

Quantum algorithms can dramatically speed up certain decision problems compared to classical methods, particularly in distinguishing properties of Boolean functions. The Deutsch-Jozsa algorithm exemplifies such an advantage by determining whether a function is constant or balanced using quantum interference. Which feature of the improved Deutsch-Jozsa algorithm guarantees that a single measurement after applying Hadamard gates to the quantum state will perfectly distinguish a constant function from a balanced function for any n? 1) The quantum states corresponding to constant and balanced functions are orthogonal, enabling perfect discrimination upon measurement. 2) Only constant functions preserve entanglement in the quantum register after one application of the function oracle. 3) Balanced functions always result in a measurement outcome with exactly half the bits set to one and half to zero. 4) The last qubit in the register acts as a control qubit, flipping the outcome for balanced functions. 5) A specific initialization of the output register ensures the function oracle acts as a classical reversible gate. 6) The algorithm measures all possible outcomes and selects the most probable result to distinguish function type. 7) The time complexity of the algorithm is exponential, ensuring that all function types are distinguishable.

✓ Correct Answer:

The correct answer is 1) The quantum states corresponding to constant and balanced functions are orthogonal, enabling perfect discrimination upon measurement..

📚 Reference Text:

constant or balanced (where balanced means that ftakes values 0 and 1 an equal number of times i.e. 2n−1times each). The problem is to decide whether fis balanced or constant. The method, described in detail in [2], involves running Uftwice (and using H O(n) times) to construct the state |f/an}bracketri}ht=1√ 2n/summationdisplay x∈Bn(−1)f(x)|x/an}bracketri}ht (4) Then |f/an}bracketri}htfor any constant function is orthogonal to the correspondin g state for any balanced function and thus we can solve our decision problem with certainty by a 3 suitable measurement on the resulting state. The quantum al gorithm always runs in timeO(n) whereas any classical algorithm (which gives the result wi th certainty in every case) will require time of O(2n) at least in some cases. Note that Deutsch’s XOR problem is the n= 1 case of the above decision problem. However the above algorithm, running Uftwice, oﬀers no advantage over the obvious classical algorithm for n= 1. Another distinction between the above two algorithms is that the XOR problem is solved only with probability 1/2 wh ereas the second algorithm is always succesful. An interesting recent innovation [11] fully uni ﬁes and considerably improves the above two algorithms: the XOR pro blem may be solved withcertainty and the state in eq. (4) may be constructed by running Ufonlyonce. The improved XOR algorithm is then precisely the n= 1 case of the improved Deutsch algorithm. The basic idea is to set the output register to the state1√ 2(|0/an}bracketri}ht−|1/an}bracketri}ht) before applying Uf. Note that by eq. (3) Uf:|x/an}bracketri}ht(|0/an}bracketri}ht − |1/an}bracketri}ht)−→/braceleftBigg |x/an}bracketri}ht(|0/an}bracketri}ht − |1/an}bracketri}ht) iff(x) = 0 −|x/an}bracketri}ht(|0/an}bracketri}ht − |1/an}bracketri}ht) iff(x) = 1 Thus Uf:1√ 2n/summationdisplay x∈Bn|x/an}bracketri}ht/parenleftBigg|0/an}bracketri}ht − |1/an}bracketri}ht√ 2/parenrightBigg −→/parenleftBigg1√ 2n/summationdisplay x∈Bn(−1)f(x)|x/an}bracketri}ht/parenrightBigg/parenleftBigg|0/an}bracketri}ht − |1/an}bracketri}ht√ 2/parenrightBigg giving the state |f/an}bracketri}htin the ﬁrst nqubits after only one application of Uf. The last qubit plays a curiously passive role in that its state is unch anged in the process. (This is reminiscent of the similarly passive role of the second re gister in Shor’s algorithm [6, 5]). The explicit description of the measurement on |f/an}bracketri}htwhich distinguishes balanced from constant functions is signiﬁcant for subsequent devel opments. We ﬁrst apply the operationHto each of the nqubits of |f/an}bracketri}ht. Denoting the resulting n-qubit operation byHnwe have, for each x∈Bn Hn:|x/an}bracketri}ht →1√ 2n/summationdisplay y∈Bn(−1)x·y|y/an}bracketri}ht (5) Note that Hn|0...0/an}bracketri}ht=1√ 2n/summationdisplay y∈Bn|y/an}bracketri}ht is the equal superposition of all the standard basis states a nd that up to an overall sign this coincides with |f/an}bracketri}htforfconstant. Since HnHn=Iit follows that Hn|f/an}bracketri}ht=|0...0/an}bracketri}ht forfconstant. Thus if fis balanced then Hn|f/an}bracketri}htmust be orthogonal to |0...0/an}bracketri}hti.e. |f/an}bracketri}htlies in the span of {|x/an}bracketri}ht:x/ne}ationslash= 0...0}. Hence to distinguish balanced from constant 4 functions we apply Hnto|f/an}bracketri}htand then read the bits to see whether they are all zero or not. The above measurement has 2nnatural outcomes (i.e. all n-bit strings) and we may ask if there are special balanced functions which yield w ith certainty the other outcomesx∈Bnin the same way that constant functions lead to the outcome 0 ...0. For eachk∈Bnconsider the function fk:Bn→Bgiven by fk(x) =k·x It is easily veriﬁed that each

Question 1423 multiple-choice

Quantum algorithms leverage group theory to solve computational problems, such as Hidden Subgroup and Hidden Translation, using properties like self-reducibility and quantum group actions. New problems, like Translating Coset, generalize these concepts and are important for advancing quantum algorithmic techniques in areas such as graph isomorphism. Which of the following best characterizes the self-reducibility property of the Translating Coset problem in finite solvable groups? 1) It allows reduction of Translating Coset only to abelian subgroups without further decomposition. 2) It depends exclusively on the existence of non-trivial automorphisms in the underlying group. 3) It requires the input states to be classical, not quantum, for recursive reduction. 4) It enables decomposition of Translating Coset instances into smaller instances in factor groups and normal subgroups, facilitating recursive solutions. 5) It restricts the reduction process to cyclic groups, excluding broader solvable groups. 6) It relies on direct product decompositions rather than quotient or subgroup structures. 7) It necessitates that the group is simple and non-solvable for self-reducibility to apply.

✓ Correct Answer:

The correct answer is 4) It enables decomposition of Translating Coset instances into smaller instances in factor groups and normal subgroups, facilitating recursive solutions..

📚 Reference Text:

3.5 is that Hidden Subgroup inZn p/multicloserightZ2is eﬃciently solvable by a quantum algorithm. To solve Hidden Translation in other groups (which include abelian groups of constant exponent), we embark in a radically new direction whose basic idea is self- reducibility .S i n c e Hidden Translation is not well-suited for this self-reducibility based approach, we deﬁne a new paradigmatic group problem. Notice that there is anatural combination of Hidden Translation withHidden Subgroup .T h i si st h e version of Hidden Translation where the functions f 0andf1are not necessarily injective, but they are certain subgroup hiding functions. Indeed, if f1hides a sub- groupHandf0(x)=f1(xu)f o rs o m e u∈Gand for every x∈G, then the set of all such elements uform a right coset of H. (In the context of graph isomorphisms, the correspondingproblemwouldbedetermining allthe bijections betweenthe vertexsets which are isomorphisms. This set is a coset of the automorphism group of one of the graphs.) The self-reducibility will be based on “averaging” over normal subgroups sothat we actually get a problem over the factor group. We will give an averaging pro- cedure which results in quantum superpositions. Therefore our new problem, called Translating Coset , is a combination of Hidden Translation andHidden Sub- groupwhere we have quantum states as input. 1Translating Coset also involves quantum group actions, that is, groups acting on a ﬁnite set of mutually orthogonalquantum states. Given two such states, |φ 0/angbracketrightand|φ1/angbracketright,t h e Translating Coset problem consists of ﬁnding their translating coset , which is deﬁned to be the stabilizer subgroup of |φ1/angbracketrightand a group element that maps |φ1/angbracketrightto|φ0/angbracketright. It turns out that with a slight modiﬁcation, our algorithm of Theorem 3.5 also works for Translating Coset inZn pwhenever many copies of the input states 1In the preliminary version [15] of the present paper, the problem Translating Coset was called Orbit Coset . This was due to the fact that the problem is actually a constructive version of testing membership in orbits of permutation groups. 4 FRIEDL, IVANYOS, MAGNIEZ, SANTHA, AND SEN are given. Moreover, we show that Translating Coset has the following self- reducibility property in any ﬁnite solvable group G: it is reducible to instances ofTranslating Coset inG/NandNfor any normal subgroup NG(Theo- rem 4.11). This is the ﬁrst general self-reducibility result for a problem subsum-ingHidden Subgroup . The proof of the result involves a new technique which is based upon constructing the uniform superposition of the orbit of a given quan- tum state ( Orbit Superposition ). The importance of generating speciﬁc super- positions for solving important algorithmic problems has been observed before; see, for instance, the paper of Aharonov and Ta-Shma [3]. For example, generating theuniform superposition of all graphs isomorphic to a given graph, which in fact is an instance of the Orbit Superposition problem of the symmetric group S nact- ing on an n-vertex graph, would allow us to solve the graph isomorphism problem. We show how Orbit Superposition is related to Translating Coset (Theo- rem 4.10). The self-reducibility of

Question 1424 multiple-choice

The hidden subgroup problem (HSP) is a foundational problem in quantum computing with deep connections to group theory, cryptography, and algorithmic complexity. Infinite groups and their subgroups play a critical role in understanding the computational limits of quantum algorithms. Which statement most accurately characterizes the complexity status of the hidden subgroup problem (HSP) for the additive group of rational numbers? 1) It is solvable in polynomial time using classical algorithms. 2) It is equivalent in complexity to factoring integers. 3) It admits a quasi-polynomial time quantum algorithm. 4) It is efficiently reducible to the abelian hidden shift problem. 5) It is shown to be solvable in subexponential time with quantum resources. 6) It is NP-hard, suggesting no efficient quantum algorithm is likely unless P=NP. 7) It is trivial due to the simplicity of rational group structure.

✓ Correct Answer:

The correct answer is 6) It is NP-hard, suggesting no efficient quantum algorithm is likely unless P=NP..

📚 Reference Text:

Title: The hidden subgroup problem for infinite groups Year: 2025 Paper ID: 057da7fcef4b201a1f801b9c429b8e10c5cbe529 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/057da7fcef4b201a1f801b9c429b8e10c5cbe529 Abstract: Following the example of Shor's algorithm for period-finding in the integers, we explore the hidden subgroup problem (HSP) for discrete infinite groups. On the hardness side, we show that HSP is NP-hard for the additive group of rational numbers, and for normal subgroups of non-abelian free groups. We also indirectly reduce a version of the short vector problem to HSP in $\mathbb{Z}^k$ with pseudo-polynomial query cost. On the algorithm side, we generalize the Shor-Kitaev algorithm for HSP in $\mathbb{Z}^k$ (with standard polynomial query cost) to the case where the hidden subgroup has deficient rank or equivalently infinite index. Finally, we outline a stretched exponential time algorithm for the abelian hidden shift problem (AHShP), extending prior work of the author as well as Regev and Peikert. It follows that HSP in any finitely generated, virtually abelian group also has a stretched exponential time algorithm.

Question 1425 multiple-choice

In the study of quantum algebra, finite-dimensional Hopf algebras and their doubles play an essential role in constructing modular tensor categories, which are central to topological quantum field theory. Key elements such as the antipode, ribbon element, and projective phases determine the algebraic and categorical properties relevant to quantum invariants. Which of the following statements correctly describes the relationship between the square of the braided antipode and the ribbon element in a ribbon Hopf algebra double D? 1) The square of the braided antipode equals the conjugation by the antipode of the unit element. 2) The square of the braided antipode is the identity map on the double D. 3) The square of the braided antipode implements left multiplication by the ribbon element. 4) The square of the braided antipode is equivalent to the adjoint action of the inverse antipode on the ribbon element. 5) The square of the braided antipode is equal to the conjugation by the ribbon element. 6) The square of the braided antipode acts as the counit map on the Hopf algebra. 7) The square of the braided antipode is given by the adjoint action by the inverse of the ribbon element, specifically Γ² = ad⁻(v⁻¹).

✓ Correct Answer:

The correct answer is 7) The square of the braided antipode is given by the adjoint action by the inverse of the ribbon element, specifically Γ² = ad⁻(v⁻¹)..

📚 Reference Text:

identities (2.31) and following. Proposition 9 For a double D(A)over a ﬁnite dimensional Hopf algebra AletµDbe as in (2.44) andS ∈End(D(A))be deﬁned as in (2.55). Then the following diagrams of isomorphisms commute: A⊗Aoβl◦S−1⊗S◦βl→ Ao⊗A Ao⊗Aβr⊗βl◦La→ A⊗Ao /arrowbt./arrowbt./arrowbt./arrowbt. D(A)S→D(A) D(A)S−1→D(A) (2.56) HereLadenotes left multiplication with a. Proof :The ﬁrst diagram is veriﬁed by direct computation: S(y.λ) =/summationdisplay ijµD(y.λfjei)⊗S(fi)S(ej)by(Pp6.3.)=/summationdisplay ijµD(λfj.eiS−2(y)) by(2.44)=/summationdisplay ijλfj(x)µ(S−1(y)ei)fi.S(ej) = µ⊳S−1(y).S(x ↽ λ) by(2.23)=βl(S−1(y)).S(βl(λ)) (2.57) The second diagram follows immediately from relations (2.30) and (2.31 ), which allow us to invert βlandβl. ✷ Let us also give a more convenient form for the braided antipode: Lemma 10 Let˜Γ :Ao⊗A/tildewidest−→A⊗Aobe given by Ao⊗AR⊗12⊗R−−−−→A⊗Ao⊗Ao⊗A⊗A⊗AoS⊗1⊗S−1⊗13−−−−−−−→A⊗Ao⊗Ao⊗A⊗A⊗Ao 1⊗·⊗·⊗1−−−−→A⊗Ao⊗A⊗Ao1⊗τ⊗1−−−→A⊗A⊗Ao⊗Ao·⊗·−−→A⊗AoS⊗1−−→A⊗Ao andΓas in (2.54). Then the following diagram commutes: Ao⊗A˜Γ→ A⊗Ao /arrowbt./arrowbt. D(A)Γ→D(A)(2.58) 21 Proof :Straightforward computation: Γ(λ.y) =/summationdisplay iS(ei)S(y).S(λ)ˆufi=/summationdisplay iS(ei)S(y)ˆuS−1(λ)fi =/summationdisplay ijS(S(fj)yei).ejS−1(λ)fi, (2.59) which is precisely the above composition. ✷ 6.) Proof of Modular Relations and the Projective Phases : For the square of the braided antipode we easily verify Γ2=ad−(v−1), (2.60) wherevis any ribbon element and ad−(y) =S−1◦ad(y)◦S. Proposition 9 and Lemma 10 put us in a position to prove the next lemma. From this we will infer one o f the modular relations and the correct projective phase. Lemma 11 We have the following relation for maps Ao⊗A/tildewidest−→Ao⊗A: βl◦S−1⊗S◦βl˜Γ = ω ⊲⊳βr⊗(βlLa) (2.61) Proof :We shall prove (2.61) by evaluating both sides on λ⊗y∈Ao⊗Aindividually and comparing results. For the right hand side we have ⊲⊳βr⊗(βlLa)(λ⊗y) =⊲⊳/parenleftBig x′λ(x′′)⊗µ⊳(ay)/parenrightBig =/angbracketleftBig (µ⊳(ay))′,x′/angbracketrightBig/angbracketleftBig S/parenleftBig (µ⊳(ay))′′′/parenrightBig ,x′′′/angbracketrightBig λ(x′′′′) (µ⊳(ay))′′⊗x′′ =/angbracketleftBig µ′,x′/angbracketrightBig/angbracketleftBig µ′′′,ayS−1(x′′′)/angbracketrightBig λ(x′′′′)µ′′⊗x′′ =/summationdisplay i/angbracketleftBig µ′,x′/angbracketrightBig/angbracketleftBig fi,x′′/angbracketrightBig/angbracketleftBig µ′′′,ayS−1(e′′ i)/angbracketrightBig λ(e′′′ i)µ′′⊗e′ i =/summationdisplay i/angbracketleftBig µ′,βr(fi)/angbracketrightBig/angbracketleftBig µ′′′,ayS−1(e′′ i)/angbracketrightBig λ(e′′′ i)µ′′⊗e′ i =/summationdisplay i/angbracketleftBig βrβr(fi)′′, ayS−1(e′′ i)/angbracketrightBig λ(e′′′ i)βrβr(fi)′⊗e′ i by(2.31)=/summationdisplay i/angbracketleftBig (S(fi)⊳a−1)′′,ayS−1(e′′ i)/angbracketrightBig λ(e′′′ i)/parenleftBig S(fi)⊳a−1/parenrightBig′⊗e′ i =/summationdisplay i/angbracketleftBig S(fi)′′,yS−1(e′′ i)/angbracketrightBig λ(e′′′ i)S(fi)′⊗e′ i (2.62) 22 The evaluation of the left hand side gives: βl◦S−1⊗S◦βl˜Γ (λ⊗y) =/summationdisplay ijβl◦S−1⊗S◦βl/parenleftBig S(S(ei)yej)⊗(fiS−1(λ)fj)/parenrightBig =/summationdisplay ijβl(S(ei)yej)⊗S◦βl(fiS−1(λ)fj) =/summationdisplay ij/angbracketleftBig fi,x′/angbracketrightBig/angbracketleftBig S−1(λ)fj,x′′/angbracketrightBig µ⊳(S(ei)yej)⊗S(x′′′) =/summationdisplay j/angbracketleftBig S−1(λ)fj,x′′/angbracketrightBig S/parenleftBig (S−1(yej)x′)⊲S−1(µ)/parenrightBig ⊗S(x′′′) by(2.28)=ω/summationdisplay ij/angbracketleftBig fi,x′′/angbracketrightBig/angbracketleftBig S−1(λ)fj,e′ i/angbracketrightBig S/parenleftBig (S−1(yej)x′)⊲µ⊳a/parenrightBig ⊗S(e′′ i) =ω/summationdisplay ij/angbracketleftBig S−1(λ)fj,e′ i/angbracketrightBig S/parenleftBig (S−1(yej)βr(fi))⊲µ⊳a/parenrightBig ⊗S(e′′ i) =ω/summationdisplay ij/angbracketleftBig S−1(λ)fj,e′ i/angbracketrightBig S/parenleftBig (S−1(yej))⊲(βrβr(fi))⊳a/parenrightBig ⊗S(e′′ i) by(2.31)=ω/summationdisplay ij/angbracketleftBig S−1(λ)fj,e′ i/angbracketrightBig S/parenleftBig (S−1(yej))⊲S(a⊲fi)⊳a/parenrightBig ⊗S(e′′ i) =ω/summationdisplay ij/angbracketleftBig S−1(λ)fj,e′ i/angbracketrightBig S2(fi)⊳(yej)⊗S(e′′ i) =ω/summationdisplay ij/angbracketleftBig S(fj)λ,e′′ i/angbracketrightBig S(fi)⊳(yej)⊗e′ i =ω/summationdisplay ij/angbracketleftBig S(fi)′′,yej/angbracketrightBig/angbracketleftBig S(fj),e′′ i/angbracketrightBig λ(e′′′ i)S(fi)′⊗e′ i =ω/summationdisplay i/angbracketleftBig S(fi)′′,yS−1(e′′ i)/angbracketrightBig λ(e′′′ i)S(fi)′⊗e′ i (2.63) Comparison of (2.62) to (2.63) proves the assertion. ✷ The projective phases of the second modular relation arise in the co mputation of the value ofSon the ribbon element. Lemma 12 Supposevis a ribbon element of a double D(A)andνis the associated fourth root ofω(see Prop.8.). Then we have for Sas deﬁned in (2.55) : S(v) =ν−1v−1S(v−1) =ν5v . (2.64) Proof :Straightforward computation: Using v=uk−1we have S(v) =/summationdisplay ijµD(uk−1fjei)S(fi).S(ej)byProp7.=/summationdisplay ijχ(eik−1fj)S(fi).S(ej) 23 =/summationdisplay ij/angbracketleftBig√α−1fj,ei√a/angbracketrightBig S(fi).S(ej) =/summationdisplay iS(fi).S/parenleftBig (ei√a)↽√α−1/parenrightBig =ν−1/summationdisplay iS(fi).√a−1S/parenleftBig ei↽√α−1/parenrightBig =ν−1/summationdisplay iS(√α−1fi).√a−1S(ei) =ν−1/summationdisplay iS(fi)kS(ei) =ν−1/summationdisplay ifiS2(ei)k=ν−1u−1k=ν−1v−1.(2.65) The second relation follows with v−1= ˆuk−1from: S(v−1) =/summationdisplay ijµD(ˆuk−1fjei)S(fi).S(ej) =ω/summationdisplay ijˆχ(fj√α−1√aS2(ei))S(fi).S(ej) =ω/summationdisplay ij/angbracketleftBig fj√α−1,S(ei)√a−1/angbracketrightBig S(fi).S(ej) =ω/summationdisplay ifi.S/parenleftBig√α−1⇀(ei√a−1)/parenrightBig =ων/summationdisplay ifi.√aS(√α−1⇀ ei) =ων/summationdisplay ifi√α−1.√aS(ei) =ν5k−1u=ν5v . (2.66) Let us now prove the second modular relation: Proposition 13 For a double D(A)with balancing, let

Question 1426 multiple-choice

Quantum circuit design is evolving with the adoption of higher-level abstractions and model-driven frameworks, enabling more accessible development of quantum algorithms. These innovations are accelerating the creation and maintenance of quantum software, even for users without deep expertise in quantum mechanics. Which framework feature most directly enables modular reuse and encapsulation when assembling complex quantum algorithms from well-defined building blocks? 1) Integrated hardware simulation 2) Direct manipulation of qubit registers 3) Low-level gate specification interfaces 4) Enhanced visualization of quantum states 5) Real-time error correction mechanisms 6) Support for composite operators 7) Native integration with classical programming languages

✓ Correct Answer:

The correct answer is 6) Support for composite operators.

📚 Reference Text:

Title: A Model-Driven Framework for Composition-Based Quantum Circuit Design Year: 2024 Paper ID: 97124eb00ddae3189bd88082a3d170d91345a71a Source: semantic-scholar URL: https://www.semanticscholar.org/paper/97124eb00ddae3189bd88082a3d170d91345a71a Abstract: Quantum programming languages support the design of quantum applications. However, to create such programs, one needs to understand the fundamental characteristics of quantum computing and quantum information theory. Furthermore, quantum algorithms frequently make use of abstract operations with a hidden low-level realization (e.g., Quantum Fourier Transform). Thus, turning from elementary quantum operations to a higher-level view of quantum circuit design not only reduces the development effort but also lowers the entry barriers for non-quantum computing experts. To this end, this article proposes a modeling language and design framework for quantum circuits. This allows the definition of composite operators to advocate a higher-level quantum algorithm design, together with automated code generation for the circuit execution. To demonstrate the benefits of the proposed approach, coined Composition-based Quantum Circuit Designer, we applied it for realizing the Quantum Counting algorithm and the Quantum Approximate Optimization Algorithm. Our evaluation results show that, compared to an existing state-of-the-art editor, the proposed approach allows for the realization of both quantum algorithms on a high level with a substantially reduced development effort. In particular, the proposed approach shows constant scaling when increasing the size of the investigated quantum circuits and a lower change criticality when evolving existing quantum circuits.

Question 1427 multiple-choice

Quantum Clifford algebras are q-deformations of classical Clifford algebras and play important roles in mathematics and mathematical physics, particularly in representation theory and quantum groups. Recent generalizations introduce new parameters to enrich their structure and applications. In the context of quantized Clifford algebras with a twist parameter k over fields of characteristic not equal to 2, what is the structure of the center of the algebra? 1) It is a polynomial algebra generated by the twist parameter k. 2) It coincides with the group algebra of a cyclic group of order k. 3) It is isomorphic to a matrix algebra over the field. 4) It forms a commutative algebra generated by the quantum group generators. 5) It is a classical Clifford algebra over the group algebra of a product of cyclic groups of order 2k. 6) It is the universal enveloping algebra of a Lie algebra associated to the quantum symmetry. 7) It consists solely of scalar multiples of the identity element.

✓ Correct Answer:

The correct answer is 5) It is a classical Clifford algebra over the group algebra of a product of cyclic groups of order 2k..

📚 Reference Text:

Title: On the structure and representation theory of q-deformed Clifford algebras Year: 2022 Paper ID: b96da04f7c4d0397eb9ee034789979b92ec91422 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b96da04f7c4d0397eb9ee034789979b92ec91422 Abstract: We provide a generalized definition for the quantized Clifford algebra introduced by Hayashi using another parameter k that we call the twist. For a field of characteristic not equal to 2, we provide a basis for our quantized Clifford algebra, show that it can be decomposed into rank 1 components, and compute its center to show it is a classical Clifford algebra over the group algebra of a product of cyclic groups of order 2 k . In addition, we characterize the semisimplicity of our quantum Clifford algebra in terms of the semisimplicity of a cyclic group of order 2 k and give a complete set of irreducible representations. We construct morphisms from quantum groups and explain various relationships between the classical and quantum Clifford algebras. By changing our generators, we provide a further generalization to allow k to be a half integer, where we recover certain quantum Clifford algebras introduced by Fadeev, Reshetikhin, and Takhtajan as a special case.

Question 1428 multiple-choice

Quantum machine learning algorithms can leverage quantum principles such as superposition and entanglement to achieve significant advantages over classical approaches in data compression, especially for data sets with hidden symmetries. These quantum methods are particularly relevant for problems involving group structure and the hidden subgroup problem. Which statement best explains why a variational quantum autoencoder utilizing the hidden subgroup quantum algorithm can achieve exponential speedup in compressing certain classical data compared to classical autoencoders? 1) It reduces the dimensionality by discarding all entangled states in the data. 2) It uses a fixed quantum circuit that does not require training or optimization. 3) It compresses data only when the underlying symmetry group is abelian. 4) It exploits quantum algorithms that identify and utilize hidden group structures, solving problems with polynomial query complexity where classical algorithms require exponential time. 5) It encodes data by mapping each input to a unique quantum basis state, avoiding feature extraction. 6) It replaces the decoder entirely with classical post-processing steps. 7) It relies exclusively on classical optimization techniques without quantum processing.

✓ Correct Answer:

The correct answer is 4) It exploits quantum algorithms that identify and utilize hidden group structures, solving problems with polynomial query complexity where classical algorithms require exponential time..

📚 Reference Text:

npj | quantum information Article Published in partnership with The University of New South Wales https://doi.org/10.1038/s41534-024-00865-2 Information compression via hidden subgroup quantum autoencoders Check for updates Feiyang Liu1,2,K a i m i n gB i a n2, Fei Meng1,2, Wen Zhang3& Oscar Dahlsten1,2,4 We design a quantum method for classical information compression that exploits the hidden subgroup quantum algorithm. We consider sequence data in a database with a priori unknown symmetries of thehidden subgroup type. We prove that data with a given group structure can be compressed with thesame query complexity as the hidden subgroup problem, which is exponentially faster than the best-known classical algorithms. We moreover design a quantum algorithm that variationally ﬁnds the group structure and uses it to compress the data. There is an encoder and a decoder, along theparadigm of quantum autoencoders. After the training, the encoder outputs a compressed data stringand a description of the hidden subgroup symmetry, from which the input data can be recovered by thedecoder. In illustrative examples, our algorithm outperforms the classical autoencoder on the meansquared value of test data. This classical-quantum separation in information compression capabilityhas thermodynamical signi ﬁcance: the free energy assigned by a quantum agent to a system can be much higher than that of a classical agent. Taken together, our results show that a possible applicationof quantum computers is to ef ﬁciently compress certain types of data that cannot be ef ﬁciently compressed by current methods using classical computers. Information compression is the ubiquitous task of reducing data size by appropriate encoding, increasing in formation storage and transmission efﬁciency1. A prominent method of compression is to employ autoencoders2,3, artiﬁcial neural networks that automatically learn to compress unlabeled data without prior knowledge of the underlyi ng patterns. Exploiting a bottleneck structure, an autoencoder can automati cally extract essential features as com- pressed data in such a manner that th e original data can be reconstructed4–6. Recently, the quantum version of autoencoders, to our knowledge ﬁrst proposed independently in refs. 7,8, has similarly been applied to encode9, compress7,8,10, classify and denoise11quantum data and been implemented experimentally12–16. Quantum autoencoders can, via unitary circuits, com- press data hidden in quantum superpositions, but it is natural to wonder ifthey can also be valuable in compressing classical data. Quantum algorithms are indeed able to extract certain features in data that are not ef ﬁciently accessible to classical computers. For instance, the quantum period ﬁnding algorithm 17–19is exponentially faster than the best current classical algorithms in identifying the period of a function. Moregenerally, the Abelian hidden subgrou p problem (HSP) represents a broad class of problems (including period ﬁnding) that do not have known ef ﬁcient classical algorithms, whereas ef ﬁcient quantum algorithms often exist 20–24.Can that quantum speedup on HSPs be turned into an advantage for information compression? To tackle this question, we combine quantumautoencoders and the quantum HSP al gorithm to create a concrete algo- rithm to compress data with symmetries of the HSP type. We prove an exponential speedup in query

Question 1429 multiple-choice

Quantum algorithms often leverage problems like the hidden shift and matrix sum to achieve computational speedups over classical approaches. The partitioning of problem instances and careful parameter selection are crucial for optimizing fidelity and success rates in these algorithms. In the context of quantum algorithms addressing the matrix sum problem, which parameter choice guarantees that efficient quantum sampling is possible with a manageable number of solutions per instance? 1) Setting $ k = 2 $ and choosing $ M $ arbitrarily large 2) Allowing $ 0 \leq \eta_{xw} \leq N $, for any $ k $ 3) Restricting $ 1 \leq \eta_{xw} \leq 4 $ by selecting $ k = c \geq 3 $ and $ M = \lfloor N^{1/k} \rfloor $ 4) Using $ k = 1 $ and $ M = N $ 5) Setting $ M = N $ without regard to $ k $ 6) Choosing $ k $ as any large integer and ignoring the value of $ M $ 7) Keeping $ \eta_{xw} $ unbounded and $ k $ fixed at 2

✓ Correct Answer:

The correct answer is 3) Restricting $ 1 \leq \eta_{xw} \leq 4 $ by selecting $ k = c \geq 3 $ and $ M = \lfloor N^{1/k} \rfloor $.

📚 Reference Text:

is close to zero unless j=s, so that a measurement of the ﬁrst register is likely to yield the hidden shift s. Ingeneral, itmaybediﬃculttoimplement(27)exactly. Inst ead, wemay only be able to perform an approximate quantum sampling tran sformation satisfying |w,x/an}⌊ra⌋k⌉tri}ht /ma√sto→/braceleftBigg |Sx w,x/an}⌊ra⌋k⌉tri}ht(x,w)∈Zgood |µx w,x/an}⌊ra⌋k⌉tri}ht(x,w)∈Zbad(30) for some states |µx w/an}⌊ra⌋k⌉tri}ht, whereZgood,Zbadform a partition of the matrix sum instances ( x,w) for which ηx w>0. The good matrix sum problem instances (x,w)∈Zgoodare those for which the quantum sampling can be done cor- rectly. Assuming the bad matrix sum instances ( x,w)∈Zbadcan be recog- nized, wecan ensurethat /an}⌊ra⌋k⌉tl⌉{tSx w|µx w′/an}⌊ra⌋k⌉tri}ht= 0for all x∈Zk N,w,w′∈ZN. Applying the approximate quantum sampling transformation followed by the Fourier transform gives the state ρ′ apx=1 Nk/summationdisplay x∈Zk N|ρ′ x,apx,x/an}⌊ra⌋k⌉tri}ht/an}⌊ra⌋k⌉tl⌉{tρ′ x,apx,x| (31) where |ρ′ x,apx/an}⌊ra⌋k⌉tri}ht:=1√ MkN/summationdisplay j∈ZN/parenleftBigg/summationdisplay (x,w)∈Zgoodωw(s−j)/radicalbig ηxw|j/an}⌊ra⌋k⌉tri}ht +/summationdisplay (x,w)∈Zbadωw(s−j)/radicalbig ηxw|νx j/an}⌊ra⌋k⌉tri}ht/parenrightBigg (32) for some states |νx j/an}⌊ra⌋k⌉tri}htwith/an}⌊ra⌋k⌉tl⌉{tj|νx j′/an}⌊ra⌋k⌉tri}ht= 0 for all x∈Zk N,j,j′∈ZN(since /an}⌊ra⌋k⌉tl⌉{tSx w|µx w′/an}⌊ra⌋k⌉tri}ht= 0 for all x∈Zk N,w,w′∈ZNand (30) is unitary). The ﬁdelity between the ideal ﬁnal state ρ′and the actual ﬁnal state ρ′ apxresulting from QUANTUM ALGORITHM FOR A GENERALIZED HIDDEN SHIFT PROBLEM 9 approximate quantum sampling is thus 1 (MN)k/summationdisplay (x,w)∈Zgoodηx w. (33) Nowηx w>1 for all ( x,w)∈Zgood, so if|Zgood|is suﬃciently large, the actual ﬁnal state is close to the ideal ﬁnal state, and hence a measurement of the ﬁrst register yields the hidden shift swith reasonable probability. As we will show in the next section, the instances with 1 ≤ηx w≤4 can be quantum sampled eﬃciently (i.e., these instances are good) . Then, letting M=⌊N1/k⌋, Lemma 2 shows that |Zgood|/Nk+1is lower bounded by a constant, and thus the ﬁdelity between ρ′andρ′ apxis lower bounded by a constant, so that the probability of successfully determi ningsis lower bounded by a constant. 4. Solution of the matrix sum problem Recall that the matrix sum problem for the generalized hidde n shift problem is the following: given x∈Zk Nandw∈ZNchosen uniformly at random, ﬁnd b∈ {0,...,M−1}ksuch that b·x=wmodN. This is a linear equation over ZNinkvariables, where the solutions are required to come from {0,...,M−1}. Such solutions can be found using integer programming, which has an eﬃcient algorithm if Mis suﬃciently large. We assumeM=⌊N1/c⌋for some positive integer c≥3. Since we can always decrease Mbyonlyconsideringasubsetoftheinputstotheﬁrstargumen tof the hiding function f, this will not constitute a loss of generality. According to Lemma 2, we take k=cso that there are between 1 and 4 solutions with probability at least some constant. Toseetheconnection tointeger programming, wenotethat th esolutions form an integer lattice. We begin by considering the equatio nb·x=wmod Nas a (k+ 1)-variable linear equation over all the integers Z. Deﬁne an extension of xby ¯x:= (x1,...,x k,N) and consider the solutions ¯b∈Zk+1 of the equation ¯b·¯x=w. For any b∈Zkthat solves the equation b·x= wmodN, there is a unique λ∈Zsuch that ¯b= (b,λ) is a solution to ¯b·¯x= w; and conversely, for any ¯b∈Zk+1that solves ¯b·¯x=w, there is a unique b∈Zk(namely, the ﬁrst kcomponents of ¯b) that solves b·x=wmodN. Hence there is a bijection between the solutions ¯b∈Zk+1to

Question 1430 multiple-choice

In quantum field theory on complex manifolds, (p, 0)-forms extend the concept of gauge fields to holomorphic settings, particularly within Kähler geometry. These forms, their equations of motion, and related dualities play a key role in understanding quantum dynamics and topological effects in such spaces. Which of the following statements correctly characterizes the duality relation for massless (p, 0)-forms on a d-dimensional Kähler manifold at one-loop order? 1) (p, 0)-forms are always self-dual, with no topological mismatch for any value of p or d. 2) The duality relates (p, 0)-forms to (d − p, 0)-forms and is exact at all quantum orders. 3) (p, 0)-forms are dual to (d − p − 2, 0)-forms, but a topological mismatch appears specifically at the one-loop quantum level. 4) The duality acts only on scalar (0, 0)-forms, leaving higher-rank forms unrelated. 5) (p, 0)-forms and their duals have identical heat kernel expansions for all values of p. 6) The duality is broken by the choice of U(1) charge and does not hold for supersymmetric models. 7) Duality relations are only valid in flat complex spaces, not in Kähler manifolds with curvature.

✓ Correct Answer:

The correct answer is 3) (p, 0)-forms are dual to (d − p − 2, 0)-forms, but a topological mismatch appears specifically at the one-loop quantum level..

📚 Reference Text:

Title: Quantum theory of massless (p, 0)-forms Year: 2011 Paper ID: 2949278ba4604b8a38f6cd3358941bb7c79533a6 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/2949278ba4604b8a38f6cd3358941bb7c79533a6 Abstract: We describe the quantum theory of massless (p, 0)-forms that satisfy a suitable holomorphic generalization of the free Maxwell equations on Kähler spaces. These equations arise by first-quantizing a spinning particle with a U(1)-extended local supersymmetry on the worldline. Dirac quantization of the spinning particle produces a physical Hilbert space made up of (p, 0)-forms that satisfy holomorphic Maxwell equations coupled to the background Kähler geometry, containing in particular a charge that measures the amount of coupling to the U(1) part of the U(d) holonomy group of the d-dimensional Kähler space. The relevant differential operators appearing in these equations are a twisted exterior holomorphic derivative ∂q and its hermitian conjugate ∂q† (twisted Dolbeault operators with charge q). The particle model is used to obtain a worldline representation of the one-loop effective action of the (p, 0)-forms. This representation allows to compute the first few heat kernel coefficients contained in the local expansion of the effective action and to derive duality relations between (p, 0) and (d − p − 2, 0)-forms that include a topological mismatch appearing at one-loop.

Question 1431 multiple-choice

Quantum Fourier Transform (QFT) circuits are pivotal in quantum computing, often constructed and analyzed using tensor network representations and algorithms for efficient simulation. The stability of tensor contraction algorithms, such as the zip-up algorithm for Matrix Product Operators (MPOs), impacts the accuracy and efficiency of quantum simulations. Which property of QFT tensor networks ensures that local singular value decompositions during MPO contraction yield globally optimal truncations, thereby guaranteeing the stability of the zip-up algorithm for any number of qubits? 1) Presence of non-unitary tensors at each site 2) Random alignment of orthogonality centers 3) Maximally entangled input states 4) Formation of isometric tensors during contraction 5) Absence of controlled phase gates in the circuit 6) Singular value spectra with all equal values 7) Use of classical Fourier transforms only

✓ Correct Answer:

The correct answer is 4) Formation of isometric tensors during contraction.

📚 Reference Text:

how would one get the QFT circuit if Eq.(F15) were not known. The ﬁrst step one may do is todeﬁne a generalized version of the order-2 phase tensors (G10) with both indices xandyof arbitrary dimension (one may not call it a phase tensor, and may not write indices in two directions, but we do so here for consistency with our pre- vious deﬁnitions). It is easy to see that the Fourier matrix Fnscaled by 2n/2is equivalent to one of such a tensor with θ=π/2n−1andx,y∈[0, 2n−1], i.e., (G11) Again, we use n=4 as an example to show a derivation of the QFT’s quantum circuit through the decomposition of the generalized order-2 phase tensor. For convenience, we consider two indices xand yas variables in the phase parameter, i.e., applying the insertion identity (G12) Then one can express xand yin terms of their binary expansions: (G13) Through our deﬁnition of the phase tensor, we can then break the sum into a product of phase tensors (G14) which is essentially breaking the exponential of the sum of phase parameters into a product of the exponential of each phase parameter. Then one observes that by bringing the index lines back, one can get Eq. (G1) . Equivalently, one observes phase tensors with parameters being multiples of 040318-25 JIELUN CHEN, STOUDENMIRE, and WHITE PRX QUANTUM 4,040318 (2023) 2πare evaluated to constant 1; thus, they can be removed from the diagram and we can directly achieve Eq. (G6) . APPENDIX H: STABILITY OF THE ZIP-UP ALGORITHM OVER THE QFTTN In this appendix, we make some arguments that the zip-up algorithm is completely stabilized for the QFTTN. While we do not have proof, the tensor diagram analysis and numerical simulation strongly suggest proof; thus, we conjecture it is true for the QFTTN with any number of qubits. We ﬁrst explain what we mean by being stabilized. For the contraction of two MPOs, The zip-up algorithm does local SVD at each step, where the singular values do not reﬂect the actual Schmidt coeﬃcients of the con- tracted MPO. Therefore, if one does truncation over the singular values, one may generally not achieve optimal truncation, and an error larger than the cutoﬀ would be introduced. However, if one can form isometric tensors by contracting each site of the two MPOs before the trunca- tion, the isometries will allow local SVDs to reﬂect the whole system’s Schmidt coeﬃcients; therefore, in this sit- uation truncating local singular values are optimal. We call this being completely stabilized. For contraction of two generic MPOs, one may not expect this property; thus, the original algorithm in Ref. [ 39] heuristically aligns the orthogonality centers to somewhat stabilize the algorithm. We now explain why the QFTTN is likely to be com- pletely stabilized. Consider the situation where one tries to contract the j0th intermediate MPO and the next phase MPO with an Hgate sandwiched in between, which can be demonstrated by the diagram (H1) where {U[k],T[n]}are sites of the j0th intermediate MPO. As introduced before, to

Question 1432 multiple-choice

Lattice-based cryptography relies on mathematical problems involving lattices and is a major candidate for post-quantum security. Advances in attack strategies and distribution analysis are crucial for understanding system vulnerabilities and designing secure cryptographic schemes. Which cryptanalytic technique specifically improves the efficiency of dual attacks in lattice-based cryptography by altering the modulus to optimize operations and reduce noise, rather than focusing on the most significant bits? 1) Sparse secret guessing 2) FFT-based search optimization 3) Matrix splitting 4) Error bias exploitation 5) Short vector search in normal form 6) Discrete Gaussian sampling 7) Modulus switching

✓ Correct Answer:

The correct answer is 7) Modulus switching.

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i.e. the distribution of |ej|forej:=/an}⌊rack⌉⊔l⌉{⊔xj,e/an}⌊rack⌉⊔ri}h⊔is biased towards elements0. For any x∈Rd, we letρσ(x):= exp(−/⌊ard⌊lx/⌊ard⌊l2/2σ2). Note that this is diﬀerent from the other (also commonly used) deﬁnition, where1 2is replaced by πin the exponent. This change is inconsequential to our results. We e xtend the deﬁnition of ρσ(·) to sets of vectors Sby lettingρσ(S):=/summationtext x∈Sρσ(x). For any latticeL ⊂Rd, we denote by DL,σthe discrete Gaussian distribution over L, deﬁned byDL,σ(x) =ρσ(x)/ρσ(L) for allx∈ L. Observing that DZn,σ(x) only depends on /⌊ard⌊lx/⌊ard⌊l, we abuse notation and for ℓ=/⌊ard⌊lx/⌊ard⌊lwriteρσ(ℓ) = exp(−ℓ2/2σ2) andDZn,σ(ℓ) =ρσ(ℓ)/ρσ(Zn). We will also make use of the modular discrete Gaussian. For any q∈N, we denote byDZdq,σthe modular discrete Gaussian distribution over Zd qdeﬁned by DZdq,σ(x) =ρσ(x+qZd) ρσ(Zd). Note that the distribution DZdq,σis isomorphic to the distribution Dd Zq,σ, a fact that we will use often implicitly. We letΦ(x) =1√ 2π/integraltextx −∞exp(−t2/2) dtbe the cumulative distribution func- tion(cdf) ofthestandardnormaldistributionand Φ−1(x) : [0,1)→Ritsinverse. 2.4 Quantum Computing Quantum Circuit Model. In the quantum circuit model, the time complexity is the circuit size, which is the total number of elementary quantum g ates. The space complexity is the number of qubits used. We will assume that th e elemen- tary quantum gates come from a ﬁxed universal set. Up to consta nt factors, the complexity does not depend on the universal set that we have chos en. Since all unitary transforms are invertible, any quantum circuit Ais reversible and we denote by A†its inverse, which is also equal to its conjugate transpose when viewed as a matrix. Given a function f:{0,1}n→ {0,1}m, we saythat aquantum circuit, imple- menting a unitary Uthat acts on n+ℓ+mqubits,computesfwith probability α 2Another signiﬁcant gain reported in [MAT22] is due to an impr ovement to the lattice sieving algorithm from [BDGL16] but discussing this is out o f scope of this work. 6 if for

Question 1433 multiple-choice

Quantum circuit optimization involves tailoring gate decomposition and qubit layout to minimize depth and error rates, especially in architectures featuring parity qubits and advanced encoding schemes. Understanding how these choices impact algorithm efficiency and gate implementation is essential for developing scalable quantum systems. In a square lattice quantum processor using parity qubit encoding, which architectural feature most directly eliminates the need for SWAP gates when implementing controlled-phase gates between distant qubits? 1) Increasing the number of data qubits per logical line 2) Using only Hadamard gates on target qubits 3) Applying negative controls by flipping all physical spins 4) Rearranging qubits to enhance nearest-neighbor connectivity 5) Implementing QFT with single-qubit rotations 6) Leveraging parity constraints that allow native multiqubit controlled-phase interactions 7) Adding extra CNOT chains for each controlled-phase gate

✓ Correct Answer:

The correct answer is 6) Leveraging parity constraints that allow native multiqubit controlled-phase interactions.

📚 Reference Text:

scheme. Here iandjdenote the control qubits and kthe target qubit. The circuit depth and the two-qubit gate count of this decomposition depend solely on the number ofparity qubits in the logical line of the target qubit (i.e., theconnectivity requirements of the logical qubit k). All required gates apart from the Hadamard gates on the target qubits canbe implemented natively (i.e., without CNOT chains) as long as the data qubit ( i) and the parity qubit ( jk) are close enough on the chip to perform the physical controlled phase gate. Similarly, a physical CPφgate between two parity qubits (ij) and ( kl) effectively represents a four-qubit gate which adds a phase if and only if si/negationslash=sjandsk/negationslash=sl. In the square lattice implementation of the all-to-all connected graph, thesederived multiqubit gates can be realized along the diagonal ofa parity constraint. If the required connectivity is not available,the qubits can be rearranged to an algorithm-speciﬁc layout toprovide the desired interactions. E. Negative controls For many applications [ 39,40], it is useful to invert control inputs of controlled multiqubit gates. That corresponds toapplying a spin ﬂip on the respective qubit before and afterthe controlled gate. In the parity encoding, a logical spin ﬂip isrealized by simultaneously ﬂipping all physical spins along alogical line [ 11]. However, as the spin ﬂips on parity qubits not involved in the operation cancel out, it is sufﬁcient to only ﬂipthe data qubit corresponding to the desired negative controland the parity qubits involved in decomposition ( 16). III. APPLICATIONS A. Quantum Fourier transform The quantum Fourier transform [ 31] is the quantum analog to the discrete Fourier transform and an important ingredi-ent for many quantum algorithms such as quantum phaseestimation [ 41] and, in further consequence, Shor’s factoring 042442-3 FELLNER, MESSINGER, ENDER, AND LECHNER PHYSICAL REVIEW A 106, 042442 (2022) FIG. 3. Circuit for performing the QFT in the LHZ scheme. CNOT gates are required to perform the logical Hadamard gates, while single- qubit Rzrotations implement logical CPHASE gates. Gates which do not contribute to the circuit depth are faded out. For all but the ﬁrst and the last line, increasing the system size would only add gates which do not contribute to the circuit depth. The diagram on the bottom visualizes the time steps occupied by each gate block of the QFT algorithm. In the layout on the left, lines depict logical lines and the gray squares andtriangles represent parity constraints. algorithm [ 2]. The unitary for the QFT on nqubits is given by UQFT=n/productdisplay i=1/bracketleftBigg H(i)n/productdisplay j=i+1CR(i,j) j−i+1/bracketrightBigg , (18) where the gates Rkare in this context deﬁned as Rk≡P2π/2k=/parenleftbigg10 0e2πi/2k/parenrightbigg (19) and the superscripts denote the logical qubits the unitary is acting on. For an implementation of the QFT on a square lat-tice device in the standard gate model, SWAP gates are required to realize the controlled-phase gates. In contrast to that, in theparity architecture, these can be completely removed at thecost of requiring CNOT gates to perform the Hadamard gates according to the decomposition (

Question 1434 multiple-choice

Quantum computing fundamentally changes how high-dimensional tensors are stored and manipulated, enabling exponential compression compared to classical systems. Understanding the differences in representation, access, and computation is essential for appreciating quantum computational advantages and limitations. Which statement best characterizes the reason quantum computers can store all 2ⁿ coefficients of a normalized n-dimensional vector in only n qubits, while classical storage requires exponentially more memory? 1) Quantum states utilize error correction codes to compress tensor data efficiently. 2) Quantum entanglement allows interdependent storage of coefficients, reducing overall memory needs. 3) Quantum gates dynamically generate tensor coefficients as needed, eliminating the need for fixed storage. 4) Quantum registers employ parallel classical bits to simulate high-dimensional vectors. 5) Superposition in quantum computing restricts basis states to a small subset, minimizing storage. 6) Each quantum measurement reveals all coefficients simultaneously, so storage of individual amplitudes is unnecessary. 7) Quantum superposition enables a normalized n-qubit state to represent all 2ⁿ basis states with complex amplitudes in a single, compact vector, allowing exponential compression compared to classical storage.

✓ Correct Answer:

The correct answer is 7) Quantum superposition enables a normalized n-qubit state to represent all 2ⁿ basis states with complex amplitudes in a single, compact vector, allowing exponential compression compared to classical storage..

📚 Reference Text:

is higher tha nO(n2) even if no explicit sum is performed on the linear combination of rank-1 tensors, which requires a trem endous amount of storage for even a modest sized n, e.g.,n= 100. 5.2 Qubits and quantum eﬃciency On a quantum computer, rank increase is not an issue. A normal ized vector of the corresponding tensor xnis kept as a quantum state, which can be viewed as a linear combin ation of a set of tensor product basis states, i.e., xn=2/summationdisplay j1,...,jn=1αj1j2...jn·ej1⊗ej2⊗···⊗ejn. (24) The 2ncoeﬃcients αj1j2...jnof the expansion can be encoded and updated in nquantum bits known as qubits. The basis vectors ej1⊗ej2⊗···⊗ejnare the canonical basis of C2n. Forn= 1, a normalized vector x1∈C2can simply be written as x1=αe1+βe2withαandβsatisfying |α|2+|β|2= 1. It follows from the superposition principle of quantum mechanics that a single qubitcan keep both the αandβcoeﬃcients simultaneously. For n>1, the 2ncoeﬃcients satisfy the normalization condition/summationtext2 j1,...,jn=1|αj1j2...jn|2= 1 and they can be stored in the state of just nqubits. Observe that an n-qubit state (24) can be in a complete superposition, i.e., h aving all coeﬃcients αj1j2...jndiﬀerent from zero, and still allow a data sparse representa tion up to some ﬁnite precision inO(n) space in memory of a classical computer if it is a rank-1 tens or (20). An additional feature of a quantum computer is that it can sto re a tensor xnwhich has tensor rank r>1 in onlynqubits, such a state is also known as an entangled state in qua ntum mechanics. The minimal representation of an entangled state, xn=r/summationdisplay i=1αi·x(i) 1⊗x(i) 2⊗···⊗x(i) n, (25) wherex(i) k∈C2,/⌊ard⌊lx(i) k/⌊ard⌊l2= 1 fori= 1,2,...,r,k= 1,2,...,n,/summationtextr i=1|αi|2= 1, andris an integer rank, is a canonical polyadic decomposition of the n-way tensor xn, see Deﬁnition 3. Notice that if (25) is not a minimal 9 representation for xn, then the actual tensor rank can be strictly smaller than r. Storing a tensor (25) up to ﬁnite precision requires O(nr) classical memory. For n= 2, every tensor x2can be represented as a sum of two terms, x2=α1·x(1) 1⊗x(1) 2+α2·x(2) 1⊗x(2) 2, (26) which is a singular value decomposition of a 2 ×2 matrix. How a quantum computer does this is beyond the scope of this pa per, and not relevant for our discussion. It is also worth pointing out that, although a quantum computer ca n store a quantum state (25), it cannot be accessed in the same way a register or memory is accessed on a classical computer. We will discuss the implication of this feature in terms of how QFT can be used later. The multiplication of a matrix operator with a vector xnof tensor rank r(25) can be carried out by ﬁrst multiplying the matrix with the tensor factors x(i) 1⊗x(i) 2⊗···⊗x(i) n,i= 1,...,r. When the matrix operation can be written in terms of Kronecker products of 2 ×2 matrices, the multiplication does not require a vector to b e explicitly formed. All we have to do is to multiply each 2 ×2 matrix in a Kronecker product with

Question 1435 multiple-choice

Representation theory of finite groups is a foundational tool in mathematics and quantum computing, providing a framework to analyze symmetries and construct efficient algorithms. Quantum Fourier transform (QFT) circuits rely on this theory, especially for exploiting group structures in computational tasks. Which of the following statements is true regarding the relationship between the irreducible representations of a finite group G and its conjugacy classes? 1) The number of irreducible representations always exceeds the number of conjugacy classes. 2) Each conjugacy class corresponds to a unique matrix element in a representation. 3) Only Abelian groups have as many irreducible representations as conjugacy classes. 4) The dimensions of all irreducible representations are equal for any group. 5) The total number of irreducible representations is determined by the group's order. 6) Irreducible representations and conjugacy classes are unrelated concepts in group theory. 7) The number of distinct irreducible representations of a finite group (up to equivalence) is equal to the number of its conjugacy classes.

✓ Correct Answer:

The correct answer is 7) The number of distinct irreducible representations of a finite group (up to equivalence) is equal to the number of its conjugacy classes..

📚 Reference Text:

leyandWeylgroups. The paper is structured as follows. Sections 2 and 3 brieﬂy su mmarize the representationtheory of ﬁnite groups, the Bratteli diagram, and adapted bases. We give our algorit hms in Section 4 along with a list of group families for whichtheprovideefﬁcientcircuitsforthe QFT.We conclude withopenproblemsinSection5. 2 Representation theory background Fourier analysis over a group Ginvolvesexpressing arbitrary functions f:G→Cas linear combinationsof speciﬁc functionson Gwhichreﬂect the group’sstructureandsymmetries. If Gis Abelian,these are preciselythe characters ofG(thehomomorphismsof GintoC). Forageneralgroup,theyarethe irreduciblematrixelements ,andtheFourier transformisthechangeofbasisfromthebasisofdeltafunct ionsto thebasisofirreduciblematrixelements. In order to be precise we need the language of (ﬁnite) group re presentation theory (see, e.g., Serre [24] for an excellentintroduction). A representation ρofaﬁnitegroup Gisahomomorphism ρ:G→U(V),whereVisa(ﬁnite) dρ-dimensionalvector space over Cwith an inner product and U (V)denotesthe groupof unitary linear operatorson V. Fixing an orthonormalbasis for V, eachρ(g)may be realized as a dρ×dρunitary matrix. When a basis has been selectedinthiswayfor V,wereferto ρasamatrixrepresentation ofG;theneachofthe d2 ρfunctions ρij(g)=[ρ(g)]ij iscalleda matrixelement (correspondingto ρ). Asρisahomomorphism,forany g,h∈G,ρ(gh)=ρ(g)ρ(h),implying thatin general, ρij(gh)=∑dρ k=1ρik(g)ρkj(h). A matrix representation ρofGonVisirreducible if no subspace (other than the trivial {0}subspace and V) is mapped into itself. This is equivalent to the statement that there is no change of basis that ﬁnds a simultaneously block diagonalization (of given shape) of all ρ(g). Otherwise the representation is said to be reducible . The irre- ducible representations will play a role in the theory analo gous to that of the characters of an Abelian group. Two 2 representations ρandσareequivalent iftheydifferonlybyachangeofbasis,sothatforsomeﬁxedu nitarymatrix U, σ(g)=U−1σ(g)U,forallg∈G. Uptoequivalence,aﬁnitegroup Ghasaﬁnitenumberofirreduciblerepresentations equal to the number of its conjugacy classes. For a group G, we letˆGdenote a collection of representations of G containingexactlyonefromeachisomorphismclassofirred uciblerepresentations. Selecting bases Bfor the representationsof ˆGresults in a set of (inequivalentirreducible)matrix repre sentations; when we wish to be explicit about this selection of bases, we d enote such a collection ˆGB. The matrix elements of thematrixrepresentations ρ∈ˆGBinfactformanorthonormalbasisforthe |G|-dimensionalvectorspaceofcomplex- valued functionson G. This implies the importantrelationshipbetween the dimen sionsof the irreduciblerepresenta- tionsofGand|G|:∑ρ∈ˆGd2 ρ=|G|.Sucha familygivesrise toa generaldeﬁnitionofFouriertra nsform. Deﬁnition1 Let f:G→C; letρ:G→U(V)be a matrix representation of G. The Fourier transform of fatρ, denotedˆf(ρ), isthematrix ˆf(ρ)=/radicalBigg dρ |G|∑ g∈Gf(g)ρ(g). We typicallyrestrict ourattentionto ˆf(ρ), whereρisirreducible. Werefertothecollectionofmatrices /an}bracketle{tˆf(ρ)/an}bracketri}htρ∈ˆGBastheFouriertransform off. Thusfismappedinto |ˆG|matrices of varyingdimensions. Thetotal numberofentriesin these m atricesis ∑d2 ρ=|G|, bythe equationmentionedabove. The Fourier transform is linear in f; with the constants used above (/radicalbig dρ/|G|) it is in fact unitary, taking the |G| complexnumbers /an}bracketle{tf(g)/an}bracketri}htg∈Gto|G|complexnumbersorganizedintomatrices. For two complex-valued functions f1andf2on a group G, there is a natural inner product /an}bracketle{tf1,f2/an}bracketri}htgiven by 1 |G|∑gf1(g)f2(g)∗. For any pair of matrix representations ρ,σ∈ˆGB, the corresponding irreducible matrix elements areorthogonalaccordingtotheinnerproduct: let ρandσbetwoelementsof ˆG; then /angbracketleftbig [ρ(·)]ij,[σ(·)]kl/angbracketrightbig =/braceleftBigg 0 if ρ/ne}ati≀nslash∼=σ 1 dρδikδjlifρ=σ.(1) Computation of the Fourier transform (with respect to a give n choice of ˆG) is equivalent to the change of basis fromthatofthepointmassestotheirreduciblematrixeleme ntsdeterminedby ˆG. Thislinearmap(ofthevectorspace offunctionson G) isinvertible,with(point-wise)inversegivenbythe Fourierinversionformula : f(s)=∑ ρ∈ˆG/radicalBigg dρ |G|tr/parenleftBig ρ(s)ˆf(ρ)−1/parenrightBig . A reducible matrix representation ρ:G→U(V)may always be decomposed into irreducible representations ; speciﬁcally,thereisabasisof Vinwhicheach ρ(g)isblockdiagonalwherethe ithblockof ρ(g)isprecisely σi(g)for someirreduciblematrixrepresentation σi. Inthiscasewewrite ρ=/circleplustextσi. Thenumberoftimesagiven σ∈ˆGappears in this decompositionis the multiplicity ofσinρ. If the irreducible representation σiappears with multiplicity wiin decompositionof

Question 1436 multiple-choice

In quantum control theory, understanding how complex systems evolve under time-dependent Hamiltonians is crucial for manipulating quantum states. Efficient algorithms are necessary for analyzing large quantum systems due to the exponential growth in computational complexity with system size. Which approach enables exponential improvements in computational efficiency for calculating pathway class amplitudes in large quantum systems? 1) Direct evaluation of all terms in the Dyson series 2) Ignoring interference effects between pathway amplitudes 3) Grouping pathway classes based solely on energy eigenvalues 4) Utilizing only time-independent Hamiltonians 5) Employing algorithms based on graph theory and algebraic topology 6) Treating each pathway as completely independent of others 7) Restricting calculations to two-level quantum systems

✓ Correct Answer:

The correct answer is 5) Employing algorithms based on graph theory and algebraic topology.

📚 Reference Text:

Title: Efficient Hamiltonian encoding algorithms for extracting quantum control mechanism as interfering pathway amplitudes in the Dyson series Year: 2024 Paper ID: c2d227d3faa8617404ee10123898d1c6d6334be7 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c2d227d3faa8617404ee10123898d1c6d6334be7 Abstract: Hamiltonian encoding is a methodology for revealing the mechanism behind the dynamics governing controlled quantum systems. In this paper, following Mitra and Rabitz \cite{abhra_1}, we define mechanism via pathways of eigenstates that describe the evolution of the system, where each pathway is associated with a complex-valued amplitude corresponding to a term in the Dyson series. The evolution of the system is determined by the constructive and destructive interference of these pathway amplitudes. Pathways with similar attributes can be grouped together into pathway classes. The amplitudes of pathway classes are computed by modulating the Hamiltonian matrix elements and decoding the subsequent evolution of the system rather than by direct computation of the individual terms in the Dyson series. The original implementation of Hamiltonian encoding was computationally intensive and became prohibitively expensive in large quantum systems. This paper presents two new encoding algorithms that calculate the amplitudes of pathway classes by using techniques from graph theory and algebraic topology to exploit patterns in the set of allowed transitions, greatly reducing the number of matrix elements that need to be modulated. These new algorithms provide an exponential decrease in both computation time and memory utilization with respect to the Hilbert space dimension of the system. To demonstrate the use of these techniques, they are applied to two illustrative state-to-state transition problems.

Question 1437 multiple-choice

Fusion categories are algebraic structures important in quantum algebra, topological quantum field theory, and mathematical physics, often constructed using representation theory of quantum groups. Modern approaches seek to analyze fusion categories through computationally accessible methods emphasizing combinatorial, geometric, and numerical perspectives. Which approach primarily broadens accessibility to the study of fusion categories for researchers without advanced knowledge of Lie theory and representation theory? 1) Focusing on combinatorial, geometric, and numerical methods 2) Requiring mastery of classical Lie algebra classification 3) Emphasizing the cohomology of Lie groups 4) Restricting analysis to topological field theory frameworks 5) Utilizing advanced representation theoretic constructions exclusively 6) Limiting study to categorical dualities in pure algebraic contexts 7) Demanding familiarity with knot invariants and quantum topology

✓ Correct Answer:

The correct answer is 1) Focusing on combinatorial, geometric, and numerical methods.

📚 Reference Text:

Title: Lie theory for fusion categories: A research primer Year: 2018 Paper ID: 97cb52c924b101490069f5ef45eba556a6927a9f Source: semantic-scholar URL: https://www.semanticscholar.org/paper/97cb52c924b101490069f5ef45eba556a6927a9f Abstract: A diverse collection of fusion categories may be realized by the representation theory of quantum groups. There is substantial literature where one will find detailed constructions of quantum groups, and proofs of the representation-theoretic properties these algebras possess. Here we will forego technical intricacy as a growing number of researchers study fusion categories disjoint from Lie theory, representation theory, and a laundry list of other obstacles to understanding the mostly combinatorial, geometric, and numerical descriptions of the examples of fusion categories arising from quantum groups. This expository piece aims to create a self-contained guide for researchers to study from a computational standpoint with only the prerequisite knowledge of fusion categories.

Question 1438 multiple-choice

In quantum field theory, studying the behavior of Yang–Mills theories on compact surfaces at large N provides deep insights into both physics and mathematics. Representation theory plays a crucial role in understanding how group symmetries manifest in the calculation of partition functions for gauge theories. Which property of 'almost flat' Young diagrams is essential for controlling the asymptotic behavior of representation dimensions and Casimir eigenvalues in large N limits of U and SU gauge theories? 1) They correspond to maximal irreducible representations with highest possible dimension. 2) They allow for non-trivial cohomology classes on non-orientable surfaces. 3) They are characterized by a fixed number of boxes independent of N. 4) Their row and column lengths scale slowly with N, keeping the diagrams nearly rectangular and ensuring dominant contributions. 5) They represent the trivial representation in the decomposition of the partition function. 6) They are invariant under the action of the Weyl group for all N. 7) They encode the topological genus of the underlying surface.

✓ Correct Answer:

The correct answer is 4) Their row and column lengths scale slowly with N, keeping the diagrams nearly rectangular and ensuring dominant contributions..

📚 Reference Text:

Title: Large N behaviour of the two-dimensional Yang-Mills partition function Year: 2019 Paper ID: 4e405787ff379678cdf7fb846c034710896629de Source: semantic-scholar URL: https://www.semanticscholar.org/paper/4e405787ff379678cdf7fb846c034710896629de Abstract: We compute the large N limit of the partition function of the Euclidean Yang–Mills measure on orientable compact surfaces with genus $g\geqslant 1$ and non-orientable compact surfaces with genus $g\geqslant 2$ , with structure group the unitary group ${\mathrm U}(N)$ or special unitary group ${\mathrm{SU}}(N)$ . Our proofs are based on asymptotic representation theory: more specifically, we control the dimension and Casimir number of irreducible representations of ${\mathrm U}(N)$ and ${\mathrm{SU}}(N)$ when N tends to infinity. Our main technical tool, involving ‘almost flat’ Young diagram, makes rigorous the arguments used by Gross and Taylor (1993, Nuclear Phys. B400(1–3) 181–208) in the setting of QCD, and in some cases, we recover formulae given by Douglas (1995, Quantum Field Theory and String Theory (Cargèse, 1993), Vol. 328 of NATO Advanced Science Institutes Series B: Physics, Plenum, New York, pp. 119–135) and Rusakov (1993, Phys. Lett. B303(1) 95–98).

Question 1439 multiple-choice

Quantum error correction codes are crucial for maintaining the integrity of quantum information, especially when physical systems exhibit symmetries such as charge conservation. Some codes, like secret-sharing codes and those using rotors or Dicke states, are specifically designed to respect these symmetries while providing robust error correction. Which property uniquely distinguishes the three-rotor secret-sharing quantum code from codes based solely on qubits or qutrits? 1) It cannot correct the loss of any subsystem in the code. 2) It encodes information using U(1)-covariant states, ensuring conservation of total charge. 3) It exclusively uses Dicke states for logical encoding. 4) It provides exact error correction only for five physical subsystems. 5) It is based on discrete symmetry groups rather than continuous ones. 6) It does not require normalization for its code words. 7) It is unable to generalize to higher numbers of physical subsystems.

✓ Correct Answer:

The correct answer is 2) It encodes information using U(1)-covariant states, ensuring conservation of total charge..

📚 Reference Text:

regimes of eitherlarge subsystem dimensions or a large number of physicalsubsystems (Table I). A. Three-rotor secret-sharing code In this subsection, we apply our criterion to a truncated version of a code introduced by Hayden et al. [12], linking that code to the well-known three-qutrit secret-sharingquantum polynomial code [44,45] . While illustratinghow to use our criterion, it also provides a covariant code which performs well in the limit of code words covering a large range of physical charge on the subsystems. 1. Rotor version of the qutrit secret-sharing code For our purposes, a quantum rotor [also, an Oð2Þor planar quantum rotor] is simply a system with a basis fjxig that is labeled by an integer x∈Zindexing representations ofUð1Þ[46]. Consider the three-rotor code given in Ref. [12] defined by the map from LtoA¼A 1⊗A2⊗ A3given as VL→A∶jxiL→X y∈Zj−3y; y−x;2ðyþxÞiA;ð35Þ where the states fjxigare eigenstates of the angular momentum operators TLandTA¼T1þT2þT3. This code can correct against the loss of any of the threesubsystems [12]. Moreover, the code is covariant with respect to the charge T: A logical state jxi Lis mapped onto a code word with the same total charge x. Interestingly, this code is a natural rotor generalization of the three-qutrit secret-sharing code [44,45] . The three-qutrit code maps the basis vectors jjiL(j¼0, 1, 2) of a logical qutrit into the code wordsP kjk; k−j; kþjiwhere the addition is modulo 3. Now, substitute each qutrit subsystem with a rotor. We obtain a code defined by the followingencoding map: ˜V L→A∶jxiL→X y∈Zjy; y−x; yþxi: ð36Þ This code is not yet covariant with respect to the charge states jxi, as the charge of the code word corresponding to jxiLis not x. However, we may apply the isometry mapping jð:Þi→j−3ð:Þion the first rotor and jð:Þi→j2ð:Þion the second, yielding the encoding map (35). [In fact, the code (36)is covariant with respect to a different physical charge generator, T0 A¼−3T1þT2þ2T3, whereas the code (35)TABLE I. Summary of the codes considered in Sec. VI: the three-rotor secret-sharing code, the five-rotor perfect code, and an n-qubit “thermodynamic code ”with code words consisting of Dicke states (and a,bchosen appropriately). Covariance Dimensions Error correction ½½3;1;2/C138/C138Z Sharp cutoff Uð1Þ Finite Approximate Smooth cutoff Uð1Þ Infinite Approximate ½½5;1;3/C138/C138Z Qudit version ZD Finite Exact Smooth cutoff Uð1Þ Infinite Approximate ½½n; alogn; blogn/C138/C138 Finite nU ð1Þ Finite ApproximatePHILIPPE FAIST et al. PHYS. REV . X 10,041018 (2020) 041018-12 is covariant with respect to the natural physical charge carried by three rotors, TA¼T1þT2þT3.] In this sense, the code (35) is a natural Uð1Þ-covariant generalization of the qutrit secret-sharing code. In the following sections, we address the problem that the code words in Eq. (35) are not normalizable, by building suitable wave packet states. We normalize the code words in two different ways: The sharp cutoff selects arange of charges to use for each rotor and discards the rest,while the smooth cutoff imposes a Gaussian envelope on each rotor, thereby keeping all the states but making them less prominent as the charge increases [47](see also related recent

Question 1440 multiple-choice

Quantum algorithms for period finding often combine quantum subroutines with classical post-processing to deduce hidden structures efficiently. Variations in the classical component can lead to significant changes in the overall computational efficiency of these algorithms. Which classical method, when used for period extraction after quantum measurement, can result in faster post-processing compared to the continued fraction approach in Shor's algorithm? 1) The classical Euclidean algorithm 2) The extended LLL lattice reduction algorithm 3) The fast Fourier transform 4) The modular exponentiation technique 5) The discrete logarithm algorithm 6) The quantum phase estimation procedure 7) The Grover search algorithm

✓ Correct Answer:

The correct answer is 1) The classical Euclidean algorithm.

📚 Reference Text:

U’: j xij yi7 ! j xij y+’(x)mod1i Thequantum partofoural gori thm consi stsof Step 0throughStep 4asdescri bedbel ow: Step 0Ini ti al i ze j 0i=j 0ij 0i2 HZ HC Step 1Appl ythei nverseFouri ertransform tothel eftregi ster,i . e.appl yF1 1toobtai n j 1i=I dx e2ix0j xij 0i=I dxj xij 0i2 HR=Z HC Step 2Appl yU’: j xij ui7 ! j xij u+’(x)mod1itoobtai n j 2i=I dxj xij ’(x)i Step 3Appl ytheFouri ertransform tothel eftregi ster,i . e.appl y F 1toobtai n j 3i=X n2ZI dx e2inxj nij ’(x)i=X n2Zj niI dx e2inxj ’(x)i2 HZ HC Thi scanbeshowntoreduceto j 3i=X ‘2Zj ‘aij (‘a)i where j (‘a)i=1=aZ 0dx e2inxj ’(x)i. Step 4M easurej 3i=X ‘2Zj ‘aij (‘a)iwi threspecttotheobservabl e O=X n2Znj nihnj toproducearandom ei genval ue ‘a. Step 5Iftheabovestepsareruntwi cetoproduceei genval ues ‘1aand‘2a,thentheprobabi l i tythatthei ntegers ‘1and‘2arerel ati vel ypri mei s (2)10: 6079,where (2)denotetheRi emannzetafuncti onat k= 2. Hence,wi thhi ghprobabi l i ty,theEucl i deanal gori thm wi l l producethedesi redanswer a. Rem ark. Pl easenotethat,unl ikeShor’sal gorithm whichusesthecl a ssicalcontinuedfraction al gorithm in its l aststep to determinetheperiod,thisal gorithm usesin its l aststep onl y themuch fastercl assicalEucl idean al gorithm to ndtheperiod1 a. 5.A CURIOUS ALG ORITH M IC DUAL OF SH OR’S ALG ORITH M Letusnow constructacl assofapproxi mati ngQHSal gori thms over ni tegroupswhi chasymptoti cal l yapproach i n thel i mi ttheQHS al gori thm AlgR=Z. To doso,foreach posi ti vei nteger Q,weapproxi matethei n ni te groupsZandR=Zrespecti vel ybythe ni tegroups 8 >< >:ZZQ=fk2ZmodQ:0k< Qg R=ZZQ=n r Q: r= 0; 1;:::;Q1o andweapproxi matethemap ’: Z! Cbytheobvi ousmap e’: ZQ! C. Theresul ti ngal gori thm AlgZQ \l i ves" i ntheHi l bertspace HZQ HCandusestheapproxi mati ngobservabl e OQ=Q1X n=0nj nihnj Theresul tforeach Qi saQHS al gori thm AlgZQ whi ch i stheal gori thmi cdualofShor’ sal gori thm. Because oftheremarkfoundattheendoftheprevi oussecti on,i tappe arstorunmuchfaster. 6.A QH S ALG ORITH M ALG PATH SON TH E SPACE PATH S Thereadershoul dbecauti onedthatthefol l owi ngal gori thm i shi ghl yspecul ati ve.Thi sal gori thm,i nthespi ri t ofFeynman,i sbasedonfuncti onali ntegral swhoseexi stenc ei sdi cul ttodetermi ne,l etal oneapproxi mate. LetPathsbethespaceofal lpaths x:[ 0; 1] ! Rni nreal n-spaceRnwhi chareL2wi threspecttothei nner product xy=1Z 0dsx(s)y(s) W emakePathsi ntoavectorspaceoverthereal s Rbyde ni ngscal araddi ti onandvectoraddi ti onrespecti vel y as 8 < :(x)(s) =x(s) (x+y)(s) =x(s)+y(s) W ewi shtosol vethefol l owi ngprobl em: Problem . Given afunctional ’: Paths! Rnandahidden subspace VofPathssuchthat ’(x+v)=’(x)8v2V, createaQHS al gorithm that ndsthehidden subspace V. LetHPathsbetheri ggedHi l bertspacewi thorthonormalbasi s fj xi: x2Pathsg wherewehavede nedthebracketproductas hxj yi=(xy) Keepi ngi nmi ndthatthespacePathscanbewri ttenasthedi sj oi ntuni on Paths=[ v2V v+V? , whereV?denotestheorthogonalcompl ementof Vi nPaths,weproceedwi ththefol l owi ngQHSal gori thm: Step 0Ini ti al i ze j 0i=j 0ij 0i2 HPaths HRn Step 1Appl ythei nverseFouri ertransform tothel eftregi ster,i . e.appl yF1 1toobtai n j 1i=Z PathsDx e2ix0j xij 0i=Z PathsDxj xij 0i Step 2Appl yU’: j

Question 1441 multiple-choice

In quantum computing, the hidden subgroup problem (HSP) is central to various algorithms that exploit group structures, and measurement strategies like the pretty good measurement (PGM) can be crucial for efficient state discrimination. Advances in algebraic techniques and quantum entanglement have informed algorithm design for both abelian and nonabelian group cases. Which group structure allows the hidden subgroup problem to be solved in poly(log p) time using quantum algorithms, provided that the parameter r is fixed? 1) The symmetric group $ S_n $ 2) The dihedral group $ D_n $ 3) The group $ \mathbb{Z}_r^p \rtimes \mathbb{Z}_p $ 4) The alternating group $ A_n $ 5) The cyclic group $ \mathbb{Z}_p $ 6) The group $ \mathbb{Z}_n^p \rtimes \mathbb{Z}_2 $ 7) The quaternion group $ Q_8 $

✓ Correct Answer:

The correct answer is 3) The group $ \mathbb{Z}_r^p \rtimes \mathbb{Z}_p $.

📚 Reference Text:

our purposes, a Gr¨ obne r basis can be computed eﬃciently. Since we consider only the cases in which there are at most 2 so lutions, elimination is also eﬃcient, giving an overall poly(log p) time algorithm for computing a list of matrix sum solutions , and hence for quantum sampling from those solutions. Collecting these results, we ﬁnd Theorem 6. The hidden subgroup problem over any group of the form Zr p⋊Zpwithrﬁxed can be solved in time poly(logp)on a quantum computer. 7 Discussion In this paper, we have studied the pretty good measurement fo r semidirect product groups of the formA⋊ZpwithAabelian and pprime. We found that the pgmis closely connected to the matrix sum problem, and we exploited this connection to ﬁnd eﬃcient quantum algorithms for certain metacyclic groups (Section 5) as well as all groups of the for mZr p⋊Zpwithrﬁxed (Section 6). The latter algorithm demonstrates that entangled measurement s may be useful for eﬃciently solving the nonabelian hsp. Aside from the fact that these particular nonabelian groups admit eﬃcient quantum algorithms, our results suggest two general directions for further inve stigation. First, in the standard approach to the nonabelian hsp, it would be helpful to have a better understanding of when en tangled measurementsarenecessaryandwhentheycanbeimplemented togiveeﬃcientalgorithms. Second, for thehspor for other problems that can be viewed as quantum state dist inguishability problems, identifying an optimal measurement (or considering a parti cularly nice measurement such as the pgm, which in general may or may not be optimal) can be used as a pri nciple for discovering new quantum algorithms. While the reduction of Lemma 1 combined with the pgmapproach outlined in Section 4 appears to eﬃciently solve the hspin most of the semidirect product groups where eﬃcient algor ithms are known, thereisonenotableexception. Thegroups Zn p⋊Z2(forwhichaneﬃcient quantumalgorithm is given in[10]) give risetoasubsetsumproblemover Zn p, which appearstobeessentially as diﬃcult as the subset sum problem over ZNarising from the dihedral group. Thus, it would be interesti ng to understand what allows the algorithm in [10] to be eﬃcient even though the matrix sum problem is (apparently) hard. Of course, there are many nonabelian groups that are semidir ect products of nonabelian groups, orthatcannotbenontrivially decomposedintosemidirectp roductsatall. The pgmapproachiswell 14 deﬁned for any group, so it would be interesting to explore th e approach in such cases, regardless of whether the pgmis optimal. Acknowledgments We thank Carlos Mochon and Frank Verstrate for helpful discu ssions of Theorem 3. AMC received support from the National Science Fo undation under Grant No. EIA- 0086038. References [1] N. Alon and J. H. Spencer. The Probabilistic Method . Wiley Interscience, New York, 2nd edition, 2000. [2] D. Bacon, A. M. Childs, and W. van Dam. Optimal measuremen ts for the dihedral hidden subgroup problem. arXiv:quant-ph/0501044. [3] R. Beals. Quantum computation of Fourier transforms ove r symmetric groups. In Proc. 29th Annual ACM Symposium on Theory of Computing , pages 48–53, 1997. [4] D. Boneh and R. J. Lipton. Quantum cryptanalysis of hidde n linear functions. In Lecture Notes

Question 1442 multiple-choice

The Schur transform is a quantum algorithmic tool used to exploit symmetries in multipartite quantum systems, enabling efficient decomposition and manipulation of quantum states. Its implementation and mathematical foundation are closely linked to the representation theory of symmetric and unitary groups. Which improvement in the new algorithm for the Schur transform most directly enables efficient handling of quantum systems with extremely large local dimensions? 1) Achieving polynomial scaling in the number of qubits, n 2) Achieving polynomial scaling in the logarithm of the local dimension, log d 3) Incorporating entanglement concentration mechanisms 4) Extending the transform to decoherence-free subspaces 5) Utilizing quantum Fourier transforms over the unitary group 6) Employing precision scaling only in terms of epsilon 7) Relying exclusively on universal compression strategies

✓ Correct Answer:

The correct answer is 2) Achieving polynomial scaling in the logarithm of the local dimension, log d.

📚 Reference Text:

An eﬃcient high dimensional quantum Schur transform Hari Krovi Quantum Engineering and Computing Physical Sciences and Systems Raytheon BBN Technologies, Cambridge, MA February 5, 2019 The Schur transform is a unitary operator that block diagonalizes the action of the symmetric and unitary groups on an nfold tensor product V⊗nof a vector space Vof dimension d. Bacon, Chuang and Harrow [5] gave a quantum algorithm for this transform that is polynomial in n,dand log/epsilon1−1, where/epsilon1is the precision. In a footnote in Harrow's thesis [18], a brief description of how to make the algorithm of [5] polynomial in logdis given using the unitary group representation theory (however, this has not been explained in detail anywhere1). In this article, we present a quantum algorithm for the Schur transform that is polynomial in n,logdandlog/epsilon1−1using a di erent approach. Speci cally, we build this transform using the representation theory of the symmetric group and in this sense our technique can be considered a \dual" algorithm to [5]. A novel feature of our algorithm is that we construct the quantum Fourier transform over the so called permutation modules , which could have other applications. 1 Introduction Schur-Weyl duality is a remarkable correspondence between the irreducible representations of the symmetric group and those of the unitary group acting on an nfold tensor product of a vector space V. This cor- respondence allows one to construct all of the so-called polynomial representations of the unitary, general linear and special linear groups. Polynomial representations of matrix groups such as unitary groups are representations whose matrix entries can be written as polynomials in the entries of the group element i.e., ρ(U)is a polynomial representation if the entries of ρ(U)are polynomial in the entries of U. Schur-Weyl duality has been generalized to many other groups and algebras including quantum groups [10, 17]. Schur-Weyl duality has numerous applications in quantum information theory. It has been used to prove that the tensor product of many copies of a density operator is close to a projector. In fact, the projector is the one corresponding to a partition in Schur-Weyl duality that is closest to the spectrum of ρ[2, 26, 21, 9]. It has also been used to prove de Finetti theorems [30], which have many applications in security proofs of quantum key distribution systems. The Schur transform was rst constructed for qubits in the work of Bacon, Chuang and Harrow in [4]. This has been extended to qudits by the same authors in [5]. A quantum circuit for the Schur transform also has numerous applications in quantum information theory. It has been applied to universal distortion-free entanglement concentration [32], universal compression [21, 22], encoding and decoding into decoherence-free subspaces [42, 29, 24, 3]. These applications and others are discussed in more detail in Harrow's thesis [18]. Recently, the Schur transform has been used as a primitive in an ecient algorithm for spectrum testing of a density operator [35] and in algorithms for sample optimal state tomography [34, 16] improving on previous algorithms [39, 14,

Question 1443 multiple-choice

The hidden subgroup problem (HSP) is a fundamental question in computational group theory and quantum computing, with applications ranging from cryptography to algorithm design. Classical algorithms for HSP are evaluated by their query complexity, which measures the number of group elements an algorithm must query to solve the problem. Which of the following best describes a recent theoretical advancement regarding deterministic query complexity for the decision version of the hidden subgroup problem in Abelian groups? 1) Proven existence of deterministic algorithms with O(√|G||H|) query complexity for the decision version 2) Classical query complexity for Abelian HSP remains at O(|G|) for all instances 3) Quantum algorithms outperform classical algorithms for Abelian HSP in terms of query complexity 4) Deterministic algorithms for Abelian HSP require at least Ω(|G|/|H|) queries 5) Only randomized algorithms achieve sublinear query complexity for Abelian HSP 6) Decision and identification versions of Abelian HSP have identical query complexity bounds 7) No deterministic algorithms have been found with query complexity below O(|G|) for Abelian HSP

✓ Correct Answer:

The correct answer is 1) Proven existence of deterministic algorithms with O(√|G||H|) query complexity for the decision version.

📚 Reference Text:

hidden subgroup problem are not known to have eﬃcient quantum algorithm s, such as the dihedral hidden subgroup problem and the symmetric hidden s ubgroup problem [22, 9]. In this paper, we focus on deterministic algorithms for the hidden su bgroup problem. Deterministic and randomized algorithms are called classical algo- rithms. A deterministic algorithm for the above problem usually compu tesf(g) 2 for enough amount of g∈Gto ﬁndH(or check whether His trivial or not). The number of g∈Gcomputed for fis called the query complexity of the algorithm. For example, a naive algorithm with query complexity O(|G|) is to compute f(g) for all g∈G. A nontrivial problem worthy of consideration is: how diﬃcult is this problem in terms of query complexity and how to ﬁn d an optimal query algorithm for it. There exist a lot of discussions abo ut the classical query complexity for some instances of the hidden subgro up problem. For example, the classical query complexity of Simon’s problem was pr oven to be Θ(√ 2n) [29, 4, 8]. Ye et al. [30] proved that a nearly optimal bound for the classical query complexity of generalized Simon’s problem ( GSP). For the order-ﬁnding problem over Z2m×Z2n, Cleve [7] proved that the deterministic query complexity is at least Ω/parenleftBig/radicalBig 2n m/parenrightBig , and the randomized query complexity is at least Ω/parenleftBig 2n/3 √m/parenrightBig . Kuperberg [22] proved the classical query complexity of the dihedral hidden subgroup problem over the dihedral group Dnis Ω(√ N). Childs [5] showed that a classical algorithm must make Ω(√ N) queries if there areNcandidate subgroups whose only common element is the identity eleme nt. Recently, Nayak [25] considered deterministic algorithms for the h idden sub- group problem. He gave a deterministic algorithm with O(/radicalbig |G|) queries to identifyHfor the Abelian hidden subgroup problem, and showed there exists a deterministic algorithm with query complexity O(/radicalbig |G|ln|G|) for the general hidden subgroup problem. Finally, he proposed an open problem: is th ere a deterministic algorithm to the hidden subgroup problem with query co mplexity O(/radicalBig |G| |H|)? 1.2. Our contributions In this paper, we will answer the above open problem. In the following , the hidden subgroup problem will be abbreviated to HSP. The one in Deﬁnition 1 will be called the identiﬁcation versionof HSP. We will also consider its decision version where the goal is to decide whether the hidden subgroup His trivial or not, instead of ﬁnding H. Our main results are the following theorems and corollaries, which not only answer the open problem proposed by Nayak [25], but also extend th e main re- sults of Ref. [30]. Theorem 1 tells that there exist instances of HSPwith query complexity far smaller than O(/radicalBig |G| |H|). Theorem 2 shows that there exist deter- ministic algorithms with query complexity O(/radicalBig |G| |H|) for the decision version of the Abelian hidden subgroup problem and deterministic algorithms with query complexity not far from O(/radicalBig |G| |H|) for the identiﬁcation version. Theorem 3 and 3 its corollary indicate

Question 1444 multiple-choice

In quantum algorithms for the hidden subgroup problem (HSP), representation theory and measurement statistics play a key role in identifying subgroups of finite groups. The distinguishability of hidden subgroups often depends on the properties of invariant subspaces and the quantum measurement distributions they produce. Which quantity provides a lower bound on the trace distance between probability distributions obtained from strong Fourier sampling on quantum states corresponding to two different subgroups H₁ and H₂ in random representation bases? 1) The dimension of the ambient representation space 2) The cardinality of the intersection H₁ ∩ H₂ 3) r(G; H₁, H₂) 4) The sum of ranks of projections onto VρH₁ and VρH₂ 5) The dimension of the subgroup generated by H₁ and H₂ 6) The overlap δ between orthogonal complements of invariant subspaces 7) The minimum trace norm over all irreducible representations

✓ Correct Answer:

The correct answer is 3) r(G; H₁, H₂).

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unitaryrepresentation of G. Let Πidenote the orthogonal projection onto Vρ Hii.e.Πi=1 |Hi|/summationtext h∈Hiρ(h). LetΠ1,2denote the orthogonal projection onto Vρ H1∩Vρ H2. It is easy to check that Vρ H1∩Vρ H2=Vρ /angbracketleftH1,H2/angbracketright, where/an}bracketle{tH1,H2/an}bracketri}htdenotes the subgroup of Ggenerated by H1andH2. Thus,Π1,2=1 |/angbracketleftH1,H2/angbracketright|/summationtext h∈/angbracketleftH1,H2/angbracketrightρ(h). DeﬁneΠ′ i= Πi−Π1,2. Π′ iis the orthogonal projection onto the subspace V′ ideﬁned as the orthogonal complement of Vρ H1∩Vρ H2 inVρ Hi.V′ 1andV′ 2have trivial intersection. Deﬁne ri= rank(Π i)andr′ i= rank(Π′ i). Deﬁne ˆh= max{|H1|r1,|H2|r2},˜h=|(|H1|r1−|H2|r2)|andδ=/bardblΠ′ 1Π′ 2/bardbl. Recallthat δ= ovlap(V′ 1,V′ 2). Consider the following three cases: 1. When√ dρ log|G|= Ω((r1+r2)3/2). Loosely speaking, r1,r2areboth small. In this case, deﬁne r(G;H1,H2;ρ) = max/braceleftBiggˆh 2/parenleftBigg Ω/parenleftBigg/radicalbig r′ 1 r1+/radicalbig r′ 2 r2/parenrightBigg −2δ1/2(1−δ2)−1/4/parenrightBigg ,˜h/bracerightBigg . 2. When√ dρ log|G|= Ω(r1)andr2 r1= Ω(log2|G|). Loosely speaking, r1is small and r2is relatively large with respect to r1. In this case, deﬁne r(G;H1,H2;ρ) = max/braceleftBiggˆh 2·Ω/parenleftbigg1√r1/parenrightbigg ,˜h/bracerightBigg . 14 3. Otherwise, deﬁne r(G;H1,H2;ρ) =˜h. Deﬁnition 2( r(G;H1,H2),r(G))LetH1,H2≤G. Deﬁner(G;H1,H2) =/summationtext ρdρ |G|·r(G;H1,H2;ρ)and r(G) = min H1,H2r(G;H1,H2). From the above deﬁnition, it iseasy tosee that r(G;H1,H2)≥w(G;H1,H2). Deﬁnition 3( PG H,B)LetBbe a set of orthonormal bases for the irreducible unitary rep resentations of G. SupposeH≤G.PG,B Hdenotestheprobability distribution ontherepresentatio n namesandcolumnindices (ρ,j)got by strong Fourier sampling the state |H/an}bracketri}htaccording to B. Thesigniﬁcance of r(G;H1,H2)arises from the following theorem. Theorem 1 Withprobabilityatleast 1−exp(−Ω(log2|G|))overthechoiceofrandomrepresentationbases Bfor Fourier sampling, /bardblPG,B H1−PG,B H2/bardbltr≥r(G;H1,H2). Using this theorem, wecan apply a‘minimum-ﬁnding-like’ al gorithm toidentify thehidden subgroup. Corollary 1 Lets(G)denote the number of distinct subgroups of G. With probability at least 2/3over the choice of random bases for representations of G, Fourier sampling O/parenleftBig logs(G) r2(G)/parenrightBig times in a random basis gives enough classical information toidentify ahidd en subgroup in G. In particular, O/parenleftbigg/parenleftBig log|G| r(G)/parenrightBig2/parenrightbigg random strong Fourier samplings sufﬁce. Proof: From Theorem 1, we get that for all pairs of subgroups H1,H2≤G, with probability at least 1−exp(−Ω(log2|G|))over the choice of random bases Bfor representations of G,/bardblPG,B H1−PG,B H2/bardbltr≥ r(G). Call a set of representation bases Bgoodif/bardblPG,B H1−PG,B H2/bardbltr≥r(G)for all pairs of subgroups H1,H2≤G. Bythe union bound on probabilities, arandom choice of repr esentation bases gives a good B with probability at least 1−s(G)exp(−Ω(log2|G|)) = 1−exp(−Ω(log2|G|)). Suppose we have such a goodB. Under the promise that the hidden subgroup is either H1orH2,Brecognises which one it is with conﬁdence atleast 1/2+r(G)/4usingBayes’s rule. UsingFact5,theconﬁdence canbebooste d toatleast 1−1 4s(G)by Fourier sampling O/parenleftBig logs(G) r2(G)/parenrightBig times with B. We can now run a classical ‘minimum-ﬁnding- like’ algorithm on the measured samples, comparing two subg roupsH1,H2≤Gat a time, to discover the actual hidden subgroup HinGwith conﬁdence at least 1−s(G) 4s(G)= 3/4. The overall conﬁdence bound becomes (1−exp(−Ω(log2|G|)))·3 4≥2 3. The second bound follows from the fact that s(G)≤2log2|G|, since anygroup of size ahas at most logagenerators. TherestofthesectionisdevotedtoprovingTheorem1. Weﬁrs tprovesomenecessarytechnicallemmas. Lemma2 LetW={a1,...,ap}∪{b1,...,bq}∪{c1,...,cr}be arandom orthonormal set of p+q+r vectors in Cd. Letai jdenote thejth coordinate of vector ai; similar notations will be used for the vectors bi,citoo. Deﬁne two d-dimensional probability vectors S,Tasfollows: Sj=1 p+r/parenleftBiggp/summationdisplay i=1|ai j|2+r/summationdisplay i=1|ci j|2/parenrightBigg ,Tj=1 q+r/parenleftBiggq/summationdisplay i=1|bi j|2+r/summationdisplay i=1|ci j|2/parenrightBigg . 15 Thenthereexists δ=θ((p+q+r)−3/2)suchthat

Question 1445 multiple-choice

In models of asymmetric dark matter (ADM), the relic abundance is typically set by an initial asymmetry between dark matter (DM) and anti-dark matter (anti-DM), analogous to the baryon asymmetry in the visible sector. Cosmological observations and direct detection experiments place important constraints on the viable parameter space for ADM scenarios. Which parameter, when increased, most effectively reduces the residual abundance of anti-dark matter and shifts the boundaries of exclusion regions in ADM models, independent of the DM mass parameter? 1) The dark matter self-interaction cross-section 2) The coupling constant gx 3) The annihilation rate coefficient 4) The mass splitting between DM and anti-DM 5) The production asymmetry parameter 6) The direct detection cross-section 7) The thermal freeze-out temperature

✓ Correct Answer:

The correct answer is 5) The production asymmetry parameter.

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F rac. 12345678910 10−510−410−310−210−112345 101 0.10.95 0.1 10−510−10DD Excl.FIG. 4. Abundance of and in the m3=2plane. The right yaxis shows the mass m =Y. Solid red contours show the fraction of the measured abundance made up by and and their anti-particles. The dashed grey lines show the fractional asymmetry between DM and anti-DM. The blue region is excluded by the CMB bound and the green by direct detection. This mass ordering coincides with the spectrum described in in Sec. V A. In Fig. 4 we show the dark matter abundance adm=adm=cof and (and their antiparticles) relative to the observed abundance cdmin them3=2plane forgx= 0:1, = 104,m'=m3=2, andc= 1. Contours of adm= cdm= 0:1;1;10 are given by solid red lines. The grey dashed lines in this gure correspond to the net residual anti-DM abundance R+R, whereR = = and similarly for . Not surprisingly, larger values of the production asymmetry parameter lead to smaller residual anti-DM abundances. In this gure we also show in blue the region of parameters that is excluded by CMB observations, as well as the region excluded by direct detection in green. These constraints will be discussed in more detail below. The ADM abundance in the gxm3=2plane is shown in Fig. 5 for = 5103,= 104, m'=m3=2, andc= 1. Again, contours of adm= cdm= 0:1;1;10 are given by solid red lines. We also plot contours of the mass with dashed grey lines. As before, the shaded green region is excluded by direct detection searches. The region excluded by CMB observations in Fig. 4 (shaded blue) coincides with larger values of the residual anti-DM abundances R +R. These residual abundances provide an annihilation mode that injects energy into the cosmological plasma during the CMB era [116], as discussed in Sec. IV D. Accounting for exclusively asymmetric annihilation and 22 m3/2 [100 T e V] gxΩadm/Ωcdm forκ= 5×10−3and/epsilon1= 10−4 Ωadm/ΩcdmmΨ 12345678910 0.05 0.1 0.15 0.2 0.25 0.310 1 0.1105 1DD ExcludedFIG. 5. Abundance of and in the gxm3=2plane. Solid red contours show the fraction of the measured abundance made up by and and their anti-particles. The dashed grey lines show the mass in GeV. The green region is excluded by direct detection. the multiple DM species, the result of Eq. (33) translates into 2fX i= ; i+ i cdm2Ri (1 +Ri)2hvii mi<2:421027cm3=s GeV: (44) The CMB exclusion shown in Fig. 4 uses f= 1, but other values in the range f= 0:2 1:0 yield similar results. The boundary of the excluded region is also nearly vertical and independent of m3=2. This can be understood in terms of an approximate cancellation of factors ofm3=2=m'in the combination 2 admhvi=m, whileRiis determined primarily by . In addition to the limits from the CMB, we have also computed the bounds from indirect detection as described in Sec. IV. These searches yield exclusions very similar to that from the CMB and are omitted from Fig. 4. Direct detection searches also place a signi cant constraint on this ADM scenario.

Question 1446 multiple-choice

Quantum computing leverages superposition and entanglement to solve certain computational problems much more efficiently than classical computers. A notable example is Shor's algorithm, which enables efficient factoring of large integers, a task crucial for modern cryptography. Which computational step in Shor's algorithm is responsible for the exponential speedup in finding the period of the function f(x) = aˣ mod N, thus enabling efficient integer factorization? 1) Classical brute-force search for divisors up to √N 2) Repeated classical matrix multiplications for the discrete Fourier transform 3) Employing sub-exponential-time classical factoring algorithms 4) Testing a randomly chosen a for immediate divisibility using Euclid’s algorithm 5) Classical evaluation of f(x) = aˣ mod N for all x 6) Using Grover's algorithm to speed up unsorted database search 7) Applying the Quantum Fourier Transform to extract the period in superposition

✓ Correct Answer:

The correct answer is 7) Applying the Quantum Fourier Transform to extract the period in superposition.

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that Uacts on the ﬁrst two qubits and that Uhas matrix elements Ui1i2 j1j2in a standard product basis of the nqubit state space. Suppose that |α/an}bracketri}hthas components ai1...in(where all indices range over the values 0 and 1). The components of the updated s tate are given by matrix multiplication: anew i1...in=/summationdisplay j1,j2Uj1j2 i1i2aj1j2i3...in. (10) This update counts as one step of quantum computation (or mor e precisely a constant number, independent of nto implement U) and the FFT decomposition amounts to an implementation of FT inO(n2) steps on a quantum computer. In contrast if eq. (10) is viewe d as a classical computation, we must perform a 4 ×4 matrix multiplication 2n−2times (for all values of the 4 stringi3...in). This ultimately gives an implementation of FT with O(n2n) classical steps, which is the standard fast Fourier transform algorithm. In summary, the quantum algorithm for determining the perio dicity of a given function f, withNinputs, begins with the computation of all values of fin superposition using one application of FT and one evaluation of f. FT is then applied to pick out the periodic structure of the resulting state. The quantum implementati on of the FFT algorithm guaran- tees that FT may be implemented in poly(log N) steps. An analogous classical computation would require O(N) invocations of fto compute a column vector of all the function values and thenO(NlogN) steps to perform the FFT. Thus the quantum algorithm repres ents an exponential speedup. 3 Quantum factoring The problem of integer factorisation is the following: give n a number N, ofn= log2Ndigits, we wish to determine a number k(not equal to 1 or N) which divides Nexactly. We now outline how this problem may be reduced to a problem of period icity determination for a suitable periodic function f. Then the quantum algorithm described in the preceding sect ion will achieve the factorisation of Nin poly(n) time i.e. polynomial in the number of digits of N. We note ﬁrst that there is no known classical algorithm which will factorise any given N in a time polynomial in the number of digits of N. For example the most naive factoring algorithm involves test-dividing Nby each number from 1 to√ N(as any composite N must have a factor in this range). This requires at least√ Nsteps (at least one step for each trial factor) and√ N= 21 2nis exponential in n. In fact using all the ingenuity of modern mathematics, the fastest known classical factoring algorithm runs in a time of order exp(n1 3(logn)2 3). To reduce the problem to a problem of periodicity we will need to use some basic results from number theory. These are further described in the appen dix of [8] and complete expo- sitions may be found in most standard texts on number theory s uch as [1, 2]. We begin by selecting a number a

Question 1447 multiple-choice

Quantum information theory examines how entanglement is affected by various operations, including the Quantum Fourier Transform (QFT), which plays a central role in algorithms like Shor's. The behavior of entanglement measures under these transformations reveals important features of quantum states relevant to computation. Which of the following statements best characterizes the effect of the Quantum Fourier Transform (QFT) on the Groverian entanglement of periodic quantum states as the number of qubits increases? 1) The QFT increases the Groverian entanglement of all periodic states regardless of system size. 2) The QFT decreases the Groverian entanglement of periodic states steadily with more qubits. 3) The QFT causes random fluctuations in Groverian entanglement for periodic states as qubits increase. 4) The QFT leads to a broad distribution of Groverian entanglement changes for periodic states. 5) The QFT leaves the Groverian entanglement of periodic states essentially unchanged as the number of qubits grows. 6) The QFT eliminates Groverian entanglement in periodic states for large systems. 7) The QFT reverses the sign of Groverian entanglement in periodic states at high qubit numbers.

✓ Correct Answer:

The correct answer is 5) The QFT leaves the Groverian entanglement of periodic states essentially unchanged as the number of qubits grows..

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it can be obtained from the corresponding regular ES state by local unitary operations , making it locally equivalent to the ES state:  1 0 0e−2q−1p Q2πi ⊗ 1 0 0e−2q−2p Q2πi ⊗···⊗ 1 0 0e−p Q2πi |ψq S/an}brack⌉tri}ht=|ψq S,p/an}brack⌉tri}ht (32) Naturally, this means that both states have the same entangl ement. As seen in Eq. (7), the relative phases added to the periodic state |ψq r,l/an}brack⌉tri}htfollow precisely the same pattern, with p=l+r(A−1)/2. Therefore, they can be ignored for all entanglement consid erations. We are left with two changes the QFT induces in periodic states: rem oving the shift and changing the period. Interestingly, although the importance of the Q FT in Shor’s algorithm is in its canceling of the shift (which enables one to extract the peri od), this change is irrelevant to the state’s entanglement, as the shift does not appear at all in the approximated formula. Therefore we are left only with the change of period. Looking again at the approximated formula for the entanglement of periodic states given in Eq. (28), we see that the values of the two branches, 1/randr/Q, are swapped by the operation r→Q/r. It is now clear why 15 Gq r=Gq Q/rand therefore why G(ψq r,l)≈G(ψq Q/r,0). The QFT operator takes each periodic state in the ascending branch to a corresponding periodic st ate in the descending branch that has the same entanglement, and vice versa (recall that t he QFT operator is its own inverse). Finally, we present numerical evidence to support the claim that the QFT does not change the entanglement of periodic states (for suﬃciently large q). To this end we deﬁne the change of the Groverian entanglement of a state ψinduced by the QFT ∆G(ψ) =G[QFT(ψ)]−G(ψ). (33) We examine this diﬀerence for periodic states as well as for r andom states. The random states are taken from a uniform distribution on the 2q-dimensional complex unit sphere. The Groverian measure of random states of qqubits exhibits a distribution that was calculated before for certain values of q[24]. Any unitary operator U(such as the QFT), when applied to a sample of these states, will produce states whose Grover ian measure exhibits the same distribution. Looking at values of ∆G(ψ), for some states it is positive and for other states it is negative. The average of ∆G(ψ)over the random states is zero, but its distribution exhibits a certain width. In Fig. 3 we show the average of the absolute value of∆G(ψ), which is an estimate of the width of the distribution of the va lues of∆G(ψ). For periodic states|∆G(ψ)|quickly decreases to zero with increasing q[Fig. 3(b)], while for random states it changes only slightly [Fig. 3(a)]. This demonstra tes the special feature of periodic states, namely, that their Groverian measure is not aﬀected by the QFT. VI. DISCUSSION The goal of the ﬁeld of quantum algorithms is not only to ﬁnd qu antum algorithms that present a speedup over classical ones, but also to estab lish a deep understanding as

Question 1448 multiple-choice

Quantum computing employs principles such as superposition and entanglement to achieve computational speedups for certain problems that are challenging for classical computers. Several quantum algorithms leverage quantum phase estimation (QPE) for tasks in machine learning and numerical analysis. Which quantum algorithm specifically utilizes quantum phase estimation to achieve exponential speedup in solving linear systems of equations under the condition of sparse and well-conditioned matrices? 1) Quantum Support Vector Machine 2) Quantum k-Means Clustering 3) Quantum Approximate Optimization Algorithm (QAOA) 4) Quantum Principal Component Analysis 5) Quantum Singular Value Thresholding 6) Harrow-Hassidim-Lloyd (HHL) Algorithm 7) Quantum Boltzmann Machine

✓ Correct Answer:

The correct answer is 6) Harrow-Hassidim-Lloyd (HHL) Algorithm.

📚 Reference Text:

Title: Quantum phase estimation based algorithms for machine learning Year: 2021 Paper ID: 427672b1b72178e7f97ecd3e1cd19adc3b44f64a Source: semantic-scholar URL: https://www.semanticscholar.org/paper/427672b1b72178e7f97ecd3e1cd19adc3b44f64a Abstract: Quantum computing is certainly one of the greatest advances in the computer science field. Thanks to the parallelism and entanglement properties, it has proved to offer several advantages compared to the classical algorithms especially in the great reduction of the processing time. Quantum phase estimation (QPE) is one of the most important algorithms for quantum computing. It is known as the eigenvalue finding module for unitary operators. The Fourier transform is the key to this procedure. It has been researched and used to solve many problems such as the order finding problem, and the factoring problem. It was also applied for quantum sampling algorithms and the calculation of the eigenvalues of unitary matrices. In this paper, we study three important quantum algorithms for machine learning that use the QPE algorithm as a subroutine: the quantum principal components analysis (PCA) for data visualization, the Harrow-Hassidim-Lloyd (HHL) algorithm for solving linear systems, and the quantum singular value thresholding (SVT) for matrix completion in recommender systems. We also discuss the advantages and limits of such algorithms compared to their classical versions. Then we discuss potential ways of amelioration of such algorithms, and end with a proposed approach for further improvement.

Question 1449 multiple-choice

In group theory, finite 2-groups with specific properties related to their generators and subgroup structures are studied to understand their classification. The relationships between the Frattini subgroup, the center, and the derived subgroup are crucial in this analysis. For a finite 2-group G where the minimal generating set has three elements, exactly one abelian maximal subgroup exists, and exactly one maximal subgroup is neither abelian nor minimal nonabelian, which of the following statements must be true? 1) The Frattini subgroup Φ is always strictly larger than the center Z. 2) The derived subgroup G′ is cyclic of order 8. 3) All maximal subgroups of G are abelian. 4) The Frattini subgroup Φ coincides with the center Z. 5) The derived subgroup G′ is trivial. 6) Every maximal subgroup of G is minimal nonabelian. 7) G has no normal elementary abelian subgroups of order 8.

✓ Correct Answer:

The correct answer is 4) The Frattini subgroup Φ coincides with the center Z..

📚 Reference Text:

is neither abelian nor minimal nonabelian. Now we cons ider a minimal nonabelian subgroup W=/an}bracketle{ta, bc/an}bracketri}ht, where [ a, bc] =z. We compute: (bc)2=b2c2=b2·a2ib2jzk=a2ib2(1+j)zk, where iis odd and 1 + jis even. We have: Φ(W) =/an}bracketle{ta2,(bc)2, z/an}bracketri}ht=/an}bracketle{ta2, a2ib2(1+j)zk, z/an}bracketri}ht=/an}bracketle{ta2, z/an}bracketri}htΦ(Z(H))2. Let Hbe a maximal subgroup ofGwhich is minimal nonabelian. Then Φ( H) = Z( H)≤Φ(G) and |H: Φ(H)|= 4. Since |G: Φ(H)|= 8, we must have also Φ( H) = Φ( G). Let K/ne}ati≀nslash=Hbe another maximal subgroup of Gwhich is minimal nonabelian. Then Z( K) = Φ( K) = Φ( G) which implies Φ( G)≤Z(G) and so Φ( G) = Z( G). LetMbe the unique maximal subgroup of Gwhich is neither abelian nor minimal nonabelian. Since |M: Φ(G)|= 4 and Φ( G) = Z( G), we have M=S∗Φ(G), where Sis minimal nonabelian, S∩Φ(G) = Φ( S)<Φ(G) and soM′=S′∼=C2with d( M)≥3. Theorem 3.2.LetGbe a2-group with d(G) = 3which has exactly one maximal subgroup Mwhich is neither abelian nor minimal nonabelian. If Gpossesses exactly one abelian maximal subgroup A, then Φ(G) = Z( G), G′∼=E4,M′∼=C2,d(M)≥3and one of the following holds: (a)IfGhas no normal elementary abelian subgroup of order 8, then Gis one of the groups given in Theorem 87.8(e) in [2]. (b)IfGhas a normal subgroup E∼=E8butΩ1(G)> E, then G=/an}bracketle{tt, t′, c|t2=t′2=c4= 1,[t, t′] =c2=z,[c, t] =u, u2= [c, t′] = [u, t] = [u, t′] = [u, c] = [z, t] = [z, t′] = 1 /an}bracketri}ht, where |G|= 25,G′= Φ(G) = Z( G) =/an}bracketle{tz, u/an}bracketri}ht∼=E4,Ω1(G) =M= /an}bracketle{tt, t′/an}bracketri}htG′∼=C2×D8,A=/an}bracketle{tt′, c/an}bracketri}htG′is abelian of type (4,2,2), and other ﬁve maximal subgroups of Gare nonmetacyclic minimal nonabelian. (c)IfGhas a normal subgroup E∼=E8,Ω1(G) =EandE/ne}ati≀nslash≤A, then G=/an}bracketle{ta, b, t |a2m+1=b4=t2= 1, a2m=z, b2=u,[a, t] =u,[b, t] =z, [u, a] = [u, t] = [a, b] = [z, t] = 1 /an}bracketri}ht, where |G|= 2m+4,m≥2,G′=/an}bracketle{tz, u/an}bracketri}ht∼=E4,Φ(G) = Z( G) = /an}bracketle{ta2, u/an}bracketri}ht∼=C2m×C2,E/ne}ati≀nslash≤Z(G),A=/an}bracketle{ta, b/an}bracketri}htis abelian of type (2m+1,4), M=/an}bracketle{tb, t/an}bracketri}ht ∗ /an}bracketle{ta2/an}bracketri}ht, where /an}bracketle{tb, t/an}bracketri}htis the nonmetacyclic minimal nonabelian group of order 24and other ﬁve maximal subgroups of Gare minimal nonabelian. FINITE 2-GROUPS 75 (d)IfGhas a normal subgroup E∼=E8,Ω1(G) =EandE≤A, then G=/an}bracketle{ta, b, d |a4=b2=d4= 1, a2=d2=z,[a, d] =z,[a, b] =c, c2= [c, d] = [c, a] = [c, b] = [b, d] = 1 /an}bracketri}ht, where |G|= 25,G′= Φ(G) = Z( G) =/an}bracketle{tz, c/an}bracketri}ht∼=E4,A=/an}bracketle{tb, d/an}bracketri}htΦ(G), M=/an}bracketle{ta, d, c /an}bracketri}ht∼=Q8×C2,E/ne}ati≀nslash≤Z(G)and other ﬁve maximal subgroups ofGare minimal nonabelian. Proof. Let Γ 1={M, A, H 1, ..., H 5}be the set of maximal subgroups ofG, where H1, ..., H 5are minimal nonabelian. By a result of A. Mann (Proposition 1.5), |G′: (A′H′ 1)|=|G′:H′ 1| ≤2 and so |G′|= 4 (since |G′|>2). If G′=/an}bracketle{tv/an}bracketri}ht∼=C4, then H′ 1=...=H′ 5=/an}bracketle{tv2/an}bracketri}ht. But then G//an}bracketle{tv2/an}bracketri}ht is

Question 1450 multiple-choice

The performance of quantum algorithms on noisy intermediate-scale quantum (NISQ) computers is limited by errors, which can arise from both environmental interactions and hardware imperfections. Effective error modeling and correction strategies are essential for improving computational fidelity, particularly in foundational algorithms like the Quantum Fourier Transform (QFT). When modeling systematic errors in the Quantum Fourier Transform on current quantum computers, which method requires reconstructing the entire quantum state and scales with the number of qubits as 4^n measurements? 1) Fault-tolerant error correction protocol 2) Quantum error mitigation using randomized compiling 3) Reduced density matrix estimation for separable states 4) Quantum process tomography with Pauli twirling 5) Measurement-based error detection using syndrome extraction 6) Quantum state tomography utilizing the isotropic index 7) Ancilla-assisted error mapping in entangled systems

✓ Correct Answer:

The correct answer is 6) Quantum state tomography utilizing the isotropic index.

📚 Reference Text:

Other areas in Physics RevistaMexicanade F ´ısica68031402 1–6 MAY-JUNE2022 Characterizing errorsforquantum Fouriertransformon IBM quantum C. Allende, A.Fonsecade Oliveira,and E. Buksman Facultadde Ingenier ´ıa, UniversidadORT Uruguay,Montevideo,Uruguay. Received26 August 2021; accepted 23 September 2021 Theperformanceoftoday’squantumcomputersareaffectedbynoise. Thiseffectcanbeanalyzedintheresultofsimplequantumalgorithms in real quantum computers. The noise can be characterized as a decoherence error or a systematic error, the last could be corrected by a unitary rotation. In this article we propose two methods to model a systematic error, in the Quantum Fourier Transform algorithm (QFT). The ﬁrst method uses the isotropic index presented in Ref. [1] and needs to reconstruct the density matrix of the experimental state, while the second method, although less general, only needs to reconstruct the reduced density matrices for each qubit. In both methods, a unitary transformationisproposed,whichapproximatestheexperimentalresulttotheexpectedtheoreticalstate. Asanexample,theQFTalgorithm is analyzed for twoqubit states, in quantum IBM Q computer ibmqsantiago. Keywords: Quantum Fouriertransform; quantum error correction; IBM quantum. DOI: https://doi.org/10.31349/RevMexFis.68.031402 1. Introduction These days, circuit model-based quantum computers require some type of active error correction, because states in theo- retically advantageous quantum algorithms acquire errors as they are prepared, manipulated by gates, and measured, re- ducing efﬁciency. For this reason, the characterization and correctionoferrorsisofvitalimportancetoobtainanaccept- able result [2]. Although there are quantum error correction methods, as Fault-tolerant quantum error protocols [3], these areexpensivefromacomputationalpointofview. Inthecur- rent situation, i.e.a noisy intermediate-scale quantum era, it is important to characterize and analyze error models, for a possible future corrections. Thecentralideaofthearticle,istousetheisotropicindex proposed in Ref. [1], that separates the model error into one part related to unitary deviation, due systematic errors, from another part which represents the loss of information. This last errors, due to decoherence, cannot be corrected without redundancy,while in principle systematic errors could be. Two methods are proposed to model systematic errors. The ﬁrst method determines the density matrix of the exper- imental output of a nqubit state, which requires about 4n projective measurements ( Quantum state tomography ) and uses the isotropic index to ﬁnd a unitary rotation, while the second method uses only 3nprojective measurements. This last method is less general and works only for algorithms in which the output expected state, of each input basis state, is completely separable in all qubits. Withthisintention,weanalyzeaparticularalgorithm: the Quantum Fourier Transform (QFT) [4], ﬁnding the unitary rotationthatmodelsthesystematicerror. The(QFT)isanim- portantpartofsomequantumalgorithms,suchasthefamous Shor’s factorization algorithm [5] and the quantum phase es- timationalgorithm[6]. However,theeffectofnoiseon(QFT) is less studied. [7,8].This article is divided into three main sections. After the introduction, in Sec. 2, we present the Quantum Fourier Transform Algorithm (QFT), and its matrix model represen- tation. InSec.3,wediscussbothcorrectionmethodsforgen- eral quantum algorithms. We explain how the unitary matrix model is determined for an nqubit state. Finally, in Sec. 4 we present, as an example, the experimental results of run- ning the QFTalgorithm for a two qubit system on IBM Q machine: ibmqsantiago, and some conclusions. 2. The quantum fouriertransformalgorithm 2.1. QFT Algorithm The Fourier transform is used in a wide range of classical problems from signal processing to complexity theory. The QFT is the quantum generalization of the Discrete Fourier Transform (DFT) where the inputs

Question 1451 multiple-choice

Tensor network theory has become integral to quantum information science, offering efficient representations for complex quantum states. Matrix Product States (MPS) are particularly valuable for modeling one-dimensional systems and underpin several important simulation algorithms and quantum circuit implementations. Which multipartite entangled states are commonly used as maximally entangled examples in the implementation of Matrix Product States on quantum circuits? 1) Bell and Dicke states 2) Cluster and AKLT states 3) EPR and singlet states 4) Werner and separable states 5) NOON and decoherence-free states 6) GHZ and W states 7) Fock and coherent states

✓ Correct Answer:

The correct answer is 6) GHZ and W states.

📚 Reference Text:

Title: Simulation of Matrix Product State on a Quantum Computer Year: 2018 Paper ID: 6519e8443b8584223390f01b9ad0a65f2afda698 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/6519e8443b8584223390f01b9ad0a65f2afda698 Abstract: The study of tensor network theory is an important field and promises a wide range of experimental and quantum information theoretical applications. Matrix product state is the most well-known example of tensor network states, which provides an effective and efficient representation of onedimensional quantum systems. Indeed, it lies at the heart of density matrix renormalization group (DMRG), a most common method for simulation of one-dimensional strongly correlated quantum systems. It has got attention from several areas varying from solid-state systems to quantum computing and quantum simulators. We have considered maximally entangled matrix product states (GHZ and W). Here, we designed the quantum circuits for implementing the matrix product states. In this paper, we simulated the matrix product states in customized IBMs (2-qubit, 3-qubit and 4-qubit) quantum systems and determined the probability distribution among the quantum states.

Question 1452 multiple-choice

Quantum technologies, particularly quantum computing, are poised to transform fields such as cryptography and artificial intelligence. Discussions about democratization in this sector examine how access, engagement, and governance can be broadened beyond traditional corporate and governmental stakeholders. Which of the following actions most directly embodies the call for reflexivity, responsiveness, and transparency essential for the genuine democratization of quantum technologies? 1) Developing proprietary quantum algorithms for commercial use 2) Releasing quantum computing platforms exclusively to academic institutions 3) Expanding cloud-based quantum computing access without accompanying educational initiatives 4) Facilitating grassroots community engagement, open-source collaborations, and publicly accountable governance structures 5) Limiting quantum hardware distribution to select research groups 6) Prioritizing private investment in quantum technology startups 7) Publishing technical papers without stakeholder consultation

✓ Correct Answer:

The correct answer is 4) Facilitating grassroots community engagement, open-source collaborations, and publicly accountable governance structures.

📚 Reference Text:

Title: Democratization of quantum technologies Year: 2022 Paper ID: a86a6b829082438e1e3d180bd553df5aa6e350eb Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a86a6b829082438e1e3d180bd553df5aa6e350eb Abstract: As quantum technologies (QT) advance, their potential impact on and relation with society has been developing into an important issue for exploration. In this paper, we investigate the topic of democratization in the context of QT, particularly quantum computing. The paper contains three main sections. First, we briefly introduce different theories of democracy (participatory, representative, and deliberative) and how the concept of democratization can be formulated with respect to whether democracy is taken as an intrinsic or instrumental value. Second, we give an overview of how the concept of democratization is used in the QT field. Democratization is mainly adopted by companies working on quantum computing and used in a very narrow understanding of the concept. Third, we explore various narratives and counter-narratives concerning democratization in QT. Finally, we explore the general efforts of democratization in QT such as different forms of access, formation of grassroot communities and special interest groups, the emerging culture of manifesto writing, and how these can be located within the different theories of democracy. In conclusion, we argue that although the ongoing efforts in the democratization of QT are necessary steps towards the democratization of this set of emerging technologies, they should not be accepted as sufficient to argue that QT is a democratized field. We argue that more reflexivity and responsiveness regarding the narratives and actions adopted by the actors in the QT field and making the underlying assumptions of ongoing efforts on democratization of QT explicit, can result in a better technology for society.

Question 1453 multiple-choice

In quantum chemistry, the behavior and construction of many-electron wavefunctions are heavily influenced by molecular symmetry operations such as rotations, reflections, and permutations. The way these operations affect orbital localization and the spin component is crucial for understanding electronic structure and energy changes in molecules. Which statement best explains why a discontinuity appears in the slope of the energy versus internuclear distance curve for a linear H₄ molecule during the transition from localized to delocalized electron bonding? 1) The nuclei instantaneously switch positions, forcing electronic reconfiguration. 2) The molecular symmetry breaks down completely at the transition point. 3) The spin component of the wavefunction becomes undefined during the transition. 4) The electronic energy drops discontinuously due to bond cleavage. 5) The group theory constraints prevent any wavefunction adaptation across the transition. 6) The mathematical form of the wavefunction changes abruptly from symmetry-adapted to localized, causing a non-physical energy slope discontinuity. 7) The physical dissociation involves perfect symmetry, leading to observable energy discontinuities.

✓ Correct Answer:

The correct answer is 6) The mathematical form of the wavefunction changes abruptly from symmetry-adapted to localized, causing a non-physical energy slope discontinuity..

📚 Reference Text:

that
= ±fn = ±rn1x \ ' \ \ \ , ....... \ 0 I e 1 I ...... ,...,_ ... J ..... ___ .,., 8 ~ cf>2x 0 ~ ~ 8

Question 1454 multiple-choice

In computational chemistry, the selection and validation of density functional theory (DFT) functionals are crucial for accurately modeling molecular interactions and predicting bulk properties such as liquid density. Benchmarking against high-level reference data and experimental measurements is a standard protocol to ensure force field reliability. Which DFT functional, when parameterizing quantum mechanical force fields for molecular dynamics simulations of chloroform, yields density predictions closest to experiment due to smaller energy errors compared to others evaluated? 1) B2PLYP-D 2) CAM-B3LYP-D 3) PBE0-D 4) B97-D 5) B2PLYP-D with no dispersion correction 6) CAM-B3LYP-D without counterpoise correction 7) B3LYP-D

✓ Correct Answer:

The correct answer is 7) B3LYP-D.

📚 Reference Text:

+0.59 0.84 +0.08 0.28 B2PLYP-D +1.03 1.34 +0.29 0.60 B97-D -0.19 0.76 -0.12 0.26 PBE0-D -0.11 0.42 -0.07 0.15 B3LYP-D +0.11 0.32 +0.02 0.15 N1 B3 Functional /angbracketleftδ/angbracketrightσDFT /angbracketleftδ/angbracketrightσDFT (kJ/mol) (kJ/mol) (kJ/mol) (kJ/mol) CAM-B3LYP-D +0.82 1.05 +0.47 1.02 B2PLYP-D +1.23 1.48 +1.02 1.39 B97-D -0.04 0.73 -0.25 0.89 PBE0-D +0.06 0.40 +0.11 0.44 B3LYP-D +0.10 0.45 -0.11 0.49 Table 1: DFT benchmark against reference CCSD(T)@cbs data, compute d forC1(top left), C2(top right),N1(bottom left) and N1(bottom right). Both /angbracketleftδ/angbracketrightandσDFT(see text for deﬁnition) refer to CP corrected energies. All functional labels refer to Table Ain the Supporting Information. simulation were compared to the ones obtained with the previ ous CAM-B3LYP-D based QMD-FFs. While details of these parameterizations are give n in Tables LandMin the Supporting Information, in Table 2 shows the diﬀerence ( δρ) between the bulk density achieved with the QMD-FFs and the experimental one. molecule /angbracketleftδ/angbracketrightCAM−B3LYP−D/angbracketleftδ/angbracketrightB3LYP−DδρCAM−B3LYP−DδρB3LYP−D (kJ/mol) (kJ/mol) (%) (%) C1 +0.59 +0.11 -17.4 -4.3 C2 +0.08 +0.02 -2.6 -0.7 Table 2: Columns 2 and 3: average error < δ >of the investigated functional with respect to reference CCSD(T)@cbs energies, computed over the 50 dimer sets extra cted byPicky. Columns 4 and 5: per- centage error δρof the ﬁnal MD density with respect to its experimental count erpart.ρwas obtained from 5 ns NPT simulations at 298 K and 1 atm, carried out with QM D-FFs parameterized on either CAM-B3LYP-D (fourth column) or B3LYP-D (last column) data. As far as the C2compound is concerned, the aforementioned slight overesti mation of the reference IPES ( i.e.less attractive interaction energies) reﬂects, as could be expected, inasmallunderestimationofthecondensedphasedensity. D espitethegoodperformances 16 of both functionals, B3LYP-D shows a better agreement with th e CCSD(T)@cbs data and it’s worth noticing that this in turn reﬂects in a better repr oduction of the experimental liquid density. Turning to chloroform ( C1), the surprisingly scarce performances of the CAM-B3LYP-D functional when compared to the reference energi es (<δ>≃0.6 kJ/mol) lead to the already mentioned unacceptable error in the pred icted density (more than 15% lower than the experiment). On the contrary, the simulation s performed with the B3LYP- D parameterized QMD-FF yielded a signiﬁcant improvement, b eing the ﬁnal density underestimated by only ∼4%, again consistently with the overestimation of the refer ence interaction energy. These data seem to indicate that the proposed protocol to eva luate DFT accuracy can give important a priori hints on the reliability of a given functional to lead to accu rate predictions of bulk properties. To conﬁrm this idea, the DFT benchmarking protocol was applied to two additional molecules, B3andN1: the former allows for extending the B3LYP-D based QMD-FFs to Bromine containing compounds, w hile the latter may serve to evaluate B3LYP-D accuracy in presence of cyano-grou ps. As for the former benchmark compounds, the results for these latter molecule s are displayed in Figure 3 (last two columns) and their most important features summar ized in Table 1. As found for chloroform, the two best IPES descriptions for th e brominated-alkane B3are

Question 1455 multiple-choice

In computational combinatorics and cryptography, the search for zero-sum subsets within sequences of vectors over finite fields is a well-studied problem. Efficient algorithms for detecting such subsets have theoretical and practical importance due to their applications in error-correction and security. Which statement best describes the main advantage of the deterministic polynomial-time algorithm for finding non-trivial zero-sum subsets in ℤₚⁿ compared to earlier more general approaches? 1) It always finds the largest possible zero-sum subset in any sequence. 2) It guarantees the discovery of zero-sum subsets without any need for randomization. 3) It can represent zero as a linear combination with arbitrary integer coefficients. 4) It works for infinite-dimensional vector spaces over ℤₚ. 5) It has lower memory requirements than randomized algorithms for the same problem. 6) It achieves better complexity bounds for the specific case of standard zero-sum subset detection in ℤₚⁿ. 7) It can handle vectors with entries from any commutative ring.

✓ Correct Answer:

The correct answer is 6) It achieves better complexity bounds for the specific case of standard zero-sum subset detection in ℤₚⁿ..

📚 Reference Text:

the following. Proposition 5 Suppose that there is an algorithm Bthat given a set of vectors from Zn p of size S ±(p,n)ﬁnds a collision. Then there is a deterministic procedure that, given S±(p,n)⌈logp⌉vectors, ﬁnds a non-trivial zero sum subset using less than S⌈logp⌉ ± applications of algorithm Band poly((S±(p,n))⌈logp⌉)other operations. Proof PutS=S±(p,n)and/lscript=⌈logp⌉. We start with ﬁnding a collision (H+ 1,H− 1) among the ﬁrst Svectors with common sum w1using algorithm B. We continue with the next Sinput vectors and ﬁnd a collision (H+ 2,H− 2)with sum w2, and so on. We then take the ﬁrst Ssubset sums w1,...,w Sand ﬁnd a pair of disjoint subsets (K+,K−)of{1,..., S}, not both empty, such that/summationtext i∈K+wi=/summationtext i∈K−wi=w. The four subsets L++=/uniontext i∈K+H+ i,L+−=/uniontext i∈K+H− i,L−+=/uniontext i∈K−H+ iand L−−=/uniontext i∈K−H− iof input vectors are pairwise disjoint, not all empty and have common sum w. Iterating this we end up with at least ppairwise disjoint subsets (not all empty) with equal sum. If one of these sets is empty, then the common sum is zero,and we can take any of the non-empty subsets. Otherwise the union of the ﬁrst pof the subsets has zero sum. The total number of applications of the collision ﬁnding algorithm BisS /lscript−1+...+S+1

Question 1456 multiple-choice

Quantum computing utilizes multi-qubit systems, where the structure of the computational basis and the behavior of quantum gates are fundamental to designing algorithms and circuits. A deep understanding of these concepts is essential for effective quantum information processing. Which statement accurately describes the action of the controlled-NOT (CNOT) gate in a two-qubit quantum circuit? 1) The CNOT gate flips the target qubit if and only if the control qubit is in the state |1⟩. 2) The CNOT gate swaps the values of the control and target qubits for all input states. 3) The CNOT gate flips both qubits simultaneously regardless of their initial states. 4) The CNOT gate always leaves the target qubit unchanged. 5) The CNOT gate acts only on single-qubit states and does not involve a control mechanism. 6) The CNOT gate applies a Hadamard transformation to the control qubit and a NOT to the target qubit. 7) The CNOT gate flips the control qubit if the target qubit is in the state |1⟩.

✓ Correct Answer:

The correct answer is 1) The CNOT gate flips the target qubit if and only if the control qubit is in the state |1⟩..

📚 Reference Text:

qubits are required for the computational space,using Eq. (2), the basis becomes 3Quantum Fourier Operators and Their Application DOI: http:/ /dx.doi.org/10. 5772/intechopen.9 4902 ∣0i/C17∣00i¼∣0i⊗∣0i¼1 0 ! ⊗1 0 ! ¼1 0 0 00 BBBBB@1 CCCCCA ∣1i/C17∣01i¼∣0i⊗∣1i¼1 0 ! ⊗0 1 ! ¼0 10 00 BBBBB@1 CCCCCA ∣2i/C17∣10i¼∣1i⊗∣0i¼0 1 ! ⊗1 0 ! ¼0 0 1 00 BBBBB@1 C CCCCA ∣3i/C17∣11i¼∣1i⊗∣1i¼0 1 ! ⊗0 1 ! ¼0 0 010 BBBBB@1 CCCCCA(13) To generalize this example for nqubits, the set of computational basis vectors can, for the sake of brevity, be labeled in base 10 as j0i,j1i,j2i,⋯,j2 n/C01i fg : (14) On the other hand, in order to highlight the qubit values, this basis can equiva- lently be expressed in base 2 as ∣k1k2⋯kni¼∣k1i⊗∣k2i⊗∣ ⋯⊗∣ kni (15) where ki∈0, 1fg fori¼1,⋯,n. In other words, k1,k2,⋯,kn fg represents the binary expansion k¼k12n/C01þk22n/C02þ⋯þkn/C0121þkn20¼Xn t¼1kt2n/C0t(16) for the kthbasis vector ∣ki/C17∣k1k2⋯kni. We have chosen this bit index ordering as it will prove convenient for the QFT formulation in the next section. An equallyacceptable (and, quite typical) bit index convention for an nqubit system could, for example, be ∣qi/C17∣q n/C01qn/C02⋯q1q0i. Eq. (15) tells us that the nqubit basis is derived from the tensor product of single qubits. This is important to keep in mind in order to avoid confusion when using the symbol ∣0i. For example, when using n¼1 qubit, ∣0iin decimal is equivalent to ∣0i in binary; however,, when using n¼3 qubits, ∣0iin decimal is equivalent to ∣000i in binary. Hence, the number of qubits nis the anchor for the relationship between Eq. (14) and Eq. (15). Assuming nqubits, there are 2nbasis vectors that can be used to construct a quantum state. Hence, all 2nbasis vectors will simultaneously evolve with their associated probabilities; again, this is the source of quantum parallelism. 4Real Perspective of Fourier Transforms and Current Developments in Superconductivity 2.3 Quantum circuits One particularly useful application of Eq. (12) arises when building up nqubit quantum circuits (i.e. schematic depictions of quantum operations on qubits). Forinstance, assume a two qubit system ∣q 1q0iwhere two unitary operators HandZact on single qubits as H∣q1i,Z∣q0i (17) and the result is desired to be combined as H∣q1i⊗Z∣q0i: (18) Eq. (12) tells us that this action is equivalent to H⊗Z ðÞ j q1i⊗jq0i/C0/C1 : (19) However, by construction, ∣q1i⊗∣q0i¼∣q1q0i. Therefore, H∣q1i⊗Z∣q0i¼ H⊗Z ðÞ ∣q1q0i (20) making it straightforward to develop multiple qubit quantum systems from unitary operators. The schematic representation of H⊗Z ðÞ ∣q1q0iis show in Figure 1 . With the groundwork laid for multiple qubits, it becomes possible to introduce more unitary operators that facilitate reversible computation. For example, thecontrolled NOT (CNOT) function can be phrased as a two qubit reversible XORoperator CNOT ¼1000 01000001 00102 66643 7775: (21) where crepresents the control bit, trepresents the target XOR function and ∣q 1q0i¼∣cti. This operator is a permutation matrix that is consistent with Table 1 in that it swaps the ∣11iand∣10iqubits. The XOR operation, by itself, can act as an irreversible controlled NOT operation. For the sake of quantum computation,

Question 1457 multiple-choice

The Fast Fourier Transform (FFT) and Quantum Fourier Transform (QFT) are fundamental algorithms in classical and quantum computing, enabling efficient computation of the discrete Fourier transform through matrix decompositions. Their implementation relies heavily on the structure and properties of certain matrices, including Kronecker products and sparse matrix representations. Which of the following statements best explains why the decomposition of the diagonal matrix Ω_n into Kronecker products of 2×2 unitary matrices is crucial for efficient quantum circuit implementation of the Quantum Fourier Transform (QFT)? 1) It allows the transformation to be performed entirely with classical gates, reducing quantum hardware requirements. 2) It enables the output state to be measured directly in the computational basis without further operations. 3) It ensures that only real-valued phase shifts are needed, simplifying circuit construction. 4) It permits the elimination of all controlled operations from the quantum circuit. 5) It reduces the overall number of quantum gates required by increasing sparsity in the circuit. 6) It allows phase shifts to be implemented using simple, local quantum gates, facilitating scalable and efficient quantum circuit design. 7) It guarantees that the Fourier transform can be reversed with a single Hadamard gate per qubit.

✓ Correct Answer:

The correct answer is 6) It allows phase shifts to be implemented using simple, local quantum gates, facilitating scalable and efficient quantum circuit design..

📚 Reference Text:

(9) where, fork= 0,1,...,n−1, A(k) n=In−k−1⊗Bk+1,andBk+1=1√ 2/bracketleftbiggIkIk Ωk−Ωk/bracketrightbigg = (Ik⊕Ωk)(H⊗Ik), (10) withΩkgiven by Deﬁnition 8. 5 Proof.We give the proof for F′ nby induction based on Theorem 1. The case n= 1 can be directly veriﬁed: F′ 1=A(0) 1= 1⊗B1=1√ 2/bracketleftbigg1 1 1−1/bracketrightbigg . Assume F′ n=A(0) nA(1) n···A(n−1) nas prescribed by Theorem 2. For the bit-reversed DFT matrix o f size 2n+1we have by Theorem 1, F′ n+1= (I2⊗F′ n)(In⊕Ωn)(H⊗In), =/parenleftBig I2⊗/bracketleftBig A(0) nA(1) n···A(n−1) n/bracketrightBig/parenrightBig (In⊕Ωn)(H⊗In), =/parenleftBig I2⊗A(0) n/parenrightBig/parenleftBig I2⊗A(1) n/parenrightBig ···/parenleftBig I2⊗A(n−1) n/parenrightBig A(n) n+1, =A(0) n+1A(1) n+1···A(n−1) n+1A(n) n+1, where we used I2⊗A(k) n=A(k) n+1fork= 0,1,...,n−1, andA(n) n+1=Bn+1. They key observation here is that each matrix A(i) nin (10) only has two nonzero elements on every row. Consequently, the matrix-vector product A(i) nvcan be computed in only O(2n) operations, resulting in an overall O(2nn) orO(NlogN) computational complexity for the Fourier transform of a ve ctor of size N= 2n. This is why the FFT is called fast. 4 Matrix Decomposition for Quantum Fourier Transform As we can see from the previous section, the reduction in comp lexity from O(N2) toO(NlogN) in the FFT algorithm essentially results from the Kronecker product s tructure that appears in the matrix factors of F′ nin (7). It is theIk⊕Ωkfactor that retains the Nfactor in the complexity of the FFT algorithm because the mul tiplication of that diagonal matrix with a vector has to be performed in O(N) operations. It is conceivable that the complexity of the computation can potentially be reduced further, at least in some special cases, if we can somehow rewrite this diagonal matri x as a Kronecker product of 2 ×2 matrices. It turns out that we can almost do that. In this section, we show that th e diagonal matrix Ik⊕Ωkcan be written as a product of ksimpler matrices, each of which is the sum of two Kronecker pr oducts of 2 ×2 matrices. This factorization yields a decomposition of the DFT matrix that enables the DFT to be performed eﬃciently on a quantum computer. Exactly how that is achieved will be discu ssed in the next section. Here we will simply refer to this decomposition as the decomposition used by the quant um Fourier transform (QFT). We begin by showing that the diagonal matrix Ωncan be written as a Kronecker product of 2 ×2 unitary matrices of the following form. Deﬁnition 9. Deﬁne Rn:=/bracketleftbiggω0 2n ω1 2n/bracketrightbigg =/bracketleftbigg1 ω2n/bracketrightbigg , (11) whereω2n:= exp(−2πı 2n). The matrix Rnsatisﬁes the identity stated in the following lemma. Lemma 3. LetRnbe deﬁned by Deﬁnition 9. Then R2j n=Rn−j, forj∈N. Proof.We start from the following identity ω2j 2n=/parenleftbigg exp/parenleftbigg−2πı 2n/parenrightbigg/parenrightbigg2j = exp/parenleftbigg−2πı 2n−j/parenrightbigg =ω2n−j. Hence, (Rn)2j=/bracketleftbigg1 ω2j 2n/bracketrightbigg =/bracketleftbigg1 ω2n−j/bracketrightbigg =Rn−j. 6 We can use this result to decompose Ωnas a Kronecker product of n Rimatrices. Lemma 4. LetΩnbe deﬁned by Deﬁnition 8. Then Ωn=R2⊗R3⊗···⊗Rn⊗Rn+1, (12) whereRnis deﬁned by Deﬁnition 9. Proof.From Lemma 1, we can use the binary representation of jto rewrite the jth diagonal element of Ωnas Ωn(j,j)

Question 1458 multiple-choice

Quantum algorithms leverage group theory and representation theory to solve problems that are classically intractable, such as certain hidden subgroup problems. The search for new quantum primitives continues to drive advances in algorithmic speedup. Which unitary transformation, arising from symmetry considerations in representation theory, has been proposed as a new primitive for quantum algorithm design—particularly for efficiently solving the Heisenberg hidden subgroup problem? 1) Quantum Fourier transform over abelian groups 2) Phase estimation transform 3) Clebsch-Gordan transform over the Heisenberg group 4) Swap test operation 5) Hadamard transform 6) Quantum walk operator 7) Grover diffusion operator

✓ Correct Answer:

The correct answer is 3) Clebsch-Gordan transform over the Heisenberg group.

📚 Reference Text:

arXiv:quant-ph/0612107v2 20 Sep 2007How a Clebsch-Gordan Transform Helps to Solve the Heisenberg Hidden Subgroup Problem D. Bacon Department of Computer Science & Engineering, University o f Washington, Box 352350 Seattle, WA 98109, USA It has recently been shown that quantum computers can eﬃcien tly solve the Heisenberg hidden subgroup problem, a problem whose classical query co mplexity is exponential. This quantum algorithm was discovered within the framework of using pretty good mea- surements for obtaining optimal measurements in the hidden subgroup problem. Here we show how to solve the Heisenberg hidden subgroup problem u sing arguments based instead on the symmetry of certain hidden subgroup states. T he symmetry we consider leads naturally to a unitary transform known as the Clebsch- Gordan transform over the Heisenberg group. This gives a new representation theoreti c explanation for the pretty good measurement derived algorithm for eﬃciently solving t he Heisenberg hidden sub- group problem and provides evidence that Clebsch-Gordan tr ansforms over ﬁnite groups are a new primitive in quantum algorithm design. Keywords : Quantum computing, quantum algorithms, hidden subgroup p roblem Communicated by : to be ﬁlled by the Editorial 1 Introduction In 1994 Peter Shor discovered that a quantum computer could eﬃc iently factor integers[1], a problem which is widely suspected to be intractable on a classical com puter[2]. Since this discovery an intrepid group of researchers have been attemp ting to discover quantum algorithms which lie beyond Shor’s factoring algorithm with mixed succe ss. On the one hand, agreatdealofsuccesshasbeen hadinachievingpolynomialspeedu psoverclassicalalgorithms within the framework of Grover’s quantum search algorithm[3]. On th e other hand, quantum algorithms which, like Shor’s algorithm, perform exponentially faster than the best classical algorithms have been harder to come by. To be sure, notable succe sses have been achieved, including Hallgren’seﬃcient quantum algorithmfor solvingPell’s equation [4] and exponential speedups in certain quantum random walks[5], but so far there have been no new eﬃcient quantum algorithms which move quantum computing’s killer application b eyond factoring integers and the resulting breaking of multiple public key cryptosyst ems. Among the most tempting problems which might be exponentially sped u p on a quan- tum computer is the graph isomorphism problem. The reasons for th is are two-fold. First, graph isomorphism belongs to a complexity class very much like that wh ich contains inte- ger factoring. In particular, the decision version of factoring is kn own to be in the com- 1 2How a Clebsch-Gordan Transform Helps to Solve ... plexityNP∩coNPwhile graph isomorphism is known to be in the similar complexity class NP∩coAM. Algorithmsin thesecomplexityclassesareunlikelytobe NP-complete. Further, both integerfactoringandgraphisomorphismarenotknowntohav eclassicalpolynomialtime algorithms despite considerable eﬀort to ﬁnd such algorithms. Thus graph isomorphism is, like integer factoring, of Goldilocks-like classical complexity, not too hard such that eﬃciently solving it would revolutionize our notion of tractable, but not so easy as to have already fallen intoP(orBPP.) The second reason for attempting to ﬁnd eﬃcient quantum algorit hms for the graph isomorphism problem is that this problem can be solved if the re

Question 1459 multiple-choice

Noncommutative sum of squares (SOS) relaxations and semidefinite programming (SDP) are essential tools in quantum information theory and operator algebra for optimizing over polynomials in noncommuting variables. Swap algebras, generated by swap operators acting on indices, are frequently used in analyzing permutation-invariant quantum systems. For a swap algebra generated by three swap operators (n=3), which statement correctly characterizes the determination of a unital *-linear functional L that vanishes on the intersection of the swap ideal and degree-2 monomials? 1) L is determined by a single complex number and three integer parameters. 2) L requires three complex numbers and one real number for its full specification. 3) L is specified entirely by three integer values corresponding to the swaps. 4) L depends only on the traces of the swap operators and their products. 5) L is uniquely determined by six real numbers corresponding to all possible swap terms. 6) L is characterized by three real numbers (ℓ_ij = L(s_ij)) and one complex number (q = L(s12s13)). 7) L requires a basis of four swap matrices with non-overlapping indices.

✓ Correct Answer:

The correct answer is 6) L is characterized by three real numbers (ℓ_ij = L(s_ij)) and one complex number (q = L(s12s13))..

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νd(h) = eigmax(h(Swap)) =ν d(h). Proof. The convergence statement follows from the Helton-McCullough Positivstellen- satz [ HM04 ] since the algebra Aswap n is finite-dimensional and hence archimedean. The convergence is finite by Theorem 3.18. 4.3.1 First relaxation for the Swap algebra We now present in some detail the SDP arising from the first noncommutative sum of squares relaxation for the Swap algebra. We proceed as in Example 4.7 and Example 4.8 and defineV1(n) = (1,s12,s13,...,s 1n,s23,...,sn−1n)∗and consider M1(L) for some∗-linear functionalL:C⟨s⟩2→Cwhich satisfies L(R⟨s⟩2)⊆R. To encode the constraint L(Iswap∩C⟨s⟩2) ={0}, we need to understand the linear space spanned by monomials of degree at most two in the swap algebra. We construct a basisB2for this space together with expansions for all the other products in terms of this basis to encode Eq. (4.10). This follows the approach outined in Item (2) of Section 4.2.1. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 36 Example 4.11. Swap Algebra, n= 3. A basis B2for the linear space spanned by mono- mials of degree at most two in the swap algebra is given by the entries of V1(3) together with one element, e.g., s12s13. All the other quadratic terms in the sijcan be expressed with these as follows: s2 ij= 1 s12s23=−1 +s12+s13+s23−s12s13 s13s12=−1 +s12+s13+s23−s12s13 s13s23=s12s13 s23s12=s12s13 s23s13=−1 +s12+s13+s23−s12s13 With this a unital ∗-linearLwithL(Iswap∩C⟨s⟩2) ={0}is determined by 3 real numbers ℓij=L(sij) and a complex number q=L(s12s13). Thus M1(L) simplifies into M1(L) = L(1)L(s12)L(s13)L(s23) L(s12)L(s2 12)L(s12s13)L(s12s23) L(s13)L(s12s13)∗L(s2 13)L(s13s23) L(s23)L(s12s23)∗L(s13s23)∗L(s2 23)  = 1 ℓ12 ℓ13 ℓ23 ℓ12 1 q−1 +ℓ12+ℓ13+ℓ23−q ℓ13 q∗1 q ℓ23−1 +ℓ12+ℓ13+ℓ23−q∗q∗1 . Conversely, each M1(L) of this form yields a unital ∗-linearLwithL(Iswap∩C⟨s⟩2) ={0}. This makes it straightforward to write down the SDP of Eq. (4.10) for the objective function L(h) forh=ℓ12+ℓ13+ℓ23in this case. maxℓ12+ℓ13+ℓ23 s.t. 1 ℓ12 ℓ13 ℓ23 ℓ12 1 q−1 +ℓ12+ℓ13+ℓ23−q ℓ13 q 1 q ℓ23−1 +ℓ12+ℓ13+ℓ23−q q 1 ⪰0 = 3. Hereℓij∈R, and as explained in Remark 4.6, we have without loss of generality assumed thatq=L(s12s13)∈R. The examples illustrate SDP relaxations using linear algebra constraints to capture Iswapwhenn= 3. The rest of this section and related appendices give the machinery needed, namely B3,B4, for anynand for 2nd swap relaxations. 4.3.2 Linear space spanned by the products of at most two swap matrices Proposition 4.12. A basis B2for the linear space spanned by monomials of degree at most two in the swap algebra Mswap n is given by I Swapiji

Question 1460 multiple-choice

In quantum algorithms that utilize group theory, the structure of regular representations and their commutants plays a crucial role in simplifying measurements and extracting hidden subgroup information. Understanding how symmetries and representation theory affect quantum state decompositions is central to analyzing algorithmic efficiency. Which statement correctly characterizes the support of operators invariant under the left regular representation in the context of group representation theory? 1) They are restricted to the subspace spanned by the trivial representation only. 2) They necessarily commute with all operators from both left and right regular representations. 3) They have nonzero support only on the left regular representation subspaces. 4) They act as scalar multiples of the identity across the entire group algebra. 5) They are required to be block-diagonal in the canonical basis of group elements. 6) They have support exclusively on the commutant of the left regular representation, which is the right regular representation. 7) They vanish on all irreducible representations except those corresponding to normal subgroups.

✓ Correct Answer:

The correct answer is 6) They have support exclusively on the commutant of the left regular representation, which is the right regular representation..

📚 Reference Text:

whichRL(g) acts as the above direct sum is also the basis in which RR(g) acts as the above direct sum. In other words, in this basis, RL(g)RR(g′) =RR(g′)RL(g) =/circleplusdisplay µDµ(g′)⊗Dµ(g). (9) Let us call the basis described above the |µ,rµ,lµ/an}bracketri}htbasis where rµ= 1,...,d µandlµ= 1,...,d µ. 4.2 Symmetry of Hidden Subgroup States Why are the left and right regular representations relevant to the hidden subgroup problem? Well it is easy to check that the hidden subgroup state ρH, for all hidden subgroups H, are invariant under conjugation by the left regular representation, DL(g)ρHDL(g−1) =ρH, (10) for allg∈ G. The reason for this is that left multiplication by a ﬁxed group element acts as a permutation on left cosets. Notice that ρHis not invariant under a similar transform using the right regular representation unless the representation being used is an element of the subgroup H. What is the consequence of the invariance of the ρHwith respect to the left regular representation? Schur’s lemma[46] tells us that if an operator is inva riant with respect to the operatorsenacting a representationofthe group, then that op eratorhas support only overthe commutant of the this representation. For our purposes the com mutant is simply the algebra of matrices which commute with the left regular representation, MDL(g) =DL(g)M. Since ρHis invariant with respect to the operators enacting the left regular representation of Gand the commutant of the left regular representation of Gis the right regular representation of G, this means that in the |µ,rµ,lµ/an}bracketri}htbasis,ρHcan be expressed as ρH=/circleplusdisplay µσµ,H⊗Idµ, (11) whereσµ,His an operator with support only on the space acted upon by the irre ducible representation of the right regular representation. What does this mean for obtaining information about Hby making measurements on ρH? For now we focus on a single measurement on a single copy of ρH. Recall that a generalized measurement is described by a set of positive operators {P1,...,P k}which sum to identity/summationtextk α=1Pα=I. Outcomes of the measurement correspond to the indices, with th e probability of getting outcome αwhen measuring state ρgiven by pα= Tr[ρPα]. So if we make a measurement with operator PαonρHthe probability of getting outcome αis pα= Tr[ρHPα]. (12) 10How a Clebsch-Gordan Transform Helps to Solve ... But, since ρHis invariant under the left regular representation operators, we c an express this probability as pα= Tr 1 |G|/summationdisplay g∈GRL(g)ρHR† L(g)Pα = Tr/bracketleftBig ρH˜Pα/bracketrightBig , (13) where ˜Pα=1 |G|/summationdisplay g∈GR† L(g)PαRL(g). (14) But˜PαisPαsymmetrized over left regular representation of the group G. This implies that ˜Pαcommutes with the left regular representation, RL(g′)†˜PαR† L(g′) =˜Pα, (15) for allginG. Thus using Schur’s lemma, ˜Pαhas support over only the commutant of the left regular representation, our good friend the right regular rep resentation. In other words, in the|µ,rµ,lmu/an}bracketri}htbasis, we ﬁnd that ˜Pα=/circleplusdisplay µQµ,α⊗Idµ. (16) This means that given some measurement Pα, the probabilities of the diﬀerent outcomes α depends only on the symmetrized version of Pα. Therefore without loss of generality we can deal with measurement which are already symmetrized and ther efore have

Question 1461 multiple-choice

Quantum machine learning explores leveraging quantum algorithms to solve complex problems in statistics and physics, such as sampling from high-dimensional continuous distributions. One area of focus is using quantum techniques for Gibbs sampling, particularly where the target function exhibits periodicity and favorable spectral properties. In quantum algorithms for Gibbs sampling from periodic, real-valued functions, which property of the function most directly influences the efficiency of quantum interpolation and differentiation? 1) The number of local minima in the function 2) The dimensionality of the input space 3) The smoothness of the energy landscape 4) The decay rate of the Fourier coefficients 5) The presence of saddle points 6) The normalization of the probability density 7) The boundary conditions of the domain

✓ Correct Answer:

The correct answer is 4) The decay rate of the Fourier coefficients.

📚 Reference Text:

Title: Gibbs Sampling of Continuous Potentials on a Quantum Computer Year: 2022 Paper ID: 20a49b7448e6338b45cf815ea97d71804dc37fc8 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/20a49b7448e6338b45cf815ea97d71804dc37fc8 Abstract: Gibbs sampling from continuous real-valued functions is a challenging problem of interest in machine learning. Here we leverage quantum Fourier transforms to build a quantum algorithm for this task when the function is periodic. We use the quantum algorithms for solving linear ordinary differential equations to solve the Fokker--Planck equation and prepare a quantum state encoding the Gibbs distribution. We show that the efficiency of interpolation and differentiation of these functions on a quantum computer depends on the rate of decay of the Fourier coefficients of the Fourier transform of the function. We view this property as a concentration of measure in the Fourier domain, and also provide functional analytic conditions for it. Our algorithm makes zeroeth order queries to a quantum oracle of the function. Despite suffering from an exponentially long mixing time, this algorithm allows for exponentially improved precision in sampling, and polynomial quantum speedups in mean estimation in the general case, and particularly under geometric conditions we identify for the critical points of the energy function.

Question 1462 multiple-choice

In representation theory and algebraic combinatorics, semistandard Young tableaux (SSYT) play a crucial role in constructing modules for symmetric groups and analyzing their branching rules. Their combinatorial structure is closely related to the conditions under which certain actions in symmetric group representations yield non-zero results. Which condition ensures that the action λ(YT)|T⟩ associated with a subset Xi of consecutive integers does NOT yield the zero vector in the module representation? 1) All elements of Xi are placed in the same row of the tableau. 2) Elements of Xi appear only in columns with strictly decreasing entries. 3) No two elements of Xi appear in the same row. 4) The boxes labeled by Xi form a vertical strip with at most one box per row. 5) Every element of Xi occupies a distinct diagonal in the tableau. 6) All elements of Xi are adjacent in the tableau, forming a connected region. 7) The boxes labeled by Xi form a horizontal strip, meaning no two elements of Xi appear in the same column.

✓ Correct Answer:

The correct answer is 7) The boxes labeled by Xi form a horizontal strip, meaning no two elements of Xi appear in the same column..

📚 Reference Text:

next that λ(YT)|T/angbracketrightcan be identiﬁed with semistandard Young tableau of shape λand content deﬁned byYT. In other words, if YT=SX1×SX2×···×SXkfor somek, then the content is |X1|1s,|X2| 2s etc. Also note that the sets Xiconsist of consecutive integers. The SSYT associated with λ(YT)|T/angbracketrightis the one where all the integers in Xiare replaced by i. This is a valid SSYT if no two integers in Xiare in the same column or equivalently, every column has at most one element of Xi. If we isolate the boxes in the SYT labeled by elements of Xiand they have the property above, such a skew Young diagram is called a horizontal strip as described in section 2.3. We now only need to show that for every Xithat if the boxes numbered with elements of Xiform a horizontal strip, then λ(YT)|T/angbracketrightis non-zero and it is zero otherwise. To show this, we focus on a single Xiand show that if there are two elements of Xiin the same column, then λ(Xi)|T/angbracketright= 0. This is done next in lemma 2. Finally, the claim of the dependence on /epsilon1follows from the fact the each of the steps in the algorithm (including the group multiplications and quantum Fourier transforms) can be done with O(polylog/epsilon1−1) elementary gates based on the results described in section 3.1 and 3.2. Lemma 2. LetAbe a set of consecutive integers {a1,...ak},SAbe the symmetric group of size |A|! permuting the elements of A. Let|T/angbracketrightbe a SYT of shape λwith entries that include the set A. Letλ(SA) =/summationtext πλ(π), whereπruns over elements of SA. Assume that Tcontains two elements of Ain the same column. Then,λ(SA)|T/angbracketright= 0, whereλ(π)is the Young orthogonal representation on SYTs. Proof. Since the elements of Aare consecutive integers, we can assume without loss of generality that there are two elements of Athat appear in the same column and in consecutive rows. Let the elements be iandj withj >i . We ﬁrst show that if j=i+ 1, thenλ(SA)|T/angbracketright= 0 and then reduce the general case to this one. So assume now that iandi+1 are in the same column (they have to be in consecutive rows). The action of the transposition ( i,i+ 1) isλ((i,i+ 1))|T/angbracketright=−|T/angbracketright. Therefore, λ(e+ (i,i+ 1))|T/angbracketright= 0, where eis the identity element. It is easy to see that for any set K={i1,i2...ir}, the symmetric group algebra element that is a sum of all possible permutations of the elements of Kcan be written as follows. SK= ((ir,i1) +···+ (ir,ir−1))...((i3,i1) + (i3,i2) +e) ((i2,i1) +e) (3.10) While this factorization is dependent on the ordering of the elements, the overall group algebra element SK is independent of it. Using this and writing A={i,i+ 1,i+ 2,...,ak,a1,a2,...,i−1}, we obtain SA=S/prime A((i,i+ 1) +e), (3.11) where inS/prime A, we collect the rest of the terms i.e., S/prime Ais a product of sums of transpositions coming from the factorization above. It is now easy to see that SA|T/angbracketright= 0 ifTcontainsiandi+ 1 in the same column. For the general case, assume inductively that when iandj−1 are in the same column, then

Question 1463 multiple-choice

Quantum algorithms have revolutionized computational approaches to problems like integer factorisation, challenging the foundations of modern cryptography. Shor’s algorithm, in particular, uses quantum mechanics to efficiently factor large numbers that underpin cryptographic schemes such as RSA. Which step in Shor's algorithm is responsible for extracting the period of the function f(k) = m^k mod N without collapsing the quantum superposition, thereby enabling efficient factorisation? 1) Measurement of the quantum state before applying any quantum operations 2) Application of classical modular exponentiation to all possible k values 3) Initialization of qubits in the ground state 4) Sequential application of the Hadamard transform 5) Calculation of the greatest common divisor using Euclid’s algorithm 6) Application of the Quantum Fourier Transform (QFT) to the modular exponentiation superposition 7) Random choice of coprime m values for input into the function

✓ Correct Answer:

The correct answer is 6) Application of the Quantum Fourier Transform (QFT) to the modular exponentiation superposition.

📚 Reference Text:

a central component of the Shor factoris ation algorithm3. This algori thm is of central importance as its successful implementation would allow the prime factoris ation of very large numbers. This is of central importance to the breaking of cryptographic codes which , due to the very large s ize of the numbers employed comprising 1024 bits or more, is very lengthy or even intractable by other computing means . Quantum algorithms offer the potential for an exponential speed up in processing time and so would make factorisation of such large numbers realistically possible3. The Shor algorithm factorises an integer N = N1N2 by finding the period, r, of a function f(k) = mk mod N where 1< m

Question 1464 multiple-choice

Quantum dynamics provides a unified framework for analyzing both quantum algorithms and physical quantum systems. Understanding the connections between algorithmic structures and natural phenomena can inform advances in computation and simulation. Which quantum algorithm's structure most closely resembles the correlated behavior of protons in water wire systems, illustrating a deep analogy between quantum entanglement and chemical processes? 1) Shor’s algorithm 2) Bernstein-Vazirani algorithm 3) Deutsch-Jozsa algorithm 4) Simon’s algorithm 5) Grover’s algorithm 6) Quantum phase estimation 7) Quantum teleportation

✓ Correct Answer:

The correct answer is 1) Shor’s algorithm.

📚 Reference Text:

Title: Synthesis of Hidden Subgroup Quantum Algorithms and Quantum Chemical Dynamics. Year: 2022 Paper ID: 44d3844a8435815d4036c7428d54b1387eab00df Source: semantic-scholar URL: https://www.semanticscholar.org/paper/44d3844a8435815d4036c7428d54b1387eab00df Abstract: We describe a general formalism for quantum dynamics and show how this formalism subsumes several quantum algorithms, including the Deutsch, Deutsch-Jozsa, Bernstein-Vazirani, Simon, and Shor algorithms as well as the conventional approach to quantum dynamics based on tensor networks. The common framework exposes similarities among quantum algorithms and natural quantum phenomena: we illustrate this connection by showing how the correlated behavior of protons in water wire systems that are common in many biological and materials systems parallels the structure of Shor's algorithm.

Question 1465 multiple-choice

In lattice gauge theories and topological quantum systems, entanglement entropy can reveal nonlocal characteristics of ground states. The quantum double model provides a concrete example of how topological order manifests through entanglement measures. Which of the following correctly expresses the topological entanglement entropy (Stopo) in the quantum double model based on a finite group G? 1) Stopo = −αL, where α is a constant and L is the boundary length 2) Stopo = log|G| 3) Stopo = −γ, where γ is the area law coefficient 4) Stopo = −1/|G| 5) Stopo = −|G| 6) Stopo = 0 7) Stopo = −log|G|

✓ Correct Answer:

The correct answer is 7) Stopo = −log|G|.

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has at the end of this procedure agree at all the edges e1,...,er. One can continue to apply this protocol to any remaining paths from a vertex in Tto one of its its leaves which have not yet been traversed. Calling the overall gauge transformation one obtains G, the net result is that G|g/angbracketrightand|g/prime/angbracketrightagree at every edge in the spanning tree, and have identical holonomies. It follows that they agree on all of Λ, and so they are in fact equal. Now letv0be the lower-left hand corner of A. Recall that by construction, gAn=hnand meanwhileφ(g)An=φ(hn). These holonomies cannot be the same up to simultaneous conjugation: since every element of Gis realized as a holonomy, this would imply that φis not an outer automorphism, in contradiction with our assumption. Thus |g/angbracketrightis gauge-inequivalent from |φ(g)/angbracketright, and so|ψ/angbracketrightis orthogonal to|χ/angbracketright. 3.4 Topological entanglement entropy An important measure of topological order is topological entanglement entropy [14, 15]. The topological entanglement entropy refers to a particular term in the von Neumann entropy of the reduced state of the vacuum. This term is well-deﬁned whenever the entanglement entropy S(ρA) for a simply connected region Awith boundary of length Lsatisﬁes an “area law,” i.e. satisﬁes S(ρA) =αL−γ+o(1), (46) whereα,γaresomenonnegativeconstants. Inthiscase,theterm −γisreferredtoasthetopological entanglement entropy Stopo, Stopo=−γ. (47) Accepted in Quantum2020-09-17, click title to verify. Published under CC-BY 4.0. 14 While this deﬁnition often works for simple models, the L→∞limit on the lattice may not be well-deﬁned for generic models.11We therefore emphasize an alternative deﬁnition, which uses any three large, mutually adjacent regions A,BandCto deﬁne Stopo=S(ρABC)−S(ρAB)−S(ρAC)−S(ρBC) +S(ρA) +S(ρB) +S(ρC)(48) as in [14]. This deﬁnition makes it clear that Stopois independent of any local contributions from “sharp” features like corners on the boundaries of the regions, since such contributions will cancel in equation (48). Moreover, if we imagine a simple model in which entanglement entropy of every region obtains an additive contribution from each boundary edge as well as an overall additive constant−γ, then the precise deﬁnition in equation (48)matches the more heuristic deﬁnition in equation (46). Assuming the system is gapped (so that the vacuum is expected to satisfy an area law in entanglement), one can more generally argue that Stopois independent of both deformations of regionsA,B,C, and also perturbations of the Hamiltonian, so long as the system remains gapped [14]. It can therefore be used to characterize topological phases of matter. After analyzing the ground state of the quantum double model in §3.2, we are well-equipped to study the topological entanglement entropy. Using the explicit form of the ground state reduced density matrix given in equation (40), we ﬁnd S(ρA) = (L−1) log|G|=Llog|G|−log|G|. (49) By deﬁnition (46)or(48), we then ﬁnd Stopo=−log|G|. Recall that the sector of the quantum double model with Av= 1may also be considered as a gauge theory: we consider the operators Av(g)as gauge transformations, and then we restrict our attention to the gauge-invariant Hilbert space, or “physical” Hilbert space, Hphys={|ψ/angbracketright∈H:Av|ψ/angbracketright=|ψ/angbracketrightfor allv}. (50) In a gauge theory, unlike in an ordinary lattice theory, there are multiple,

Question 1466 multiple-choice

Quantum computation leverages both qubits and qudits, with advanced gate operations enabling increased algorithmic efficiency. Multi-value-controlled gates (MVCGs) are a key innovation in utilizing the full state space of qudits. Which statement best describes how a multi-value-controlled gate (MVCG) in a d-dimensional qudit system differs fundamentally from the Muthukrishan-Stroud (MS) gate? 1) The MVCG applies a fixed operation regardless of the control state, whereas the MS gate applies distinct operations for each control state. 2) The MVCG can perform a unique unitary operation for every possible control state, while the MS gate only reacts to a single designated control state. 3) The MS gate operates exclusively on superconducting circuits, while the MVCG is limited to trapped ions. 4) The MVCG requires entanglement between qudits, while the MS gate operates only on separable states. 5) The MS gate is reversible, whereas the MVCG is inherently irreversible. 6) The MVCG is implemented using classical multiplexers, while the MS gate uses quantum logic. 7) The MS gate enables error correction, while the MVCG does not support error correction.

✓ Correct Answer:

The correct answer is 2) The MVCG can perform a unique unitary operation for every possible control state, while the MS gate only reacts to a single designated control state..

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into qubit systems to create a hybrid qudit system offers the potential of improvement to quantum computation. 2.2.4 Qudit Multi-Level Controlled Gate For a qubit controlled gate, the control qubit has only two states so it is a “do-or-don ’t”gate. Qudits, on the other hand, have multiple accessible states and thus a qudit-controlled gate canperform a more complicated operation [ 46]. The Muthukrishan- Stroud gate (MS gate) for a qudit applies the speci ﬁed operation on the target qudit only if the control qudit is in a selected one ofthedstates, and leaves the target unchanged if the control qudit is in any other d−1 states. Hence, the MS gate is essentially a “do- or-don ’t”gate generalized to qudits and does not fully utilize the d states on the control qudit [ 118]. To fully utilize the dstates on the control qudit, people have developed the quantum multiplexer to perform the controlled U operations in a qudit system as shown in Figure 7 , where the MS gate and shifting gates are combined to apply different operationsto the target depending on different states on the control states[87]. Here we discuss the multi-value-controlled gate (MVCG ) for qudits, which applies a unique operation to the target qudit foreach unique state of the control qudit [ 106]. For a d-dimensional qudit system, a two-qudit multi-value- controlled gate is represented by a d 2×d2matrix MVCG /equals⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝U 000 / 0 0U10/ 0 00 U2/ 0 ««« 1 0 000 / Ud−1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (61) where each U i(i/equals0,1,... ,d−1) is a unique unitary single-qudit operation. The Uioperation is applied to the target qudit when the control qudit is in |Es〉state. In the later sections, Section 3.2.1 and Section 3.2.2 the controlled gates are MVCG and improve the ef ﬁciency of the qudit algorithm. MVCG can be built in many physical systems and one example in a photonicsystem is introduced in Section 5.1 . 2.3 Geometrically Quantifying Qudit-Gate Efﬁciency In a quantum computer, each qudit can remain coherent for a limited amount of time (decoherence time). After this time, the FIGURE 7 | d-valued Quantum Multiplexer for the second qutrit and its realization in terms of Muthukrishan-Stroud gates (the control Uoperation that only act on one speci ﬁc control state). The gate labeled +1 is the shifting gate that increases the state value of the control qudit by 1 (mod d). Depending on the value of the top control qudit, one of Uiis applied to the second qudit, for i∈{0,1,...d−1}. FIGURE 8 | Schematic view of the quantum circuit for the parity determining algorithm. FTis the Fourier transform and Ufkis the gate that does one of the two permutations and the last box represents the measurement. Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504 8Wang et al. Qudits and High-Dimensional Quantum Computing quantum information is lost due to the outside perturbations and noises. In the computation process, quantum gates take certainamount of time to alter the states of the qudits. The decoherencetime of

Question 1467 multiple-choice

In group theory and ring theory, the verbally Noetherian property investigates whether verbal subgroups and verbal ideals are finitely generated, relating group properties to the structure of their group algebras. Varieties of groups, defined by identities, are foundational in classifying group behaviors and understanding transfer of properties between groups and their algebras. Which of the following statements is true regarding the construction of a quotient of a free group of countably infinite rank and its group algebra over a field of characteristic 2? 1) The quotient group is not verbally Noetherian and its group algebra is verbally Noetherian. 2) The verbal subgroup used is always infinitely generated. 3) The group algebra is verbally Noetherian for all fields, regardless of characteristic. 4) The quotient group is verbally Noetherian, but its group algebra over a field of characteristic 2 is not verbally Noetherian. 5) The Specht property is necessary for the construction. 6) The construction is only possible for finite rank free groups. 7) The verbal Noetherian property always passes from groups to their group algebras.

✓ Correct Answer:

The correct answer is 4) The quotient group is verbally Noetherian, but its group algebra over a field of characteristic 2 is not verbally Noetherian..

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if a relatively free group F/ V is not verbally Noetherian then so is the group algebra K(F / V) for every K (if N is a non-finitely generated verbal subgroup in FI V then the ideal generated by the set (N — 1) is a non-finitely generated verbal ideal in K(F/ V)). No other ways to get examples of non-(verbally Noetherian) group algebras of relatively free groups of countably infinite rank over a Noetherian ring were known. The following problem is equivalent to the one posed by Plotkin and Vovsi (see [8, Problem 4.2.8]). Does there exist a verbal subgroup U in a free group F of countably infinite rank such that U is finitely generated (as a verbal subgroup) and satisfies the followingconditions: (i) FI U is verbally Noetherian; (ii) over some field K the group algebra K(F/ U) is not verbally Noetherian? We resolve this by proving the following theorem. Let(x, y) = xy—yx, {x, y, z) = ((x, y), z) — xyz — yxz — zxy + zyx and let ab = b~xab. Define aUv) = axya~~yx and a(x-y-z) = axyza-yxzar"yazyx. Let a"v = (au)v. THEOREM 1. Let K be an arbitrary field of characteristic 2, F the free group of countably infinite rank on free generators x\, X2, • •., U the verbal subgroup of F generated (as a verbal subgroup) by the elements (1) [[*1,*2,.X3L[*4.*5,*6]]. (2) [x ux2,x3](M'Xi-x<')(Xl-x*). Then the group F/ U is verbally Noetherian but the group algebra K(F/ U) is not verbally Noetherian. https://doi.org/10.1017/S1446788700002056 Published online by Cambridge University Press [3] Solution of a problem of Plotkin and Vovsi 331 REMARK. In fact, some extra calculations show that Theorem 1 remains valid if one replaces the element (2) with [x ux2, x3YXi>X5-X6). 2. The techniques developed in order to prove Theorem 1 can be applied to varieties of groups. Let x\, x2,... be free generators of the free group F. For any v — v(x\,... ,x n) 6 F, v = 1 is said to be an identity (or a law or an identical relation) of a group G if v(g u ... , g n) = 1 for all g u ... , g n € G. The class of all groups satisfying a given set of identities is called a variety of groups. We refer to Neumann [6] for terminology and basic facts related to identities and varieties of groups. A variety of groups V is called finitely based if V can be defined by a finite set of identities. A group variety V is called Specht (or hereditarily finitely based) if all subvarieties of V including V itself are finitely based (equivalently: if each group in V has a finite basis for its identities). Many varieties of groups are known to be Specht; in particular, this applies to the variety N CA of all (nilpotent of class at most c)-by-Abelian groups for each c (Cohen [2] for c = 1, Bryant and Newman [1] for c = 2,

Question 1468 multiple-choice

In quantum computing and mathematical physics, operators on group algebras such as C(Sn) play a central role, with properties like Hermiticity and unitarity ensuring meaningful quantum evolution and measurement. Quantum phase estimation (QPE) is a key algorithm for extracting eigenvalues of unitary operators, leveraging representation theory and inner product structures. Which statement best describes a consequence of the Hermitian nature of the operator Tµ with respect to the inner product g on the group algebra C(Sn)? 1) All eigenvalues of Tµ are guaranteed to be real, allowing the construction of unitary operators Uµ with well-defined phases. 2) Tµ necessarily commutes with all elements of the symmetric group Sn. 3) The operator Tµ must be diagonalizable only in the standard basis of C(Sn). 4) Tµ cannot be used to construct unitary operators relevant for quantum algorithms. 5) The eigenvalues of Tµ are always integer-valued. 6) Tµ maps each permutation in Sn to its inverse. 7) The inner product g becomes degenerate when restricted to the image of Tµ.

✓ Correct Answer:

The correct answer is 1) All eigenvalues of Tµ are guaranteed to be real, allowing the construction of unitary operators Uµ with well-defined phases..

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the ﬁrst column has length no greater than Nis irrelevant. This equality Λ(n) =k∗(n)will be used in section 4 where we describe a classical gravity analog of the projector detection task. There is a potential issue in that strictly speaking the validity of the semi-classical super-gravity solution in AdS5/CFT4 requires n∼N2. It is possible that this is not an issue for the identiﬁcation of the highly supersymmetric LLM geometry itself, but only in questions related to ﬂuctuations of the LLM geometry. Another possible scenario for physically justifying the semi-classical LLM calculation in section 4 is that the LLM plane may be relevant to a dimensionally reduced model of AdS/CFT, e.g. of the kind considered in the context of spin matrix theory [ 47]. 2.1.3 Unitary operators We introduce the following operators on C(Sn). Consider the linear conjugation operator S:C(Sn)→C(Sn)maps a linear combination A=/summationtext iciσi∈C(Sn)toS(A) :=/summationtext iciσ−1 i. We easily check that S2= idand callSan involution. Let δ:C(Sn)×2→Cbe the symmetric bilinear pairing that acts on two group algebra elements A=/summationtext iciσiand B=/summationtext jc/prime jσj, as δ(A,B) =/summationdisplay i,jcic/prime jδ(σiσj) (2.14) – 6 – JHEP05(2023)191withδ(s) = 1ifσ= idand0otherwise. Note that δis also extended by linearity to C(Sn) in the following. We then deﬁne a sesquilinear form on C(Sn)×2by g(A,B) =δ(S(A),B), (2.15) where ¯x∈Cstands for the complex conjugation. One checks that gis nondegenerate and therefore induces an inner product on C(Sn). We recall a useful proposition and its proof from [ 32]. Proposition 1. For anyµ/turnstileleftn,Tµis a Hermitian operator acting on C(Sn)endowed with its inner product g. Proof.We want to verify the identity g(A,TµB) =g(TµA,B)forA=/summationtext iciσiandB=/summationtext jc/prime jσj. We have g(A,TµB) =/summationdisplay i,j¯cic/prime jδ(S(σi)Tµσj) =/summationdisplay i,j¯cic/prime jδ(S(Tµσj)σi) =/summationdisplay i,j¯cic/prime jδ(S(σj)S(Tµ)σi) =/summationdisplay i,j¯cic/prime jδ(S(σj)Tµσi) =/summationdisplay i,j¯cic/prime jδ(S(Tµσi)σj) =g(TµA,B) (2.16) where at an intermediate stage we use S(AB) =S(B)S(A),S(Tµ) =Tµ, andδ(S(A)) = δ(A). Tµ, as a hermitian operator, determines a unitary operator Uµ=e2πi χmaxµTµwith phases given by the normalized characters /hatwideχR(Tµ)/χmax µ:Uµ=e2πi χmaxµTµPR=e2πi χmaxµ/hatwideχR(Tµ)PR. It is therefore natural to use QPE to estimate the phases of such unitaries. The next task uses QPE with C- Uµ-gates to identify the corresponding normalized characters /hatwideχR(Tµ)and from that addresses new questions. 2.2 Quantum phase estimation We review in this section the QPE algorithm and its run time complexity. We use Nielsen and Chuang’s notation, see section 5.2 in [ 7]. Readers with a background in quantum information theory may wish to skip this section and simply glance at the last paragraph of this section. Those with an AdS/CFT and mathematical physics background but little quantum information, like ourselves at the start of this project, will hopefully ﬁnd it useful. Consider a unitary operator Uwith an eigenstate |ψ/angbracketrightof eigenvalue e2πiλ, for an unknown λ. QPE is a quantum algorithm that aims at approximating the value of λ. It uses oracles (black boxes) giving access to controlled- U2j(C−U2jor shortly CU) operations, for some j, and capable of preparing an initial state |ψ/angbracketright. As in many quantum algorithms, the fast Fourier transform deﬁnes a key subroutine of QPE. Finally the last

Question 1469 multiple-choice

In computational algebraic number theory, algorithms often combine random sampling, classic number-theoretic procedures, and heuristic assumptions to efficiently construct special algebraic integers or solve norm equations, especially for cryptographic applications. Coprimality, powersmoothness, and polynomial time guarantees are key concepts in designing robust procedures for both prime and composite moduli. Which of the following statements accurately describes Lemma 5.5's mathematical result regarding the number of solutions to a linear equation modulo a composite integer N? 1) It asserts that the number of solutions is always prime when N is composite. 2) It states that the number of solutions is equal to Euler's totient function of N. 3) It claims that solutions exist only if N is a product of powersmooth numbers. 4) It provides a bound on the number of solutions based on the largest prime factor of N. 5) It determines that the number of solutions equals the number of divisors of N. 6) It states that the number of solutions depends on whether N is square-free. 7) It gives a formula for the number of solutions based on the Chinese Remainder Theorem, expressing it as a product over solutions modulo the prime power factors of N.

✓ Correct Answer:

The correct answer is 7) It gives a formula for the number of solutions based on the Chinese Remainder Theorem, expressing it as a product over solutions modulo the prime power factors of N..

📚 Reference Text:

( n(C+Dj),N) = 1 and n(C+Dj) is powersmooth; ii) ( n(C),N) = 1; iii) ( n(D),N) = 1; iv) (N,ImR(A¯BC¯D)) = 1 and v) 4 q(4p2n(ABCD)−n(AC)+pn(AD)) is a square modulo N. 1Randomly generate M > plogǫpwithǫ >8 that is powersmooth and coprime toN. 2Setm=⌊/radicalBig M p(q+1)⌋and sample random integers z,t∈[−m,m]2. 3SetM′=M−p(z2+qt2). 4If(z2+qt2,N)/ne}ationslash= 1or(M′,N)/ne}ationslash= 1orM′is not a prime then 5Go back to Step 2. 6IfCornacchia (M′) =⊥then 7Go back to Step 2. 8x,y=Cornacchia (M′). 9C←x+yi,D←z+ti. 10If(N,ImR(A¯BC¯D))/ne}ationslash= 1then 11Go back to Step 2. 12If4q(4p2n(ABCD)−n(AC)+pn(AD))is not a square modulo Nthen 13Go back to Step 2. 14Return C,D. Heuristic 5.2. We assume that M′,z2+qt2,ImR(A¯BC¯D)and4q(4p2n(ABCD)− (n(AC)−pn(AD))2)appearing in Algorithm 1 behave like random integers of the same size. Lemma 5.3. LetNbe as in Theorem 5.1, assuming Heuristic 5.2, Algorithm 1 returns a solution in polynomial time. Proof.SinceM′,z2+qt2,ImR(A¯BC¯D)and4q(4p2n(ABCD)−((n(AC)−pn(AD))2) are assumed to behave like random integers, then Step 4 succeeds with probabil- ity greater than 1 /O(log3p), and Step 10 succeeds with probability greater than 1/O(logp). Sinceqis of sizeO(log2p) and #Cℓ(Z[i]) is approximately O(logp), Title Suppressed Due to Excessive Length 23 Step 6 succeeds with probability greater than 1 /O(logp). Finally, Step 12 suc- ceeds with probability greater than 1 /O(logp). In total, the success probability is greater than 1 /O(log6p). Sinceǫis chosen to be greater than 8, there will be enough ( z,t) pairs from Step 2 to ensure Algorithm 1 succeeds. Therefore, Algorithm 1 will return a solution in polynomial time. ⊓ ⊔ We also provide a generalization of the StrongApproximation algorithm to allow for composite N(note that in this algorithm we don’t have restriction on the number of divisors of N). The sub-index psrefers to powersmooth. Algorithm 2: StrongApproximationps(N,µ0) Input:An odd integer Nandµ0∈Rjsuch that ( n(µ0),N) = 1. Output: λ∈Zsuch that ( λ,N) = 1 and µ∈Rwith powersmooth norm such thatµ=λµ0modNO0. 1Writeµ0as (t+si)jwitht,s∈Z. 2LetS={ℓsuch that ℓ|Nand/parenleftbign(µ0) ℓ/parenrightbig =−1}. 3Randomly generate # Smany prime pi’s such that pi pN4. 9Denote one of the square root ofF n(µ0)byλ. 10Randomly generate c,dsuch that λp(2tc+2qsd)≡(F−λ2p(t2+qs2))/NmodN. 11SetM= (F−p((λt+cN)2+q(λs+dN)2))/N2. 12IfMis not a prime or Cornacchia (M′) =⊥then 13Go back to Step 10 14a,b=Cornacchia (M). 15Return λµ0+N(a+bi+(c+di)j),λ. Heuristic 5.4. We assume that Mappearing in Algorithm 2 behaves like a random integer of the same size. Lemma 5.5. Consider a linear equation N1x+N2y=N3modNwheregcd(N,N1) = d1,gcd(N,N2) =d2and(d1,d2) = 1. Then this equation hasN d1d2solutions in (Z/NZ)2. Proof.This can be seen by checking how many solutions there are for equat ion N1x+N2y=N3modℓei iwithℓei ibeing a prime power divisor of Nand then using Chinese Reminder Theorem. ⊓ ⊔ 24 M. Chen, M. Imran, G.Ivanyos, P.Kutas, A.Leroux, C. Petit Lemma 5.6. LetNbe an odd integer, assuming Heuristic 5.4, Algorithm 2 returns a solution in polynomial time with overwhelming pro bability. Proof.The #S×#Slinear equations behave like a random system of linear equations of the

Question 1470 multiple-choice

Distributed quantum computing enables the scaling of quantum computers by networking multiple quantum processing units (QPUs) using quantum communication channels. Efficient entanglement generation between nodes is fundamental to the performance and reliability of these systems. Which architectural feature in a distributed quantum computing network most directly increases the probability of successful entanglement generation between nodes? 1) Using only local qubits for computation at each node 2) Deploying multiple communication qubits per node and running parallel entanglement attempts 3) Increasing the number of algorithmic fidelity benchmarks 4) Relying exclusively on superconducting qubits 5) Implementing single-photon detectors with low efficiency 6) Avoiding mid-circuit measurements in distributed quantum circuits 7) Connecting nodes using only microwave links over long distances

✓ Correct Answer:

The correct answer is 2) Deploying multiple communication qubits per node and running parallel entanglement attempts.

📚 Reference Text:

The simulation of distributed quantum algorithms Sreraman Muralidharan1 1Wells Fargo, New York, NY, USA Abstract Distributed quantum computing (DQC) provides a way to scale quan- tum computers using multiple quantum processing units (QPU) connected through quantum communication links. In this paper, we have built a dis- tributed quantum computing simulator and used the simulator to investi- gate quantum algorithms such as the quantum Fourier transform, quan- tum phase estimation, quantum amplitude estimation, and generation of probability distribution in DQC. The simulator can be used to easily gen- erate and execute distributed quantum circuits, obtain and benchmark DQC parameters such as the fidelity of the algorithm and the number of entanglement generation steps, and use dynamic circuits in a distributed setting to improve results. We show the applicability of dynamic quan- tum circuits in DQC, where mid-circuit measurements, local operations, and classical communication are used in place of noisy inter-processor (non-local) quantum gates. 1 Introduction To make quantum computers advantageous in industry, the scalability of the hardware needs to be significantly increased [1, 2, 3, 4]. Due to the hardware limitations, increasing the qubits in the same device, increases the cross talk, decoherence that causes the effective gate and the measurement errors to in- crease [5]. Distributed quantum computing (DQC) [6, 7] offers an interesting approach, where instead of scaling up, we use multiple quantum processing units (QPUs) with fewer qubits connected through quantum communication links to increase computational capabilities [8, 9]. The QPUs can be separated by short distances, i.e., chiplets connected through microwave links [5] or long distances connected through optical fibers/photonic links [10]. The nodes of the DQC (Fig. 1) consist of the local qubits necessary for computation and communica- tion qubits [11], which are used to entangle the nodes and apply quantum gates between them [12, 13, 14, 10]. The communication qubits can be entangled us- ing a probabilistic entanglement generation procedure that relies on successful detector click patterns [15, 16, 17]. The generation process is repeated until suc- cessful click patterns are obtained, indicating that the qubits are entangled. En- tanglement generation between remote nodes has been experimentally realized 1arXiv:2402.10745v3 [quant-ph] 26 Feb 2025 (a) (b) Figure 1: Schema for the DQC, a) Showing multiple quantum processing units. b) A two-node quantum computer showing entangled (dark blue) communica- tion qubits and local (blue) processing qubits in superconducting qubits [17], ions [16, 10], and NV centers [15] respectively. We will briefly study the probability of success of this procedure. Suppose the channel’s total efficiency (including qubit-node coupling) is p. Forkrepetitions (trials), the probability of obtaining at least one entangled pair with one communication qubit/node is given by P= (1−(1−p)k). To increase the probability of success, we can have multiple communication ( m) qubits per node, where the entanglement generation procedure is run for each pair in parallel. The probability that we have at least one entangled pair is given byPs= 1−(1−P)m. The entangled pair is used to apply a remote quantum gate between local qubits in the

Question 1471 multiple-choice

In integrable systems and conformal field theory, W-algebras and their realization via the Miura map play a crucial role in understanding symmetries and conserved quantities. The interplay between gauge symmetries, Poisson structures, and Lie algebra representations is central to the construction of these models. Which of the following statements correctly describes the role of the generalized Miura map in the context of finite W-algebras associated with the direct sum Lie algebra sl₂ ⊕ u(1)? 1) It projects the W-algebra generators onto the center of the universal enveloping algebra. 2) It defines an isomorphism between the finite W-algebra and the Cartan subalgebra of sl₂. 3) It provides an injective Poisson homomorphism from the finite W-algebra into the Kirillov Poisson algebra of sl₂ ⊕ u(1). 4) It eliminates all gauge degrees of freedom by mapping to abelian subalgebras only. 5) It maps the finite W-algebra generators to quadratic Casimir invariants of u(1). 6) It identifies the W-algebra structure with the Virasoro algebra alone. 7) It transforms the Poisson brackets into commutative multiplication for all generators.

✓ Correct Answer:

The correct answer is 3) It provides an injective Poisson homomorphism from the finite W-algebra into the Kirillov Poisson algebra of sl₂ ⊕ u(1)..

📚 Reference Text:

t o be second class. The result is that both constraints in (3.84) are ﬁrst class. The point however is that the gauge symmetry induced by the second constraint in eq. (3.84 ) can be completely ﬁxed by adding the gauge ﬁxing condition J1 2,1 2 (1)= 0 (3.85) which then leaves us with exactly the same set of constraints and gauge invariances as before. We can now describe the generalized Miura map for this case. A n arbitrary element oft++g0is given by J0≡ h+s e 1 f s −h0 0 0 −2s  (3.86) Note thatg0≃sl2⊕u(1) which means that the Poisson relations in K(g0) between the generatorsh,e,f ands(viewed as elements of C∞(g0)) are given by {h,e}=e {h,f}=−f {e,f}= 2h (3.87) 30 andscommutes with everything. As shown in the previous section t he equation that we have to solve in order to get explicit formulas for the Miur a map is is the following equation for h∈ˆH(where ˆHis again the group of gauge transformations generated by the two ﬁrst class constraints (3.84)). hJ0h−1= J0,0 (1) 0 1 J1 2,−1 2 (2) −2J0,0 (1)0 J1,−1 (1)J1 2,−1 2 (1)J0,0 (1) (3.88) The unique solution of this equation is given by h= 1 0 0 0 1 0 3 2s+1 2h e 1  (3.89) Inserting this back into eqn.(3.88) one ﬁnds certain expres sions forJ0,0 (1),J1 2,−1 2 (1),J1 2,−1 2 (2) andJ1,−1 (1)in terms of the functions {e,f,h,s }. In terms of the generators (3.81) these read H= 2h−2s E= (3s−h)e F=4 3f C=−4 3(h2+ 3s2+fe) (3.90) It is easy to check, using the relations (3.87) that these sat isfy the algebra (3.82). Therefore we ﬁnd that indeed (3.90) provides an injective Po isson homomorphism from W(2) 3into the Kirillov Poisson algebra of the Lie algebra g0=sl2⊕u(1) . In the appendix we consider more examples of ﬁnite Walgebras and their quantum versions. In particular we construct all ﬁnite Walgebras associated to sl4(to get the classical versions of the algebras displayed in the appendi x just replace commutators by Poisson brackets and forget about all terms of order ¯ h2or higher). 4. FiniteWSymmetries in Generalized Toda Theories 31 It is well known [21] that inﬁnite Walgebras arise as algebras of conserved cur- rents of Toda theories. The inﬁnite Walgebras related to arbitrary sl2embeddings are related to the conserved currents of so-called generali zed Toda theories [22, 7]. Since ﬁnite Walgebras are essentially a dimensional reduction of inﬁnit eWalgebras, one expects that there are one-dimensional analogues of the ordinary two-dimensional Toda actions whose conserved currents are related to a ﬁnite Walgebra. Indeed, the construction of these one-dimensional Toda actions is comp letely straightforward, and has the desired properties. One starts with the action for a free particle moving on the gr oup manifold G= SL(n). The metric on Gis given by extending the Cartan-Killing metric ( ta,tb) = Tr(tatb) on the Lie algebra gall overGin a left-right invariant way. This leads to the familiar action S[g] =1 2/integraldisplay dtTr/parenleftigg g−1dg dtg−1dg dt/parenrightigg

Question 1472 multiple-choice

Commitment schemes are fundamental cryptographic protocols that allow one party to lock in a value while keeping it hidden, with properties crucial for privacy and security in applications such as zero-knowledge proofs and secure voting. The interplay between hiding, binding, and knowledge soundness determines their robustness against adversaries. In an interactive sigma protocol for proving knowledge of the opening of a commitment, which property is achieved by using a rewinding-based extractor to obtain two valid responses to different challenges for the same commitment? 1) Unconditional hiding of the committed value 2) Computational indistinguishability between different commitments 3) Perfect completeness of the protocol 4) Resistance to chosen-message attacks 5) Non-malleability of the commitment 6) Simulation soundness for zero-knowledge 7) Knowledge soundness of the proof system

✓ Correct Answer:

The correct answer is 7) Knowledge soundness of the proof system.

📚 Reference Text:

as in a normal execution of the commitment scheme. Given that P∗breaks the binding property as described above, this allows us (except with negligible probability) to compute M, a multiple of the order of h. Now choose anytthat is relatively prime to Mand outputht−1modMandt. Summarizing, we have: Theorem 1. Under the root assumption, the above scheme is an unconditionally hiding and computationally binding commitment scheme. If the root assumption is(t(k),zepsilon<(k))-secure, then the binding property is/parenleftBig t(k),/epsilon1(k)l(k) 1−2−γk/parenrightBig -secure for some constantγ. 5 AssociatedProtocols 5.1 Proving You Know How to Open The following protocol can be used by Pto show that he can open a given commitment c=gxhr. We will assume that xis in [−T..T] whereT(>0) is a public constant. Tcan be chosen arbitrarily large, and is only used to control the size of the prover’s random choices, this allows an honest prover to ensure that the protocol hidesthe value of x, whenever −T≤x≤T. In any application of the scheme, one simplychooses Tlargeenoughtoaccommodateanychoiceof xanhonestprover would need to make in the given scenario. The protocol guarantees an honestveriﬁer that −TC(k)(2 k+2)≤x≤TC(k)(2k+2). To prove xis in some other (smaller) interval, other techniques exist, see e.g. [4]. 1.Pchoosesy∈[0..TC(k)2k[,s∈[0..C(k)2B+2k[ at random and sends d= gyhstoV. 2.Vchooses at random e∈[0..C(k)[ and sends to P. 3.Psendsu=y+ex,v=s+er∈Z.Vchecks that guhv=dceand that [−TC(k)..TC(k)(2k+1)]. AStatistically-HidingIntegerCommitmentScheme 135 Deﬁne a relation generator Ras follows: run G(1k)t og e tG, chooseh∈H withC(k)-rough order, set g=hαand output descr(G),g,h. Then deﬁne the relationR={(c,(µ,x,r))|c,b∈G, c=µgxhr,µlG=1,x∈[−TC(k)(2k+ 2)..TC(k)(2k+2)]}. We now analyze to what extent the protocol above satisﬁes our deﬁnition of knowledge soundness, in particular for which knowledge error functions is the deﬁnition satisﬁed. Accordingly, let κ() be any knowledge error function, such thatκ(k)≥4/C(k) for allk. We then must deﬁne an extractor M. Let a poly- nomial time prover P∗be given and let viewbe any view P∗may have after having produced a commitment c. Now, it can be shown that since there are C(k) diﬀerent challenges, then if zepsilonκ(k)≥4/C(k), standard rewinding techniques allow us to obtain in expected polynomial time a situation where, fora givend,P ∗has correctly answered two diﬀerent values eande/primewith numbers u,vandu/prime,v/prime,sow egetgu−u/primehv−v/prime=ce−e/prime.Let Rewindbea(probabilistic)pro- cedure that creates e,e/prime,u,v,u/prime,v/primein this way. A concrete algorithm for Rewind is given in Appendix A. It runs in expected time 56 /zepsilone/primeandsupposethat( e−e/prime)divides both (u−u/prime) and (v−v/prime). We now see that the element µ=gu−u/prime e−e/primehv−v/prime e−e/primec−1 satisﬁes that µe−e/prime= 1. Sincee−e/prime

Question 1473 multiple-choice

Quantum computing architectures utilizing nuclear and electron spins are being developed to improve scalability and gate fidelity. Both liquid-state NMR and solid-state spin-based approaches face challenges with decoherence and qubit control, but differ in operation speed and scalability potential. Which quantum computing scheme achieves higher fidelity for implementing the Quantum Fourier Transform but faces scalability limitations due to spectral crowding and increased decoherence as qubit count grows? 1) Liquid-state NMR using serial pulse sequences 2) Solid-state silicon donor spins with isotopic engineering 3) Parallel QFT in liquid-state NMR 4) Paired electron-nuclear spin architecture with SWAP gates 5) Selective-pulse scheme in liquid-state NMR 6) Kane’s silicon-based proposal with single-spin measurement 7) Superconducting qubits with microwave control

✓ Correct Answer:

The correct answer is 5) Selective-pulse scheme in liquid-state NMR.

📚 Reference Text:

in NMR) [7] F=Tr(ρthρexp) /radicalbig Tr(ρ2 th)/radicalig Tr(ρ2exp)/radicaligg Tr(ρ2exp) Tr(ρ2 init) The ﬁrst term in the expression measures the correla- tion between the experimental deviation density matrix ρexpand the theoreticaldeviationdensity matrix ρth(ob- tained by “applying” the unitary operator corresponding to the ideal QFT transformation, to the initial density matrixρinit). The second term is the weighting factor to take into account the overall signal loss due to decoher- ence during the experiment. The ﬁdelities measured for the serial, parallel and selective-pulse versions of the QFT are 79%, 80% and 85% respectively. The reduction in ﬁdelity is mainly due to imperfections in pulse calibrationas well assystem de- coherence. It is not surprising that the serial and parallelversions are equally accurate for the case of two qubits. The savings in time and the increase in accuracy of the parallel QFT will be realised only for a larger number of qubits. The better performance of the selective scheme is due to the fact that such a direct implementation of the square-root of the controlled-NOT gate does not require refocusing schemes [26, 27]. However, in systems with a larger number of qubits such selective pulse schemes might not be feasible, the major stumbling blocks in such cases being decoherence during the pulses and the over- lap of transitions in crowded spectra. V. OTHER SPIN-BASED ARCHITECTURES Recently, several approaches have been suggested for the design of solid-state spin-based quantum computers. Kane’s proposal [15] using single donor spins in Si, ad- dresses the problem of scalability but has the disadvan- tagesinherentinsingle-spinmeasurements. Laddet. al.’s solid-stateNMR quantum computing device on the other hand, is made entirely of silicon, with the qubits being spin-1/2 nuclei located in isolated atomic chains [16]. Suter et. al. [17] proposed an alternative architecture with each logical qubit being represented by two phys- ical qubits - an active electron spin to manipulate quan- tum information and a passive nuclear spin to store in- formation. A logical qubit is addressed using magnetic ﬁeld gradients and SWAP gates, realised as a cascade of three transition-selective pulses, are used to convert be- tween active and passive states. A basic two-qubit gate relies on the dipolar interaction between electron spins and requires four additional SWAP gates, two to switch between active and passive states and two back-SWAPS to switch oﬀ the interaction between the neighbouring qubits. The hyperﬁne interaction is of the order of a few MHz and the electron dipolar interaction strength is around 10-50 MHz. An estimate of the actual time-cost of the QFT for such a solid-state spin quantum computer yields Tser=nδ+n−1/summationdisplay j=0n/summationdisplay k=j+1(τjk+2∗τSWAP) =nδ+2n∆+κn−1/summationdisplay j=0n/summationdisplay k=j+12j−k =nδ+2n∆+κ(n−1+2−n)≈O(n) (10) whereδis the time-cost of each single-qubit Hadamard rotation,κ=π/d,dis the strength of the dipolar inter- action and ∆ is the time unit of one SWAP gate. Since the gate times for this implementation are very fast, a greater number of logical operations compared to liquid- state NMR computers, can be performed within the sys- tem decoherence limit. These solid-state proposals are also scalable to a very large number

Question 1474 multiple-choice

In quantum computing and computational group theory, the Hidden Subgroup Problem (HSP) is a foundational challenge whose solution underpins quantum algorithms for factoring, discrete logarithms, and graph isomorphism. Algorithms for HSP often rely on properties of cosets, subgroup normality, and the behavior of functions that hide subgroups within finite groups. When extending algorithms for the Hidden Subgroup Problem from abelian to non-abelian groups, which obstacle arises due to the possible lack of subgroup normality, and what is its direct consequence for the reduction strategy? 1) The subgroup may be infinite, leading to unbounded query complexity. 2) Cosets may overlap, making collision detection unreliable. 3) The quotient group may be undefined, preventing partitioning into cosets and invalidating the reduction approach. 4) The function hiding the subgroup may no longer be constant on cosets, causing incorrect outputs. 5) The group operation may not be associative, disrupting algorithmic steps. 6) The algorithm may require exponentially more queries due to loss of abelian structure. 7) The intersection of subgroups may not be trivial, resulting in ambiguous subgroup identification.

✓ Correct Answer:

The correct answer is 3) The quotient group may be undefined, preventing partitioning into cosets and invalidating the reduction approach..

📚 Reference Text:

as left cosets of H1inGare subsets of left cosets of HinG, and the 10 function fis constant on left cosets of H. Moreover, the function f1hides the subgroup H/H1 ofG/H1, as the left cosets of HinGcorrespond to left cosets of H/H1inG/H1. Finally, the function f1may be evaluated with one query to the oracle for f. Using this reduction, we may ﬁnd the hidden subgroup Husing Algorithm 2. The correct- ness of the algorithm follows by observing that in any iterat ion, if f1is injective, then H1equals the hidden subgroup. If f1is not injective, by Theorem 5.2, Find-Collision( G1,f1) returns a colli- sion g1H1,g2H1∈G/H1forf1. Note that g1,g2is a collision for f. We also have g−1 1g2/\⌉}atio\slash∈H1, so along with g−1 1g2, the set Sgenerates a larger subgroup of H. Thus, the size of the subgroup H1 increases by a factor of at least 2 in every iteration a collis ion is found, and the algorithm termi- nates after at most log2miterations. In every iteration of Algorithm 2, the ratio of t he order of G1 and the hidden subgroup H/H1equals n/m. The query complexity of the algorithm now follows from Theorem 5.2. It is not clear how to extend Algorithm 2 to the non-abelian ca se, since the group H1may not be normal in general, and the corresponding quotient gro up may not be deﬁned. We present a different algorithm, Algorithm 4, that works for some abel ianandsome non-abelian instances (which we describe after Theorem 6.3). Algorithm 4 builds on Algorithm 3, which is a variant of Algorithm 1 and is also based on Lemma 5.1. Algorithm 3: Find-New-Collision( G,H1,f) Input : group Gof order n, with n>1; subgroup H1≤G; oracle for f:G→Sthat hides a subgroup containing H1 Output : no-new-collision , orcollision a,b∈Gsuch that a−1b/\⌉}atio\slash∈H1 1Letkbe the integer lsuch that n∈(2l, 2l+1]; 2ifG is abelian then m1←max{2,|H1|}; 3else m1←|H1|; 4k0←⌈log2m1⌉−1 ; 5while k≥k0do 6 Find, if there is one, a subgroup G1≤Gwith the largest order in/bracketleftbig n/2k+1,n/(⌊2k⌋+1)/bracketrightbig such that G1∩H1={e}, where eis the identity element ofG; /* the expression ⌊2k⌋is required to correctly handle the case k=k0=−1*/ 7 ifsuch a subgroup G 1exists then 8 Find a generating pair S1,S2forG1which minimizes max {|S1|,|S2|}; 9 ifG1H1=Gthen g←e, the identity element of G; 10 else g←any element in G\(G1H1); 11 LetR←S−1 1∪S2∪/parenleftbig S−1 1{g}/parenrightbig ; 12 Query fat all the elements in R; 13 iff(z) = f(y)for some z ,y∈R such that z−1y/\⌉}atio\slash∈H1then return collision z,y; 14 k←k−1 ; 15returnno-new-collision Theorem 6.2. Let(G,f)be an instance of the Hidden Subgroup Problem, H the subgroup t hat f hides, and H 1a subgroup of H. Let the orders of G ,H,H1be n,m,m1, respectively. Suppose G has a subgroup G 0 of order n 0such that n 0≥n/m, and G 0intersects H 1only in the identity element. Then Find-New- Collision(G ,H1,f ) (Algorithm 3) returns “no new collision” if H 1=H, and returns a collision a ,b∈G 11 such that a−1b/\⌉}atio\slash∈H1otherwise. Further, if n 0≤κn/m for some κ≥1, the algorithm has query complexity as stated

Question 1475 multiple-choice

Quantum Process Tomography (QPT) is a crucial technique for characterizing quantum operations and diagnosing error mechanisms in quantum computing platforms such as Nuclear Magnetic Resonance (NMR) quantum processors. Understanding distinct error types is essential for improving the fidelity and scalability of quantum devices. Which type of error in quantum information processing is characterized by systematic, non-uniform unitary evolution across different members of an ensemble, often arising from spatial variations such as magnetic field gradients in NMR experiments? 1) Random measurement error 2) Incoherent error 3) Coherent error 4) Decoherent error 5) Fault-tolerant error 6) Statistical sampling error 7) Hardware-induced thermal error

✓ Correct Answer:

The correct answer is 2) Incoherent error.

📚 Reference Text:

from th e superoperator implied by the model in terms of speciﬁc defects in the model. Additional err ors introduced during the QPT procedure itself further complicate the analysis of the QPT results. In this paper we explore these issues via a concrete case study, in which QPT is per formed on a previ- ously reported three-qubit quantum Fourier transform (QFT) imp lemented using a nuclear magnetic resonance (NMR) quantum information processor [8]. The dynamics of an isolated quantum system are described by the Sc hr¨ odinger equation, which gives rise to an N×Nunitary operator, where Nis the dimension of system’s Hilbert space. Open quantum systems, however, generally interact with a ninaccessible environment and thereby undergo decoherence [5]. Furthermore, in the case o f expectation value mea- surements as in liquid-state NMR, each element of the statistical en semble may undergo a slightly diﬀerent unitary operation which, though nonrandom, is diﬃc ult to distinguish from decoherence [9, 10]. The statistics of measurements on open quan tum systems are generally described by an N×Ndensity operator, and the evolution of a density operator under a n incoherent distribution of Hamiltonians and/or interactions with an e nvironment, although non-unitary, remains linear and is described by a superoperator. T he goal of quantum process tomography is to determine this superoperator. Methods for implementing QPT have been presented Refs. [5, 6, 7, 1 2, 13], and two- qubit NMR implementations of QPT have previously been reported [14, 15]. In the present paper, QPT is carried out with the aim of validating the mathematical m odel of the system- plus-apparatus used to design the NMR implementation of a multi-qub it, entangling unitary 2 operation, andtoidentifythetypesandstrengthsoftheerrors thatoccurred. Wehavefound it useful to classify the errors as coherent, incoherent, and dec oherent, because each class is related to speciﬁc short-comings in the experimental implementatio n. Coherent errors are systematic errors in the net unitary operation that was actually imp lemented. Incoherent errors refer to unwanted unitary evolution which is not uniform acr oss the ensemble of spin systems in the NMR sample, so that even though each member of the ensemble undergoes strictly unitary evolution, the evolution of the ensemble averages a ppears non-unitary. The eﬀects of incoherence are reversible, at least in principle, and know ledge of the coherent and incoherent errors can be used to design better quantum gates (u nitary operations). In con- trast, decoherent errors are due to unknown interactions with a n inaccessible environment, so they are not reversible and can be eliminated only by relatively cost ly changes to the apparatus or the way in which information is encoded within it. An important beneﬁt of QPT is that it poses a rigorous test of the ac curacy of the math- ematical model of the system-plus-apparatus used to design and interpret the experiments. This is done by comparing the experimental results of QPT to the res ults of computer sim- ulations of the complete QPT procedure, based upon this same mode l. Simulations based

Question 1476 multiple-choice

Quantum error-correcting codes are crucial for protecting quantum information against errors and often leverage mathematical structures such as group symmetries to enhance their robustness. The scaling of error suppression and the properties of code states play a key role in determining the effectiveness of these codes. Which property distinguishes thermodynamic quantum codes from W-state codes in terms of error suppression as the number of physical systems n increases? 1) Thermodynamic codes have full-rank reduced states, achieving error scaling proportional to 1/n, while W-state codes have rank-deficient reduced states with error scaling proportional to 1/n. 2) Thermodynamic codes exhibit rank-deficient reduced states, leading to error scaling proportional to 1/√n, while W-state codes have full-rank reduced states with scaling proportional to 1/n. 3) Thermodynamic codes and W-state codes both have rank-deficient reduced states, resulting in identical error scaling for increasing n. 4) W-state codes achieve optimal error scaling proportional to 1/n due to their covariance with all logical unitaries, while thermodynamic codes scale as 1/√n. 5) Thermodynamic codes saturate the optimal bound for error scaling because their reduced states are always pure, whereas W-state codes are always mixed. 6) W-state codes achieve better error suppression than thermodynamic codes for small n, but worse scaling for large n due to their basis state structure. 7) Both code types utilize regular representations of finite groups to achieve equivalent robustness against errors for any value of n.

✓ Correct Answer:

The correct answer is 1) Thermodynamic codes have full-rank reduced states, achieving error scaling proportional to 1/n, while W-state codes have rank-deficient reduced states with error scaling proportional to 1/n..

📚 Reference Text:

¼ 1−ð1=nÞfor all x, and it follows thatﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1−F2ðρx 1;j⊥ih⊥jÞp ≤ﬃﬃﬃﬃﬃﬃﬃﬃ 2=np ≕ϵ.F o rx≠x0 we have, according to Eq. (31),ρx;x0 1¼1 njxihx0j; ð61Þ and, thus, kρx;x0 1k1¼1=n≕ν. The corresponding reduced states on the other physical subsystems are the same by symmetry of the code word (59). Then, Proposition 4 asserts that this code has an error parameter that is at most ϵworst≤ﬃﬃﬃ 2p þdLﬃﬃﬃnp : ð62Þ That is, for fixed dL, the code becomes a good error- correcting code in the limit n→∞. In contrast to the thermodynamic codes presented above, thisW-state code does not saturate our bound on ϵworst, which is inversely proportional to nrather than the square root of n. The reason for this discrepancy is the same as for the difference between a sharp and a smooth cutoff for the three-rotor code, discussed in Sec. VI A 3 . Again, here, asnbecomes large, the local reduced state grows close to the rank-deficient state j⊥ih⊥j, which is a regime where the infidelity is particularly sensitive to small perturbations. In contrast, for instance, our thermodynamic codes of Sec. VI C have reduced states that are full rank, allowing the code to achieve the same scaling as our accuracy bound as n→∞. While this code does not achieve the same 1=nscaling as the thermodynamic codes, it does exhibit covariance with respect to the full logical unitary group UðdLÞ. VIII. ERROR-CORRECTING CODES FOR GENERAL GROUPS In this section, we construct families of codes that are covariant with respect to any group Gadmitting a left- and right-invariant Haar measure, encompassing, in particular, codes that are based on rotors, oscillators, and qudits. Our construction is based on quantum systems that transformas the regular representation ofG. Orthonormal basis states fjgig g∈Gfor this representation are labeled by group elements; if the group has an infinite number of elements, then the quantum system is infinite dimensional. Regular representations of a group are a useful tool to developmodels in quantum information and quantum matter: Examples include Kitaev ’s quantum double model [2], dyonic models [64], generalized color codes [65], and generalized cluster states [66]. Our quantum error- correcting codes can also be seen as extensions to general groups of continuous variable codes [51,67 –69]. A qubit can transform as the regular representation of the group Z 2, and a qudit as the regular representation of ZD. An oscillator provides a regular representation of the(noncompact) group R, with the group acting by translation in either its position basis fjxigor its momentum basis fjpig. Similarly, a rotor provides a regular representationCONTINUOUS SYMMETRIES AND APPROXIMATE QUANTUM … PHYS. REV . X 10,041018 (2020) 041018-19 of the group Uð1Þ, with orthonormal basis states fjeiϕig; when Fourier transformed, it can transform as a regularrepresentation of Z, where the basis states are the eigen- states of angular momentum fjlig l∈Z. The present constructions are inspired by the three-rotor code of Ref. [12]. Our constructions differ from the other constructions of Ref. [12] for both infinite- and finite- dimensional groups. It is shown

Question 1477 multiple-choice

In modern cryptography, RSA and related protocols rely on the difficulty of certain group-theoretic problems, which often involve modular arithmetic and assumptions about the hardness of extracting roots and factoring integers. Security guarantees are tightly linked to the way group structures and computational assumptions interact. Which construction ensures that the multiplicative group of integers modulo an RSA modulus n = pq has the property that only elements with order 2 exist in both (Z/pZ)× and (Z/qZ)×, thereby aiding in establishing equivalence between generalized and standard strong RSA assumptions? 1) Choosing n = pq with both p and q as safe primes 2) Selecting n = pq with p and q such that p−1 and q−1 are divisible by large primes 3) Constructing n = pq such that gcd(p−1, q−1) = 2 4) Using n = pq where p and q are consecutive primes 5) Setting n = pq with p−1 and q−1 both even but not relatively prime 6) Picking n = pq with one of p or q congruent to 1 mod 4 7) Choosing n = pq such that both p−1 and q−1 are powers of two

✓ Correct Answer:

The correct answer is 3) Constructing n = pq such that gcd(p−1, q−1) = 2.

📚 Reference Text:

Gby exponentiations to l(k)!. The price is loss of eﬃciency, since l(k)!∼√ 2πl(k)l(k)+1/2e−l(k)/greatermuchlG. However, the cost to exponentiate to power l(k)! is still polynomial in k. The second assumption is a generalization of the strong RSA assumption – we require that extracting non-trivial roots is hard, even if one is allowed tomultiply the input by an element of relatively small known order. We may thinkof this as root extraction in a factor group: when the adversary algorithm getsan input element Y, this represents an element ¯Yin the quotient group G/U, and the adversary’s task actually is to extract a non-trivial ( e’th) root of ¯Yin G/U. He must, however, demonstrate that his answer when raised to the e’th 130 IvanDamg˚ ardandEiichiroFujisaki power represents the same element as does Y. We require he does this by also producingµ. If we specialize to the RSA case, i.e., G=(Z/nZ)×for an RSA modulus n, it may seem that the root assumption as we deﬁned it here would make an even stronger requirement than the standard strong RSA assumption [2,14] – since the adversary in our case is given lGand does not have to ﬁnd a root of Y, but ofµYfor anyµ∈U. This is not the case, however. We now show that RSAmodulicanbeconstructedsuchthatourassumptionsaresatisﬁedforthesegroups, based only on the strong RSA assumption in its standard form. Supposewe make ak-bit modulus n=pqsuch that gcd( p−1,q−1) = 2. We choose C(k) assomesuper-polynomialfunctionmuchlessthan2 k,forinstance C(k)=2k/10, and we setl(k)=k. We construct p,qsuch that the factor of ( p−1)(q−1) with prime factors less than C(k)i si nO(k) (this factor is lG, whereG=(Z/nZ)× ). We then set G=(Z/nZ)×and descr(G)={n,lG}. Now, the root assumption (in its asymptotic form) turns out to be equivalent to the standard strong RSAassumption.Firstnotethatitmakeslittlediﬀerencewhether l Gisknownsinceit can be guessed with signiﬁcant probability. Then suppose algorithm Aon input Y,n,lGﬁndsX,e,µsuch thatY=µXe,µlG= 1. Now, if there is non-negligible probability that µ/negationslash=±1, we can use µ,lGto factorn, namely we ﬁrst factor lGand then we can ﬁnd an element ˜ µof known prime order s.I fs=2 ,˜µis a non-trivial square root of 1 and gcd(˜ µ−1,n) is a factor in n. But ifsis odd, it cannot divide both p−1 andq−1 and therefore ˜ µmust be congruent to 1 modulo one of porqand be diﬀerent from 1 modulo the other. Hence, also in this case, gcd(˜ µ−1,n) is a non-trivial factor of n. On the other hand, if µ=±1 with non-negligible probability, we can solve the strong RSA problem: given inputh∈(Z/nZ)×, we choose a random bit band give ( −1)bhas input to A. SinceAreceives the same input distribution as usual, it outputs a non-trivial root of ( −1)bYor−(−1)bYwith good probability. Since A’s choice of the sign cannotbecorrelatedtoourchoiceof b,weobtainarootof Ywithnon-negligible probability. Note that a special case of this construction of nis whenn=pqis a safe primeproduct,i.e.,( p−1)/2,(q−1)/2areprimes,butevidentlytheconstruction covers a much larger class of moduli. 3 SomeDeﬁnitions We will often use the concepts of zero-knowledge and computational/statisticalindistinguishability. For deﬁnitions of these, refer to [16]. The deﬁnitions beloware all in the exact security style. It is straightforward to derive asymptotic

Question 1478 multiple-choice

Computational vibrational spectroscopy combines quantum mechanical and molecular mechanics approaches to efficiently simulate vibrational properties of molecules in complex environments. Semiclassical methods play a key role in balancing accuracy and computational cost, especially for systems with strong intermolecular interactions. Which statement most accurately captures the principal advantage of using the QVP method with a QM/MM-derived potential energy surface in vibrational frequency calculations for solvated chromophores? 1) It computes all possible vibrational mode couplings explicitly, including intramolecular effects. 2) It requires empirical parameterization of the potential energy surface for each new molecular system. 3) It treats all nuclear motions fully quantum mechanically, regardless of system size. 4) It mandates solving the full vibrational Schrödinger equation for every MD snapshot. 5) It is limited to small, isolated clusters and cannot be scaled to macromolecular systems. 6) It dynamically incorporates solvent effects without the need for empirical surfaces, enabling efficient and accurate vibrational predictions for diverse systems. 7) It ignores solvent fluctuations and only models vibrational frequencies in the gas phase.

✓ Correct Answer:

The correct answer is 6) It dynamically incorporates solvent effects without the need for empirical surfaces, enabling efficient and accurate vibrational predictions for diverse systems..

📚 Reference Text:

cm−1 between the two methods is attributed to nuclear quantum effects. The difference between the QVP1 and DAF results also illustrates the use of the composite approach to effectively correct errors in the low-level QM/MM representation of the absolute vibrational frequency. 5. CONCLUSIONS We have presented a nuclear quantum vibrational perturbation (QVP) method for efficiently determining the instantaneous vibrational frequency of a chromophore in molecular dynamics simulations. The QVP approach is a semiclassical approach that combines quantum mechanical treatment of nuclear motions of specific vibrational modes with classical dynamics simulations. The method treats intermolecular interactions as a perturbation to the vibrational Hamiltonian on a predefined reference state. Computational efficiency is achieved as a result of three key ingredients, including (a) the use of potential optimized discrete variable representation (PODVR) of the vibrational wave functions for all levels of interest, (b) the application of perturbation theory to evaluate the energy shifts due to solvent dynamic fluctuations, and (c) the representation of the potential energy surface by a combined QM/MM approach for condensed-phase and macromolecular systems. Furthermore, we found that first-order perturbation is sufficiently accurate for the present HCl–water cluster system, a small but challenging test case due to strong bond-dissociation character associated with the vibrational mode as well as hydrogen bonding interactions. The use of the PODVR representation and perturbation theory to determine the energy shifts avoids the need to solve the vibrational Schrödinger equation. Thus, the computational cost of the QVP method is equivalent to that needed to determine the potential energies at the PODVR points. The use of a combined QM/MM approach incorporates naturally the effects of solvent dynamics on the potential energy surface, which can be systematically improved in accuracy and is computationally efficient. It also avoids the need to develop empirical potential energy surface and offer the accompanying dipole moment surface for every new system. Since time-dependent vibrational frequencies can be determined on the fly in molecular dynamics simulations, the mode-specific linear and 2D-IR spectra can be extracted from the frequency time correlation function. Although the QVP method is validated using the HCl–water clusters, we emphasize that the approach can easily be extended to macromolecular systems since the perturbation theory calculation is only concerned with one-electron integrals in a combined QM/MM potential. Thus, as long as it is possible to carry out QM/MM simulations at the desired QM level, the QVP1 method can be used to determine on-the-fly vibrational frequencies at the same or better accuracy.Xue et al. Page 11 J Chem Theory Comput . Author manuscript; available in PMC 2018 February 01. Author Manuscript Author Manuscript Author Manuscript Author Manuscript Intra- and intermolecular mode coupling can be important, but it is not specifically included in the present QVP method except the effect due to intermolecular interactions in the QM/MM potential energy surface. This is a subject of future research, though one can straightforwardly extend the present single mode approach to two or more coupled modes (as illustrated in the text), making use of the PODVR numerical

Question 1479 multiple-choice

Quantum error correction codes that respect continuous symmetries, such as charge conservation, face unique challenges in preserving entanglement and correcting local errors. The effectiveness of these covariant codes is closely linked to the properties of operators representing conserved quantities within the system. Which property of covariant quantum error correcting codes fundamentally limits their worst-case entanglement fidelity when correcting a single erasure? 1) Small dimension of the code subspace 2) Low number of physical subsystems 3) Minimal redundancy in code design 4) Limited coupling between logical and physical systems 5) Narrow range of logical charge operator eigenvalues 6) Lack of exact charge conservation 7) Restricted spread of the logical and local charge operators' eigenvalues

✓ Correct Answer:

The correct answer is 7) Restricted spread of the logical and local charge operators' eigenvalues.

📚 Reference Text:

code words, then theaccuracy of the code is bad; specifically, we can showthat, for any two logical states jxi L;jx0iL,w eh a v e (Supplemental Material [37], Lemma 28) ϵworstðN∘EÞ≥1 2δ½dN∘EðjxihxjLÞ;dN∘Eðjx0ihx0jLÞ/C138: ð25Þ By combining the above, we establish our simplified main result. Theorem 1. The performance of the covariant code Eð·Þ¼Vð·ÞV†under the above assumptions, quantified by the worst-case entanglement fidelity, is bounded as follows: ϵworstðN∘EÞ≥1 2nΔTL max iΔTi; ð26Þ where ΔTL(respectively, ΔTi) is the difference between the maximum and minimum eigenvalue of TL(respec- tively, Ti). One can also derive a bound on the figure of merit ϵe introduced in Eq. (9a): ϵeðN∘EÞ≥1 nmaxfsðTLÞ;s0ðTLÞ=2g max iΔTi; ð27Þ where sðTLÞands0ðTLÞare measures of the spread of the eigenvalues of TL. These are defined more precisely asCONTINUOUS SYMMETRIES AND APPROXIMATE QUANTUM … PHYS. REV . X 10,041018 (2020) 041018-9 sðTLÞ¼d−1 LkTL−μ1k1, where μis the median of the eigenvalues of TLcounted with multiplicities, and s0ðTLÞ¼d−1 LkTL−trðTLÞ1L=dLk1. These quantities, unlike ΔTL, take contributions from all eigenvalues of TL.I n sðTLÞ, the median appears as the optimal solution to sðTLÞ¼d−1 Lmin μkTL−μ1k1. The argument of the norm ins0ðTLÞis simply the charge TLwith a global shift that makes the operator traceless. Equation (27) is proven as a special case of Theorem 2 below. The bound (27) is intuitively sensitive to the “average amount of logical charge ”in absolute value, up to an arbitrary charge offset; this sensitivity makes sense, since the entanglement fidelity for the maximally mixed inputstate“only samples the average logical state. ”On the other hand, the worst-case entanglement fidelity picks up the worst possible situation, noticing that there are two states with maximally different charges; the bound (26) reflects that the code performs the worst for those input states. In short, a covariant code with respect to a local charge may not perform well for correcting a single erasure at aknown location, unless either it encodes the informationinto large physical systems, with a large range of possible charge values (max iΔTi→∞), or it encodes the informa- tion into many physical systems ( n→∞). It is also possible to prove the bound (27) using a different approach (cf. Supplemental Material, Sec. B [37]), by studying the correlations induced by the code words between the local charges and the logical charge. Theessence of this alternative argument goes as follows. If thecode is covariant with respect to a continuous symmetry, then the global physical charge must be exactly correlated with the logical charge, and this constraint imposes aglobal constraint on the correlation functions of the localcharges with the logical charge. On the other hand, for an approximate error-correcting code, the local reduced states of the code words must be nearly uncorrelated with thelogical state to ensure that hardly any information leaks to the environment. Combining quantitative versions of these two statements provides an alternative proof of the bound(27)withs 0ðTLÞin the numerator (up to a constant factor). We now present a theorem that generalizes Theorem 1 in the following ways. It allows for the code to only approximately conserve charge,

Question 1480 multiple-choice

Representation theory plays a central role in understanding the symmetries and structure of quantum systems, particularly within linear optics and unitary group actions. Irreducibility and decomposition into invariant subspaces are crucial for analyzing how operators act on spaces of quantum states. Which of the following statements is true regarding the action of photon creation and annihilation operators a†_i a_j in the group Un,m on the Fock basis of quantum states? 1) They preserve the number of photons in each individual mode but cannot connect different basis vectors. 2) They only generate reducible representations for all values of n and m. 3) They act as scalar multiplications on all Fock basis vectors, leaving states unchanged. 4) They can only move photons within a single mode and not between modes. 5) They move photons between modes and act transitively on the Fock basis, demonstrating that the representation is irreducible. 6) They annihilate all Fock basis vectors with more than one photon. 7) They generate only the center of the Lie algebra, corresponding to scalar multiples of the identity.

✓ Correct Answer:

The correct answer is 5) They move photons between modes and act transitively on the Fock basis, demonstrating that the representation is irreducible..

📚 Reference Text:

V∗is equivalent to the adjoint representation ofU(d)on its complexified Lie algebra gl(d). As a U(d)-representation (and therefore also a G-representation), V ⊗ V∗ decomposes as follows: V ⊗ V∗∼=gl(d) = sl(d)⊕CI, (B3) where Iis the d×didentity matrix. Both sl(d)andCIare nonzero representations of G(since d≥2), so by the above they must be irreducible G-representations. We proved above that the derived subalgebra [g,g]is a nonzero G-subrepresentation ofsl(d); since sl(d)is irreducible, we must have sl(d) = [ g,g]⊆g.In particular, taking the skew-Hermitian part, we see that the real Lie algebra of Gcontains su(d), and by applying the matrix exponential, SU(d)⊆G. ■ 4. Proof of Proposition 4 Proposition 5. Letn, m≥2. The group Un,mof linear optical unitaries is a 1-design in U(H). Proof. Recall from (26) the isomorphism U(m)→Un,m, and let D=diag(d1, . . . , d m)∈U(m). For any Fock basis vector |n1, . . . , n m⟩ ∈ H ,Dacts by D:|n1, . . . , n m⟩ 7→ dn1 1···dnmm|n1, . . . , n m⟩.In the language of representation theory, this tells us that |n1, . . . , n m⟩is aweight vector with weight (n1, . . . , n m). In particular, His a highest weight representation for U(m)∼=Un,m with highest weight (n,0, . . . , 0). Therefore, Hcontains a copy of V(n,0,...,0), the irreducible representation of U(m)with highest weight (n,0, . . . , 0). One may compute using the Weyl dimension formula (see e.g. Ref. 54 Corollary 24.3) that dimV(n,0,...,0)=n+m−1 n =dn,m= dim H,so the two spaces must be equal. ■ To avoid appealing to general character theory, one may instead show directly that, for any modes i, j, the operators a† iaj are in un,m, the Lie algebra of the linear optical unitaries. These operators move a photon from mode jto mode iand are easily seen to act transitively on the Fock basis vectors. This fact, combined with the fact that His a weight representation forUn,mwith one-dimensional weight spaces spanned by the Fock basis vectors, shows that His irreducible. 24 5. Discussion of Theorem 8 We note the following characterization of the vector |ζ⟩in Theorem 8. This was mentioned in the proof of Lemma 1 in the Supplementary Material to Ref. 8. One can show that the U(2)-module H ⊗ H has a unique irreducible subrepresentation with highest weight 0(necessarily 1-dimensional), and |ζ⟩is a nonzero vector in that subrepresentation. This gives a decomposition into subrepresentations, H ⊗ H =W⊕C|ζ⟩. (B4) (Where Wis generally not irreducible.) In view of Corollary 1, we may view Theorem 8 in the m= 2case as saying that, for any operator Vthat fails to preserve the decomposition (B4) into subrepresentations, the addition of VtoUn,mturns H ⊗ H into an irreducible representation. Perhaps more intuitively, we may rephrase the above as saying that |ζ⟩is (up to scalar) the unique element of H ⊗ H with the property that |ζ⟩is an eigenvector of U⊗Ufor all U∈Un,m. (Equivalently, |ζ⟩is annihilated by the action of sun,m, (the traceless subalgebra of) the Lie algebra

Question 1481 multiple-choice

In quantum computing, variational hybrid quantum-classical algorithms play a critical role in leveraging noisy intermediate-scale quantum (NISQ) devices. These algorithms combine classical optimization with quantum circuit evaluations to solve complex computational tasks in the presence of hardware noise. Which statement best describes the demonstrated behavior of variational quantum compiling algorithms when exposed to incoherent noise such as measurement, gate, and Pauli channel errors? 1) They fail to find any valid gate sequence due to noise in cost evaluations. 2) They require error correction protocols to maintain accuracy in parameter optimization. 3) Their optimal parameters shift significantly, reducing the fidelity of the compiled circuit. 4) Their optimal variational parameters remain unaffected, preserving the accuracy of the compiled gate sequence. 5) They can only compile circuits for noise-free quantum devices and are unsuitable for NISQ hardware. 6) They show resilience exclusively to coherent noise, not incoherent noise sources. 7) Their robustness is limited to classical optimization steps rather than quantum circuit evaluations.

✓ Correct Answer:

The correct answer is 4) Their optimal variational parameters remain unaffected, preserving the accuracy of the compiled gate sequence..

📚 Reference Text:

Title: Noise resilience of variational quantum compiling Year: 2019 Paper ID: 1386c99f3aaae52b60071064256cd168f5b0fc10 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/1386c99f3aaae52b60071064256cd168f5b0fc10 Abstract: Variational hybrid quantum-classical algorithms (VHQCAs) are near-term algorithms that leverage classical optimization to minimize a cost function, which is efficiently evaluated on a quantum computer. Recently VHQCAs have been proposed for quantum compiling, where a target unitary U is compiled into a short-depth gate sequence V. In this work, we report on a surprising form of noise resilience for these algorithms. Namely, we find one often learns the correct gate sequence V (i.e. the correct variational parameters) despite various sources of incoherent noise acting during the cost-evaluation circuit. Our main results are rigorous theorems stating that the optimal variational parameters are unaffected by a broad class of noise models, such as measurement noise, gate noise, and Pauli channel noise. Furthermore, our numerical implementations on IBM’s noisy simulator demonstrate resilience when compiling the quantum Fourier transform, Toffoli gate, and W-state preparation. Hence, variational quantum compiling, due to its robustness, could be practically useful for noisy intermediate-scale quantum devices. Finally, we speculate that this noise resilience may be a general phenomenon that applies to other VHQCAs such as the variational quantum eigensolver.

Question 1482 multiple-choice

Quantum algorithms such as Shor’s and Grover’s exploit group-theoretic structures to achieve performance advantages over classical algorithms. Understanding how these algorithms relate to hidden subgroup problems reveals deeper connections within quantum computational theory. In the context of quantum algorithms, which of the following statements accurately characterizes the hidden subgroup structure associated with Grover’s algorithm? 1) Grover’s algorithm finds the order of an element in a finite cyclic group using the quantum Fourier transform. 2) The algorithm directly solves the hidden subgroup problem for general non-abelian groups by employing entangled measurements. 3) Grover’s speedup relies exclusively on properties of abelian groups and periodic functions. 4) The marked item in Grover’s algorithm is identified by measuring the quantum Fourier transform of a quotient group. 5) Grover’s algorithm exhibits invariance under the stabilizer subgroup of the symmetric group SN, with the stabilizer subgroup corresponding to all permutations that fix the marked item. 6) Grover’s algorithm amplifies amplitudes using quantum walks on coset spaces of hidden subgroups. 7) The hidden subgroup in Grover’s algorithm is the group of all database entries, regardless of the marked item.

✓ Correct Answer:

The correct answer is 5) Grover’s algorithm exhibits invariance under the stabilizer subgroup of the symmetric group SN, with the stabilizer subgroup corresponding to all permutations that fix the marked item..

📚 Reference Text:

Shor’s oracle . Shor’s algorithm consists in ﬁrst executing the quantum subroutine QRand(/tildewideϕ), thereby producing a random character γy/Q:mmodQ/ma√sto→my Qmod1 oftheﬁnite cyclicgroup ZQ. Thetransversal τusedinpushinghasbeenengineered to assure that the character γy/Qis suﬃciently close to a character γd/P:kmodP/ma√sto→kd Pmod1 of the hidden quotient group Z/PZ=ZP. In this case ”suﬃciently close” means that /vextendsingle/vextendsingle/vextendsingle/vextendsingley Q−d P/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1 2P2, which that d/Pis a continued fraction convergent of y/Q, and thus can be found found by the classical polynomial time continued fraction algorithm. 6.Description of Grover’s algorithm Now let us turn to Grover’s algorithm. We begin with a brief description . Consideranunstructureddatabaseof N= 2nrecordslabeledwithoutrepetitions with the labels 0,1,2,...,N−1. We are given the oracle f:{0,1}n−→{0,1}, where f(x) =  1 ifj=j0(“Yes”) 0 otherwise (“No”) , calledGrover’s oracle , and asked to solve the following search problem: Search Problem for an Unstructured Database. Find the unknown record labeled asj0with the minimum amount of computational work, i.e., with th e min- imum number of queries of the oracle f, and with bounded probability of error . LetHbe the Hilbert space with orthonormal basis |0/an}bracketri}ht,|1/an}bracketri}ht,|2/an}bracketri}ht,...,|N−1/an}bracketri}ht, 3Atransversal for an epimorphism αϕ:Z−→ZQis an injection τϕ:ZQ−→Zsuch that αϕ◦τϕis the identity map on ZQ, i.e., a map that takes each element of ZQonto a coset representative of the element in Z. IS GROVER’S ALGORITHM A QUANTUM HIDDEN SUBGROUP ALGORITHM ? 7 whereN= 2n. Then Grover’s oracle is essentially given as the unitary transfor- mation I|j0/angbracketright:H −→ H |j/an}bracketri}ht /ma√sto−→ (−1)f(j)|j/an}bracketri}ht where I|j0/angbracketright=I−2|j0/an}bracketri}ht/an}bracketle{tj0| is inversion in the hyperplane orthogonal to |j0/an}bracketri}ht. LetHdenote the Hadamard transform on the Hilbert space H. Then Grover’s algorithm is given as: Grover’s Algorithm STEP0.(Initialization) |ψ/an}bracketri}ht←−H|0/an}bracketri}ht=1√ NN−1/summationdisplay j=0|j/an}bracketri}ht k←−0 STEP1.Loop until k=/floorleftbigg π 4sin−1(1/√ N)/floorrightbigg ≈/floorleftBig π 4√ N/floorrightBig |ψ/an}bracketri}ht←−Q|ψ/an}bracketri}ht=−HI|0/angbracketrightHI|j0/angbracketright|ψ/an}bracketri}ht k←−k+1 STEP2.Measure|ψ/an}bracketri}htwith respect to the standard basis |0/an}bracketri}ht,|1/an}bracketri}ht, ... ,|N−1/an}bracketri}htto obtain the unknown state|j0/an}bracketri}htwith probability ≥1−1 N. 7.The symmetry hidden within Grover’s algorithm But where is the hidden symmetry in Grover’s algorithm ? LetSNbe the symmetric group on the symbols 0,1,2,3,...,N−1 . Then Grover’s algorithm is invariant under the hidden subgroup Stabj0={g∈SN:g(j0) =j0}⊂SN, called the stabilizer subgroup forj0, i.e., Grover’s algorithm is invariant under the group action Stabj0×H −→ H /parenleftBig g,/summationtextN−1 j=0aj|j/an}bracketri}ht/parenrightBig /ma√sto−→/summationtextN−1 j=0aj|g(j)/an}bracketri}ht Moreover, if the hidden subgroup Stabj0is known, then so is the integer j0, and vice versa. 8 SAMUEL J. LOMONACO, JR. AND LOUIS H. KAUFFMAN Thus, Grover’s algorithm is an algorithm that solves the following hidde n sub- group problem, which we will henceforth refer to as Grover’s hidden subgroup problem : Grover ’s Hidden Subgroup Problem. Given a map SNϕ−→S from the the symmetric group SNinto a target set S={0,1,2,...,N−1}with hidden subgroup structure given by the commutative diagram SNϕ−→S νj0ց ր ι SN/Stabj0, whereνj0:SN−→S/Stab j0is the natural surjection of SNonto the coset space S/Stab j0, and where ι:SNϕ−→S (j j0)Stabj0/ma√sto−→j is theunknown relabeling (bijection) of the coset space SN/Stabj0onto the set S. Find the hidden subgroup Stabj0with bounded probability of error. Let (ij)∈SNdenote the permutation that interchanges integers iandj, and leavesall other integersﬁxed. Thus, ( ij) is a transpositionif i/ne}ationslash=j, and the identity permutation

Question 1483 multiple-choice

Modular tensor categories are central to the mathematical formulation of topological quantum field theories and quantum computation, providing algebraic structures that capture braiding and fusion in low-dimensional topology. Quantum groups, particularly at roots of unity, serve as foundational sources for constructing these categories due to their rich representation theory and connections to knot invariants. Which algebraic structure, defined as a q-deformation of the universal enveloping algebra of a simple complex finite-dimensional Lie algebra, provides solutions to the quantum Yang-Baxter equation and enables the construction of modular tensor categories relevant for topological quantum field theories? 1) Quantum group (as defined by Drinfeld and Jimbo) 2) Group algebra of a finite group 3) Symmetric group algebra 4) Universal enveloping algebra without deformation 5) Affine Hecke algebra 6) Clifford algebra 7) Brauer algebra

✓ Correct Answer:

The correct answer is 1) Quantum group (as defined by Drinfeld and Jimbo).

📚 Reference Text:

Title: From Quantum Groups to Unitary Modular Tensor Categories Year: 2005 Paper ID: b839a3fc50f07506ec8a37ad6dc2dfd9f14c992f Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b839a3fc50f07506ec8a37ad6dc2dfd9f14c992f Abstract: Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties nec- essary to produce 3-dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to un- derstand and describe these examples as explicitly as possible, especially those that are "physically feasible." We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasiz- ing explicit computations. We outline the development of the theory of modular tensor categories from quantum groups with an eye towards new applications to quantum computing that motivate our point of view. In this article, we take quantum group to mean the "classical" q-deformation of the universal enveloping algebra of a simple complex finite dimensional Lie algebra as in the book by Lusztig (L), rather than the broader class of Hopf algebras the term sometimes describes. 1.1. Background. The representation theory of quantum groups has proven to be a useful tool and a fruitful source of examples in many areas of mathematics. The general definition of a quantum group (as a Hopf algebra) was given around 1985 by Drinfeld (D) and independently Jimbo (Ji) as a method for finding solutions to the quantum Yang-Baxter equation. These solutions led to new representations of Artin's braid group Bn and connections to link invariants. In fact, special- izations of the famous polynomial invariants of Jones (J), the six-authored paper (HOMFLY) and Kauffman (Kf) have been obtained in this way. Reshetikhin and Turaev (RT) used this connection to derive invariants of 3-manifolds from modular Hopf algebras, examples of which can be found among quantum groups at roots of unity (see (RT) and (TW1), much simplified by constructions in (A)). When Witten (Wi) introduced the notion of a topological quantum field theory (TQFT)

Question 1484 multiple-choice

Quantum algorithms often exploit group-theoretic structures, such as cosets, stabilizers, and character theory, to solve problems like the Hidden Subgroup and Hidden Translation problems efficiently. Understanding the relationships between quantum states under group actions is essential for designing these algorithms. When the translating coset between two quantum states |φ₀⟩ and |φ₁⟩ in a group G is non-empty, which statement best describes the relationship between their stabilizer subgroups? 1) The stabilizer of |φ₀⟩ is a direct product with the stabilizer of |φ₁⟩. 2) The stabilizer subgroups of |φ₀⟩ and |φ₁⟩ are identical subgroups of G. 3) The stabilizer subgroups of |φ₀⟩ and |φ₁⟩ are conjugate subgroups in G. 4) The stabilizer of |φ₀⟩ is trivial, while the stabilizer of |φ₁⟩ is the whole group G. 5) The stabilizer subgroups generate the same coset as the identity element. 6) The stabilizers intersect only at the identity element. 7) The stabilizer subgroups have no elements in common.

✓ Correct Answer:

The correct answer is 3) The stabilizer subgroups of |φ₀⟩ and |φ₁⟩ are conjugate subgroups in G..

📚 Reference Text:

u i v a l e n t l y , |G·φ/angbracketright=1 √ |G|/summationtext x∈G|x·φ/angbracketright.T h e translating coset of two states |φ0/angbracketrightand|φ1/angbracketrightof Γ is the set {u∈G:|u·φ1/angbracketright= |φ0/angbracketright}. The translating coset of |φ0/angbracketrightand|φ1/angbracketrightis either empty or a left coset uG|φ1/angbracketright (or equivalently a right coset G|φ0/angbracketrightu)f o rs o m e u∈G. If the latter case occurs, |φ0/angbracketright and|φ1/angbracketrighthave conjugate stabilizers: G|φ0/angbracketright=uG|φ1/angbracketrightu−1.Translating Coset is a generalization of Stabilizer : Translating Coset (G,α,t) Input:tcopies of two quantum states |φ0/angbracketright,|φ1/angbracketright∈Γ. Output: •reject ifG(|φ0/angbracketright)∩G(|φ1/angbracketright)=∅; •u∈Gsuch that |u·φ1/angbracketright=|φ0/angbracketrightand O(log |G|) generators for G|φ1/angbracketright otherwise. For a function fonG, deﬁne the superposition |f/angbracketright=1 √ |G|/summationtext g∈G|g/angbracketright|f(g)/angbracketright,a n d forx∈G, deﬁne the function x·f:g/mapsto→f(gx). Let Γ( f)={|x·f/angbracketright:x∈G}.T h e n a group element xacts naturally on |f/prime/angbracketright∈Γ(f) by mapping it to the superposition |x·f/prime/angbracketright. We call this group action the translation action . The mapping |x/angbracketright|f/prime/angbracketright/mapsto→ |x/angbracketright|x·f/prime/angbracketrightis realized by right multiplying the ﬁrst register of |f/prime/angbracketrightbyx−1. Proposition 2.2. Suppose Gis a ﬁnite group and let t= poly(log |G|).T h e n Hidden Subgroup (G)(resp., Hidden Translation (G)) can be solved with a call HIDDEN TRANSLATION AND TRANSLATING COSET 9 toStabilizer (G,τ,t)(resp., Translating Coset (G,τ,t)), where τdenotes the translation action. Proof.L e tfbe an instance of Hidden Subgroup . Then the stabilizer of |f/angbracketrightis the group hidden by f.L e t(f0,f1) be an instance of Hidden Translation .T h e n the translating coset of |f0/angbracketrightand|f1/angbracketrightis a singleton whose element is the translation of (f0,f1). 3. Hidden translation in Zn p.Inthissection, weshowthat Hidden Transla- tion(G) can be solved in quantum polynomial time in the special case when G=Zn p for any ﬁxed prime number p>2. In this section we use the additive notation for the group operation, and x·y∈Zpstands for the standard inner product for x,y∈Zn p. SinceZn 2/multicloserightZ2is isomorphic to the abelian group Zn 2×Z2, one already has a quantum polynomialtimealgorithmfor Hidden Translation inZn 2by reducing it to Hidden Subgroup inZn+1 2by the method of [13]. For the convenience of the reader we present our method using intermediate mea- surements. However, the measurements can always be eliminated (see [2]), giving a unitary and therefore reversible algorithm, possibly with errors. The quantum part of our algorithm consists of performing FourierSampling over the abelian group Zn p×Z2. It turns out that from the samples we will only use elements of the form ( y,1). The important property of these elements yis that they arenotorthogonal to the hidden translation. Some properties of the distribution of the samples are stated for general abelian groups in the following lemma. Lemma 3.1. LetGbe a ﬁnite abelian group. Let f=(f0,f1),f:G×Z2→S be an instance of Hidden Translation (G)having a translation u/negationslash=0.T h e n FourierSamplingf(G×Z2)outputs an element in G×{1}with probability 1/2.M o r e - over, the probability of sampling the element (y,1)depends only on χy(u),a n di s0if and only if y∈u⊥. Proof. The state vector of FourierSamplingf(G×Z2) before the ﬁnal observa- tion is 1 2|G|/summationdisplay x∈G/summationdisplay y∈G/summationdisplay c=0,1χy(x)/parenleftbig 1+(−1)cχy(u)/parenrightbig |y/angbracketright|c/angbracketright|f0(x)/angbracketright. Therefore the probability of sampling ( y,1) is proportional to |1−χy(u)|2,

Question 1485 multiple-choice

In quantum computing, the Quantum Fourier Transform (QFT) is implemented as a circuit composed of Hadamard gates and controlled phase gates, and its entanglement properties can be analyzed using tensor network representations. Tensor networks enable the visualization and simplification of complex quantum operations, such as the QFT, by breaking them into interconnected tensor components. Which statement accurately describes how the singular values of a submatrix of the QFT circuit are related to quantum entanglement across a bipartition in the tensor network representation? 1) The singular values are always equal to one, indicating no entanglement across any bipartition. 2) The singular values correspond to the eigenvalues of the QFT circuit's full unitary matrix. 3) The singular values of the QFT submatrix are precisely the Schmidt coefficients that quantify entanglement across the bipartition. 4) The singular values are determined solely by the number of Hadamard gates present in the circuit. 5) The singular values represent the probabilities of measurement outcomes in the computational basis. 6) The singular values of the QFT submatrix are unrelated to any entanglement measures in the system. 7) The singular values are a function of the swap gates used to reverse qubit order in the QFT circuit.

✓ Correct Answer:

The correct answer is 3) The singular values of the QFT submatrix are precisely the Schmidt coefficients that quantify entanglement across the bipartition..

📚 Reference Text:

fourth diagram to the last diagram uses Eq. (F8) again and writes x=22x1+2x2+x3and y=22y1+2y2+y3. 040318-23 JIELUN CHEN, STOUDENMIRE, and WHITE PRX QUANTUM 4,040318 (2023) which is identical to Eq. (20). The Schmidt coeﬃcients can thus be calculated by solving the eigenvalues of this matrix. One may wonder how this is connected to our analy- sis in Sec. II Bthat the Schimdt coeﬃcients of Qnat cut j are the same as singular values of Fn,j(the top 2j×2n−j submatrix of Fn). We demonstrate this by showing that MBM† B=2nFn,jF† n,j. We ﬁrst claim that the following ten- sor diagram represents the 2m×2ltop-left submatrix of√ 2nFn: (F15) That is, the tensor network is evaluated to exp (i2πxy/2n) where x=2l−1x1+···+ xland y=2m−1y1+···+ ym. We refer to this tensor network as the “generalized QFTTN.” It is easy to see that the tensor network of MBM† B can be simpliﬁed to this tensor network contracted with its conjugate transpose with m=jand l=n−j, or equiva- lently MBM† B=2nFn,jF† n,j; see Fig. 8for an example. One can achieve this simpliﬁcation by iteratively applying the identity to all order-4 and order-3 phase tensors, i.e., (F16) Therefore, for any nand j, the lower partition MBhas the same set of nonzero singular values as those of√ 2nFn,j, linking back to results in Sec. II B. = P(−π 8) P(−π 16) P(−π 32) P(−π 4) P(−π 8) P(−π 16) P(−π 2) P(−π 4) P(−π 8)P(π 8) P(π 16) P(π 32)P(π 4) P(π 8) P(π 16)P(π 2) P(π 4) P(π 8) P(−π 8) P(−π 16) P(−π 32) P(−π 4) P(−π 8) P(−π 16) P(−π 2) P(−π 4) P(−π 8)P(π 8) P(π 16) P(π 32)P(π 4) P(π 8) P(π 16)P(π 2) P(π 4) P(π 8) FIG. 8. Example of simpliying MBM† Bto 2nFn,jF† n,jwith n=6 andj=3. The diagram on the left comes from step 2 in Fig. 7.APPENDIX G: EMERGENT QFT QUANTUM CIRCUIT In this appendix, we show that the quantum circuits of the QFT can be derived from the tensor network structure introduced in Eq. (F15) , which is a generalized QFTTN that represents any submatrix of the QFT. To demonstrate this, we consider the 4-qubit example, where l=m=4. Equation (F15) becomes (G1) We introduce a few identities to remove phase gates with parameters 2 kπ, with kbeing an integer. Firstly, an order-3 phase gate will reduce to identity with an extra plus tensor, (G2) where the plus tensor is deﬁned as (G3) The order-4 phase tensor will reduce to two crossing identity tensors (G4) and any order-4 phase tensor can eat up a plus tensor to become an order-3 tensor: (G5) 040318-24 QUANTUM FOURIER TRANSFORM... PRX QUANTUM 4,040318 (2023) This allows us to reduce Eq. (G1) to (G6) It is also easy to see that the order-3 phase tensors have the identity (G7) and thus Hgates and copy tensors naturally arise from such decomposition. Ignoring the scalar, we then have the diagram (G8) and by bending yindex lines to the horizontal direction, we obtain the QFT quantum circuit (G9) We also consider

Question 1486 multiple-choice

In quantum computing, the physical connectivity between qubits and the choice of gate sets significantly influence the efficiency and speed of quantum gate operations and algorithms. Understanding how hardware constraints and control techniques affect quantum speed limits is essential for advancing computational performance. Which of the following statements best explains why adding next-to-nearest-neighbor (NNN) couplings can enhance quantum speed limits (QSLs) for multi-qubit gates, but does not fully eliminate the QSL gap between certain gates on superconducting circuits and neutral atom platforms? 1) NNN couplings enable error correction, which is the main factor in reducing gate times. 2) Superconducting circuits allow for arbitrary qubit connectivity, so NNN couplings have no effect. 3) Neutral atom platforms cannot utilize OCT, limiting their QSL improvements. 4) NNN couplings primarily benefit single-qubit gates, not multi-qubit gates. 5) While NNN couplings improve connectivity and reduce QSLs, other hardware-specific constraints in superconducting circuits prevent their QSLs from matching those of neutral atom platforms for all gate types. 6) The use of standard gate sets (SGS) negates any advantage from enhanced couplings. 7) Circuit run times depend only on algorithm selection, not on physical qubit interactions.

✓ Correct Answer:

The correct answer is 5) While NNN couplings improve connectivity and reduce QSLs, other hardware-specific constraints in superconducting circuits prevent their QSLs from matching those of neutral atom platforms for all gate types..

📚 Reference Text:

g 13(t) and g24(t)i nt h es a m e form as the already present couplings g12(t),g23(t),g34(t), and g41(t). This scenario is inspired by the pseudo-2D ar- chitecture for neutral atoms, which also exhibits identicalnearest-neighbor (NN) and next-to-nearest-neighbor (NNN)couplings and where having these couplings is advantageous.Figures 5(c) and 5(d) show the results for the two cases with only NN couplings (triangles) and with NN plus NNNcouplings (stars). Besides observing much better convergenceproperties for the latter case, we also obtain improved QSLs ofT ZZZ QSL,NNN=20 ns and TZZZZ QSL,NNN=60 ns for the ZZZ( γ) and ZZZZ( γ) gate, respectively. However, despite improvements in this scenario, the QSL of the ZZZZ( γ) gate does not get close to the QSL of the ZZZ( γ) gate as it does for neu- tral atoms. The presence of NNN couplings might thereforebe just a partial explanation of the QSL differences for theconstraint gates between neutral atoms and superconductingcircuits. Despite the scenario with NNN couplings not reﬂect-ing the actual architecture for superconducting circuits, wenevertheless add the circuit run times for the QAOA circuitsin the PM using these faster constraint gates to Fig. 1(c) for reference purposes as purple pentagons. VI. CONCLUSIONS In this study, we have determined the QSLs for several common two-qubit and two speciﬁc multi-qubit quantumgates for two promising quantum computing platforms thatallow for a 2D arrangement of qubits—neutral atoms andsuperconducting circuits. We have used OCT to determine theQSLs, as it provides a generally applicable tool that warrantsa fair comparison of both platforms. On the level of individual quantum gates, our study allows assessing how close gate protocols from the literature arecompared to their fundamental QSLs or, in other words, howmuch can (theoretically) still be gained in time if known gateprotocols are replaced by numerically optimized ones. We ﬁnd the QSLs for all investigated two-qubit gates, encompassing CNOT, CZ, and SW AP, to be very similar within each plat-form and close to the reference gate times for a CZ gate incase of neutral atoms [ 20] and a Sycamore gate in case of superconducting circuits [ 8]. On the level of quantum algorithms, our study has more- over allowed us to determine the “QSLs” for entire quantumcircuits. To this end, we have assumed a 2D grid architecturefor both platforms and qubit connectivity that allows physical quantum gates only between neighboring qubits. However, we have assumed these gates to be executable at the QSL.This has allowed us to calculate the circuit run times at the“QSL” for two paradigmatic quantum algorithms—the QFTand a single step of the QAOA. We ﬁnd that the correspondingweighted circuit run times scale comparably with respect tothe system size. Furthermore, we observe this to be inde-pendent of the chosen gate set used to translate the quantumalgorithms into executable quantum circuits, i.e., independentof whether the SGS or QGS is used. We observe platform independently that the QGS yields circuit run times and gatecounts that are roughly half compared to those in the SGS. Onthe one hand, this demonstrates that further speedup of circuitrun

Question 1487 multiple-choice

In the study of representation theory and convex geometry, the concept of the gap constant is crucial for understanding the stability of group actions and the geometry of weights. This constant has significant implications for optimization, computational complexity, and invariant theory. Which statement most accurately describes the gap constant $ p(\lambda) $ associated with a highest weight $ \lambda $ in a group representation? 1) It is the maximal norm among all weights of the representation. 2) It equals the trace of the representation's action on the tensor. 3) It is the minimal eigenvalue of the weight matrix. 4) It is the sum of the Euclidean norms of all weights in the convex hull. 5) It measures the largest singular value of the group element acting on the vector space. 6) It is the number of distinct weights occurring in the representation. 7) It is the minimal Euclidean norm attained over the convex hull of weights associated with the representation.

✓ Correct Answer:

The correct answer is 7) It is the minimal Euclidean norm attained over the convex hull of weights associated with the representation..

📚 Reference Text:

lower bound by studying the geometry of weights underlying the representation that underlies the shifting trick (Lemma 2.1): Deﬁnition 5.1 (Gap constant) .Letbe a highest weight and `jj. We deﬁne the gap constant by pq:min"kxk2 `:S pSym`pVqbVq;0RconvpSq;xPconvpSq* ; where we recall that pWqdenotes the set of weighs that occur in a representation W(see Section 2.1). The argument in the following proof is essentially from [Kir84b]. Proposition 5.2. LetXPTenpn0;n1;:::;ndqbeanonzerotensor, beahighestweight,and `jj. If p:{`RpXqthen mintkqpk2:qPpXqu¥ pq: Proof.It suﬃces to show that, for all rXsPXandUPK, d¸ i1kpiqprXsqUpiqdiagpppiq ni;:::;ppiq 1qpUpiqq:k2 F¥ 2pq: By the shifting trick, Eq. (11), this is equivalent to 1 `2d¸ i1kpiq WprZsqk2 F¥ 2pq; (28) whereZ:Xb`bUvPW:Sym`pVqbV. By assumption, RpXqpXq. According to Lemma 2.3 and Theorem 2.4, this means that any G-invariant polynomial vanishes on Z, so 0PGZ. By the Hilbert-Mumford criterion, there exists a 1-parameter subgroup of the form exppAtq,whereApAp1q;:::;ApdqqqisatupleofHermitianmatrices,suchthat exppAtqZÑ0for tÑ8. Without loss of generality we may assume that the Apiqare diagonal matrices for i1;:::;d (otherwise conjugate each by an appropriate unitary Ypiq, and replace XbypUp1qb:::bUpdqq, 40 which leaves the left-hand side of Eq. (28) invariant). Thus, exppAtqPTfor alltPC. We now expandZin terms of weight vectors, Z° !P pWqZ!, so that exppAtqZ¸ !P pWqe°d i1!piqapiqZ!; wherewewrite apiqforthediagonalentriesof Apiq,i1;:::;d;thedotintheexponentdenotesthe standardinnerproducton Rni. Sinceweknowthat exppAtqZÑ0,itfollowsthat°d i1!piqapiq 0 wheneverZ!0. This implies that 0RconvpSq, whereS:t!:Z!0u. Ontheotherhand,notethatbythedeﬁnitionofthemomentmap(Eq.(9))andtheorthogonality of the weight space decomposition, we have trrpiq WprZsqdiagpbpiqqsBt0xZ;exppdiagpbpiqqtqZy xZ;ZyBt0¸ !PSjZ!j2 kZk2et!piqbpiq ¸ !PSjZ!j2 kZk2p!piqbpiqq for allbpiqPRni. This implies that if we orthogonally project each component of WprZsqonto the diagonal, we obtain° !PSjZ!j2 kZk2!PconvpSq. As the Frobenius norm of a matrix is never smaller than the`2-norm of is diagonal, we obtain 1 `2d¸ i1kpiq WprZsqk2 F¥1 `2mintkxk2 2:xPconvpSqu¥ 2pq; which establishes Eq. (28). Next, we lower-bound the gap constant. First, we will need the following elementary lemma. Lemma 5.3. SupposeS tv1;:::;vmu PZNis a set of integer vectors s.t. 0RconvpSq. Let the bit complexityofentriesof vi’sbeatmost b. Denoteby pSqdenotetheEuclideandistanceof 0toconvpSq. Then pSq¥exppOpNplogpNqbqqq Proof.Since 0RconvpSq, by Farkas’ lemma (e.g. see [ Sch98]), there exists a vector wPRNs.t. xw;viy¥1for alli. By ([Sch98], Corollary 3.2b, Theorem 10.1), there exists such a rational w with bit complexity bounded of each entry bounded by OpNplogpNqbqq. Hence there exists a vectorw1PRN(normalizationof w)s.t. kw1k1andxw1;viy¥exppOpNplogpNqbqqqforall i. Now consider any element xPconvpSq. Thenxw1;xy¥exppOpNplogpNqbqqq. Hence by Cauchy-Schwarz inequality and using the fact that kw1k1, kxk¥exppOpNplogpNqbqqq This completes the proof. We are now ready to lower bound the gap-constant. 41 Lemma 5.4. Letbe a highest weight and `jj. Then, pq¥ pn1;:::;nd;`q:exp O pn1ndqlogp`max jnjq Proof.Consider a minimizer of pqgiven by some S pSym`pVqbVqs.t.0RconvpSq. We will apply Lemma 5.3. Here the dimension Nn1nd. We know that for a degree e1 polynomial representation ofGLpnq, the weights that appear in the representation must satisfy 1ne1andi¥0for alli. If we have a rational representation which is of the form{dete2, then the weights ~that appear are of the form ~p1e2;:::;ne2q, where 1ne1andi¥0for alli. Hence ~iPre2;maxte1;e2usfor alli. Now Sym`pVqbV isarepresentationof GLpn1q GLpndq,wherethe epjq 1`njpjq 1°nj1 i1pjq iande2pjq 1 for the action of component j. Thus the weights in Shave entries in r`;`maxjnjs. Hence b¤logp`maxjnjq. Therefore by Lemma 5.3,

Question 1488 multiple-choice

In quantum computation and computational group theory, encoding group elements and subgroups as quantum states enables efficient algorithms for problems such as the hidden subgroup problem. Concepts like coset superpositions, purification, and the structure of group algebras play a fundamental role in these applications. Which of the following correctly characterizes a pure quantum state that purifies the mixed subgroup state corresponding to subgroup H in group G within the group algebra CG ⊗ V? 1) It is a uniform superposition over G, where the attached states |v(x)⟩ are equal for x in the same coset and orthogonal for x in different cosets. 2) It is a product state over G and V, with no entanglement between subsystems. 3) It is a superposition only over elements of subgroup H, with identical attached states for all group elements. 4) It is a mixed state constructed by averaging over all coset representatives. 5) It is a uniform superposition over G, with the attached states |v(x)⟩ being identical for all x in G. 6) It is a superposition over G, with attached states |v(x)⟩ chosen randomly and independently for each x. 7) It is a pure state over V only, with no reference to group elements.

✓ Correct Answer:

The correct answer is 1) It is a uniform superposition over G, where the attached states |v(x)⟩ are equal for x in the same coset and orthogonal for x in different cosets..

📚 Reference Text:

subspace Uof a vector space V, we choose a basis of Vconsisting of a basis of Uand further vectors. Then the elements aγ1 1aγ2 2...aγss((γ1,...γ s)∈/producttexts i=1Zpi) are a left transversal of LinG.F o r an element x∈G, the representative of the coset in this transversal can be computed as follows. First we ﬁnd the smallest non-negative integer γ1such that xa−γ1 1∈M1 by computing the base a1discrete logarithm of xmodulo M1. This can be done by solving an instance of the hidden subgroup problem in Z2 p1. Speciﬁcally, we deﬁne the function (β,γ) /mapsto→| xγ/angbracketright|a−β 1M1/angbracketright. The function can be evaluated with the aid of computing the uniform superposition |M1/angbracketrightusing the exact version [ 16] of Watrous’s method [ 29]. The values are ppairwise orthogonal states, and the hidden subgroup is{(δ,γ) :xδa−γ 1∈M1}. We use the exact hidden subgroup algorithm of [ 16]t o ﬁnd a generator of this group. From this, γ1can be obtained in an obvious way. Now we proceed with xa−γ1 1to compute γ2, and so on. We set αL(x)=aγ1 1...aγrrand βL(x)=αL(x)−1x. IfLis a normal subgroup of G, we can encode the coset xLbyαL(x). This makes the factor group G/La black-box group: The elements are encoded by the elements of the 123 14 Page 8 of 23 M. Imran, G. Ivanyos transversal {α(x);x∈G}, and the multiplication oracle is obtained as a composition of the multiplication oracle for Gwith the computation of the function αL. 2.3 Subgroup states and purifications Let Gbe a ﬁnite group and let Hbe a subgroup of G. We consider elements of the group algebra CGas pure quantum states. (The “natural” scalar product (/summationtext xαx|x/angbracketright,/summationtext yαy|y/angbracketright)=/summationtextαβxy−1makes CGa Hilbert space where the group elements form an orthonormal basis.) A (left) coset superposition of HinGis the uniform superposition |aH/angbracketright= 1√|H|/summationtext h∈H|ah/angbracketrightwhere a∈G. The (left) subgroup state of HinGis the mixed state with the density matrix /Xi1G,H=1 |G|/summationdisplay a∈G|aH/angbracketright/angbracketleftaH|=1 |G:H|/summationdisplay a∈X|aH/angbracketright/angbracketleftaH|, where Xis any left transversal (a set of representatives of the left cosets) of HinG. Apuriﬁcation of/Xi1G,His any pure state |ψ/angbracketright∈ CG⊗Vfor some Hilbert space V such that /Xi1G,His the relative trace of |ψ/angbracketright/angbracketleftψ|with respect to the second subsystem. For general facts about puriﬁcation of mixed states, in particular for the connection with Schmidt decompositions, we refer the reader to Section 2.5 of [ 30]. The following lemma gives a characterization of puriﬁcations of subgroup states. Lemma 1 The pure state |ψ/angbracketright∈ CG⊗V is a puriﬁcation of the subgroup state /Xi1G,H if and only if it can be written as |ψ/angbracketright=1√|G|/summationdisplay x∈G|x/angbracketright|v(x)/angbracketright, where the states |v(x)/angbracketrightand|v(y)/angbracketrightare equal if x and y are in the same left coset of H and orthogonal otherwise. Proof The “if” part follows easily from that the conditions on |v()/angbracketrightimply |ψ/angbracketright= 1√ k/summationtext a∈X|aH/angbracketright|v(a)/angbracketright. To see the “only if” part, recall that a Schmidt decomposition of a state |ψ/angbracketright∈ CG⊗Vis of the form |ψ/angbracketright=/summationtextm i=1λi|ui/angbracketright|vi/angbracketrightwhere m=|G|,|u1/angbracketright,...,|um/angbracketrightis an arbitrary orthonormal basis of CGin which the relative trace of |ψ/angbracketright/angbracketleftψ|w.r.t. to the second subsystem is diagonal (with entries λ1,...,λ m) and the system of

Question 1489 multiple-choice

LWE-based cryptosystems rely on the complexity of lattice problems for their security, particularly facing both classical and quantum attack strategies. Cost optimization for dual attacks involves analyzing metrics related to computational effort and algorithmic parameters. Which metric specifically represents the number of coordinates guessed during the dual attack process, directly impacting the guessing phase workload? 1) rop 2) β (BKZ block size) 3) ζ (zeta) 4) d (lattice dimension) 5) t (FFT coordinates modulo p) 6) red 7) guess

✓ Correct Answer:

The correct answer is 3) ζ (zeta).

📚 Reference Text:

params: LWEParameters, red_cost_model=red_cost_model_default , log_level=1, ): """ Optimizes cost of dual attack as presented in [Matzov22]_. :param params: LWE parameters :param red_cost_model: How to cost lattice reduction The returned cost dictionary has the following entries: - ‘‘rop‘‘: Total number of word operations ( ≈CPU cycles). - ‘‘red‘‘: Number of word operations in lattice reduction an d short vector sampling. - ‘‘guess‘‘: Number of word operations in guessing and FFT. - ‘‘β‘‘: BKZ block size. - ‘‘ζ‘‘: Number of guessed coordinates. - ‘‘t‘‘: Number of coordinates in FFT part mod ‘p‘. - ‘‘d‘‘: Lattice dimension. """ params = params.normalize() forpinearly_abort_range (2, params.q): fork_enum inearly_abort_range (0, params.n, 5): fork_fftinearly_abort_range (0, params.n - k_enum[0], 5): with local_minimum( 40, params.n, log_level=log_level + 4 ) as it: forbetainit: cost = self.cost( beta, params, p=p[0], k_enum=k_enum[0], k_fft=k_fft[0], red_cost_model=red_cost_model , ) it.update(cost) Logging.log( "dual", log_level + 3, f"t: {k_fft[0]}, {repr(it.y)}" , ) k_fft[1].update(it.y) Logging.log( "dual", log_level + 2, f "ζ: {k_enum[0]}, {repr(k_fft[1].y)}" ) k_enum[1].update(k_fft[1].y) Logging.log( "dual", log_level + 1, f "p:{p[0]}, {repr(k_enum[1].y)}" ) p[1].update(k_enum[1].y) Logging.log( "dual", log_level, f "{repr(p[1].y)}" ) return p[1].y classQMATZOV(MATZOV): 53 @classmethod defcost( cls, beta, params, m=None, p=2, k_enum=0, k_fft=0, beta_sieve=None, red_cost_model=red_cost_model_default , ): """ Theorem 7.6 """ ifmisNone: m = params.n k_lat = params.n - k_fft - k_enum # p.15 # We assume here that β_sieve ≈β N = cls.Nf( params, m, beta, beta_sieve ifbeta_sieve elsebeta, k_enum, k_fft, p, ) rho, T_sample, _, beta_sieve = red_cost_model.short_vect ors( beta, N=N, d=k_lat + m, sieve_dim=beta_sieve ) H = cls.Hf(params.Xs) T_guess = ((27 / 8 / e) ** (k_enum / 4)) * sqrt( 2 ** (k_enum * H) * p ** k_fft * cls.T_tablef(N) ) cost = Cost(rop=T_sample + T_guess, problem=params) cost["red"] = T_sample cost["guess" ] = T_guess cost["beta"] = beta cost["p"] = p cost["zeta"] = k_enum cost["t"] = k_fft cost["beta_" ] = beta_sieve cost["N"] = N cost["m"] = m cost.register_impermanent( {"β’": False, "ζ": False, "t": False}, rop=True, p=False, N=False ) return cost defrunall( schemes=( Kyber512, Kyber768, Kyber1024, LightSaber, Saber, FireSaber, TFHE630, TFHE1024, ), 54 # schemes=(Kyber512, Kyber768, Kyber1024, LightSaber, Sa ber, FireSaber), nns=( "list_decoding-naive_classical" , "list_decoding-classical" , "list_decoding-naive_quantum" , "list_decoding-ge19" , ), ): results = {} forscheme inschemes: results[scheme] = {} print(f"{repr(scheme)}" ) fornninnns: cost = MATZOV()(scheme, red_cost_model=RC.MATZOV.__cla ss__(nn=nn)) results[scheme][(nn, "classical" )] = cost print(f" nn: {nn}, cost: {repr(cost)}" ) cost = QMATZOV()(scheme, red_cost_model=RC.MATZOV.__cl ass__(nn=nn)) print(f" nn: {nn}, qcost: {repr(cost)}" ) results[scheme][(nn, "quantum" )] = cost cost = MATZOV()(scheme, red_cost_model=RC.ADPS16) print(f" C0, cost: {repr(cost)}" ) results[scheme][( "C0","classical" )] = cost cost = MATZOV()(scheme, red_cost_model=ChaLoy21()) print(f" Q0, cost: {repr(cost)}" ) results[scheme][( "Q0","classical" )] = cost cost = QMATZOV()(scheme, red_cost_model =ChaLoy21()) print(f"Q0, qcost: {repr(cost)}" ) results[scheme][( "Q0","quantum" )] = cost return results defresults_table(results, fmt=None): import tabulate rows = [] defpp(cost): return round (log(cost[ "rop"], 2), 1) forscheme, costs inresults.items(): row = [ scheme.tag, pp(costs[( "list_decoding-classical" ,"classical" )]), pp(costs[( "list_decoding-naive_classical" ,"classical" )]), pp(costs[( "C0","classical" )]), pp(costs[( "list_decoding-ge19" ,"classical" )]), pp(costs[( "list_decoding-naive_quantum " ,"classical" )]), pp(costs[( "Q0","classical" )]), pp(costs[( "list_decoding-naive_quantum " ,"quantum" )]), pp(costs[( "Q0","quantum" )]), ] rows.append(row) iffmtisNone: return rows else: return tabulate.tabulate( results_table(results), headers=[ "Scheme" , "CC",

Question 1490 multiple-choice

Symmetry principles play a foundational role in quantum information theory, influencing how quantum states are classified and manipulated. Certain symmetry labels remain invariant under specific group actions, which is crucial for applications such as quantum error correction and state tomography. Which symmetry group's action is responsible for the invariance of purity in quantum states during reversible evolutions in a d-dimensional Hilbert space? 1) The unitary group U(d) 2) The cyclic group Zₙ 3) The symmetric group Sₙ 4) The parity group Z₂ 5) The alternating group Aₙ 6) The reflection group R 7) The general linear group GL(d)

✓ Correct Answer:

The correct answer is 1) The unitary group U(d).

📚 Reference Text:

deﬁned as yi=f(j ii) =Pn k=1zikmod 2. Deﬁning the spin-ﬂip operatorP=Nn j=1Xj, whereXjis the Pauli- xoperator acting on the j-th qubit. One can readily see that while Pj ii6=j ii, the parity is invariant under P, i.e.f(Pj ii) =f(j ii). For a concrete example, f(Pj01i) = 1 =f(j10i). In other words, the states are not invariant under the symmetry, but the labels are. •States:Bitstring product states j ii=jzi1zi2:::zini2(C2) n, wherezik2f0;1gandneven •Labels:Parityyi=f(j ii) =Pn k=1zikmod 2 •Group: Z2=f1;pg •Representation: R:G7!GL((C2) n), where 1j ii=j ii,j ii=Pj ii Qubit reﬂection parity: Consider a problem of classifying states according to their qubit-reﬂection parity. Deﬁning the qubit-reﬂection operator R:=R1;nR2;n1:::Rbn=2c;bn=2c+1, whereRj;j0swaps qubits jandj0, and writing i=j iih ij, the states will have label yi= 0(yi= 1) ifiis an eigenstate of Rwith eigenvalue 1(1). Here, one can readily verify that RiRy=i. Qubit permutations: Learning problems with permutation symmetries abound. Examples include learn- ing over sets of elements, modeling relations between pairs (graphs) or multiplets (hypergraphs) of entities, problems deﬁned on grids (such as condensed matter systems), molecular systems, evaluating genuine multi- partite entanglement, or working with distributed quantum sensors. Consider for instance a problem where ann-qubitis a graph state encoding the topology of an underlying graph. One can create such state by starting with the state j+i n, and applying a unitary U(a;b)for each edge (a;b)in the graph. Here U(a;b)=ei ((j0ih0j)a 11b+(j1ih1j)a Zb)is an Ising-type interaction. By conjugating the state with an element of Sn, one obtains a new quantum state whose interaction graph is isomorphic to the original one. •States:Quantum states on qubits, where the qubit labeling index do not matter. •Labels: (Here, any label will work, since the states themselves are invariant). •Group:G=Sn, the symmetric group on nletters •Representation: R:G7!GL((C2) n), where the 2-cycle (j;j0)j ii= SWAPj;j0j ii. Note that since any permutation in Sncan be expressed as a product of swaps, this deﬁnes our representation on all permutations. Translation invariance: LetHa Hamiltonian and consider the problem of classifying energies yiof a set of eigenstates j ii. Suppose H=Pn j=1hj;j+1, wherehj;j+1is a nearest-neighbor interaction and we impose periodic boundary conditions so that n+ 11. ThenHcommutes with the translation operator g: (C2) n!(C2) n, which translates the state j ito the right by gsites (e.g. 0=1and 2j01101i=j01011i). We can then use Prop. 5 to argue that the energy label yiis invariant under the group of translations, so f(j ii) =f( gj ii)for all translations g. •States:Eigenstatesj iiof a Hamiltonian Hon a ring of nqubits •Labels:Eigenenergies yi, i.e.Hj ii=yij ii •Group:G=Zn, the cyclic group of order n. •Representation: :G7!GL((C2) n), where gtranslates to the right by gsites 41 Example 12: Continuous symmetries Unitary transformations and purity: Consider a problem of classifying pure states from mixed states. The dataset here is composed of states with label yi= 0(yi6= 0) ifiis pure (mixed). Since the purity is a spectral property, then the labels in Sare invariant under the action of any unitary. Note that here f(UiUy) =f(i), but in general UiUy6=i. •States:Statesi2D(H). •Labels:Pureyi= 0and mixed yi6= 0. •Group:G=U(d),

Question 1491 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) leverage lattice theory and group embeddings to enable efficient computation over both discrete and continuous Abelian groups. These techniques are foundational for advancements in quantum cryptography and computational mathematics. Which property of the reduced basis is crucial for ensuring that the dual lattice structure can be efficiently computed in quantum algorithms for the Hidden Subgroup Problem (HSP) over continuous groups like R^m? 1) The orthogonality of all basis vectors 2) The invariance of the basis under group automorphisms 3) The boundedness of the condition number of the basis 4) The presence of integer-valued basis coefficients 5) The existence of a finite generating set for the basis 6) The symmetry of the basis with respect to reflection 7) The requirement that all basis vectors have unit length

✓ Correct Answer:

The correct answer is 3) The boundedness of the condition number of the basis.

📚 Reference Text:

improved analysis of [BK93] in Section 4.3. It can be shown that the condition number of a reduced basis is bounded so that the dual lattice, which is the hidden subgroup, can be computed. These provide the main ingredients in the full proof. Theorem 6.1. There is a polynomial time quantum algo- rithm for solving the HSP over Rm. In Section 6.2 we outline the steps of the algorithm and indicate how to derive the probability expression. First we show that the known cases of the Abelian HSP can be re- duced to the new continuous case in Deﬁnition 1.1 over the HSP instance Rm. 6.1 Application: Reduction to G=Rm Our HSP algorithm is applicable to Abelian groups of the formRk×Zl×(R/Z)s×H, whereHis ﬁnite. We call such groups “elementary”. The reduction to G=Rk×Zlis straightforward. In the case of interest, the hidden subgroup Lis a full-rank lattice in G⊆Rk×Rlsuch thatλ1(L∩Rk)/greaterorequalslant λandd(L)/lessorequalslantdfor some ﬁxed numbers λandd. We now describe the further reduction to the group ˜G=Rk+l. The main idea can be illustrated in the one-dimensional case, where the parameter λhas no meaning. We embed G=Zinto ˜G=Rin the standard way, set ν= 2−qfor someq/greaterorequalslant2, and deﬁne the R-oraclegin terms of the Z- oraclefas follows: ˛˛g(x)¸ =c0˛˛strν(t)¸ ⊗˛˛f(s)¸ + c1˛˛strν(t−1)¸ ⊗˛˛f(s+ 1)¸ , (6.2) wheres=⌊x⌋, t =x−s, c 0= cos`π2t´ , c1= sin`π 2t´ . It is clear that gis a continuous function. If fis a periodic function, then gis also periodic with the same period. To construct the state |g(x)/angbracketright using the original oracle f, we compute sandt, use them as parameters in the following sequence of operations, and “uncompute” sandt: |0/angbracketright /mapsto→X zcz|z/angbracketright /mapsto→X zcz|z/angbracketright ⊗˛˛f(s+z)¸ /mapsto→X zcz˛˛strν(t−z)¸ ⊗˛˛f(s+z)¸ , wherez∈ {0, 1}. The last step, |z/angbracketright /mapsto→ | strν(t−z)/angbracketright, re- quires that we discriminate between the states |strν(t)/angbracketrightand |strν(t−1)/angbracketright. This is easy because the supports of those states on the ν-grid do not overlap. Let us now consider the general case, G=Rk×Zl. The groupGis embedded in ˜G=Rk+lby scaling the Zfactors byλ. This is to guarantee that λ1(˜L)/greaterorequalslantλ, where ˜Lis the image ofLunder the embedding. The other condition on the new hidden subgroup reads: d(˜L)/lessorequalslant˜d, where ˜d=dλl. The generalization of Eq. (6.2) is straightforward: ˛˛g(x,x 1,...,x l)¸ =X z1,...,z l∈{0,1} lO j=1|ψ(xj,zj)/angbracketright! ⊗˛˛f(x, s(x 1,z1),...,s(x l,zl)¸ ,(6.3) 300 wheres(x,z ) =⌊x/λ⌋ +z,|ψ(x,z)/angbracketright= cos`π 2t´˛˛strν(t)¸ , witht=x/λ−s(x,z ). Note that the terms in the above sum are mutually or- thogonal vectors. It can be shown that ghas parameters ˜a2=a2+l`π 2νλ(1 +ν)´2, ˜r2=r2+l(2νλ)2and/epsilon1. 6.2 An HSP algorithm for the group Rm Letfbe an (a,r,ε) oracle function for some full-rank lat- ticeL⊆Rmsuch thatλ1(L)/greaterorequalslantλandd(L)/lessorequalslantd(see Def- inition 1.1). The core part of our algorithm is a sampling subroutine that generates an approximation to a random point of the reciprocal lattice L∗. It works under certain assumptions about the oracle parameters. Letω:R→Cbe some Lipschitz function with unit L2- norm supported on the interval [0, 1]. For example, ω(x) =(√2 sin(πx) for x∈[0,1], 0 otherwise.(6.4) Let us also choose a suﬃciently large number ∆ = 2q1and a suﬃciently small number

Question 1492 multiple-choice

Quantum phase estimation is a foundational technique in quantum computing, crucial for algorithms that analyze eigenvalues of unitary operators. Optimizing resource requirements, especially quantum query complexity and qubit usage, is essential for scalable quantum computing. Which of the following statements about coherent iterative phase estimation is correct when the rounding promise on eigenvalues is not present? 1) The algorithm cannot estimate any phase without the rounding promise and always fails. 2) Error from lacking the rounding promise propagates to all bits of the phase estimate, leading to high inaccuracy. 3) The estimator approximates the desired quantum map with limited error, and mistakes are confined mostly to the least significant bit. 4) In the absence of the rounding promise, query complexity becomes exponential in the desired accuracy parameter δ. 5) Garbage qubits increase significantly without the rounding promise, making further computation unmanageable. 6) The estimator requires additional classical post-processing steps only when the rounding promise is violated. 7) The algorithm cannot be applied to problems like thermal state preparation without a rounding promise.

✓ Correct Answer:

The correct answer is 3) The estimator approximates the desired quantum map with limited error, and mistakes are confined mostly to the least significant bit..

📚 Reference Text:

10−m·δ2 8(140) This solution to the inequality relies on the fact that only δampactually enters the query complexity, so if classical resources are cheap but query complexity is expensive then we can make the classical computer do as much work as possible by making mlarger. Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 27 Finally, we compute the query complexity. An invocation of e2πiˆλ(k) jrequires 2n−k−1 invocations of U=e2πiˆλ. By Lemma 10, the number of queries made by the unitary Usvt toUsignalis the degree of the polynomial. By Lemma 11, the polynomial Aη→δamp(x2)has degree 2·Mη→δ, so the query complexity is: 2n−k−1·2·Mη→δamp (141) A really nice feature of the coherent iterative phase estimator we present is that it produces no garbage qubits. All singular value transformation is performed on the ﬁnal output qubit. It does still produce extra phase shifts between the eigenstates, which in some applications may still need to be be uncomputed. However, in applications where phase diﬀerences between eigenstates do not matter, like thermal state preparation, we expect that this uncomputation step can be skipped. To ﬁnish the discussion of coherent iterative phase estimation, we stitch the iterative estimator we just deﬁned into a regular phase estimator. In doing so, we also remark on what happens when no rounding promise is present. A summary of the construction is presented in Figure 4. Corollary 13. Improved phase estimation. The coherent iterative phase estimator from Theorem 12 can be combined with Lemma 7 to make a phase estimator with phases with query complexity at most: O/parenleftBig 2nα−1log(δ−1)/parenrightBig (142) assumingαis bounded away from 1by a constant. Furthermore, if no rounding promise is given, then the estimator δ-approximates in diamond norm a map: |0n/angbracketright|ψj/angbracketright→/parenleftBig ξ|ﬂoor(2nλj)/angbracketright+ζ|λ/prime j/angbracketright/parenrightBig |ψj/angbracketright (143) for some complex amplitudes ξ,ζandλ/prime j=ﬂoor(2nλj)−1mod 2nis an erroneous estimate. The performance is the same, except that 0<α< 1can be any constant. Proof.Writeηk:=1 2−1 2k/parenleftBig 1 2+α 2/parenrightBig ifk≥1andη0:=α 2ifk= 0, and, following Lemma 7, demand an accuracy of δamp,k:= (1−10−m)(δ2−k−1)2/8for thek’th bit. Recall from Lemma 11 that Mηk→δamp∈O/parenleftBig η−1 klog(δ−1 amp)/parenrightBig . Then, the overall query complexity is: n−1/summationdisplay k=02n−k·Mηk→δamp,k∈O/parenleftBiggn−1/summationdisplay k=02n−kη−1 klog(δ−1 amp,k)/parenrightBigg (144) =O/parenleftBiggn−1/summationdisplay k=02n−kη−1 klog(2k+1δ−1)/parenrightBigg (145) Whenk= 0we haveη0=α 2, and when k >0we haveηk>1−α 4which is bounded from Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 28 below by a constant. We can use this to split the sum: ≤O/parenleftBigg 2n−0η−1 0log(20+1δ−1) +n−1/summationdisplay k=12n−kη−1 klog(2k+1δ−1)/parenrightBigg (146) ≤O/parenleftBigg 2nα−1log(δ−1) +n−1/summationdisplay k=12k(k+ log(δ−1))/parenrightBigg (147) ≤O/parenleftBig 2nα−1log(δ−1) + 2(2n−n−1) + (2n−2) log(δ−1)/parenrightBig (148) ≤O/parenleftBig 2nα−1log(δ−1)/parenrightBig (149) This completes the runtime analysis. Now we turn to the case when no rounding promise is present. Notice that when k≥1, the regions where λjis assumed not to appear are guaranteed by the previous estimators deﬁnitely outputting 1for previous bits regardless of the promise. Thus, the only bit that can be wrong is the ﬁrst bit. When an eigenvalueλjfalls into a region disallowed by the rounding promise, the ﬁrst bit will be some superposition ξ|0/angbracketright+ζ|1/angbracketright. Flipping the ﬁnal bit of an

Question 1493 multiple-choice

Quantum algorithms have significantly advanced the study of computational group theory, particularly by enabling efficient solutions to problems involving black-box groups and the Hidden Subgroup Problem. These developments rely on group structure, encoding conventions, and the use of oracles within the quantum computing framework. In quantum computation, which class of groups admits a polynomial-time quantum algorithm that can decompose the group into a direct sum of cyclic groups of prime power order, assuming group elements have unique encoding through a black-box oracle? 1) Abelian black-box groups 2) Non-Abelian simple groups 3) Arbitrary finite groups 4) Infinite Abelian groups 5) Solvable black-box groups without unique encoding 6) Symmetric groups of degree greater than five 7) Nilpotent groups with non-unique encodings

✓ Correct Answer:

The correct answer is 1) Abelian black-box groups.

📚 Reference Text:

encoding length . The groups will be given by generators, and therefore the input size of a group is the product of the encoding length and the number of generators. Note that the encoding of group elements need no t to be unique, a single group element may be represented by several strings. If the encoding is not unique, one also needs an oracle for identity tests. Typical examples of groups which ﬁt in this m odel are factor groups G/Nof matrix groupsG,whereNis a normal subgroup of Gsuch that testing elements of Gfor membership inNcan be accomplished eﬃciently. Also, every binary string of lengthndoes not necessarily corresponds to a group element. If the black box is fed such a s tring, its behavior can be arbitrary on it. Since we will deal with black-box groups we shall shortly des cribe them in the framework of quantum computing (see also [21] or [27]). For a general intr oduction to quantum computing the reader might consult [13] or [23]. We will work in the quantum Turing machine model. For a group Gof encoding length n,the black-box will be given by two oracles UGand its inverse U−1 G,both operating on 2 nqubits. For any group elements g,h∈G,the eﬀect of the oracles is the following: UG|g/a\}bracketri}ht|h/a\}bracketri}ht=|g/a\}bracketri}ht|gh/a\}bracketri}ht, and U−1 G|g/a\}bracketri}ht|h/a\}bracketri}ht=|g/a\}bracketri}ht|g−1h/a\}bracketri}ht. The quantum algorithms we consider might make errors, but th e probability of making an error should be bounded by some ﬁxed constant 0 < ε <1/2. Let us quote here two basic results about quantum group algor ithms respectively in Abelian and in solvable black-box groups. 3 Theorem 1 (Cheung and Mosca [8]). Assume that Gis an Abelian black-box group with unique encoding. Then the decomposition of Ginto a direct sum of cyclic groups of prime power order can be computed in time polynomial in the input size by a quantum a lgorithm. Theorem 2 (Watrous [27]). Assume that Gis a solvable black-box group with unique encoding. Then computing the order of Gand testing membership in Gcan be solved in time polynomial in the input size by a quantum algorithm. Moreover, it is possib le to produce a quantum state that approximates the pure state |G/a\}bracketri}ht=|G|−1/2/summationtext g∈G|g/a\}bracketri}htwith accuracy ε(in the trace norm metric) in time polynomial in the input size +log(1/ε). When we address the HSP, we will suppose that a function f:{0,1}n→ {0,1}mis given by an oracle, such that for some subgroup H≤Gthe function fis constant on the left cosets of H and takes diﬀerent values on diﬀerent cosets. We will say that fhidesthe subgroup H.The goal is to ﬁnd generators for Hin time polynomial in the size of Gandm, that is we assume that m is also part of the input in unary. The following theorem resu mes the status of this problem when the group is Abelian. Theorem 3 (Mosca [21]). Assume that Gis an Abelian black-box group with unique encoding. Then the hidden subgroup problem can be solved in time polynom ial in the input size by a quantum algorithm.

Question 1494 multiple-choice

In quantum information theory, the analysis of randomized measurement protocols often leverages the statistical properties of unitary groups and operator norms. The Clifford group plays a pivotal role in efficient quantum computations due to its symmetry properties. Which formula correctly expresses the inverse measurement channel M⁻¹(ρ) for a quantum state ρ in dimension d, when using a measurement channel defined by averaging over the Clifford group? 1) M⁻¹(ρ) = dρ + Tr(ρ)I 2) M⁻¹(ρ) = (Tr(ρ)I − ρ)/d 3) M⁻¹(ρ) = (d−1)ρ + Tr(ρ)I 4) M⁻¹(ρ) = (d+1)ρ − Tr(ρ)I 5) M⁻¹(ρ) = (Tr(ρ)I + dρ)/(d−1) 6) M⁻¹(ρ) = (dρ − Tr(ρ)I)/(d+1) 7) M⁻¹(ρ) = ρ − Tr(ρ)I

✓ Correct Answer:

The correct answer is 4) M⁻¹(ρ) = (d+1)ρ − Tr(ρ)I.

📚 Reference Text:

µ, while the variance is related to a third moment. Observation 58. M(ρ) = Tr 1/parenleftigg (ρ⊗I)/parenleftiggd/summationdisplay b=1E U∼µ/bracketleftig U†⊗2|b⟩⟨b|⊗2U⊗2/bracketrightig/parenrightigg/parenrightigg (305) Var (ˆoi) = Tr/parenleftigg /parenleftbig ρ⊗M−1(Oi)⊗M−1(Oi)/parenrightbig/parenleftiggd/summationdisplay b=1E U∼µ/bracketleftig U†⊗3|b⟩⟨b|⊗3U⊗3/bracketrightig/parenrightigg/parenrightigg −Tr(Oiρ)2.(306) Proof. For the measurement channel M(ρ), we have: M(ρ):=d/summationdisplay b=1E U∼µ/bracketleftbig ⟨b|UρU†|b⟩U†|b⟩⟨b|U/bracketrightbig (307) =d/summationdisplay b=1E U∼µ/bracketleftbig Tr/parenleftbig ρU†|b⟩⟨b|U/parenrightbig U†|b⟩⟨b|U/bracketrightbig , (308) = Tr 1/parenleftigg (ρ⊗I)/parenleftiggd/summationdisplay b=1E U∼µ/bracketleftig U†⊗2|b⟩⟨b|⊗2U⊗2/bracketrightig/parenrightigg/parenrightigg . (309) For the variance Eq.(303), we need to consider: E/parenleftbig ˆo2 i/parenrightbig:=d/summationdisplay b=1E U∼µ/bracketleftig ⟨b|UρU†|b⟩Tr/parenleftbig OiM−1/parenleftbig U†|b⟩⟨b|U/parenrightbig/parenrightbig2/bracketrightig . (310) We first observe that Tr( AM(B)) =/summationtextd b=1E U∼µ/bracketleftbig ⟨b|UBU†|b⟩⟨b|UAU†|b⟩/bracketrightbig = Tr(M(A)B) for allA,B∈L/parenleftbig Cd/parenrightbig . This implies that Tr/parenleftbig AM−1(B)/parenrightbig = Tr/parenleftbig M−1(A)B/parenrightbig for allA,B∈L/parenleftbig Cd/parenrightbig . 40 Therefore: E/parenleftbig ˆo2 i/parenrightbig =d/summationdisplay b=1E U∼µ/bracketleftig ⟨b|UρU†|b⟩Tr/parenleftbig M−1(Oi)U†|b⟩⟨b|U/parenrightbig2/bracketrightig (311) =d/summationdisplay b=1E U∼µ/bracketleftig Tr/parenleftbig ρU†|b⟩⟨b|U/parenrightbig Tr/parenleftbig M−1(Oi)U†|b⟩⟨b|U/parenrightbig2/bracketrightig (312) = Tr/parenleftigg /parenleftbig ρ⊗M−1(Oi)⊗M−1(Oi)/parenrightbig/parenleftiggd/summationdisplay b=1E U∼µ/bracketleftig U†⊗3|b⟩⟨b|⊗3U⊗3/bracketrightig/parenrightigg/parenrightigg . (313) We will now consider the unitary probability distribution that corresponds to the uniform distribution over the Clifford group, that we defined in Eq. (188) . It is important to remember that the any unitary of the Clifford group can be implemented with O(n2/log(n))elementary gates [72] and that there are efficient algorithms to sample uniformly from the Clifford group [73, 74]. Since the uniform distribution over the Clifford group is an exact 3-design, its first three mo- ments coincide with those of the Haar measure. Thus, we need to insert in Eq. (305) the formula for the second moment over the Haar measure to find the expression of the measurement channel M(ρ)and then invert it. Observation 59. The measurement channel is: M(ρ) =1 d+ 1(Tr(ρ)I+ρ). (314) Thus, its inverse is: M−1(ρ) = (d+ 1)ρ−Tr(ρ)I. (315) Proof. We have: M(ρ) =d/summationdisplay b=1Tr1/parenleftbigg (ρ⊗I)E U∼µ/bracketleftig U⊗2|b⟩⟨b|⊗2U†⊗2/bracketrightig/parenrightbigg (316) =d/summationdisplay b=1Tr1/parenleftbigg (ρ⊗I)I+F d(d+ 1)/parenrightbigg (317) =1 d+ 1(Tr(ρ)I+ρ), (318) where in the second equality we utilized Eq.(95) and in the second equality the partial-swap-trick (152). Therefore, we have that M−1(ρ) = (d+ 1)ρ−Tr(ρ)I, since it can be easily verified that this expression satisfies M−1(M(ρ)) =ρ. To bound the variance we need to compute a third moment over the Haar measure of the unitary group, due to the 3-design property of the Clifford group. Observation 60. The variance is bounded by Var (ˆoi)≤3 Tr/parenleftbig O2 i/parenrightbig . Proof. By utilizing Observation 58, we have: Var (ˆoi) = Tr/parenleftigg /parenleftbig ρ⊗M−1(Oi)⊗M−1(Oi)/parenrightbig/parenleftiggd/summationdisplay b=1E U∼µ/bracketleftig U†⊗3|b⟩⟨b|⊗3U⊗3/bracketrightig/parenrightigg/parenrightigg −Tr(Oiρ)2.(319) 41 Using Proposition 3, we know that E U∼µ/bracketleftig U†⊗3|b⟩⟨b|⊗3U⊗2/bracketrightig =E U∼µ/bracketleftig U⊗3|b⟩⟨b|⊗3U†⊗3/bracketrightig . Therefore by applying Eq.(95), we can express the third moment as: E U∼µ/bracketleftig U†⊗3|b⟩⟨b|⊗3U⊗3/bracketrightig =1 d(d+ 1)(d+ 2)/parenleftigg/summationdisplay π∈S3Vd(π)/parenrightigg (320) =1 d(d+ 1)(d+ 2)/parenleftigg +++++ /parenrightigg ,(321) where we used the Tensor Network representation of permutations defined in Section 6. Substi- tuting this expression into the first term of Eq.(319), up to a global factor (( d+ 1)(d+ 2))−1, we have : Tr/parenleftigg /parenleftbig ρ⊗M−1(Oi)⊗M−1(Oi)/parenrightbig/parenleftigg/summationdisplay π∈S3Vd(π)/parenrightigg/parenrightigg = Tr  +++++ +++++ρℳ−1(O)BBBℳ−1(O)ℳ−1(O)ℳ−1(O)ρ  +++++   = Tr(ρ) Tr/parenleftbig M−1(Oi)/parenrightbig Tr/parenleftbig M−1(Oi)/parenrightbig + Tr(ρ) Tr/parenleftig M−1(Oi)2/parenrightig + Tr/parenleftbig ρM−1(Oi)/parenrightbig Tr/parenleftbig M−1(Oi)/parenrightbig + Tr/parenleftig ρM−1(Oi)2/parenrightig + Tr/parenleftig ρM−1(Oi)2/parenrightig + Tr/parenleftbig ρM−1(Oi)/parenrightbig Tr/parenleftbig M−1(Oi)/parenrightbig . (322)

Question 1495 multiple-choice

In group theory and ring theory, abelian groups can sometimes support various ring structures, leading to classifications such as CR-groups and AR-groups. The existence and properties of indecomposable torsion-free abelian groups with restricted ring structures are of significant interest in algebra. Which of the following statements about indecomposable torsion-free abelian groups and their ring structures is true? 1) All indecomposable torsion-free abelian groups support only commutative ring structures. 2) AR-groups are always CR-groups and cannot be decomposable. 3) Every torsion-free abelian group of finite rank admits a non-zero commutative ring structure. 4) There exist indecomposable torsion-free abelian groups of finite rank greater than two that support a non-zero associative ring structure in certain quotient groups. 5) Absolute annihilators can only exist in groups with torsion elements. 6) CR-groups necessarily admit non-associative ring structures. 7) Decomposable abelian groups cannot be AR-groups.

✓ Correct Answer:

The correct answer is 4) There exist indecomposable torsion-free abelian groups of finite rank greater than two that support a non-zero associative ring structure in certain quotient groups..

📚 Reference Text:

Title: A simple solution of Stratton and Webb’s problem Year: 2022 Paper ID: 380ffa8858a6bf6fd45dbf0df41c87db64e29ffa Source: semantic-scholar URL: https://www.semanticscholar.org/paper/380ffa8858a6bf6fd45dbf0df41c87db64e29ffa Abstract: Abstract The paper contains a simple solution of Stratton and Webb’s problem related to the so-called absolute annihilators of abelian groups researched also by Feigelstock. Namely, there are provided constructions of indecomposable torsion-free abelian groups G of any finite rank greater than two for which the quotient group modulo the subgroup generated by squares of all possible rings on G supports a non-zero associative ring structure. They are much less complicated than those obtained before and show that groups satisfying the mentioned condition are not as rare phenomena as might be supposed. Moreover, most of these constructions led to new examples of CR- and AR-groups, i.e. abelian groups supporting only commutative and only associative rings, respectively. In particular, the first examples of indecomposable as well as decomposable AR-groups which do not satisfy the condition CR are presented. The first example of an indecomposable torsion-free group satisfying the condition CR only in the class of associative rings is also obtained.

Question 1496 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) are pivotal for solving many computational tasks, particularly in non-abelian groups where measurement strategies are more complex. The efficiency of these strategies depends heavily on the structure of the group and the choice of measurement basis in the quantum Fourier transform. Which statement best explains the fundamental limitation of weak Fourier sampling in non-abelian HSP quantum algorithms? 1) It cannot distinguish between subgroups of abelian groups with identical orders. 2) It fails to differentiate a hidden subgroup from its conjugates due to identical probability distributions. 3) It always requires exponentially many quantum queries for any non-abelian group. 4) It only works when the hidden subgroup is trivial or the whole group. 5) It reconstructs all hidden subgroups but not their normal cores. 6) It is unable to sample matrix indices associated with irreducible representations. 7) It is less effective than classical random sampling for subgroup identification.

✓ Correct Answer:

The correct answer is 2) It fails to differentiate a hidden subgroup from its conjugates due to identical probability distributions..

📚 Reference Text:

the non-abelian HSPis O(log|G|); however, their algorithm takes 2O(log2|G|)quantum operations. Ivanyos et al. [IMS03] and Friedl et al. [FIM+03] apply abelian Fourier transform methods to give polynom ial quantum algorithms for the HSP for some special classes of no n-abelian groups. Given the success of Fourier sampling in solving the abelian HSP,one can similar ly ask whether Fourier sampling over the non- abelian group Ghelps in solving the HSP over G. The Fourier transform over a (in general, non-abelian) groupGgives us asuperposition over (ρ,i,j)whereρisan irreducible unitary representation of Gandi,j are the row and column indices of the matrix ρ. The choice of basis for ρgives us a degree of freedom in deﬁning the Fourier transform over G. This is in contrast to the abelian case, where all represent ations are one-dimensional and hence only their names ρmatter. The algorthim starts out with a tensor product of t=O(log|G|)superpositions over random cosets of the hidden subgroup H. Exploiting the symmetries in these states, one can show that (see e.g. [Kup03, Ip03, MRS 05]) the optimal measurement to recover Hconsists of applying the Fourier transform to each coset sta te, measuring the names of the tirreducible representations, followed by a joint POVMon the column spac es of the resulting tstates. In strong Fourier sampling, onemeasures eachof the tcolumn spaces using anorthonormal basis, i.e.,oneperform s atensor product oftcompletevonNeumannmeasurements insteadofajointPOVM.I nweakFouriersampling , one measures the names of the trepresentations only. Hallgren, Russell and Ta-Shma [HRTS03] showed that polynom ially many iterations of weak Fourier sampling give enough information to reconstruct normal hid den subgroups. More generally, they show that the normal core c(H)of the hidden subgroup H(i.e. the largest normal subgroup of Gcontained inH) can be reconstructed via the weak method. Grigni, Schulman , Vazirani and Vazirani [GSVV04] and Gavinsky [Gav04] extended the weak method to ﬁnd a hidden subgroupHinGif[G:κ(G)H] = (log|G|)O(1). Here,κ(G)is the Baer subgroup of Gdeﬁned asκ(G) =/intersectiontext K:K≤GN(K), whereN(K) denotes the normaliser of KinG. The main shortcoming of the weak method is that it gives exac tly the same probability distribution if the hidden subgroup is Hor a conjugate gHg−1ofH. This leads us to consider the strongmethod. Theamount ofadditional information about thehidd en subgroup Hthat canbe extractedbymeasuringthecolumnspaceinanorthonormalba sisdepends,ingeneral,ontheparticularbasis. Inarecent paper, Moore, Russell and Schulman [MRS05]showe d that for the symmetric group Sn, for any choice of bases for the representations, there are order two subgroups that require exponential number of strong Fourier samplings in order to distinguish them from t he identity subgroup. Grigni et al. [GSVV04] studythe randomstrong methodwherearandom measurement basisisusedforeachrepr esentationρ. They deﬁne a group-theoretic parameter αdepending on GandHand show that if αis exponentially large, the additional advantage of the random strong method over th e weak method is exponentially small. In particular, this is case when G=SnandH≤Sn,|H|= 2O(nlogn). 1.1 Ourcontributions In this paper, we analyse the power of the random strong metho d and show, for the ﬁrst time, that under certain (different) general conditions on Gpolynomially many iterations of the random strong method do give enough classical information

Question 1497 multiple-choice

Quantum mechanics represents physical systems using vectors in Hilbert spaces, and the behavior of composite systems, measurement, and information transfer is governed by mathematical principles unique to quantum theory. Concepts such as tensor products, entanglement, and the limitations of state manipulation underpin quantum computing and information protocols. Which of the following statements most accurately describes the No-Cloning Theorem in quantum information science? 1) It states that measuring a quantum system always produces identical copies of its state. 2) It allows for the duplication of classical information stored in quantum systems using unitary operations. 3) It asserts that any entangled state can be decomposed into tensor products of individual states. 4) It implies that any quantum state can be cloned if the system is isolated from the environment. 5) It guarantees that two orthogonal quantum states can always be cloned simultaneously. 6) It proves that no unitary operation can produce an exact copy of an arbitrary unknown quantum state. 7) It suggests that partial measurement of a quantum state enables perfect duplication of its information.

✓ Correct Answer:

The correct answer is 6) It proves that no unitary operation can produce an exact copy of an arbitrary unknown quantum state..

📚 Reference Text:

by jwii=juii jvii: (2.9) Arbitrary vectorsjy1i2H1andjy2i2H2are expressed as jy1i=å iaijuii jy2i=å ibijvii (2.10) with their tensor product jfi2Hgiven by jfi=å i;j aibj juii jvji: (2.11) The map :(y1;y2)7!y1 y2 (2.12) is linear in both y1andy2. We can also form a tensor product of operators. Let U1be an operator that acts on jy1i2 H1, and let U2be an operator that acts on jy2i2H2. The action of U1 U2onjfi2His deﬁned by (U1 U2)jfi= (U1 U2)(jy1i jy2i) =U1jy1i U2jy2i: (2.13) Operators may be speciﬁed in the matrix form relative to some orthonormal basis as U1=å j;kajkjjihkjwhere ajk=hjjU1jki; (2.14) U2=å j;kbjkjjihkjwhere bjk=hjjU2jki; (2.15) where hjj=jji†: (2.16) 8 Their tensor product U=U1 U2can also be expressed in matrix form. Deﬁnition 2.1.1. The tensor product of rsandlmmatrices A=0 BBBBBBB@a11a12::: a1s a21a22::: a2s ............ ar1ar2::: ars1 CCCCCCCAand B=0 BBBBBBB@b11b12::: b1m b21b22::: b2m ............ bl1bl2::: blm1 CCCCCCCA is an rlsmmatrix deﬁned as A B=0 BBBBBBB@a11B a 12B::: a1sB a21B a 22B::: a2sB ............ ar1ar2B::: arsB1 CCCCCCCA: IfAandBare unitary, then so is A B. Not all rlsmmatrices can be represented as tensor products of an rsmatrix and an lmmatrix, since an rlsmmatrix possesses rslm degrees of freedom, while an rsmatrix and an lmmatrix have a total of rs+lm, and for large enough r;s;l;m rslm rs+lm: (2.17) It is possible to decompose the state of an nqubit system into a tensor product of its component states if each of the nqubits forms a closed system. However, if the qubits are allowed to interact, it may not be possible to write the composite state as a product. In this case, the qubits become entangled, i.e., their properties become correlated regardless of spatial separation. A separable state of nqubits depends on 2 nreal parameters, while the entangled state has 2 (2n1) degrees of freedom. 9 Postulate 4 (State Measurement). Quantum measurements are described by a collection fMmg of measurement operators that act on the state space of a measured system, where mis an index that corresponds to a possible measurement result. If the system is in the state jyiimmediately prior to the measurement, then the probability of measuring a particular outcome mis given by Pr(m) =hyjM† mMmjyi (2.18) and the post-measurement state of the system is jy0i=Mmjyip Pr(m): (2.19) Measurement operators satisfy a completeness relation å mM† mMm=I: (2.20) In essence, the act of measurement “collapses” the wave function, returning one result that depends on the measurement basis and destroying all the other information about the state. The goal of a quantum algorithm is to produce the most “telling” measurement result, i.e., the one from which the most information can eventually be extracted. 10 2.2 The No-Cloning Theorem One of the limitations of a quantum computing model arises from the No-Cloning Theorem proved by Dieks [20], and Wootters and Zurek [21]. Theorem 2.2.1. It is impossible to build a machine that performs unitary transformations and is able to enact the evolution jyi jfi!j yi jyi (2.21) for an arbitrary state jyiand some target state jfi. Proof. Suppose there exists a unitary transformation

Question 1498 multiple-choice

In nuclear magnetic resonance (NMR) quantum simulation experiments, precise control over spin states is achieved using specially engineered radiofrequency pulses. These pulses enable the implementation of quantum gates and facilitate reconstruction of quantum states through tomography. When performing full quantum state tomography for a hybrid system composed of two qubits and a qutrit (d = 2×2×3), how many total variables are required to completely characterize the 12×12 density operator, and how are the populations isolated during measurement? 1) 36 variables; populations isolated by applying hard pulses on each qubit 2) 72 variables; populations measured directly without gradient pulses 3) 24 variables; populations isolated by simultaneous shaped pulses 4) 120 variables; populations isolated using a shaped pulse and selective excitation 5) 143 variables; populations isolated by applying a z-gradient followed by spin-selective rotations 6) 96 variables; populations measured via repeated π/2 pulses on all spins 7) 12 variables; populations isolated using multiplet pattern analysis

✓ Correct Answer:

The correct answer is 5) 143 variables; populations isolated by applying a z-gradient followed by spin-selective rotations.

📚 Reference Text:

spins resonate at the frequencies ω1,ω2andω3, withJijcorresponding to the scalar coupling constant between spins i,j. fourth qubits is performed by a/parenleftbigπ 2/parenrightbig3,4 ypulse (which ef- fectively mimics a single-qutrit Fourier gate). All pulses on qubit 1 are Gaussian pulses of 13 ms duration and the simultaneous excitation of qubits 3 and 4 is achieved using a 7 ms shaped pulse ( Seduce.100 ). All the pulses on qubit 2 are hard pulses. Since qubits 3 and 4 are only 36 Hz apart it is diﬃcult to individually address them in an experiment, and hence the controlled-rotation gates on these two qubits have been implemented without ap- plying individual refocusing pulses on them. C. Tomographic reconstruction Partial tomography of the QQQQ system was per- formed in order to reconstruct the desired QQT density matrix. First, a set of operators were designed to recon- struct the full QQT density operatorin d×ddimensional Liouville space (in this case d= 2⊗2⊗3). This set of operators was then extended to the analogous set of to- mography operations in a 16 ×16 dimensional operator space of the four-qubit NMR emulator. The complete characterizationof a 12 ×12dimensional QQT density operator requires the determination of 143 variables. The 11 diagonal elements (populations) are 5 (d)Qubit 4 ( ω4) J14 J24 J241−2 3−4 5−6 7−8 9−10 11−12 13−14 15−16(c)Qubit 3 ( ω3) J13 J231−3 2−4 9−11 10−12 5−7 6−8 13−15 14−16(b)Qubit 2 ( ω2) J12 J23 J23 J24 J24 J241−5 2−6 9−13 3−7 10−14 4−8 11−15 12−16(a)Qubit 1 ( ω1) J12 J13 J13 J14 J141−9 3−11 2−10 5−13 4−12 7−15 6−14 8−16 FIG. 3. Multiplet pattern: (a) Of qubit 1, (b) Of qubit 2, (c) Of qubit 3 and (d) Of qubit 4 of the four-qubit NMR em- ulator. Each multiplet is centered around its chemical shif t value (black lines representing ω1,ω2,ω3andω4respectively), which is further split by scalar coupling interaction. The r atio of coupling constants J12:J13:J14,J12:J23:J24,J13:J23 andJ14:J24is the same as in the 5-Fluorotryptophan molecule. The line thickness is proportional to the spectra l line intensity. Labels below each line refer to the correspo nd- ing transition as marked in Figure 1(a). obtainedbyapplyinganappropriatez-gradienttokilloﬀ- diagonal elements, followed by spin-selective rotations to project the diagonal elements onto experimentally mea- surable parts of the density matrix. The remaining 132 elements are obtained by a set of 19 operators: III′, YII′, XII′, IYI′, IXI′, IIY′, IYY′, IXY′, YIY′, XIX′, XXI′, YYI′, XYI′, YXI′, XYY′, XXY′, YXX′, YYX′and IIΛ′- Gradz-IIΥ′, where I is the identity operator, X(Y) refers to a single spin operator and primed operators corre- spond to qutrit operations. These operators can be im- plemented by applying the corresponding spin-selective π 2pulses (or a no-operation for the identity operator). The last tomography experiment consists of a zgradient pulse sandwiched between two ypulses represented by Λ and Υ applied on the qutrit, of ﬂip angles ( −π 2) and (π 4) respectively. A corresponding set of 19 operations can be used to determine the oﬀ-diagonal elements of the four-qubit QQQQ system: IIII, YIII,

Question 1499 multiple-choice

Metasurfaces are intricately nanostructured optical devices that can manipulate light at subwavelength scales, offering new possibilities for quantum information processing. Programmable metasurfaces, in combination with photonic technologies, are advancing miniaturized hardware for quantum computing. Which device enables dynamic selection and excitation of specific quantum algorithms encoded on a metasurface without physically altering the hardware? 1) Photonic crystal fiber 2) Avalanche photodiode 3) Spatial light modulator 4) Superconducting nanowire 5) Piezoelectric actuator 6) Beam splitter 7) Quantum dot array

✓ Correct Answer:

The correct answer is 3) Spatial light modulator.

📚 Reference Text:

Nanophotonics2024;13(6):927–936 Research Article RandyStefanTanuwijaya,HongLiang,JiaweiXi,WaiChunWong,TszKitYung,WingYimTamand JensenLi* Metasurface for programmable quantum algorithms with classical and quantum light https://doi.org/10.1515/nanoph-2023-0844 ReceivedNovember25,2023;acceptedFebruary9,2024; publishedonlineFebruary21,2024 Abstract :Metasurfaces have recently opened up applica- tions in the quantum regime, including quantum tomog-raphy and the generation of quantum entangled states.Withtheircapabilitytostoreavastamountofinformationby utilizing the various geometric degrees of freedom ofnanostructures,metasurfacesareexpectedtobeusefulforprocessing quantum information. Here, we propose andexperimentally demonstrate aprogrammablemetasurfacecapableofperformingquantumalgorithmsusingbothclas-sicalandquantumlightwithsinglephotons.Ourapproachencodes multiple programmable quantum algorithms andoperations,suchasGrover’ssearchalgorithmandthequan-tum Fourier transform, onto the same metalens array onametasurface.Aspatiallightmodulatorselectivelyexcitesdifferentsetsofmetalensestocarryoutthequantumalgo-rithms,whiletheinterferencepatternscapturedbyasingle-photon camera are used to extract information about theoutput state at the selected output directions. Our pro-grammable quantum metasurface approach holds promis-ingpotentialasacost-effectivemeansofminiaturizingcom-ponents for quantum computing and information process- ing. Keywords: programmablemetasurface;quantuminforma- tion;quantumoptics;quantumalgorithms *Corresponding author: Jensen Li , Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, HongKong,P.R.China, E-mail:jensenli@ust.hk .https://orcid.org/0000-0002-2099-8942 Randy Stefan Tanuwijaya ,Hong Liang ,Jiawei Xi,Wai Chun Wong ,Tsz Kit Yung and Wing Yim Tam , DepartmentofPhysics,TheHongKong University of Science and Technology, Clear Water Bay, Kowloon, HongKong,P.R.China,E-mail: rstanuwijaya@connect.ust.hk (R.S.Tanuwijaya), hliangap@connect.ust.hk (H.Liang), jxi@connect.ust.hk (J.Xi), wcwongaw@connect.ust.hk (W.C.Wong), tkyungaa@connect.ust.hk (T. K. Yung), phtam@ust.hk (W. Y. Tam). https://orcid.org/0000-0002- 4192-1258 (H.Liang). https://orcid.org/0000-0002-5275-626X (W.Y.Tam)1 Introduction While superconducting circuits and trapped ions are two promisingphysicaltechnologiesforimplementingquantumcomputing, photonic implementation remains a versatile platform [ 1]–[3]. Photonic implementation not only allows for the investigation of fundamental questions in quan-tum mechanics [ 4]–[6] but also provides a practical way to demonstrate the advantages of quantum algorithms in solving problems that are signiﬁcantly slower on classical computers[ 7]–[9].Infact,classicalopticsalreadypossesses keyelementsofquantumcomputing,suchassuperposition,interference, and to a large extent, non-separable states analogous to entanglement without quantum non-locality [10],[11]. It becomes possible to construct non-separable statesandperformquantumalgorithmsbyutilizingthedif- ferent internal degrees of freedom of single photons, evenusing classical light [ 12],[13]. This approach has already been adopted in real applications, such as quantum key distributionstoincreasekeyrates[ 14]. Tremendous efforts have been devoted to simulating variousquantumalgorithmsusingclassicallightasaneas-ily accessible testbed. For instance, quantum algorithms involving up to two-qubit operations can be constructed usingcommonopticalcomponents,suchasbeamsplitters,polarization beam splitters, lenses, and Dove prisms, by leveraging the two paths and polarization degrees of free- dom[12],[13].Recentexperimentshaveexploredtheuseof vortex beams generated or manipulated by a spatial light modulator(SLM)toachievequantumalgorithmswithupto4 qubits, utilizing 16 orbital angular momenta (OAM) [ 15], [16].Additionally,moreefficientandprogrammableutiliza- tion of the SLM has demonstrated the implementation of the Deutsch–Jozsa algorithm with an effective complexity of 20 qubits [ 17],[18]. These principles can also be applied at the single-photon level when using a quantum source, wherepracticaldemonstrationsoftenemploytheheralding technique to enhance detection purity and signal-to-noiseratio[18]–[20].Moreover,byworkingwithphotoncounting in the quantum regime, it becomes possible to investigate OpenAccess. ©2024theauthor(s),publishedbyDeGruyter. ThisworkislicensedundertheCreativeCommonsAttribution4.0InternationalLicense. 928—R.S.Tanuwijayaetal.:Metasurfaceforprogrammablequantumalgorithms two-photonphysicswithinthesamelinearopticalsystem.It canfurtherexpandthewaystoconstructqubitsandquan-tum algorithms, including boson sampling for evaluatingthepermanentsofmatrices[ 21]andentangledclusterstates forone-wayquantumcomputation[ 22]. On the other hand, metasurfaces, which consist of a single layer of nanostructures, have proven to be highlyuseful in multiplexing or demultiplexing optical informa-tion. They have been applied successfully in generatingvortex beams or holograms with ﬁne resolution [ 23]–[28]. As we move towards the quantum regime, we have beenwitnessing early demonstrations of metasurfaces in quan-tumtomography,thegenerationofhigh-dimensionalentan-glement, quantum imaging, and the manipulation of two- photon interference [ 29]–[35]. Furthermore, due to their capability to store a signiﬁcant amount of informationregarding light–matter interaction, metasurfaces offer thepotential for processing quantum optical information andimplementingquantumalgorithms.Somestudieshavepro-posedusingmetamaterialstosimulatequantumalgorithmsin the classical domain

Question 1500 multiple-choice

The braid group and its representations play a central role in topological quantum computation, knot theory, and the mathematical structures underlying quantum gates. Unitary and faithful representations have significant implications for how quantum information is manipulated and preserved. Which condition is necessary for the Jones representation of the braid group to be unitary and thus physically meaningful in quantum computation? 1) The complex parameter A must be real and the Temperley-Lieb algebra (TLA) elements must be non-Hermitian. 2) The phase angle φ must be irrational and the parameters a, b arbitrary. 3) The generators must be redefined using combinatorial Bell matrices. 4) The integer m must always be odd and unrelated to the algebraic relations. 5) The representation must be non-faithful with arbitrary phase factors. 6) The Jones representation requires A to be a unit complex number and the TLA elements to be Hermitian. 7) The entanglement must be increased under local operations and classical communication.

✓ Correct Answer:

The correct answer is 6) The Jones representation requires A to be a unit complex number and the TLA elements to be Hermitian..

📚 Reference Text:

the braid group, and to determine the correct combi- nation of the braid generators that gives the required matrix. We ﬁnd that a very simple way to achieve this 1Not to be confused with the Bell matrix of combinatorial math - ematics (after E.T. Bell). 2A more precise statement is that for bipartite states entang le- ment is preserved under LOCC (local operations and classica l communication).task is by means of the Jones representation of the braid group, which we describe below. Unitary Jones representation of B3.– In his con- struction of certain polynomial invariants, the Jones polynomials, for knots and links, Jones [11] provided a new representation of the braid group based on what is essentially the TLA. The TLA, more speciﬁcally denoted byTLm(d), is deﬁned, for an integer mand a complex numberd, to be the algebra generated by the unit ele- mentIand the elements h1,h2,...,h m−1satisfying the relations hihj=hjhi,|i−j|>1; hihi±1hi=hi,1≤i

Question 1501 multiple-choice

In computational quantum mechanics, constructing localized basis functions efficiently is crucial for modeling electronic structures in large systems. Randomized algorithms often use statistical sampling methods to achieve coverage of regions with significant electron density. Which aspect of a randomized SCDM algorithm ensures, with high probability, that all distinct regions of high electron density are represented in the selected localized basis set? 1) An upper bound on the number of sampled columns provided by a probabilistic theorem under assumptions about overlap and support 2) Enforcing explicit geometric proximity between grid points during column selection 3) Using a fixed number of columns regardless of electron density distribution 4) Selecting columns solely based on their spatial coordinates 5) Applying a deterministic selection of columns from the entire grid 6) Ignoring regions with low electron density in the sampling process 7) Relying on the largest possible threshold parameter to maximize region flexibility

✓ Correct Answer:

The correct answer is 1) An upper bound on the number of sampled columns provided by a probabilistic theorem under assumptions about overlap and support.

📚 Reference Text:

the density matrix corresponds to the vacuum, then the electron density is very small and hence the probability of picking the column is very low. In statistics this corresponds to lever- age score sampling, see, e.g., [26]. This randomized version of the SCDM algorithm is outlined in Algorithm 3. Algorithm 3 Computing an approximately localized collection of basis vectors Given: Kohn-Sham orbitals ;electron density ;concentration ;and failure probability 1:Sample (ne= ) logne=elements fromf1;:::;Ngbased on the discrete distribu- tion Pr (fjg) =(j)=ne and denote this set eC 2:Compute the column pivoted QR factorization eC;: =QR 3:Form approximately localized basis e = Q To complete our discussion of Algorithm 3 we must justify the sub-sampling procedure used to select eC. In order to do so we introduce a simple model for the column selection procedure based on the idea that columns \similar" to the ones selected by Algorithm 1 will work well to compute an approximately localized basis. Our underlying assumption is that a set of columns will serve to approximately localize the basis if it contains at least one column in each region where one of the iis large. Because the iconstructed via the SCDM procedure decay exponentially, this is analogous to saying that any column associated with a grid point close enough to 7 the \true" grid point used will suce. Though, by avoiding explicit use of spatial relations of grid points our algorithm and its parameters are not dependent on the physical geometry. To codify this postulate, we let Iif1;:::;Ngbe the smallest non-empty set such that (3.1)X j2Iij i(j)j2 ; where 2(0;1). If multiple such sets exist we select the one that maximizesP j2Iij i(j)j2. Now, we may write our assumption more concretely: a column ci2 f1;:::;Ngsuces to approximately construct iif it is contained in Ii. Taking to be suciently small would enforce adequate sampling to ensure the columns selected by Algorithm 2 are selected to be part of eC:However, in practice this is not necessary for the construction of a localized basis and by choosing a larger we allow for other columns near the peak of ito act as good surrogates. Under this assumption, to approximately localize the basis, we must simply ensure thateCcontains at least one distinct column in each of the sets I1;I2;:::;Ine;i.e.we need a one to one matching between sets and columns. Theorem 3.1 provides an upper bound on the required cardinality of eCto ensure it may be used to approximately localize the basis with high probability. We do require an additional mild assumption that ensures the sets Iido not simultaneously overlap signi cantly and have small support. Theorem 3.1. Letbe the largest constant such that there exist disjoint subsets Is iIi; i= 1;:::;n eeach satisfying X j2Is ij i(j)j2X j2Iij i(j)j2: The seteCconstructed by sampling m(ne= ) logne= elements with replacement from f1;:::;Ngbased on the discrete distribution Pr (fjg) =(j)=ne contains an unique element in each Iifori= 1;:::;n ewith probability 1: Proof . LetFbe the event that eCdoes not contain a

Question 1502 multiple-choice

In probabilistic analysis of quantum algorithms for hidden shift problems, understanding the distribution and concentration of solution counts is critical for algorithmic reliability. Key results involve number-theoretic bounds and inequalities that quantify the likelihood of various solution multiplicities. Which of the following statements is true regarding the variance σ² of the random variable η_xw (the solution count), when N is prime and k ≥ 3? 1) The variance σ² becomes unbounded as N increases. 2) The variance σ² is always greater than π²/6. 3) The variance σ² does not depend on the factorization properties of N. 4) The variance σ² is minimized when k = 2. 5) The variance σ² is large if M is increased arbitrarily. 6) The variance σ² is approximately π²/12 for prime N. 7) The variance σ² is approximately 1 for prime N.

✓ Correct Answer:

The correct answer is 7) The variance σ² is approximately 1 for prime N..

📚 Reference Text:

1+M/qchoices of bi∈ {0,...,M−1}that are divisible by q, and hence the number of b,csuch that gcd( b1−c1,...,bk−ck) =qis upper bounded byMk[(M+q)/q]k. Therefore, for ﬁxed k≥3 andMk/N= 1+O(1/N), we have E x,w[(ηx w)2]≤µ+Mk N2M−1/summationdisplay q=1q/parenleftbiggM+q q/parenrightbiggk (47) =µ+Mk N2M−1/summationdisplay q=1k/summationdisplay j=0/parenleftbiggk j/parenrightbiggMj qj−1(48) ≤µ+Mk N2/bracketleftBigg O(M2logM)+k/summationdisplay j=3/parenleftbiggk j/parenrightbigg Mj∞/summationdisplay q=11 qj−1/bracketrightBigg (49) ≤µ+Mk N2/bracketleftBigg O(M2logM)+π2 6k/summationdisplay j=3/parenleftbiggk j/parenrightbigg Mj/bracketrightBigg (50) =µ+π2 6+o(1). (51) whereinthenexttolaststepwehaveusedthefactthat/summationtext∞ q=1q−(j−1)≤π2/6 for anyj≥3. Asµ=Mk/N= 1+o(1), we ﬁnd σ2=Ex,w[(ηx w)2]−µ2≤ π2/6+o(1). Since the variance is small, Chebyshev’s inequality shows t hat the prob- ability of deviating far from the mean number of solutions is small: Pr(|ηx w−µ| ≥∆)≤σ2 ∆2. (52) Putting ∆ = 4 and using the fact that ηx wmust be an integer, we ﬁnd Pr(ηx w≥5)≤π2/96+o(1). To see that we are unlikely to have no solutions, we need a slig htly stronger bound than the Chebyshev inequality. Since ηx w∈N, we have Pr(ηx w= 0)≤σ2/(µ2+σ2) [1, p. 58]. Now, noting that the gcd in (45) is at least 1, we have E x,w[(ηx w)2]≥µ+Mk(Mk−1) N2= 2+o(1) (53) so thatσ2≥1 +o(1). Therefore, we ﬁnd Pr( ηx w= 0)≤π2/12 +o(1). Combining these results, we see that Pr(1 ≤ηx w≤4)≥1−3π2/32+o(1)≥ 0.0747 +o(1), so that the probability is lower bounded by a constant fo r suﬃciently large N. /square While the above bounds apply to arbitrary values of N, they are not tight, and better bounds can be obtained using knowledge of t he factor- ization of N. For example, if Nis prime, σ2∼1. Fork= 2, the above QUANTUM ALGORITHM FOR A GENERALIZED HIDDEN SHIFT PROBLEM 15 argument is not suﬃcient except for special values of N(such as Nprime); indeed, if Nhas an unbounded number of distinct prime factors, then it ap - pears that the variance of ηx wmight be unbounded. However, for this case, one can simply decrease Mand usek= 3 copies, as mentioned previously. References [1] N. Alon and J. H. Spencer, The Probabilistic Method , 2nd ed., Wiley Interscience, New York, 2000. [2] D. Bacon, A. M. Childs, and W. van Dam, Optimal measurements for the dihedral hidden subgroup problem , arXiv:quant-ph/0501044. [3] ,From optimal measurement to eﬃcient quantum algorithms for the hid- den subgroup problem over semidirect product groups , Proceedings of the 46th An- nual IEEE Symposium on Foundations of Computer Science, 200 5, arXiv:quant- ph/0504083. [4] R. Boneh and R. Lipton, Quantum cryptanalysis of hidden linear functions , Advances in Cryptology – Crypto’95, Lecture Notes in Computer Scienc e, vol. 963, Springer- Verlag, Berlin, 1995, pp. 424–437. [5] I.BoroshandL.B.Treybig, Bounds on positive integral solutions of linear diophantin e equations , Proceedings of the American Mathematical Society 55(1976), 299–304. [6] H. Cohen, A Course in Computational Algebraic Number Theory , Springer-Verlag, Berlin, 1993. [7] W. van Dam, S. Hallgren, and L. Ip, Quantum algorithms for some hidden shift problems , Proceedings of the ACM-SIAM Symposium on Discrete Algorit hms, 2003, pp. 489–498, quant-ph/0211140. [8] M. Ettinger and P. Høyer, A quantum observable for the graph isomorphism problem , arXiv:quant-ph/9901029. [9] ,On quantum algorithms for

Question 1503 multiple-choice

Quantum algorithms have dramatically improved efficiency for certain algebraic problems, including those involving structure discovery in semigroups and groups. Period finding and discrete logarithms are central to cryptography and computational algebra, but present significant classical challenges, especially in large or black-box settings. Which step in an efficient quantum algorithm for finding the period of an element in a black-box semigroup crucially relies on the quantum Fourier transform followed by classical postprocessing to extract the period from measurement outcomes? 1) Preparing a superposition state over all possible indices 2) Implementing modular exponentiation using repeated squaring 3) Performing a binary search to identify the index where the cycle starts 4) Measuring the quantum state and applying continued fractions to recover the period 5) Checking whether repeated application of the element returns to itself for cycle detection 6) Verifying that an element belongs to the tail before the cycle 7) Computing the discrete logarithm with classical search after identifying the period

✓ Correct Answer:

The correct answer is 4) Measuring the quantum state and applying continued fractions to recover the period.

📚 Reference Text:

algorithms.) Lemma 2.1. There is an efﬁcient quantum algorithm to determine the index and the period of an element gof a black-box semigroup. Proof. First we ﬁnd the period, as follows. Create the statePM jD1jjijgji=p M for some sufﬁciently large M(it sufﬁces to take M >N2CN). Note that we can computegjefﬁciently even for exponentially large jusing repeated squaring, so this state can be made in polynomial time. Next, we discard the second register. To understand what happens when we do this, suppose we measure the second register. If we obtain an element in the tail of g, then the ﬁrst register is left in a computational basis state, which is useless. However, with probability at least .MtC1/=M1N=M , we obtain an element in the cycle of g, and we are left with anr-periodic state 1p LL1X jD0jx0Cjri 408 A. M. Childs and G. Ivanyos g g2g3gt−1gt=gt+r gt+1 gt+2gt+r−2gt+r−1 Figure 1. The semigroup hgi. for some unknown x02¹t;tC1;:::;tCr1º, whereLis eitherb.Mt/=rc ord.Mt/=re(depending on the value of x0). This is precisely the type of state that appears in Shor’s period-ﬁnding algorithm (see for example [5, Algorithm 5]). After Fourier transforming this state over ZMand measuring, we obtain the out- comek2ZMwith probability Pr.k/D1 LM L1X jD0e2ik.x 0Cjr/=M 2 Dsin2.krL M/ LM sin2.kr M/: A simple calculation (see for example [5, equations (57)–(60)]) shows that the probability of obtaining a closest integer to one of the rinteger multiples of M=r is at least4=2. By efﬁcient classical postprocessing using continued fractions, we can recover rwith constant probability by sampling from such a distribution [19]. Since we are in the cycle of gwith overwhelming probability, the overall procedure succeeds with constant probability (which could be boosted by standard techniques). Given the period of g, we can ﬁnd its index by an efﬁcient classical procedure. Observe that we can efﬁciently decide whether a given element gjis in the tail or the cycle of g: simply compute grby repeated squaring and multiply by gjto computegjCr. IfgjCrDgj, thengjis in the cycle of g; otherwise it is in the tail ofg. Let .gj/´´ 1ifgjCrDgj(i.e.,gjis in the cycle of g), 0otherwise (i.e., gjis in the tail of g): The list. .g/; .g2/;:::; .gN//consists oft1zeros followed by NtC1 ones, so we can ﬁnd tinO.logN/iterations by binary search. Quantum computation of discrete logarithms in semigroups 409 3 Computing discrete logarithms We now show how to efﬁciently compute discrete logarithms in semigroups on a quantum computer. Theorem 3.1. There is an efﬁcient quantum algorithm to compute loggxon input x;g2S(or to determine if no such value exists). Proof. First, we use Lemma 2.1 to compute the index tand the period rofg. Then we determine whether xis in the tail or the cycle of g. As described in the proof of Lemma 2.1, this can be done efﬁciently by determining whether xgrDx. Ifxis in the tail of g, then we compute p, the smallest positive integer such that .xgp/D1. This can be done efﬁciently by using binary search to ﬁnd the ﬁrst 1 in the list. .xg/;

Question 1504 multiple-choice

In quantum computing, group actions on quantum states are used to uncover hidden symmetries and solve problems like the Hidden Subgroup Problem (HSP). Techniques involving multiple independent copies of quantum states are crucial for error reduction and algorithmic efficiency. In the context of quantum algorithms for finite abelian groups given as black-box groups, what is the primary reason for generating t independent copies of a quantum state when seeking generators for its stabilizer subgroup? 1) To enable self-reducibility and error reduction by allowing parallel group actions on independent superpositions 2) To bypass the no-cloning theorem by creating entangled pairs for measurement 3) To ensure the stabilizer subgroup is abelian by repeated group actions 4) To allow for classical simulation of quantum states through repeated measurement 5) To compress the representation of the group action for faster computation 6) To guarantee that the algorithm always outputs the full subgroup, not just generators 7) To make the orbit of the state trivially uniform under group action

✓ Correct Answer:

The correct answer is 1) To enable self-reducibility and error reduction by allowing parallel group actions on independent superpositions.

📚 Reference Text:

n o t eb y α tthe group action of Gon Γt={|φ/angbracketright⊗t:|φ/angbracketright∈Γ} deﬁned by αt(x,|φ/angbracketright⊗t)=|x·φ/angbracketright⊗t. Observe that one can construct a quantum oracle forαtusingtqueries to a quantum oracle for α. W en e e dt h en o t i o no f αtfor the following reason. Below, we deﬁne problems involving group actions on quantum superpositions where the input superpositionscannot, in general, be cloned (that is, itmay be impossible to make further copies of the input state from just one). However, it will be possible to generate multiple independent copies of the input superpositions by a separate process before the start of o ur algorithm. Hence, in the interests of reducing the error of our algorithm, we start it oﬀ with several independent copies of the input superpositions. Our self-reducibility arguments will reduce the main 8 FRIEDL, IVANYOS, MAGNIEZ, SANTHA, AND SEN problem into a bunch of problems involving actions of smaller groups on quantum superpositions. To solve each of these subproblems with small error, we will require that the self-reduction process leave a suﬃcient number of independent copies of the input superpositions for a subproblem. This is easy to ensure since we startwith a large number of independent copies of the input superpositions to the original problem. However, in order to achieve this goal, the self-reduction process needs to act on several independent superpositions simultaneously by the same group element. The groupaction α tcaptures this notion. This notion will be crucial for our induction arguments. Also note that the stabilizer and the translating coset, deﬁned later, arethe same for group actions αandα t. Thestabilizer of a state |φ/angbracketright∈Γ is the subgroup G|φ/angbracketright={x∈G:|x·φ/angbracketright= |φ/angbracketright}.G i v e n |φ/angbracketright∈Γ, the problem Stabilizer (G,α,t) consists of ﬁnding O(log |G|) generators for the subgroup G|φ/angbracketright,g i v e ntcopies of |φ/angbracketright. Proposition 2.1. LetGbe a ﬁnite abelian group given as a black-box group with encoding length /lscriptand letαbe a group action of G.W h e n t=Ω ( l o g ( |G|)log(1/ε)), thenStabilizer (G,α,t)c a nb es o l v e di nq u a n t u mt i m e poly(/lscript)log(1/ε)with error ε. Proof.L e t|φ/angbracketright⊗tbe the input of Stabilizer .L e tfbe the quantum function onGdeﬁned by |f(x)/angbracketright=|x·φ/angbracketrightfor every x∈G.O b s e r v e t h a t fis an instance of the natural extension of Hidden Subgroup to quantum functions and it hides the stabilizer G|φ/angbracketright. The algorithm for Stabilizer is simply the standard algorithm for the abelian Hidden Subgroup witherror ε. Inthestandardalgorithm,everyqueryisoftheform |x/angbracketrightG|0/angbracketrightS.W es i m u l a t et h e ith query |x/angbracketrightG|0/angbracketrightSusing the ith copy of |φ/angbracketright. The second register of the query is swapped with |φ/angbracketright, and then we let xact on it. We remark that the standard algorithm for abelian Hidden Subgroup outputs O(log |G|) generators for the hidden subgroup. Note that in general the input superposition |φ/angbracketright⊗tgets destroyed by the above algorithm. Theorbitof a state |φ/angbracketright∈Γ is the subset G(|φ/angbracketright)={|x·φ/angbracketright:x∈G}. Deﬁne |G·ϕ/angbracketright=1 √ |G(|φ/angbracketright)|/summationtext |ϕ/prime/angbracketright∈G(|ϕ/angbracketright)|ϕ/prime/angbracketright.E q

Question 1505 multiple-choice

In quantum information theory, multipartite entanglement describes how quantum states are correlated across multiple subsystems, with certain classes of states exhibiting maximal entanglement properties relevant for quantum computation and cryptography. The mathematical characterization and construction of such states often rely on advanced tools from geometry and group theory. Which property uniquely characterizes an absolutely maximally entangled (AME) state among multipartite quantum states? 1) Every pair of parties shares a Bell state. 2) All single-party reduced density matrices are pure. 3) The global state is separable across all bipartitions. 4) Only the full system is maximally mixed. 5) Entanglement entropy is zero for all subsystems. 6) Only adjacent subsystems of size two are maximally mixed. 7) Every possible subsystem of half or fewer parties has a maximally mixed reduced density matrix.

✓ Correct Answer:

The correct answer is 7) Every possible subsystem of half or fewer parties has a maximally mixed reduced density matrix..

📚 Reference Text:

called locally maximally mixed states or 1-un iform states. Various authors in the past (see, for example [GB98, HS00, Sc04, GZ14, HCRLL12, FF PP08, HGS17]) have also considered the more constrained problem of requiring in addition that certain compo site subsystems are maximally mixed [Sc04]. LME states are called m-uniform when all m-party reduced density matrices are maximally mixed. States for which all subsystems with dimension less than or equal to the dimension of the complementary subsystem are called absolutely maximally entangled (AME) or maximally multipartite entangled states (MMES) [FFPP08]. We will discuss and provide examples of these more s pecial states in section 3.1, examples 7 and 8. 2 the GHZ state |Ψ/an}b∇acket∇i}ht=1√ 2(|0/an}b∇acket∇i}ht⊗|0/an}b∇acket∇i}ht⊗|0/an}b∇acket∇i}ht+|1/an}b∇acket∇i}ht⊗|1/an}b∇acket∇i}ht⊗|1/an}b∇acket∇i}ht)∈ H2⊗H2⊗H2 (1.3) and its generalizations, various quantum error-correcting code s tates, cluster states, and graph states. In this paper, we consider the following basic questions: 1. For which dimensions ( d1,...,dn) do LME states exist?2 2. How can we characterize the space of these states? What is the dimension, geometry, etc... ? 3. Can we give explicit constructions of states in SLME? These questions exhibit a remarkable mathematical richness: it tur ns out that they are related to natural questions in representation theory, symplect ic and algebraic geometry, and geometric invariant theory, as reviewed for example in [Walt14], [SOK14] (see additional referencebelow). Makinguseoftoolsfromalloftheseareas,wea reabletoprovideacomplete answer to (1), and new results for questions (2) and (3). Explicit constructions We begin in section 2 with some explicit constructions, reviewing the ca sen= 2 and consid- ering in detail the case n= 3 with dimensions 2 ≤d2≤d3. In these cases, we can explicitly construct all states in SLME. Forn= 2 these states exist if and only if ( d1,d2) = (N,N) while for(d1,d2,d3) = (2,d2,d3)weﬁndthatthese statesexist ifandonlyif( d2,d3) = (N,N) withN≥2 or (d2,d3) = (Nk,(N+1)k) withNk≥2. Except for the cases (2 ,N,N) with N≥4, we ﬁnd that there is a unique LME state up to local unitary transf ormations (i.e. up to a change of basis for each subsystem). For the case (2 ,N,N) withN≥3 the space of LME states up to local unitary transformations has real dimension 2(N−3). It is equivalent to the space of collections of Nunit vectors in R3adding to zero, with sets of vectors related by anSO(3) rotation considered as equivalent.3 2It has been suggested previously that the inequalities di≤/producttext j/negationslash=idjprovide necessary and suﬃcient conditions. We will see that these conditions are necessary but not suﬃcient. 3Remarkably, this space is also equivalent to N-tuples of points on CP1, with no more than N/2 of the points coinciding, up to M¨ obius transformations. This is a very conc rete realization of the Kempf-Ness equivalence between symplectic and algebraic quotients which we disc uss in subsequent sections. 3 Connection to representation theory In section 3, we point out a general way to construct LME states u sing data from the representation theory of ﬁnite or connected compact groups. G iven any group Hwith unitary irreducible representations R1,...,Rnof dimensions d1,...,dnwhose tensor product contains the trivial representation, let

Question 1506 multiple-choice

Computational complexity theory investigates the power of different computation models, including those based on permutations of distinguishable particles. These models can be classical or quantum, and their relationships to standard complexity classes help clarify the boundaries of efficient computation. Which statement best characterizes the computational complexity of simulating a restricted quantum model involving exactly solvable particle scattering on a line, given certain measurements and inputs? 1) It can always be simulated classically in polynomial time using log-space algorithms. 2) Classical simulation is infeasible unless the polynomial hierarchy collapses. 3) It is equivalent to the class L and thus very efficient. 4) The model is known to be NP-complete in all cases. 5) It is universally as hard as general BQP problems. 6) Simulation is feasible using probabilistic algorithms in BPP. 7) The model is trivial and can be solved by deterministic algorithms in constant time.

✓ Correct Answer:

The correct answer is 2) Classical simulation is infeasible unless the polynomial hierarchy collapses..

📚 Reference Text:

Title: The computational complexity of ball permutations Year: 2016 Paper ID: 4828bafbc030d4f2affa0d9b601cc6053e88e18e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/4828bafbc030d4f2affa0d9b601cc6053e88e18e Abstract: We define several models of computation based on permuting distinguishable particles (which we call balls) and characterize their computational complexity. In the quantum setting, we use the representation theory of the symmetric group to find variants of this model which are intermediate between BPP and DQC1 (the class of problems solvable with one clean qubit) and between DQC1 and BQP. Furthermore, we consider a restricted version of this model based on an exactly solvable scattering problem of particles moving on a line. Despite the simplicity of this model from the perspective of mathematical physics, we show that if we allow intermediate destructive measurements and specific input states, then the model cannot be efficiently simulated classically up to multiplicative error unless the polynomial hierarchy collapses. Finally, we define a classical version of this model in which one can probabilistically permute balls. We find this yields a complexity class which is intermediate between L and BPP, and that a nondeterministic version of this model is NP-complete.

Question 1507 multiple-choice

In computational chemistry, accurately modeling molecular interactions in advanced materials requires force fields derived from quantum mechanical calculations. The development and extension of these force fields involve trade-offs between computational cost and the accuracy of representing complex interactions. Which methodological advancement is most directly aimed at improving the accuracy of simulations involving halogenated hydrocarbons in quantum mechanics derived force fields? 1) Implementing the Electrostatic Halogen (ESH) model to capture halogen-specific interactions 2) Increasing the cut-off distance in Lennard-Jones potentials 3) Relying solely on Density Functional Theory (DFT) for all energy calculations 4) Excluding dispersion corrections from the force field parameterization 5) Restricting the force field to only two-body interaction terms 6) Using classical mechanics parameters without quantum mechanical benchmarking 7) Applying only repulsive interactions in the potential energy function

✓ Correct Answer:

The correct answer is 1) Implementing the Electrostatic Halogen (ESH) model to capture halogen-specific interactions.

📚 Reference Text:

a sound and accurate QMD-FF parameterization procedure, for computer simulations of advanced materials and complex systems. Based on the ﬁndi ngs of the present study, three complementary tasks should be pursuited in forthcomi ng work. First, given the good results achieved for the simple molecules investigate d in this work, the protocol should be extended to larger molecules, implementing novel solutions to diminish the consequent computational burden (in particular in the QM le vel), without a signiﬁcant loss of accuracy. The second task is connected to the former, and concerns the choice of the QM level adopted to sample the dimer IPES. Despite comp utational feasibility 33 suggests to resort to DFT based techniques, the proposed (an d necessary) benchmarking against CCSD(T) data becomes rapidly infeasible with the gr owing molecular dimensions. The search and development of new dispersion corrections an d/or QM methods able to balance accuracy and computational cost is in this context o f the foremost importance. Finally, as mentioned throughout the result discussion, th e need of more ﬂexible potential FF functions clearly emerges from this study. In the case of h alogenated hydrocarbons, the implementation of an ESH model, whose parameters could b e inserted in the QMD- FF and optimized together with the standard LJ and charges, c ould certainly improve the ﬁnal results. More in general, the adoption of more complex f unctions, especially those describing the repulsive branches of the interaction curve s, can also be a key strategy to dramatically improve the future QMD-FF performances. In this framework, further research could involve the implementation also of many body terms, beyond the pure two- body approximation here adopted. On the one hand, this could allow for a more sound representation of the interactions that take place in the bu lk, but on the other hand the computational eﬀort to account for such terms during simulat ions could serioulsy under- mine the possibility to extend the approach to large molecul es and complex systems. Supporting Information Additional data and several details about the reported calcu lations not included in this paper. 34 References [1] Allen, M. P.; Tildesley, D. J. Computer Simulation of Liqu ids; Clarendon: Oxford, 1987. [2] Frenkel, D.; Smith, B. Understanding Molecular Simulati ons; Academic Press: San Diego, 1996. [3] Ferrario, M.; Ciccotti, G.; Binder, K. Computer Simulat ions in Condensed Matter Systems: From Materials to Chemical Biology Volume 2; Lectu re Notes in Physics, Vol. 704; Springer Berlin Heidelberg: Berlin, Heidelberg, 20 06. [4] Mackerell, A. D. Empirical force ﬁelds for biological mac romolecules: Overview and issuesJ. Comp. Chem. 2004,25, 1584–1604. [5] Naidoo, K.; Brady, J.; Field, M.; Gao, J.; Hann, M. Modellin g Molecular Structure and Reactivity in Biological Systems; Special Publication s; The Royal Society of Chemistry, 2006. [6] Rappe, A. K.; Casewit, C. J.; Colwell, K. S.; Goddard, W. A.; Skiﬀ, W. M. UFF, a Full Periodic Table Force Field for Molecular Mechanics an d Molecular Dynamics Simulations J. Am. Chem. Soc. 1992,114, 10024–10035. [7] Jorgensen,

Question 1508 multiple-choice

Quantum computing has significantly impacted computational group theory by providing efficient algorithms for problems such as the hidden subgroup problem (HSP). The difficulty of solving HSP varies depending on the structure of the underlying group, with cryptographic implications tied to these distinctions. Which of the following statements accurately describes a major advancement in quantum algorithm research for the hidden subgroup problem involving non-abelian groups? 1) Efficient quantum algorithms for HSP are only available for cyclic groups. 2) Solving HSP for non-abelian groups is believed to be impossible on quantum computers. 3) All non-abelian groups admit exponential-time quantum algorithms for HSP, but none admit polynomial-time solutions. 4) Non-abelian HSP solutions have no relevance to cryptography. 5) Quantum algorithms for HSP in abelian groups do not generalize to any non-abelian cases. 6) Specific families of non-abelian groups have been identified for which HSP can be solved in polynomial time using quantum algorithms. 7) Quantum algorithms for HSP are universally efficient for all finite groups regardless of structure.

✓ Correct Answer:

The correct answer is 6) Specific families of non-abelian groups have been identified for which HSP can be solved in polynomial time using quantum algorithms..

📚 Reference Text:

Title: Polynomial-Time Solution to the Hidden Subgroup Problem for a Class of non-abelian Groups Year: 1998 Paper ID: 2494248a3a029f621cdb1fb6300a71e3eda6d5f6 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/2494248a3a029f621cdb1fb6300a71e3eda6d5f6 Abstract: We present a family of non-abelian groups for which the hidden subgroup problem can be solved efficiently on a quantum computer.

Question 1509 multiple-choice

In quantum information theory, the implementation of quantum channels using programmable quantum processors is constrained by mathematical principles from representation theory. The structure and multiplicity of irreducible representations in tensor products play a key role in determining the feasibility of such implementations. Which statement about the appearance of one-dimensional irreducible representations (irreps) in the direct sum decomposition of the tensor product of two group representations is correct? 1) A one-dimensional irrep can appear with unlimited multiplicity in any tensor product decomposition. 2) The multiplicity of a one-dimensional irrep is always equal to the product of the dimensions of the representations. 3) One-dimensional irreps cannot appear at all in the tensor product decomposition of two finite-dimensional representations. 4) In the direct sum decomposition of the tensor product, any one-dimensional irrep appears with multiplicity at most one. 5) The multiplicity of one-dimensional irreps in tensor product decompositions is always greater than one for non-Abelian groups. 6) If both representations are one-dimensional, their tensor product decomposes into multiple one-dimensional irreps. 7) The structure of the multiplicity space allows for arbitrary linear combinations of one-dimensional irreps in the decomposition.

✓ Correct Answer:

The correct answer is 4) In the direct sum decomposition of the tensor product, any one-dimensional irrep appears with multiplicity at most one..

📚 Reference Text:

that there is a k∈{1,...,K} such thatnk>1. Then, there would be elements in JUVof the form Bϕ=|ϕk/angbracketright/angbracketleftϕk|⊕0∈JUV where|ϕk/angbracketright/angbracketleftϕk|∈D (H/prime k)is a rank-one projector and dim( 0) =d2−nk. By the Choi- Jamiołkowski isomorphism, there is a corresponding channel T∈ TUV. We consider a Kraus representation of the channel T. SinceTis a completely positive map and rank(Bϕ) = 1, the channel Tcan be written as T(·) =X(·)X∗with one Kraus operator X∈B(H2). Since, additionally, Tis trace-preserving, we know that XX∗=1=X∗X, i.e. the Kraus operator is a unitary and Tis a unitary channel T(·) =W(·)W∗ withW∈U2. Sincethereareinﬁnitelymanypurestates, theprocessorwouldhavetoimplementinﬁnitely many corresponding unitary channels. This contradicts the No-Programming Theorem according to which there is no processor that implements inﬁnitely many unitary channels exactly with program dimension dP<∞. This shows that a processor PwithdP<∞ cannot exist if bk= 1andnk>1for somek∈{1,...,K}. Note that the above argument fails forbk>1, since then there might be no rank- 1elements inK. However, from a representation-theoretic perspective this situation does not arise because a one-dimensional irrep in the direct sum decomposition of U⊗¯Valways has multiplicity ≤1as the following proposition shows. Proposition 21. LetUbe an irrep onH1of a compact group G. LetVbe another representation of GonH2with dimension d2≤d1. In a direct sum decomposition of ¯U⊗V, the one-dimensional irreps λappear with multiplicity nλ≤1. Proof.LetχUbe the character of the irrep U,χVthe character of V, andλthe character of the one-dimensional irrep λ. We use the following scalar product (see Lemma 8) /angbracketleftχ,ψ/angbracketright=/integraldisplay Gχ(g)ψ(g)dµ(g), whereµis the Haar measure on G. Here,ψis the character of an arbitrary representation ofG. Note that if Uis an irrep, then this scalar product of the corresponding characters Accepted in Quantum2021-06-18, click title to verify. Published under CC-BY 4.0. 11 gives the multiplicity of Uin the representation corresponding to ψ(see Corollary 9). We want to show that the multiplicity of λin¯U⊗Vis≤1. Note that the character of ¯U⊗V is¯χU·χV. Let ˆGbe the set of irreps of Gand let χV=/summationdisplay α∈ˆGnαχα bethedecompositionof Vintoirreduciblerepresentationsonthelevelofcharacters. Then, /angbracketleftλ,¯χU·χV/angbracketright=/integraldisplay Gλ(g)/parenleftbigχU(g)χV(g)/parenrightbigdµ(g) =/integraldisplay Gλ(g)χU(g)χV(g)dµ(g) =/angbracketleftλ·χU,χV/angbracketright. =/summationdisplay α∈ˆGnα/angbracketleftλ·χU,χα/angbracketright. Sinceλis the character of a one-dimensional irrep, it is equal to the representation itself and|λ(g)|2= 1forallg∈G. Thus,/angbracketleftλ·χU,λ·χU/angbracketright= 1andtherepresentationcorresponding toλ·χUis again irreducible. Note that the representation corresponding to λ·χUhas the same dimension as U. The scalar product /angbracketleftλ·χU,χα/angbracketrightthus gives the multiplicity of λ·χUinχα. Letαbe such thatnα>0. If the dimension of the representation corresponding to χαis smaller than d2, then/angbracketleftλ·χU,χα/angbracketright= 0by Lemma 8. Thus, /angbracketleftλ,¯χU·χV/angbracketright= 0. If the dimension is d2, thenχV=χαfor dimensional reasons and Vis irreducible. The multiplicity of an irrep in another irrep can be at most equal to 1which proves the assertion. Remark 22. We can also give a direct algebraic argument for the impossibility of a one- dimensional irrep of Gto occur in ¯U⊗Vwith multiplicity larger than one, if Uis an irrep andVis an irrep, too, and has the same dimension as U. Namely, observe that all pure states in the multiplicity space Kof a one-dimensional irrep have to be maximally entangled due to Schur’s Lemma (their reduced states on both factors have to be maximally mixed). However, there is no two- or

Question 1510 multiple-choice

Quantum Markov semigroups (QMS) and their generators play a crucial role in describing the evolution of open quantum systems, especially when symmetries and group-theoretic structures are involved. Hörmander systems and projective representations of Lie groups connect stochastic processes with quantum dynamics in both finite and infinite-dimensional settings. In the context of quantum Markov semigroups on Hilbert spaces, which property specifically allows the Lindblad generator LV(x) = Σj [aj²x + xaj² − 2ajxaj] (with self-adjoint aj) to be interpreted as generating a primitive Markov semigroup on the space of essentially bounded functions over a Lie group G? 1) The commutativity of the operators aj 2) The existence of a full Laplacian in the Lie algebra 3) The restriction to finite-dimensional Hilbert spaces 4) The invariance of LV under unitary conjugation 5) The construction of independent Poisson processes for each aj 6) The ability to represent Kraus operators as elements of SU(2) 7) The identification of iaj as tangent vectors at the identity in the unitary group U, spanning a Hörmander system that defines the tangent space of G

✓ Correct Answer:

The correct answer is 7) The identification of iaj as tangent vectors at the identity in the unitary group U, spanning a Hörmander system that defines the tangent space of G.

📚 Reference Text:

t=0 =d dtu(g)∗u(gj(t))∗xu(gj(t))u(g)/divides.alt3 t=0 =−i([aj;x])(g): Therefore we get (LV⊗idB(H))○(x) =−/summation.disp ji2([aj;[aj;x]]) =(/summation.disp ja2 jx+xa2 j−2ajxaj): It means that the Lindblad generator of the transferred QMS is given by LV(x)=/summation.disp ja2 jx+xa2 j−2ajxaj: Considering the more general case of Hörmander systems instead of the Laplacian is motivated by simple examples relevant to quantum information. One such example is the Lindblad generator with Kraus operatorsxandz. These are transferred from a Hörmander system for the group SU(2). However, the third direction is missing and hence it is not a Laplace-Beltrami Laplacian, which would involve the whole orthogonal basis of the Lie algebra. Conversely, if a Lindblad generator of a QMS onB(H)has the form given by the previous equation for some self-adjoint elements aj∈B(H), then we can consider the anti-self-adjoint operators iajas tangent elements of the Lie group U(H)at the identity IH. Therefore they generate a Hörmander system. Furthermore, we can consider the Lie-subgroup Gof U(H)with tangent space at identity spanned by this Hörmander system. The corresponding generator LVis therefore the generator of a primitive Markov semigroup (St)t≥0onL∞(G). We summarize this discussion in the next theorem. Theorem II.3. Letg/uni21A6u(g)be a projective representation of a compact Lie group Gon some ﬁnite dimensional Hilbert space H. Then the Lindblad generator of the transferred QMS (Tt=e−tLV)t≥0as 4 deﬁned by Equation (3)is given by LV()=/summation.disp ja2 j+a2 j−2ajaj; (8) where theajare deﬁned by Equation (7). Conversely, letLbe the Lindblad generator of a QMS on B(H) which takes the form (8)for some self-adjoint elements ajinB(H). Then there exists a compact Lie group G, a continuous projective representation u∶G→U(H)and a Hörmander system V={V1;:::;Vm}in the Lie algebra ofGsuch that∶x/uni21A6(g/uni21A6u(g)∗xu(g))satisﬁes (L(x))=(LV⊗idB(H))○(x)∀x∈B(H): C. Jumps Let now Gbe a ﬁnite group and let (kt(g;h))g;h∈Gbe a right-invariant density kernel onG. Let us denote by (gt)t≥0the stochastic process onGinduced by this kernel. The corresponding Markov semigroup admits a transition matrix Lsuch that St=e−tLfor allt≥0. In view of Equation (2), the connection between the Markov kernel and the transition matrix is therefore given by: kt(g;h)=/divides.alt0G/divides.alt0e−tL(g;h); g;h ∈G: Writingch=−L(h−1;e)for allh≠e, we then have by right invariance that for all f∈L∞(G), L(f)(g)=−/summation.disp h∈Gch[f(hg)−f(g)]: Thanks to the right-invariance, we can deﬁne a family of independent Poisson processes /parenleft.alt1(~Nh t)t≥0/parenright.alt1h∈Gwith intensitychsuch that for any function f∈L∞(G): f(gt)−f(gt−)=/summation.disp h∈G(f(hgt−)−f(gt−))/parenleft.alt1~Nh t−~Nh t−/parenright.alt1: Deﬁne the compensated Poisson process with intensity chand jumps 1/slash.left√ch,h≠e: Nh t=1√ch(~Nh t−cht): Writingdf(gt)∶=f(gt)−f(gt−)anddNh t∶=Nh t−Nh t−, we can rewrite the previous equation as the stochastic differential equation: df(gt)=/summation.disp h∈G/slash.right{e}ch(f(hgt−)−f(gt−))dt (9) +/summation.disp h∈G/slash.right{e}√ch(f(hgt−)−f(gt−))dNh t =/summation.disp h∈G/slash.right{e}(f(hgt−)−f(gt−))d~Nh t: We are now ready to build a QMS from this Markov chain. Letg/uni21A6u(g)be a projective representation of G on some ﬁnite dimensional Hilbert space H. We want to ﬁnd a stochastic differential equation for (u(gt))t≥0. To this end, take y∈B(H)and deﬁnefy∶h∈G/uni21A6Tr[yu(h)]. Applying Equation (9) to fywe ﬁnd dfy(gt)=/summation.disp h∈G/slash.right{e}chTr[y(u(hgt−)−u(gt−))]dt +/summation.disp h∈G/slash.right{e}√chTr[y(u(hgt−)−U(gt−))]dNh t: From this we deduce du(gt)=/summation.disp h∈G/slash.right{e}ch(u(h)−IH)u(gt−)dt +/summation.disp h∈G/slash.right{e}√ch(u(h)−IH)u(gt−)dNh t =/summation.disp h∈G/slash.right{e}(u(h)−IH)u(gt−)d~Nh t: (10) This equation is well-known in the theory of quantum stochastic calculus, see [13], [14]. Theorem II.4. Let(St=e−tL)t≥0be a Markov semigroup on a ﬁnite group Gwith right-invariant Markov kernel. Write

Question 1511 multiple-choice

In quantum computing, qudits are generalizations of qubits with d-dimensional Hilbert spaces, enabling more complex gate operations and potentially increased efficiency in circuit design. The mathematical structure and universality of qudit gate sets are crucial for advancing quantum algorithms and error correction. Which property of the SUM gate in universal qudit gate sets ensures that the system cannot be decomposed into simpler subsystems, thereby guaranteeing universality? 1) The SUM gate is diagonal in the computational basis 2) The SUM gate is imprimitive 3) The SUM gate is a member of the Clifford group 4) The SUM gate always commutes with the Hadamard gate 5) The SUM gate acts only on single qudits 6) The SUM gate preserves tensor product structure 7) The SUM gate is equivalent to the SWAP gate in all dimensions

✓ Correct Answer:

The correct answer is 2) The SUM gate is imprimitive.

📚 Reference Text:

system is the sum gate SUM p,t h e Hadamard gate Hpand the gates Q[i]pfori∈[p−1].T h e proof is analogous to that of the qutrit set. The Hadamard Hpand the Q[i]gates are combined to form a dense subgroup of SU(p)and SUM pis shown to be imprimitive. Implementing Theorem 1.3 in Ref. 23; the set is a universal gate set. These universal gate sets for the qudit systems, with fewer numbers ofgates in each set compare to that in the previous examples, havethe potential to perform qud it quantum algorithms on the topological quantum computer. 2.2 Examples of Qudit Gates In this section we introduce the qudit versions of many importantquantum gates and discuss some of the gates ’advantages compared to their qubit counterparts. The gates discussed arethe qudit versions of the π/8 gate in Section 2.2.1 , the SWAP gate inSection 2.2 and the multi-level controlled gate in Section 2.2 InSection 2.2.3 , we also introduce how to simplify the qubit Toffoli gate by replacing one of the qubit to qudit. This gives ideasabout improving the qubit circuits and gates by introducingqudits to the system. 2.2.1 Qudit Versions π/8Gate The qubit π/8 gate Thas an important role in quantum computing and information processing. This gate has a widerange of applications because it is closely related to the Cliffordgroup but does not belong to the group. From the Gottesman-Knill theorem [ 64] it is shown that the Clifford gates and Pauli measurements only do not guarantee universal quantumcomputation (UQC). The π/8 gate, which is non-Clifford and from the third level of the Clifford hierarchy, is the essential gateto obtaining UQC [ 20]. This gate can be generalized to a d dimensional qudit system, where, throughout the process, dis assumed to be a prime number greater than 2 [ 71]. To de ﬁne the Clifford group for a d-dimensional qudit space, weﬁrst de ﬁne the Pauli Zgate and Pauli Xgate. The Pauli Zgate and Pauli Xgate are generalized to ddimension in the matrix forms [ 11, 67, 124, 130 ]. FIGURE 2 | (A) is the qudit SWAP circuit using CXdandKdgates [ 58, 131 ].(B)is the qudit SWAP circuits with the CXd,t h e CX† dand the Kdgates. Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504 4Wang et al. Qudits and High-Dimensional Quantum Computing Xd/equals⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝00/ 01 10/ 00 01/ 00 ««1«« 00/ 10⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,Z d/equals⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝10 0 / 0 0ω 0/ 0 00 ω2/ 0 «« « 1 0 00 0 / ωd−1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (18) for ωthedthroot of unity Eq. 14 . The function of the Zgate is adding different phase factors to each basis states and that of the X gate is shifting the basis state to the next following state. Usingbasis states the two gates are Z d⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉:/equalsωj⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉Xd⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉:/equals⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉+1,j∈{0,1,2,... ,d−1} (19) In general, we de ﬁne the displacement operators as products of the Pauli operators, D(x|z)/equalsτxzXx dZz d,τ:/equalse(d+1)πi/d, (20) where (x|z)correspond to the xandyin the exponent of τ,Xand Z. This leads to the de ﬁnition of the Weyl-Heisenberg group

Question 1512 multiple-choice

Quantum computation relies on the precise manipulation of qubits using unitary operations, which are represented as matrices that govern the evolution of quantum states. Understanding the properties and roles of single-qubit gates is essential for building and analyzing quantum algorithms. Which single-qubit gate is specifically responsible for converting a computational basis state into an equal superposition of |0⟩ and |1⟩, thus enabling quantum parallelism? 1) Hadamard gate 2) Pauli-X (NOT) gate 3) Pauli-Z gate 4) Phase shift S gate 5) Identity gate 6) Pauli-Y gate 7) Phase shift T gate

✓ Correct Answer:

The correct answer is 1) Hadamard gate.

📚 Reference Text:

Usuch that U(jyi jfi) =jyi jyi: (2.22) By linearity of U U(ajyi jfi) =a(U(jyi jfi)) =ajyi jyi: (2.23) Letjci=ajyi. Then from Eq (2.22) it follows that U(jci jfi) =jci jci=ajyi ajyi =a2jyi jyi (2.24) which contradicts Eq (2.23). 2.3 Quantum Gates To perform a quantum computation, it is necessary to: 1. Prepare the quantum computer in a well-deﬁned initial state, for example, j00:::0i. 2. Manipulate the quantum-computer wave function via unitary transformations. 3. Perform a measurement in the computational basis at the end of an algorithm. The evolution of an n-qubit system is described by a unitary matrix of degree 2n, but any 11 such matrix can be decomposed into a product of unitary operations on one or two qubits, and these operations are the elementary quantum gates of the quantum circuit model of computation. It follows from linearity that the action of a gate is completely speciﬁed by its action on the com- putational basisfj0i;j1ig. 2.3.1 Single-qubit gates A quantum gate acting on a single qubit is represented by a 2 2 unitary matrix. The simplest single-qubit transformation is the identity Iwhose action on the computational basis is deﬁned by I:j0i7!j 0i;j1i7!j 1i (2.25) with the matrix representation I=j0ih0j+j1ih1j=0 B@1 0 0 11 CA (2.26) The quantum NOT gate implements the mapping X:j0i7!j 1i;j1i7!j 0i (2.27) and is represented by the matrix X=j1ih0j+j0ih1j=0 B@0 1 1 01 CA=sx: (2.28) Similarly, Y:j0i7!j 1i;j1i7!j 0i (2.29) 12 is represented by Y=j0ih1jj1ih0j=0 B@01 1 01 CA=isy; (2.30) and Z:j0i7!j 0i;j1i7!j 1i (2.31) has a representation Z=j0ih0jj1ih1j=0 B@1 0 011 CA=sz: (2.32) The matrices sx;sy;szare Pauli matrices used in quantum mechanics to describe the spin of a spin-1/2 particle in three spatial dimensions. TheZgate is a special case (with q=p) of the phase shift gate Pthat alters the relative phase of the amplitudes of a qubit: P:j0i7!j 0i;j1i7!eiqj1i (2.33) P=j0ih0j+eiqj1ih1j=0 B@1 0 0eiq1 CA (2.34) Other special cases of interest include the Sgate with q=p 2whose matrix is given by S=0 B@1 0 0i1 CA (2.35) and the Tgate with q=p 4: T=0 B@1 0 0eip 41 CA (2.36) 13 A variation of the phase gate is the discrete phase gate Rkgiven by Rk=0 B@1 0 0e2pi 2k1 CA (2.37) A gate used to create superposition states is the Hadamard gate Hspeciﬁed by H=1p 2(j0ih0j+j0ih1j+j1ih0jj1ih1j): (2.38) The action of Hon the computational basis is given by H:8 >>< >>:j0i7!j0i+j1ip 2 j1i7!j0ij1ip 2(2.39) with a representation H=1p 20 B@1 1 111 CA (2.40) In the quantum circuit notation, a qubit is denoted by a single line, called a quantum wire, which represents a qubit that is not evolving, i.e., a qubit acted upon by the identity operator I. If a qubit is initialized into a particular state, the state is written on the left of the appropriate quantum wire, and if a qubit is to be subjected to a particular unitary transformation, a corresponding gate symbol is placed along the quantum wire. For example, the process of initializing a qubit to a state j0i,

Question 1513 multiple-choice

Quantum photonic circuits increasingly utilize advanced metasurface technologies to enable programmable manipulation of single photons for algorithmic processing. Materials and fabrication choices directly impact transmission efficiency, scalability, and the performance of quantum operations in these platforms. Which design improvement is most likely to increase transmission efficiency in a metasurface-based quantum photonic circuit? 1) Replacing metallic components with dielectric metasurfaces 2) Increasing the number of shared optical elements between algorithms 3) Using thicker metal films for improved robustness 4) Operating at lower photon energies to reduce losses 5) Eliminating nanoslot patterning during fabrication 6) Employing classical light sources for input state preparation 7) Relying solely on geometric metasurfaces without material changes

✓ Correct Answer:

The correct answer is 1) Replacing metallic components with dielectric metasurfaces.

📚 Reference Text:

ourcurrent sample, as the center lens is shared between twoalgorithms, there can be some output directions that are 934—R.S.Tanuwijayaetal.:Metasurfaceforprogrammablequantumalgorithms chosen not to read at decoding for a speciﬁc algorithm. So thereisadedicatedbalancebetweenprogrammabilityandtransmissionefficiency,whichcanbefurtherimprovedbyoptimizingthedesignmatrixinminimizingthenumberofusedoutputdirectionsoradoptingdielectricmetasurfacestoincreasetheoveralltransmissionefficiency. 3 Conclusions Inconclusion,wehavedevelopedametalensarrayplatformcapable of implementing quantum algorithms with singlephotons. The functionality of the platform has been vali-dated using both classical and quantum light sources. Byselectively exciting subsets of metalenses and interpretingthe interference patterns at speciﬁc output directions, wehave demonstrated the programmability of quantum algo-rithms and operations, exempliﬁed by the demonstrationof Grover’s search (GS) and Quantum Fourier Transform(QFT)inthiswork.Suchaplatformholdspromiseforminia-turizing quantum optical circuits and algorithms, openinguppossibilitiesformetasurfaceapplications.Bythedemon-strationusingasingle-photonquantumsource,ourschemepresentspromisingprospectsforfutureworks.Speciﬁcally,ourschemecanbefurtherextendedtoprobemoreinforma-tion,suchasaboutthemetasurfaceoperationbyleveragingmultiphoton quantum interference [ 29]. For example, by illuminating a pair of lenses using two photons and mea-suringthecoincidenceofanytwooutputdirections,wecan probeoutthephasedifferencebetweenthecompetingpro- cesswithinthetransfermatrix,sharingasimilarprincipleasinbosonsampling[ 7],[21].Whilethescalabilitytoalarger number of qubits through this approach requires furtherexploration, it already offers the opportunity to ‘factorize’certain components within a larger-scale quantum com-putation. The metasurface can effectively reduce the sizeandeliminatetheneedforopticalalignmentinthefactor-ized part. While we have demonstrated resource sharingbetween two operators on the same metasurface, furtheroptimization of resource management can be explored infuturedevelopments.Additionally,thecurrentworkutilizesageometricmetasurfacewithrobustoperation,relyingonmetallic components rather than local resonances. How-ever, it may be advantageous to adopt dielectric metasur-faces in the foreseeable future to achieve higher transmis-sionefficiency,therebyenablingtheuseoflargermetalensarraysforquantumoperations.4 Methods 4.1 Sample fabrication Prior to fabrication, the glass substrate is immersed in concentrated sulfuricacidfor24handthencleanedsuccessivelywithdistilledwaterand acetone. A 50nm-thicksilverﬁlmis then depositedonto the sub- strate by an E-beam Evaporator (AST 600 EI Evaporator), operating at a deposition rate of 1 ̊A∕s. Finally, a focus ion beam (FEI Helios G4 UX,30kV,41pA)isusedtowritethedesignednanoslotpatternonthe preparedsilverﬁlm. 4.2 Experimental and measurement details The full experimental setup is presented in the Supplemen- taryMaterials .Weusea2-mm-thicktype-IIBBOanda200mW405nm laser (CrystaLaser DL-405-400) to generate the polarization-entangledphoton pairs in a state of 1/√2(|HV⟩ +|VH⟩). The photons are split into the signal arm and heralding arm using a prism. To compensate for the translational and longitudinal walk-off effects of the photonpairs, a half-wave plate (HWP), with an optical axis at 45 ◦with the horizontalaxis,andaBBOwithhalfofthethicknessofthemainBBO areinstalledinbotharms.Ontheheraldingarm,apolarizerselectivelyﬁlters horizontally (H) polarized photons, such that the heralding process only triggers imaging for those photons with vertical (V) polarization. The heralding photons are then detected using a singlephoton counting module (SPCM) (Excelitas-SPCM-800-14-FC), which triggerstheSinglePhotonAvalancheDiode(SPAD)camera(PiImaging SPAD 512S). On the signal arm, the photons are then coupled into a10msingle-modeﬁbertotheimagingsetup.Sincethepropagationin theﬁberaltersthepolarizationofthesignalphotons,aQWPandHWP are used for polarization correction. The orientations of these waveplatesarecalibratedbyusingacorrelator(UQD-Logic-16)toobtainthe maximumcoincidencebetweenthesignalarm(SPCM1)andheralding arm(SPCM2intheﬂipmirrorpath).Alternatively,wecanreplacetheinput of the signal ﬁber with the light from 810nm laser (OBIS LX 808nm 150mW Laser), polarized with a V-polarizer as our classical source. The SLM (Holoeye Pluto 2.1), only responds to V-polarized pho- tons, generates the input state and selectively excites the metalens. To control the amplitude from the phase-SLM, some light is divertedand blocked by an iris positioned at the focal plane of a lens with f 1=40cm. This lens and a 10 ×objective are used to image the SLM planetothemetasurfaceplane.TheﬁrstQWP,withafastaxisoriented atthediagonal,convertsV-polarizedlighttoLCP.Uponinteractingwith the metasurface, the photons diffract into different output directions.The second QWP, with a fast axis oriented at the anti-diagonal, and an H-polarizer are used to convert the polarization back to linear polarizationthatisorthogonaltotheinputlightinthesignalarm.A20 × objective(beforethesecondQWP)isusedtomapthefocalplaneofthe metasurfacetothecamera.Theinterferencepatternscorrespondingto differentoutputdirectionsareformedatthefocalplaneofthemetalensarrayandcapturedbytheSPADcamera. For our classical result, the same camera is used in the photon counting mode with an exposure

Question 1514 multiple-choice

In quantum machine learning for group-theoretic problems, recovering the hidden subgroup generator often involves sophisticated model design and optimization techniques. Overcoming issues like barren plateaus is essential for effective training and solution discovery. Which strategy is primarily employed to prevent a quantum-inspired encoder from consistently outputting the trivial generator (s = 0) and enable discovery of non-trivial subgroup generators during training? 1) Directly solving linear congruences using classical algorithms 2) Applying post-training quantization of parameters 3) Using a fixed-point iteration method without model adjustments 4) Incorporating a pre-training phase to push outputs away from the trivial solution 5) Increasing the model depth to arbitrary levels 6) Randomizing the cost function after each training epoch 7) Eliminating the decoder from the autoencoder architecture

✓ Correct Answer:

The correct answer is 4) Incorporating a pre-training phase to push outputs away from the trivial solution.

📚 Reference Text:

have to optimize theparameters to increase the probability of ﬁnding the correct hidden sub- group of f. This optimization is carried out by passing the results to the decoder, calculating the average cost function, and then using gradientdescent. In fact, the parameter ~θcan be used to describe the group structure. We then encode ~θ,t h eg e n e r a t o ro f H, and the data points ( c,f(c)) for c∈G/ Hinto a classical message σ classical ; this classical message is regarded as the compressed data. Pre-training of the encoder . In practical training, we discovered that ﬁnding the generator of the hidden subgroup by directly applying Euclid ’s algorithm to solve linear congruences (e.g. Eq. ( 33)) is inef- fective. Instead, we employ a pre-training method to make the auto-encoder functional. During interm ediate training steps, the correct Fourier transformation remains und iscovered, and thus many incor- rect bit strings are sampled (all bit strings are possible in general). This results in the congruence equations having only the trivial solution, i.e., all bits being zero. Consequently, the classical post-processing con- tinuously outputs the wrong generator (e.g. the secret key in Simon ’s algorithm) of the hidden subgroup as s= 0. With this incorrect gen- erator, the cost function remains lar ge, and gradient descent often fails to improve the situation, leading to a barren plateau phenomenon withthe output stuck at s=0 . To circumvent such barren plateau s, we introduce a pre-training technique to assist the encoder in ide ntifying a non-trivial generator s, although this sis not guaranteed to be the correct hidden subgroup ’sg e n - erator. The autoencoder training is thus split into two stages. First, the encoder is pre-trained to output a non-trivial s(i.e.,s≠0). In the second step, the hidden subgroup ’s generator sis sent to the decoder to reconstruct the data sequence and be evaluated by the overall cost function shown in Eq.(17). The autoencoder is then trained as a whole to obtain the correct generator s. During each iteration in the ove rall training, we apply the pre- training technique described belo w to ensure the encoder outputs a non- trivial generator s≠0. To help the encoder output a non-trivial generator s,w ei n c o r p o r a t ea new parameter set S!, where each S itakes values in the interval [0, 1]. The set will be used to calculate the continuous bit string s¼sm1sm2...smq,w h e r e m1+m2+…mq=n, as follows: smi¼Smiþmi/C01þ/C1/C1/C1þ mq×2mi/C01 þSmi/C01þmi/C01þ_þmq×2mi/C02þ... þS1þmi/C01þ/C1/C1/C1þ mq×20ð32Þ By Eq. ( 32) we will get each continuous smi, which depends on the corre- sponding Z2migroups. To make sure sis valid, we choose the nearest integer for each smiinsbefore feeding it to the decoder. Now we train both ~θand ~Stoﬁnd a non-trivial generator s.T a k e Simons ’problem as an example. Here m1=m2=⋯=mq=1 ,a n d q=n,s o we will write smiassiin Simon ’s problem for simplicity (see Supplementary Note C.2 for a detailed introduction

Question 1515 multiple-choice

In quantum chemistry and molecular simulation, the choice and validation of density functional theory (DFT) functionals are crucial for accurate force field parameterization. Statistical measures such as average error and standard deviation are used to assess the reliability of computational methods for modeling molecular energies and vibrational frequencies. Which statement best explains why relying solely on average error can be misleading when evaluating the accuracy of a DFT functional for force field parameterization? 1) Average error provides information only about the computational speed of the functional. 2) Average error always reflects the true accuracy, regardless of variability in results. 3) Average error alone can mask large individual overestimations and underestimations that cancel each other out, giving a false impression of reliability. 4) Average error is sufficient because standard deviation does not relate to consistency. 5) Average error is only relevant for universal force fields, not molecule-specific ones. 6) Average error does not require consideration if the functional includes dispersion corrections. 7) Average error can be replaced by evaluating only high-frequency vibrational modes.

✓ Correct Answer:

The correct answer is 3) Average error alone can mask large individual overestimations and underestimations that cancel each other out, giving a false impression of reliability..

📚 Reference Text:

given by PBE0-D and B3LYP-D functionals, but their behavi or is reversed, because they respectively slightly overestimate (+0.11 kJ /mol,) and underestimate (- 0.11 kJ/mol) the reference energies. Conversely, the CAM-B3 LYP-D description again severely overestimates the energies with respect to the CCS D(T) values. Similar results were obtained for the trichloro acetonitrile ( N1), where again both PBE0-D and B3LYP- D give the best results. It is worth recalling that both the av erage error and the standard deviation should be considered in the evaluation of each ben chmarked functional. This becomes apparent, for instance, when handling the B97D-D fu nctional: despite for N1it shows the best /angbracketleftδ/angbracketright, it is evident from both Figure 3 and the computed σDFTthat the small /angbracketleftδ/angbracketrightvalue comes from the compensation of severely underestimat ed and overestimated energies rather than from a consistent agreement. In conclu sion, considering the more consistent results achieved on the ﬁrst tested molecules, t he B3LYP-D functional was 17 adopted in all Pickyparameterizations presented in this work. 3.3 QMD-FF results for halogenated hydrocarbons Another aim of the present work is to further validate the prot ocol for QMD-FF param- eterization here adopted and sketched in Figure 2. It is wort h recalling that most of the QMD-FF approaches tend to favor accuracy over transferabil ity. In other words, rather than attempting to ﬁnd ”universal” parameters or at least pa rameters shared by a very broad class of similar molecules ( e.g.small organic liquids, bio-molecules, polymers, etc.), QMD-FFs can be exploited to account in some detail for chemic al diversity, speciﬁcally tailoring FF parameters on each target molecule. For this re ason, rather than evaluating the same set of parameters on a very broad class of benchmark m olecules, one can focus on a relatively small number of molecules, here the one displ ayed in Figure 1, validating the procedure that leads to a speciﬁc parameter set for each i ndividual molecule. 3.3.1 Intramolecular Parameterization As shown in the scheme displayed in Figure 2, the proposed QMD- FF parameterization starts for each molecule by building an intra-molecular FF t hrough the Joyceproce- dure. The Joycestandard protocol (see References [22,33] and Supporting Information for additional details), was applied to all the molecules in the benchmark set displayed in Figure 1. An example of the achieved accuracy is shown for 2- iodo-propane ( I2) in Figure 4. The comparison between QM and QMD-FF computed vibr ational frequencies, displayed in the central panel, highlights the accuracy of t he intramolecular FF descrip- tion achieved for each normal mode. Nonetheless, it might be i nteresting to notice that once again33,82the best agreement with QM frequencies is achieved for high f requency (localized) and low frequency (aﬀected by the torsional pro ﬁles) modes, while a some- what smaller accuracy is obtained for intermediate frequen cies, where coupling eﬀects can play some role. However, by looking at Figure 4 top panel, t he projection coeﬃ- cients of each QMD-FF normal mode over its QM counterpart app

Question 1516 multiple-choice

Quantum computing leverages group theory to design algorithms and quantum circuits that manipulate quantum information efficiently. The quantum representation of finite groups enables these symmetries to be expressed as unitary operations within a quantum system. Which statement accurately characterizes the guarantee provided by the quantum representation of finite groups (QRFG)? 1) It allows only abelian groups to be mapped to quantum circuits using unitary matrices. 2) It ensures that every finite group can be represented as unitary operations in a quantum system. 3) It restricts group representations to classical reversible logic circuits. 4) It requires explicit enumeration of all group elements for quantum implementation. 5) It is incompatible with variational quantum algorithms for circuit construction. 6) It applies exclusively to infinite groups with continuous symmetry. 7) It mandates that numerical simulations must use proprietary quantum hardware.

✓ Correct Answer:

The correct answer is 2) It ensures that every finite group can be represented as unitary operations in a quantum system..

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Title: Quantum representation of finite groups Year: 2022 Paper ID: d1f1a028ee1ae33d2414c6ac77a4efdf10609c0f Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d1f1a028ee1ae33d2414c6ac77a4efdf10609c0f Abstract: The concept of quantum representation of finite groups (QRFG) has been a fundamental aspect of quantum computing for quite some time, playing a role in every corner, from elementary quantum logic gates to the famous Shor's and Grover's algorithms. In this article, we provide a formal definition of this concept using both group theory and differential geometry. Our work proves the existence of a quantum representation for any finite group and outlines two methods for translating each generator of the group into a quantum circuit, utilizing gate decomposition of unitary matrices and variational quantum algorithms. Additionally, we provide numerical simulations of an explicit example on an open-access platform. Finally, we demonstrate the usefulness and potential of QRFG by showing its role in the implementation of some quantum algorithms and quantum finite automata.

Question 1517 multiple-choice

Quantum computing devices can be built from two-level systems (qubits) or generalized to d-level systems (qudits), with photonic integration enabling scalable architectures. Qudit-based processing units expand computational resources by encoding more information per quantum unit. Which feature distinguishes a large-scale silicon-photonic qudit quantum processing unit from traditional qubit-based devices in terms of algorithm implementation and hardware flexibility? 1) It relies exclusively on trapped ions for physical realization. 2) It can only execute binary logic gates due to its structure. 3) It requires physical replacement of chip components for each new algorithm. 4) It limits measurement to single-photon detection without post-processing. 5) It supports only two-level quantum states for initialization and manipulation. 6) It enables reprogramming of both software logic circuits and hardware photonic configurations to run multiple multi-valued quantum algorithms without altering the physical chip. 7) It restricts feed-forward operations, preventing advanced quantum algorithm execution.

✓ Correct Answer:

The correct answer is 6) It enables reprogramming of both software logic circuits and hardware photonic configurations to run multiple multi-valued quantum algorithms without altering the physical chip..

📚 Reference Text:

the major enabling capability of quantum computing tech- nologies. For example, several mil estones in qubit-based quantum computing have been all realised i n programmable quantum devices of photons49,50,t r a p p e di o n s51,52, superconductors5,6and semiconductors53. However, limited to the best to our knowledge, such a qudit-based quantum computing device that can be fullyrecon ﬁgured and reprogrammed to implement different tasks has not been realised to date, in any quantum system. Likely, it requires an integrated platform35,54,55, capable of initialising, manipulating and measuring qudit states and gates, in a fully controllable and highlyprogrammable manner. Realising a programmable qudit-based quantum processor therefore presents a signi ﬁcant step to transi- tion the technological advances of controlling qudit states and logic gates to the implementations of quantum tasks and quantum com- putational algorithms, in d-ary. In this work, we demonstrate a programmable qudit-based quantum processing unit ( d-QPU) on a large-scale silicon-pho- tonic quantum chip. The initialisation, manipulation and mea- surement of arbitrary single-qudit and two-qudit states, and multi-value quantum-controlled logic gates can be implemented on the single d-QPU chip. Such a fully monolithic integration of all necessary functionalities allows the implementation of a top- down hierarchy of programmable qudit-based quantum compu- tation, as shown in Fig. 1. Different quantum tasks and compu- tational algorithms are implemented, all in quaternary, by recompiling the qudit logic circuit in the software level, and then executing the circuit by reprograming the con ﬁgurations of the d- QPU chip in the hardware level. We then benchmark the enhancement of quantum computational parallelism, by per- Fig. 1 The top-down hierarchy of qudit-based quantum computation. Users can de ﬁne different quantum tasks and implement different quantum algorithms in d-ary, e.g., the generalised Deutsch-Jozsa (DJ), Bernstein- Vazirani (BV), quantum phase estimation (QPEA) and Shor ’s fast factorization algorithms. In the software level, a multi-qudit quantum logical circuit for executing the algorithm is compiled with single-qudit gates (e.g, Zd,Xdand Hd) and multi-value controlled-unitary (MVCU d) gates. In the hardware level, the logical circuit is physically implemented by an integrated photonic quantum device, i.e, the programmable d-QPU, and the gate operations are realised by optical waveguide devices, such as entangled photon sources, phase shifters, beamsplitters and interferometers. Multiple quantum tasks and algorithms can be executed, without the need of altering the device, only by reprogramming the conﬁgurations of waveguide circuits. The outcome of the hardware is given by photon coincidence counts, which are recorded and analysed by classical electronics and classical computer. Experimental outcomes can be feed- forwarded into the d-QPU for the implementation of Kitaev ’s version of quantum Fourier transform algorithms.ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-28767-x 2 NATURE COMMUNICATIONS | (2022) 13:1166 | https://doi.org/10.1038/s41467-022-28767-x | www.nature.com/naturecommunications forming the generalised Deutsch-Jozsa and Bernstein-Vazirani algorithms, quaternary phase estimation and order ﬁnding algo- rithms. Our results show a proof-of-principle demonstration of qudit-based quantum computer with integrated optics, thatallows improvement of the capacity, accuracy and ef ﬁciency of quantum computing. Results Scheme of multiqudit quantum processor .F i g u r

Question 1518 multiple-choice

In quantum chemistry, multi-electron systems are often modeled using equations that account for both spatial and spin variables, with independent particle interpretations simplifying the complex interactions between electrons. Spin coupling schemes play a crucial role in determining the energy and properties of such systems. Which statement best describes the role of the parameter L in the Spin-Orbital Generalized Independent particle (SOGI) equations for multi-electron systems? 1) L determines the spatial symmetry of each orbital, independent of electron spin interactions. 2) L sets the strength of electron-nucleus attraction for all electrons in the system. 3) L specifies the dimensionality of the Hamiltonian operator applied to the wavefunction. 4) L dictates the scheme by which electron spins are coupled, affecting energy levels and computational optimization. 5) L controls the normalization condition for the total electronic wavefunction. 6) L defines the number of electrons included in the system’s configuration interaction. 7) L governs the convergence rate of the nonlinear algebraic equations in electronic structure calculations.

✓ Correct Answer:

The correct answer is 4) L dictates the scheme by which electron spins are coupled, affecting energy levels and computational optimization..

📚 Reference Text:

the integrals (if> I H I 0;/'if>) and (if> I 0;/if>) and these occur as coefficients; we will refer to these equations as the spin SOGI equations, since L determines the spin coupling scheme. When making variations in if>, we can take the sums over i andj of Eq. (17a) back into the integral to give I= ((if> I H I Ou~'Lif> )/(if> I Ou~'Lif>)) + L fk(cf>k I c/>k)+A. L Ltn2• ( 17b) k n From this expression we obtain the following set of equations, using exactly the arguments of Sec. I of Paper II: HL(k)cf>k=fkkc/>k k=1, 2, •• ·, N, (18) where JlL(k)cf>k={Dkkhcpk+ L {vI k)Dikhhcf>;+ L {s I hI k)Du/·'cf>,+ L {s I hI t)(v I k)Dtkusvkcf>,. j,#k s,ur'ok uvstr'ok + L {sf I g I kj)Dkju81kcf>,.+t L' {st I g J ij)(v I k)Dijku•tvkcf>,.-E L {v J k)Dkuvkcf>u}. (19) jstu:~k iistuv~k u,~k Here we have redefined D/ to be the coefficient of cf>;*(i)cf>i(i) in f (dx;'), D;1ik to be the coefficient of c/>;*(i)cf>k *(k)c/>;(i)cf>z('k) in f (dx;k"), etc. This redefinition of the D matrices is the only difference between the spatial SOGI equations and the GI equations. If we define :JC;i=-(if>! HI 0;/'if>) and :Jl;j= (if> I 0;/'if>), the spin SOGI equations are CL: Lt;(:JC;p+:JCp;) ](L LtkLu:Jlkz)-(L Lt;Llj:JC;j) [L Ltk(:npk+:nkp) ]-2A.LP(L LliLtj:Jl;j)2=0 p= 1, • oojl', 1. kl ii k ~i (20) Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp1076 R. C. LADNER AND W. A. GODDARD III -7.432 ·-7.434 - -7.436 -7.438 ,; 2-7.440 • >-"' 8J -7442 z w -7.444 -7.446 -7448 -7 450';-;,----!;;------!;;,---';--~;c;--~--!;---!c-~-±---::':-----c~--,J -30 -25 -20 -:5 -10 -5 0 10 i5 2C 25 30 2/2 (OJ FIG. 2. Energy as a function on representation parameter, 2, for the 22S state of Li. These equations are simple nonlinear algebraic equa tions. Since they involve X;; and ';YL;i for all i and j, and since the evaluation of all of these quantities fre quently requires a great deal of effort, it will often be expedient to find the optimum L by actually calculat ing the energy for various values of the parameters of L until a minimum is found. B. The Independent Particle Interpretation and Other Properties of SOGI Wavefunctions Each HL(k) operator in (18) is equivalent to the Hamiltonian of an electron moving in the (nonlocal) field due to electrons in the N -1 other orbitals. Since c/Jk is an eigenfunction of this operator, we can interpret each ¢k as the eigenstate of an electron moving in the field due to N -1 other electrons. That is, the SOGI orbitals can be given an independent particle inter pretation (IPI) just as the GI orbitals were.2 In the independent particle interpretation we inter pret each spatial orbital of theN-electron wavefunction as the eigenstate of an electron moving in the average (self-consistent) potential due to electrons in the other N -1 orbitals. The criteria we use for such an inde pendent particle interpretation are the following: (i) There

Question 1519 multiple-choice

In representation theory of symmetric groups, induced representations and their decompositions play a central role in understanding the structure of group algebras and their spectral properties. The interplay between partitions, Young diagrams, and block matrices is crucial for analyzing these representations. When decomposing the induced representation $\mathrm{ind}_{S(n-1)}^{S(n-2)}(j_a)$, which of the following statements about the associated block matrices $Q(a)$ is true? 1) Each eigenvalue of $Q(a)$ must be distinct and nonzero. 2) All block matrices $Q(a)$ have only positive eigenvalues. 3) The number of zero eigenvalues of $Q(a)$ is always equal to the number of coset representatives. 4) Every irreducible representation of $S(n-1)$ corresponds to a unique eigenvalue of $Q(a)$. 5) Zero eigenvalues in $Q(a)$ are forbidden by the complete reducibility property. 6) At most one eigenvalue of $Q(a)$ can be zero, counted with multiplicity. 7) The multiplicity of zero eigenvalues in $Q(a)$ equals the dimension of the induced representation.

✓ Correct Answer:

The correct answer is 6) At most one eigenvalue of $Q(a)$ can be zero, counted with multiplicity..

📚 Reference Text:

brieﬂy some object appearing in the structure of the algebra A0 n(d), in particular the properties of the induced representation indS(n1) S(n2)(ja). The irreducible representations of the group S(n2)are characterized by the partitions a= (a1, ...,ak)ofn2, which describe also the corresponding Young diagram Y(a). The representation indS(n1) S(n2)(ja)is completely and simply reducible i.e. we have [31]. Proposition 6. indS(n1) S(n2)(ja) =M nyn, (20) where the sum is over all partitions n= (n1, ...,nk)of n1,such that their Young diagrams Y (n)are obtained from Y (a)by adding, in a proper way, one box. Deﬁnition 7. [32] Let j:H!M(n,C)be a matrix representation of a subgroup H of the group G .Then the matrix form of the induced representation p=indG H(j)of a group G induced by an irrep. jof the subgroup H G has the following block matrix form 8g2Gpbj ai(g) = (bjij(g1 aggb)), where g a,a=1, ...,[G:H]are representatives of the left cosets G /H and bjij(g1 aggb) =jij(g1 aggb)if g1 aggb2H, 0 if g1 aggb/2H. 10 Before main considerations for the appendix let us introduce some notation. Notation 8. Any permutation s2S(n)deﬁnes, in a natural and unique way, two natural numbers a ,b2f1, 2, ..., ng n=s(a), b=s(n) Thus we may characterize any permutation by these two numbers in the following way ss(a,b)sab. Note that in general a ,b may be different except the case, when one of them is equal to n ,because in this case we have a=n,b=n. When a =n=b,thens(n) =n and we will use abbreviation s=s(n,n)sn2S(n1)S(n). From Proposition 6 and Deﬁnition 7 it follows that the induced representation indS(n1) S(n2)(ja)may be described in two bases. The ﬁrst one, is the basis of the matrix form of the induced representation of the form fea i(a):a=1, ..., n1, i=1, ..., dim jag, (21) where the index a=1, ..., n1 describes the the cosets S(n1)/S(n2)and the the index i=1, ..., dim jais the index of a matrix form of ja. The second one is a basis of the reduced form of indS(n1) S(n2)(ja), which is of the form n fn jn:yn2indS(n1) S(n2)(ja),jn=1, ..., dim yno . (22) The next important objects are the following matrices Deﬁnition 9. For any irreducible representation jaof the group S (n2)we deﬁne the block matrix Qd n1(a)Q(a) = ( ddabja ij[(an1)(ab)(bn1)]) = ( Qab ij(a)), (23) where a ,b=1, ..., n1, i,j=1, ..., dim jaand the blocs of the matrix Q (a)are labeled by indices (a,b)whereas the elements of the blocks are labeled by the indices of the irreducible representation ja= (ja ij)of the group S (n2)and Q(a)2M((n1)wa,C). The matrices Q(a)are hermitian and their structure and properties are described in the [21], where it has been shown, that the eigenvalues lnof the matrix Q(a)are labeled by the irreducible representations yn2indS(n1) S(n2)(ja) and the multiplicity of lnis equal to dim yn. The essential for properties of the irreducible representations of the algebraA0 n(d)is the fact, that at most one (up to the multiplicity) eigenvalue lnof the matrix Q(a)may be equal to zero [20, 21]. The structure of the

Question 1520 multiple-choice

Quantum computing in the noisy intermediate-scale quantum (NISQ) era faces significant challenges due to limited error correction and hardware imperfections. Hybrid approaches such as digital-analog quantum computing (DAQC) are being explored to improve the efficiency and fidelity of key quantum algorithms. Which statement best explains how digital-analog quantum computing (DAQC) enhances the implementation of the Quantum Fourier Transform (QFT) on NISQ devices? 1) It replaces all digital gates with analog simulation to eliminate noise entirely. 2) It combines analog blocks for robust simulation of interactions with digital gates for universality, reducing resource requirements and improving fidelity as qubit count increases. 3) It relies solely on quantum error correction protocols to address noise issues in QFT circuits. 4) It avoids using any post-processing methods to mitigate errors during computation. 5) It increases the number of discrete gate operations to improve programmability, regardless of noise sensitivity. 6) It employs classical computation to simulate quantum dynamics rather than using quantum hardware. 7) It uses only digital gate operations for full control, sacrificing robustness to hardware imperfections.

✓ Correct Answer:

The correct answer is 2) It combines analog blocks for robust simulation of interactions with digital gates for universality, reducing resource requirements and improving fidelity as qubit count increases..

📚 Reference Text:

PHYSICAL REVIEW RESEARCH 2, 013012 (2020) Digital-analog quantum algorithm for the quantum Fourier transform Ana Martin,1,*Lucas Lamata,1,2Enrique Solano,1,3,4and Mikel Sanz1,† 1Department of Physical Chemistry, University of the Basque Country, Apartado 644, 48080 Bilbao, Spain 2Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, 41080 Sevilla, Spain 3IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain 4International Center of Quantum Artiﬁcial Intelligence for Science and Technology (QuArtist) and Department of Physics, Shanghai University, 200444 Shanghai, China (Received 21 June 2019; published 6 January 2020) Quantum computers will allow calculations beyond existing classical computers. However, current technology is still too noisy and imperfect to construct a universal digital quantum computer with quantum error correction.Inspired by the evolution of classical computation, an alternative paradigm merging the ﬂexibility of digitalquantum computation with the robustness of analog quantum simulation has emerged. This universal paradigmis known as digital-analog quantum computing. Here, we introduce an efﬁcient digital-analog quantumalgorithm to compute the quantum Fourier transform, a subroutine widely employed in several relevant quantumalgorithms. We show that, under reasonable assumptions about noise models, the ﬁdelity of the quantum Fouriertransformation improves considerably using this approach when the number of qubits involved grows. Thissuggests that, in the noisy intermediate-scale quantum era, hybrid protocols combining digital and analogquantum computing could be a sensible approach to reach useful quantum supremacy. DOI: 10.1103/PhysRevResearch.2.013012 I. INTRODUCTION Almost four decades ago, a new paradigm, based on laws of quantum mechanics, had been put forward by Manin [ 1] and Feynman [ 2]. The new paradigm employed quantum features to speed up calculations, and it was called quan-tum simulation or quantum computation (QC). There existseveral computational tasks for which QC offers exponentialspeedups over their classical counterparts [ 3,4]. If we had a fully functional error corrected quantum computer, we wouldbe able to solve problems that not even the largest classicalsupercomputers can. But, nowadays, we are far from thispoint. The ﬁrst series of commercial digital quantum proces-sors based on superconducting circuits have been introducedby companies, such as IBM, Rigetti, Google, and Alibaba.These devices belong to the so-called noisy intermediate-scale quantum (NISQ) era in which their performance stillfaces multiple technical constraints. These constraints pose agreat challenge when one tries to solve real-world problems,limiting its size to the small scale [ 5,6]. Quantum error mitigation (QEM) techniques have been proposed as a possible method to bypass the NISQ-era hardware limitations and improve the calculation ofmean values of observables in problems comprising short- *Corresponding author: ana.martinf@ehu.eus †Corresponding author: mikel.sanz@ehu.es Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.depth circuits [ 7]. These techniques, in general, postpro- cess the information in order to mitigate the effects of thenoise [ 8–11]. Parra-Rodriguez et al. introduced, in Ref. [ 12], an alternative hybrid quantum computation approach whichcould reduce the limitations of NISQ devices. This uni-versal paradigm, called digital-analog quantum computation (DAQC),

Question 1521 multiple-choice

In group theory, finite p-groups with prescribed relations among their generators and commutator subgroups can be classified using techniques from both algebra and linear algebra. For 2-groups, matrix invariants over the field F₂ play a crucial role in distinguishing isomorphism types. Which condition guarantees that two finite 2-groups with the same commutator subgroup and quotient structure are isomorphic? 1) There exists an invertible matrix over F₂ relating their characteristic matrices. 2) The orders of their generators are equal. 3) Both groups have identical Frattini subgroups. 4) Their lower central series have the same length. 5) Their group presentations share the same number of generators. 6) The groups have the same order. 7) Their commutator subgroups are trivial.

✓ Correct Answer:

The correct answer is 1) There exists an invertible matrix over F₂ relating their characteristic matrices..

📚 Reference Text:

wemaylet ¯bθ=ax212n−1bcx23. Bycalculations, (¯bpm)θ= (¯b2)θ= (ax212n−1bcx23)2=ax212nb2c2x23[c,b]x23=ax212nb2. Hencewe also have Equation 3.2. It follows that Equation 3.3 and 3.6 hold. Therefore we have the following theorem. 5 Theorem 3.4. Suppose that Gand¯Gare ﬁnite 2-groups such that G3≤Φ(G′)∼=C2 andG/Φ(G′)∼=M2(n,m,1), wheren≥3. Let two characteristic matrices of Gand¯G bew(G) = (wij)andw(¯G) = (¯wij)respectively. Then G∼=¯Gif and only if there exists X=/parenleftbiggx11x12 x212n−mx22/parenrightbigg , an invertible matrix over F2, such that /parenleftbigg¯w11 ¯w21/parenrightbigg =/parenleftbiggx11x122n−m x21x22/parenrightbigg/parenleftbiggw11 w21/parenrightbigg and /parenleftbigg¯w12 ¯w22/parenrightbigg =X/parenleftbiggw12 w22/parenrightbigg . In addition, if m= 1, thenw22= 1. Theorem 3.5. LetGbe a ﬁnite p-group such that G3≤Φ(G′)∼=CpandG/Φ(G′)∼= Mp(n,m,1)wheren >1forp= 2andn≥m. Then Gis one of the following non-isomorphic groups: (A1)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea4=b2=c4= 1,[a,b] =c,[c,a] = [c,b] =c2/an}bracketri}ht; (A2)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea4=b4= 1,c2=a2,[a,b] =c,[c,a] = [c,b] =c2/an}bracketri}ht; (A3)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8=b2= 1,c2=a4,[a,b] =c,[c,a] = [c,b] =c2/an}bracketri}ht; (B1)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea2n+1=b2= 1,c2=a2n,[a,b] =c,[c,a] = 1,[c,b] =c2/an}bracketri}ht, wheren≥3; (B2)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea2n=b2=c4= 1,[a,b] =c,[c,a] = 1,[c,b] =c2/an}bracketri}ht, wheren≥3; (B3)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea2n=b4= 1,c2=b2,[a,b] =c,[c,a] = 1,[c,b] =c2/an}bracketri}ht, wheren≥3; (C1)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8= 1,c2=a4=b4,[a,b] =c,[c,a] = 1,[c,b] = 1/an}bracketri}ht; (C2)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8=b4= 1,c2=a4,[a,b] =c,[c,a] = [c,b] = 1/an}bracketri}ht; (C3)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8= 1,c2=a4=b4,[a,b] =c,[c,a] = 1,[c,b] =c2/an}bracketri}ht; (C4)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8=b4= 1,c2=a4,[a,b] =c,[c,a] = 1,[c,b] =c2/an}bracketri}ht; (C5)/an}bracketle{ta,b,c/vextendsingle/vextendsinglea8=b4= 1,c2=a4,[a,b] =c,[c,a] =c2,[c,b] = 1/an}bracketri}ht; (D1)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpn= 1,cp=apn,[a,b] =c,[c,a] = 1,[c,b] =ctp/an}bracketri}ht, where n≥3forp= 2,n≥2andt∈F∗ p; (D2)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpn= 1,cp=apn,[a,b] =c,[c,a] =cp,[c,b] = 1/an}bracketri}ht, wheren≥3 forp= 2andn≥2; (D3)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpn= 1,cp=apn,[a,b] =c,[c,a] = 1,[c,b] = 1/an}bracketri}ht, wheren≥3 forp= 2andn≥2; (D4)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn=bpn=cp2= 1,[a,b] =c,[c,a] =cp,[c,b] = 1/an}bracketri}ht, wheren≥3 forp= 2andn≥2; (D5)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn=bpn=cp2= 1,[a,b] =c,[c,a] = 1,[c,b] = 1/an}bracketri}ht, wheren≥3 forp= 2andn≥2; (E1)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpm= 1,cp=apn,[a,b] =c,[c,a] = 1,[c,b] =ctp/an}bracketri}ht, where n > m≥2andt∈F∗ p; (E2)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpm= 1,cp=apn,[a,b] =c,[c,a] =cp,[c,b] = 1/an}bracketri}ht, where n > m≥2; 6 (E3)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn+1=bpm= 1,cp=apn,[a,b] =c,[c,a] = 1,[c,b] = 1/an}bracketri}ht, where n > m≥2; (E4)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn=bpm+1= 1,cp=bpm,[a,b] =c,[c,a] = 1,[c,b] =cp/an}bracketri}ht, where n > m≥2; (E5)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn=bpm+1= 1,cp=bpm,[a,b] =c,[c,a] =ctp,[c,b] = 1/an}bracketri}ht, where n > m≥2andt∈F∗ p; (E6)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn=bpm+1= 1,cp=bpm,[a,b] =c,[c,a] = 1,[c,b] = 1/an}bracketri}ht, where n > m≥2; (E7)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn=bpm=cp2= 1,[a,b] =c,[c,a] = 1,[c,b] =cp/an}bracketri}ht, wheren > m≥2; (E8)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn=bpm=cp2= 1,[a,b] =c,[c,a] =cp,[c,b] = 1/an}bracketri}ht, wheren > m≥2; (E9)/an}bracketle{ta,b,c/vextendsingle/vextendsingleapn=bpm=cp2= 1,[a,b] =c,[c,a] = 1,[c,b] = 1/an}bracketri}ht, wheren > m≥2. Proof Case 1: m= 1. Ifp >2, thencp= [a,bp] = 1, a contradiction. Hence p= 2. It follows that n≥2. Ifn= 2, then |G|= 25. By checking the list of groups of order 25, we get the groups of Type (A1)–(A3). In the following, we may assume that n≥3. Suppose that Gand¯G are two groups described in the theorem. By Theorem 3.4, ¯ w22=w22= 1, and ¯G∼=G if and only if there exists X=/parenleftbigg1x12 x212n−11/parenrightbigg , an invertible matrix over F2, such that /parenleftbigg¯w11 ¯w21/parenrightbigg =/parenleftbigg1 0 x211/parenrightbigg/parenleftbiggw11 w21/parenrightbigg and /parenleftbigg¯w12 1/parenrightbigg =/parenleftbigg1x12 0 1/parenrightbigg/parenleftbiggw12 1/parenrightbigg . Letx12=w12. Then we have ¯ w12= 0. Ifw11= 0, then ¯ w11= 0 and ¯ w21=w21. If w11= 1, then, letting x21=w21, we have ¯ w21= 0. To sum up, characteristic matrices of non-isomorphic groups are: (b1)/parenleftbigg1 0 0 1/parenrightbigg (b2)/parenleftbigg0 0 0 1/parenrightbigg (b3)/parenleftbigg0 0 1 1/parenrightbigg . Hence we get the groups of Type (B1)–(B3). Case 2: p=n=m= 2. In this case, |G|= 26. By checking the list of groups of order 26, we get the groups of Type (C1)–(C5). Case 3: n≥3 forp= 2

Question 1522 multiple-choice

In group theory, the study of abelian and nilpotent p-groups involves examining extensions, commutator identities, and classification results, especially regarding infinite groups and countability. Techniques such as induction on Ulm length and commutator calculus are often employed to analyze structural properties. Which of the following is a critical reason why certain results about abelian p-groups do not generalize from countable to uncountable groups? 1) The lower central series does not exist for uncountable groups. 2) Split extensions are impossible for uncountable groups. 3) Ulm’s theorem fails for uncountable groups, limiting classification techniques. 4) Restricted direct products cannot be formed with uncountable groups. 5) Commutator identities do not hold in uncountable groups. 6) Divisibility is never present in uncountable p-groups. 7) Cyclic subgroups cannot exist in uncountable groups.

✓ Correct Answer:

The correct answer is 3) Ulm’s theorem fails for uncountable groups, limiting classification techniques..

📚 Reference Text:

I. A an abelian 2-group, B a nilpotent 2-group (Theorem 3.6); II. p odd, A an abelian/7-group of finite exponent and B a nilpotent /?-group (Theorems 3.1 and 3.2); III. p odd, A an abelian ^-group of infinite exponent and B a divisible (abelian)/7-group (Theorem 3.5); IV. p odd, A a countable abelian /?-group of infinite exponent and B a non-divisible nilpotent ^-group (Theorems 5.3 and 5.4). The major portion of the paper is needed for cases III and IV. The countability condition in IV arises in the proof of Theorem 5.2, concerning the splitting off from a reduced /?-group of "large" factors of a basic subgroup, the underlying reason being that Ulm's theorem is false for uncountable groups in general. Our methods are a blend of abelian group theory and the commutator calculus of metabelian groups. By far the longest proof is that of Theorem 4.2 (the special case where A is a restricted direct product of cyclic /^-groups and B is cyclic of order p). There, it is not the use of deep theorems that is needed but rather, detailed calculations using certain commutator identities repeatedly, together with induction arguments in two directions. The extension of Theorem 4.2 to general abelian groups in Section 5 is usually a relatively routine induction on the Ulm length, but not always (see for instance Step 2 in the proof of Theorem 5.3). Throughout, we have resisted the temptation to consider best possible bounds for the nilpotency class arising; we think that this would complicate matters without good enough cause. 2. Notation and preliminaries. We shall write all groups multiplicatively, even when the concern is exclusively with abelian groups. The cyclic group of order n is denoted by C IV and the quasicyclic/?-groups by Coo. For any subset I of a group, (X) means the subgroup generated by X\ for a p-group G, ®>m(G) means (x\x e G and x pm =1). The lower central series of a group G is G = Yl(G) ^ y 2(G) = G' S . . . S y„(G) i= . . . , as usual. The split extension of A by B is written A x3 B. We need the following well-known commutator identity in metabelian groups. In it, [x, ny] denotes the iterated commutator [JC, y,y,...,y], with n occurrences of y. (2.1) For any elements x, y of a metabelian group G and any positive integer n, https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press NILPOTENT EXTENSIONS 1027 [x,/1] = [x,y]^\x, 2y]^> . . . [JC, (n - l)yV»- ]'[x,ny]. Also easy is (2.2) Let G be a group generated by an abelian normal subgroup A and an element b. Then (i) G is nilpotent if and only if there exists n such that that [a, nb] = 1 for all a in A ; (u)for every integer r, [a r, b] = [a, b] r. The first of these results follows from the fact that for / ^ 2, y, + 1(G) is generated

Question 1523 multiple-choice

In the study of integrable models in string theory, modifications such as the η-deformation of the AdS5 × S5 background introduce new mathematical structures and alter the properties of the worldsheet S matrix. Quantum group symmetry and modern computational techniques play crucial roles in understanding the spectrum and scattering processes in these theories. Which of the following statements correctly describes a fundamental consequence of the η-deformation on two-particle asymptotic states in the worldsheet S matrix of AdS5 × S5 string theory? 1) Two-particle states remain unchanged and local at all loop orders. 2) Integrability is lost at tree level due to particle production. 3) The S matrix becomes non-factorizable into two-particle processes at tree level. 4) Redefinition of two-particle states in a non-local, η-dependent manner is required at loop level due to quantum group symmetry. 5) Quantum integrability is unaffected by the choice of asymptotic state basis. 6) Logarithmic S matrix terms are unrelated to the analytic structure of the theory. 7) Rational terms in the S matrix can be computed only at tree level.

✓ Correct Answer:

The correct answer is 4) Redefinition of two-particle states in a non-local, η-dependent manner is required at loop level due to quantum group symmetry..

📚 Reference Text:

Title: On the asymptotic states and the quantum S matrix of the η-deformed AdS5 × S5 superstring Year: 2014 Paper ID: 80666f4176d0ff1275a84a4fa438470450031ba7 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/80666f4176d0ff1275a84a4fa438470450031ba7 Abstract: A bstractWe investigate the worldsheet S matrix of string theory in η-deformed AdS5 × S5. By computing the six-point tree-level S matrix we explicitly show that there is no particle production at this level, as required by the classical integrability of the theory. At one and two loops we show that integrability requires that the classical two-particle states be redefined in a non-local and η-dependent way. This is a significant departure from the undeformed theory which is probably related to the quantum group symmetry of the worldsheet theory. We use generalized unitarity to carry out the loop calculations and identify a set of integrals that allow us to give a two-loop Feynman integral representation of the logarithmic terms of the two-loop S matrix. We finally also discuss aspects of the calculation of the two-loop rational terms.

Question 1524 multiple-choice

Quantum Fourier Transform (QFT) is a fundamental algorithm in quantum computing, enabling exponential speedup for certain problems, such as integer factorization. Its circuit utilizes specific quantum gates to process information in superposition, mapping input states to frequency representations. Which component is essential for constructing the QFT circuit as the quantum analog of the FFT’s Butterfly operation, and is responsible for creating superpositions in quantum algorithms? 1) Hadamard gate 2) Pauli-X gate 3) Toffoli gate 4) SWAP gate 5) Controlled-NOT gate 6) Measurement gate 7) Phase estimation gate

✓ Correct Answer:

The correct answer is 1) Hadamard gate.

📚 Reference Text:

operation is unitary maintains the wavefunction superposition state but transforms it to the Fourier coefficients corresponding to input wavefunction , thus: 11 10 01 0011 10 01 00 X X X X  (7) where the Xn are complex coefficients corresponding to the complex Fourier components at that qubit location in the output array. However, since they comprise the overall wavefunction they will not be directly acce ssible to measurement. Rather the probability of detection , by a single measurement , will be given by |Ψ|2. If the input wavefunction is periodic, e.g. arises from the successful application of the Shor modular exponentiation procedure, the output wavefunction will have multiple peak s in its probability distribution at the output register location s corresponding t o the period icity of the output comb function, as described in Section 2 . Thus repeated application of the QFT will yield more detection events at these location s and hence al low determination of the periodicity, 1/r, of the comb function as required to factor the initial input sequence, N. Thus the Q FT is more powerful than the FFT in that it can process 2n inputs in parallel with effectively the same complexity of hardware structure (and so is exponentially faster in computation), as described below. However, the FFT yields N complex frequency components at its output whereas the QFT produces a probability distribution only whi ch collapses to a single detection event upon m easurement. This, however, may be very useful and used to solve problems with exponential complexi ty if applied appropriately, as has been shown in its application to the factoring problem. The building block for the QFT is the Ha damard gate . This can be written:   1 111 21H (8) This is the basic operation comprising the Hadamard transform and will act on a single qubit state to give: 1 210 211 01 0 1 0 1 0 xx xx x x  H (9) Thus it can be seen that the Hadamard transform performs a 2 -point QFT by implementing the basic FFT building block of the Butterfly operation i.e. the sub traction and addition of the two input signals, albeit in a superposition state1. Thus t he FFT decomposition of the QFT can be made using the Hadamard gate as a basic building element together with controlled phase rotation gates described by matrices of the form7,13:      njn eR 22 .000. ....0 .1000 .0100 .001  (10) This allows the QFT structure for a n-qubit input to be represente d compactly as shown in Figure 2 . Figure 2 . n-qubit QFT quantum gate array structure The decomposition is thus an FFT structure acting on a binary n-qubit input to yield a complex -valued n-qubit output (in reverse bit order). However, a n implementation problem arises in the phase resolution required by the final term in the Rn matrices for large n, as this scales as 2n. Clearly, for a

Question 1525 multiple-choice

In quantum chemistry, the modeling of electron spin states and their couplings is fundamental for accurately describing molecular systems and reaction mechanisms. Computational methods have evolved to balance physical accuracy with tractability, especially in larger molecules with many electrons. Which approach allows significant simplification in spin-coupling calculations for large electron systems by assuming that core and certain bond electrons remain paired in singlet states? 1) Restricting spin-coupling optimization to only unpaired and reactive electrons 2) Assigning a parameter for each possible spin-coupling state without any assumptions 3) Treating all electrons as individually unpaired regardless of their location 4) Ignoring spin-coupling effects entirely in all calculations 5) Optimizing both spatial and spin parameters for every electron configuration 6) Pairing electrons only in the outermost shell and optimizing the rest 7) Applying spin-coupling optimization uniformly across all molecular orbitals

✓ Correct Answer:

The correct answer is 1) Restricting spin-coupling optimization to only unpaired and reactive electrons.

📚 Reference Text:

we will discuss briefly a set of calcula tions which we have not yet carried out in order to illustrate simplifications in applying SOGI which arc expected to be appropriate for larger numbers of elec trons. Consider the reaction LiH + H~Li +Hz. The system is a five-electron doublet and the appropri ate representation of Ss is five dimensional. If we were to allow a complete! y general coupling of the electron Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jspQUANTUM THEORY OF MANY-ELECTRON SYSTEMS. V 1087 spins, we would have to simultaneously optimize four parameters controlling the spin coupling as well as the five spatial orbitals; all of this for each nuclear con figuration. However, on physical grounds, and on the basis of the calculations on Li and LiH, we expect the core electrons to be very nearly singlet paired for all nuclear geometries. From this we can see that only the mixing of the first and third standard spin basis functions (Fig. 9) is important since all others corre spond to breaking the pairing of the Li core. Thus by introducing only one spin-coupling parameter, we can expect to obtain an accurate description of this reac tion. This approach becomes increasingly more impor tant as the system becomes larger. For the reaction C!L+H~CHa+H2or CH4+H~CH 3-+H2, there are 132 different spin-coupling states. Hence to allow for a general spin-coupling scheme, we would have to in troduce 131 new nonlinear parameters. If we assume, however, that the electrons in the unbroken bonds and in the core of the carbon atom remained paired, we need introduce only one new parameter to allow a general coupling among the electrons in the bonds being broken and formed. IV. CONCLUSIONS We have developed a way of eliminating the arbi trariness in the choice of spin function in the GI method. This method has been applied to three-electron atoms and three-and four-electron molecules. For three electron atoms it was found that the G 1 wavefunction is very nearly optimum and that only the spin-depend-ent properties changed significantly when the spin coupling was optimized. A very good value for the spin density at the nucleus was obtained for the ground state of Li, and for the 2 2 P state the results were much improved over the G 1 values. Similarly for LiH the G 1 wavefunction had nearly the optimum spin coupling and only a minor improvement occurred in the energy in going to SOGI. For the unstable molecu lar systems H3 and H4, however, we found that optimi zation of the spin-coupling was necessary in order to properly and consistently describe the system. Thus these calculations indicate that the SOGI method may allow a proper description of the Fermi contact portion of the hyperfine interaction of atoms and molecules. More importantly however is that the SOGI method may allow a good description of the changes which occur as bonds are formed, broken, and distorted during chemical reactions. ACKNOWLEDGMENTS

Question 1526 multiple-choice

In algebraic group theory, the interplay between subgroups, maximal tori, and normalizers is fundamental to understanding the structure of reductive groups. Rigidity results often restrict the possible configurations of subgroups within a larger group. If a connected subgroup D of an algebraic group G is normalized both by a maximal torus of G and by a subgroup M, which of the following best describes the possible structure of D under these conditions? 1) D must be a parabolic subgroup of G 2) D can be any proper nontrivial subgroup of G 3) D must be either the trivial subgroup or all of G 4) D must coincide with the unipotent radical of G 5) D must be a non-connected subgroup 6) D must be a maximal torus itself 7) D must be a simple subgroup of G

✓ Correct Answer:

The correct answer is 3) D must be either the trivial subgroup or all of G.

📚 Reference Text:

== L2(3Y or U3(2)Y for some positive integers x, y. Moreover, if [C"' E] = 1 then ICalp = JCc(e) 0JP, and hence as above, [C, E] = 1; consequently [K, E] = 1, which is false. So [C"' E] * 1. Since M normalizes Ca and E, we have 1 *[Ca, E]:,,;:; C0 n E, and hence E.;;; Ca as M normalizes no proper subgroup of E. Thus (ii) or (iii) holds. LEMMA 2.2. Suppose that M (a) normalizes a connected subgroup D of G, such that D is normalized by a maximal torus of G. Then D is 1 or G. Proof. Suppose that D is not 1 or G, and take D maximal with respect to the hypotheses of the lemma. Let T be a maximal torus in Na(D). If Q = Ru(D) * 1 then M (a).;;; No(Q):,,;:; P, where P is the canonical parabolic subgroup of G determined by Q as in [5]. But then M normalizes the parabolic Pc,, so M must be parabolic, a contradiction. 26 A. M. COHEN, M. W. LIEBECK, J. SAXL AND G. M. SEITZ Thus D is reductive, so D = D'Z with D' semisimple and Z = Z(D)0• The maximality of D implies that D = (DC0(D))0• We claim that DT :s:; DCo(D): for DT is reductive and so corresponds to a subsystem of roots relative to T, from which the claim is clear. Thus T :s:; D, and hence Z = Cr(D)0• Since M normalizes Z, our hypothesis on M forces Zu = 1. Moreover, we may take T to be a-stable, by [26, I, 2.9]. Suppose that D' =I= 1. Since T :s:; D, components A 1 and A2 of the root system of D' (relative to T) are generated by root subgroups of G, so cannot have a-fixed point groups of the form L2(3) or U3(2). Thus by Lemma 2.1 we have [D', E] = 1. Note also that Z =I= l, since otherwise T :s:; D = D', giving E :s:; C0(D') 0 :s:; C0(T)0 = T0, contrary to hypothesis. Also Z(D) :s:; C(T) = T. We have now D=D'Z, D'=l=l, Z=l=l. We next claim that C0(D') = Z(D')Z. To see this, let D' = H1 .•• Hm be the expression for D' as a commuting product of its homogeneous factors H; (see the beginning of this section for the definition). By the maximality of D, for each i, C0(H;)0 is the product of those Hj with j =I= i, and also C0(Z)0 =D. Moreover, the rank of D' is less than the rank of G, as Z =I= 1. Let b be the root system of D'. The possibilities for b satisfying the above conditions are not hard to determine, using the lists of all closed subsystems of the root system of G given in [10, Tables 7-11]. Indeed, the possibilities are as follows: G G2 E. E6 E1 Es fl. none B3 (p odd) D5, D4, 4A1> 2A1 + A3 E6, A6,

Question 1527 multiple-choice

In representation theory, Fock spaces provide a framework for understanding the actions of quantum groups and Lie superalgebras. The structure and decomposition of modules, such as those for the Lie superalgebra osp(2|2n), are often analyzed using canonical bases and the behavior of symmetric tensors. Which mathematical concept is essential for explicitly determining the irreducible composition factors of symmetric tensor powers of the natural osp(2|2n)-module? 1) Canonical and dual canonical bases of an appropriately constructed Fock space 2) Weyl character formula for classical Lie algebras 3) Schur–Weyl duality for symmetric groups 4) Crystal bases for quantum groups of type B 5) Kac–Moody algebra root multiplicities 6) Superdimension formula for typical representations 7) Young tableaux classification for tensor products

✓ Correct Answer:

The correct answer is 1) Canonical and dual canonical bases of an appropriately constructed Fock space.

📚 Reference Text:

Title: A Fock space approach to representation theory of osp(2|2n) Year: 2005 Paper ID: 84e8ba36317aefe006e05c6902bbe112af726234 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/84e8ba36317aefe006e05c6902bbe112af726234 Abstract: A Fock space is introduced that admits an action of a quantum group of type A supplemented with some extra operators. The canonical and dual canonical basis of the Fock space are computed and then used to derive the finite-dimensional tilting and irreducible characters for the Lie superalgebra osp(2|2n). We also determine all the composition factors of the symmetric tensors of the natural osp(2|2n)-module.

Question 1528 multiple-choice

Kitaev’s quantum double models are central to understanding topological quantum order and its applications to quantum error correction. These models generalize the toric code to arbitrary finite groups, supporting both Abelian and non-Abelian anyonic excitations. In two-dimensional Kitaev quantum double models based on arbitrary finite groups, which statement correctly characterizes the ground state’s local properties in a contractible region? 1) Local gauge fluxes completely determine the reduced state in any region. 2) Wilson loops form a complete commuting set of observables for all finite groups. 3) Trivial holonomies fully determine local properties of gauge-invariant ground states. 4) The ground state is always unique for non-Abelian group constructions. 5) Topological entanglement entropy always depends on the "log dim R" term. 6) Non-Abelian anyons cannot appear in models based on non-Abelian groups. 7) Local observables can distinguish between all ground states in contractible regions.

✓ Correct Answer:

The correct answer is 3) Trivial holonomies fully determine local properties of gauge-invariant ground states..

📚 Reference Text:

Kitaev’s quantum double model as an error correcting code Shawn X. Cui1, Dawei Ding2, Xizhi Han2, Geoﬀrey Penington2, Daniel Ranard2, Brandon C. Rayhaun2, and Zhou Shangnan2 1Departments of Mathematics, Physics and Astronomy, Purdue University, West Lafayette, IN 47907 2Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305 Kitaev’s quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We oﬀer an explicit proof that this is the case for arbitrary ﬁnite groups. Actually a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region. Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. We contrast this result with the fact that local properties of gauge-invariant states are not generally determined by specifying all of their non-Abelian ﬂuxes — that is, the Wilson loops of lattice gauge theory do not form a complete commuting set of observables. We also note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a diﬀerent context can be adapted to provide another proof of the error correcting property of Kitaev’s model. Finally, we compute the topological entanglement entropy in Kitaev’s model, and show, contrary to previous claims in the literature, that it does not depend on whether the “log dim R” term is included in the deﬁnition of entanglement entropy. Contents 1 Introduction 2 2 Background 3 2.1 Error correcting codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Topological quantum order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Kitaev’s ﬁnite group lattice model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Ground state space of Kitaev’s model . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Main results 8 3.1 The toric code is a QECC: a warm-up . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 States with locally zero energy density are locally indistinguishable . . . . . . . . . 10 3.3 Wilson loops are not a complete set of observables . . . . . . . . . . . . . . . . . . 12 3.4 Topological entanglement entropy . . . . .

Question 1529 multiple-choice

Quantum algorithms for Hidden Subgroup Problems (HSPs) often require adapting abstract algebraic problems into forms that can be efficiently executed on physical quantum computers. The technique known as "pushing" enables this adaptation by translating HSPs defined over large or infinite groups into manageable, finite groups. Which statement best describes the role of an epimorphism in the "pushing" technique for quantum Hidden Subgroup algorithms? 1) It establishes the correspondence between quantum states and classical bits. 2) It ensures that only abelian groups are considered for quantum computation. 3) It maps the original, possibly infinite or large group onto a smaller, finite group suitable for efficient quantum algorithm implementation. 4) It provides a mechanism for error correction in quantum circuits. 5) It transforms the transversal map into a homomorphism. 6) It determines the basis used for the quantum Fourier transform. 7) It restricts the subgroup structure to cyclic groups only.

✓ Correct Answer:

The correct answer is 3) It maps the original, possibly infinite or large group onto a smaller, finite group suitable for efficient quantum algorithm implementation..

📚 Reference Text:

ij/parenrightBig/angbracketrightBig /radicalbigg/angbracketleftBig Φ/parenleftBig γ† ij/parenrightBig |Φ/parenleftBig γ† ij/parenrightBig/angbracketrightBig∈H/hatwideG⊗HS Step 5. Outputγijand stop. 4.Pushing HSPs for the generic QHS algorithm QRand For certain hidden subgroup problems (HSPs) ϕ:G−→S, the corresponding generic QHS algorithm QRand either is not physically implementable or is too ex- pensive to implement physically. For example, the HSP ϕis usually not physically implementable if the ambient group is inﬁnite (e.g., Gis the inﬁnite cyclic group Z), and is too expensive to implement if the ambient group is too large (e .g.,G is the symmetric group S10100). In this case, there is a standard generic way of ”tweaking” the HSP to get around this problem, which we will call pushing . Deﬁnition 2. Letϕ:G−→Sbe a map from a group Gto a setS. A map /tildewideϕ:/tildewideG−→Sfrom a group /tildewideGto the setSis said to be a pushofϕ, written /tildewideϕ=Push(ϕ), provided there exists an epimorphism µ:G−→/tildewideGfromGonto/tildewideG, and a transversal τ:/tildewideG−→Gofµsuch that /tildewideϕ=ϕ◦τ. IS GROVER’S ALGORITHM A QUANTUM HIDDEN SUBGROUP ALGORITHM ? 5 If the epimorphism µand the transversal τare chosen in an appropriate way, then execution of the generic QHS subroutine with input /tildewideϕ=Push(ϕ) , i.e., execution of QRand(/tildewideϕ) , will with high probability produce an irreducible representation /tildewideγof the group /tildewideG which is suﬃciently close to an irreducible representation γof the group G. If this is the case, then there is a polynomial time classical algorithm which up on input /tildewideγ produces the representation γ. Obviously, much more can be said about pushing. But unfortunately that would take us far aﬁeld from the objectives of this paper. For more information on pushing, we refer the reader to [24]. 5.Shor’s algorithm Shor’s factoring algorithm is a classic example of a QHS algorithm creat ed from the push of an HSP. LetNbe the integer to be factored. Let Zdenote the additive group of integers, andZ× Ndenote the monoid of integers under multiplication modulo N(i.e., the ring of integers modulo Nignoring addition.) Shor’s algorithm is a QHS algorithm that solves the following HSP ϕ:Z−→ Z× N m/ma√sto−→ammodN with unknown hidden subgroup structure given by the following comm utative dia- gram Zϕ−→Z× N νց ր ι Z/PZ, whereais an integer relatively prime to N, wherePis the hidden integer period of the mapϕ:Z−→Z× N, wherePZis the additive subgroup all integer multiples of P(i.e., the hidden subgroup), where ν:Z−→Z/PZis the natural epimorpism of of the integers onto the quotient group Z/PZ(i.e., the hidden epimorphism), and whereι:Z/PZ−→Z× Nis the hidden monomorphism. An obstacle to creating a physically implementable algorithm for this HS P is that the domain Zofϕis inﬁnite. As observed by Shor, a way to work around this diﬃculty is to push the HSP. In particular, as illustrated by the following commutative diagram Zϕ−→ Z× N µցտτրϕ=Push(ϕ) =ϕ◦τ ZQ, 6 SAMUEL J. LOMONACO, JR. AND LOUIS H. KAUFFMAN a push/tildewideϕ=Push(ϕ) is constructed by selecting the epimorphism µ:Z−→ZQof Zonto the ﬁnite cyclic group ZQof orderQ, where the integer Qis the unique power of 2 such that N2≤Q<2N2, and choosing the transversal3 τ:ZQ−→Z mmodQ/ma√sto−→m, where 0≤m

Question 1530 multiple-choice

Quantum algorithms for the hidden subgroup problem (HSP) play a crucial role in computational group theory and underpin several well-known quantum speedups. Techniques for solving HSP differ significantly between abelian and non-abelian groups, with recent advances addressing non-abelian cases using new sampling strategies. Which statement correctly characterizes the role of the parameter r in the random strong Fourier sampling method for solving the hidden subgroup problem in non-abelian groups? 1) r measures the number of conjugacy classes in a group and determines the optimal classical post-processing time. 2) r quantifies how abelian a group is, with higher values indicating closer similarity to abelian groups. 3) r determines the size of the minimal generating set required to solve HSP using standard Fourier sampling. 4) r specifies the dimension of the largest irreducible representation relevant for HSP algorithms. 5) r characterizes when random strong Fourier sampling is effective, with larger r indicating that polynomially many samples suffice to solve HSP. 6) r represents the probability of successfully identifying a subgroup with a single Fourier sample. 7) r identifies the total number of non-commuting elements within the group relevant to quantum algorithms.

✓ Correct Answer:

The correct answer is 5) r characterizes when random strong Fourier sampling is effective, with larger r indicating that polynomially many samples suffice to solve HSP..

📚 Reference Text:

Title: On the Power of Random Bases in Fourier Sampling: Hidden Subgroup Problem in the Heisenberg Group Year: 2005 Paper ID: a350e96ab77ce30a5ea1d3f7335294a0eb4b05fa Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a350e96ab77ce30a5ea1d3f7335294a0eb4b05fa Abstract: The hidden subgroup problem (HSP) offers a unified framework to study problems of group-theoretical nature in quantum computing such as order finding and the discrete logarithm problem. While it is known that Fourier sampling provides an efficient solution in the abelian case, not much is known for general non-abelian groups. Recently, some authors raised the question as to whether post-processing the Fourier spectrum by measuring in a random orthonormal basis helps for solving the HSP. Several negative results on the shortcomings of this random strong method are known. In this paper however, we show that the random strong method can be quite powerful under certain conditions on the group G. We define a parameter r(G) and show that O((log |G| / r(G))2) iterations of the random strong method give enough classical information to solve the HSP. We illustrate the power of the random strong method via a concrete example of the HSP over finite Heisenberg groups. We show that r(G) = Ω(1) for these groups; hence the HSP can be solved using polynomially many random strong Fourier samplings followed by a possibly exponential classical post-processing without further queries. The quantum part of our algorithm consists of a polynomial computation followed by measuring in a random orthonormal basis. As an interesting by-product of our work, we get an algorithm for solving the state identification problem for a set of nearly orthogonal pure quantum states.

Question 1531 multiple-choice

Quantum algorithms such as Grover’s and Shor’s are often analyzed using group theory and the framework of hidden subgroup problems (HSPs). The structural properties of the underlying groups determine the applicability and limitations of quantum HSP techniques. Which property of the symmetric group SN specifically prevents standard non-abelian hidden subgroup algorithms from efficiently finding the hidden subgroup Stabj0 corresponding to Grover’s search target? 1) SN is a cyclic group of order N. 2) The subgroups Stabj0 are all abelian. 3) SN has only normal subgroups of index two. 4) The subgroups Stabj0 are maximal subgroups. 5) The order of SN is a prime number. 6) The subgroup Stabj0 is invariant under all automorphisms of SN. 7) The subgroups Stabj0 are not normal and are mutually conjugate, so only the trivial normal subgroup can be found.

✓ Correct Answer:

The correct answer is 7) The subgroups Stabj0 are not normal and are mutually conjugate, so only the trivial normal subgroup can be found..

📚 Reference Text:

1 if i=j. Proposition 1. The set {(0j0),(1j0),(2j0),...,((N−1)j0)} is a complete set of distinct coset representatives for the h idden subgroup Stabj0 ofSN, i.e., the coset space SN/Stabj0is given by the following complete set of mutually distinct cosets. SN/Stabj0={(0j0)Stabj0,(1j0)Stabj0,(2j0)Stabj0,...,((N−1)j0)Stabj0} Proof.Since (kj0)Stabj0= (ℓj0)Stabj0⇐⇒(ℓj0)−1(kj0)∈Stabj0⇐⇒k=l, it follows that (0j0)Stabj0,(1j0)Stabj0,(2j0)Stabj0,...,((N−1)j0)Stabj0 are mutually distinct cosets of Stabj0inSN. It now follows from Lagrange’s theorem that the above collection of mutually distinct cosets is comp lete. /square IS GROVER’S ALGORITHM A QUANTUM HIDDEN SUBGROUP ALGORITHM ? 9 8.A comparison of Grover’s and Shor’s algorithms Now let us compare Shor’s algorithm with Grover’s. LetSbe the set of integers S={0,1,2,...,N−1}, whereN= 2n, and letj0∈Sdenote the unknown label to be found by Grover’s algorithm. Shor’s algorithm solves the HSP ϕ:Z−→Z× Nwith hidden subgroup structure Zϕ−→Z× N νց ր ι Z/PZ, whereZ× Ncan be thought of as the result of the unknown (”malicious”) relabe ling ι:k+PZ/ma√sto−→akmodN ofZ/PZ. In likemanner, Grover’salgorithmsolvesan HSP, namely, the HSP ϕ:SN−→S with hidden subgroup structure SNϕ−→S νց ր ι SN/Stabj0, whereS={0,1,2,...,N−1}denotes the set resulting from the unknown (”mali- cious”) relabeling (bijection) ι: (j j0)Stabj0/ma√sto−→j ofSN/Stabj0. For Shor’s algorithm, Shor’s oracle /tildewideϕ:ZQ−→Z× Nis created by pushing the HSPϕ:Z−→Z× Nusing Zϕ−→Z× N µցտτր/tildewideϕ Z/QZ, thereby producing /tildewideϕ=Push(ϕ) =ϕ◦τwith the transversal τ:kmodQ/ma√sto−→k. In likemanner, for Grover’salgorithm, Grover’soraclecan be creat ed by pushing the HSPϕ:SN−→Susing SNϕ−→S µցտτր/tildewideϕ SN/Stab0, thereby producing /tildewideϕ=Push(ϕ) =ϕ◦τwith the transversal τ: (0j)Stab0/ma√sto−→SN of the natural surjection µ. 10 SAMUEL J. LOMONACO, JR. AND LOUIS H. KAUFFMAN Although it is not immediately apparent, the resulting push /tildewideϕ(forj0/ne}ationslash= 0) is actually Grover’s oracle relabelled by the injection ι:SN/Stabj0−→S. For /tildewideϕ=ϕ◦τ= (ι◦ν)◦τ=ι◦(ν◦τ) and (ν◦τ)[(0j)Stab0] =  (0j0)Stabj0ifj=j0 Stabj0 otherwise which is informationally the same as Grover’s oracle f(j) =  1 ifj=j0 0 otherwise Hence, we can conclude that Grover’s algorithm is an quantum algorit hm very much like Shor’s algorithm, in that it is a quantum algorithmthat solvest he Grover hidden subgroup problem. 9.However However,... this appearstobe wherethe similaritybetweenthesetw oalgorithms ends. For, the standard non-abelian QHS algorithm on SNfor the HSP ϕ(or/tildewideϕ) can not ﬁnd the hidden subgroup Stabj0for each of the following two reasons: •Since the subgroups Stabjare not normal subgroups of SN, it follows from the work of Hallgren et al [11] that the standard non-abelian hidden sub- group algorithm will ﬁnd the largest normal subgroup of SNlying inStabj. But unfortunately, the largest normal subgroup of SNlying inStabj. is the trivial subgroup of SN. •The subgroups Stab0,Stab1, ... ,StabN−1are mutually conjugate sub- groups of SN. We should also mention that this hidden subgroup approach can not p ossibly lead to a quantum algorithm that is faster than Grover’s.. For Zalka[3 1] has shown that Grover’s algorithm is asymptotically optimal. IS GROVER’S ALGORITHM A QUANTUM HIDDEN SUBGROUP ALGORITHM ? 11 A Comparison of Two Quantum Algorithms Shor’s Algorithm Grover’s Algorithm Similarities Shor’s algorithm solves an HSP, namely: Zϕ−→Z× N νց ր ι Z/PZGrover’s algorithm solves an HSP, namely: SNϕ−→S νց ր ι SN/Stabj0 Pushingϕusing Zϕ−→Z× N µցտτր/tildewideϕ Z/QZ produces /tildewideϕ=Push(ϕ) =ϕ◦τ which is Shor’s oraclePushingϕusing SNϕ−→S µցտτր/tildewideϕ SN/Stab0

Question 1532 multiple-choice

Quantum phase estimation is a fundamental procedure in quantum computing for determining the eigenvalues of unitary operators, enabling critical applications such as factoring and quantum simulation. Various algorithms exist to optimize this process for hardware limitations and accuracy. Which phase estimation algorithm is notable for being the first to use iterative measurements and Hadamard gates to encode phase information, making it suitable for quantum devices with limited qubits? 1) Lloyd’s algorithm using inverse quantum Fourier transform 2) Constant precision quantum phase estimation algorithm 3) Standard quantum phase estimation algorithm with maximum qubit usage 4) Shor’s factoring algorithm 5) Grover’s search algorithm 6) Bayesian phase estimation algorithm 7) Kitaev’s algorithm

✓ Correct Answer:

The correct answer is 7) Kitaev’s algorithm.

📚 Reference Text:

physical limitations. The experimental results sho w that the accuracy of ﬁnding the correct phase decreases as the number of qubits and quantum operations increase. To mitigate the problem, we propose modiﬁed solutions of these QPE algorithms by minimizing the number of control gates and phase shift operators. This paper is categorized as follows. Section II describes the basic quantum operations and various phase estimation algorithms such as Kitaev’s algorithm, the iterative algor ithm to estimate the phase, Lloyd algorithm for phase estimation based on inverse quantum Fourier transform (IQFT), and the constant precision algorithm. In section III, the simulati on and experimental results for each method are provided and compared. Finally, Section V summarizes the results and conclusions from this work. II. P HASE ESTIMATION Phase estimation is a technique that is used to estimate the eigenvalues |λ/angbracketrightof a unitary matrix Uwith its known eigenvector |ψλ/angbracketright[3], U|ψλ/angbracketright=λ|ψλ/angbracketright, (1) where the eigenvalues of the unitary matrix are λ=e2πiϕn. The phase of the unitary matrix can be written as ϕn=0.x1x2x3...xn where nis the number of qubits used for phase estimation. The estimated variable (ˆϕ)can be expressed as a binary representation,|0/angbracketright H K• H✌✌✌ |ψ/angbracketright U2k−1|ψ/angbracketright Fig. 2: Controlled U circuit ˆϕ=x1 21+x2 22+x3 23+···+xn 2n(2) Fig.1 illustrates a quantum computing circuit that incorpo - rates this problem of phase determination. The circuit cons ists of one qubit and an eigenstate, a Hadamard gate (H) and a rotation gate (U). The output of the circuit contains the pha se 1√ 2/parenleftbig |0/angbracketright+ei2πϕ|1/angbracketright/parenrightbig described in the top of the Fig.1. The goal of phase estimation is to ﬁnd the eigenvalues of an unitary matrix and then apply these eigenvalues to estima te the unknown phase of the unitary operator. However, with only this information available from the circuit output it i s impossible to ﬁnd the correct phase due to the superposition state on the value. There are different methods used to calculate the phase for an unitary matrix expressed in this circuit constructio n. Section II-A describes sequential post processing techniq ues to calculate the unknown phase. Section II-B introduces an iterative technique to implement the Kiteav’s algorithm wi th higher accuracy. Section II-C describes applying the inver se quantum Fourier transform (IQFT) to derive the unknown phase information and Section II-D discusses the arbitrary precision QPE that reduces the number of shift operators for phase estimation thereby decreasing the depth of the quantu m circuits. These techniques along with their simulated and experimentally implemented results will be discussed in de tail below. A. Kitaev’s algorithm Kitaev’s algorithm is the ﬁrst algorithm that was introduce d to estimate the phase of an unitary matrix. In this technique a set of Hadamard gates are applied to the input qubits. The outputs of the Hadamard gates connected with the controlled - U2k−1result in an output represented by a phase shift op- erator. Applying a controlled- Uoperator ktimes transforms the control qubit to1√ 2/parenleftbig |0/angbracketright+e−i2πϕk2k−1|1/angbracketright/parenrightbig . At each test phase ϕk=2k−1ϕcan be calculated. By doing

Question 1533 multiple-choice

Quantum mechanics has become increasingly important in modeling large biological systems, revealing effects that classical methods cannot capture. Advances in computational strategies have made it feasible to apply quantum methods to complex biomolecules, impacting areas like drug design and structural biology. Which computational strategy enables linear scaling in quantum mechanical calculations for large biomolecular systems by partitioning them into smaller, manageable subsystems? 1) Divide and conquer approach 2) Classical molecular mechanics simulation 3) Monte Carlo integration 4) Hartree-Fock self-consistent field method 5) Quantum annealing optimization 6) Density functional perturbation theory 7) Ab initio random structure search

✓ Correct Answer:

The correct answer is 1) Divide and conquer approach.

📚 Reference Text:

Title: Using Quantum Mechanical Approaches to Study Biological Systems Year: 2014 Paper ID: 4c3901e99f570050792c3933c85797d9b4356495 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/4c3901e99f570050792c3933c85797d9b4356495 Abstract: Conspectus Quantum mechanics (QM) has revolutionized our understanding of the structure and reactivity of small molecular systems. Given the tremendous impact of QM in this research area, it is attractive to believe that this could also be brought into the biological realm where systems of a few thousand atoms and beyond are routine. Applying QM methods to biological problems brings an improved representation to these systems by the direct inclusion of inherently QM effects such as polarization and charge transfer. Because of the improved representation, novel insights can be gleaned from the application of QM tools to biomacromolecules in aqueous solution. To achieve this goal, the computational bottlenecks of QM methods had to be addressed. In semiempirical theory, matrix diagonalization is rate limiting, while in density functional theory or Hartree–Fock theory electron repulsion integral computation is rate-limiting. In this Account, we primarily focus on semiempirical models where the divide and conquer (D&C) approach linearizes the matrix diagonalization step with respect to the system size. Through the D&C approach, a number of applications to biological problems became tractable. Herein, we provide examples of QM studies on biological systems that focus on protein solvation as viewed by QM, QM enabled structure-based drug design, and NMR and X-ray biological structure refinement using QM derived restraints. Through the examples chosen, we show the power of QM to provide novel insights into biological systems, while also impacting practical applications such as structure refinement. While these methods can be more expensive than classical approaches, they make up for this deficiency by the more realistic modeling of the electronic nature of biological systems and in their ability to be broadly applied. Of the tools and applications discussed in this Account, X-ray structure refinement using QM models is now generally available to the community in the refinement package Phenix. While the power of this approach is manifest, challenges still remain. In particular, QM models are generally applied to static structures, so ways in which to include sampling is an ongoing challenge. Car–Parrinello or Born–Oppenheimer molecular dynamics approaches address the short time scale sampling issue, but how to effectively use QM to study phenomenon covering longer time scales will be the focus of future research. Finally, how to accurately and efficiently include electron correlation effects to facilitate the modeling of, for example, dispersive interactions, is also a major hurdle that a broad range of groups are addressing The use of QM models in biology is in its infancy, leading to the expectation that the most significant use of these tools to address biological problems will be seen in the coming years. It is hoped that while this Account summarizes where we have been, it will also help set the stage for future research directions at the interface of quantum mechanics and biology.

Question 1534 multiple-choice

In algebraic number theory and quantum computing, the structure of unit groups in number fields, especially cyclotomic fields, is crucial for efficient algorithm design. Cyclotomic units and their associated lattices have important applications in cryptography and computational mathematics. Which statement accurately describes a key advantage of the quantum algorithm for computing the unit group of cyclotomic fields Q(ζm)? 1) It requires exponential time for all abelian number fields. 2) It uses O(m^5) qubits, making it less efficient than classical methods. 3) It is only applicable to fields with dihedral Galois groups. 4) It cannot compute a basis for the unit group in polynomial time. 5) It depends solely on the class number being small. 6) It computes a basis for the unit group in polynomial time using O(m^2 log m) qubits, leveraging the structure of cyclotomic units. 7) It does not benefit from the existence of sublattices with short bases.

✓ Correct Answer:

The correct answer is 6) It computes a basis for the unit group in polynomial time using O(m^2 log m) qubits, leveraging the structure of cyclotomic units..

📚 Reference Text:

ﬁe lds with automor- phisms in order to speed-up the protocol, e.g. it has been don e by XTR in cryptosystems based on discrete logarithms in ﬁnite ﬁelds [ 39]. 4 A new algorithm using cyclotomic units WhenK=Q(ζm)for an arbitrary integer mwe deﬁne the group of cyclotomic units to be the subgroup CofK∗generated by −1,ζmandζj m−1withj= 1,2,...,m−1, intersected with the group of units O∗ K. We follow the notations of [18, Sec. 3], in particular mandkdon’t have the same meaning as in the other sections. Factor m=pα1 1pα2 2···pαk kand, for each index i, putmi=m/pαi i. Forj= 1,...,m−1we set vj=/braceleftBigg 1−ζj m,if for alliwe havemi∤j, 1−ζj m 1−ζmim,otherwise for the unique isuch thatmi|j. 4.1 Unconditional results Lemma 41 (Theorem 4.2 in [35]) The lattice M:= LogCadmits the sys- tem of generators {bj= Log(vj)|j= 1,2,...,m−1}. 11A similar speed-up was obtained in the case of Z[ζk]-LLL in [32]: an example took 20 s over Z[ζk]compared to 75 s over Z. The special case of cyclotomic ﬁelds in quantum algorithms f or unit groups 17 Lemma 42 (Lemma 3.5 in [18]) For any integer j∈ {1,2,...,m−1}we have/vextenddouble/vextenddouble1−ζj m/vextenddouble/vextenddouble 2=O(√m). Hence/⌊ard⌊lBM/⌊ard⌊l2=O(√m). Theorem 43 Algorithm 1 computes a basis of the unit group of Q(ζm)inpoly(m) time and uses O(m2logm)qubits. Proof. The only quantum step of Algorithm 1 is the dual lattice sampl er, whose parameters are δ=/⌊ard⌊lBL/⌊ard⌊l/λ∗ 1(L)andη= 1/k2where k= log2(√mLip(f))+log2(detL)) is given in step 1 of the algorithm. In this section mis not necessarily equal to the rankm′ofZ[ζ]∗butO(m′) =O(m)so that we use mandm′interchangeably. By Lemma 32, the sampler used O(Qm)qubits with Qas below. We write a≪bfor a=O(b). We use Th 36: logLip(f)≪m2+mlogDand thenk≪m2+mlogD. We also use Lemma 42: log1/(δλ∗ 1(L))≪log(1//⌊ard⌊lBM/⌊ard⌊l)≪logmandlogD= (m−2)logm≪mlogm. Q≪mlog(mlog1 η))+log1 δλ∗ 1(L)+log(Lip(f)) ≪(mlogm+mloglogk)+logm+m2+mlogD ≪mlogm+mloglogD+logm+m2+mlogD ≪m2+mlogD ≪m2logm. This is to be compared to the O(m5+m4logD)qubits used by number ﬁelds in general (Cor 37) and O(m5logm)used by cyclotomic ﬁeld if the Algorithm 3 were used without taking proﬁt of the cyclotomic units (Ex 1) . Remark 44 Our improvement can be used whenever the lattice of units has a sublattice admitting a short basis and this is not limited to t he cyclotomic ﬁelds and, more generally, abelian ﬁelds. For instance, Kihel [31 ] proposed a family of ﬁelds with dihedral Galois group which have a full-rank subg roup of units which are short. Although one expects a speed-up in all these cases, the cyclotomic case is even more special. Indeed, the parameter δ≤ /⌊ard⌊lBM/⌊ard⌊l/λ∗ 1(L)depends in a large extent on the index [L:M]. By [36, Ex 8.5] (see also [17, Th 2.8]), in the case of cyclotomic ﬁelds of prime conductor, [L:M] =h+(m). By Conjecture 1 this is small, so the impact of the full-rank subgroup is more impo rtant in this case. 4.2 Consequences of a recent conjecture for cyclotomic ﬁeld s One denotes h(m)the class group of Q(ζm),h+(m)the class group of its max- imal real subﬁeld, i.e. K:=Q(ζm+ζ−1 m), and one sets h−(m) =h(m)/h+(m). A folklore conjecture states that h−(m)is large but easy to compute whereas h+(m)is small but hard

Question 1535 multiple-choice

Quantum computing introduces novel algorithms and hardware architectures that differ fundamentally from classical computing, presenting challenges and opportunities for programmers with traditional backgrounds. Simulation and error correction are key areas for practical quantum application development. Which quantum algorithm is specifically designed for factoring large numbers efficiently, thus threatening the security of widely used cryptographic systems? 1) Grover's algorithm 2) Quantum Fourier transform 3) Shor's algorithm 4) Quantum teleportation 5) Deutsch-Jozsa algorithm 6) Superdense coding 7) Variational quantum eigensolver

✓ Correct Answer:

The correct answer is 3) Shor's algorithm.

📚 Reference Text:

Title: Quantum Computing for Programmers Year: 2022 Paper ID: abb93f4963ee00c2e5a65982af8eae89e30be4ba Source: semantic-scholar URL: https://www.semanticscholar.org/paper/abb93f4963ee00c2e5a65982af8eae89e30be4ba Abstract: This introduction to quantum computing from a classical programmer's perspective is meant for students and practitioners alike. Over 25 fundamental algorithms are explained with full mathematical derivations and classical code for simulation, using an open-source code base developed from the ground up in Python and C++. After presenting the basics of quantum computing, the author focuses on algorithms and the infrastructure to simulate them efficiently, beginning with quantum teleportation, superdense coding, and Deutsch-Jozsa. Coverage of advanced algorithms includes the quantum supremacy experiment, quantum Fourier transform, phase estimation, Shor's algorithm, Grover's algorithm with derivatives, quantum random walks, and the Solovay–Kitaev algorithm for gate approximation. Quantum simulation is explored with the variational quantum eigensolver, quantum approximate optimization, and the Max-Cut and Subset-Sum algorithms. The book also discusses issues around programmer productivity, quantum noise, error correction, and challenges for quantum programming languages, compilers, and tools, with a final section on compiler techniques for transpilation.

Question 1536 multiple-choice

Commitment schemes are cryptographic protocols that allow one party to commit to a chosen value while keeping it hidden, with the ability to reveal the committed value later. Two central security properties are statistical hiding and computational binding, which play crucial roles in ensuring privacy and integrity. Which property guarantees that, even with unlimited computational resources, an adversary cannot distinguish between commitments to different values? 1) Computational binding 2) Zero-knowledge soundness 3) NP-witness integrity 4) Statistical hiding 5) Probabilistic key generation 6) Modified Bellare-Goldreich definition 7) Commitment randomization

✓ Correct Answer:

The correct answer is 4) Statistical hiding.

📚 Reference Text:

typedeﬁnitions from the exact-security style ones. We then deﬁne the type of commitment scheme we will look at. Our commit- ments will be statistically (unconditionally) hiding and computationally binding . Concretely, a commitment scheme consists of a probabilistic polynomial timekey generator H, which on input 1 koutputs a public key pkand a witness w. We letLHbe the set of public keys that Hcan produce as an output, wis AStatistically-HidingIntegerCommitmentScheme 131 an NP-witness to the fact that pk∈LH. The scheme also deﬁnes an algorithm committhat takes as inputs pk, a stringsto be committed to, and a random stringr, both of lengths that are ﬁxed polynomials in k. The output is a com- mitment tos,commitpk(s,r) and a string u. Finally we have an algorithm verify that takes inputs pk, commitment c, and strings s,u, where the output (denoted verifypk(c,s,u)) isacceptorreject. Such a scheme can be used by a committer Cand a receiver Ras follows: in theset-upphase,RrunsHto getpk,w, sendspkand useswto give a zero- knowledge interactive proof [16] that pk∈LH.Ccan commit to sby running commitonpk,sand random input r, sendingc=commitpk(s,r)t oRand keepingrsecret. Opening takes place by revealing s,rtoR, who can then check that verifypk(c,s,r)=accept. We then require the following: Hiding: Forpk∈LH,uniformr,r/primeandanys,s/prime,wehavethatthedistributions ofcommitpk(s,r) and commitpk(s/prime,r/prime) are (statistically) indistinguishable (as deﬁned in [16]). Binding: We say that binding is ( t(k),zepsilon<(k))-secureif it holds that for any C running in time at most t(k), the probability that Con inputpkcomputes s,r,s/prime,r/primesuch that commitpk(s,r)= commitpk(s/prime,r/prime) ands/negationslash=s/prime, is at most zepsilon<(k). We will also need to consider proofs of knowledge in the following. For this, we use a modiﬁcation of the deﬁnition of Bellare and Goldreich, in the version for computationally convincing proofs of knowledge [6]. Let a binary relation R be given, where a prover Pand a veriﬁer Vare both probabilistic polynomial time interactive Turing machines. Intuitively, the prover’s claim is that for agiven common input c, he knowswsuch that (c,w)∈R. To deﬁne this in our setting, we cannot use the original deﬁnition in [6] without change. This is because it asks that the soundness of the protocol holdsforall(large enough) instances c. In our scheme, this is more than we can reasonably ask for: in our case, one may think of cas a commitment and was thestringPcanusetoopen c.Furthermore,thescenarioisthat Pseesthepublic key of the scheme, produces the commitment and then tries to prove he knowshow to open it. But this proof is only computationally convincing in our case, soacheatingprovermayhavesomechanceofproducing,basedonthepublickey,acommitment he cannot open, but where the proof nevertheless is successful withlarge probability. This can typically happen if the prover manages to computesome trapdoor information associated with the public key. This informationcan always be guessed with non-zero probability and so the problem cannot becompletely avoided, but we can at least require that it occurs with only smallprobability. In our deﬁnition of soundness, therefore, we ﬁrst let the prover seea public piece of information, he then produces cand conducts the proof. A cheating prover P ∗wins if the standard soundness requirement fails for this c, and we are satisﬁed with the proof system if within

Question 1537 multiple-choice

Quantum walks are quantum analogs of classical random walks, where a particle moves on a lattice with its position and direction encoded in quantum registers. Efficient circuit implementations of quantum walks are crucial for leveraging quantum parallelism in practical algorithms. In a discrete-time quantum walk on an N-dimensional lattice, which operation is responsible for updating the particle's position based on its velocity, ensuring proper movement on a circular lattice via modular arithmetic? 1) The controlled-NOT (CNOT) gate 2) The Quantum Fourier Transform (QFT) 3) The measurement operation in the computational basis 4) The scattering unitary acting only on the velocity register 5) The spatial convolution operator without modular arithmetic 6) The right shift operator acting independently of the velocity register 7) The propagation operation that maps |v,x⟩ to |v,x+v⟩ modulo N

✓ Correct Answer:

The correct answer is 7) The propagation operation that maps |v,x⟩ to |v,x+v⟩ modulo N.

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length in gates from an input to an output. 1 2 EFFICIENT AND SCALABLE QUANTUM WALK ALGORITHMS the particle’s position x∈ {0,...,N−1}then labels a basis element |x/an}⌊ra⌋ketri}htof anN-dimensional positionHilbert space. At each time step, the particle can move one lattice po sition to the “left”x/ma√sto→x−1, or to the “right”, x/ma√sto→x+ 1, where the addition/subtraction is modulo N. This directional motion is encoded in a two dimensional velocityHilbert space with basis elements labeled |v/an}⌊ra⌋ketri}ht,v=−1,+1, for left and right moving particle respectively. The QW Hilbert space His given as H=C2⊗CN with basis elements labeled |v,x/an}⌊ra⌋ketri}ht:=|v/an}⌊ra⌋ketri}ht⊗|x/an}⌊ra⌋ketri}ht. The state of a QW is an element ψ∈ Hof unit norm /⌊ard⌊lψ/⌊ard⌊l= 1. A QW evolves through two consecutive unitary actions on its sta te. (i) Scattering operation that acts on the velocity space, i.e., is of th e formS⊗Ix, where Sis a unitary matrix on C2, the velocity space, typically [8], S=/bracketleftbigg ieiαsinθ eiαcosθ eiαcosθ ieiαsinθ/bracketrightbigg . andIxis the identity operator on the position space. (ii) Propagation operation that moves the particle in the direction of the velocity, σ:|v,x/an}⌊ra⌋ketri}ht /ma√sto→ |v,x+v/an}⌊ra⌋ketri}ht, where the addition is modulo N. Denoting ˆS=S⊗Ix, (1) a stepTof QW evolution is thus: T=σˆS. This is the model of QW that we are going to be concerned with. The rest of this paper is organized as follows. In Section 2 we describ e our QW algorithm that, forn-qubit position space, implements the right/left shifts (increment/ decrement) in propagation σbysinglerightshift(increment)and2 nCNOTgatesthatcontrolthedirection. We recall the quantum Fourier transform (QFT) and compare the s hift implementation using the QFT with the one based on generalized CNOT gates, common in applications. In Section 3 we introduce the QW whose state, during propagation, evolves through spatial convolution, generalizing the shift. We also describe a strategy for eﬃcient implementation of this family of QWs. In Section 4 we run our basic QW algorithm with QFT based shift on some of the IBM quantum computers for a few lattice sizes and numb er of steps of evolution, and examine their performances. Section 5 is the conclusion. 2.An algorithm for QW Let us ﬁrst express the vectors and operators in the natural ba ses for the position and velocity spaces. Considering the position space by itself, we can use the natural ordering EFFICIENT AND SCALABLE QUANTUM WALK ALGORITHMS 3 induced on the basis by values of xand write |x/an}⌊ra⌋ketri}htas a coordinate vector |x/an}⌊ra⌋ketri}ht= 0 0 ... ... 0 1 0 ... 0 (0) (1) ... ... (x) ...(2) where the coordinate indices are displayed in parentheses and all th e component values are 0, except at the index x, whose component value is 1. Similarly, for the joint velocity- position space, we write |v,x/an}⌊ra⌋ketri}htas a coordinate vector of length 2 N, with the ﬁrst Nelements representing positions corresponding to v= 0 and the last N, those corresponding to v= 1, |v,x/an}⌊ra⌋ketri}ht= 0 0 ... ... 0 1 0 ... 0 (0) (1) ... ... (Nv+x) .... Denote by Xthe right shift on the position space alone (generalizing the symbol f

Question 1538 multiple-choice

Quantum machine learning utilizes the mathematical framework of symmetry groups to optimize classification tasks, particularly distinguishing quantum states based on their structural properties. Understanding the differences between discrete and continuous (Lie) symmetry groups is fundamental for designing robust quantum algorithms. In quantum machine learning, which property uniquely distinguishes continuous symmetry groups like SU(2) from discrete groups such as {1, X}, making them essential for tasks involving unitary invariance of quantum states? 1) They allow only a finite number of symmetry operations. 2) They preserve labels only under bit-flip transformations. 3) They lack the ability to generate smooth paths between group elements. 4) They possess an associated Lie algebra enabling differentiation and the exponential map for connecting group elements. 5) They can only be used to classify classical data mapped to quantum states. 6) Their elements cannot be parametrized by real numbers. 7) They do not support invariance under unitary transformations.

✓ Correct Answer:

The correct answer is 4) They possess an associated Lie algebra enabling differentiation and the exponential map for connecting group elements..

📚 Reference Text:

classiﬁ- cation task of Fig. (2)(b), where we want to classify single- qubit pure states (blue circles on the surface of the Bloch sphere) with label yi= 0from single-qubit mixed states (orange triangles in a shell inside of the Bloch sphere) with labelyi= 1. It is easy to see that now the labels are in- variant under the action of any unitary. This corresponds Figure2. Examples of discrete and continuous symmetry groups. a) In this QML task the goal is to classify real valued datax. The data xiwith labelyi= 0(blue circles) is sampled from the interval [ 4; 4], while the data xiwith label yi= 1(orange triangles) is sampled from ( 2; 4][[ 4; 2). As indicated by the pink dashed arrow, the symmetry operation x!xpreserves the label. We encode this classical data in a quantum state by initializing a single qubit to the state j+i and performing a rotation about the y-axis with an angle xi. As schematically depicted, the quantum states iin the dataset are pure states living on the surface of the Bloch sphere. Here, the symmetry group that preserves the labels of the quantum states isGb ip=f11;Xgof Eq. (6) (see pink dashed arrow over the Bloch sphere). b) In this QML task the goal is to classify single- qubit pure states from single-qubit mixed states. The data i with labelyi= 0(blue circles) correspond to pure states living on the surface of the Bloch sphere, while the data xiwith label yi= 1(orange triangles) are mixed states living in a shell inside of the Bloch sphere. Since the purity is a spectral property, it gets preserved by the action of any unitary. As such, the symmetry group preserving the labels is Guni=fU2SU(2)g of Eq. (7) (see pink dashed arrow over the Bloch sphere). Note that here the data is quantum mechanical in nature, as it does not correspond to classical data encoded in quantum states. to the group Guni=fU2SU(2)g; (7) whereSU(2)denotes the special unitary group of degree 2(all22unitary matrices with determinant 1). Here, 6 Guniis continuous as it is the set of all (inﬁnitely many) unitaries acting on one qubit. More speciﬁcally, we note that any element in Gunican be expressed as U=c011+ (c1X+c2Y+c3Z), wherec02RandP3 i=0c2 i= 1are the real coordinates parametrizing the manifold of 22 unitaries.2 Continuous groups necessarily have uncountably many elements, in stark contrast to the often ﬁnite discrete groups. But since continuous groups are also manifolds, theycontainadditionalstructurelackedbydiscretegroups: Wecanconstructsmoothpathsinthegroup, justaswecan construct paths on surfaces. We can then take derivatives along paths in the group, just as we can take derivatives of paths on surfaces. From here, analogously to solving ordinary diﬀerential equations, the exponential map al- lows us to “integrate” and lift from derivatives over time to paths in the group. Speciﬁcally, the structure that stores the information of “directional derivatives of con- tinuous group paths” is called the Lie algebra gassociated to the continuous group, which we henceforth refer to as aLie group . Exponentiation of every element Xof the Lie algebra leads to

Question 1539 multiple-choice

In quantum information theory, constructing informationally complete measurements is crucial for fully characterizing quantum states. Group representations and their properties play a central role in designing such measurement schemes. Which condition must be satisfied by the state ν to guarantee that a POVM constructed from group representations Ug⊗U∗g with only inequivalent irreducible components is informationally complete? 1) The state ν must be a maximally mixed state. 2) The trace Tr[Pσ|ν⟩⟨ν|] must vanish for all σ. 3) ν must be an eigenvector of every projector Pσ. 4) Tr[Pσ|ν⟩⟨ν|] must be equal for all σ. 5) Tr[Pσ|ν⟩⟨ν|] must be nonzero for every σ. 6) ν must commute with all elements of the group representation. 7) The state ν must be pure and separable.

✓ Correct Answer:

The correct answer is 5) Tr[Pσ|ν⟩⟨ν|] must be nonzero for every σ..

📚 Reference Text:

integral in Eq. (11). When Ug⊗U∗ ghas only inequivalent irreducible representations on H⊗H, upon denoting by Pσthe projectors over the invariant subspaces, one has F=d/summationdisplay σTr[Pσ|ν/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tν|] Tr[Pσ]Pσ. (12) Informationally complete measurements and groups represe ntation 5 As a consequence, the POVM {Πg}is info-complete iﬀ the state νsatisﬁes the condition Tr[Pσ|ν/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tν|]/ne}ationslash= 0∀σ. (13) In this case the inverse of Fis easily calculated as follows F−1=d−1/summationdisplay σTr[Pσ] Tr[Pσ|ν/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tν|]Pσ, (14) and the canonical dual Θ gis obtained by the identity |Θg/an}bracketri}ht/an}bracketri}ht=F−1|UgνU† g/an}bracketri}ht/an}bracketri}ht, namely |Θg/an}bracketri}ht/an}bracketri}ht=d−1/summationdisplay σTr[Pσ] Tr[Pσ|ν/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tν|]Pσ|UgνU† g/an}bracketri}ht/an}bracketri}ht =d−1Ug⊗U∗ g/summationdisplay σTr[Pσ] Tr[Pσ|ν/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tν|]Pσ|ν/an}bracketri}ht/an}bracketri}ht, (15) where we used the property [ Ug⊗U∗ g,Pσ] = 0. By Eq. (15) one can notice that the canonical dual is covariant itself, namely Θ g=UgξU† g, whereξis given by |ξ/an}bracketri}ht/an}bracketri}ht=d−1/summationdisplay σTr[Pσ] Tr[Pσ|ν/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tν|]Pσ|ν/an}bracketri}ht/an}bracketri}ht. (16) At this stage we can make some general remarks. Among the invaria nt subspaces there is always the span of1√ d|I/an}bracketri}ht/an}bracketri}ht, and thus Eq. (15) has always the term d−1Ug⊗U∗ g|I/an}bracketri}ht/an}bracketri}ht=d−1|I/an}bracketri}ht/an}bracketri}ht. (17) The other invariant subspaces depend on the representation Ug, but we can prove that for any UIR Ugsuch thatUg⊗U∗ ghas inequivalent irreducible representations, there always exists a suitable νsuch that the POVM UgνU† gis info-complete. In fact, upon writing the projectors Pσin terms of their eigenvectors Pσ=/summationdisplay j|Ψ(σ) j/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tΨ(σ) j|, (18) from/summationtext σPσ=I, it follows that {Ψ(σ) j}is an orthonormal basis for the Hilbert-Schmidt operators. By identifying Ψ(0) 0≡I√ d, from the orthogonality one has Tr[Ψ(σ) j] =√ dδσ0. Then, one can ﬁnd suitable phases {θµ}and a real constant αsuch that the Hermitian operator ν=I d+α/summationdisplay µ/negationslash=0(eiθµΨ(µ) j+e−iθµΨ(µ) j†) (19) is a density matrix satisfying condition (13). Notice that in Eq. (19) w e just need a single label j=j(µ) for eachµ. In the last part of this Section, we want to notice that all POVM’s of t he form (10) are equivalent to a generalized Bell measurement [19] on a tensor-p roduct space H⊗H, where the second space describes an ancilla prepared in the state ντ. In fact, one has UgνU† g= TrA[(I⊗ντ)|Ug/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tUg|], (20) where Tr Adenotes the partial trace over the ancilla space. In general, the p rojectors on the maximally entangled states |Ug/an}bracketri}ht/an}bracketri}htare not orthogonal. The above considerations allow to understand the construction of the quantum universal de tectors introduced in Ref. [22]. Informationally complete measurements and groups represe ntation 6 4. Examples In this section we will provide some examples of info-complete POVM’s, showing their underlying group structure. 4.1.Zd×Zd This ﬁrst example involves a minimal info-complete POVM, namely a POVM havingd2 elements, andwill giveussomegeneralinsight inthecaseofproject ive representations of abelian groups. Consider the group Zd×Zd, and the following d-dimensional projective UIR Um,n=d−1/summationdisplay k=0e2πi dkm|k/an}bracketri}ht/an}bracketle{tk⊕n|, (21) wherem,n∈[0,d−1],and⊕denotessummodulo d. Thecompositionandorthogonality relations of the set are given by Um,nUp,qU† m,n=e2πi d(np−mq)Up,q, (22) Tr/bracketleftbig U† p,qUm,n/bracketrightbig =dδmpδnq. (23) We will now look for an info-complete POVM of the form Ξm,n=1 dUm,nνU† m,n. (24) The properties of the projective UIR help us to ﬁnd the conditions f or informational completeness, and to evaluate the dual frame directly, as an alter native way to the general method developed

Question 1540 multiple-choice

Finite p-groups, where the group order is a power of a prime p, play a crucial role in algebra due to their rich subgroup and commutator structures. Matrix methods are frequently used to classify and distinguish non-isomorphic groups within certain families of p-groups. Which condition must be satisfied for two finite p-groups defined by generators and commutator relations to be isomorphic via their characteristic matrices, when the matrices are over the field Fp? 1) Their characteristic matrices must have the same trace. 2) Their characteristic matrices must be related by an invertible diagonal matrix over Fp. 3) The determinant of each characteristic matrix must be 1 modulo p. 4) Each entry of the characteristic matrices must be congruent modulo p. 5) Their characteristic matrices must be symmetric and have equal rank. 6) Their characteristic matrices must be similar via conjugation by an upper triangular matrix. 7) There must exist an invertible matrix X over Fp such that one characteristic matrix transforms into the other via X.

✓ Correct Answer:

The correct answer is 7) There must exist an invertible matrix X over Fp such that one characteristic matrix transforms into the other via X..

📚 Reference Text:

=bpm/an}bracketri}ht, wheren≥3 forp= 2andn≥m≥2; (L9)/an}bracketle{ta,b,c,d/vextendsingle/vextendsingleapn=bpm=cp2=dp= 1,[a,b] =c,[c,a] = 1,[c,b] =d,[d,a] = [d,b] = 1/an}bracketri}ht, wheren≥3forp= 2andn≥m≥2. ProofIfm= 1, then 1 = [ a,bp] =cp[c,b](p 2). It follows that Φ( G′)≤G3, a contra- diction. Hence m≥2. Ifp=n=m= 2, then |G|= 27. By checking the list of groups of order 27, we get the groups of Type (K1)–(K10). In the following, we ma y assume thatn≥3 forp= 2. Suppose that Gand¯Gare two groups described in the theorem. By Theorem 5.3, G∼=¯Gif and only if there exists X=/parenleftbiggx110 x21x22/parenrightbigg , an invertible matrix over Fp, such that w(¯G) =Xw(G)diag(x−1 11x−1 22,x−1 11x−2 22). (5.6) By suitably choosing x21, that is, using an elementary row operation, we can sim- plifyw(G) to be one of the following types: (a)/parenleftbiggw11w12 0w22/parenrightbigg wherew11/ne}ationslash= 0, (b)/parenleftbigg0w12 w210/parenrightbigg wherew12/ne}ationslash= 0, and (c)/parenleftbigg0 0 w21w22/parenrightbigg .In the following, we as- sume that both w(G) andw(¯G) are such matrices. By Equation 5.6, it is easy to check that (i) diﬀerent types give non-isomorphic groups, (i i)G∼=¯Gif and only if there exists X= diag(x11,x22), an invertible matrix over Fp, such that w(¯G) = Xw(G)diag(x−1 11x−1 22,x−1 11x−2 22). By Table 5.1, we get the groups of type (L1)–(L9). /square w(G) Cases X Remark 1 w(¯G) Group Remark 2 (a) w22/negationslash= 0 diag( w−1 11w22,w11)/parenleftbigg 1w−2 11w12 0 1/parenrightbigg (L1) s=−w−2 11w12 (a) w22= 0 diag(1 ,w11)/parenleftbigg 1w−2 11w12 0 0/parenrightbigg (L2) if w12/negationslash= 0 (L3) if w12= 0t=w2 11w−1 12 (b) w21/negationslash= 0 diag( w21,z) w12=νz2/parenleftbigg 0ν 1 0/parenrightbigg (L4) (b) w21= 0 diag(1 ,z) w12=νz2/parenleftbigg 0ν 0 0/parenrightbigg (L5) (c) w21/negationslash= 0,w22/negationslash= 0 diag( w21,w−1 21w22)/parenleftbigg0 0 1 1/parenrightbigg (L6) (c) w22= 0,w21/negationslash= 0 diag( w21,1)/parenleftbigg 0 0 1 0/parenrightbigg (L7) (c) w21= 0,w22/negationslash= 0 diag(1 ,w22)/parenleftbigg 0 0 0 1/parenrightbigg (L8) (c) w21=w22= 0/parenleftbigg 0 0 0 0/parenrightbigg (L9) Table 5.1: Theorem 5.4 where n≥3 forp= 2 5.2 The type of CG(G′)/Φ(G′)G3is(pn−1,pm,p) Ifn=m, then the type of CG(G′)/Φ(G′)G3is also (pn,pm−1,p). The problem is reduced to the case in last subsection. Hence we may assume th atn > min this section. In this case, [ a,c]/ne}ationslash= 1 and [ b,c] = 1. Let [ a,c] =y. ThenG3=/an}bracketle{ty/an}bracketri}ht. Since apn∈Φ(G′)G3, we may assume that apn=cw11pyw12. By a similar reason, we may assume that bpm=cw21pyw22. Letw(G) = (wij). Then we get a 2 ×2 matrix over Fp. 20 w(G) is called a characteristic matrix of G. Notice that w(G) will be changed if we change the generators a,b. We also call a,ba set of characteristic generators of w(G). Theorem 5.5. Suppose that pis an odd prime, Gis a ﬁnite p-group such that G3∼= Cp,Φ(G′)G3=C2 p,G/Φ(G′)G3∼=Mp(n,m,1)and the type of CG(G′)/Φ(G′)G3is (pn−1,pm,p)wheren > m. Letw(G) = (wij)be a characteristic matrix of G. Then the following conclusions hold: (1)Imin≥2. That is, Gdoes not have an A1-subgroup of index p; (2)Imax≥m; (3)Imax= 2if and only if m= 2and one of the following conditions holds: (i) w12/ne}ationslash= 0andw11|w(G)| /ne}ationslash=w2 12;(ii)w12= 0,w11/ne}ationslash= 0andw22/ne}ationslash= 0;(iii)w2 21+4w22is a quadratic non-residue modular p. ProofWe getImaxandIminby investigating maximal subgroups of G. LetN= /an}bracketle{tb,ap,c,y/an}bracketri}htandMi=/an}bracketle{tabi,bp,c,y/an}bracketri}ht.

Question 1541 multiple-choice

In multipartite quantum systems and invariant theory, certain integer sequences describe subsystem dimensions or tensor ranks, with structural properties reflected in functions like discriminants. Exceptional cases can arise where these functions attain minimal values, often with implications for entanglement classification. Which sequence of integers yields the minimal value of the discriminant function ∆ for n = 3 subsystems, under the constraints that all entries are at least 2 and ∆ > 0? 1) (2, 2, 3) 2) (2, 3, 3) 3) (2, 2, 2) 4) (2, 4, 4) 5) (3, 3, 3) 6) (4, 4, 4) 7) (2, 3, 4)

✓ Correct Answer:

The correct answer is 5) (3, 3, 3).

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5.3, so ∆ >0 and the recursion in Theorem 5.1 terminates 49 on case (3) with d′ n≤1 2d′ 1···d′ n−1and (d′ 1,...,d′ n)/ne}ationslash= (1,...,1,2,d′,d′). For ∆>0 at least three of the d′ is must be greater than 1. Since ∆( /vectord) is unchanged if we remove any 1s, the conditions of the following Lemma are satisﬁed for {d′ i}with 1s removed. Since the cases (2,2,2,2) and (3,3,3) do not have trivial stabilizer, the Lemma implies that ∆ >3. Lemma C.1. Suppose 2≤d1≤ ··· ≤dn≤1 2d1···dn−1forn≥3and∆(/vectord)>0. Then (d1,...,dn) = (3,3,3)with∆ = 2,(d1,...,dn) = (2,2,2,2)with∆ = 3, or∆>3. Proof.As we explained in section 5.2, ∆ considered as a function of dnis a downward parabola that increases monotonically from dn=dn−1to its maximum at dn=1 2d1···dn−1. Thus, fordn≤1 2d1···dn−1we have that ∆(d1,...,dn)≥∆(d1,...,dn−1,dn−1) =d1···dn−1dn−1−d2 1−···−d2 n−1−d2 n−1+n−1 ≥(d1···dn−2−n)d2 n−1+n−1 ≥(2n−2−n)d2 n−1+n−1 Forn>4, the expression in parentheses in the last line is positive, so we have ∆>3. Forn= 4, the expression in parentheses in the last line vanishes, so we hav e that ∆ ≥3. For equality, the last inequality requires d1=d2= 2, the penultimate inequality requires d1=d2=d3, and the ﬁrst inequality requires that d4=d3. Thus we have ∆ >3 or (d1,...,d 4) = (2,2,2,2) with ∆ = 3 in this case. Forn= 3, ∆>0 requires that d1≥3 since for ( d1,d2,d3) = (2,d2,d3), the maximum ∆, achieved for d3=d2, is -2. From the penultimate line above, the expression in parenthes is will then be positive (and consequently ∆ will be at least 18) unless d1= 3. Ford1= 3, the second line above gives ∆ ≥(d2 2−9)+2. This will be at least 9 unless d2= 3. Finally, for (d1,d2,d3) = (3,3,d3), we ﬁnd that ∆ = 2 for the case (3 ,3,3) and ∆>3 otherwise. References [BZ17] I. Bengtsson, K. Zyczkowski, Geometry of quantum states , second edition, Cam- bridge University Press, Cambridge, 2017. MR3752196 [BGOWW] P. B¨ urgisser, A. Garg, R. Oliveira, M. Walter, A. Wigderson , Alternating min- imization, scaling algorithms, and the null-cone problem from invariant theory, in9th Innovations in Theoretical Computer Science , Art. 24, 20 pp, LIPIcs. Leibniz Int. Proc. Inform., 94, Schloss Dagstuhl. Leibniz-Zent. In form., Wadern. MR3761760 50 [BRV] J. Bryan, Z. Reichstein and M. Van Raamsdonk, Existence of L o- cally Maximally Entangled Quantum States via Geometric Invariant The - ory, Ann. Henri Poincar´ e 19(2018), no. 8, 2491–2511. MR3830220 https://doi.org/10.1007/s00023-018-0682-6 [Br02] J.-L. Brylinski, Algebraic measures of entanglement, in Mathematics of quantum computation , 3–23, Comput. Math. Ser, Chapman & Hall/CRC, Boca Raton, FL, 2002. MR2007941 [BB02] J.-L. Brylinski and R. Brylinski, Universal quantum gates, in Mathematics of quantum computation , 101–116, Comput. Math. Ser, Chapman & Hall/CRC, Boca Raton, FL, 2002. MR2007944 [DC71] J. A. Dieudonn´ e and J. B. Carrell, Invariant theory, old and new , Academic Press, New York, 1971. MR0279102 [DVC00] W. D¨ ur, G. Vidal and J. I. Cirac, Three qubits can be entan gled in two in- equivalent ways, Phys. Rev. A (3) 62(2000), no. 6, 062314, 12 pp. MR1804183, https://doi.org/10.1103/PhysRevA.62.062314

Question 1542 multiple-choice

Quantum error correction codes are essential for protecting quantum information from noise and enabling reliable quantum computation and communication. Their design and implementation are deeply influenced by underlying symmetry requirements and limitations, especially concerning transversal gates and covariant properties. Which statement correctly characterizes the main limitation imposed by the Eastin-Knill theorem on quantum error correction codes that admit universal transversal gates? 1) Such codes can achieve perfect error correction in any finite-dimensional system without restriction. 2) These codes allow for error correction independently of the symmetry group acting on the subsystems. 3) The codes must be constructed solely from qubits and cannot use higher local dimensions. 4) Strong error correction in these codes is only possible if the local subsystem dimension is large or the total number of subsystems is large. 5) Universal transversal gates can be implemented in any quantum code regardless of its covariance properties. 6) These codes are always able to perfectly transmit quantum reference frames over noisy channels. 7) The Eastin-Knill theorem permits unrestricted access to Heisenberg-limited measurements in all codes.

✓ Correct Answer:

The correct answer is 4) Strong error correction in these codes is only possible if the local subsystem dimension is large or the total number of subsystems is large..

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results Theorem 6 and Proposition 5 expose serious limitations on thisapproach. If a code admits a universal set of transversalgates, then it is necessarily covariant with respect to a tensor product representation of Uðd LÞ. Moreover, such codes achieve good accuracy only for quite sizable valuesof the local dimension dof each physical subsystem or the total number nof subsystems, as discussed in Sec. VII. While originally derived in the context of fault-tolerant quantum computing, the Eastin-Knill theorem has a variety of other applications, for example, to quantum referenceframes and quantum clocks [12,25] (cf. also recent related work [24]) and to the holographic dictionary relating bulk and boundary physics in the AdS/CFT correspondence[12]. When applied to these settings, our results provide limitations on transmission of reference frames over noisy channels and help to clarify the relationship between bulkand boundary time evolution for the AdS/CFT quantumcode. Our lower bounds on infidelity also apply to therecently discovered quantum codes arising in one-dimen- sional translation-invariant spin chains [8]. Our work builds on Ref. [12], where covariant quantum codes arise in the study of reference frames, i.e., asym- metric states which convey “physical ”information [10,95] . As shown in Ref. [12], exact error correction of reference frames is impossible for finite-dimensional systems, yet in the real world reference frames are always finite dimen- sional and communication channels are always imperfect.Nevertheless, in practice we routinely share reference frames over noisy channels, easily reaching agreement about which direction is “up”or what time it is; further- more, quantum technologists can distribute entanglement between nodes of a quantum network, which is possible only if the nodes share a common phase reference. Ourresults clarify, quantitatively, why accurate communication of reference information is achievable in practice. A quantum reference frame of sufficiently high dimension-ality becomes effectively classical, quite robust against the ravages of environment noise. Examples of such systems include highly excited oscillators and rotors, Bose-Einsteincondensates, superconductors, and other macroscopicphases of quantum matter. In metrology, quantum error correction provides a promising tool for improving sensitivity by protecting a probe system against a noisy environment [25,96 –99]. However, there is a delicate balance to achieve betweenerror correcting against the noise while still being sensitive to the physical observable Hone wishes to measure. In order to correct against errors, one needs to encode in anappropriate code space. Furthermore, in order to measure H, it needs to act nontrivially within that code space. The ability to measure Hdirectly by local observations corre- sponds, in the language of this paper, to covariance of the code with respect to the physical charge H. In other words, adapting our setup to one from quantum metrology isstraightforward: The goal now is to estimate the continuousparameter ωinH¼ωT Aas accurately as possible while at the same time being able to correct against relevant noise. Recent work shows that it is possible to measure Hat the Heisenberg limit using an error-correcting code if Hcannot be expressed as a linear combination of correctable noiseoperators [100–104]. But if the physical charge T Ais

Question 1543 multiple-choice

Quantum algorithms for the hidden subgroup problem (HSP) often rely on sophisticated measurement strategies to extract information from quantum states associated with group structures. Representation theory and entanglement play a fundamental role in the design and power of these algorithms. Which property of the Schur decomposition makes it particularly essential for distinguishing hidden subgroups in certain non-Abelian groups using quantum algorithms? 1) It transforms quantum states exclusively into classical probability distributions. 2) It only provides separable subspaces suitable for independent single-copy measurements. 3) It introduces entangled subspaces across multiple quantum copies, enabling measurements that are necessary for some non-Abelian groups. 4) It eliminates the need for representation theory in subgroup identification. 5) It guarantees efficient classical post-processing of measurement results for any group. 6) It restricts measurement outcomes to Abelian group structures. 7) It ensures that rank estimation is always trivial in quantum collision problems.

✓ Correct Answer:

The correct answer is 3) It introduces entangled subspaces across multiple quantum copies, enabling measurements that are necessary for some non-Abelian groups..

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subspaces, even when it alone fa ils to solve the hsp, so we can use weakFourier-Schursamplingtodecomposethe spaceevenfur ther. The Schurdecomposition has the additional complication that the reﬁned subspaces are no lo nger simply tensor products of single-copy subspaces, but this may actually be an advantage sin ce entangled measurements are known to be necessary for some groups. In principle, strong Fourier-Schur sampling is guaranteed to provid e enough information to solve the hsp, simply because the hidden subgroup states are always distinguisha ble with k= poly(log |G|) copies. However, it would be interesting to ﬁnd a new eﬃcient quant um algorithm for some hspbased on strong Fourier-Schur sampling. Perhaps a ﬁrst step in th is direction would be to analyze the performance of measurement in a r andom basis, as has been studied extensively in the case of weak Fourier sampling [24,40,43,4 8]. Moving away from our original motivation of the hidden subgroup pro blem, the quantum collision problem may be of independent interest. As discussed in the la tter part of Section 4.1, our results on the quantum collision sampling problem can be viewed as a n exploration of spectrum estimation with k=o(d2) copies, but much remains unknown about that regime. Many open problems also remain regarding the black box version of th e quantum collision problem: Can the running time be reduced? What if both input and out put registers are in unknown bases? Is the inverse oracle essential? And is black box ran k estimation really no easier than rank estimation by sampling? Acknowledgments We thank Scott Aaronson, Andris Ambainis, Masahito Hayashi, Keiji Matsumoto, Pranab Sen, and Umesh Vazirani for helpful discussions. We also thank Patrick H ayden for organizing a Bellairs ResearchInstitute workshopon representationtheoryin quantum information, at which the seeds for this work were planted. This work was supported in pa rt by the National Science Foundation under grant PHY-456720, by the Army Research Oﬃce under grant W9111NF-05- 1-0294, and by the European Commission under Marie Curie grant AS TQIT. References [1] S. Aaronson and Y. Shi, Quantum lower bounds for the collision and the element disti nctness problems , J. ACM 51(2004), 595–605, quant-ph/0111102, quant-ph/0112086. [2] D. Bacon, A. M. Childs, and W. van Dam, From optimal measurement to eﬃcient quantum algo- rithms for the hidden subgroup problem over semidirect prod uct groups , Proc. 46thIEEESymposium on Foundations of Computer Science, 2005, pp. 469–478. [3] ,Optimal measurements for the dihedral hidden subgroup prob lem, to appear in Chicago Journal of Theoretical Computer Science, quant-ph/050104 4. [4] D. Bacon, I. L. Chuang, and A. W. Harrow, Eﬃcient quantum circuits for Schur and Clebsch- Gordan transforms , quant-ph/0407082. [5] ,The quantum Schur transform: I. Eﬃcient qudit circuits , To Appear in Proc. 18th ACM- SIAM Symposium on Discrete Algorithms, 2007. 18 [6] E. Bagan, M. A Ballester, R. Munoz-Tapia, and O. Romero-I sart,Purity estimation with separable measurements , Phys. Rev. Lett. 95(2005), quant-ph/0509087, 110504. [7] J. Baik, P. Deift, and K. Johansson, On the distribution of the

Question 1544 multiple-choice

Quantum error mitigation techniques are essential for improving the accuracy of quantum computations, especially when implemented on current noisy intermediate-scale quantum (NISQ) devices. Two principal methods for correcting errors after quantum algorithms like the Quantum Fourier Transform (QFT) involve different approaches to state reconstruction and measurement efficiency. When correcting errors in quantum states following the application of the Quantum Fourier Transform, which approach leverages the separability of theoretical output states for basis inputs to reduce the number of required measurements to 3n instead of 4ⁿ for n qubits? 1) Using ancilla-assisted error detection to identify and discard faulty runs 2) Applying global optimization of the density matrix using maximum likelihood estimation 3) Employing full quantum state tomography with measurements along all possible axes 4) Estimating local errors by measuring each qubit individually along X, Y, and Z axes and extracting principal eigenvectors 5) Implementing randomized benchmarking across multiple basis states 6) Utilizing error-correcting codes based on stabilizer formalism for each basis input 7) Repeating the QFT with varied initial states to statistically average out errors

✓ Correct Answer:

The correct answer is 4) Estimating local errors by measuring each qubit individually along X, Y, and Z axes and extracting principal eigenvectors.

📚 Reference Text:

to obtain !and^½. In the particular case where ^½is a pure state, i.e.,^½= jÁihÁj, one could partially correct the error with a unitary matrix that takes jÁitothe state jÃi. Rev.Mex.Fis. 68031402 CHARACTERIZINGERRORSFOR QUANTUMFOURIER TRANSFORM ON IBM QUANTUM 3 FIGURE2.Representations of the states in Bloch’sSphere. Unfortunately, generally the state ^½is not a pure state, so itisnecessarytoﬁndthepurestate jÁi,closestto ^½,andwith maximal alignment with it, i.e.,jÁishould satisfy Eq. ( 14) 9jÁi=maximum of A(^½;jÁi); (14) where Ais the alignment index. These states can be graphi- cally seen in Fig. 2. In order to ﬁnd jÁithat satisﬁed the requirement, opti- mization methods were used. Once jÁiwas found, a rotation matrix Uthat takes jÁito theoretical jÃi, is easy to write out,foranybasisinitialstate. Itisacorrespondingmatrix Ui andjÁiifor each input canonical basis state jii. Hence, the complete model is deﬁned as a control unitary rotation Uc, andnancillary qubits, that simultaneously correct the nba- sis states, and so any input state. An example of Ucfor two qubitsHilbertspace,isgiveninSec.4(seeFig.3). Although this method is more exact, it has a high computational cost since the resulting state needs to be measured in all 4nlo- cal projections, where nis the number of qubits. In Sec. 3.3 wewillanalyzeanothermethodthatmanagestoestimatethe necessary correction by measuring only 3nprojections. 3.3. Errormodel using only 3nprojection’s Since determining the density matrix ½expusing the QST methodiscomputationallyexpensive,anothermethodispro- posedtoavoidthisstep. Theideaistoﬁndthepurestate jÁi, closest to the ½expmeasuring only those projections relevant for the correct functioning of the algorithm. This is, those that maximally differentiate each of the states resulting from the application of the QFT, on each of the 2ninitial basis states. Fortunately, in the case of the QFT algorithm, the theo- retical output state for each input basis state is separable on allqubits. Althoughthemixedstate ½expisgenerallynot,we assumed the hypothesis that the error is expected to affect in such a way that the resulting state stays close to the theoreti- cal one, and we shall discuss if this assumption proved right experimentally. We assume that a good estimate in this caseis to ﬁnd the reduced density matrix of ½expin each qubit i, ½i exp,i.e.,measuringtheprojections X; Y; Z,ofthisqubit,in total3nmeasurements. Once we have obtained the reduced density matrices of each qubit i, the eigenvector with the highest eigenvalue is extracted jÁiiwithoutoptimization. Inthecaseofaonequbit matrix state, the decomposition in Eq. ( 13),^½is a pure state and is alwaysthe most aligned. Similarlytothepreviousmethod,theunitaryrotationma- trices are determined for each input basis state U=U1 U2:::Un, that takes the state jÃitojÁi. Finally the total Uc matrix is deﬁned similarly to previousmethod as in Fig. 3. Asanexample,theresultingunitarymatrixfortwoqubits on IBM Q computer ibmsantiago , are shown in the next section and the results are compared to those of the ﬁrst method. 4. Analyzing experimental results The procedure will be explained for one of the basis input state of the algorithm j01i, the other basis state were treated similarly. Running the QFT algorithm in IBM Q computer ibmsantiago , we estimate the resulting probabilities of measuring the projections in all basis. The mean values are obtained from 8192shots for each projection, and shown in TablesI and II. TABLEI.Experimental results for the basis input

Question 1545 multiple-choice

Topological quantum computing utilizes exotic quasiparticles in two-dimensional materials, where both anyons and symmetry defects play key roles in encoding and manipulating quantum information. Mathematical frameworks such as modular tensor categories and their extensions are essential for describing the fusion and braiding behaviors that underpin quantum gate operations. In the context of topological quantum computing, which mechanism enables abelian anyon systems, such as the Z2 toric code, to realize non-abelian statistics and potentially achieve universal quantum computation? 1) Introducing higher-dimensional anyons without symmetry considerations 2) Implementing local spin measurements within the abelian phase 3) Utilizing time-reversal symmetry alone to modify statistics 4) Incorporating symmetry defects via G-crossed braided extensions of modular tensor categories 5) Increasing system temperature to induce phase transitions 6) Coupling the system to external magnetic fields to break symmetry 7) Restricting quantum gate operations to only those generated by abelian anyon braiding

✓ Correct Answer:

The correct answer is 4) Incorporating symmetry defects via G-crossed braided extensions of modular tensor categories.

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arXiv:1811.02143v1 [math.QA] 6 Nov 2018SYMMETRY DEFECTS AND THEIR APPLICATION TO TOPOLOGICAL QUANTUM COMPUTING COLLEEN DELANEY AND ZHENGHAN WANG ABSTRACT . We describe the mathematical theory of topological quantu m computing with symmetry defects in the language of fusion categories and un itary representations. Sym- metry defects together with anyons are modeled by G-crossed braided extensions of uni- tary modular tensor categories. The algebraic data of these categories afford projective unitary representations of the braid group. Elements in the image of such representations correspond to quantum gates arising from exchanging anyons and symmetry defects in topological phases of matter with symmetry. We provide some small examples that high- light features of practical interest for quantum computing . In particular, symmetry defects show the potential to generate non-abelian statistics from abelian topological phases and to be used as as a tool to enlarge the set of quantum gates acces sible to an anyonic device. 1. I NTRODUCTION Topological order of a 2-dimensional topological phase of m atter can be modeled by a unitary modular tensor category C. If a 2-dimensional topological phase of matter has a symmetry group G, thenCinherits an action of G. A nontrivial action of GonCis a form of symmetry breaking, called weak symmetry breaking in [17] . It follows that the associ- ated topological defects, which manifest the topological p roperties of symmetry breaking patterns, provide a window to the world of symmetry and topol ogy. A practical reason to study such symmetry defects is as a new mechanism to genera te non-abelian statistics from a purely abelian topological phase. For example, we sho w that the symmetry defects arising from the electromagnetic duality of the Z2toric code behave almost identically to Majorana zero modes or Ising anyons. The symmetry defects we consider here are extrinsic point-l ike objects that are in gen- eral conﬁned. Like anyons, they lead to topological degener acy and braid group represen- tations, though only projectively. In this paper, we invest igate the mathematical modeling of symmetry defects and their application to topological qu antum computing. In Section 2 we review the algebraic description of symmetry defects via G-crossed extensions of unitary modular tensor categories following [2], where this mathematical model for topological order with symmetry was ﬁrst introduc ed. Some examples are pre- sented to illustrate the theory and are further developed in Section 3, where we describe the (projective) unitary representations associated to unita ryG-crossed extensions and discuss their interpretation as quantum gates for a topological qua ntum computer built by braiding symmetry defects and anyons. In Section 4, we give a mathemat ical analysis of an exam- ple of topological order with Z2symmetry that shows allowing projective measurement and braiding with symmetry defects can enlarge an anyonic ga te set with ﬁnite image in the unitary group to one that is universal for quantum comput ation. This feature of two layers of the Ising theory with a layer-exchange symmetry/parenleftig Ising(1)⊠Ising(1)/parenrightig× Z2was 1 2 COLLEEN DELANEY AND ZHENGHAN WANG proposed in

Question 1546 multiple-choice

In algebraic geometry and representation theory, theta functions and their zero loci play a fundamental role in understanding the structure of algebraic varieties and the behavior of associated maps between rings and fields. The interplay between Laurent expansions, decomposition theorems, and Zariski topology reveals deep properties of these varieties. Which statement best characterizes the implications of having 1 in the set S(β) for the subset NRβ within an algebraic variety endowed with a theta function structure? 1) NRβ must be equal to the entire variety. 2) NRβ contains only the identity element. 3) NRβ is always a finite set. 4) NRβ is guaranteed to be dense in the Zariski topology. 5) NRβ contains all points where Θβ does not vanish. 6) NRβ is contained within a proper Zariski-closed subset, restricting its generality. 7) NRβ is independent of the choice of S(β).

✓ Correct Answer:

The correct answer is 6) NRβ is contained within a proper Zariski-closed subset, restricting its generality..

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that C,,, e u?. Thus Σ C9tηXδ(Q)Xη(R) e uT. We argue similarly if ^(l) 73 S. §11.5. The function ^P Let P eAk. Then ίP: R2->k induces a map ΐP (x) 1: R'2 = i?2 (x) i?2 -> i22. If ^eiSs its image under iP® 1 is denoted by (0|X = P). If P = ^>(a?), then (β(X9 Y)\X = P) is just the Laurent series 0(#, Y). We abbreviate ^i,...,i to λx and let ψ be the element λι(XY)λι(XY~1) of i?2 For PeAfc let ΘP — (ψ\X = P). #P, like ΐPJ is determined up to multiplication by a unit in 0. If 0eR2 and QGA, we say that Θ{Q) = 0 if ΐe(0) = 0. Note that θ(φ(x)) = 0 if and only if 0(a?) = 0. We shall need a simple result, Propo- sition II.5.2, concerning the zeroes of ΘP, which follows from: LEMMA II.5.1. ΘP(Q) = 0i/ and only if either λ\(PQ) = 0 or λ\{PQ~ι) = 0. Proof. (P, Q) and (PQ, PQ"1) are in A'k = Ak x Ak and so give homomorphisms R{2) -> k. i{P>Q) = iP®iQ and i{PQiPQ-X)g{X, Y) — iiP}Q)θ(XY, XY~ι). Thus: and the result follows. https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press ABELIAN VARIETIES 87 PROPOSITION II.5.2. Suppose P,Q,Re Ak and ΘP{R) = 0. Then, either ΘPQ-X(QR) = 0 or θPQ {0,2} and each Ca is a unit. Proof. Apply iP ® 1 to both sides of Theorem II.4.7. PROPOSITION II.5.4. The reduction of ΘP(Y) is a non-zero polynomial in Yj and Yj1 (1 < j < g), which does not involve Yό or Yj1 if j e S(P). Proof. If a: {1,2, . , g) -»{0,2} is chosen so that or1®) = S(P), then -X2-1XP) is a unit. So by Proposition II.5.3 ΘP Φ 0. Suppose now j e S(P). Then, if as = 0, 5β does not involve Y3 or Yj1 while if as = 2, Z2_α(P) zJί. The result follows. §11.6. The decomposition theorem Throughout this section we assume k algebraically closed. Our goal is the following "decomposition theorem": Suppose PeAk. Then P = QJ2 where Q — φ(z, 1, , 1) for some z e fc* and 1 g S(iu). We begin the proof with a criterion which guarantees that 1 g S(JB). Suppose i? e Ak and (^2> , ^ff) e (ί*)^"1. We say that (ΰ2, , %) is in ΛΓβ if there exists u — (u19 - - -,Ug) eU° such that ut lifts ΰι for i > 1, and θR(u) = 0. PROPOSITION II.6.1. // leS(β), tfee^i iV^ is contained in a proper Zariski-closed subset of (k*)g~ι. Proof. Let ΘB be the reduction of ΘR. By Proposition II.5.4, ΘR is a non-zero polynomial in Yό and Γj1 for /> 1. If (ΰ2, - - ,ΰg) eNR then We next derive some simple results on the zeroes of power

Question 1547 multiple-choice

Bilinear equations play a central role in integrable systems and representation theory, particularly through their connection with τ-functions and quantum groups. The construction of these equations often involves advanced algebraic tools such as intertwining operators, Verma modules, and comultiplication in Hopf algebras. Which mathematical feature ensures that the set of solutions to bilinear equations involving τ-functions remains equally large for the quantum group SLq(2) when the deformation parameter q is generic, compared to the classical case q = 1? 1) The presence of non-trivial central extensions in quantum groups 2) The robustness of algebraic structures under deformation of q 3) The replacement of finite-dimensional representations with infinite-dimensional ones 4) The use of vertex operators solely associated with the trivial representation 5) The restriction to only classical symmetries in representation theory 6) The absence of comultiplication in Hopf algebras 7) The exclusion of Verma modules from the construction

✓ Correct Answer:

The correct answer is 2) The robustness of algebraic structures under deformation of q.

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the form (1). 3 Section 2 describes the generic procedure to deduce bilinear relatio ns. Explicit examples of these equations for G=SLq(2) are considered in Section 3. Section 4 demonstrates that, despite the possible subtlety of the notion of “group element” for q uantum groups, the set of solutions to bilinear equation for SLq(2) is as big for q/ne}ationslash= 1 as it is for q= 1. Section 5 describes application of the general procedure to collection of all f undamental representations ofG=SL(N). The corresponding equations are very close to those of the sta ndard Toda- lattice hierarchy. It also contains some comments on quantum case . 2 From intertwining operators to bilinear equations We suggest the following construction in terms of intertwining opera tors as a general source of bilinear equations for the τ-function (1). One can easily recognize the standard free- fermion derivation of Hirota equations for KP/Toda τ-functions as a particular example (for levelk= 1 Kac-Moody algebras ˆGk=1,Vis a fundamental representation, Wis the sim- plest fundamental representation corresponding to the very lef t root of the Dynkin diagram). Construction below involves a lot of arbitrariness. In order to make the consideration more transparent, we formulate our construction explicitly for ﬁnite-d imensional Lie algebras and theirq-counterparts. In the case of ( q)-Aﬃne algebras this actually coincides with that used in [8, 9]. Bilinearequationswhichwearegoingtoderivearerelating τ-functions(1)forfourdiﬀerent Verma modules V,ˆV, V′,ˆV′. GivenV, V′, every allowed choice of ˆV,ˆV′provides a separate set of bilinear identities. Of course, not all of these sets are actua lly independent and can be parametrizedbysourcemodules VandV′andbyaweightofﬁnite-dimensionalrepresentation. Also diﬀerent choices of positive root systems and their ordering in ( 1) provides equations in somewhat diﬀerent forms. A more invariant description of the minim al set of bilinear equations for given Gwould be clearly interesting to ﬁnd. 1. Our starting point is embedding of Verma module ˆVinto the tensor product V⊗W, 4 whereWis some irreducible ﬁnite-dimensional representation of G.2OnceVandWare speciﬁed, there is only ﬁnite number of choices for ˆV. Now we deﬁne right vertex operator of the W-type as homomorphism of G-modules: ER:ˆV−→V⊗W. (2) This intertwining operator can be explicitly continued to the whole rep resentation once this is constructed for its vacuum (highest-weight) state: ˆV=/braceleftigg |nα/an}bracketri}htˆV=/productdisplay α>0(∆(T−α)nα|0/an}bracketri}htˆV/bracerightigg , (3) where comultiplication ∆ provides the action of Gon the tensor product of representations3, and |0/an}bracketri}htˆV= /summationdisplay {pα,iα}A{pα,iα}/parenleftigg/productdisplay α>0(T−α)pα⊗(T−α)iα/parenrightigg |0/an}bracketri}htV⊗|0/an}bracketri}htW. (4) For ﬁnite-dimensional W’s, this provides every |nα/an}bracketri}htˆVin a form of ﬁnitesums of states |mα/an}bracketri}htV with coeﬃcients, taking values in elements of W. 2. The next step is to take another triple, deﬁning left vertex oper ator4, ¯E′ L:ˆV′−→W′⊗V′, (5) such that the product W⊗W′contains unitrepresentation of G. The projection to this unit representation π:W⊗W′−→I (6) is explicitly provided by multiplication of any element of W⊗W′by π=W/an}bracketle{t0|⊗W′/an}bracketle{t0| /summationdisplay {iα,i′α}π{iα,i′ α}/parenleftigg/productdisplay α>0(T+α)iα⊗(T+α)i′ α/parenrightigg  (7) 2In thecaseofAﬃne algebra,oneshould useevaluationrepresenta tion–zerochargerepresentationinduced from ﬁnite-dimensional one – see the deﬁnition of vertex operator in [8]. 3For Lie algebra it is just ∆( T) =T⊗I+I⊗T. 4Note the change of ordering at the r.h.s., this is slightly

Question 1548 multiple-choice

Quantum algorithms are increasingly being explored to perform numerical differentiation and integration, tasks that are computationally intensive for classical computers. Spectral methods and efficient quantum circuit design play a crucial role in advancing these algorithms for scientific computing over complex domains. Which of the following innovations specifically enables efficient element-wise multiplication in quantum algorithms for numerical integration? 1) Use of Grover's search for locating integration points 2) Implementation of quantum phase estimation for function evaluation 3) Application of amplitude amplification to increase measurement accuracy 4) Development of a new encoding technique for function values 5) Utilization of quantum error correction for stability 6) Deployment of classical-to-quantum data conversion circuits 7) Introduction of quantum teleportation for data transfer

✓ Correct Answer:

The correct answer is 4) Development of a new encoding technique for function values.

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Title: A spectral quantum algorithm for numerical differentiation and integration Year: 2025 Paper ID: f1f3d2dd8197c29d9a2ef7d70d35d5ae3721ad14 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/f1f3d2dd8197c29d9a2ef7d70d35d5ae3721ad14 Abstract: This paper presents low-complexity quantum algorithms for numerical differentiation and integration in partially bounded domains. The algorithms are based on a spectral approach that allows taking advantage of the computationally efficient quantum Fourier transform algorithm. Full derivations and gate-level circuit examples are presented. As part of the integration algorithms, we also develop an encoding technique to efficiently perform element-wise multiplication operations and unitary operators to perform partial summations of information encoded in state amplitudes.

Question 1549 multiple-choice

In quantum information theory and representation theory, efficient algorithms for the Schur transform are essential for decomposing quantum states into irreducible components, which is crucial for quantum protocols and symmetry-based computations. These algorithms frequently leverage combinatorial structures like Standard Young Tableaux and advanced quantum operations. Which step in the DualSchur algorithm serves to block diagonalize a quantum register into permutation modules before further decomposition into irreducible representations of symmetric or unitary groups? 1) Converting quantum basis entries to Gelfand-Tsetlin patterns 2) Applying the RSK correspondence to tableau pairs 3) Mapping quantum register entries to type and transversal elements 4) Computing Manhattan distances between tableau entries 5) Performing a quantum Fourier transform on permutation modules (QFTPermMod) 6) Labeling output registers by irreducible representation indices 7) Inductive proof of operator vanishing for consecutive tableau rows

✓ Correct Answer:

The correct answer is 5) Performing a quantum Fourier transform on permutation modules (QFTPermMod).

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SA|T/angbracketright= 0. Now, suppose that iandjare in the same column and in consecutive rows. Consider the element k, wherekis the largest element between iandjsuch thatkis in a diﬀerent row fromjand all elements between kandjare in the same row as j. For example, if j−1 is in a diﬀerent row fromj, thenk=j−1. This, in particular, means that kandk+ 1 are in diﬀerent rows. If they are in the same column, then we are done. So assume that they are not in the same column. Then we have Let (k,k+ 1) be a transposition and let |(k,k+ 1)T/angbracketrightbe the SYT with kandk+ 1 interchanged. Since they are not in the same row or column, ( k,k+ 1)Tis also a SYT. We have (k,k+ 1)|T/angbracketright=aT k|T/angbracketright+bT k|(k,k+ 1)T/angbracketright, (3.12) (k,k+ 1)|(k,k+ 1)T/angbracketright=bT k|T/angbracketright−aT k|(k,k+ 1)T/angbracketright, (3.13) 15 whereaT kis the inverse of the Manhattan distance in Tbetweenkandk+ 1 andbT k=/radicalBig 1−(aT k)2. Using these equations, we have |T/angbracketright= (Ae+B(k,k+ 1))|(k,k+ 1)T/angbracketright, (3.14) whereAk=1 bT kandBk=aT k bT k. Therefore, we have SA|T/angbracketright=SA(Ake+Bk(k,k+ 1))|(k,k+ 1)T/angbracketright= (Ake+Bk(k,k+ 1))SA|(k,k+ 1)T/angbracketright, (3.15) where the last equality follows from the fact that both eand (k,k+ 1) commute with SA. Now note that in (k,k+ 1)T,k+ 1 andk+ 2 are not in the same row or column. Continuing this process we get SA|T/angbracketright=(Ake+Bk(k,k+ 1))(Ak+1e+Bk+1(k+ 1,k+ 2))...(Aj−1e+Bj−1(j−1,j)) SA|(j−1,j)...(k,k+ 1)T/angbracketright. (3.16) It is easy to see that |(j−1,j)...(k,k+ 1)T/angbracketrightis a SYT where iandj−1 are in the same column and consecutive rows. By the induction hypothesis, we have SA|(j−1,j)...(k,k+ 1)T/angbracketright= 0. 4 Dual algorithm for the Schur transform We are now ready to describe our dual algorithm for the Schur transform using the above tools. It involves essentially two main steps. The rst is a block diagonalization into permutation modules and the second is a block diagonalization of each permutation module into irreps using the above transform. The algorithm is as follows. DualSchur (n,d,/epsilon1 ) Inputs: A quantum register Awith the computational basis given by n-tuples|e1,...,en/angbracketright, where ej∈[d]. Outputs: Quantum registers |λ,i,j/angbracketright, whereλis the irrep label of the symmetric or unitary groups, iis the irrep label of the symmetric group in the Young orthonormal basis and jis the irrep register of the unitary group in the Gelfand-Tsetlin basis. Runtime: O(poly (n,logd,log/epsilon1−1)). 1. Map the entries of any basis vector that are greater than nto entries inside nand keep track of this mapping. For example, |e/angbracketright=|e1,...,en/angbracketrightis now|pe,˜e/angbracketright, where ˜eis a vector with entries in [n]andpeis the map. 2. Convert|˜e/angbracketrightto|T,t/angbracketright, whereTis the type and tis the transversal element of the subgroup YTinSn. 3. Conditioned on T, apply QFTPermMod to|t/angbracketrightand obtain the basis |λ,i,j/angbracketright, wherejis an SSYT with entries in [d]andiis an SYT. 4. The basis at this point is |λ,i,(T,j)/angbracketright. Use RSK to convert the pair (T,j)to a SSYT as described in section 2.4. Theorem 3. Given annfold tensor product of ddimensional Hilbert spaces and accuracy /epsilon1, the quantum algorithm DualSchur runs in time O(poly logd,n,log 1//epsilon1) and performs the strong Schur transform i.e., performs the change of basis from the computational basis to

Question 1550 multiple-choice

Quantum Fourier Transform (QFT) is a fundamental quantum algorithm with applications in phase estimation and quantum simulation. Radix-d generalization extends QFT beyond binary qubits to qudits, enhancing its versatility for higher-dimensional quantum systems. Which mathematical operation is essential for decomposing the DFT matrices in radix-d Quantum Fourier Transform circuits, allowing efficient recursive construction of large QFT circuits from smaller components? 1) Kronecker product 2) Hadamard product 3) Tensor contraction 4) Singular value decomposition 5) Matrix inversion 6) Gram-Schmidt orthogonalization 7) Jordan canonical form

✓ Correct Answer:

The correct answer is 1) Kronecker product.

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choice gives the same QFT circuit that is dis cussed in [14, section 5.1]. The gate count of for each type of gate used in a QFT is summariz ed in Table 1. The total number of elementary gates required to implement the QFT is O(n2), which deﬁnes the complexity of the QFT algorithm. It represents an exponential improvement over the O(2nn) complexity of the FFT algorithm. Table 1: Gate count for the QFT circuit on nqubits. Matrix Count Gate type Pn ⌊3n/2⌋CNOT A(0) n···A(n−1) nn Hadamard n(n−1)/2 controlled- R 6 Generalization to Radix- dQuantum Fourier transform The QFT decomposition presented in section 4 can be easily ge neralized for DFT matrices of dimension dn×dn for any integer d>2. The generalization relies on a radix- dFFT decomposition and the base- drepresentation of an integerj j= [j1j2···jn−1jn] =j1·dn−1+j2·dn−2+···+jn−1·d1+jn·d0, whereji={0,1,...,d−1}fori= 1,...,n. Hence, Lemmas 1 and 2 generalize in a straightforward manne r to base-d. 6.1 Radix- ddecomposition of the discrete Fourier transform matrix In this section, we use An=Adnto denote matrices of dimension dn. Deﬁnition 10. DeﬁneΩn∈Cdn×dn,Rn∈Cd×das the following matrices: Ωn= Ωdn:= ω0 dn+1 ω1 dn+1 ... ωdn−1 dn+1 , R n= ω0 dn ω1 dn ... ωd−1 dn , whereωdi= exp(−2πı di). If the DFT matrix is of dimension dn×dn, then it can be factored in d×dblock partitioning, also known as the radix- dfactorization of the DFT matrix [18, 15]. This is summarized in the following two theorems which naturally extend the results of the radix-2 framework of sec tion 3.2. 13 Theorem 4 (See [18]) .LetF′ n=PnFnbe thedn×dnbase-dreversed DFT matrix. Then F′ nadmits the following factorization: F′ n= (Id⊗F′ n−1)(In−1⊕Ωn−1⊕···⊕Ωd−1 n−1)(Fd⊗In−1), (33) whereΩn−1is given by Deﬁnition 10 and Fdis thed×dDFT matrix deﬁned in Deﬁnition 7. Equation (33) essentially decomposes F′ ninto the Kronecker product of the dn−1×dn−1DFT matrix F′ n−1 and ad×dblock matrix with block size dn−1×dn−1. The full radix- dfactorization of the DFT matrix follows from Theorem 4 via the same simple induction argument used in the proof of Theorem 2. Theorem 5. Thedn×dnDFT matrix Fncan be factored as: Fn=PnF′ n=PnA(0) n...A(n−1) n, (34) where, fork= 0,...,n−1, A(k) n=In−k−1⊗Bk+1,andBk+1= (Ik⊕Ωk⊕···⊕Ωd−1 k)(F1⊗Ik). (35) withΩkdeﬁned in Deﬁnition 10 and F1=Fdthed×dDFT matrix given by Deﬁnition 7. Deﬁnition 11. The base-dreversal permutation matrix Pn∈Cdn×dnis deﬁned as the permutation matrix which satisﬁes: Pn(v1⊗···⊗vn) =vn⊗···⊗v1, wherevk∈Cd, fork= 1,...,n. Deﬁnition 11 shows that the permutation matrix can be implem ented in ⌊n/2⌋SWAPs ofddimensional vectors in a Kronecker product of nsuch vectors. 6.2 Decomposition of the diagonal matrix In this section we decompose the diagonal matrix Ik⊕Ωk⊕ ··· ⊕Ωd−1 kintokcontrolled-Runitaries, with R deﬁned in Deﬁnition 10, following an analogous strategy to s ection 4. We start again with a decomposition of Ωn as a Kronecker product of d×d Rimatrices. Lemma 6. LetRnbe deﬁned by Deﬁnition 10. Then Rdj n=Rn−j, forj∈N. Proof.The proof proceeds analogously to Lemma 3. Lemma 7. LetΩnandRnbe deﬁned by Deﬁnition 10. Then Ωn=R2⊗R3⊗···⊗Rn⊗Rn+1. Proof.The proofproceeds analogously toLemma4, replacingthebin aryrepresentation with base- drepresentation and using Lemma 6. The generalization of Theorem 3 to the decomposition

Question 1551 multiple-choice

In the study of finite p-groups, particularly those arising from central extensions and cohomological constructions, matrix methods and quadratic residue analysis play a significant role in group classification and understanding subgroup structure. Connections between linear algebra and group theory are especially evident in determining isomorphism classes. Which condition ensures that two finite p-groups with characteristic matrices over Fp are isomorphic? 1) Their characteristic matrices are conjugate by an invertible matrix over Fp. 2) Their characteristic matrices have the same determinant modulo p. 3) Their characteristic matrices differ by addition of a scalar multiple of the identity. 4) Their characteristic matrices have equal trace modulo p. 5) Their characteristic matrices are diagonalizable over Fp. 6) Their characteristic matrices are symmetric. 7) Their characteristic matrices have all entries as quadratic residues modulo p.

✓ Correct Answer:

The correct answer is 1) Their characteristic matrices are conjugate by an invertible matrix over Fp..

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0. Conversely, if w11=w22=w12+w21= 0, then, by using above argument, N∈ A2 andMi∈ A2. HenceImin/ne}ationslash= 1. 25 (5) Ifn > m= 1 and p >2, then 1 = [ a,bp] =cp. IfImin/ne}ationslash= 1, then Mi/ne}ationslash∈ A1. By calculation, [ abi,c] =y−1xiand (abi)pn=apn=xw11yw12. Hence, ∀i,w11=−iw12. It follows that w11=w12= 0. Conversely, if w11=w12= 0, then, by using the above argument, N/ne}ationslash∈ A1and Mi/ne}ationslash∈ A1. HenceImin/ne}ationslash= 1. (6) Ifn=m= 1 and p >3, then 1 = [ a,bp] =cp. IfImax/ne}ationslash= 2, then N∈ A1 andMi∈ A1. Sincey∈ /an}bracketle{tb,c/an}bracketri}ht,w22/ne}ationslash= 0. By calculation, [ abi,c] =y−1xiand (abi)p= apbip=xw11+iw21yw12+iw22. Hence, ∀i,w11+iw21/ne}ationslash=−iw12−i2w22. It follows that (w12+w21)2−4w11w22is a quadratic non-residue modular p. Conversely, if ( w12+w21)2−4w11w22/ne}ationslash∈F2 p, then, by using the above argument, N∈ A1andMi∈ A1. HenceImax/ne}ationslash= 2. (7) Ifp >2,n= 2 and m= 1, then 1 = [ a,bp] =cp. By calculation, Mi∈ A1or A2. IfImax= 2, then N∈ A2. It follows that y∈ /an}bracketle{tb,c/an}bracketri}ht. Hencew22/ne}ationslash= 0. Conversely, if w22/ne}ationslash= 0, then N∈ A2andMi∈ A1orA2. HenceImax= 2. (8) Ifn=m= 2, then, by (1), Imin≥2. IfImax= 2, then N∈ A2andMi∈ A2. Since|G:/an}bracketle{tb,c/an}bracketri}ht|=p2, we have w22/ne}ationslash= 0. By calculation, [ c,abi] =x−iyand (abi)p2= xw11+iw21yw12+iw22. Since|G:/an}bracketle{tc,abi/an}bracketri}ht|=p2, Equation w11+iw21=−i(w12+iw22) aboutihas no solution. Hence ( w12+w21)2−4w11w22is a quadratic non-residue modular p. Conversely, if ( w12+w21)2−4w11w22is a quadratic non-residue modular p, then, by the above argument, N∈ A2andMi/ne}ationslash∈ A2. HenceImax= 2. (9) Ifn=m= 2, then, by (1), Imin≥2. IfImax= 2, then N∈ A2andMi∈ A2. LetNj=/an}bracketle{tajb,ap,c,x,y/an}bracketri}ht. ThenN0=NandNj=Mj−1forj/ne}ationslash= 0. Since M0∈ A2, |G:/an}bracketle{ta,c/an}bracketri}ht|=p2and hence w11/ne}ationslash= 0. Let A=/an}bracketle{tap,bp,c,x/an}bracketri}ht,Br=/an}bracketle{tajbcr,ap,cp,x/an}bracketri}htand Cst=/an}bracketle{tajbasp,catp,cp,x/an}bracketri}htwhere 0≤r,s,t≤p−1. Then A,BrandCstare all maximal subgroups of Nj. First of all, Ais the unique abelian maximal subgroup of Nj. Since w11/ne}ationslash= 0, Br∈ A1. Since [ catp,ajbasp] =x−1yt+jand (ajbasp)p2=xjw11+w21yjw12+w22and (catp)p=xtw11ytw12+1,Cst∈ A1if and only if the following equation set about jandt has no solution. /braceleftBigg jw12+w22=−(j+t)(jw11+w21) (6.1.1) tw12+1 =−(j+t)tw11 (6.1.2)(6.1) Adding up (6.1.1) and (6.1.2), we have w11(j+t)2+(w12+w21)(j+t)+w22+1 = 0 (6 .1.3) If Equation (6.1.3) about j+thas no solution, then ( w12+w21)2−4w11(w22+1) is a quadratic non-residue modular p. Hence (i) holds. If Equation (6.1.3) about j+thas a uniquesolution, then ( w12+w21)2−4w11(w22+ 1) = 0. In this case, Cst∈ A1if and only if Equation (6.1.2) about thas no solution. 26 That is, j+t=−w−1 11w12 (6.1.4) By (6.1.3) and (6.1.4), w21=w11(w22+1) and hence (ii) holds. If Equation (6.1.3) has two solutions, then there exists a so lution such that w11(j+ t)+w12/ne}ationslash= 0. It follows that Equation set (6.1) has a solution, and hen ce there exists j,tsuch that Cst/ne}ationslash∈ A1. Conversely, if (i) or (ii) holds, then, by using the above arg ument, it is easy to see that Equation set (6.1) has no solution. Hence Cst∈ A1andImax= 2. /square Theorem 6.2. Suppose that Gand¯Gare ﬁnite p-groups such that Φ(G′)≤G3∼=C2 p, G3≤Z(G)andG/G3∼=Mp(n,m,1), wherep >2,n≥m≥2. Let two characteristic matrices of Gand¯Gbew(G) = (wij)andw(¯G) = (¯wij)respectively. Then G∼=¯G if and only if there exists Y=/parenleftbiggy11y12 y21pn−my22/parenrightbigg , an invertible matrix over Fp, such thatw(¯G) =Y1w(G)Ytandv(¯G) =Yv(G), whereY1=/parenleftbiggy11y12pn−m y21y22/parenrightbigg . ProofSupposethat a,band ¯a,¯bare two set of characteristic generators of w(G),v(G) andw(¯G),v(¯G). Letθbe an isomorphism from ¯GontoG. We have Φ( ¯G)θ= Φ(G)

Question 1552 multiple-choice

Quantum mechanically derived force fields (QMD-FFs) aim to achieve high accuracy in molecular simulations by leveraging data from advanced electronic structure methods. The choice of density functional and benchmarking protocols critically affects their ability to reproduce experimental condensed phase properties across different molecules. Which benchmarking approach provides the most robust validation for selecting an appropriate density functional when parameterizing force fields for condensed phase simulations? 1) Comparing functionals using only experimental density values of bulk liquids 2) Evaluating functionals using a single optimized dimer geometry 3) Benchmarking functionals against high-level CCSD@cbs data over many thermodynamically accessible dimer geometries 4) Using molecular orbital energies from Hartree-Fock calculations 5) Matching force field parameters to vibrational spectra alone 6) Assessing functionals by their computational cost and speed 7) Validating functionals by reproducing gas-phase dipole moments

✓ Correct Answer:

The correct answer is 3) Benchmarking functionals against high-level CCSD@cbs data over many thermodynamically accessible dimer geometries.

📚 Reference Text:

In the light of the good results recently achieved with the benzene liq- uid,40this preliminary parameterization was carried out adoptin g the same combination 12 of DFT functional (CAM-B3LYP), basis set (6-311+G(2d,2p)) an d dispersion correction (D3 Grimme’s correction78) adopted therein to describe the QM IPES. As far as the C2is concerned, dichloromethane condensed phase was reprodu ced rather well by the PickyQMD-FF. For instance, NPT simulations at 298 K and 1 atm yielde d an average density of 1297 kg/m3and a vaporization enthalpy of 28.5 kJ/mol, which well compa res with the experimental79values (1327 kg/m3and 28.8 kJ/mol), and increases the accu- racy17,59of the empirical FFs (1258 or 1200 kg/m3and 26.5 or 23.4 kJ/mol with GAFF or OPLS). Unfortunately, a very diﬀerent situation raised during the p arameterization of chloro- form (C1). In fact, after the ﬁfth parameterization cycle, the MC den sity was stabilized around 1260 kg/m3, against an experimental80value of 1479 kg/m3, showing an error ( > 15%) even worse ( ∼8%) than the one found with empirical FFs. This could be inter preted as a failure of the chosen functional in describing the chlor oform IPES. Once again,40the problem is then shifted on assessing the quality of the IPES g iven by a certain func- tional. A straightward route to solve this dilemma could sta nd in carrying out a number ofPicky parameterizations based on diﬀerent functionals and choos e the one giving the best agreement with the target properties. This procedu re has however at least two drawbacks. First, it is very time-consuming, given the plet hora of DFT functionals avail- able and the fact that the whole parameterization protocol h as to be repeated for each of the benchmarked ones. Next, and probably more important fr om our point of view, a FF validation against experimental measures would somehow renege on the intention of maintaining the a priorinature of QMD-FFs, that could instead be exploited when such experimental information is scarce, hardly available or to tally missing. An alternative route can be devised though, resorting to the D FT benchmarking pro- posed in Ref. [40]. Therein, the performance of diﬀerent DFT functionals wit h respect to reference CCSD(T)@cbs data was connected to the accuracy of the macroscopic bulk phase properties predicted by the MD simulations carried ou t with QMD-FFs parame- terized over the tested functionals. The results reported i n Ref. [40] showed that a sound validation of a given functional should not be assessed by co mparing with higher level data for only one or few dimer arrangements, but instead care fully evaluated over a sig- 13 niﬁcant number of geometries, possibly sampled taking into account their thermodynamic accessibility. Accordingly, a CCSD(T)@cbs database was bui lt40therein for the investi- gated molecule, by computing the interaction energies of th e 50 dimers sampled in the ﬁrstPickycycle. Following this idea, a Pickysampling is here performed on the ﬁnal conﬁguration of the initial MC run (see Section 2.1) of four t est molecules, chosen among Figure

Question 1553 multiple-choice

Deliberate self-harm (DSH) among adolescents is a significant global mental health concern, frequently associated with complex psychological and social factors. Prevention strategies are often categorized into primary, secondary, and tertiary levels to address the wide spectrum of needs. Which prevention strategy for adolescent deliberate self-harm primarily focuses on building resilience and psychosocial assets to reduce the initial likelihood of self-harming behaviors? 1) Primary prevention 2) Tertiary prevention 3) Secondary prevention 4) Crisis intervention 5) Pharmacological treatment 6) Acute medical care 7) Postvention support

✓ Correct Answer:

The correct answer is 1) Primary prevention.

📚 Reference Text:

- tivistic approach represents the dominant research para - digm in medical and social science research, its limita - tions should be duly acknowledged. Methodologically, it would be illuminating if qualitative research adopting constructionist orientations could be carried out in fu - ture. Fifth, comparative studies should be conducted to examine deliberate self-harm in different communities. As commented by Madge et. al. , “there is little reliable comparative international information on its extent or characteristics” (p.667).93 In view of the rising suicide figures in different non-Western communities such as Hong Kong,94 comparisons regarding deliberate self- harm among different communities would be important. Finally, research on intervention programs for deliber - ate self-harm should be carried out. For tertiary preven - tion, relative efficacy of different intervention approach - es should be examined. For secondary prevention, iden - tification of adolescents with self-harming behavior that deserves professional attention at an earlier stage is im - portant. This requires the development of screening in - struments with good psychometric properties. Similarly, primary prevention programs that reduce young people’s likelihood of engaging in deliberate self-harm is of para - mount importance. Based on the perspective of positive youth development, it is argued that strengthening young people’s psychosocial competence and developmental assets in young people would reduce their likelihood to engage in deliberate self-harm behavior.95, 96 Summary Deliberate self-harm is a widespread yet often hidden problem in adolescents, especially females, which shows both similarities and differences internationally.31 Two main characteristics of DSH noted are negative emotion and self-derogation. Reasons for non-suicidal self-injury include efforts to resist suicidal thoughts, expression of self-anger or disgust, resolve times of dissociation, influ - ence others, or seek help from others.42, 43 There is an as - sociation of drug abuse (including alcohol abuse) and eating disorders with self-injurious behavior.48 There are various subgroups of self-cutters and most are not at high risk for suicide.97 However, all acts of self-injury should be taken seriously by healthcare professionals and comprehensive therapy offered to any adolescent with a history of self-injury or DSH. Programs for sui - cide prevention should be implemented for youth who are at risk for suicide, including those who self-injure themselves.2, 3, 22, 98 -100 More research on deliberate self- harm should be carried out with reference to the research gaps identified in this paper. References 1. Greydanus DE, Patel DR, Pratt HD, editors: Essential Adolescent Medicine. New York: McGraw-Hill; 2006 2. Greydanus DE, Calles Jr. JL: Suicide in children and adolescents. Prim Care: Clinics in Office Practice 2007; 34: 259－274 3. Greydanus DE, Ryan SA, Calles Jr. JL: Adolescent Suicide. American Academy of Pediatrics PREP Audio 2007; 2 4. Mann JJ, Apter A, Bertolote J, Beautrais A, Currier D, Haas A, et al: Suicide prevention strategies: a systematic review. JAMA 2005; 294: 2064－2074 5. Barker G: What about boys: a literature review in the health and development of adolescent boys. WHO Department of Child and Adolescent Health and Development. Geneva: Switzerland; 2000 6. Brown P: Choosing

Question 1554 multiple-choice

The Hidden Subgroup Problem (HSP) is fundamental in quantum computing, serving as the framework for efficient quantum algorithms applied to algebraic and computational problems. Techniques such as coset sampling and Fourier sampling are central to quantum approaches for solving HSP in various group structures, including abelian, dihedral, and symmetric groups. Which aspect of the inductive approach to the Hidden Subgroup Problem enables the reduction of problems over complex groups to more tractable cases, specifically facilitating efficient oracle construction? 1) Utilizing classical brute-force search methods within simple groups 2) Applying weak Fourier sampling exclusively to abelian subgroups 3) Restricting subgroup analysis to only non-normal subgroups 4) Employing coset sampling limited to cyclic groups 5) Transforming the group structure to semidirect products 6) Ignoring factor groups during subgroup identification 7) Breaking down the problem via normal subgroups and factor groups to enable efficient oracle use

✓ Correct Answer:

The correct answer is 7) Breaking down the problem via normal subgroups and factor groups to enable efficient oracle use.

📚 Reference Text:

Title: The Hidden Subgroup Problem Year: 2010 Paper ID: 2a2c9f2e021778dfdede7563d82131b3c898e436 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/2a2c9f2e021778dfdede7563d82131b3c898e436 Abstract: We give an overview of the Hidden Subgroup Problem (HSP) as of July 2010, including new results discovered since the survey of arXiv:quant-ph/0411037v1. We recall how the problem provides a framework for efficient quantum algorithms and present the standard methods based on coset sampling. We study the Dihedral and Symmetric HSPs and how they relate to hard problems on lattices and graphs. Finally, we conclude with the known solutions and techniques, describe connections with efficient algorithms as well as miscellaneous variants of HSP. We also bring various contributions to the topic. We show that in theory, we can solve HSP over a given group inductively: the base case is solving HSP over its simple factor groups and the inductive step is building efficient oracles over a normal subgroup N and over the factor group G/N. We apply this analysis to the Dedekindian HSP to get an alternative abelian HSP algorithm based on a change of the underlying group. We also propose a quotient reduction by the normal group obtained via Weak Fourier Sampling. We compute the exact expression of Strong Fourier Sampling over the dihedral group, showing how the previous reduction is natural and matches the standard one. We also give a reduction of rigid graph isomorphism problem to HSP over the alternating group. For this group and other simple groups, we propose maximal subgroup reduction as a possible approach. We also analyse Regev's algorithm for the poly(n)-uniqueSVP, prove how the degree of the polynomial is related to the oracle complexity used and we suggest several variants.

Question 1555 multiple-choice

In quantum computing, circuit depth and gate efficiency are critical factors impacting the feasibility of algorithms on current hardware. Parity-based architectures such as the Lechner-Hauke-Zoller (LHZ) scheme enable complex quantum operations using only nearest-neighbor interactions. Which of the following best explains why the LHZ architecture can achieve circuit depth for quantum Fourier transform (QFT) circuits nearly matching ideal all-to-all connectivity, despite its restriction to nearest-neighbor interactions? 1) It implements all CNOT gates as simultaneous operations within each qubit register. 2) It uses additional physical qubits to compensate for limited connectivity. 3) Logical CPHASE gates are decomposed into single-qubit Rz rotations, allowing many operations to be merged and reducing depth. 4) The architecture leverages error-correcting codes to minimize gate requirements. 5) Only gates between crossings to adjacent lines contribute significantly to circuit depth, regardless of other constraints. 6) Quantum addition is performed in the computational basis, simplifying circuit structure. 7) CNOT chains are completely parallelized without any sequential constraints.

✓ Correct Answer:

The correct answer is 3) Logical CPHASE gates are decomposed into single-qubit Rz rotations, allowing many operations to be merged and reducing depth..

📚 Reference Text:

7). The corresponding circuit for a possible low-depth implementation is shown in Fig. 3. The CNOT chains for implementing the logical Hadamard gates can be parallelized apart from a small contribution: Forthe logical qubit (0) (light blue line in the ﬁgure), the ﬁrst CNOT chain along the line, the Hadamard gate on the data qubit and the second chain until qubit (0 ,1) need to be performed before qubit (0 ,1) is free to be used in the next line. In general, for any line i, only the gates between the crossing to the previous line [qubit ( i−1,i)] and the crossing to the next line [qubit ( i,i+1)] contribute to the circuit depth. The necessary logical CPHASE gates appear in the circuit as single-qubit Rz rotations (marked with Z). Note that many of them can be merged into other Rzrotations and are therefore not shown in Fig. 3. Following that procedure, the operations on the ﬁrst qubit (light blue) block n+3 steps. The second and the (n−1)th ones take ﬁve steps and the last one (dark blue) n+2, because there is no CPHASE gate required. All otheroperations in between occupy six time steps. This construction results in a total circuit depth of 8 n−9. A comparison of the required resources in the parity archi- tecture and in the standard gate model is given in Table II. The circuit depth for the QFT in the LHZ scheme is up toa constant the same as in an all-to-all connected setup. Thisis remarkable in that the LHZ scheme can be implementedin current experiments and does not rely on impracticablehardware requirements. B. Quantum addition Another prominent problem in various quantum algorithms is quantum addition as the basic building block of algebraicmanipulation of quantum registers. An efﬁcient circuit forthe addition of two quantum registers based on the QFT,proposed by Draper [ 33], performs the addition in the Fourier TABLE II. Comparison of the required resources for the QFT implemented on an all-to-all connected device, a square lattice with nearest-neighbor interactions (requiring SWAP operations), and par- ity mapped on a square lattice with nearest-neighbor interactions. The numbers for the gate model implementations are taken from Ref. [ 23], while the numbers for the LHZ encoding are analytically deduced. All operations have been decomposed into single-body rotations RxandRz,H,a n d CNOT gates. Resource all-to-all square lattice LHZ qubits nn1 2n(n+1) CNOT n(n−1)3 2n(n−1) 2 n(n−1) single-qubit n2n2 1 2n(n+3) total gates n(2n−1)1 2n(5n−3)1 2n(5n−1) circuit depth 8 n−10 10 n−13 8 n−9 042442-4 APPLICATIONS OF UNIVERSAL PARITY QUANTUM … PHYSICAL REVIEW A 106, 042442 (2022) FIG. 4. Implementation proposal for the quantum addition of two registers R1 (labeled with numbers) and R2 (labeled with letters) in the LHZ scheme. The embedding consists of three standard LHZschemes, two encoding the interactions within register R1 and R2, respectively, and one encoding the interregister interactions. The solid lines represent logical lines for register R1 and the dashedlines the logical lines for qubits in register R2. The qubits encoding interactions

Question 1556 multiple-choice

Quantum computational complexity is fundamental to understanding the efficiency of algorithms for problems involving hidden algebraic structures, such as polynomials and subgroups. Key distinctions exist between the query complexities of the Hidden Subgroup Problem (HSP) and its generalizations, such as the Hidden Symmetry Subgroup Problem (HSSP). When considering the affine group over a finite field Fq, what is the quantum query complexity lower bound for the Hidden Symmetry Subgroup Problem (HSSP), and which classical algorithm's complexity does it match? 1) Ω(log q), matching classical binary search 2) Ω(q), matching brute-force search 3) Ω(q^2), matching exhaustive enumeration 4) Ω(q^1/4), matching collision finding 5) Ω(1), matching constant-time lookup 6) Ω(q^1/2), matching Grover's search 7) Ω(q log q), matching radix sort

✓ Correct Answer:

The correct answer is 6) Ω(q^1/2), matching Grover's search.

📚 Reference Text:

HPPwas investigated by Childs, Schulman, and Vazirani [14]. They showed the following. Fact 3. Ifn≥2anddare constants, then for an 1−o(1)fraction of the hidden polynomials, HPP(Fq,n,d)has polylogarithmic query complexity. Here, like in the case of the HQPP, polynomials are determined up to constant terms and scalar factors. We are not aware of any results regarding the quantum computational complexity even in the univariate quadratic case. R¨ otteler showed that multivariate quadratic (not hidden) Boolean functions can be identiﬁed using a linear number of quantum queries [20]. For the HPGP, Decker, Draisma, and Wocjan [15] showed the following. Fact 4. (a)HPGP(Fq,n,d)can be reduced in polynomial time to HPGP(Fq,1,d)for every constant n. (b)For every dthere exists a ﬁnite set Edof primes such that if dis constant and the characteristic of Fqis not in Ed,t h e n HPGP(Fq,1,d)can be solved in quantum polynomial time. 3. A general reduction of the HSSP to the HSP. How much greater is the complexity of an HSSPcompared to the complexity of the related HSP?T oa n a l y z e this, we ﬁrst give a simple example, which shows that the query complexity of the HSSPcan be exponentially higher than th e query complexity of the related HSP. Then, more interestingly, we will establish a general condition on the group actionunder which the HSSPcan be reduced in polynomial time to the related HSP. 3.1. HSSP with exponential query complexity. While the quantum query complexity of the HSPis polylogarithmic in the size of the group, we show in this section that the query complexity of an HSSPc a nb ei nt h eo r d e ro f |G| 1/4.M o r e precisely, we show that Grover’s searc h problem can be reduced to some speciﬁc HSSP. For a prime power q,t h egeneral aﬃne group Aﬀqof invertible aﬃne transforma- tions over Fqis deﬁned as the semidirect product Fq/multicloserightF∗ q,w h e r eF∗ qdenotes the multi- plicative group of Fq. The natural action of Aﬀ qonFqis deﬁned as ( b,a)◦x=ax+b. For every c∈Fq, the stabilizer of cis the subgroup Hc={((1−a)c,a):a∈F∗ q}, which has two orbits, {c}and{d∈Fq:d/negationslash=c}, and thus Aﬀ qis 2-transitive. Clearly, Hcis a closed subgroup. We set H={Hc:c∈Fq}. HIDDEN SYMMETRY SUBGROUP PROBLEMS 1993 Proposition 3.1. The query complexity of HSSP(Aﬀq,Fq,◦,H)isΩ(q1/2). Proof. Grover’s search over Fqcan be trivially reduced to this HSSP. Indeed, if the oracle input is fc, deﬁned by fc(x)=δc,x,w h e r eδc,xis the Kronecker delta, then fchidesHcas symmetry subgroup. From any generator ( b,a)o fHcone recovers c simply by computing (1 −a)−1b. Hence, the query complexity of the HSSPis at least the query complexity Ω( q1/2) of Grover’s search [21]. 3.2. A reduction scheme of the HSSP to the HSP. In this section, we describe a rather natural framework for reducing the HSSPto the related HSP. Es- sentially, the same idea was used in [18] for reducing certain hidden shift problems to the HSP in the aﬃne group over prime ﬁelds. We assume that we are given a black-box function foverM,

Question 1557 multiple-choice

Quantum algorithms have shown notable advantages in solving combinatorial problems involving group structures, especially those relevant to cryptography and coding theory. Difference sets within abelian groups are foundational to constructing certain error-correcting codes and cryptographic primitives. Which type of difference set is constructed from quadratic residues in finite fields and is directly related to the quantum algorithm for the shifted Legendre function? 1) Paley difference set 2) Hadamard difference set 3) Singer difference set 4) Quadratic difference set 5) Reed-Solomon difference set 6) Walsh difference set 7) Galois difference set

✓ Correct Answer:

The correct answer is 1) Paley difference set.

📚 Reference Text:

Title: Quantum Algorithms for Abelian Difference Sets and Applications to Dihedral Hidden Subgroups Year: 2016 Paper ID: 1f0479fb6b4b68ea535539814d942c70ab8f47c3 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/1f0479fb6b4b68ea535539814d942c70ab8f47c3 Abstract: Difference sets are basic combinatorial structures that have applications in signal processing, coding theory, and cryptography. We consider the problem of identifying a shifted version of the characteristic function of a (known) difference set and present a general algorithm that can be used to tackle any hidden shift problem for any difference set in any abelian group. We discuss special cases of this framework which include a) Paley difference sets based on quadratic residues in finite fields which allow to recover the shifted Legendre function quantum algorithm, b) Hadamard difference sets which allow to recover the shifted bent function quantum algorithm, and c) Singer difference sets which allow us to define instances of the dihedral hidden subgroup problem which can be efficiently solved on a quantum computer.

Question 1558 multiple-choice

Finite-dimensional Hopf algebras and their Drinfeld doubles play a fundamental role in quantum algebra, especially in the study of integrals, cointegrals, and the properties of the antipode. Understanding the interplay between modulus, comodulus, and canonical elements is crucial for applications in representation theory and quantum topology. In the Drinfeld double D of a finite-dimensional Hopf algebra A, which statement best describes the properties of the canonical element p = μ·S⁻¹(x), where μ is a left integral and x is a left cointegral? 1) p is both a left and right cointegral, and the modulus α_D of D is trivial (α_D = 1) 2) p is only a left cointegral, and the modulus α_D of D has finite order greater than one 3) p is only a right cointegral, and the modulus α_D of D is a nontrivial root of unity 4) p is neither a left nor right cointegral, and α_D equals the comodulus a 5) p is a left cointegral, but α_D is nontrivial and different from one 6) p is both a left and right cointegral, but the modulus α_D is a nontrivial group-like element 7) p is a right cointegral, and α_D = a⁻¹, the inverse of the comodulus

✓ Correct Answer:

The correct answer is 1) p is both a left and right cointegral, and the modulus α_D of D is trivial (α_D = 1).

📚 Reference Text:

the coinvariance of the coaction and /tildewidest−→is given by the left action on M. It is instructive to apply this to the situation where M=D∗with actions h.λ:=λ⊳S(h) and coaction δ(λ) =λ⊗1∆∈End(D) =D⊗D∗. The isomorphism J⊗D/tildewidest−→D∗then implies that J={λ:λ⊗1∆(y) = 1λ(y)}- the space ofright integrals - is one dimensional and every nonzero element induces a nondegene rate bilinear form. Analogous statements are found if we use left actions or consider Hopf modules of the dual algebra. Let us ﬁx once and for all a left integra lµand a left cointegral xwith the properties 1⊗µ∆(h) = 1µ(h), hx =ǫ(h)x,andµ(x) = 1. (2.22) As in [Rd] we use notations for the following isomorphisms: βl, βr:D/tildewidest−→D∗with βl(h) =µ⊳handβr(h) =h⊲µ (2.23) βl,βr:D∗/tildewidest−→Dwith βl(λ) =x ↽ λandβr(λ) =λ ⇀ x . (2.24) 13 They intertwine the right and left actions as in βl(kh) =βl(k)⊳handβr(kh) =k⊲βr(h). (2.25) It is obvious that xhis again a left cointegral for any h∈D. Hence by uniqueness of x, we ﬁndα∈Ch(D) and for the dual situation a∈G(D) such that α(h)x=xhandµ⊗1∆(h) =aµ(h). (2.26) SinceDis ﬁnite dimensional, both the modulusαand thecomodulus aare of ﬁnite order and ω:=α(a) (2.27) is a root of unity. Note, that in the following we use the opposite comu ltiplication for D∗ so that, e.g., S−1=S∗(=γin [Rd]). The antipode acts on the integrals as follows: S−1(µ) =a⊲µ=ωS(µ) =ωµ⊳a (2.28) S(x) =α ⇀ x=ωS−1(x) =ωx ↽ α (2.29) Thecompositionsofisomorphismsin(2.23)and(2.24)aregivenbythe followingformulae. Each one can be given on D∗or the adjoint one on Dusingβ∗ l=βr: βrβl(λ) =S−1(λ) βrβl(h) =S(h) (2.30) βrβr(λ) =S(a⊲λ) βlβl(h) =S−1(h)·a (2.31) βlβl(λ) =S(α−1λ) βrβr(h) =S−1(α ⇀ h) (2.32) βlβr(λ) =α·S−1(λ⊳a)βlβr(h) =a·S(h ↽ α) (2.33) From (2.28)-(2.30) we can derive further useful relations betwee n adjoints: βl(S(h)) =ωa−1⊲S(βr(h)) and βl(S−1(h)) =a−1⊲S−1(βr(h)) (2.34) βl(α ⇀ h) =βr(S2(h)) resp µ((α ⇀ k)h) =µ(hS2(k)) (2.35) βl(S(λ)) =α−1⇀ S(βr(λ)) and βl(S−1(λ)) =ωα−1⇀ S−1(λ).(2.36) βr(S−2(λ)) =βl(a⊲λ) resp S2⊗1∆′(x) = ∆(x)(a⊗1) (2.37) Combining these identities we ﬁnd Radfords formula for S4: S4=ad∗(α)◦ad(a−1) (2.38) Since in a double D(A) the adjoint action of G(Ao) coincides with the coadjoint action ofCh(A) onAwe ﬁnd from that (2.38) and the corresponding equation on Aothat the group like element α⊗a−1implements S4onD(A). The same is true for the element g deﬁned in (2.17). In fact we have the following result of Drinfeld for d oubles: 14 Proposition 4 [Dr1] g=α.a−1 3.) Integrals and Canonical Traces of Doubles : In the construction of representations of the modular group the integrals of the deﬁning algebra play an impo rtant role. The integralµDand the cointegral xDof a double D(A) clearly have to be related to the integrals and cointegrals of A. In this section we are also interested in ﬁnding the moduli αDandaD. Comparing Proposition 4 to (2.38) we are led to expect that αD= 1 andaD is the same as g. We shall prove triviality of the comodulus ﬁrst: Proposition 5 For left integrals µandxas in (2.22) deﬁne the canonical element in D(A)byp=µ.S−1(x). Then 1.S(p) =p 2.pis both a right and left cointegral in D(A)andαD= 1 3. LetP∈End(A)be the image of punder the map

Question 1559 multiple-choice

Quantum algorithms for lattice problems often rely on properties of lattices, sublattices, and the ability to distinguish quantum states associated with different structures. Careful parameter selection and analysis of inner product bounds are crucial for the success of these algorithms, especially when tackling the Hidden Subgroup Problem (HSP). Which parameter choice ensures that, in high-dimensional lattice-based quantum algorithms, the inner product between quantum states associated with distinct ideals becomes exponentially small unless the ideals differ only by a unit? 1) Setting $ s = n^2D $ and $ \nu = 1/(n^3D) $ 2) Choosing $ s = D/\sqrt{n} $ and $ \nu = n/D^2 $ 3) Selecting $ s = nD^2 $ and $ \nu = 1/(n^2D^2) $ 4) Defining $ s = 22n\sqrt{n}D $ and $ \nu = 1/(4n(s\sqrt{n})^{2n}) $ 5) Using $ s = n^3D $ and $ \nu = D/n^2 $ 6) Letting $ s = D^2 $ and $ \nu = 1/(2n^2D) $ 7) Assigning $ s = n\sqrt{D} $ and $ \nu = 1/nD $

✓ Correct Answer:

The correct answer is 4) Defining $ s = 22n\sqrt{n}D $ and $ \nu = 1/(4n(s\sqrt{n})^{2n}) $.

📚 Reference Text:

Lemma 5.5. Suppose we are given lattices LandL/prime, and sublatticesI⊆LandI/prime⊆L/prime. Assume there is a 1-1 corre- spondenceh:I→I/primes.t., for any (x,x/prime)∈˜L×˜L/prime, 299 1. if (x,x/prime)∈C:={(u,v )∈I×I/prime:v=h(u)}, then /bardblx−x/prime/bardbl ≤ε, 2. otherwise /bardblx−x/prime/bardbl ≥2ν√n. IfI/subsetnoteqlLandI/prime/subsetnoteqlL/prime, then for any n≥5,/angbracketleftf(L)|f (L/prime)/angbracketright ≤ 3/4if in addition s≥4πn3max{λn(L),λ n(L/prime)}. Lemma 5.6. Letν/lessorequalslantλ 2√nands/greaterorequalslantcn(√n/λ)n−1dfor a certain constant c, and let us restrict the encoding L/mapsto→ |f(L)/angbracketright to lattices with d(L)/lessorequalslantdandλ1(L)/greaterorequalslantλ. On such lattices,fhas a Lipschitz constant a=√πns 4ν+ 1. Next we can choose the parameters to show that the hid- den subgroup property holds. Since the inner product is at most a constant <1, a tensor product of ncopies will reduce the inner product to be exponentially small. For simplicity, we state the theorem only for the free part of the unit group LogO∗≤Rs+t−1, and for elements of norm 1. Theorem 5.7. Lets= 22n√nD,ν= 1/(4n(s√n)2n). Lett1,t2∈Rn. Letγ= (γ 1,...,γ n)bet2−t1−u, where u is the unit closest to t2−t1in Log O∗. Then the function fisa-Lipschitz with constant a=√πn s 4ν+ 1. The inner product /angbracketleftet1O|et2O/angbracketrightis at most 3/4 if for some i, we either have ln` 1−(s√n)n−12ν√n´ ≥γiorγi≥ ln` 1 + (s√n)n−12ν√n´ . The idea is as follows: ﬁx an ideal et1O. We want to bound its inner product with ideals et2O. First we show that when the two lattices have points inside the ball of ra- diuss√nwhere they overlap exactly, then their inner prod- uct must be small (unless the two lattices coincide, i.e. when t2−t1is a unit). Then we show that this inner product does not get bigger when we perturb one of the lattices slightly. Finally, we show that when two lattices are not close to hav- ing an exact point of intersection, then their inner product is small as well. Finally, it can be shown that a good approximation of these states can be computed when evaluated on rational numbers. First an approximate basis is computed using the algorithm in the last section, and then the superposition is created over the points. 6. THE HIDDEN SUBGROUP PROBLEM ON A CONTINUOUS GROUP The HSP algorithm for Rmcan be thought of in the usual structure, however the analysis is diﬃcult, and we combine phase estimation and our new continuous deﬁnition of the HSP to get it to work. The analysis works in continuous space and is discretized in a general way to derive the al- gorithm. The algorithm creates a superposition of points in Rmwith a suﬃciently broad wavefunction w. We cannot measure in the Fourier basis of the continuous group, but we show that phase estimation can be used to approximate this to measure a point uof the reciprocal lattice L∗with the probability distribution qu=1d(L)2Z (Rm/L)2/angbracketleftf(x/prime)|f(x)/angbracketrighte2πi/angbracketleftx−x/prime,u/angbracketrightdxdx/prime.(6.1)This distribution is not close to uniform but we are able to show that the probability of staying in any sublattice is bounded. For rounding, the samples deviate from the lattice points by, roughly, the inverse width of the wavefunction w. We show that the reconstruction of a lattice from an approximate generating set can be done using an

Question 1560 multiple-choice

In quantum measurement theory, group-theoretic techniques and frame theory are used to design and analyze measurement schemes that enable complete reconstruction of quantum states. Covariant POVMs derived from unitary irreducible group representations play a crucial role in this approach. Which of the following statements about informationally complete quantum measurements is true? 1) Informationally complete POVMs must always be constructed from commuting observables. 2) The invertibility of the operator formed from POVM elements ensures that any operator’s expectation value can be reconstructed from measurement outcomes. 3) Frame theory is only applicable to classical signal processing, not to quantum measurements. 4) Covariant POVMs cannot be derived from unitary irreducible group representations. 5) The Weyl-Heisenberg group is irrelevant to continuous variable quantum information. 6) Operator frames are strictly non-redundant and do not allow for error correction. 7) The isomorphism between Hilbert-Schmidt operators and bipartite vectors does not facilitate quantum measurement analysis.

✓ Correct Answer:

The correct answer is 2) The invertibility of the operator formed from POVM elements ensures that any operator’s expectation value can be reconstructed from measurement outcomes..

📚 Reference Text:

harmonic oscillator and the spin. In this paper, we present a more general treatment of the proble m based on group- theoretic techniques. We will see that informationally complete meas urements can be achieved through POVM’s derived from unitary irreducible represen tations of a group on the Hilbert space of the system. With the help of frame theory, w e will also provide a constructive way to evaluate the data-processing function for estimating ensemble averages of arbitrary operators. The paper is organized as follows. In Sec. II we prove the equivalenc e of the informational completeness of a measurement and the invertibility o f an operator constructed with the POVM. This proof makes use of frame theory [16, 17], and also shows how to obtain the data-processing function for arbitrary o perator. In Sec. III we derive the conditions of informational completeness for POVM’s tha t are covariant with respect to a (compact) group that has unitary irreducible repres entation on the Hilbert space of the system. We devote Sec. IV to explicit examples of infor mationally complete POVM’s with diﬀerent covariance group. In the example of the Weyl-H eisenberg group, we recover the results of Ref. [18]. Some concluding remarks are giv en in Sec. V. 2. Info-complete POVM’s and frame of operators In the following, we will make extensive use of the isomorphism [19] bet ween the Hilbert space of the Hilbert-Schmidt operators A,BonH, with scalar product /an}bracketle{tA,B/an}bracketri}ht= Tr[A†B], and the Hilbert space of bipartite vectors |A/an}bracketri}ht/an}bracketri}ht,|B/an}bracketri}ht/an}bracketri}ht ∈ H ⊗H , with /an}bracketle{t/an}bracketle{tA|B/an}bracketri}ht/an}bracketri}ht ≡ /an}bracketle{tA,B/an}bracketri}ht, and |A/an}bracketri}ht/an}bracketri}ht=d/summationdisplay n=1d/summationdisplay m=1Anm|n/an}bracketri}ht⊗|m/an}bracketri}ht, (1) where|n/an}bracketri}htand|m/an}bracketri}htare ﬁxed orthonormal bases for H,d= dimH, andAnm=/an}bracketle{tn|A|m/an}bracketri}ht. Notice the identities A⊗B|C/an}bracketri}ht/an}bracketri}ht=|ACBτ/an}bracketri}ht/an}bracketri}ht, Informationally complete measurements and groups represe ntation 3 A⊗B†|C/an}bracketri}ht/an}bracketri}ht=|ACB∗/an}bracketri}ht/an}bracketri}ht, (2) whereτand∗denote transposition and complex conjugation with respect to the ﬁxed bases in Eqs. (1). Aninformationallycompletemeasurement foraquantumsystem with Hilbertspace His described by a POVM {Πi}, Πi≥0 and/summationtext iΠi=I, that allows to obtain the expectation value of any operator Oof the system in the state ρas follows /an}bracketle{tO/an}bracketri}ht ≡Tr[ρO] =/summationdisplay ifi(O)Tr[ρΠi], (3) wherefi(O) is the data processing function of the outcome iwhich depends on the operatorO. Such a POVM will be referred shortly to as ”info-complete” POVM. S ince Eq. (3) holds for any state ρ, it holds generally at the operator level without the ensemble average, namely one has the expansion for operators O=/summationdisplay ifi(O)Πi. (4) Eq. (4) states that the set of positive operators Π ispan the linear space of operators of the system. Spanning sets of operators have been already used in quantum tomography [20]: their characterization of spanning sets of operators is natur ally accomplished in the context of frame theory [16, 17]. An operator frame{Ξi}is simply a set of operators Ξ ithat span a normed linear space of operators, namely there are two constants a, andbwith 0< a≤b <∞ such that for all operators Aone hasa||A||2≤/summationtext i|ci(A)|2≤b||A||2, whereci(A) are the coeﬃcients of the expansion of Aover the set. Here, for simplicity, we will consider the (Hilbert) space of Hilbert-Schmidt operators on H, whence

Question 1561 multiple-choice

The Quantum Fourier Transform (QFT) is a central operation in quantum computing, vital for algorithms such as Shor's factoring and quantum phase estimation. Efficient circuit decomposition of QFT is crucial for its implementation on physical quantum hardware. Which circuit optimization specifically enables the QFT to be implemented with a non-selective Hadamard gate and a sequence of multi-qubit gates, making it advantageous for platforms like NMR quantum computing? 1) Replacing all controlled-phase gates with Toffoli gates 2) Using only single-qubit rotations and eliminating entangling gates 3) Applying selective Hadamard gates to each qubit individually 4) Removing SWAP gates from the output stage 5) Utilizing measurements to collapse superpositions before phase encoding 6) Decomposing the QFT circuit into a global Hadamard transformation and multi-qubit gates, leveraging roots of controlled-NOT gates 7) Implementing the QFT through repeated application of Pauli-X gates

✓ Correct Answer:

The correct answer is 6) Decomposing the QFT circuit into a global Hadamard transformation and multi-qubit gates, leveraging roots of controlled-NOT gates.

📚 Reference Text:

of the Hilbert space and nthe number of qubits. In this basis, the QFT can be represented as a unitary operator F, which transforms the basis states |a/an}bracketri}htinto F|a/an}bracketri}ht=1√qq−1/summationdisplay c=0e2πiac/q|c/an}bracketri}ht. (1) The states |c/an}bracketri}hthave the same form as |a/an}bracketri}ht. When applied to an arbitrary state |ψ/an}bracketri}ht=/summationtext aAa|a/an}bracketri}ht, the QFT yields F|ψ/an}bracketri}ht ≡ Fq−1/summationdisplay a=0Aa|a/an}bracketri}ht −→q−1/summationdisplay c=0Cc|c/an}bracketri}ht (2) where the coeﬃcients Ccare the discrete Fourier trans- form of the input coeﬃcients Aa. The basis transformation of Eqn. 1 can be written in terms of individual qubits as F|a/an}bracketri}ht=n−1/productdisplay j=0⊗|p(φj)/an}bracketri}htj (3) 2 where each qubit is in a state |p(φj)/an}bracketri}ht= (|0/an}bracketri}ht+ ei2πφj|1/an}bracketri}ht)/√ 2. The phases are determined by φj=n−1−j/summationdisplay k=0ak2j+k−n Equation 3 serves as the basis for implementing the QFT by one- and two-qubit operations. One implemen- tation [19, 20, 21] uses single-qubit Hadamard rotations gatesHjand two-qubit controlled-phase gate Bj,kthat act on the qubits jandkand are given by Bj,k= 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0eiθjk  whereθjk=π2j−kis a conditional phase shift applied only if both qubits are in the state |1/an}bracketri}ht. In terms of these gates, the quantum circuit for nqubits is QFTn= (H1B1,2...B1,n)(H2B2,3...B2,n)... ...(Hn−1Bn−1,n)(Hn) (4) with the sequence of operations being performed from right to left. With this implementation, the bit values of the result appear in reversedorder. If a sequence reversal is required, this can be achieved by a sequence of SWAP operations on pairs of qubits. We shall denote this decomposition of the QFT, as “serial”. For nqubits, it requires a total of n Hjgates, n(n−1)/2Bj,kgates andn/2 SWAP operations, leading to a computational complexity of O(n2). Individual Hadamard operations are qubit-selective and hence costlier than a total Hadamard operator that is applied on all qubits simultaneously. It would be thus desirable to have a decomposition of the QFT that in- volves a non-selective Hadamard transformation [9]. A more useful (for NMR) decomposition of the QFT can be obtained by noting that the Hadamard opera- tor is self-inverse and that a Hadamard rotation of the controlled-phase gate can be decomposed as a root of a controlled-NOT gate , HkBjkH−1 k=eiπ/2k−j+1(UCNOT)1/2k−j j,k wherejis the control and kthe target qubit. The global phase factor does not inﬂuence measurement results and is henceforth ignored. Further, using [Hi,Bj,k] = 0,i/ne}ationslash=j,k, the sequence of operations in Eqn. (4) can be modiﬁed to QFTn= [HT][U1U2...Un−2Un−1] (5) where HT=H1H2H3...Hn−1Hn /B4/CP/B5 /CY /D2 /CX /CY /D2 /A0 /BD /CX /CY /BE /CX /CY /BD /CX /C0 /BU/BD/B8/BE/D6 /BU/BD/B8/D2/B9/BD /BU/BD/B8/D2/C0 /BU/BE/B8/D2/B9/BD /BU/BE/B8/D2/C0 /BU/D2/B9/BD/B8/D2/C0/D6 /D6 /D6 /D6 /D6/B4/CQ/B5 /CY /D2 /CX /CY /D2 /A0 /BD /CX /CY /BE /CX /CY /BD /CX/C0/CC /CD/D2 /A0 /BD /CD/D2 /A0 /BE /CD/BD FIG. 1: (a) Quantum circuit for the serial implementation of the QFT using qubit-selective Hadamard rotations and two- qubit controlled-phase gates. (b) Circuit for the parallel im- plementation of the QFT using a non-selective Hadamard ro- tation on all qubits, and multiqubit gates. The readout of th e QFT is performed in reverse order on the qubits, achieved by SWAP operations (not

Question 1562 multiple-choice

Quantum circuit design for algorithms like the Quantum Fourier Transform (QFT) is critical for achieving hardware-efficient implementations on current quantum computers, especially those constrained by nearest-neighbor connectivity. The optimization of two-qubit gates, such as CNOT and SWAP, directly impacts the scalability and performance of these quantum algorithms. In the context of linear nearest-neighbor (LNN) architectures, which circuit design improvement most effectively reduces the total number of CNOT gates required for an n-qubit Quantum Fourier Transform while maintaining scalability for larger n? 1) Replacing all CNOT gates with single-qubit gates 2) Increasing the number of SWAP gates between non-adjacent qubits 3) Eliminating phase gates from the QFT circuit 4) Using n(n−1)/2 SWAP gates, each decomposed into three CNOT gates 5) Synthesizing the QFT circuit directly with CNOT gates, bypassing SWAP gates, to achieve a CNOT count of n²+n−4 6) Doubling the depth of the QFT circuit to minimize gate errors 7) Restricting the QFT implementation to only three qubits per circuit

✓ Correct Answer:

The correct answer is 5) Synthesizing the QFT circuit directly with CNOT gates, bypassing SWAP gates, to achieve a CNOT count of n²+n−4.

📚 Reference Text:

of equations4, and Shor’s factoring algorithm1, to name a few. Therefore, the cost optimization of QFT would result in the efficiency improvement of these quantum algorithms. There have been studies aimed at reducing circuit costs of QFT8,14–22. Among them are studies related to the number of CNOT gates in QFT, including the following:OPEN 1Department of Electrical and Computer Engineering and Center for Quantum Information Processing, University of Seoul, 163 Seoulsiripdae-ro, Dongdaemun-gu, Seoul 02504, Republic of Korea. 2First Quantum Inc., 2F-210, Sparkplus, 180, Bangbae-ro, Seocho-gu, Seoul 06586, Republic of Korea. 3Physics Department, Charles E. Schmidt College of Science, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431-0991, USA. *email: dahn@ uos.ac.kr 2 Vol:.(1234567890) Scientific Reports | (2023) 13:8638 | https://doi.org/10.1038/s41598-023-35625-3 www.nature.com/scientificreports/1. When constructing an n-qubit QFT circuit using the basic gate library, n(n−1) CNOT gates are required, provided that qubit reordering is allowed8. Qubit reordering implies that the sequence of qubits can be altered before and after the execution of the circuit. 2. In Ref.14, the authors incorporated n(n−1)/2 extra SW AP gates to develop an n-qubit linear nearest-neighbor (LNN) QFT circuit, which accommodates qubit reordering. (i) To synthesize a single SW AP gate using the basic gate library, three CNOT gates are required8. (ii) Consequently, the total number of CNOT gates required for the n-qubit LNN QFT circuit presented in Ref.14 is 5n(n−1)/2 . (iii) By employing SW AP gates in the construction of LNN QFT circuits, the primary term representing the quantity of CNOT gates increases by a factor of 2.5. 3. Previous research efforts, as documented in case studies, have investigated techniques to minimize the amount of SW AP gates required in the LNN architecture when assembling n-qubit LNN QFT circuits15–18. These studies aimed to optimize the circuit design and improve overall efficiency. In this paper, we propose a new n-qubit LNN QFT circuit design that directly utilizes CNOT gates, unlike previous studies14–18 that utilized SW AP gates. Our approach offers a significant advantage by synthesizing a more compact QFT circuit using CNOT gates instead of SW AP gates, as the implementation of each SW AP gate requires three CNOT gates. Upon qubit reordering, our n-qubit LNN QFT circuit requires n2+n−4 CNOT gates, which are 40% of those in Ref.14 asymptotically. Furthermore, we demonstrate that our circuit design significantly reduces the number of CNOT gates compared to the best-known results for 5- to 10-qubit LNN QFT circuits17,18. In the following analysis, we compare our QFT circuit with the conventional QFT circuit8 when used as inputs for the Qiskit transpiler23, which is required for implementation on IBM quantum computers that neces- sitate NN architecture10. Our findings confirm that using our QFT circuit as input requires fewer CNOT gates in comparison to the conventional QFT circuits. This evidence indicates that our QFT circuit design could serve as a foundation for synthesizing QFT circuits that are compatible with NN architecture, potentially leading to more efficient implementations. Furthermore, we present experimental results from implementing the QPE using 3-qubit QFTs on actual

Question 1563 multiple-choice

Representation theory and combinatorics play a central role in quantum algorithms for knot invariants, utilizing algebraic objects such as Young tableaux, braid groups, and Hecke algebras. These structures facilitate the connection between symmetric group actions and the computation of polynomials that classify knots. Which statement accurately describes the process by which the representation π(k,ℓ)_n of the Hecke algebra Hn(q) acts on a vector space associated with (k,ℓ) tableaux? 1) It acts on a vector space whose basis is labeled by (k,ℓ) tableaux, encoding the combinatorial structure of the diagrams. 2) It acts only on vector spaces spanned by irreducible characters of the symmetric group, regardless of tableau restrictions. 3) It defines a representation on the set of all possible partitions with no row or level constraints. 4) It uses basis vectors labeled by braid word length rather than tableaux. 5) It acts on a space whose basis is indexed by group elements of Bn exclusively. 6) Its basis vectors correspond to knots rather than tableaux or diagrams. 7) It only acts on the trivial representation when k ≠ n and ℓ is finite.

✓ Correct Answer:

The correct answer is 1) It acts on a vector space whose basis is labeled by (k,ℓ) tableaux, encoding the combinatorial structure of the diagrams..

📚 Reference Text:

numbered boxes yields, at each 10The Jones Polynomial: Quantum Algorithms and Applications in Quantum Complexity Theory ∅ /d15/d15 /d28/d28/d57/d57/d57/d57/d57 /d0/d0/d2/d2/d2/d2/d2 /d28/d28/d58/d58/d58/d58/d58 /d0/d0/d2/d2/d2/d2/d2/d2 /d3/d3/d7/d7/d7/d7 /d15/d15 /d15/d15 /d120/d120/d113/d113/d113/d113/d113/d113/d113/d113 Fig. 3.3. The Young graph Λ(2,5) 0,4. Comparison with Figure 3.2 reveals that this is the subgrap h of Λ0,4obtained by removing vertices whose diagrams either have mo re than two rows, or level greater than 3 = 5 −2. step, another ( k,ℓ)-diagram. These tableaux can be identiﬁed with paths in Λ(k,ℓ) 0,ngoing from ∅toλ. We also deﬁne T(k,ℓ) nandT(k,ℓ)in a similar manner as above. In the Section 5, we will see that the (2 ,ℓ)-diagrams and tableaux are what is needed to give represent ation-theoretic expressions for the Jones polynomials J(e2πi/ℓ) of closures of braids, while the ( k,ℓ)-diagrams are relevant for the one-variable HOMFLYPT polynomials H(k)(e2πi/ℓ). Unitary Jones-Wenzl representations of Bn The representations of Bnwe will introduce are parameterized by integers ℓ>k > 0, where ℓspeciﬁes a primitive ℓ’th root of unity which we will denote q=e2πi/ℓ. The irreducible components of the representations we will describe reduce t o the usual irreps of Snwhen k=nandℓ→ ∞, so thatq→1. For the rest of this section, we ﬁx ℓ > k > 0 and setq=e2πi/ℓ, describing a corresponding representation of Hn(q) which induces a unitary representation of Bn. This representation is more easily described by a change of variables which leads to an equivalent presentation of Hn(q). Rewriting the quadratic relation (3.4) as (gi+ 1)(gi−q) = 0, we may deﬁne idempotents ei= (q−gi)/(1 +q) (meaning that e2 i=ei) for which gi=q(1−ei)−ei=q−(1 +q)ei. (3.7) Using these idempotents, the relations (3.4–3.6) may be exp ressed as e2 i=ei (3.8) eiejei−τei=ejeiej−τej, |i−j|= 1 (3.9) eiej=ejei, |i−j|>1, (3.10) where we follow the usual convention in setting τ≡[2]−2 ℓ. For each integer d, deﬁne aℓ(d) =[d+ 1] ℓ [2]ℓ[d]ℓ, (3.11) noting that aℓ(d) +aℓ(−d) = 1 =aℓ(1). Given any n-box tableau tand an integer 1 ≤i

Question 1564 multiple-choice

Quantum algorithms for the Hidden Subgroup Problem (HSP) in finite abelian groups leverage specialized Boolean functions, matrix representations, and amplification techniques to efficiently identify subgroup structure. These approaches are foundational for problems such as factoring and discrete logarithms in quantum computation. Which statement best characterizes how the algorithm handles the case when the group order m is an odd prime? 1) It requires an exponential number of queries due to complex subgroup structure. 2) It necessitates constructing multiple Boolean functions for different cosets. 3) It requires O(n log m) queries and considers all possible values of j. 4) It simplifies by considering only j=0 and requires O(n) queries. 5) It uses a Smith normal form matrix in every iteration regardless of group order. 6) It employs amplitude amplification exclusively for non-prime group orders. 7) It reduces the problem to non-cyclic subgroup detection for increased efficiency.

✓ Correct Answer:

The correct answer is 4) It simplifies by considering only j=0 and requires O(n) queries..

📚 Reference Text:

The required number of elementary gates is ( nlogm)O(1). Remarks. (1) Whenmis an odd prime and u62Hthen we have d= 1 therefore considering the casej= 0 only is sucient. Also, in each step KorLgets enlarged by a factor p. Therefore, for a prime mwe have an algorithm using O(n) queries. (2) The key problem in the approach proposed by Brassard and Hoyer [6] to extend their method from Zm 2toZn mis existence and ecient computability of a Boolean function : Zn m!f0;1gsuch thatis identically zero on the subgroup what is denoted by Lin our context while it takes value one on exactly half (or another, prescribed suciently large fraction) of the elements of the (unknown) H?unlessK=H?. Using that, one can use the method of Subsection 2.2 to obtain an element of H?nKor conclude K?=Hwith certainty. Notice that if u62Hthen the function f(j;;) :Zn mf0;1gde ned in Line 6 of the algorithm has analogous properties for the appropriate j: it is identically zero on Kf0;1g while nonzero on exactly the half of the pairs from H?f0;1g. Thus, method for tailoring the "total squared amplitude" can be considered as a solution of a modi ed version of the problem of Brassard and Hoyer. Note that amplitude ampli cation using f(j;;) may happen to nd an element of H?nLnot only for the appropriate j. The current pseudo-code adds a new element found by the largest "successful" j. In fact, at this point of the algorithm it cannot be determined in general which one of the functionsf(j;;) takes nonzero value exactly on the half of H?. 3.2 Extending to the HSP in Zn mk We describe how to reduce the hidden subgroup problem of Zn mkto that of Zn m. Assume that fhides the subgroup H. We will use an iteration during which we maintain a subgroup H0of H. In each round H0is increased until we conclude that H=H0. InitiallyH0=f0g. It will be convenient to represent H0by a matrix in Smith normal form (together with multipliers, in particular the left multiplier for the basis change of Zn) whose columns are a basis for the latticeLH0. We start each round with computing the subgroup K0=fx2Zn mk:mx2H0g. This can be eciently done using the Smith normal form representation of H0. (Indeed, if the Smith normal form matrix for H0is diag(d1;:::;dn) the matrix for K0is diag(d1 gcd(d1;m);:::;dn gcd(dn;m)). WhenK0=H0we can stop as in that case Hcannot be larger than H0. Otherwise, we con- sider the factor group K=K0=H0. The group Kis isomorphic toLn i=1Zgcd(di;m)and one can nd a map 0:Zn m!K0such that the map x7! 0(x) +H0is a homomorphism from Zn montoK. Using the simultaneous diagonal form for H0andK0such a 0can be de ned by mapping (x1;:::;xm)Tto (x0 1d1 gcd(d1;m);:::;x0 ndn gcd(dn;m))T, wherex0 iis the least positive inte- ger congruent with ximodulo gcd( di;m). The composition f 0de nes a function hiding M. Imran and G. Ivanyos 783 a subgroup SofZn msuch that the subgroup generated at 0(S) andH0generatesK0\H. We compute generators for Susing the hidden subgroup

Question 1565 multiple-choice

In quantum information science, accurately modeling quantum processes requires careful consideration of both mathematical representations and physical constraints such as complete positivity and trace preservation. Ensuring these properties is essential for characterizing and simulating quantum operations reliably. Which procedure is commonly used to transform a non-completely-positive, non-trace-preserving quantum superoperator into the closest physical (CPTP) superoperator? 1) Renormalizing the superoperator matrix entries proportionally 2) Adding random unitary noise until CPTP conditions are met 3) Discarding non-physical eigenvectors and rebuilding from scratch 4) Averaging the original superoperator with the identity map 5) Applying a single projection onto the set of trace-preserving maps 6) Multiplying eigenvalues by a constant to enforce positivity 7) Alternately enforcing complete positivity and trace preservation in an iterative process

✓ Correct Answer:

The correct answer is 7) Alternately enforcing complete positivity and trace preservation in an iterative process.

📚 Reference Text:

a single operation. When a second operation S2is applied following the ﬁrst S1, however, we must take the spatial correlations produced by the ﬁrst into account in computing the correct overall superoperator. In terms of the f oregoing equations, this may be expressed as S2S1=/integraldisplay Vd/vector r2/parenleftBig U2(/vector r2)⊗U2(/vector r2)/parenrightBig/integraldisplay Vd/vector r1/parenleftBig U1(/vector r1)⊗U1(/vector r1)/parenrightBig 8 =/integraldisplay Vd/vector r2/integraldisplay Vd/vector r1/parenleftBig U2(/vector r2)U1(/vector r1)/parenrightBig ⊗/parenleftBig U2(/vector r2)U1(/vector r1)/parenrightBig (14) /ne}ationslash=/integraldisplay Vd/vector r/parenleftBig U2(/vector r)U1(/vector r)/parenrightBig ⊗/parenleftBig U2(/vector r)U1(/vector r)/parenrightBig =S, whereSis the true superoperator of the combined operations. Clearly, a s imilar result could be obtained if we were integrating over time, although that is no t relevant to the experiments described in this paper. A simulation of an example of the lack of composibility of superoperators describing incoherent errors is shown in Fig. 3. It follows that the N2×N2supermatrix produced by our QPT procedure (Fig. 2) cannot be expected to precisely correspond to any physical process, an d it will usually even fail to deﬁne a completely positive and trace-preserving superoperator . In this event, an improved estimate of the supermatrix of the quantum process that is realize d by the QFT implemen- tation can be obtained by making the smallest possible change to the s upermatrix derived from QPT so as to make it completely positive and trace-preserving ( CPTP). One can, of course, always convert a Kraus operator sum into an e quivalent supermatrix via the “ col” operation introduced above, col/parenleftBig/summationdisplay kAkρiniA† k/parenrightBig =/parenleftBig/summationdisplay kAk⊗Ak/parenrightBig col(ρini). (15) To go the other direction, i.e. to take a supermatrix Sand convert it into an equivalent Kraus operator sum, we ﬁrst rearrange its elements into a Hermitia n supermatrix called the Choi supermatrix [13], T=N−1/summationdisplay ij=0(Eij⊗IN)S(IN⊗Eij), (16) whereINis theN×Nidentity matrix and Eijis a matrix of zeros except for a 1 in the (i,j)th position. Then, if T=/summationtext kλkvkv† kis the spectral decomposition of Tand we assume that all the eigenvalues λk≥0, our Kraus operators Akmay be shown to be col(Ak) =/radicalbig λkvk(k= 1,...,N). (17) It follows from the above that a superoperator Sis completely positive if and only if its Choi supermatrix is positive semi-deﬁnite. In the event that an exp erimentally determined supermatrix Sobsdoes not have a positive semi-deﬁnite Choi supermatrix, it has been shown [13] that the completely positive superoperator S′closest to it in the least-squares sense may be obtained simply by setting the negative eigenvalues of its Choi sup ermatrix to zero to 9 get a new matrix T′, and mapping it back to S′via Eqn. (16) with SandTswapped. This procedure, however, will result in a supermatrix S′that does not preserve the trace of the density matrix. That condition can be reimposed by subtracting ∆T′≡N−1col(IN)col†(IN)T′−IN×N (18) fromT′, with the result that T′is no longer positive semi-deﬁnite. It has further been shown, however, that iterating on these two procedures generates a se quence of supermatrices which converges to the Choi supermatrix corresponding to the supero perator that is closest to S and isbothtrace preserving and completely positive.

Question 1566 multiple-choice

In abstract algebra, the decomposition of products of functions into sums over specific combinatorial maps often involves modular arithmetic and analysis of coefficients' properties, such as invertibility. Understanding the criteria for coefficients to be units or elements of an ideal is essential in module theory and representation theory. Which condition ensures that the coefficients in a decomposition involving sums over maps δ and η are units (invertible elements) in the underlying ring? 1) The values δj and ηj are always zero for all indices j. 2) The intersection of sets S and S is empty for all cases. 3) The coefficients Cδ,η are computed without reference to modular arithmetic. 4) At least one value of δj or ηj is equal to 2 for every j. 5) The union of sets S and S is always the entire index set. 6) Only those pairs (δ,η) where δj and ηj inside S ∪ S are chosen so that the coefficients are invertible, while values outside these sets ensure the coefficients lie in the ideal. 7) All coefficients Cδ,η are elements of a maximal ideal of the ring.

✓ Correct Answer:

The correct answer is 6) Only those pairs (δ,η) where δj and ηj inside S ∪ S are chosen so that the coefficients are invertible, while values outside these sets ensure the coefficients lie in the ideal..

📚 Reference Text:

rjj = 2. Proo/. Let ^(X) = Σ ^iχi and Λr(X) = £ C^. The coefficient we are studying is just b(δ+η)/2C(δ_v)/2. shows that (fy + J^)/2 = ^mod. 2, and that (δj - ηj)/2 = γj mod. 2. Also (^ + ηs)j2 and (^ - ^)/2 are both in {0, ±1} except for the single exceptional case δ3 — ηs = (^^ + ^)/2 = 2. The result now follows from remark (c) of §11.2. THEOREM Π.4.6. λβ{XY)λγ(XY~ι) = Σs,v CδJδ(X)θv(Y). Here δ and v range over all maps {1, -,g) —> {0, ±1,2} satisfying (a) ami (b) below, and Cδ)V = (^mί) (Π ^^ )> ί^β product ranging over all j such that δj = % = 2.' (a) whenever βj = ^ ί/^en ^y α^d ^y are in {0,2}. ΓfeeT/ are equal when γj = 0 aticί unequal when γ3 = 1. (b) whenever βj Φ γά then δj and ηά are in { — 1,1}. ^^. = 0 awd unequal when γ^ = 1 Proo/. ^(IY)^(XY-1) e R'2 and so may be written as Σa>, CiiVθδ(X) θη(Y). Lemma II.4.4 shows that only δ and 27 satisfying (a) and (b) can occur in this decomposition. Comparing coefficients of XδYv and using Lemma II.4.5 we get the result. Taking every βj and γά equal to 1 in Theorem II.4.6 we get: THEOREM Π.4.7. where a ranges over all maps {1,2, •••,#}—> {0,2} and the Ca are units. We now prove Theorem Π.4.1. Suppose S(Q) Π S(R) = 0, and let S = S(Q) U S(/2). It suffices to show that the hypotheses of Lemma II.4.3 are satisfied. So, by Theorem II.4.6 we must show that 2 CδjVXδ(Q)Xη(R) https://doi.org/10.1017/S0027763000017943 Published online by Cambridge University Press 86 HORACIO TAPIA-RECILLAS is a unit when β-\l) = r~\l) = S and is in Jί when /Γ^l) ~fi S or ^(l) 73 S. Suppose first that ^(l) = γ-\ί) = S. For / 6 S(Q) let 3, = 2 and τ]j = 0, for y e S(β) let 3, = 0 and ηs = 2 and for / g S let fy = ^ = 0. Then <5,5? satisfy the conditions of Theorem II.4.6 and Cδ,ηXδ(Q)Xη(R) is a unit. Suppose we have any pair δ,η appearing in the expansion of λβ{XY)λγ{XY~ι). If Xδ(Q) is to be a unit we must have δj = 2 (and ^ = 0) for j e S(Q). If X,GR) is a unit, ^ = 2 (and 3, = 0) for j e S(R). Finally if C^η is a unit, δ3 = ^ = 0 for j g >S. So CδiηXδ(Q)Xv(R) is a unit for a single pair and 2] CδyVXδ(Q)Xη(R) is a unit. Suppose next that β~ι(ϊ) 73 S. Take an index j e S such that y5^ = 0. If Tj = 1 then δj and ^ are in {±1} and Xδ(Q)Xη(R) ei, If ^ = 0 then either 3^ = ηά = 0 so that Z/Q)X,(i2) e ^#, or 5^ = ηs = 2 so

Question 1567 multiple-choice

In both classical and quantum signal processing, transformations such as the Discrete Fourier Transform (DFT), Fast Fourier Transform (FFT), and Quantum Fourier Transform (QFT) are fundamental for analyzing frequency components of signals. Efficient computation and implementation of these transforms rely on matrix decompositions and quantum gate operations. When implementing the Quantum Fourier Transform (QFT) for a two-qubit quantum system, which of the following statements accurately describes its mathematical and circuit structure in comparison to the classical Fast Fourier Transform (FFT)? 1) QFT uses only Pauli-X and Pauli-Z gates without any need for matrix decomposition. 2) QFT can be decomposed into two sparse matrices, mirroring the FFT decomposition, and is implemented with Hadamard, controlled phase, and swap gates. 3) FFT and QFT are unrelated in terms of their mathematical structure or computational efficiency. 4) QFT does not enable transformation between time and frequency domains for quantum states. 5) QFT requires exponentially more quantum gates than the classical FFT requires arithmetic operations. 6) Applying QFT to |01⟩ cannot yield the same result as applying FFT to the vector (0,1,0,0). 7) Matrix sparsity is not exploited in either FFT or QFT for computational efficiency.

✓ Correct Answer:

The correct answer is 2) QFT can be decomposed into two sparse matrices, mirroring the FFT decomposition, and is implemented with Hadamard, controlled phase, and swap gates..

📚 Reference Text:

section 2.1 we have derived a form for the DFT matrix for N=4 and is given by equation 10. [22] showed how the DFT matrix can be decomposed into two sparse matrices in the following way: F4=U1U2 (46)  1 1 1 1 1i−1−i 1−1 1−1 1−i−1i = 1 0 1 0 0 1 0 1 1 0i20 0 1 0i2  1 1 0 0 0 0 1i 1i20 0 0 0 1i3  whereU1andU2are the two sparse matrices used in the FFT computations. Moreover, we know that the QFT for a two qubit system can be constructed by applying a Hadamard gate on the rst qubit ( H1) followed by a controlled phase (R)gate, then another Hadamard gate on the second qubit (H2)and nally a swap gate (S). In matrix representation, this is given by:QFT =SH2RH1 [22] showed that we can decompose the QFT into two sparse matrices in the following way: U2=H1 (47) U1=SH2R (48) QFT =U1U2 (49) Thus we can see that the FFT and the QFT are closely related. We now consider applying the FFT on the signal (0,1): 19 SciPost Physics Codebases Submission Python: 1#FFT computation of a signal 2from scipy.ﬀtpack import ﬀt 3import numpy as np 4x = np.array([0,1,0,0]) 5y = ﬀt(x) 6print (y) Figure 16 A FFT computation of the signal (0,1,0,0) This produces the results: [ 1.-0.j 0.-1.j -1.-0.j 0.+1.j]. Next, using the Qiskit's Built-in QFT function, we nd the QFT of the state |1/angbracketright=|01/angbracketright Python: 1from qiskit import QuantumCircuit 2from qiskit.circuit.library import QFT 3qc = QuantumCircuit(2) 4qc.x(0) 5#apply QFT 6qc.append(QFT(2),[0,1]) 7results = qi.Statevector.from instruction(qc) 8sim = QasmSimulator() 9options = {’method’: ’statevector’ } 10execute(qc, sim, backend options=options) 11results.draw(’latex’, preﬁx=’Statevector1:’) Figure 17 A generation of Qiskit’s QFT on the state |01/angbracketright which gives the output state: 1 2|00/angbracketright−i 2|01/angbracketright−1 2|10/angbracketright−i 2|11/angbracketright= 1 i −1 −i  Which is the same result we got using the FFT. 20 SciPost Physics Codebases Submission 5 USING THE QFT FOR SIGNAL PROCESSING The QFT is analogous to the DFT by transforming information between the time and fre- quency domain. This means that we can use the QFT to nd the frequencies of some sample input encoded on the basis states of the system. This is the basis of the study of signal processing. 5.1 Using the QFT to Find a Musical Note 5.1.1 Generating a Musical Note To begin this experiment, rst we will generate a single musical note. We have selected the note A4 otherwise known as \A440" as it is a pitch standard of a sound oscillating at exactly 440Hz. Figure 18 The Musical Note A440 5.1.2 Sampling Audio Audio is any sound or noise that the human ear is capable of hearing measured in units of cycles per second or Hertz (Hz). Since audio by nature is an analog signal, we must convert this analog signal to a digital signal by sampling at a rate fast enough to capture the wave. In signal processing, the Nyquist frequency is a characteristic of a sampler,

Question 1568 multiple-choice

Quantum algorithms for non-abelian hidden subgroup problems are central to post-quantum cryptography, particularly due to their connections with the security of lattice-based cryptosystems. The dihedral Hidden Subgroup Problem (dHSP) and its quantum solutions, such as Kuperberg’s algorithm, play a key role in evaluating cryptographic hardness. Which statement best describes the implications of an efficient quantum algorithm for all instances of the dihedral Hidden Subgroup Problem (dHSP) on lattice-based cryptography? 1) It would have no effect on the security of any post-quantum cryptosystem. 2) It would only impact classical symmetric encryption schemes. 3) It would make Shor’s algorithm obsolete for factoring large numbers. 4) It would allow the direct calculation of discrete logarithms in any group. 5) It would reduce the complexity of solving the abelian Hidden Subgroup Problem. 6) It could compromise the security foundations of lattice-based post-quantum cryptography by solving problems like uSVP efficiently. 7) It would improve the efficiency of Grover’s algorithm for unstructured search.

✓ Correct Answer:

The correct answer is 6) It could compromise the security foundations of lattice-based post-quantum cryptography by solving problems like uSVP efficiently..

📚 Reference Text:

reduced, but in the worst case, the algorithm is Regev’s algorithm. In 2016, Roetteler [ 71] ﬁrst raised a quantum algorithm that can solve a special type of dHSP problem in polynomial time and space complexities: When N=2 m−1, more than O(2m2)instances are easy to solve among dHSP problem on the dihedral group DN, that is, the total number of easily solved instances increases exponentially with m. In recent years, signiﬁcant progress has been made in lattice-based cryptogra- phy among the post-quantum public-key cryptography. Many lattice-based public-key encryption schemes have been proposed in light of the fact that some lattice problems[72–74] such as unique shortest vector problem (uSVP) are the foundation of the trap- door one-way function. However, uSVP can be reduced to a kind of non-abelian hidden subgroup problem [ 5]: the dihedral hidden subgroup problem. Therefore, the study on quantum algorithm for the dihedral hidden subgroup problem has great signiﬁcance for the security of lattice-based cryptography. 8.1 Kuperberg’s quantum algorithm for dHSP In 2003, Kuperberg [ 76] reduced the dHSP to ﬁnding the slope dwhen N=2 nand H=/angbracketleft(1,d)/angbracketright. Suppose the black box f:DN→Rfor hidden His given and Uf (i.e., the quantum circuit for implement f) is at hand, where Ris the range of f. That is,fon each coset of His constant. Now, a quantum algorithm for determining the parity of d, due to Kuperberg, is depicted in Fig. 5[51] and described as the following eight steps: •Step 1 Initialize the register |ϕ0/angbracketright=| 0/angbracketright|0/angbracketright|0/angbracketright 123 178 Page 18 of 26 J. Suo et al. •Step 2 Prepare the initial quantum state |ϕ1/angbracketright=2n−1/summationdisplay x=01/summationdisplay y=0|x/angbracketright|y/angbracketright|f(x,y)/angbracketright •Step 3 Measure the third register |ϕ2/angbracketright=1√ 2(|x0/angbracketright|y0/angbracketright+| x0+(−1)y0d/angbracketright|y0+1/angbracketright) =/braceleftBigg1√ 2(|x0/angbracketright|0/angbracketright+| x0+d/angbracketright|1/angbracketright) y0=0 1√ 2(|x0−d/angbracketright|0/angbracketright+|(x0−d)+d/angbracketright|1/angbracketright)y0=1 because x0,y0are any value |ϕ2/angbracketrightwhich can be generalized to |ϕ2/angbracketright=1√ 2(|x/angbracketright|0/angbracketright+| x+d/angbracketright|1/angbracketright) •Step 4 Apply QFT to the ﬁrst register |ϕ3/angbracketright=1√ 2n+12n−1/summationdisplay z=0(e2πizx 2n|z/angbracketright|0/angbracketright+e2πiz(x+d) 2n|z/angbracketright|1/angbracketright) =1√ 2n+12n−1/summationdisplay 0e2πizx 2n|z/angbracketright⊗(|0/angbracketright+e2πizd 2n|1/angbracketright) •Step 5 Measure the ﬁrst register, ignore the global phase e2πiz0x 2nand the ﬁrst register, and the second register collapses to |ϕz0/angbracketright=1√ 2/parenleftBig |0/angbracketright+e2πiz0d 2n|1/angbracketright/parenrightBig •Step 6 Apply Kuperberg’s sieve, continue the combines operation, and gain the target quantum state |ϕ2n−1/angbracketright=/parenleftbigg |0/angbracketright+e2πi2n−1d 2n|1/angbracketright/parenrightbigg =/parenleftBig |0/angbracketright+eπid|1/angbracketright/parenrightBig =/parenleftBig |0/angbracketright+(−1)d|1/angbracketright/parenrightBig •Step 7 Apply Hadamard gate H|ϕ2n−1/angbracketright=/parenleftbigg1+(−1)d 2/parenrightbigg |0/angbracketright+/parenleftbigg1−(−1)d 2/parenrightbigg |1/angbracketright 123 Quantum algorithms for typical hard problems… Page 19 of 26 178 Table 5 The complexity of DHSP Authors Year Time complexity Space complexity Query complexity Quantum Classical Kuperberg et al. [ 76] 2003 2O(√logN)2O(√logN)–2O(√logN) Regev et al. [ 5] 2004 2O(√ log(logN)logN)2O(√NlogN)–2O(√ log(logN)logN) Kuperberg et al. [ 70] 2011 2O(√logN)2O(√logN)2O(√logN)2O(√logN) Time, space and query complexity are in the table •Step 8 Measure the second register. If 0 is observed, dis even; otherwise, dis odd. After the parity (i.e., the least signiﬁcant bit) of dis found, Kuperberg suggested to use the sieving idea to ﬁnd all bits of diteratively. The following steps are Kuperberg’s sieve idea, reformulated by Regev [ 5]. – When dis even. Then, consider the black box f/prime:DN/2→Rgiven by f(a,b):= f(a,2b). Note that this function hides the subgroup H/prime=/angbracketleft(1,d/prime)/angbracketrightofDN/2with d/prime=d/2.

Question 1569 multiple-choice

Computational vibrational spectroscopy often utilizes hybrid quantum/classical methods to simulate molecular spectra in solution, accounting for solvent effects and electronic structure. Accurate prediction of vibrational frequencies, such as the carbonyl stretch in acetone, depends on both the choice of quantum model and treatment of the solvent environment. Which of the following statements best describes the impact of using the AM1 Hamiltonian within QM/MM simulations on the predicted vibrational frequency of acetone's carbonyl stretch in aqueous solution? 1) It causes the simulated C=O stretch frequency to be overestimated by approximately 100 cm−1 compared to experiment. 2) It leads to negligible error in the vibrational frequency because AM1 is highly accurate for carbonyl groups. 3) It results in a blue shift of the C=O stretch frequency in solution relative to gas phase values. 4) It provides the same frequency as high-level quantum calculations, making reference corrections unnecessary. 5) It underestimates the vibrational frequency by about 50 cm−1, which is easily compensated by solvent effects. 6) It systematically underestimates the C=O stretch frequency by nearly 300 cm−1, and this error propagates to solution-phase simulations. 7) It causes spectral broadening that is significantly narrower than experimental full width at half maximum (fwhm) values.

✓ Correct Answer:

The correct answer is 6) It systematically underestimates the C=O stretch frequency by nearly 300 cm−1, and this error propagates to solution-phase simulations..

📚 Reference Text:

in Figure S2 along with key geometrical parameters indicated. 4.2. Acetone in Aqueous Solution To illustrate the general applicability of the QVP method in condensed-phase simulations, we carried out “on-the-fly” vibrational frequency calculations for the carbonyl stretching mode of acetone in aqueous solution. QM/MM potential54,104 is employed, in which the chromophore is approximated by the semiempirical Austin Model 1 (AM1)105 Hamiltonian, and the solvent is represented by the TIP3P model for water. A single solute molecule is embedded in a cubic box of 467 water molecules using the isothermal–isobaric (NPT) ensemble at 25 °C and 1 atm. The time-step in the simulation is 1 fs, and a convergent QVP spectrum of C ＝O vibration is obtained using a 50 ps trajectory. All computations are carried out using CHARMM. Here, the instantaneous vibrational frequency of the carbonyl stretch in water is determined using the composite approach described in the previous section, where the reference state is determined at a high level of theory, whereas the perturbation energy is obtained at the combined QM/MM level to account for solvent effects. In particular, the C ＝O vibrational wave functions for the first three vibrational states of an isolated acetone molecule in the gas phase are determined using the potential energy curve from CCSD(T)-F12/cc-pVTZ-F12 calculations along the C ＝O normal mode coordinate. The 0 → 1 transition frequency is determined to be 1776 cm−1, in reasonable agreement with the experimental value of 1738 cm−1 in the vapor phase, and the small difference of +38 cm−1 will be fully transferred to the solution result. Figure 6 displays the computed vibrational spectra for the C ＝O stretching mode of acetone in water using the Fourier transform of the dipole autocorrelation function (DAF; blue curve) and that from the cumulant expansion of the QVP1 frequency fluctuation autocorrelation function (red curves), along with the experimental results (black curve).57 The peak position at 1741 cm−1 from the QVP1 method represents a solvent-induced red shift of −35 cm−1, which may be compared with a solvent effect of −41 cm−1, corresponding Xue et al. Page 10 J Chem Theory Comput . Author manuscript; available in PMC 2018 February 01. Author Manuscript Author Manuscript Author Manuscript Author Manuscript to a maximum peak position at 1697 cm−1. The agreement indicates that the AM1 model used in combined QM/MM simulations can capture the large bulk of solvation effects. In addition, the calculated fwhm (14.8 cm−1) is in good accord with the experiment (15.1 cm −1).57,58 On the other hand, the maximum peak position is found at 1414 cm−1 by Fourier transform of DAF, which is red-shifted by −30 cm−1 from the corresponding gas-phase value. The AM1 Hamiltonian severely underestimates the carbonyl stretch frequency of acetone by 294 cm−1 compared with experimental results, which is fully translated into the condensed-phase results by direct application of the molecular dipole autocorrelation function. Since the Fourier transform of DAF includes anharmonicity and captures the same solvent effect as that in the QVP1 calculations, the small difference of 5

Question 1570 multiple-choice

Quantum algorithms such as Shor's, Simon's, and Deutsch's XOR algorithm are notable for their use of quantum logic and Hilbert space structure to solve problems that are challenging for classical computation. These approaches leverage the mathematical properties of subspaces and projective geometry inherent in quantum mechanics. Which statement best describes the foundational quantum advantage in algorithms framed as hidden subgroup problems? 1) Classical algorithms efficiently distinguish overlapping Boolean propositions by direct computation. 2) Quantum algorithms rely solely on entanglement between qubits to encode global properties. 3) Quantum logic enforces a strict true/false dichotomy analogous to classical computation. 4) Quantum algorithms encode global properties as subspaces in Hilbert space, using projective geometry to distinguish them via measurements. 5) The speedup in quantum algorithms arises from sequentially querying all possible function inputs. 6) Quantum measurements are limited to identifying individual basis states, not subspaces. 7) The computational advantage is due to random sampling of classical truth tables.

✓ Correct Answer:

The correct answer is 4) Quantum algorithms encode global properties as subspaces in Hilbert space, using projective geometry to distinguish them via measurements..

📚 Reference Text:

Title: Quantum computation from a quantum logical perspective Year: 2006 Paper ID: 0c527d2cf1c2c227de1962f35f4685add6ec0a24 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/0c527d2cf1c2c227de1962f35f4685add6ec0a24 Abstract: It is well-known that Shor's factorization algorithm, Simon's period-finding algorithm, and Deutsch's original XOR algorithm can all be formulated as solutions to a hidden subgroup problem. Here the salient features of the information-processing in the three algorithms are presented from a different perspective, in terms of the way in which the algorithms exploit the non-Boolean quantum logic represented by the projective geometry of Hilbert space. From this quantum logical perspective, the XOR algorithm appears directly as a special case of Simon's algorithm, and all three algorithms can be seen as exploiting the non-Boolean logic represented by the subspace structure of Hilbert space in a similar way. Essentially, a global property of a function (such as a period, or a disjunctive property) is encoded as a subspace in Hilbert space representing a quantum proposition, which can then be efficiently distinguished from alternative propositions, corresponding to alternative global properties, by a measurement (or sequence of measurements) that identifies the target proposition as the proposition represented by the subspace containing the final state produced by the algorithm.

Question 1571 multiple-choice

In the representation theory of compact quantum group Hopf $*$-algebras, the study of coideal $*$-subalgebras and their doubles provides important insight into quantum symmetries and the structure of representations. Quantum $SL(2,\mathbb{R})$ serves as a central example in quantum harmonic analysis, particularly regarding decomposition of its regular representation. Which of the following best describes the outcome of decomposing the regular representation of the quantum group $SL(2,\mathbb{R})$ within the framework of doubles of coideal $*$-subalgebras? 1) The regular representation decomposes into characters corresponding to the center of the algebra. 2) The regular representation remains irreducible under all coideal double constructions. 3) Only finite-dimensional representations appear in the decomposition. 4) The decomposition yields only one-dimensional trivial representations. 5) The regular representation decomposes into a direct sum (or integral) of irreducible representations, revealing the fundamental "building blocks" of the representation category. 6) All representations obtained are necessarily equivalent to each other. 7) The decomposition is undefined due to the lack of a $*$-structure in quantum $SL(2,\mathbb{R})$.

✓ Correct Answer:

The correct answer is 5) The regular representation decomposes into a direct sum (or integral) of irreducible representations, revealing the fundamental "building blocks" of the representation category..

📚 Reference Text:

Title: Induction for representations of coideal doubles, with an application to quantum $SL(2,\mathbb{R})$ Year: 2024 Paper ID: f2fd55f130c1388edb1b0aa0e7117ef9aba4cb32 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/f2fd55f130c1388edb1b0aa0e7117ef9aba4cb32 Abstract: We investigate the theory of induction in the setting of doubles of coideal $*$-subalgebras of compact quantum group Hopf $*$-algebras. We then exemplify parts of this theory in the particular case of quantum $SL(2,\mathbb{R})$, and compute the decomposition of the regular representation for quantum $SL(2,\mathbb{R})$ into irreducibles.

Question 1572 multiple-choice

Quantum computing leverages unique phenomena such as entanglement and measurement to achieve computational advantages over classical methods. Measurement-based quantum computing (MBQC) is a model where computation is performed through sequences of measurements on highly entangled cluster states, and graphical tools like ZX-calculus are used to analyze and translate quantum processes between models. In the MBQC reformulation of Simon’s hidden subgroup algorithm for an n-qubit instance, which of the following correctly describes the structure of the cluster state required for computation? 1) A linear chain of n nodes with n−1 edges 2) A fully connected graph of n nodes with n(n−1)/2 edges 3) A star graph with n+1 nodes and n edges 4) A cluster state graph consisting of 2n nodes and n^2 edges 5) A ring topology with n nodes and n edges 6) A grid of n×n nodes with 2n(n−1) edges 7) A tree structure with n nodes and n−1 edges

✓ Correct Answer:

The correct answer is 4) A cluster state graph consisting of 2n nodes and n^2 edges.

📚 Reference Text:

Title: Simon algorithm in measurement-based quantum computing Year: 2024 Paper ID: 932200014774c95cee849726769bd22442d60727 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/932200014774c95cee849726769bd22442d60727 Abstract: Simon's hidden subgroup algorithm was the first quantum algorithm to prove the superiority of quantum computing over classical computing in terms of complexity. Measurement-based quantum computing (MBQC) is a formulation of quantum computing that, while equivalent in terms of computational power, can be advantageous in experiments and in displaying the core mechanics of quantum algorithms. We present a reformulation of the Simon algorithm into the language of MBQC -- in detail for two qubits and schematically for $n$ qubits. We utilize the framework of ZX-calculus, a graphical tensor description of quantum states and operators, to translate the circuit description of the algorithm into a form concordant with MBQC. The result for the two-qubit Simon algorithm is a ten-qubit cluster state on which single-qubit measurements suffice to extract the desired information. Additionally, we show that the $n$-qubit version of the Simon algorithm can be formulated in MBQC as cluster state graph with $2n$ nodes and $n^2$ edges. This is an example of the MBQC formulation of a quantum algorithm that is exponentially faster than its classical counterpart. As such, this formulation should aid in understanding the core mechanics of such an established algorithm and could serve as a blueprint for experimental implementation.

Question 1573 multiple-choice

Quantum circuit architecture search is an emerging area that combines probabilistic graph-based algorithms with quantum computing to optimize circuits for data analysis and machine learning tasks. Efficiently navigating possible circuit configurations is essential for practical and scalable quantum algorithm design. Which of the following advantages best distinguishes a graph-based quantum architecture search algorithm using Monte Carlo methods from classical search techniques like genetic algorithms and simulated annealing? 1) It guarantees the discovery of the globally optimal circuit for any task 2) It eliminates the need for evaluating circuits on benchmark datasets 3) It ensures that only discrete gates are used in circuit construction 4) It automatically adapts to mixed quantum states without further modification 5) It prevents the use of importance sampling during search 6) It avoids cycles and redundant states, resulting in more efficient exploration and shorter circuit designs 7) It compares unfavorably to decision trees and shallow neural networks in machine learning tasks

✓ Correct Answer:

The correct answer is 6) It avoids cycles and redundant states, resulting in more efficient exploration and shorter circuit designs.

📚 Reference Text:

three used datasets and the overall performance. Note, that the overall quality is similar to the the results obtained with decision trees or shallow neural networks. IV. SUMMARY In this paper we have proposed a quantum architecture search algorithm based on Monte Carlo graph search and measures of importance sampling. Each trajectory in this graph leads to a quantum circuit which can be evaluated according to whether it achieves a specific task. Our model also allows for the optimization of 11 FIG. 7. Example realisations of the Cellular automata 30, 90, 110 and 184. The images show the evolution of the pattern over time (from top to bottom). The two examples at the left side have been initialized with a zero vector and a one at the middle, the examples at the right side have been initialized with a random binary pattern. FIG. 8. Quantum circuits for vectors encoding RX cellular automata mixed discrete and continuous gates and sev- eral experiments demonstrate the applicability for different tasks, such as matrix factorization, producing cellular automata vectors, and sim- ple machine learning models. A comparison with classical approaches such as greedy sam- pling, genetic algorithms, particle filter or sim- ulated annealing was carried out and demon- strates that the graph model performs more ef- ficiently, since cycles and redundancies are ex- plicitly avoided. The shortest path from the start node to the target node provides efficient algorithms in terms of circuit length. A future challenge is to overcome the computation time required by the graph model as the number of nodes is increased. One immediate and relevantnext step is to generalize the method described here to apply to mixed states and completely positive maps. Acknowledgments This work was supported, in part, by the Quantum Valley Lower Saxony, the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–Project-ID 274200144–SFB1227, under Germanys Excel- lence Strategies EXC-2123 QuantumFrontiers and EXC-2122 PhoenixD. 12 FIG. 9. Optimized quantum circuits for the wine, zoo and iris dataset. (Images have been generated using quantum-computing.ibm.com/composer ) [1] R. Rietsche, C. Dremel, S. Bosch, L. Steinacker, M. Meckel, and J. M. Leimeister, Electronic Markets (2022). [2] J. Preskill, Quantum 2, 79 (2018). [3] S. Endo, S. C. Benjamin, and Y. Li, Physical Review X 8, 031027 (2018). [4] S. P. Jordan, K. S. Lee, and J. Preskill, Science 336, 1130 (2012). [5] P. Bargassa, T. Cabos, A. C. O. Choi, T. Hes- sel, and S. Cavinato, Physical Review D 104, 096004 (2021). [6] J. Davids, N. Lidstr¨ omer, and H. Ashrafian, “Artificial intelligence in medicine using quan- tum computing in the future of healthcare,” inArtificial Intelligence in Medicine , edited by N. Lidstr¨ omer and H. Ashrafian (Springer In- ternational Publishing, Cham, 2022) pp. 423– 446.[7] Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. Johnson, M. Kieferov´ a, I. D. Kivlichan, T. Menke, B. Peropadre, N. P. Sawaya, et al. , Chemical reviews 119, 10856 (2019). [8] V. Marx, Nature Methods 18, 715 (2021). [9] M. Mannone, V. Seidita, and A.

Question 1574 multiple-choice

In representation theory and quantum algorithms, the study of symmetric groups and their irreducible representations is central to understanding the difficulty of certain computational problems, including those relevant to cryptography. The behavior of character values and the structure of specific subsets of partitions influences both theoretical bounds and practical security. Which statement accurately describes the significance of the subset Λc of partitions in the symmetric group Sn for analyzing strong Fourier sampling in quantum algorithms? 1) Λc consists of partitions with either a large first row or column, and its size and maximal dimension are both small, enabling tighter upper bounds on relevant irreducible representations. 2) Λc is the set of all balanced Young diagrams, which have the largest dimension among all irreducible representations. 3) Λc includes all partitions of n, making it the largest possible subset for analyzing representation growth. 4) Λc contains only the trivial and sign partitions, which do not impact the complexity of Fourier sampling. 5) Λc is defined by partitions with minimal first rows and columns, ensuring maximal diversity of character values. 6) Λc partitions grow exponentially in both number and dimension as n increases, complicating computational analysis. 7) Λc is used primarily to classify abelian subgroups within Sn for group-theoretic cryptosystems.

✓ Correct Answer:

The correct answer is 1) Λc consists of partitions with either a large first row or column, and its size and maximal dimension are both small, enabling tighter upper bounds on relevant irreducible representations..

📚 Reference Text:

are relatively few irreps, i.e., |S|/lessmuchDand|/hatwideG|/lessmuch| G|, we can simply upper bound Δby|S|and upper bound| L|by|/hatwideG|. We discuss the proof of this theorem in Section 5. Most details are relegated to the Appendix A. 4 Applications of the Main Theorem In this section, we present applications of Theorem 1 to analyze strong Fourier sampling over certain non-abelian groups, including the symmetric group and the wreath product corresponding to the McEliece-type cryptos ystems. Another application to the HSP over the groups GL2(Fq)is omitted for lack of space. McEliece and Niederreiter Cryptosystems 771 4.1 Strong Fourier Sampling over Sn We focus now on the case where Gis the symmetric group Sn. Recall that each irrep of Snis in one-to-one correspondence to an integer partition λ=(λ1,λ2,..., λt)ofnoften g i v e nb ya Young diagram oftrows in which the ithrow contains λicolumns. The conju- gate representation of λis the irrep corresponding to the partition λ/prime=(λ/prime 1,λ/prime 2,..., λ/prime t/prime), obtained by ﬂipping the Young diagram λabout the diagonal. As in [15], we shall apply Roichman’s upper bound [19] on normalized characters: Theorem 2 (Roichman’s Theorem [19]). There exist constant b >0and04, for every π∈Sn, and for every irrep λof S n, /vextendsingle/vextendsingle/vextendsingle/vextendsingleχλ(π) dλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftbigg max/parenleftbigg q,λ1 n,λ/prime 1 n/parenrightbigg/parenrightbiggb·supp(π) where supp (π)=#{k∈[n]|π(k)/negationslash=k}is the support of π. This bound works well for unbalanced Young diagrams. In particular, for a constant 00, /vextendsingle/vextendsingle/vextendsingle/vextendsingleχλ(π) dλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤e−α·supp(π). (4) On the other hand, both |Λc|and the maximal dimension of representations in Λcare small, as shown in the following Lemma of [15]. Lemma 2 (Lemma 6.2 in [15]). Let p (n)denote the number of integer partitions of n. Then|Λc|≤2cn·p(cn), and d μd2 S. First, we need to bound the maximal normalized character χ S(H).B y( 4 ) ,w e have χμ(H)≤e−αmfor all μ∈/hatwideSn\S. Hence, χ S(H)≤e−αm. To bound the second term in the upper bound of Theorem 1, as Δ≤|S|, it sufﬁces to bound: |S|·d2 S D≤2cn·p(cn)·n2cn ndn≤eO(√ n)·n(2c−d)n≤n−γn/2 772 H. Dinh, C. Moore, and A. Russell for sufﬁciently large n, so long as γnne−nby Stirling’s approxima- tion, |

Question 1575 multiple-choice

Quantum state compression is an active area of research aiming to reduce the resource requirements for storing or transmitting quantum information, often leveraging variational algorithms and group theoretic methods. Certain approaches focus on states invariant under the action of unknown Abelian groups, using hybrid quantum-classical protocols. Which step in a variational quantum algorithm for compressing translationally invariant quantum states specifically utilizes classical post-processing to identify the hidden subgroup generator? 1) Applying the parameterized quantum Fourier transform to the input state 2) Measuring quantum outputs and collecting bit strings 3) Minimizing the average infidelity via quantum measurements 4) Constructing encoding and decoding quantum circuits 5) Using classical post-processing to infer subgroup generators 6) Evaluating compression quality using fidelity metrics 7) Optimizing quantum circuit parameters with gradient descent

✓ Correct Answer:

The correct answer is 5) Using classical post-processing to infer subgroup generators.

📚 Reference Text:

CPTP map Dwill reconstruct the state, DðeρÞ¼XjCj/C01 i;j¼0∣^ciih^cj∣hci∣eρ∣cji:ð43Þ Variational quantum state compression . Mathematically, we can for- mulate the following quantum state compression problem where ourvariational algorithm can be used to compress it. De ﬁne the generalized translational operator as T h0∣xi¼∣x/C3h0i; ð44Þ where/C3i st h eg r o u po p e r a t i o ni nt h eu n k n o w nA b e l i a ng r o u p G.As o u r c e will randomly generate quantum state ∣φ/C11/C10n2S(multiple copies are used for evaluating the cost in training , see Supplementary Note E for more explanations) according to a probability density μ∣ψi, where the state ∣φ/C11 ishttps://doi.org/10.1038/s41534-024-00865-2 Article npj Quantum Information | (2024) 10:74 11 promised to be translational invariant under the action of Th0. Given access to the quantum states generated fro mt h i sr a n d o ms o u r c eb u tw i t h o u t knowledge of the Abelian group structure Gand the hidden subgroup H,t h e goal is to design EandDthat compress and decompress the quantum states. Then our variational HSP algorithm can be applied, with a slight modi ﬁcation in the cost function. Instead o f reconstruction error of classical bit strings, our new optimization goal is to minimize the average in ﬁdelity between the input state to the reconstructed state, CF¼1/C0Z ψ/C10 ∣D°Eð∣ψ/C11 ψ/C10 ∣Þ∣ψ/C11 dμ∣ψi:ð45Þ The variational algorithm for this quantum states compression problem takes 5 steps. We list the sketch of the protocol here. (i) First, apply thevariational QFT ansatz to the one copy of the random state ∣φ/C11 . (ii) Measure the state after the parametrized QFT i n the computational basis to get a bit string j. Repeat and get a set of bit strings j. (iii) Use steps 3,4,5 of Algorithm 1 in the main text to solve for a tentative generator h 0of the hidden subgroup. (iv) Use h0to construct the hidden subgroup Hand then use Eq. ( 42)a n d Eq. ( 43) to construct the encoding map Eand the decoding map D.( v ) Evaluate the in ﬁdelity between the input state ∣φ/C11 φ/C10 ∣and the reconstructed stateD°Eð∣φ/C11 φ/C10 ∣Þ. (vi) Finally, use gradient descent to update the para- meters in the QFT ansatz to minimize the the in ﬁdelity cost Eq. ( 45). Data availability T h es o u r c ec o d ea n dd a t a s e tc a nb ef o u n do nG i t h u b https://github.com/ unknownfy/parameterized-Abeli an-hidden-subgroup-algorithm . Received: 14 September 2023; Accepted: 15 July 2024; References 1. Goldberg, A. V. & Sipser, M. Compression and ranking. SIAM J. Comput. 20, 524–536 (1991). 2. Ackley, D. H., Hinton, G. E. & Sejnowski, T. J. A learning algorithm for Boltzmann machines. Cogn. Sci. 9, 147–169 (1985). 3. Kingma, D. P. & Welling, M. An introduction to variational autoencoders. Found. Trends Mach. Learn. 12, 307–392

Question 1576 multiple-choice

In quantum chemistry, the application of group theory is essential for classifying electronic states and understanding molecular symmetry. Computational methods such as Gaussian integrals are frequently used to evaluate multi-electron wavefunctions in molecules with complex symmetry. Which scenario specifically enables the formation of a spatially degenerate Eu state in a molecule with D4h symmetry, given the need for correct orbital symmetry relations? 1) Both φx and φy orbitals belong to the same irreducible representation of D4h and couple with identical spin states. 2) The product of orbitals is antisymmetric under all D2h symmetry operations and singlet spin coupling is enforced. 3) Only φx is used to construct the wavefunction, ignoring the transformation properties of φy. 4) Orbitals are chosen such that the nuclear configuration belongs to a subgroup lower than D2h. 5) φx and φy orbitals transform separately as different components of the Eu irreducible representation and are combined in the wavefunction product to maintain degeneracy. 6) The symmetry operations of C4 are omitted, and only permutation symmetry is considered in constructing the wavefunction. 7) Spin coupling is maximized without regard for the spatial transformation properties of the orbitals.

✓ Correct Answer:

The correct answer is 5) φx and φy orbitals transform separately as different components of the Eu irreducible representation and are combined in the wavefunction product to maintain degeneracy..

📚 Reference Text:

(1969). 32 The integrals for this calculation were carried out with the Dunning version of the Murray Geller Gaussian integrals program. 33 S. Huzinaga, ]. Chern. Phys. 42, 1293 (1965). 70.52878° Four-Electron Triplet States: The SOGI spin varia tion for the four-electron singlet state was considered above; here we will discuss the spin variation for triplet states. In this case P'=3 and the general transforma tion L is a function of two parameters Z _and T _and is given by L(A, T) ( cosA/2 sinA/2 sinT/2 = -sinA/2 cosA/2 sinT/2 0 -cosT/2 sinA/2 cosT/2) cosA/2 cosT/2 . sinT/2 (32) In this case A= 0 corresponds to G 1 ; A= 180° and T=O corresponds to GF; and A=180° and T=180° corresponds to G2. Spatial Symmetry Restrictions: We will discuss the symmetry properties of the orbitals for all three states first, beginning with the 3A20 and 1B10 states, since they are closely related. It is only necessary to consider the generators of D4h (the symmetry of the nuclear con figuration) and, in each case, we will examine the effect of the C4 element of D4,. and determine if the orbitals can be related in such a way that they are symmetry functions only for the subgroup D2,.. The orbitals used to construct the 8A20 and 1B10 states have the same symmetry; the different spatial symmetry of the many-electron wavefunctions arises through the difference in the spin coupling. These orbitals are sketched in Fig. 7 (b). The product function =cp12cf>3cf>.1 has the following properties: 64=-(12) (34), C2il>= (12) (34), (33a) (33b) (33c) Downloaded 21 Dec 2005 to 131.215.225.171. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp1086 R. C. LADNER AND W. A. GODDARD III From (32c) we see that the state must be "g." From Sec. II.A, we know that the state will be A2u if Un~L[ (12) (34) ]= -1 and B1u if Un~L[(12) (34) ]= + 1. For a 3A2u state, this gives the condition that lj11[2•121L((12 (34) )= -cos2!Z +t sin2!Z sin2!T-( 4V2/3) sin2!Z cos!T sin!T -isin2!Zcos2!T=-1. (34) This condition is satisfied if or (2V2/3) sinT+t cosT=1, (35b) or sinT=2VZ/3 and cosT=! (35c) or finally 1=70.52878·. •0 (35d) (there is no condition on Z). For the 1B1g state, we have the condition that U11[2•21L((12) (34) )=+1, but this condition is always satisfied since (12) (34) is an element of the invariant subgroup and is represented by the unit matrix in this representation. We have not discussed the case of a spatially de generate state, but an example should make the pro cedure clear. Consider the orbitals sketched in Fig. 7 (c). The following relations are easily seen: ty = -axCP4x and Ivc/>2vc/>3vc/>4v· For the many-electron functions 'l'x=O'Lxx and 'l'v=O'Lyx to form an Eu state, it is necessary that amount of G1 coupling. (3) For the 3Eu state the or bitals must have D4h symmetry unless Z=O (i.e., G1).

Question 1577 multiple-choice

Quantum computing leverages group theory to solve computational problems efficiently, with the Hidden Subgroup Problem (HSP) serving as a foundation for several breakthrough algorithms. Understanding the structural properties of groups and their impact on quantum algorithms is essential for advancing this field. Which of the following is a crucial technical condition for efficiently solving the Hidden Subgroup Problem (HSP) in a black-box group via reduction to a known group, as outlined in quantum algorithmic frameworks? 1) The existence of a normal subgroup in every instance of the group 2) The requirement that all group elements have order two 3) The ability to sample uniformly from all subgroups of the group 4) The presence of an efficiently computable group inverse operation 5) The guarantee that the group is cyclic and Abelian 6) The existence of a centralizer for every element in the group 7) An efficiently implementable bijection between the black-box group and a group with a known quantum polynomial-time HSP solution, alongside periodicity conditions

✓ Correct Answer:

The correct answer is 7) An efficiently implementable bijection between the black-box group and a group with a known quantum polynomial-time HSP solution, alongside periodicity conditions.

📚 Reference Text:

polynomial time. Now, let us ﬁrst consider the case p/ne}ationslash= 2. By using l, we can obtain an element Y′of the form xαpr−1yβ whereβ/ne}ationslash≡0 modp: Y′=X−lY=xαpr−1yb′−bl where αpr−1=a−l(−l−1) 2bpr−1−al+a′−a′blpr−1. Notice that b′−blcannot be a multiple of pbecause, since a′≡almodp, this would contradictthe hypothesis ab′/ne}ationslash≡a′bmodp. Thus, /an}bracketle{tXpr−1,Y′/an}bracketri}ht=/an}bracketle{txpr−1,y/an}bracketri}ht∼=Zp×Zp is an Abelian group (see Proposition 8) and /an}bracketle{txtpr−1y/an}bracketri}htis a subgroup of it. We thus use Abelian Fourier sampling over /an}bracketle{tXpr−1,Y′/an}bracketri}htand output a set of generators for H. Ifp= 2, then, by a similar argument, Y′=X−lYis of the form xαpr−2yβwhereβ/ne}ationslash≡0 modp. Then /an}bracketle{tXpr−2,Y′/an}bracketri}ht=/an}bracketle{txpr−2,y/an}bracketri}ht∼=Zp2×Zp is an Abelian group (here, we use our convention that the case p=r= 2 is excluded) and /an}bracketle{txtpr−1y/an}bracketri}htis a subgroup of it. We thus use Abelian Fourier sampling over /an}bracketle{tXpr−2,Y′/an}bracketri}htand output a set of generators of H. 5 Algorithm Solving the HSP over Zm pr⋊Zp We ﬁnally present a quantum algorithm solving the HSP algorithm over t he groups of the form Zm pr⋊Zp, forpprime, where Zpacts separately on each coordinate of Zm pras inPp,r. Formally, Zm pr⋊Zpis the group generated by m+ 1 elements x1,...,x mandy, where /an}bracketle{tx1,...,x m/an}bracketri}ht∼=Zm prandyxi=xpr−1+1 iyfor each i∈ {1,...,m}. 7 We will now show that, although Zm pr⋊Zpis not an Abelian group, applying the Abelian Fourier transform to it (i.e., the Fourier transform over to the group Zm pr×Zp) is suﬃcient to get enough information to ﬁnd the hidden subgroup. We ﬁrst state a general useful proposition. Proposition 10. LetGbe a black-box group, Ha hidden subgroup of G and fanH-periodic function. If there exists a group G′over which a quantum polynomial-time solution for the HSP is known and a bijection π:G→G′verifying the following conditions (i)π(H)is a subgroup of G′; (ii)f◦π−1isπ(H)-periodic; (iii) there is a polynomial-size quantum circuit that, for a nyg′∈G′, maps|g′/an}bracketri}htto|wg′/an}bracketri}ht, wherewg′is a string representing π−1(g′)(in the black-box representation of G); thenHcan be found by a quantum computer in polynomial-time. Proof.The algorithm for the HSP over G′is used with, as input, the π(H)-periodic function f◦π−1. This gives, in polynomial time, a set of generators for H′. This set is used to create random elements of H′using standard methods [2, 3], which are then mapped using π−1to obtain almost uniformly random elements of H. A polynomial number of such elements is, with high probability, a gene rating set for H. The above proposition is stated in the most general context of Gbeing a black-box group but, in this case, the condition (iii) is problematic. Indeed, even if such a bijectio nπ−1exists, it seems very diﬃcult to implement it when the explicit form of the generators for Gare unknown. For Zm pr⋊Zp, we do not know how to do this in the black-box framework and we need to have some know ledge of the form of the generators. We will solve the HSP over these groups when the input is given as a set of generators for Zm prand one generator forZp. This means that we can isolate the left part and the right part of th e semi-direct product. More precisely, our result is as

Question 1578 multiple-choice

The hidden subgroup problem (HSP) is a fundamental abstraction in computational group theory and quantum computing that underpins several quantum algorithms. Solutions to the quantum hidden subgroup problem (QHSP) rely heavily on group representation theory and quantum Fourier transforms to reveal concealed subgroup structures. In the quantum hidden subgroup algorithm, what is the principal reason that the problem simplifies for abelian groups, enabling efficient quantum algorithms like Shor’s? 1) Abelian groups have a trivial group structure that allows classical algorithms to solve HSP rapidly without quantum resources. 2) The subgroup structure of abelian groups is uniquely determined by their order, removing the need for quantum measurement. 3) The Fourier basis for abelian groups consists entirely of conjugacy classes, making computation straightforward. 4) All irreducible representations of abelian groups are one-dimensional, so the quantum Fourier transform and measurement are greatly simplified. 5) Abelian groups guarantee uniform probability distributions over measurement outcomes, allowing instant identification of hidden subgroups. 6) The structure of abelian groups eliminates the need for unitary operators in the quantum algorithm. 7) Quantum algorithms for abelian groups operate independently of group representations due to commutativity.

✓ Correct Answer:

The correct answer is 4) All irreducible representations of abelian groups are one-dimensional, so the quantum Fourier transform and measurement are greatly simplified..

📚 Reference Text:

above diagram the hidden subgroup structure of the map ϕ:G−→S. Remark 1. The underlying intuition motivating this formal deﬁnition is as follows: Given a natural surjection (or epimorphism) ν:G−→G/Kϕ, an ”archvillain with malice of forethought” hides the algebraic structure of νby intentionally renaming all the elements of G/Kϕ, and ”tossing in for good measure” some extra elements to form a set Sand a map ϕ:G−→S. The hidden subgroup problem can be stated as follows: Problem 1 (Hidden Subgroup Problem (HSP) ).Given a map ϕ:G−→S with hidden subgroup structure, determine a hidden subgrou pKϕofG. An algo- rithm solving this problem is called a hidden subgroup algorithm . We will call a map with hidden subgroup structure a hidden subgroup problem (HSP). 1By saying that this diagram is commutative, we mean ϕ=ιϕ◦ν. This concept generalizes in an obvious way to more complicated diagrams. IS GROVER’S ALGORITHM A QUANTUM HIDDEN SUBGROUP ALGORITHM ? 3 The corresponding quantum form of this HSP is stated as follows: Problem 2 (Hidden Subgroup Problem: Quantum Version ).Let ϕ:G−→S be a map with hidden subgroup structure. Construct a quantum implementation of the mapϕas follows: LetHGandHSbe Hilbert spaces deﬁned respectively by the orthonormal ba ses {|g/an}bracketri}ht|g∈G}and{|s/an}bracketri}ht|s∈S}, and lets0=ϕ(1), where1denotes the identity of the ambient group G. Finally, letUϕbe a unitary transformation such that Uϕ:HG⊗HS−→ H G⊗HS |g/an}bracketri}ht|s0/an}bracketri}ht /ma√sto−→ | g/an}bracketri}ht|ϕ(g)/an}bracketri}ht, Determine the hidden subgroup Kϕwith bounded probability of error by making as few queries as possible of the blackbox Uϕ. A quantum algorithm solving this problem is called a quantum hidden subgroup (QHS) algorithm. 3.The generic QHS algorithm QRand Letϕ:G−→Sbe a map from a group Gto a setSwith hidden subgroup structure. We assume that all representations of Gare equivalent to unitary representations2. Let/hatwideGdenote a complete set of distinct irreducible unitary representations ofG. Using multiplicative notation for G, we let 1 denote the identity ofG, and lets0denote its image in S. Finally, let /hatwide1 denote the trivial representation of G. Remark 2. IfGis abelian, then /hatwideGbecomes the dual group of characters. The generic QHS algorithm is given below: Quantum Subroutine QRand(ϕ) Step 0. Initialization |ψ0/an}bracketri}ht=/vextendsingle/vextendsingle/vextendsingle/hatwide1/angbracketrightBig |s0/an}bracketri}ht∈H/hatwideG⊗HS Step 1. Application of the inverse Fourier transform F−1 GofGto the left register |ψ1/an}bracketri}ht=1/radicalbig |G|/summationdisplay g∈G|g/an}bracketri}ht|s0/an}bracketri}ht∈HG⊗HS, where|G|denotes the cardinality of the group G. 2This is true for all ﬁnite groups as well as a large class of inﬁ nite groups. 4 SAMUEL J. LOMONACO, JR. AND LOUIS H. KAUFFMAN Step 2. Application of the unitary transformation Uϕ |ψ2/an}bracketri}ht=1/radicalbig |G|/summationdisplay g∈G|g/an}bracketri}ht|ϕ(g)/an}bracketri}ht∈HG⊗HS Step 3. Application of the Fourier transform FGofGto the left register |ψ3/an}bracketri}ht=1 |G|/summationdisplay γ∈/hatwideG|γ|/summationdisplay g∈GTrace/parenleftBig γ(g)†|γ/an}bracketri}ht/parenrightBig |ϕ(g)/an}bracketri}ht=1 |G|/summationdisplay γ∈/hatwideG|γ|Trace/parenleftbig |γ/an}bracketri}ht/vextendsingle/vextendsingleΦ/parenleftbig γ†/parenrightbig/angbracketrightbig /parenrightbig ∈H/hatwideG⊗HS, where|γ|denotes the degree of the representation γ, whereγ†denotes the contragradient representation (i.e., γ†(g) =γ/parenleftbig g−1/parenrightbigT=γ(g)T), where Trace/parenleftbig γ†(g)|γ/an}bracketri}ht/parenrightbig =/summationtext|γ| i=1/summationtext|γ| j=1γ(g)ji|γij/an}bracketri}ht, andwhere/vextendsingle/vextendsingle/vextendsingleΦ/parenleftBig γ† ij/parenrightBig/angbracketrightBig =/summationtext g∈Gγji(g)|ϕ(g)/an}bracketri}ht. Step 4. Measurement of the left quantum register with respect to the ort honormal basis /braceleftBig |γij/an}bracketri}ht:γ∈/hatwideG, 1≤i,j≤|γ|/bracerightBig . Thus, with probability Probϕ(γij) =|γ|2/angbracketleftBig Φ/parenleftBig γ† ij/parenrightBig |Φ/parenleftBig γ† ij/parenrightBig/angbracketrightBig |G|2, γijis the measured result, and the quantum system ”collapses” to the state |ψ4/an}bracketri}ht=|γij/an}bracketri}ht/vextendsingle/vextendsingle/vextendsingleΦ/parenleftBig γ†

Question 1579 multiple-choice

Quantum algorithms often utilize group-theoretic symmetries, such as those from the symmetric group Sn and special unitary groups SU(d), to efficiently simulate and optimize quantum states. Techniques like Schur transforms and variational methods play a crucial role in preparing and manipulating these states. What is the primary theoretical significance of Theorem 5 regarding the simulation of gradients for SU(d)-symmetric k-local Hamiltonians in quantum circuits? 1) It proves that all classical algorithms can simulate such gradients with exponential speedup. 2) It ensures that the quantum query complexity for finding ground states is always constant. 3) It demonstrates that computing certain Fourier coefficients related to gradients can be efficiently simulated on a quantum computer, with complexity scaling polynomially in relevant parameters. 4) It establishes that the Schur transform is unnecessary for state preparation in these simulations. 5) It shows that only one measurement is required to determine expectation values for any SU(d)-symmetric Hamiltonian. 6) It guarantees that variational quantum algorithms cannot benefit from group-theoretic symmetries. 7) It claims that tensor product reordering has no effect on the basis expansion of Young-basis elements.

✓ Correct Answer:

The correct answer is 3) It demonstrates that computing certain Fourier coefficients related to gradients can be efficiently simulated on a quantum computer, with complexity scaling polynomially in relevant parameters..

📚 Reference Text:

the same reason, it is easy to see that reordering ten- sor products of |0···0/angbracketrightand|s/angbracketrightin Eq. (5)yields a diﬀerent expansion of Young-basis elements for the same Snirreps. Figure 7illustrates two more cases: |s/angbracketright⊗| 0/angbracketright⊗| s/angbracketright⊗| 0/angbracketright and|s/angbracketright⊗| 00/angbracketright⊗| s/angbracketright. 1 4123 4123 46 5 523 1 4123 4123 45 623 12 3 12 3123 1235 1 3124 3124 36 5 524 1 3124 3124 35 624 512 FIG. 6. Decomposing the initial state by content vectors and spin labels. 020327-10 SPEEDING UP LEARNING QUANTUM... PRX QUANTUM 4,020327 (2023) 134 2134 26 5 5 134 2134 25 651 2124 3134 26 5 534 1 234 1 21 2135 243 413 245613 21 2 FIG. 7. Reordering tensor products yields diﬀerent Young- basis expansions. This method can be generalized to SU (d)−Snduality. For instance, when d=3 to initialize states for three-row Young diagrams, let us consider the upper, down, strange states u,d,sof3.L e t |/Psi10/angbracketright=| u···u/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright kmany/angbracketright⊗(ud−du)⊗···⊗ (ud−du)/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright [(n−k)/2] many. This state lies in the (n+k,n−k,0)irrep. Tensoring with SU (3)singlet |s/angbracketright=(uds−usd+dsu−dus+sud− sdu)/√ 6,|/Psi1init/angbracketright=|/Psi10/angbracketright⊗| s/angbracketright⊗lis a state from the (n+ k+l,n−k+l,l)-irrep. Its expansion can still be tracked by the branching rule as in Figs. 6and7.A n S6-CQA quan- tum circuit with state initialization described above can be seen in Fig. 8. C. Quantum superpolynomial speedup For variational algorithms, typically one would make many measurements with updated parameters {θμ}bysome classical gradient descent scheme: θμ(t+1)=θμ(t)−/summationdisplay νημ(t)A−1 μν(θ(t))∂ ∂θν/angbracketleftH/angbracketrightθ(t),( 6 ) where the learning rate tensor Aμν(θ(t))is often taken as the identity matrix while ημ=ηis the learning rate. The quantity /angbracketleftH/angbracketrightθ(t)is the expectation value to minimize and (∂/∂θ ν)/angbracketleftH/angbracketrightθ(t)is the derivative with respect to θν. With an explicitly parameterized unitaries such as in our case, we can utilize the quantum circuits to measure the gradient of the expectation. Here, we refer taking one measurement at time tas a query andquery complexity as the total time Tin order to converge. The query complexity can be analyzed by recent development of the quantum neural tangent kernel [79]. Though it would be an interesting case to consider the bound on the query complexity to converge, in this work we focus only on showing that the circuit complexity per query can be eﬃciently simulated on quantum circuits while this is not known in classical regime. Theorem 5: Let U(p) CQA denote the CQA ansätze with p alternating layers and let H ∈C[Sn]be a SU(d)- symmetric k-local Hamiltonian with most N terms. Then for any S nirrep Sλ, the Fourier coeﬃcients: ∂ ∂θμ/angbracketleftαT/prime,μS|U(p)† CQA(θ)HU(p) CQA(θ)|αT,μS/angbracketright,( 7 ) where T ,T/primeare standard tableaux of λ(Figs. 3and4) and μSrecords the multiplicity of Sλ(Sec. IV A), can be simu- lated in O (pN(θn4+k2))withθbeing the largest absolute values of parameters. The proof is also given in the Appendix. Precisely, we assume that there exists an eﬃcient Schur transform (QST) FIG. 8. S6-CQA circuit with state initialization in Fig. 6. 020327-11 HAN ZHENG et al. PRX QUANTUM 4,020327 (2023) [47,51,80] with a polynomial overhead to prepare |αT,μS/angbracketright. Calculating the Fourier coeﬃcients over Snis a

Question 1580 multiple-choice

Quantum state discrimination relies on advanced mathematical techniques from representation theory, especially when analyzing measurement protocols for systems of multiple qubits or higher-dimensional particles. Schur-Weyl duality and combinatorial formulas play a key role in understanding the efficiency and limitations of these procedures. Which combinatorial concept directly determines the dimension of an irreducible representation of the symmetric group associated with a partition of n in the context of decomposing tensor power spaces? 1) Young tableaux count 2) Hook-length formula 3) Character orthogonality 4) Kostka numbers 5) Cycle types 6) Gelfand-Tsetlin patterns 7) Littlewood-Richardson coefficients

✓ Correct Answer:

The correct answer is 2) Hook-length formula.

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dUε=ε. (125) Letλbe ﬁxed for the moment. Let the invariant subspace Vλhave a decomposition into irreducible components of SnasVλ=/circleplustext iVi. The operator Aλcan then, due to Schur-Weyl duality, be written as Aλ=mn,λ/summationdisplay i,j=1cijYij, (126) for operators Yijsuch that for each pair i/ne}ationslash=j, the operator Yijcommutes with the action of the permutation group and maps the irreducible representation VjtoViand each Pi:=Yiiis the orthogonal projection onto Vi. Moreover, tr {Yij}=δ(i,j)dim(Fλ). The numbers cijsatisfycii≥0 for alli∈[mn,λ] and|cij| ≤/radicalbig |cii|·|cjj|, sinceAλ≥0. Now choose a state ρwith spectrum tsatisfying t=λ, and observe that/integraltext dU(UρU†)⊗n=/summationtext λuλPλfor appropriate number 0 ≤uλ≤1, so that for every i∈[mλ,n]: ε≥tr{Aλ/integraldisplay U(d)(UρU†)⊗n}dU (127) =/summationdisplay i,jcijtr{Yij/integraldisplay U(d)(UρU†)⊗n}dU (128) =/summationdisplay icii/integraldisplay U(d)tr{Yii(UρU†)⊗n}dU (129) =/summationdisplay icii/integraldisplay U(d)tr{U†⊗nYiiU⊗nρ⊗n}dU (130) ≥ciim−1 λ,ntr{Pλρ⊗n} (131) ≥ciim−1 λ,ntr{Pλ,λρ⊗n} (132) ≥ciim−1 λ,ndim(Fλ) |Tλ|t⊗n(Tλ). (133) We now need to estimate the termdim(Fλ) |Tλ|. Withh(i,j) denoting Hook-lenghts, the dimensions of the irreducible subspaces of any representation of Snon (Cd)⊗n(d >0) obey the following estimates. n!/producttextn i=1(λi+d+1)!≤n!/producttext (i,j)∈λh(i,j)= dimFλ≤n!/producttextd i=1λi!(λ∈Yd,n). (134) This implies that dim(Fλ) |Tλ|≥d/productdisplay i=1λi! (λi+d+1)!≥d/productdisplay i=11 (n+d+1)d= (n+d+1)−d2. (135) Thus, ε≥ciim−1 λ,n(n+d+1)−d2t⊗n(Tλ)≥ciim−1 λ,n(n+d+1)−d2(2n)−d, (136) 16 leading to cii≤(2dn)3d2εfor alli∈[mλ,n]. Now take any vector x=/summationtext ixi∈Vλsuch that each xi∈Vi, and/an}bracketle{tx,x/an}bracketri}ht= 1. Then /an}bracketle{tx,Aλx/an}bracketri}ht=/summationdisplay i,jcij/an}bracketle{tx,Yijx/an}bracketri}ht (137) =/summationdisplay i,jcij/an}bracketle{txi,Yijxj/an}bracketri}ht (138) ≤/summationdisplay i,j|cij| (139) ≤/summationdisplay i,j/radicalBig |cii|·|cjj| (140) ≤mn,λ(2dn)3d2ε (141) ≤(2dn)4d2ε, (142) and this proves that Aλ≤(2dn)4d2εPλ∀λ∈Yd,n (143) so that we ﬁnally get the desired A≤(1−ε·(2dn)4d2) /BD. (144) VI Example As an example, we consider d= 2. Here, the set of states can be identiﬁed with the unit ball in R3, and this is also the only reason why we restrict ourselves to that case. A ny density operator can be written in the Bloch sphere representation as ξ=1 2( /BD+/summationtext3 i=1xiσi), where σ1,σ2,σ3∈ B(C2) are the usual Pauli matrices, x1,x2,x3∈Randρ∈ S(C2) is equivalent to/summationtext3 i=1x2 i≤1. letσ=1 2( /BD+1 2σ3), andS={ρ:x3≤1/4}. Let us assume that, by using (for some large number nand some small ε >0), the POVM M:={ /BD−Pn}∪{Pf,λ}(f,λ)∈Λε. The measurement outcome associated to /BD−Pnshall be denoted e. But then for every ρs∈S, tr{Pn,sρ⊗n s} ≥1−(2n)82−nln(2)ε2. (145) This implies the following. Assume that a state ξ∈S∪{σ}is being sent, and that we try to ﬁnd out which state it was. Let ˜ rρdenote the distribution corresponding to the eigenvalues of the pin ching of ρ to the eigenbasis of σandrρthe spectrum of ρ. Deﬁne the sets B(f,λ) :={ρ∈S∪{σ}:/bardbl¯f−˜rρ/bardbl ≤ε/logicalanddisplay /bardbl¯λ−rρ/bardbl ≤ε}. (146) After application of our measurement Mto the system, we know the following: P(ξ=σ|M=e)≥1−2−nminρ∈SD(σ/bardblρ)−Θ(n,ε))(147) P(ξ∈B(f,λ)|M= (f,λ))≥1−(2n)82−nln(2)ε2. (148) So, in the end, we can (asymptotically, with high probability) not only d istinguish σfromS, but to some extent also the elements of Sfrom one another (with accuracy ε, and a drawback being that states with the same spectrum and the same pinching to the eigenbasis of σcannot be distinguished). In our example for d= 2 this means that we get the z-coordinate of the true state, and its spectrum (=distance 17 from the origin). Conclusion. It is an open question whether an even ﬁner decomposition of Vfinto a more subtle choice of irreducible subspaces

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